RSD characterization for n=4 agents (automatically generated proof). We prove that that the Random Serial Dictatorship is the only mechanism satisfying -- Symmetry (Anonymity + Neutrality); -- SD-Strategy-Proofness; -- Ex-Post Pareto Optimality. Some nuances: -- Symmetry allows us to fix the preference of the first agent and hence to reduce the number of profiles to consider. -- SD-Strategy-Proofness is equivalent to the combination of 3 properties under the swap of the two adjancent objects in agent's ranking: a) upper-invariance b) lower invariance c) swap monotonicity. We don't use swap monotonicity in the proof. -- Instead of Ex-Post Pareto Optimality we use a slightly weaker property. Namely, if an agent does not receive a certain object in any Pareto-optimal allocation, then the mechanism must never allocate this object to this agent. Let's start! Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > b > c > d agent 4: a > b > c > d Application of Anonymity & No-waste Constraint gives a b c d agent 1: 6/24 6/24 6/24 6/24 agent 2: 6/24 6/24 6/24 6/24 agent 3: 6/24 6/24 6/24 6/24 agent 4: 6/24 6/24 6/24 6/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > a > c > d agent 4: b > a > c > d The allocation takes the form a b c d agent 1: */24 */24 6/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > b > d agent 4: a > c > b > d The allocation takes the form a b c d agent 1: 6/24 */24 */24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: a > b > d > c The allocation takes the form a b c d agent 1: 6/24 6/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > b > c > d agent 4: b > a > c > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 6/24 6/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > b > c > d agent 4: a > c > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 6/24 */24 */24 6/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > b > c > d agent 4: a > b > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 6/24 6/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > a > c > d agent 4: b > a > c > d Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 6/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 0/24 6/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 0/24 6/24 6/24 agent 2: 4/24 8/24 6/24 6/24 agent 3: 4/24 8/24 6/24 6/24 agent 4: 4/24 8/24 6/24 6/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > b > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > a > d > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: 12/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > c > d agent 4: b > a > c > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 6/24 6/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > a > c > d agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 8/24 */24 6/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > a > c > d agent 4: b > a > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 4/24 8/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > b > d agent 4: c > a > b > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 6/24 0/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 6/24 0/24 6/24 agent 2: 4/24 6/24 8/24 6/24 agent 3: 4/24 6/24 8/24 6/24 agent 4: 4/24 6/24 8/24 6/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 0/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: d > a > b > c agent 3: d > a > b > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 6/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > b > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 6/24 */24 6/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > b > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 8/24 6/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > b > d agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 4/24 */24 8/24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > a > d agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 18/24 0/24 0/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 18/24 0/24 0/24 6/24 agent 2: 2/24 8/24 8/24 6/24 agent 3: 2/24 8/24 8/24 6/24 agent 4: 2/24 8/24 8/24 6/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > a > d agent 4: b > c > a > d The allocation takes the form a b c d agent 1: 18/24 */24 */24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: d > b > a > c agent 3: d > b > a > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 18/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 2/24 */24 */24 6/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 8/24 6/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 8/24 8/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > a > d agent 4: b > c > a > d Application of Ex-post Efficiency gives a b c d agent 1: 18/24 0/24 0/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 18/24 0/24 0/24 6/24 agent 2: 2/24 8/24 8/24 6/24 agent 3: 2/24 8/24 8/24 6/24 agent 4: 2/24 8/24 8/24 6/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > b > d agent 4: a > c > b > d The allocation takes the form a b c d agent 1: 6/24 */24 0/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > a > c agent 4: b > d > a > c The allocation takes the form a b c d agent 1: 18/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 2/24 */24 */24 6/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > a > d agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 8/24 */24 6/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > a > d agent 4: b > c > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 8/24 8/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > b > d agent 4: a > c > b > d Application of Ex-post Efficiency gives a b c d agent 1: 6/24 */24 0/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 6/24 12/24 0/24 6/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 6/24 12/24 0/24 6/24 agent 2: 6/24 4/24 8/24 6/24 agent 3: 6/24 4/24 8/24 6/24 agent 4: 6/24 4/24 8/24 6/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > b > c agent 4: a > d > b > c The allocation takes the form a b c d agent 1: 6/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > b > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 4/24 */24 6/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > b > d agent 4: a > c > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 6/24 */24 */24 6/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > b > d agent 4: a > c > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 6/24 */24 8/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > b > c agent 4: a > d > b > c Application of Ex-post Efficiency gives a b c d agent 1: 6/24 12/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 6/24 12/24 6/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 6/24 12/24 6/24 0/24 agent 2: 6/24 4/24 6/24 8/24 agent 3: 6/24 4/24 6/24 8/24 agent 4: 6/24 4/24 6/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > a > c agent 4: b > d > a > c The allocation takes the form a b c d agent 1: 18/24 0/24 6/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: a > d > c > b agent 4: a > d > c > b The allocation takes the form a b c d agent 1: 6/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > b > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 4/24 6/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > b > c agent 4: a > d > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 6/24 */24 6/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > b > c agent 4: a > d > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 6/24 */24 */24 8/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > a > c agent 4: b > d > a > c Application of Ex-post Efficiency gives a b c d agent 1: 18/24 0/24 6/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 18/24 0/24 6/24 0/24 agent 2: 2/24 8/24 6/24 8/24 agent 3: 2/24 8/24 6/24 8/24 agent 4: 2/24 8/24 6/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: c > d > a > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 18/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > a > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 2/24 */24 6/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > a > c agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 8/24 6/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > a > c agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 8/24 */24 8/24 Consider the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: c > d > a > b agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: 18/24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 18/24 6/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 18/24 6/24 0/24 0/24 agent 2: 2/24 6/24 8/24 8/24 agent 3: 2/24 6/24 8/24 8/24 agent 4: 2/24 6/24 8/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: c > d > b > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: d > c > a > b agent 3: d > c > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 18/24 6/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 2/24 6/24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > d > a > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 6/24 8/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 8/24 8/24 Consider the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: c > d > b > a agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 8/24 8/24 8/24 agent 3: 0/24 8/24 8/24 8/24 agent 4: 0/24 8/24 8/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: b > d > c > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: d > c > b > a agent 3: d > c > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 8/24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > d > b > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 8/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: c > d > b > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 8/24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: b > d > c > a agent 4: b > d > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 8/24 8/24 8/24 agent 3: 0/24 8/24 8/24 8/24 agent 4: 0/24 8/24 8/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: a > d > c > b agent 4: a > d > c > b The allocation takes the form a b c d agent 1: 6/24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: b > c > d > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 8/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 8/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 8/24 */24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: a > d > c > b agent 4: a > d > c > b Application of Ex-post Efficiency gives a b c d agent 1: 6/24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 6/24 18/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 6/24 18/24 0/24 0/24 agent 2: 6/24 2/24 8/24 8/24 agent 3: 6/24 2/24 8/24 8/24 agent 4: 6/24 2/24 8/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: a > c > d > b The allocation takes the form a b c d agent 1: 6/24 18/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: a > d > c > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 2/24 8/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > d > c > b agent 4: a > d > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 6/24 2/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > c > b agent 4: a > d > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 6/24 */24 */24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: a > c > d > b Application of Ex-post Efficiency gives a b c d agent 1: 6/24 18/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 6/24 18/24 0/24 0/24 agent 2: 6/24 2/24 8/24 8/24 agent 3: 6/24 2/24 8/24 8/24 agent 4: 6/24 2/24 8/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: b > c > d > a The allocation takes the form a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: a > b > d > c The allocation takes the form a b c d agent 1: 6/24 6/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 2/24 */24 8/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: a > d > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 6/24 2/24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > d > b agent 4: a > c > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 6/24 */24 8/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: b > c > d > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 8/24 8/24 8/24 agent 3: 0/24 8/24 8/24 8/24 agent 4: 0/24 8/24 8/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > b > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 8/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 8/24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: b > c > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 8/24 8/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > b > d > a agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 8/24 8/24 8/24 agent 3: 0/24 8/24 8/24 8/24 agent 4: 0/24 8/24 8/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > a > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: d > b > c > a agent 3: d > b > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 8/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 8/24 8/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > a > d > b agent 4: c > a > d > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 8/24 8/24 agent 3: */24 */24 8/24 8/24 agent 4: */24 */24 8/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > a > d > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: 12/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 8/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 8/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > a > d > c agent 4: b > a > d > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 0/24 12/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 0/24 12/24 0/24 agent 2: 4/24 8/24 4/24 8/24 agent 3: 4/24 8/24 4/24 8/24 agent 4: 4/24 8/24 4/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: a > b > d > c The allocation takes the form a b c d agent 1: 6/24 6/24 12/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > a > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 8/24 8/24 agent 3: */24 */24 8/24 8/24 agent 4: */24 */24 8/24 8/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > a > d > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 4/24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > a > d > c agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 8/24 4/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > a > d > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 4/24 8/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: a > b > d > c Application of Ex-post Efficiency gives a b c d agent 1: 6/24 6/24 12/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 6/24 6/24 12/24 0/24 agent 2: 6/24 6/24 4/24 8/24 agent 3: 6/24 6/24 4/24 8/24 agent 4: 6/24 6/24 4/24 8/24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 4/24 8/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: a > d > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 6/24 */24 4/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > b > d > c agent 4: a > b > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 6/24 6/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: b > a > d > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 4/24 8/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 12/24 4/24 8/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 8/24 agent 3: */24 */24 */24 8/24 agent 4: 0/24 12/24 4/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > a > d > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 4/24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 2/24 */24 8/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > a > d > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 4/24 8/24 agent 3: */24 */24 */24 8/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 12/24 4/24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: b > a > c > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 12/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > a > d > c agent 4: b > a > d > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 4/24 8/24 agent 3: */24 */24 */24 8/24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 4/24 8/24 agent 3: */24 */24 */24 8/24 agent 4: */24 */24 */24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 4/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: c > a > d > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 8/24 agent 3: */24 */24 */24 8/24 agent 4: */24 */24 */24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 */24 */24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 8/24 agent 4: */24 2/24 */24 8/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 8/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 16/24 0/24 8/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 16/24 0/24 8/24 agent 3: 0/24 4/24 12/24 8/24 agent 4: 0/24 4/24 12/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 8/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > b > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 16/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 16/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 8/24 agent 4: 0/24 */24 */24 8/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 12/24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 4/24 12/24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 */24 agent 3: 0/24 */24 */24 8/24 agent 4: 0/24 0/24 */24 8/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 */24 agent 3: 0/24 */24 */24 8/24 agent 4: 0/24 0/24 16/24 8/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 12/24 4/24 8/24 agent 3: 0/24 12/24 4/24 8/24 agent 4: 0/24 0/24 16/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 8/24 agent 4: */24 2/24 */24 8/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 8/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 12/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 4/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 16/24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 16/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: c > a > d > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 8/24 agent 4: 0/24 2/24 */24 8/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 8/24 agent 4: 0/24 2/24 14/24 8/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 5/24 8/24 agent 3: */24 */24 5/24 8/24 agent 4: 0/24 2/24 14/24 8/24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 5/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 2/24 14/24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 14/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > d > b agent 4: c > a > d > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 */24 agent 2: */24 */24 5/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 2/24 14/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 2/24 14/24 8/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 5/24 8/24 agent 3: */24 */24 5/24 8/24 agent 4: 0/24 2/24 14/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 16/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 12/24 4/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 8/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 5/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 2/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 14/24 8/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 16/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 16/24 8/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 12/24 4/24 8/24 agent 3: 0/24 12/24 4/24 8/24 agent 4: 0/24 0/24 16/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 16/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 8/24 agent 4: 0/24 */24 12/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 12/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 4/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 16/24 8/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 16/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 16/24 0/24 8/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 16/24 0/24 8/24 agent 3: 0/24 4/24 12/24 8/24 agent 4: 0/24 4/24 12/24 8/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > b > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 16/24 */24 8/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 8/24 agent 3: 0/24 12/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 12/24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 4/24 12/24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: c > a > d > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 12/24 */24 agent 4: */24 */24 12/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 4/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 12/24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > b > d agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: b > a > d > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 4/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > a > d > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 */24 agent 3: */24 */24 12/24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > a > d > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > b > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 12/24 12/24 0/24 agent 3: 0/24 12/24 12/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > b > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 12/24 12/24 0/24 agent 3: 0/24 12/24 12/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 2/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 12/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 2/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 2/24 0/24 22/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 1/24 agent 3: */24 */24 12/24 1/24 agent 4: 0/24 2/24 0/24 22/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 1/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 2/24 */24 22/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > c > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 22/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > b > d > c agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 1/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 0/24 agent 3: */24 */24 12/24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 12/24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > b > d > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 5/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 12/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 12/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 12/24 12/24 0/24 agent 3: 0/24 12/24 12/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 12/24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 12/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 12/24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 4/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 12/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 12/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 12/24 12/24 0/24 agent 3: 0/24 12/24 12/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 12/24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 0/24 */24 agent 3: */24 4/24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > d > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 0/24 agent 3: */24 */24 12/24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > a > d > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 12/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > b > d agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > a > d > c agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 12/24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 12/24 12/24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 */24 agent 3: 0/24 12/24 12/24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 12/24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 0/24 agent 3: 0/24 */24 12/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 12/24 0/24 agent 3: 0/24 12/24 12/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 12/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 12/24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 */24 0/24 agent 2: */24 12/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 0/24 */24 0/24 agent 2: */24 12/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 0/24 */24 0/24 agent 2: */24 12/24 */24 0/24 agent 3: 0/24 12/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 0/24 */24 0/24 agent 2: */24 12/24 */24 0/24 agent 3: 0/24 12/24 12/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 12/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 12/24 12/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 12/24 */24 0/24 agent 2: */24 12/24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > b > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 0/24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 12/24 0/24 */24 agent 3: */24 0/24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > c > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > c > b agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 0/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 12/24 */24 0/24 agent 2: */24 12/24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: d > a > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > a > b > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > a > c > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 12/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > a > b > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > b > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > d > b > c agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > c > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > a > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > b > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > d > c > b agent 4: b > c > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: d > a > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > b > c > a We already know that the allocation has the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > a > c > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > a > b > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: d > b > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: d > b > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > a > b > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > b > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: d > a > b > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > b > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > a > b > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: d > b > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: d > a > b > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: d > a > b > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > a > c > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > a > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > b > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > a > b > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: d > a > b > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: d > a > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > a > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > b > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > d > c > b agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 12/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > d > a > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > b > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > d > c > b agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: d > a > b > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > b > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > a > b > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > a > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > d > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > a > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > b > c > a We already know that the allocation has the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > c > b > a We already know that the allocation has the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > a > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > d > a > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > b > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > c > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > d > c > b agent 4: b > a > c > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > c > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 */24 0/24 0/24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > a > b > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: d > a > b > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > a > c > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: d > a > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: d > a > b > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > a > c > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > a > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > a > c > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: d > b > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > a > b > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > a > b > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: d > a > c > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: d > a > c > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > d > b > c agent 4: b > a > c > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > c > a > b We already know that the allocation has the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > b > c > a We already know that the allocation has the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > a > b > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: d > c > b > a We already know that the allocation has the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: d > a > c > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: d > b > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 */24 agent 2: 0/24 */24 0/24 */24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > d > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > a > c > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > a > b > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > d > c > b agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > a > b > c We already know that the allocation has the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > d > b > c agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > b > a > c We already know that the allocation has the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: d > a > b > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: d > a > c > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: d > a > b > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: d > b > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > d > a > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > b > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > d > c > b agent 4: b > c > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > a > d > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > d > b > c agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: d > a > b > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > b > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: c > b > a > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 0/24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > d > c > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: b > c > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 12/24 */24 0/24 agent 2: */24 12/24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > c > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > c > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > b > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 0/24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 12/24 0/24 */24 agent 3: */24 0/24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > c > b agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > d > b agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > d > c agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > c > b agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 0/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 12/24 */24 */24 agent 2: */24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > c > d > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 12/24 */24 0/24 agent 2: */24 12/24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > c > a > b We already know that the allocation has the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: d > b > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: d > a > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: d > b > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: d > b > c > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > b > a > c We already know that the allocation has the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > a > c We already know that the allocation has the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: d > b > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: d > a > b > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: d > a > c > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: d > b > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > a > c > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 0/24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: a > d > c > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > b > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > a > c > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > a > c > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > a > c > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > a > b > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > d > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 1/24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > b > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > d > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > b > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: d > b > a > c agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 0/24 */24 0/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 0/24 24/24 0/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > a > b > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: d > b > a > c agent 3: d > b > c > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 */24 24/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 24/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 0/24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: d > b > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > d > a > b agent 3: d > a > b > c agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > c > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: d > b > c > a agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: d > c > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 0/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 24/24 0/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > c > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > a > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > a > c agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 24/24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 24/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: a > d > c > b agent 4: c > a > d > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 0/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > c > a agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 */24 24/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > d > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > b > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 */24 0/24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > b > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 */24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 24/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > b > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > b > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > d > c > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > d > c > b agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > d > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > d > b > c agent 4: b > a > d > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > a > b > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > a > c > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > a > b > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > d > b > c agent 4: b > a > c > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > a > b > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 12/24 0/24 agent 3: */24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > a > c agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > d > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: d > b > a > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: d > b > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > d > a > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: d > a > c > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > a > c > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > b > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: c > d > b > a agent 3: d > a > c > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: d > a > c > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > d > a > b agent 4: d > b > c > a We already know that the allocation has the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > d > a > c agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > c > a > b We already know that the allocation has the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > a > d agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > c > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: c > d > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 12/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: d > b > a > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > a > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: d > b > c > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: d > a > c > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: d > a > b > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > d > c > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > b > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: b > d > a > c agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > d > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > c > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > a > c > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > d > c agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: b > c > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > a > b > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > b > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: a > d > c > b agent 4: b > c > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 0/24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > a > b > c We already know that the allocation has the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: d > a > b > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: a > d > b > c agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > a > c > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: d > a > c > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 */24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > d > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > b > d > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > a > c > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > a > d > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: a > c > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 */24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > a > c > d agent 4: c > d > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > d > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > a > d > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > a > d > c agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: c > d > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > a > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > a > c > d agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 */24 0/24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > b > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > b > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > c > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > b > d agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > d > c agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > d > c agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > d > a agent 3: d > a > b > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 */24 0/24 agent 4: 0/24 */24 0/24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > b > d > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: b > c > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > c > d > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > d > a agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: b > c > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > d > a agent 4: c > b > d > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: b > d > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > d > b agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: 0/24 */24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > d > a > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 0/24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > b > d > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > a > d > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 12/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 0/24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 12/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > b > a agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: */24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: d > c > a > b agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: 0/24 0/24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: d > b > a > c agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: b > c > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 */24 */24 0/24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 */24 */24 24/24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > d > a agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 0/24 24/24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 */24 0/24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > d > c agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: c > a > d > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > a > d > b agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 0/24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > a > d > c agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 */24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > a > c > b We already know that the allocation has the form a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: */24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 0/24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: d > a > b > c agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: c > b > a > d agent 3: d > a > b > c agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 */24 */24 0/24 agent 3: */24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > a > c > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 0/24 24/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > a > d > b agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: c > b > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: 12/24 */24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: 0/24 */24 0/24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 */24 agent 3: 0/24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > a > b > d agent 4: d > a > c > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 0/24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > c > a Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 */24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > d > a agent 3: c > b > d > a agent 4: d > a > b > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 0/24 */24 agent 3: 0/24 0/24 24/24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: c > a > b > d agent 3: d > b > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > c > a agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 0/24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Assume agent 3 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > b > a > d agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: */24 */24 24/24 0/24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > d > b agent 4: d > b > c > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 0/24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: 0/24 0/24 */24 24/24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: b > d > c > a agent 4: c > a > b > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 0/24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > c > a agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 24/24 0/24 */24 0/24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 0/24 */24 24/24 Consider the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: d > c > b > a Application of Ex-post Efficiency gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 */24 0/24 agent 3: 0/24 */24 24/24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > d > c agent 3: c > d > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: 24/24 0/24 */24 */24 agent 2: 0/24 24/24 */24 */24 agent 3: 0/24 */24 24/24 0/24 agent 4: 0/24 */24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: d > c > b > a agent 4: d > c > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > b > a agent 4: c > d > b > a The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: d > c > a > b Application of Ex-post Efficiency gives a b c d agent 1: */24 */24 0/24 0/24 agent 2: */24 */24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 12/24 0/24 0/24 agent 2: 12/24 12/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 24/24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > a > c > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 0/24 0/24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: c > d > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: 12/24 12/24 */24 */24 agent 2: 12/24 12/24 */24 */24 agent 3: */24 0/24 24/24 */24 agent 4: */24 0/24 0/24 24/24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: d > c > a > b agent 4: d > c > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 0/24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: c > d > a > b agent 4: c > d > a > b The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: d > b > a > c Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: */24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 */24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 0/24 agent 2: 12/24 0/24 12/24 0/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 0/24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > d > a > c agent 3: c > a > b > d agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 */24 0/24 agent 4: */24 0/24 */24 24/24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > c > d agent 3: b > d > a > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 */24 0/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > d > b agent 3: b > d > a > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 12/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: d > b > a > c agent 4: d > b > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > c > b > d agent 3: b > d > a > c agent 4: b > d > a > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Anonymity & No-waste Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 */24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Application of Lottery Constraint gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. Assume agent 2 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > a > b > c The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 24/24 0/24 */24 agent 3: */24 0/24 24/24 0/24 agent 4: */24 0/24 0/24 */24 Assume agent 2 has flipped objects b and c. Hence, at the profile agent 1: a > b > c > d agent 2: a > b > d > c agent 3: b > c > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: 12/24 */24 0/24 */24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 2 has flipped objects c and d. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 */24 */24 12/24 agent 3: */24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 3 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: c > b > a > d agent 4: c > b > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: 0/24 */24 */24 */24 agent 4: */24 */24 */24 */24 Assume agent 4 has flipped objects a and b. Hence, at the profile agent 1: a > b > c > d agent 2: a > d > b > c agent 3: b > c > a > d agent 4: b > c > a > d The allocation takes the form a b c d agent 1: */24 */24 */24 */24 agent 2: */24 */24 */24 */24 agent 3: */24 */24 */24 */24 agent 4: 0/24 */24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: a > d > c > b agent 3: b > c > a > d agent 4: c > b > a > d Application of Ex-post Efficiency gives a b c d agent 1: 12/24 */24 */24 */24 agent 2: 12/24 0/24 0/24 12/24 agent 3: 0/24 */24 */24 */24 agent 4: 0/24 0/24 */24 */24 Consider the profile agent 1: a > b > c > d agent 2: b > c > a > d agent 3: c > b > a > d agent 4: d > a > b > c Application of Ex-post Efficiency gives a b c d agent 1: */24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 */24 Application of Lottery Constraint gives a b c d agent 1: 24/24 0/24 0/24 0/24 agent 2: 0/24 24/24 0/24 0/24 agent 3: 0/24 0/24 24/24 0/24 agent 4: 0/24 0/24 0/24 24/24 Let's apply Strategy-Proofness. Assume agent 1 has flipped objects