Fedor
Sandomirskiy

We study when mediation expands the set of equilibrium predictions in finite games. With multiple Nash equilibria, mediation can help players coordinate on which one to play. We ask when mediation does more: when correlated equilibria extend beyond mixtures of Nash equilibria. For regular games, we show a sharp dichotomy: either the convex hull of Nash equilibria is a proper subset of the set of correlated equilibria, or that set is a singleton. Thus mediation either has real bite, or the equilibrium prediction is maximally robust. The dichotomy extends to coarse correlated equilibria, Bayesian games, communication equilibria, and Bayes correlated equilibria.

We ask when a normal-form game yields a single equilibrium prediction, even if players can coordinate by delegating play to an intermediary such as a platform or a cartel. Delegation outcomes are modeled via coarse correlated equilibria (CCE) when the intermediary cannot punish deviators, and via the set of individually rational correlated profiles (IRCP) when it can. We characterize games in which the IRCP or the CCE is unique, uncovering a structural link between these solution concepts. Our analysis also provides new conditions for the uniqueness of classical correlated and Nash equilibria that do not rely on the existence of dominant strategies. The resulting equilibria are robust to players' information about the environment, payoff perturbations, pre-play communication, equilibrium selection, and learning dynamics. We apply these results to collusion-proof mechanism design.

We study stochastic choice across decision problems, each represented as a menu of action labels paired with observable outcome vectors. We propose a consistency condition for behavior in decision problems composed of two separable components: choice probabilities must agree with those obtained when each component is considered in isolation. Together with monotonicity and continuity, this separability requirement characterizes the family of random coefficients logit rules.

Blackwell’s theorem, connecting majorization to the existence of signals inducing a desired distribution of posterior means, has numerous applications in economics. We give a new proof of this theorem via an explicit construction. Our approach provides a concrete way to generate signals: we demonstrate that any distribution inducible by some signal can also be induced by a “downward-uniform signal,” which simply imposes a stochastic lower bound on the realized state. We further study properties of these signals, indicating their suitability in static and dynamic economic environments.

We study whether a given Nash equilibrium can be improved within the set of correlated equilibria for arbitrary objectives. Our main contribution is a sharp characterization: in a generic game, a Nash equilibrium is an extreme point of the set of correlated equilibria if and only if at most two agents randomize. Consequently, any sufficiently mixed Nash equilibrium involving at least three randomizing agents can always be improved by correlating actions or switching to a less random equilibrium, regardless of the underlying objective. We show that even if one focuses on objectives that depend on payoffs, excess randomness in equilibrium implies improvability. We extend our analysis to symmetric games, incomplete information games, and coarse correlated equilibria, revealing a fundamental tension between the randomness in Nash equilibria and their optimality.

Agents receive private signals about an unknown state. The resulting joint belief distributions are complex and lack a simple characterization. Our key insight is that, when conditioned on the state, the structure of belief distributions simplifies: feasibility constrains only the marginal distributions of individual agents across states, with no joint constraints within a state. We apply this insight to multi-receiver persuasion, identifying new tractable cases and introducing optimal transportation and duality tools.

We study solution concepts for normal-form games. We obtain a characterization of Nash equilibria and logit quantal response equilibria, as well as generalizations capturing non-expected utility. Our axioms reflect that players are responsive to payoffs induced by the play of others and, whenever several games are played simultaneously, players may consider each separately.

This paper links matching markets with aligned preferences to optimal transport theory. We show that stability, efficiency, and fairness emerge as solutions to a parametric family of optimal transport problems. The parameter indexes a planner's attitude towards inequality. This link offers insights into structural properties of matchings and trade-offs between objectives, showing how stability can lead to welfare inequalities, even among similar agents. Our model captures supply-demand imbalances in contexts like spatial markets, school choice, and ride-sharing. We also show that large markets with idiosyncratic preferences can be well approximated by aligned preferences, expanding the applicability of our results.

We investigate inherent stochasticity in individual choice behavior across diverse decisions. Each decision is modeled as a menu of actions with outcomes, and a stochastic choice rule assigns probabilities to actions based on the outcome profile. Outcomes can be monetary values, lotteries, or elements of an abstract outcome space. We characterize decomposable rules: those that predict independent choices across decisions not affecting each other. For monetary outcomes, such rules form the one-parametric family of multinomial logit rules. For general outcomes, there exists a universal utility function on the set of outcomes, such that choice follows multinomial logit with respect to this utility. The conclusions are robust to replacing strict decomposability with an approximate version or allowing minor dependencies on the actions' labels. Applications include choice over time, under risk, and with ambiguity.

This paper revisits a classical question in economics: how do individual preferences and incomes of consumers shape aggregate behavior? We develop a method that reduces the hard problem of aggregation to simply computing a weighted average. The method applies to populations with homothetic preferences. The key idea is to handle aggregation in the space of logarithmic expenditure functions.

We demonstrate the power of this method by (i) characterizing classes of preferences invariant with respect to aggregation, i.e., such that any population of heterogeneous consumers with preferences from the class behaves as if it were a single aggregate consumer from the same class; (ii) characterizing classes of aggregate preferences generated by popular preference domains such as linear or Leontief; (iii) describing indecomposable preferences, i.e., those that do not correspond to aggregate behavior of any non-trivial population; (iv) representing any preference as an aggregation of indecomposable ones.

We discuss connections and applications of our findings to robust welfare analysis, information design, stochastic discrete choice, pseudo-market mechanisms, and preference identification.

We consider the problem of revenue-maximizing Bayesian auction design with several i.i.d. bidders and several items. We show that the auction-design problem can be reduced to the problem of continuous optimal transportation introduced by Beckmann. We establish the strong duality between the two problems and demonstrate the existence of solutions. We then develop a new numerical approximation scheme that combines multi-to-single-agent reduction and the majorization theory insights to characterize the solution.

A sender communicates with a receiver through a sequence of mediators. The sender is the only informed agent and the receiver is the only one taking an action. All the agents have their own utility functions, which depend on the receiver's action and the state. For any number of mediators, the sender's optimal value is characterized. For one mediator, the characterization has a clear geometric meaning of constrained concavification of the sender's utility, optimal persuasion requires the same number of signals as without mediators, and the presence of the mediator is never profitable for the sender. Surprisingly, the second mediator may improve the sender's utility; however, optimal persuasion with several mediators may require more signals.

We study efficiency in general collective choice problems when agents have ordinal preferences and randomization is allowed. We establish the equivalence between welfare maximization and ex-ante efficiency for general domains. We relate ex-ante efficiency with ex-post efficiency, characterizing when the two notions coincide. Our results have implications for well-studied mechanisms including random serial dictatorship and a number of specific environments, including the dichotomous, single-peaked, and social choice domains.

Whether or not a given social network aggregates information depends not only on the topology of the network and information available to agents but also on the order in which they make their decisions. We consider a model with Bayes-rational agents and identify the topological property sufficient for aggregation for most of the orders. We use this property and the insights from the theory of expander graphs to show that learning is possible without opinion leaders and that it can be robust to the elimination of large groups of agents.

In a private private information structure, agents' signals contain no information about the signals of their peers. We study how informative such structures can be, and characterize those that are on the Pareto frontier, in the sense that it is impossible to give more information to any agent without violating privacy. In our main application, we show how to optimally disclose information about an unknown state under the constraint of not revealing anything about a correlated variable that contains sensitive information.

We study the set of possible joint posterior belief distributions of a group of agents who share a common prior regarding a binary state, and who observe some information structure. For two agents we introduce a quantitative version of Aumann's Agreement Theorem and show that it is equivalent to a characterization of feasible distributions due to Dawid et al. (1995). For any number of agents, we characterize feasible distributions in terms of a "no-trade" condition. We use these characterizations to study information structures with independent posteriors. We also study persuasion problems with multiple receivers, exploring the extreme feasible distributions.

Ann likes oranges much more than apples; Bob likes apples much more than oranges. Tomorrow they will receive one fruit that will be an orange or an apple with equal probability. Giving one half to each agent is fair for each realization of the fruit. However, agreeing that whatever fruit appears will go to the agent who likes it more gives a higher expected utility to each agent and is fair in the average sense: in expectation, each agent prefers his allocation to the equal division of the fruit, i.e., he gets a fair share.

We turn this familiar observation into an economic design problem: upon drawing a random object (the fruit), we learn the realized utility of each agent and can compare it to the mean of his distribution of utilities; no other statistical information about the distribution is available. We fully characterize the division rules using only this sparse information in the most efficient possible way, while giving everyone a fair share. Although the probability distribution of individual utilities is arbitrary and mostly unknown to the manager, these rules perform in the same range as the best rule when the manager has full access to this distribution.

A mixed manna contains goods (that everyone likes), bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others. If all items are goods and utility functions are homothetic, concave (and monotone), the Competitive Equilibrium with Equal Incomes maximizes the Nash product of utilities: hence it is welfarist (determined utility-wise by the feasible set of profiles), single-valued and easy to compute.

We generalize the Gale-Eisenberg Theorem to a mixed manna. The Competitive division is still welfarist and related to the product of utilities or disutilities. If the zero utility profile (before any manna) is Pareto dominated, the competitive profile is unique and still maximizes the product of utilities. If the zero profile is unfeasible, the competitive profiles are the critical points of the product of disutilities on the efficiency frontier, and multiplicity is pervasive. In particular the task of dividing a mixed manna is either good news for everyone, or bad news for everyone.

We refine our results in the practically important case of linear preferences, where the axiomatic comparison between the division of goods and that of bads is especially sharp. When we divide goods and the manna improves, everyone weakly benefits under the competitive rule; but no reasonable rule to divide bads can be similarly Resource Monotonic. Also, the much larger set of Non Envious and Efficient divisions of bads can be disconnected so that it will admit no continuous selection.

Siblings who inherited several apartments would not be satisfied by an allocation giving them an apartment with probability 50% or envy-free up to one apartment. We suggest a new approach to fair division with valuable items, which bridges the modern "divisible" and "indivisible" literature: sharing minimization. The problem of sharing minimization among fair Pareto-optimal allocations turns out to be algorithmically tractable for almost all instances.

Boltzmann distributions are used in statistical mechanics to describe how the states of a system are distributed at a given temperature. We give a novel characterization of this family as the unique one satisfying independence for uncoupled systems. The theorem boils down to a statement about endomorphisms of the convolution semi-group of finitely supported probability measures on the natural numbers, or, alternatively, about endomorphisms of the multiplicative semi-group of polynomials with non-negative coefficients.

This is the first explicit algorithm for computing market equilibria of "non-convex" exchange economies that have a disconnected equilibrium set. We avoid the "black box" of the cell-enumeration technique used in the literature by a novel approach based on enumerating all the faces of the Pareto frontier via a simple 2-agent reduction. The results are applied to approximately fair division of indivisible chores.

Through a sequence of examples, we survey the main results of "Feasible Joint Posterior Beliefs" [Arieli, Babichenko, Sandomirskiy, Tamuz 2021]. A group of agents share a common prior distribution regarding a binary state, and observe some information structure. What are the possible joint distributions of their posteriors? We discuss feasibility of product distributions, correlation of posteriors in feasible distributions, extreme feasible distributions and the characterization of feasibility in terms of a "no-trade" condition.

A population of voters must elect representatives among themselves to decide on a sequence of possibly unforeseen binary issues. Voters care only about the final decision, not the elected representatives. While an issue-by-issue vote by all voters would maximize social welfare, we are interested in how well the preferences of the population can be approximated by a small committee.

The recent literature on fair Machine Learning manifests that the choice of fairness constraints must be driven by the utilities of the population. However, virtually all previous work makes the unrealistic assumption that the exact underlying utilities of the population (representing private tastes of individuals) are known to the regulator that imposes the fairness constraint. In this paper we initiate the discussion of the mismatch, the unavoidable difference between the underlying utilities of the population and the utilities assumed by the regulator. We demonstrate that the mismatch can make the disadvantaged protected group worse off after imposing the fairness constraint and provide tools to design fairness constraints that help the disadvantaged group despite the mismatch.

In this note, we show that Pareto-optimal and almost-fair (Proportional up to 1 item) allocations of a mixture of indivisible goods and bads always exist and can be computed in strongly-polynomial time. The technique is based on the trading-cycle algorithm for finding divisible Pareto improvements from the paper "Efficient Fair Division with Minimal Sharing" joint with Erel Segal-Halevi and on an extension of Barman-Krishnamurthy rounding.

We compare the Egalitarian rule (aka Egalitarian Equivalent) and the Competitive rule (aka Competitive Equilibrium with Equal Incomes) to divide bads (chores). They are both welfarist: the competitive disutility profile(s) are the critical points of their Nash product on the set of efficient feasible profiles. The C rule is Envy Free, Maskin Monotonic, and has better incentives properties than the E rule. But, unlike the E rule, it can be wildly multivalued, admits no selection continuous in the utility and endowment parameters, and is harder to compute. Thus in the division of bads, unlike that of goods, no rule normatively dominates the other.

We consider repeated zero-sum games with incomplete information on the side of Player 2 with the total payoff given by the non-normalized sum of stage gains. In the classical examples the value of such an N-stage game is of the order of N or square root of N as N goes to infinity. Our aim is to find what is causing another type of asymptotic behavior of the value observed for the discrete version of the financial market model introduced by De Meyer and Saley. For this game Domansky and independently De Meyer with Marino found that the value remains bounded as N goes to infinity and converges to the limit value. This game is almost-fair, i.e., if Player 1 forgets his private information the value becomes zero. We describe a class of almost-fair games having bounded values in terms of an easy-checkable property of the auxiliary non-revealing game. We call this property the piecewise property, and it says that there exists an optimal strategy of Player 2 that is piecewise constant as a function of a prior distribution. Discrete market models have the piecewise property. We show that for non-piecewise almost-fair games with an additional non-degeneracy condition the value is of the order of square root of N.

We discuss the difference Schrödinger equation with the potential given by cotangent, the so-called Maryland model. We obtain explicit renormalization formulas relating its solutions for large coordinates to solutions of the same equation with renormalized parameters and bounded coordinate. These formulas are similar to the renormalization formulas from the theory of Gaussian exponential sums.

The famous theorem of R. Aumann and M. Maschler states that the sequence of values of an N-stage zero-sum game with incomplete information on one side converges as N tends to infinity, and the error term is bounded by a constant divided by the square root of N if the set of states K is finite. The paper deals with the case of infinite K. It turns out that for a countably-supported prior distribution with heavy tails the error term can decrease arbitrarily slowly. The slowest possible speed of decrease for a given distribution is determined in terms of an entropy-like family of functionals. Our approach is based on the well-known connection between the behavior of the maximal variation of measure-valued martingales and asymptotic properties of repeated games with incomplete information.