Blackwell's celebrated theorem is a cornerstone of information economics, establishing a necessary and sufficient condition for when a distribution F of posterior means can be induced from a prior G: namely, if and only if F majorizes G. While this result provides a deep understanding of the structure of feasible belief distributions, it is non-constructive and does not explain how to design a signal that achieves a given F.

Correlated equilibria arise naturally when agents communicate or rely on intermediaries such as recommendation systems.We study when a given Nash equilibrium can be improved within the set of correlated equilibria for general objectives.Our key insight is a detail-free criterion: any Nash equilibrium with three or more randomizing agents is generically improvable.We refine this insight to specific classes of games and objectives, including Pareto and utilitarian welfare, and provide constructive methods to obtain improvements.Our findings underscore the ubiquity of improvable Nash equilibria and the crucial role of correlation in enhancing strategic outcomes.

We investigate stochasticity in choice behavior across diverse decisions. Each decision is modeled as a menu of actions with associated outcomes, and a stochastic choice rule assigns probabilities to actions based on the outcome profile. We characterize rules whose predictions are not affected by whether or not additional, irrelevant decisions are included in the model. Our main result is that such rules form the parametric family of mixed-logit rules.

We study matching markets with aligned preferences and establish a connection between common design objectives---stability, efficiency, and fairness---and the theory of optimal transport. Optimal transport gives new insights into the structural properties of matchings obtained from pursuing these objectives, and into the trade-offs between different objectives. Matching markets with aligned preferences provide a tractable stylized model capturing supply-demand imbalances in a range of settings such as partnership formation, school choice, organ donor exchange, and markets with transferable utility where bargaining over transfers happens after a match is formed.

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.

We consider a model of Bayesian persuasion with one informed sender and several uninformed receivers. The sender can affect receivers' beliefs via private signals and the sender's objective depends on the combination of induced beliefs.

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.

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 \emph{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.

Bayes-rational agents reside on a social network. They take binary actions sequentially and irrevocably, and the right action depends on an unobservable state. Each agent receives a bounded private signal about the realized state and observes the actions taken by the neighbors who acted before. How does the network topology affect the ability of agents to aggregate the information dispersed over the population by means of the private signals?

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. The disutility of a voter is proportional to the fraction of issues, where his preferences disagree with the decision. 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. We show that a k-sortition (a random committee of k voters with the majority vote within the committee) leads to an outcome within the factor 1+O(1/√ k) of the optimal social cost for any number of voters n, any number of issues m, and any preference profile. For a small number of issues m, the social cost can be made even closer to optimal by delegation procedures that weigh committee members according to their number of followers. However, for large m, we demonstrate that the k-sortition is the worst-case optimal rule within a broad family of committee-based rules that take into account metric information about the preference profile of the whole population.

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. Our main result is that, for the two-agent case, a quantitative version of Aumann's Agreement Theorem provides a necessary and sufficient condition for feasibility. For any number of agents, a related "no-trade" condition likewise provides a characterization of feasibility. We use our characterization to construct joint belief distributions in which agents are informed regarding the state, and yet receive no information regarding the other's posterior. We study a related class of Bayesian persuasion problems with a single sender and multiple receivers, and explore the extreme points of the set of feasible distributions.