We study binary coordination games with random utility played in networks. A typical equilibrium is fuzzy - it has positive fractions of agents playing each action. The set of average behaviors that may arise in an equilibrium typically depends on the network. The largest set (in the set inclusion sense) is achieved by a network that consists of a large number of copies of a large complete graph. The smallest set (in the set inclusion sense) is achieved on a lattice-type network. It consists of a single outcome that corresponds to a novel version of risk dominance that is appropriate for games with random utility.
Stationary social learning in a changing environment , with Raphaël Levy and Nicolas Vieille , 2021 , arXiv:2201.02122
We consider social learning in a changing world. Each period, new-born agents observe a ﬁnite sample of past actions and can acquire a signal about the current state before acting. When the state of the world is close to persistent, a consensus in which almost all the population chooses the same action typically emerges. The consensus action is not perfectly correlated with the state though, because the society exhibits some inertia whenever the state changes. The possibility that the state changes drastically limits the value of social learning. Indeed, when signals are too precise – in particular, with perfect signals – actions within a sample are too correlated and even observing unanimous samples is not informative enough to allow herding on past behavior.
We study a reputational war-of-attrition bargaining over a pie with heterogeneous parts, incomplete information over preferences, and behavioral types. To screen across preference types, each player may demand that the opponent chooses from an arbitrary menu of offers. When there is one-sided uncertainty about preferences (and two-sided about the behavioral type), there is a unique limit equilibrium, in which the player with known preferences proposes a menu of all allocations that give him at least his worst complete information bargaining payoff. The outcome is ex ante and ex post efficient. Being able to commit to a menu instead of a single-offer increases equilibrium payoffs of the player with known preferences. Multiple equilibria are possible with two-sided incomplete information about preferences.
This paper characterizes stable stationary equilibria in large population dynamic games. Each player has a type which changes over time. A player’s flow payoff as well as the evolution of her type depends on the distribution of population types and population strategies. A stationary equilibrium is called stable if, after perturbing the equilibrium strategies slightly, revision dynamics converge back to the equilibrium. We derive simple sufficient (and almost necesarry) conditions for stability. These conditions involve eigenvalues of a one-dimensional familiy of matrices. Moreover, in order to check whether an equilibrium is stable, it is enough to consider sine wave perturbations of the equilibrium.
Smooth Stable Matching , 2011
We analyze a continuous version of the Gale-Shapley matching problem. Men and women are represented by a d-dimensional vector of characteristics (such as intelligence, beauty, wealth, etc.) and their preferences over matches with the opposite sex depend only on the respective characteristics. We assume that preferences are monotonic. We show that each differentiable and pairwise stable matching has to satisfy a system of partial di§erential equations. For generic values of parameters, there exists at most one smooth (i.e., analytic) stable matching.
This paper analyzes asynchronous repeated games with private and rich monitoring. We assume that strategies have finite past, i.e., in each period, continuation strategies must be measurable with respect to finite partitions of past histories. This class includes Önite automata and bounded recall strategies. Additionally, we assume that the monitoring has an infinite number of signals. We show that any equilibrium with Önite past and generic infinite monitoring has to satisfy a version of the belief-free property: in each period t; the set of best responses does not depend on the information received before period t; with a possible exception of the information received in the first periods of the game. Under an additional payo§ smoothness assumption, the equilibrium strategy are essentially pastindependent: each periodís action depends only on the formation received immediately prior to the choice of the action.
A joint distribution of an infinite collection of random variables \teta(x) for x \in X is exchangeable, if the joint distributions of any two finite tuples of variables of the same length are equal. A famous result by de Finetti shows that each random variable \teta(x) can be decomposed as an outcome of two kinds of independent shocks: an aggregate shock that affects all variables in the same way and a collection of i.i.d. idiosyncratic shocks that affect each variable separately. In this paper, we present a generalization of the de Finettis Theorem. We assume that all tuples of variables of a given length are divided into finitely many classes of analogy. A joint distribution of all random variables is invariant if the distributions of analogous tuples of variables are equal. Under the finite dimensionality assumption on the system of analogies, we show that each random variable \teta(x) can be decomposed into finitely many independent shocks. These may include the aggregate shock that a§ects all variables, idiosyncratic shocks that affect each variable separately, and shocks that affect the non-trivial subset of variables.
We present a model of two sociological phenomena: the tendencies to form groups and to favor others who are similar. Individuals divide society into friends and enemies. Individual payoffs depend on their own choices and on the choices of others. We assume different types of complementarities: Other things equal, each individual prefers to be friendly towards those who are friendly toward her (second-degree complementarity) and toward those who are friendly toward those … who are friendly toward her (higher-degree complementarity). With second degree complementarities, but no higher-degree externalities, individuals want to reciprocate friendship. Any additional amount of higher-degree complementarities pushes individuals to form groups. Next, we assume everybody may make mistakes that make him confuse individuals who are similar to each other. To minimize the cost of the mistakes, individuals want to keep their friends as different from their enemies as possible. Combined with group formation, individuals would like to be friendly toward others who are similar to them. Although individuals act as if they have preferences for similar others, in reality, their behavior is a best response to the equilibrium behavior of others.
Learning Through Theories , 2007
This paper examines the relationship between language and knowledge in a learning model. An agent describes the world through theories. A theory consists of universal propositions called patterns, and it is formulated in some language. We look at two characteristics of a good theory. A theory is informative if it ensures correct predictions. A theory is brief if it consists of one pattern. We offer different characterizations of informative theories. In particular, we identify languages for which there is no trade-off between both characteristics: Any informative theory logically implies a theory that is informative as well as brief. The results are illustrated with speciÖc problems of reasoning under uncertainty.
Small Group Coordination , 2005 , Job Market Paper
This paper studies a coordination process of a large population. We assume that subgroups of the population differ in the intensity of their coordination problems. Specifically, small (but not large) groups are able to coordinate and simultaneously take a coalitional best response an action, which improves the welfare of all the members of the group. When agents interact on a network, in the long-run they always play an action which is either payoff- or risk-dominant. The outcome depends on the network. In general, the ability of small groups to coordinate may decrease the chances of selecting the efficient outcome. When players interact on many networks at the same time, the dynamics may uniquely select an action, which is neither payoff nor risk dominant. This happens when such an action is sufficiently attractive for small groups. We characterize situations when efficiency is helped or obstructed by the presence of small groups. As an application, we discuss how networks of interactions may affect the probability of self-segregation and polarization.