1. Abstract Two players bargain over a single indivisible good and a transfer, with one-sided incomplete information about preferences. Both players can offer arbitrary mechanisms to determine the allocation. We show that there is a unique perfect Bayesian equilibrium outcome. In the equilibrium, one of the players proposes a menu that is optimal for the uninformed player among all menus, such that each type of the informed player receives at least her payoff under complete information. The optimal menu can be implemented with at most three allocations. Under a natural assumption on the uninformed player's beliefs, the optimal menu coincides with the Myerson's neutral solution to the bargaining problem in this environment.
  2. Abstract We define the distance between two information structures as the largest possible difference in value across all zero-sum games. We provide a tractable characterization of distance and use it to discuss the relation between the value of information in games versus single-agent problems, the value of additional information, informational substitutes, complements, or joint information. The convergence to a countable information structure under value-based distance is equivalent to the weak convergence of belief hierarchies, implying, among other things, that for zero-sum games, approximate knowledge is equivalent to common knowledge. At the same time, the space of information structures under the value-based distance is large: there exists a sequence of information structures where players acquire increasingly more information, and \varepsilon>0 such that any two elements of the sequence have distance of at least \varepsilon. This result answers by the negative the second (and last unsolved) of the three problems posed by J.F. Mertens in his paper “Repeated Games”, ICM 1986.
  3. Abstract We develop a tractable, dynamic, and strategic model of many-to-many matching with payoff externalities across links. The joint dynamic surplus or certain second properties of individual utilities, like payoff externalities, can typically be identified. We characterize a class of interior equilibria as solutions to an optimization problem with an objective function that consists of welfare minus an inefficiency loss term. In equilibrium, too few matches are formed. We compare transferable and nontransferable versions of the model; the equilibria of the two versions are equivalent up to a re-scaling of parameters. We describe the asymptotic limits of disappearing frictions.
  4. Abstract We analyze a large roommate problem (i.e., marriage matching in which the marriage is not restricted solely to matchings between men and women) with non-transferable utility. It is well known that while a roommate problem may not have a stable proper matching, each roommate problem does have an stable improper matching. In a random utility model with types from Dagsvik (2000) and Menzel (2015), we show that all improper stable matchings are asymptotically close to being a proper stable matching. Moreover, the distribution of types in stable matchings (proper or not) converges to the unique maximizer of an expression that is a sum of two terms: the average “welfare” of the matching and the Shannon entropy of the distribution. In the noiseless limit, when the random component of the utility is reduced to zero, the distribution of types of matched pairs converges to the outcome of the transferable utility model.
  5. Abstract We develop a theory of how the value of an agent’s information advantage depends on the persistence of information. We focus on strategic situations with strict conflict of interest, formalized as stochastic zero-sum games where only one of the players observes the state that evolves according to a Markov operator. Operator Q is said to be better for the informed player than operator P if the value of the game under Q is higher than under P regardless of the stage game. We show that this defines a convex partial order on the space of ergodic Markov operators. Our main result is a full characterization of this partial order, intepretable as an ordinal notion of persistence relevant for games. The analysis relies on a novel characterization of the value of a stochastic game with incomplete information.
  6. Abstract We characterize perfect public equilibrium payoffs in dynamic stochastic games in the case where the length of the period shrinks, but players' rate of time discounting and the transition rate between states remain fixed. We present a meaningful definition of the feasible and individually rational payoff sets for this environment, and we prove a folk theorem under imperfect monitoring. Our setting differs significantly from the case considered in previous literature (Dutta (1995), Fudenberg and Yamamoto (2011), and Hörner et al. (2011)) where players become very patient. In particular, the set of equilibrium payoffs typically depends on the initial state.
  7. Abstract We analyze discounted repeated games with incomplete information, such that the players' payoffs depend only on their own type (known-own payoff case). We describe an algorithm for finding all equilibrium payoffs in games for which there exists an open set of belief-free equilibria of Hörner and Lovo (2009). This includes generic games with one-sided incomplete information and a large and important class of games with multi-sided incomplete information. When players become sufficiently patient, all Bayesian Nash equilibrium payoffs can be approximated by payoffs in sequential equilibria in which information is revealed finitely many times. The set of equilibrium payoffs is typically larger than the set of equilibrium payoffs in repeated games without discounting, and larger than the set of payoffs obtained in belief-free equilibria. The results are illustrated in bargaining and oligopoly examples.
  8. Abstract We consider a dynamic economy in which agents are repeatedly matched and decide whether or not to form profitable partnerships. Each agent has a physical color and a social color. An agent's social color acts as a signal, conveying information about the physical color of agents in his partnership history. Before an agent makes a decision, he observes his match's physical and social colors. Neither the physical color nor the social color is payoff relevant. We identify environments where equilibria arise in which agents condition their decisions on the physical and social colors of their potential partners. That is, they discriminate.
  9. Abstract We prove an anti-folk theorem for repeated games with private monitoring. We assume that the strategies have a finite past (they are measurable with respect to finite partitions of past histories), that each period players' preferences over actions are modified by smooth idiosyncratic shocks, and that the monitoring is sufficiently connected. In all repeated game equilibria, each period play is an equilibrium of the stage game. When the monitoring is approximately connected, and equilibrium strategies have a uniformly bounded past, then each period play is an approximate equilibrium of the stage game.
  10. Abstract How can we know in advance whether simplifying assumptions about beliefs will make a difference in the conclusions of game-theoretic models? We define critical types to be types whose rationalizable correspondence is sensitive to assumptions about arbitrarily high-order beliefs. We show that a type is critical if and only if it exhibits common belief in some non-trivial event. We use this characterization to show that all types in commonly used type spaces are critical. On the other hand, we show that regular types (types that are not critical) are generic, although perhaps inconvenient to use in applications.
  11. Abstract Recent literature on testing experts shows that it is difficult, and often impossible, to determine whether an expert knows the stochastic process that generates data. Despite this negative result, we show that often exist contracts that allow a decision maker to attain the first-best payoff in the following sense: in the case in which the expert knows the stochastic process, the decision maker achieves the payoff she would obtain if there were no incentive problems; while in the case in which the expert does not know the stochastic process, she achieves the payoff she would obtain in the absence of any expert. More precisely, this kind of full-surplus extraction is always possible in infinite-horizon models in which future payoffs are not discounted. If future payoffs are discounted (but the discount factor tends to 1), the possibility of full-surplus extraction depends on a constraint involving the forecasting technology
  12. Abstract This paper presents a rational theory of categorization and similarity-based reasoning. I study a model of sequential learning in which the decision maker infers unknown properties of an object from information about other objects. The decision maker may use the following heuristics: divide objects into categories with similar properties and predict that a member of a category has a property if some other member of this category has this property. The environment is symmetric: the decision maker has no reason to believe that the objects and properties are a priori di§erent. In symmetric environments, categorization is an optimal solution to an inductive inference problem. Any optimal solution looks as if the decision maker categorizes. Various experimental observations about similarity-based reasoning coincide with the optimal behavior in my model.
  13. Abstract This paper proposes two (ordinal and cardinal) generalizations of Harsanyi and Selten (1988) risk-dominance to multi-player, multi-action games. There are three reasons why generalized risk-dominance (GR-dominance) is interesting. Extending the logic of riskdominance, GR-dominant actions can be interpreted as best responses to conjectures that satisfy a certain type of symmetry. Second, in a local interaction game of Ellison (1993), if an action is risk-dominant in individual binary interactions with neighbors, it is also GRdominant in the large game on a network. Finally, we show that GR-dominant actions are stochastically stable under a class of evolutionary dynamics. The last observation is a corollary to new abstract selection results that applies to a wide class of so-called asymmetric dynamics. In particular, I show that a (strictly) ordinal GR-dominant proÖle is (uniquely) stochastically stable under the approximate best-response dynamics of Kandori, Mailath, and Rob (1993). A (strictly) cardinal GR-dominant equilibrium is (uniquely) stochastically stable under a class of payo§-based dynamics that includes Blume (1993). Among others, this leads to a generalization of a result from Ellison (2000) on the 1/2-dominant evolutionary selection to all networks and the unique selection to all networks that satisfy a simple, sufficient condition.
  14. Abstract This paper studies repeated games with incomplete information on one side and equal discount factors for both players. The payoffs of the informed player I depend on one of two possible states of the world, which is known to her. The payoffs of the uninformed player U do not depend on the state of the world (that is, U knows his payoffs), but player I's behavior makes knowledge of the state of interest to player U. We define a finitely revealing equilibrium as a Bayesian perfect equilibrium where player I reveals information in a bounded number of periods. We define an ICR profile as a strategy profile in which (a) after each history the players have individually rational payoffs and (b) no type of player I wants to mimic the behavior of the other type. We show that when the players are patient, all Nash equilibrium payoffs in the repeated game can be approximated by payoffs in finitely revealing equilibria, which themselves approximate the set of all ICR payoffs. We provide a geometric characterization of the set of equilibrium payoffs, which can be used for computations.
  15. Abstract This note provides simple necessary and su¢ cient conditions for the comparison of information structures in zero-sum games. This solves an open problem of (Gossner and Mertens 2001). The conditions are phrased in terms of Blackwell garbling of information of each of the players.
  16. Abstract In games with incomplete information, conventional hierarchies of belief are incomplete as descriptions of the players’ information for the purposes of determining a player’s behavior. We show by example that this is true for a variety of solution concepts. We then investigate what is essential about a player’s information to identify behavior. We specialize to two player games and the solution concept of interim rationalizability. We construct the universal type space for rationalizability and characterize the types in terms of their beliefs. Infinite hierarchies of beliefs over conditional beliefs, which we call ∆-hierarchies, are what turn out to matter. We show that any two types in any two type spaces have the same rationalizable sets in all games if and only if they have the same ∆-hierarchies.