Changes in risk and strategic interaction

In this paper we study the comparative statics of Nth degree stochastic dominance shifts in a large class of non-cooperative games. We consider symmetric equilibria as well as asymmetric equilibria in which the risk changes are idiosyncratic and not necessarily of the same stochastic order. Furthermore, we establish conditions for risk changes to produce multiplier effects on equilibrium strategies. Finally, we evaluate the comparative statics of stochastic dominance shifts in supermodular games, which may feature multiple equilibria and non-convex strategy sets.     

Observable implications of Nash and subgame-perfect behavior in extensive games

We provide necessary and sufficient conditions for observed outcomes in extensive game forms, in which preferences are unobserved, to be rationalized first, weakly, as a Nash equilibrium and then as the unique subgame-perfect equilibrium. Thus, one could use these conditions to find that play is (a) consistent with subgame-perfect equilibrium, or (b) not consistent with subgame-perfect behavior but is consistent with Nash equilibrium, or (c) consistent with neither.     

Repeated two-person zero-sum games with unequal discounting and private monitoring

We consider discounted repeated two-person zero-sum games with private monitoring. We show that even when players have different and time-varying discount factors, each player’s payoff is equal to his stage-game minmax payoff in every sequential equilibrium. Furthermore, we show that: (a) in every history on the equilibrium path, the pair formed by each player’s conjecture about his opponent’s action must be a Nash equilibrium of the stage game, and (b) the distribution of action profiles in every period is a correlated equilibrium of the stage game. In the particular case of public strategies in public monitoring games, players must play a Nash equilibrium after any public history.     

Two definitions of correlated equilibrium

Correlated equilibrium constitutes one of the basic solution concepts for static games with complete information. Actually two variants of correlated equilibrium are in circulation and have been used interchangeably in the literature. Besides the original notion due to Aumann (1974), there exists a simplified definition typically called canonical correlated equilibrium or correlated equilibrium distribution. It is known that the original and the canonical version of correlated equilibrium are equivalent from an ex-ante perspective. However, we show that they are actually distinct – both doxastically as well as behaviourally – from an interim perspective. An elucidation of this difference emerges in the reasoning realm: while Aumann’s correlated equilibrium can be epistemically characterized by common belief in rationality and a common prior, canonical correlated equilibrium additionally requires the condition of one-theory-per-choice. Consequently, the application of correlated equilibrium requires a careful choice of the appropriate variant.     

Estimation of finite sequential games

I propose a new estimation method for finite sequential games that is efficient, computationally attractive, and applicable to a fairly general class of finite sequential games that is beyond the scope of existing studies. The major challenge is the computation of high-dimensional truncated integration whose domain is complicated by strategic interaction. This complication resolves when unobserved off-the-equilibrium-path strategies are controlled for. Separately evaluating the likelihood contribution of each subgame-perfect equilibrium that generates the observed outcome allows the use of the GHK simulator, a widely used importance-sampling probit simulator. Monte Carlo experiments demonstrate the performance and robustness of the proposed method.     

Non-cooperative games on hyperfinite Loeb spaces

We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large non-anonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover the case when the action sets are taken to be the unit interval, results now known to be false when they are based on more familiar measure spaces such as the Lebesgue unit interval. We also emphasize three criteria for the modelling of such game-theoretic situations—asymptotic implementability, homogeneity and measurability—and argue for games on hyperfinite Loeb spaces on the basis of these criteria. In particular, we show through explicit examples that a sequence of finite games with an increasing number of players or sample points cannot always be represented by a limit game on a Lebesgue space, and even when it can be so represented, the limit of an existing approximate equilibrium may disappear in the limit game. Thus, games on hyperfinite Loeb spaces constitute the ‘right' model even if one is primarily interested in capturing the asymptotic nature of large but finite game-theoretic phenomena.     

Bounded memory and permanent reputations

A recent result in repeated incomplete information games is that after an arbitrarily long history, any equilibrium of the continuation game must be an equilibrium of the complete information game. This result implies that reputation is a short-run phenomenon. We study a particular class of reputation games and show that bounded memory may lead to permanent reputations. In fact, for a particular range of parameters, a bounded memory player may never be able to learn anything at all.     

MACROECONOMIC POLICY EVALUATION AND OPTIMAL CONTROL THEORY: A CRITICAL REVIEW OF SOME RECENT DEVELOPMENTS

Abstract. This survey addresses the recent literature on the application of optimal control theory and game theory to macroeconomic policy evaluation and design. This literature focuses on strategic interactions between governments and private agents engaged in dynamic non-cooperative games and emphasizes such issues as precommitment, credibility and reputation as important and endemic aspects of optimal policy design. A dominant theme is the problem of dynamic inconsistency and the inferiority of an equilibrium in single-stage full-information games without commitment. This may be alleviated in repeated games involving reputation effects and threat strategies designed to coerce rivals into pursuing particular strategies and to sustain a sequential equilibrium. The type and characteristics of a sequential equilibrium also depend importantly upon the information structure conditioning players'beliefs under uncertainty. Included here are separating and pooling equilibria, involving notions of learning, signalling and information revelation. Additional dimensions exist in stochastic systems which introduce other forms of uncertainty. Optimal control and game theory have also been applied to international macroeconomic policy design, with emphasis on the potential costs and benefits of non-cooperative and cooperative behaviour between countries and problems of international policy coordination.     

Implementability via protective equilibria

We present a notion of non-cooperative strategic equilibrium for games generated by social choice functions, and fully characterize the class of those functions which are directly implementable under this equilibrium concept. Correct preference revelation turns out to always be such an equilibrium for the games generated by this class of implementable functions.     

Dynamic coordination among heterogeneous agents

We study a dynamic model of coordination with timing frictions and payoff heterogeneity. There is a unique equilibrium, characterized by thresholds that determine the choices of each type of agent. We characterize equilibrium for the limiting cases of vanishing timing frictions and vanishing shocks to fundamentals. A lot of conformity emerges: despite payoff heterogeneity, agents’ equilibrium thresholds partially coincide as long as a set of beliefs that would make this coincidence possible exists. However, the equilibrium thresholds never fully coincide. In case of vanishing frictions, the economy behaves almost as if all agents were equal to an average type. Conformity is not inefficient. In the efficient solution, agents follow others even more often.     

A dominance solvable global game with strategic substitutes

Global games emerged as an approach to equilibrium selection. For a general setting with supermodular payoffs, unique selection of equilibrium has been obtained through iterative elimination of strictly dominated strategies. For the case of global games with strategic substitutes, uniqueness of equilibrium has not been proved by iterative elimination of strictly dominated strategies, making the equilibrium less appealing. In this work we provide a condition for dominance solvability in a simple three-player binary-action global game with strategic substitutes. This opens an unexplored research agenda on the study of global games with strategic substitutes.     

Escape dynamics and equilibria selection by iterative cycle decomposition

This paper explores the medium-run behaviour of bounded rational players in repeatedly played games when they occasionally experiment or make mistakes. The formal analysis introduces a hierarchical structure of limit sets to characterize the most possible medium-run behaviour over gradually increased time intervals. The paper refines the notion of stochastic stability and offers a precise measure of the speed at which stochastically stable equilibria occur. Finally, the paper applies the results to a 3×3 symmetric game of Young (1993).     

Potential games with incomplete preferences

This paper studies potential games allowing the possibility that players have incomplete preferences and empty best-response sets. We define four notions of potential games, ordinal, generalized ordinal, best-response, and generalized best-response potential games, and characterize them using cycle conditions. We study Nash equilibria of potential games and show that the set of Nash equilibria remains the same when every player’s preferences are replaced with the smallest generalized (best-response) potential relation or a completion of it. Similar results are established about strict Nash equilibria of ordinal and best-response potential games. Lastly, we examine the relations among the four notions of potential games as well as pseudo-potential games.     

Endogenous mechanisms and Nash equilibrium in competitive contracting games

We model strategic competition in a market with asymmetric information as a noncooperative game in which each firm competes for the business of a buyer of unknown type by offering the buyer a catalog of products and prices. The timing in our model is Stackelberg: in the first stage, given the distribution of buyer types known to all firms and the deducible, type-dependent best responses of the agent, firms simultaneously and noncooperatively choose their catalog offers. In the second stage the buyer, knowing his type, chooses a single firm and product-price pair from that firm’s catalog. By backward induction, this Stackelberg game with asymmetric information reduces to a game over catalogs with payoff indeterminacies. In particular, due to ties within catalogs and/or across catalogs, corresponding to any catalog profile offered by firms there may be multiple possible expected firm payoffs, all consistent with the rational optimizing behavior of the agent for each of his types. The resolution of these indeterminacies depends on the tie-breaking mechanism which emerges in the market. Because each tie-breaking mechanism induces a particular game over catalogs, a reasonable candidate would be a tie-breaking mechanism which supports a Nash equilibrium in the corresponding catalog game. We call such a mechanism an endogenous Nash mechanism. The fundamental question we address in this paper is, does there exist an endogenous Nash mechanism—and therefore, does there exist a Nash equilibrium for the catalog game? We show under fairly mild conditions on primitives that catalog games naturally possess tie-breaking mechanisms which support Nash equilibria.     

Transitions between equilibria in bilingual games under logit choice

We study the effect of introducing a bilingual option on the long run equilibrium outcome in a class of two-strategy coordination games with distinct payoff and risk dominant equilibria under the logit choice rule. Existing results show that in the class of two-strategy games under consideration, the inefficient risk dominant equilibrium is selected in the long run under noisy best response models. We show that if the cost of the bilingual option is sufficiently low then the efficient payoff dominant equilibrium will be selected in the long run under the logit choice rule.     

Behavioral equilibrium and evolutionary dynamics in asset markets

This paper analyzes a dynamic stochastic equilibrium model of an asset market based on behavioral and evolutionary principles. The core of the model is a non-traditional game-theoretic framework combining elements of stochastic dynamic games and evolutionary game theory. Its key characteristic feature is that it relies only on objectively observable market data and does not use hidden individual agents’ characteristics (such as their utilities and beliefs). A central goal of the study is to identify an investment strategy that allows an investor to survive in the market selection process, i.e., to keep with probability one a strictly positive, bounded away from zero share of market wealth over an infinite time horizon, irrespective of the strategies used by the other players. The main results show that under very general assumptions, such a strategy exists, is asymptotically unique and easily computable.     

Markov Perfect equilibria in repeated asynchronous choice games

This paper examines the issue of multiplicity of Markov Perfect equilibria in alternating move repeated games. Such games are canonical models of environments with repeated, asynchronous choices due to inertia or replacement. Our main result is that the number of Markov Perfect equilibria is generically finite with respect to stage game payoffs. This holds despite the fact that the stochastic game representation of the alternating move repeated game is “non-generic” in the larger space of state dependent payoffs. We further obtain that the set of completely mixed Markov Perfect equilibria is generically empty with respect to stage game payoffs.     

A PRIMER ON GLOBAL GAMES APPLIED TO MACROECONOMICS AND FINANCE

This paper shows how to solve global games applied to macroeconomics and finance. We ascertain the roles of public and private information for the determination of a unique equilibrium, and discuss the informative role of market prices. We examine the impact of public information on social welfare, comparing models with and without complementarities at the aggregate level.     

Solving dynamic public insurance games with endogenous agent distributions: Theory and computational approximation

We make two contributions in this paper. First, we extend the characterization of equilibrium payoff correspondences in history-dependent dynamic policy games to a class with endogenously heterogeneous private agents. In contrast to policy games involving representative agents, this extension has interesting consequences as it implies additional nonlinearity (i.e., bilinearity) between the game states (distributions) and continuation/promised values in the policymaker’s objective and incentive constraints. The second contribution of our paper is in addressing the computational challenges arising from this payoff-relevant nonlinearity. Exploiting the game’s structure, we propose implementable approximate bilinear programming formulations to construct estimates of the equilibrium value correspondence. Our approximation method respects the property of upper hemicontinuity in the target correspondence. We provide small-scale computational examples as proofs of concept.     

Coherent Dempster–Shafer equilibrium and ambiguous signals

This paper reappraises the Dempster–Shafer equilibrium, a novel solution concept for signaling games introduced by Eichberger and Kelsey (2004), and suggests a new refinement approach. It is demonstrated that if the types of the Sender–but not messages–are assumed to be ex-ante unambiguous, then the Receiver’s conditional Choquet preference derived by the Dempster–Shafer updating rule coincides with subjective expected utility. This property of the pessimistic updating rule narrows the pooling, but not separating, Dempster–Shafer equilibrium to be behaviorally equivalent to the perfect Bayesian equilibrium. Moreover, if one refines the separating Dempster–Shafer equilibrium à la Ryan (2002a) by imposing the belief persistence axiom, then no deviations from the perfect Bayesian equilibrium are feasible. To eliminate Ryan’s type of behavior, a less stringent refinement based on the notion of coherent beliefs is elaborated.