Equilibrium in behavior strategies in infinite extensive form games with imperfect information

Summary We prove the existence of equilibrium in behavior strategies for extensive form games when the game has infinite actions. The result is derived under the assumption that the behavior strategies satisfy the bounded measurability condition. The condition implies that the behavior strategies are restricted to those which can be viewed as continuous functions from the set of initial histories to the space of probability distributions over action spaces which satisfy the Lipschitz bound.I am grateful for the helpful comments of an anonymous referee.     

Mixed-strategy equilibria and strong purification for games with private and public information

This paper shows the existence of mixed-strategy equilibria for games with private and public information under general conditions. Under the additional assumptions of finiteness of action spaces and diffuseness and conditional independence of private information, a strong purification result is obtained for the mixed strategies in such games. As a corollary, the existence of pure-strategy equilibria follows. I am very grateful to Yeneng Sun, Nicholas C. Yannelis and M. Ali Khan for helpful discussions and suggestions. I also wish to thank an anonymous referee whose comments led to many improvements in the paper.     

Nash equilibrium in games with incomplete preferences

Summary. This paper investigates Nash equilibrium under the possibility that preferences may be incomplete. I characterize the Nash-equilibrium-set of such a game as the union of the Nash-equilibrium-sets of certain derived games with complete preferences. These games with complete preferences can be derived from the original game by a simple linear procedure, provided that preferences admit a concave vector-representation. These theorems extend some results on finite games by Shapley and Aumann. The applicability of the theoretical results is illustrated with examples from oligopolistic theory, where firms are modelled to aim at maximizing both profits and sales (and thus have multiple objectives). Mixed strategy and trembling hand perfect equilibria are also discussed.Received: 22 September 2003, Revised: 24 June 2004, JEL Classification Numbers: D11, C72, D43.I would like to thank Jean-Pierre Benôit, Juan Dubra, Alejandrio Jofre, Debraj Ray, Kim-Sau Chung and the seminar participants at NYU and at the Universidad de Chile for their comments. I am most grateful to Efe Ok, for his comments, criticism, suggestions and questions.     

Waiting times and equilibrium selection

Summary. This paper shows how graphs can be used to calculate waiting times in models of equilibrium selection. It also shows how reducing the state space can simplify the calculations of both waiting times and selected equilibria. The results are applied to potential games and games with strategic complementarities.Received: 31 December 2002, Revised: 25 October 2003, JEL Classification Numbers: C72, C73.Alan Beggs: I am grateful to an anonymous referee for helpful comments.     

Alternating-Offer Bargaining with Two-Sided Incomplete Information

I study alternating-offer bargaining games with two-sided incomplete information about the players' discount rates. For both perfect Bayesian equilibrium and a rationalizability-style notion, I characterize the set of expected payoffs which may arise in the game. I also construct bounds on agreements that may be made. The set of expected payoffs is easy to compute and incorporate into applied models. My main result is a full characterization of the set of perfect Bayesian equilibrium payoffs for games in which the distribution over the players' discount rates is of wide support, yet is in a weak sense close to a point mass distribution. I prove a lopsided convergence result: each player cannot gain from a slight chance that she is a strong type, but the player can suffer greatly if there is a slight chance that she is a weak type.     

Type correlated equilibria for games with payoff uncertainty

Summary Aumann's notion of correlated equilibrium is extended to games with payoff uncertainty. A type correlated equilibrium is a correlated equilibrium for Harsanyi's game in player-types. An equivalent definition is a probability distribution over types and actions which is consistent with the prior distribution over types, such that when each player observes its type and action, the observed action is optimal and no further information about other players' types is obtained. Any such equilibrium can be implemented by a type-independent correlation device when players' observations may be type-dependent. The type correlated equilibrium correspondence is shown to be upperhemicontinuous with respect to player information.Support from NSF grant IRI-8609208 is gratefully acknowledged. I am grateful to Maxwell Stinchcombe for comments on an earlier draft of this paper. Any remaining errors are my own.     

Knightian games and robustness to ambiguity

This paper introduces a notion of robustness to ambiguous beliefs for Bayesian Nash equilibria. An equilibrium is robust if the corresponding strategies remain approximately optimal for a class of games with ambiguous beliefs that results from an appropriately defined perturbation of the belief structure of the original non-ambiguous belief game. The robustness definition is based on a novel definition of equilibrium for games with ambiguous beliefs that requires equilibrium strategies to be approximate best responses for all measures that define a player's belief. Conditions are derived under which robustness is characterized by a newly defined strategic continuity property, which can be verified without reference to perturbations and corresponding ambiguous belief games.     

Robustness of equilibria in anonymous local games

This paper studies the robustness of symmetric equilibria in anonymous local games to perturbations of prior beliefs. Two priors are strategically close on a class of games if players receive similar expected payoffs in equilibrium under the priors, for any game in that class. I show that if the structure of payoff interdependencies is sparse in a well-defined sense, the conditions for strategic proximity in anonymous local games are strictly weaker than the conditions for general Bayesian games of Kajii and Morris (1998) [11] when attention is restricted to symmetric equilibria. Hence, by exploiting the properties of anonymous local games, it is possible to obtain stronger robustness results for this class.     

On the limit perfect public equilibrium payoff set in repeated and stochastic games

This paper provides a dual characterization of the existing ones for the limit set of perfect public equilibrium payoffs in a class of finite stochastic games (in particular, repeated games) as the discount factor tends to one. As a first corollary, the folk theorems of Fudenberg et al. (1994), Kandori and Matsushima (1998) and Hörner et al. (2011) obtain. As a second corollary, it is shown that this limit set of payoffs is a convex polytope when attention is restricted to perfect public equilibria in pure strategies. This result fails for mixed strategies, even when attention is restricted to two-player repeated games.     

A STATISTICAL THEORY OF EQUILIBRIUM IN GAMES*

This paper describes a statistical model of equiliobrium behaviour in games, which we call Quantal Response Equilibrium (QRE). The key feature of the equilibrium is that individuals do not always play responses to the strategies of their opponents, but play better strategies with higher probability than worse strategies. we illustrate several different applications of this approach, and establish a number of theoretical properties of this equilibrium concept. We also demonstrate an equililance between this equilibrium notion and Bayesian games derived from games of complete information with perturbed payoffs     

Meaning and Credibility in Cheap-Talk Games

I define neologism-proofness, a refinement of perfect Bayesian equilibrium in cheap-talk games. It applies when players have a preexisting common language, so that an unexpected message′s literal meaning is clear, and only credibility restricts communication. I show that certain implausible equilibria are not neologism-proof; in some games, no equilibrium is. Journal of Economic Literature classification numbers: D83 D82 C73.     

An experimental study of information and mixed-strategy play in the three-person matching-pennies game

Summary. Recent experiments on mixed-strategy play in experimental games reject the hypothesis that subjects play a mixed strategy even when that strategy is the unique Nash equilibrium prediction. However, in a three-person matching-pennies game played with perfect monitoring and complete payoff information, we cannot reject the hypothesis that subjects play the mixed-strategy Nash equilibrium. Given this support for mixed-strategy play, we then consider two qualitatively different learning theories (sophisticated Bayesian and naive Bayesian) which predict that the amount of information given to subjects will determine whether they can learn to play the predicted mixed strategies. We reject the hypothesis that subjects play the symmetric mixed-strategy Nash equilibrium when they do not have complete payoff information. This finding suggests that players did not use sophisticated Bayesian learning to reach the mixed-strategy Nash equilibrium. Received: August 9, 1996; revised version: October 21, 1998     

The local best response criterion: An epistemic approach to equilibrium refinement

The standard refinement criteria for extensive form games, including subgame perfect, perfect, perfect Bayesian, sequential, and proper, reject important classes of reasonable Nash equilibria and accept many unreasonable Nash equilibria. This paper develops a new refinement criterion, based on epistemic game theory, that captures the concept of a Nash equilibrium that is plausible when players are rational. I call this the local best response (LBR) criterion. This criterion is conceptually simpler than the standard refinement criteria because it does not depend on out-of-equilibrium, counterfactual, or passage to the limit arguments. The LBR is also informationally richer because it clarifies the epistemic conditions that render a Nash equilibrium reasonable. The LBR criterion appears to render the traditional refinement criteria superfluous.     

Infinitely repeated games with public monitoring and monetary transfers

This paper studies infinitely repeated games with imperfect public monitoring and the possibility of monetary transfers. It is shown that all public perfect equilibrium payoffs can be implemented with a simple class of stationary equilibria that use stick-and-carrot punishments. A fast algorithm is developed that exactly computes the set of pure strategies equilibrium payoffs for all discount factors.     

Dynamic demand and noncompetitive pricing strategies

This paper studies optimal noncompetitive pricing strategies when the evolution of demand is the result of intertemporal considerations. Two different hypotheses of price expectations (myopia and perfect foresight) are treated. The major implication is that the slight modification from an instantaneous to a very fast consumer reaction may completely modify a monopolist's price strategy. More precisely, the price strategy should be volatile if the equilibrium demand is convex, independent whether the consumers act myopically or employ rational expectations. On the other hand, asynchronous dynamics (e.g., due to competitive fringe supply or different segments of demand) cannot explain even damped price oscillations. The equilibrium price strategy of the noncompetitive supplier exceeds the static rule if consumers employ myopic expectations; rational expectations may lead to prices above or below the static rule depending on the rate of discount.I am grateful for the helpful and elaborate comments from three anonymous referees.     

Recursive structure and equilibria in games with private monitoring

Summary. In each stage of a repeated game with private monitoring, the players receive payoffs and privately observe signals which depend on the players' actions and the state of world. I show that, contrary to a widely held belief, such games admit a recursive structure. More precisely, I construct a representation of the original sequential problem as a sequence of static games with incomplete information. This establishes the ground for a characterization of strategies and, hence, of behavior in interactive-decision settings where private information is present. Finally, the representation is used to give a recursive characterization of the equilibrium payoff set, by means of a multi-player generalization of dynamic programming. Received: February 11, 2002; revised version: July 22, 2002 RID="*" ID="*" I am very grateful to In-Koo Cho, Larry Epstein, Denis Gromb, Stephen Morris, Paolo Siconolfi, Lones Smith and Max Stinchcombe for several insights and suggestions. A referee's comments helped improving the exposition. Finally, I wish to thank the participants to the seminars at MEDS, NYU, Columbia University, Caltech, UCLA, University of Rochester, University of Texas-Austin, Northwestern Summer Microeconomics Conference 98, Summer in Tel Aviv 98, and NASM98.     

Fully endogenous mechanism selection on finite outcome sets

Summary This paper defines a choice process over social outcomes in which agents choose the institutional rules ormechanisms themselves without outside interference. Truly endogenizing the mechanism selection process in this way, however, involves facing an infinite regress problem in which outcomes are chosen by games which are themselves chosen by games, ad infinitum. This paper allows the possibility of such an infinite regress which we callfully endogenous mechanism selection.We introduce the notion ofFree Choice which restricts the class of mechanisms in the regress to those which prevent agents from being locked in to an equilibrium outcome by the actions of others. Under this condition, the infinite regress is shown to get truncated with the number of selection iterations endogenously determined. It turns out that the outcomes resulting from a Free Choice-constrained regress are (Weakly) Pareto optimal; in particular, these outcomes solve a weighted Rawlsian Maxmin criterion. We also show that these outcomes are invariant to the equilibrium concept used to evaluate games in the regress.This paper is based on the author's dissertation from the University of Minnesota (November, 1989).I am very grateful for the guidance, advice, and encouragement from my advisor, Marcel K. Richter, and for the many helpful suggestions from David Levine. I have also benefited from conversations with Nabil Al-Najjar, Gerhard Glomm, Leonid Hurwicz, James Jordan, Ramon Marimon, Andrew McClennan, Ariel Rubinstein, and William Thomson.     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.     

Exploitable actions of believers in the “law of small numbers” in repeated constant-sum games

In repeated fixed-pair constant-sum games with unique equilibria in mixed strategies, such as matching pennies, the subgame perfect equilibrium is repeating the stage-game mixed-strategy equilibrium action. In such games rational players avoid strategies that are exploitable, in that current actions either deviate systematically from the equilibrium action probabilities or fail to be serially independent of past actions. I revisit classic experiments and find that subjects’ actions are sometimes exploitable because they are serially dependent. Subjects have difficulty in producing serially independent actions and in recognizing serially dependent sequences due to a bias called local representativeness.