Evolution in Finitely Repeated Coordination Games

I apply a version of Kandori et al. (1993, Econometrica, 61, 29–56) and Young's (1993, Econometrica, 61, 57–84) evolutionary dynamic to finitely repeated coordination games. The dynamic is modified by allowing mutations to affect only off path beliefs. I find that repetition within a match leads agents to sacrifice current payoffs in order to increase payoffs in later stages. As a consequence, evolution leads to (almost) efficiency. Journal of Economic Literature Classification Number: C79.     

Finitely Many Players with Bounded Recall in Infinitely Repeated Games

We study the set of limit points of equilibrium payoffs in n-player repeated games, with bounded recall, when the memory capacities of all the players grow to infinity. Two main issues are explored: (i) whether differential information enables players to play correlatively, and (ii) the extent to which boundedly rational players can learn others′ behavior patterns and conceal their own. Journal of Economic Literature Classification Number: 026.     

Nash Equilibria of Repeated Games with Observable Payoff Vectors

We study a model of repeated games with imperfect monitoring where the payoff vector is observable. In this situation, any profitable deviation is detectable by all the players but the identity of the deviator may be unknown. We design collective punishments directed against the set of potential deviators. A particular class of signals is studied for which a characterization of the set of equilibrium payoffs is obtained. Journal of Economic Literature Classification Numbers: C73.     

A Property of Nash Equilibria in Repeated Games with Incomplete Information

We consider two-person non-zero-sum infinitely repeated games with lack of information on one side. The characterization of Nash equilibrium payoffs obtained by Hart allows for complex strategies, which are actually required by some equilibrium payoffs in some games. We show that appropriate one-shot public communication mechanisms make Nash equilibrium payoffs achievable by means of simple strategies. Furthermore, these mechanisms satisfy a notion of self-fulfillment.Journal of Economic LiteratureClassification Numbers: D82, C72.     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.     

On the Existence of Undominated Nash Equilibria in Normal Form Games

In this paper the existence problem of undominated Nash equilibrium in normal form games is analyzed. It is shown that an undominated Nash equilibrium exists, if (a) strategy sets are convex polytopes inRⁿand (b) utility functions are affine with respect to each player's own strategy. It is shown by counterexamples that, first, it is not sufficient to have concave utility functions instead of affine under condition (b) even when condition (a) is satisfied, and, second, it is not sufficient to have just compact and convex strategy sets instead of polytopes in condition (a) even when condition (b) is satisfied.     

On the Number of Pure Strategy Nash Equilibria in Random Games

How many pure Nash equilibria can we expect to have in a finite game chosen at random? Solutions to the above problem have been proposed in some special cases. In this paper we assume independence among the profiles, but we allow either positive or negative dependence among the players' payoffs in a same profile. We provide asymptotic results for the distribution of the number of Nash equilibria when either the number of players or the number of strategies increases. We will show that different dependence assumptions lead to different asymptotic results. Journal of Economic Literature Classification Number C72.     

Subgame Perfect Equilibria in Stage Games

It is well known that a stage game with infinite choice-sets, unless it contains a public coordination-device in each stage, may have no subgame perfect equilibria. We show that if a game with public coordination-devices has a subgame perfect equilibrium in which two players in each stage use non-atomic strategies, then the game without coordination devices also has a subgame perfect equilibrium. Journal of Economic Literature Classification Numbers: C6, C7, D8.     

Finitely Repeated Voluntary Provision of a Public Good

This paper theoretically explores the voluntary provision of a public good when either one of the following holds: (i) agents’ utility is the sum of their monetary earnings and a nonmaterial component, or (ii) agents’ exhibit satisficing behavior. We show that a small degree of either nonmaterial payoffs or satisficing behavior can generate large contributions in a finitely repeated game, even if the incentive to free‐ride on others’ contributions calls for negligible public good provision in the static game. The equilibrium is characterized by a sharp decline in contributions toward the end of the game. Several comparative results regarding group size and technology are consistent with laboratory data obtained in experimental studies. The model also predicts the restart effect observed in previous experiments.     

On the Irrelevance of Risk Attitudes in Repeated Two-Outcome Games

We study equilibrium and maximin play in supergames consisting of the sequential play of a finite collection of stage games, where each stage game has two outcomes for each player. We show that for two-player supergames in which each stage game is strictly competitive, in any Nash equilibrium of the supergame, play at each stage is a Nash equilibrium of the stage game provided preferences over certain supergame outcomes satisfy a natural monotonicity condition. In particular, equilibrium play does not depend on risk attitudes. We establish an invariance result for games with more than two players when the solution concept is subgame perfection. Journal of Economic Literature Classification Numbers: C72, C9.     

Evolutionary Stability in the Finitely Repeated Prisoner 's Dilemma Game

The dynamic evolutionary stability of mutual defection is proven for the repeated prisoner 's dilemma game where payoffs are cumulative and the number of repetitions is known. This agrees with the classical result that the only Nash equilibrium outcome is to defect at all stages of this repeated game. Moreover, it is shown that, for any initial polymorphic population, the evolutionary dynamic converges to a unique Nash equilibrium strategy that depends on the original polymorphism. Both these results confirm earlier conjectures concerning the application of evolutionary game theory to the repeated prisoner 's dilemma.Journal of Economic LiteratureClassification Numbers: C70, C72.     

Renegotiation in Repeated Games with Side-Payments

We consider repeated games with side-payments: players have an endowment of wealth in each period in which transfers can be made. We show that if endowments are large enough and the common discount factor high enough, then a strongly renegotiation–proof equilibrium (SRP) in the sense of Farrell and Maskin exists. As the discount factor goes to 1, the set of SRP payoffs converges to the set of efficient, individually rational payoffs. These results provide a justification for the efficiency principle when agreements are not enforceable. Journal of Economic Literature Classification Numbers: C73, D23, L14.     

Repeated Games with Bounded Entropy

We investigate the asymptotic behavior of the maxmin values of repeated two-person zero-sum games with a bound on the strategic entropy of the maximizer's strategies while the other player is unrestricted. We will show that if the bound η(n), a function of the number of repetitions n, satisfies the condition η(n)/n → γ (n → ∞), then the maxmin value Wⁿ(η(n)) converges to (cav U)(γ), the concavification of the maxmin value of the stage game in which the maximizer's actions are restricted to those with entropy at most γ. A similar result is obtained for the infinitely repeated games. Journal of Economic Literature Classification Numbers: C73, C72.     

Repeated Games with Almost-Public Monitoring

In repeated games with imperfect public monitoring, players can use public signals to coordinate their behavior, and thus support cooperative outcomes, but with private monitoring, such coordination may no longer be possible. Even though grim trigger is a perfect public equilibrium (PPE) in games with public monitoring, it often fails to be an equilibrium in arbitrarily close games with private monitoring. If a PPE has players' behavior conditioned only on finite histories, then it induces an equilibrium in all close-by games with private monitoring. This implies a folk theorem for repeated games with almost-public almost-perfect monitoring. Journal of Economic Literature Classification Numbers: C72, C73.     

Evolutionary Stability of Pure-Strategy Equilibria in Finite Games

Sufficient conditions for pure-strategy Nash equilibria of finite games to be (Lyapunov) stable under a large class of evolutionary dynamics, the regular monotonic selection dynamics, are discussed. In particular, it is shown that in almost all finite extensive-form games, all the pure-strategy equilibria are stable. In such games, all mixed-strategy equilibria close to pure-strategy equilibria are also stable.Journal of Economic LiteratureClassification Numbers: C70, C72.     

Markov Equilibria of Stochastic Games with Complementarities

The existence of Markov equilibria for stochastic games with a continuum of states is a complex issue for which no general result holds as yet. In this article, the problem is solved for a class of stochastic games that satisfy assumptions of complementarity and monotonicity. The proof of existence relies on results from lattice programming. In the Markov equilibria singled out by the Theorem of Existence, the policies and continuation values are increasing and Lipschitz continuous functions of the state variable.Journal of Economic LiteratureClassification Numbers: C62, C73.     

Sowing Doubt Optimally in Two-Person Repeated Games

Consider a two-person repeated game, where one of the players, P1, can sow doubt, in the mind of his opponent, as to what P1's payoffs are. This results in a two-person repeated game with incomplete information. By sowing doubt, P1 can sometimes increase his minimal equilibrium payoff in the original game. We prove that this minimum is maximal when only one payoff matrix, the negative of the payoff matrix of the opponent, is added (the opponent thus believes that he might play a zero-sum game). We obtain two formulas for calculating this maximal minimum payoff. Journal of Economic Literature Classification Numbers: C7, D8.     

Perfect and Proper Equilibria of Large Games

This paper deals with the existence and other related issues of perfect and proper equilibria of games with a continuum of players. A sufficient condition for the existence of a perfect (proper) equilibrium as an almost everywhere limit of a sequence of ε-perfect (ε-proper) equilibria is given. An example shows that almost everywhere convergence need not obtain if the condition is violated. Extension to the case where the set of actions available to the players can differ is discussed.Journal of Economic LiteratureClassification Number: C79.     

Quantal Response Equilibria for Extensive Form Games

This article investigates the use of standard econometric models for quantal choice to study equilibria of extensive form games. Players make choices based on a quantal-choice model and assume other players do so as well. We define an agent quantal response equilibrium (AQRE), which applies QRE to the agent normal form of an extensive form game and imposes a statistical version of sequential rationality. We also define a parametric specification, called logit-AQRE, in which quantal-choice probabilities are given by logit response functions. AQRE makes predictions that contradict the invariance principle in systematic ways. We show that these predictions match up with some experimental findings by Schotter et al. (1994) about the play of games that differ only with respect to inessential transformations of the extensive form. The logit-AQRE also implies a unique selection from the set of sequential equilibria in generic extensive form games. We examine data from signaling game experiments by Banks et al. (1994) and Brandts and Holt (1993). We find that the logit-AQRE selection applied to these games succeeds in predicting patterns of behavior observed in these experiments, even when our prediction conflicts with more standard equilibrium refinements, such as the intuitive criterion. We also reexamine data from the McKelvey and Palfrey (1992) centipede experiment and find that the AQRE model can account for behavior that had previously been explained in terms of altruistic behavior. This revised version was published online in August 2006 with corrections to the Cover Date.     

Communication in Repeated Games with Private Monitoring

The paper considers repeated games where each player can be observed by only a subset of the other players, and where players can make public announcements about the behavior of the players they observed. We address the following question: What is the minimal level of observability that is required to obtain efficient outcomes? The main result is that the limit set of sequential equilibrium payoffs, when the discount factor tends to one, contains the set of individual rational payoffs whenever each player is observed by at leasttwoother players.Journal of Economic LiteratureClassification Number: C73.