Cooperative equilibria in finite horizon noncooperative supergames

It is well known that cooperative outcomes can be supported by noncooperative equilibrium strategy combinations in games that are infinitely repeated; however, the example of the repeated prisoner's dilemma has led people to conclude that such cooperative outcomes cannot be sustained if the game is repeated finitely often. In this paper it is proved that the belief about finitely repeated games is false. Conditions are given that allow the existence of cooperative outcomes in finitely repeated games that are supported by subgame perfect noncooperative equilibrium strategies.     

Rational cooperation in the finitely repeated prisoners' dilemma

A common observation in experiments involving finite repetition of the prisoners' dilemma is that players do not always play the single-period dominant strategies (“finking”), but instead achieve some measure of cooperation. Yet finking at each stage is the only Nash equilibrium in the finitely repeated game. We show here how incomplete information about one or both players' options, motivation or behavior can explain the observed cooperation. Specifically, we provide a bound on the number of rounds at which Fink may be played, when one player may possibly be committed to a “Tit-for-Tat” strategy.     

Disappearing private reputations in long-run relationships

For games of public reputation with uncertainty over types and imperfect public monitoring, Cripps et al. [Imperfect monitoring and impermanent reputations, Econometrica 72 (2004) 407-432] showed that an informed player facing short-lived uninformed opponents cannot maintain a permanent reputation for playing a strategy that is not part of an equilibrium of the game without uncertainty over types. This paper extends that result to games in which the uninformed player is long-lived and has private beliefs, so that the informed player's reputation is private. The rate at which reputations disappear is uniform across equilibria and reputations also disappear in sufficiently long discounted finitely repeated games.     

Cooperation without Reputation: Experimental Evidence from Prisoner's Dilemma Games

This paper investigates cooperative play in prisoner's dilemma games by designing an experiment to evaluate the ability of two leading theories of observed cooperation: reputation building and altruism. We analyze both one-shot and finitely repeated games to gauge the importance of these theories. We conclude that neither altruism nor reputation building alone can explain our observations. The reputation model is inconsistent with play in both one-shots and finitely repeated games while the model with altruism is unable to explain observed play in the finitely repeated games.Journal of Economic LiteratureClassification Numbers: C72, C92.     

Measured anxiety affects choices in experimental “clock” games

We measure anxiety by skin conductance response (SCR) in an economic setting. In “clock” games, six agents receive private signals when an asset's price exceeds its fundamental value. They can sell for immediate value or wait to sell at a higher value. Waiting is risky because the price crashes to a lower value when three agents sell. Anxiety could lead people to sell too quickly when the game is played dynamically over time, compared to a static version with precommitted selling. Empirically, delays are shorter in dynamic games than in payoff-equivalent static games, and are associated with anxiety as measured.     

Egalitarianism and Efficiency in Repeated Symmetric Games

We analyze the symmetric equilibria of repeated symmetric games where there is a conflict of interests over equilibria—the battle-of-the-sexes or the hawk–dove game are key examples. If one restricts attention to symmetric equilibria, efficient equilibria must be egalitarian. For finitely repeated games, and generic discount factors, there is a unique outcome path which ensures efficiency within the class of symmetric equilibria. This is also true for the infinitely repeated games if the players are sufficiently impatient. Journal of Economic Literature Classification Numbers: C72, C73.     

UNIQUENESS OF EQUILIBRIUM PAYOFFS IN FINITELY REPEATED GAMES WITH IMPERFECT MONITORING*

This paper examines when a finitely repeated game with imperfect monitoring has a unique equilibrium payoff vector. This problem is nontrivial under imperfect monitoring, because uniqueness of equilibrium (outcome) in the stage game does not extend to finitely repeated games. A (correlated) equilibrium is equilibrium minimaxing if any player's equilibrium payoff is her minimax value when the other players choose a correlated action profile from the actions played in the equilibrium. The uniqueness result holds if all stage game correlated equilibria are equilibrium minimaxing and have the same payoffs. The uniqueness result does not hold under weaker conditions.     

Edgar Allan Poe's riddle: Framing effects in repeated matching pennies games

Framing effects have a significant influence on the finitely repeated matching pennies game. The combination of being labelled “a guesser,” and having the objective of matching the opponent's action, appears to be advantageous. We find that being a player who aims to match the opponent's action is advantageous irrespective of whether the player moves first or second. We examine alternative explanations for our results and relate them to Edgar Allan Poe's “The Purloined Letter.” We propose a behavioral model which generates the observed asymmetry in the players' performance.     

Finitely repeated games with monitoring options

We study finitely repeated games where players can decide whether to monitor the other players? actions or not every period. Monitoring is assumed to be costless and private. We compare our model with the standard one where the players automatically monitor each other. Since monitoring other players never hurts, any equilibrium payoff vector of a standard finitely repeated game is an equilibrium payoff vector of the same game with monitoring options. We show that some finitely repeated games with monitoring options have sequential equilibrium outcomes which cannot be sustained under the standard model, even if the stage game has a unique Nash equilibrium. We also present sufficient conditions for a folk theorem, when the players have a long horizon.     

On Renegotiation-Proof Equilibria in Finitely Repeated Games

This paper studies renegotiation-proof equilibria of Beno?̂t and Krishna (1993) in finitely repeated games with more than two players. We provide a simple characterization of the set of average payoffs from renegotiation-proof equilibria: the limiting set of average equilibrium payoffs is either efficient or dimensionally small as the horizon of the repeated game goes to infinity. Two sufficient conditions for the former to occur are provided. An example shows that those conditions cannot be weakened.Journal of Economic LiteratureClassification Numbers: C72, C73.     

Reputation building in a monetary policy game

This paper examines the notion of reputation building on the part of the central bank as a means of eliminating socially suboptimal inflation rates that arise in monetary policy games. The framework developed here explicitly models the behavior of wage setters, and it is shown that in the Nash equilibrium these private agents do not attain their desired outcome. Hence, wage setters have an incentive for engaging in a reputation-building game with the central bank. In this game, wage setters are allowed to select “optimally” a reputation based wage strategy, thereby making the strategy choice endogenous. This framework thus lays the groundwork for models in which the private sector behaves as a strategic player. It is shown that there exists a wage contract resembling an indexing arrangement which eliminates the suboptimal inflation rate. Finally, a discussion on the ways of restricting the number of permissible solutions to this game is presented.     

Finitely repeated games with social preferences

A well-known result from the theory of finitely repeated games states that if the stage game has a unique equilibrium, then there is a unique subgame perfect equilibrium in the finitely repeated game in which the equilibrium of the stage game is being played in every period. Here I show that this result does in general not hold anymore if players have social preferences of the form frequently assumed in the recent literature, for example in the inequity aversion models of Fehr and Schmidt (Quartely Journal of Economics 114:817–868, 1999) or Bolton and Ockenfels (American Economic Review 100:166–193, 2000). In fact, repeating the unique stage game equilibrium may not be a subgame perfect equilibrium at all. This finding should have relevance for all experiments with repeated interaction, whether with fixed, random or perfect stranger matching.     

A folk theorem for repeated games with unequal discounting

We introduce a “dynamic non-equivalent utilities” (DNEU) condition and the notion of dynamic player-specific punishments for a general repeated game with unequal discounting, both naturally generalizing the stationary counterparts in Abreu et al. (1994). We show that if the DNEU condition, i.e., no pair of players have equivalent utility functions in the repeated game, is satisfied, then any feasible and strictly sequentially individually rational payoff sequence allows dynamic player-specific punishments. Using this result, we prove a folk theorem for unequal discounting repeated games that satisfy the DNEU condition.     

Equilibrium play in matches: Binary Markov games

We study two-person extensive form games, or “matches,” in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in “point games.” We call these matches binary Markov games. We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game; and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. An application to tennis is provided.     

The Explanatory Power of Game Theory in International Politics: Syrian–Israeli Crisis Interactions, 1951–87

The present paper assesses the usefulness of game theory in explaining crisis interactions between Israel and Syria. We begin with the simplest game-theoretic tool for analyzing strategic situations: the one-shot 2×2 game. By analyzing the various episodes of the protracted Syrian--Israeli conflict as one-shot 2×2 games, we avoid the multiplicity of equilibria of the infinitely repeated game and the difficulty of specifying an endpoint of a finitely repeated game. The pure strategy Nash equilibria of these one-shot games are treated as theoretical predictions and are compared with the observed outcomes.     

Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.     

Existence of nontrivial equilibria in repeated games with imperfect private monitoring

We consider finitely repeated games with imperfect private monitoring, and provide several sufficient conditions for such a game to have an equilibrium whose outcome is different from repetition of Nash equilibria of the stage game. Surprisingly, the conditions are consistent with uniqueness of the stage game equilibrium. A class of repeated chicken is shown to satisfy the condition.     

Decompositions and potentials for normal form games

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader's payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.     

Why Any Seatbelt Mandate is an Infinitely Iterated Prisoner’s Dilemma: A Health Economics Perspective

Abstract

Among the most controversial of public health measures are those that restrict people’s freedom of choice presumably for the purpose of protecting their own health and safety. Mandatory use of automobile seatbelts can inspire allusions to the “tyranny of health” and the ‘health police.” This study revisits state seatbelt legislation, which is in force in all but one state of the USA. In modeling seatbelt mandates as indefinitely repeated Prisoner’s Dilemma (PD) games, the study qualifies and overcomes the limitations of conventional interest group and expressive voting theories. It indicates why and how utility-maximizing players, particularly carmakers, insurance companies, consumer advocates and regulators, may choose to cooperate, even in instances where mutual defection would bring each of them higher payoffs. The study suggests why state seatbelt mandates endure, despite ongoing concerns that they undermine individual autonomy or that their social costs outweigh the benefits of law-enforcement. The evolution and mediating effects of reputation and reciprocity in otherwise highly contestable healthcare decisions are consequently explored. Finally, the study identifies the implications of indefinitely repeated games on the continuing nature of strategic relationships and for mitigating conflict over traffic safety regulation, regardless of time and place.     

农村社区公共物品的农户自愿供给——基于声誉理论的分析

农户的声誉不仅能给农户带来直接效用,它还具备信息效应与资本效应,对农户参与农村社区公共物品供给博弈的均衡路径产生重要影响。在独立博弈下,声誉的直接效用能使农户采取合作行动;在关联博弈和重复博弈中,声誉的信息及资本效应对农户的偷懒行为有很强的约束,将导致博弈进入合作均衡。因此基于声誉损益的考虑,由农户自愿供给农村社区内的公共物品会是一个有效的结果。