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.     

Inflation: The Invisible Foot of Macroeconomics*

We take a simple, well-known macroeconomic model and treat it as a game between two players—the government and an all-embracing union. The payoffs have the form of a prisoner's dilemma. The equilibrium outcome produces unwanted inflation and is not Pareto optimal. This is despite the fact that all participants are assumed to have full information. This result is shown to be quite robust to the form of the model and is not affected if one of the players is forced to announce its strategy in advance. This we call the invisible foot of macroeconomics.     

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.     

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.     

Which hat to wear? Impact of natural identities on coordination and cooperation

As the workforce becomes increasingly diverse, motivating individuals from different backgrounds to work together effectively is a major challenge facing organizations. In an experiment conducted at a large public university in the United States, we manipulate the salience of participants' multidimensional natural identities and investigate the effects of identity on coordination and cooperation in a series of minimum-effort and prisoner's dilemma games. By priming a fragmenting (ethnic) identity, we find that, compared to the control, participants are significantly less likely to choose high effort in the minimum-effort games, leading to less efficient coordination. In comparison, priming a common organization (school) identity significantly increases the choice of a rational joint payoff maximizing strategy in a prisoner's dilemma game.     

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.     

FINITELY REPEATED PRISONERS' DILEMMA WITH SMALL FINES: THE PENANCE CONTRACT*

This paper investigates finitely repeated prisoners' dilemma games with explicit contractual devices. It shows that full collusion can be achieved in a unique manner by incentivizing the players' final‐period play with small fines. The incentivizing modality is the penance contract, by which a player is penalized if and only if he deviates from the penance strategy in the final period. Using this contractual agreement brings the penance strategy profile into unique subgame perfect equilibrium in the entire game and achieves full collusion without being invalidated by renegotiation.     

Maximum‐revenue versus Optimum‐welfare Export Taxes: a Delegation Game*

In the Eaton and Grossman (1986 ) Bertrand duopoly model of strategic export taxes, both countries may be better off if they both delegate to policymakers who maximize tax revenue rather than welfare. However, both countries delegating to policymakers who maximize tax revenue is not a Nash equilibrium unless the degree of product substitutability is sufficiently high. For a wide range of values for the degree of product substitutability, the game is a prisoner's dilemma where both countries are better off delegating to policymakers who maximize tax revenue but both will delegate to policymakers who maximize welfare in the Nash equilibrium.     

In-class Simulations of the Iterated Prisoner's Dilemma Game

The author developed a simple computer program for the in-class simulation of the repeated prisoner's dilemma game with student-designed strategies. He describes the basic features of the software and presents two examples for the use of the program in teaching the problems of cooperation among profit-maximizing agents.     

Reputation and imperfect information

A common observation in the informal literature of economics (and elsewhere) is that is multistage “games,” players may seek early in the game to acquire a reputation for being “tough” or “benevolent” or something else. But this phenomenon is not observed in some formal game-theoretic analyses of finite games, such as Selten's finitely repeated chain-store game or in the finitely repeated prisoners' dilemma. We reexamine Selten's model, adding to it a “small” amount of imperfect (or incomplete) information about players' payoffs, and we find that this addition is sufficient to give rise to the “reputation effect” that one intuitively expects.     

Patents,imitation and welfare

We consider the effects of product and process patents on profits and welfare. In a duopoly model, we show that if the cost of imitation is not very large, prisoner's dilemma occurs under process patent, thus creating lower profit of each firm under process patent than under product patent. Welfare is higher under process (product) patent for very small (not very small) cost of imitation. Although the possibility of cross-licensing never makes lower welfare under process patent for all costs of imitation, welfare is never lower under product patent under infinitely repeated game.     

Global dynamics in repeated games with additively separable payoffs

This paper studies the global dynamics of a class of infinitely repeated two-player games in which the action space of each player is an interval, and the one-shot payoff of each player is additively separable in actions. We define an immediately reactive equilibrium (IRE) as a pure-strategy subgame perfect equilibrium such that each player's action is a stationary function of the opponent's last action. We completely characterize IREs and their dynamics in terms of certain indifference curves. Our results are used to show that in a prisoners' dilemma game with mixed strategies, gradual cooperation occurs when the players are sufficiently patient, and that in a certain duopoly game, kinked demand curves emerge naturally.     

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.     

Conversion of the Cuban Economy to Soviet Orthodoxy

Our hypothesis is that both the transition phases and the diversity of trajectories of post-communist countries are the result of a significant difference in actors' strategic behavior. If we apply game theory to the socioeconomic context of post-communist evolution, this difference reflects the two main models of cooperation, namely the prisoner's dilemma and the stag hunter. The prisoner's dilemma, which becomes the dominant strategy under the conditions of high social heterogeneity and broken informational channels, implies that it is profitable not to cooperate. Under the stag hunter model — a model involving a common goal and a common project — cooperative strategies are more advantageous. The various post-communist countries in different transition phases can be approximated to either one of those two games — the prisoner's dilemma or the stag hunter. The alternation of the games is conditional on the existence of external and internal social anchors.     

Are smarter groups more cooperative? Evidence from prisoner's dilemma experiments, 1959–2003

Are more intelligent groups better at cooperating? A meta-study of repeated prisoner's dilemma experiments run at numerous universities suggests that students cooperate 5–8% more often for every 100-point increase in the school's average SAT score. This result survives a variety of robustness tests. Axelrod [Axelrod, R., 1984. The Evolution of Cooperation. Basic Books, New York] recommends that the way to create cooperation is to encourage players to be patient and perceptive; experimental evidence suggests that more intelligent groups implicitly follow this advice.     

Annual Index

The authors present a constructivist approach for teaching game theory, on the basis, in part, of Axelrod's research approach. Using the Axelrod tournament multi-user system (ATMUS) software, students create strategies for a repeated prisoner's dilemma (RPD). Later, these strategies are matched with those of their classmates' in a classroom tournament while the instructor interactively and graphically demonstrates the behavior of the strategies. A two- to three-week instructional implementation strategy is provided to highlight effective use of the ATMUS software, according to constructivist learning principles, to ensure that students are engaged in critical thinking regarding RPD.     

When a Precedent of Donation Favors Defection in the Prisoner's Dilemma

Abstract. In this paper, we examine the question of whether a collective activity can influence cooperation in a subsequent repeated one‐shot prisoner's dilemma (PD) game. We conduct two series of experiments. The first consists of control experiments in which 30 periods of a PD game are played, with a random rematching of the pairs in every period. In a second series of experiments, subjects first play a donation game (DG) and then the PD game. In the DG, they collectively discuss the amount of a donation to a given charity, before putting the question to an individual and anonymous vote. Cooperation levels in the PD games preceded by the DG are significantly lower than those observed in the control experiment.     

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.     

A folk theorem for minority games

We study a particular case of repeated games with public signals. In the stage game an odd number of players have to choose simultaneously one of two rooms. The players who choose the less crowded room receive a reward of one euro (whence the name “minority game”). The players in the same room do not recognize each other, and between the stages only the current majority room is publicly announced. We show that in the infinitely repeated game any feasible payoff can be achieved as a uniform equilibrium payoff, and as an almost sure equilibrium payoff. In particular we construct an inefficient equilibrium where, with probability one, all players choose the same room at almost all stages. This equilibrium is sustained by punishment phases which use, in an unusual way, the pure actions that were played before the start of the punishment.     

To obey or not to obey? Can game theory explain human behavior in the context of coronavirus disease?

The objective of the current study is to explain noncompliance to social distancing rules in Western societies in the absence of a stringent law enforcement mechanism and vaccines. In the first part of the analysis, an evolutionary game theory mechanism of two players is developed. The theoretical model assumes the existence of the prisoner's dilemma due to personal inconveniences associated with mask wearing, hand washing, and lockdowns. The model demonstrates that in the absence of sufficient law enforcement mechanism, and regardless of the initial strategy undertaken, one of the three potential equilibria solutions is the convergence of the system to defection of both players. In the second part of the analysis, based on the freedom-house measures, we provide empirical evidence supporting the notion that law enforcement efficiency is higher in autocratic countries. We show the perseverance of higher projected infection rates per 100,000 persons in democratic countries even 8 months after the outbreak of the COVID-19 pandemic. Given the well-known inclination to cooperate more often than expected by game theory, this real-life outcome of noncompliance is remarkable. Moreover, the recent protests against lockdowns in China might reflect a shift from one equilibrium point (cooperation) to another (noncompliance).