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.     

Testing threats in repeated games

Under most game-theoretic solution concepts, equilibrium beliefs are justified by off-equilibrium events. I propose an equilibrium concept for infinitely repeated games, called “Nash Equilibrium with Tests” (NEWT), according to which players can only justify their equilibrium beliefs with events that take place on the equilibrium path itself. In NEWT, players test every threat that rationalizes a future non-myopic action that they take. The tests are an integral part of equilibrium behavior. Characterization of equilibrium outcomes departs from the classical “folk theorems”. The concept provides new insights into the impact of self-enforcement norms, such as reciprocity, on long-run cooperation.     

EVOLUTIONARY DRIFT IN PREFERENCE AND EQUILIBRIUM SELECTION*

This paper studies the situation where myopic players repeatedly face a society with two actions where their common fitness exhibits economies of scale. Both states in which all players choose the same actions are equilibria. In each period, players adjust their actions based on their preferences, which are in turn shaped by natural selection. The preferences of the players need not match the underlying fitness. When rare mutations are introduced into the evolutionary process, their preferences may drift without affecting equilibrium behaviour. This paper shows that these drifts influence the results of equilibrium selection.     

A resolution of N-person prisoners' dilemma

Summary. The purpose of this paper is to show that cooperation is possible in N-person prisoners' dilemma if players have incomplete information on the order of their moves. We consider a modified version of N-person prisoners' dilemma in which players sequentially move in the order determined by Nature, and during the play they get some information about others. We provide an information partition and a condition of payoffs for which cooperation is attained in an equilibrium. Further, for the case that full cooperation is not attained, we examine the largest partial cooperation which is achieved in an equilibrium. Received: March 5, 1995; revised version August 11, 1996     

Contracting with externalities and outside options

This paper proposes a model of multilateral contracting where players are engaged in two parallel interactions: they dynamically form coalitions and play a repeated normal form game with temporary and permanent decisions. We show that when outside options are independent of the actions of other players all Markov perfect equilibrium without coordination failures are efficient, regardless of externalities created by interim actions. Otherwise, in the presence of externalities on outside options, all Markov perfect equilibrium may be inefficient. This formulation encompasses many economic models, and we analyze the distribution of coalitional gains and the dynamics of coalition formation in four illustrative applications.     

Learning from private information in noisy repeated games

We study the perfect type-contingently public ex-post equilibrium (PTXE) of repeated games where players observe imperfect public signals of the actions played, and both the payoff functions and the map from actions to signal distributions depend on an unknown state. The PTXE payoffs when players are patient are determined by the solutions to a family of linear programming problems. Using this characterization, we develop conditions under which play can be as if the players have learned the state. We provide a sufficient condition for the folk theorem, and a characterization of the PTXE payoffs in games with a known monitoring structure.     

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.     

Bilateral commitment

We consider non-cooperative environments in which two players have the power to gradually and unilaterally rule out some of their actions. Formally, we embed a strategic-form game into a multi-stage game, in which players can restrict their action spaces in all but the final stage, and select among the remaining actions in the last stage. We say that an action profile is implementable by commitment if this action profile is played in the last stage of a subgame-perfect equilibrium path. We provide a complete characterization of all implementable action profiles and a simple method to find them. It turns out that the set of implementable profiles does not depend on the length of the commitment process. We show, furthermore, that commitments can have social value in the sense that in some games there are implementable action profiles that dominate all Nash equilibria of the original game.     

Cooperation in Stochastic OLG games

This paper builds on Cremer's (Quart. J. Econ. 101 (1986) 33) seminal analysis which shows that (almost) complete cooperation can be achieved as an equilibrium in a game played by overlapping generations of players if the institution in which players cooperate is infinitely lived.We analyze a similar model in which the costs of cooperation are subject to random shocks. Even if these random shocks are very small, the range of parameters for which cooperation can be sustained is decreased considerably in comparison to the deterministic case. Furthermore, we show how the efficient outcome can be approximated if the level of cooperation can be varied continuously and the cooperation technology has decreasing or constant returns to scale, while this is not possible in the case of increasing returns to scale.     

The folk theorem for repeated games with observation costs

The folk theorem literature has been relaxing the assumption on how much players know about each other's past action. Here we consider a general model where players can “buy” precise information. Every period, each player decides whether to pay a cost to accurately observe the actions chosen by other players in the previous period. When a player does not pay the cost, he obtains only imperfect private signals. Observational decisions are unobservable to others. Known strategies such as trigger strategies do not work since they fail to motivate players to pay for information. This paper shows that the folk theorem holds for any level of observation costs. Unlike existing folk theorems with private monitoring, ours imposes virtually no restriction on the nature of costless imperfect signals. The theorem does not use explicit or costless communication, thereby having implications on antitrust laws that rely on evidence of explicit communication. The main message is that accurate observation alone, however costly, enables efficient cooperation in general repeated games.     

Dynamic beauty contests: Learning from the winners to win?

I build a dynamic game consisting of a continuum of players to investigate the effects of previous winners’ actions on the spreading of subsequent players’ actions. In each stage, besides the private signal, each player also observes actions taken by the winners of all previous stages as public signals. A unique equilibrium of the game is found and characterized. I then define the variances of three forms of gap: the gap between the average play and the underlying fundamental value, the gap between a generic player’s action and the average play, and the gap between a generic player’s action and the winner’s play. By checking their dynamics in the equilibrium, it is shown that the accumulation of private signals always reduces the first variance and the accumulation of the public signals always reduces the second variance. However, the effects of accumulated public (private) signals on the first (second) variance are rather ambiguous. Based on a theoretical finding that expresses the third variance as a weighted sum of the other two, I conduct an empirical study on the Miss Korea pageant during 1994–2013. I find a descending trend in the variance of the gap between the average face and the underlying “true beauty” face over these years. Moreover, this process is accompanied by ascending trends in the other two variances, indicating that contestants’ faces have been converging to the “true beauty” overall but diverging from each other over the two decades, which is consistent with our theoretical results.     

The effects of costless pre-play communication: Experimental evidence from games with Pareto-ranked equilibria

Cheap talk is shown to facilitate coordination on the unique efficient equilibrium in experimental order-statistic games. This result is roughly consistent with theoretical predictions according to which cheap talk promotes efficient Nash play. The evidence concerning the mechanisms that theory appeals to is mixed: Frequent agreement of messages and actions is consistent with messages being viewed as self-committing. Risk in the underlying game and the absence of self-signaling messages may explain why message profiles are not unanimous. Time-varying message profiles can be interpreted as evidence for players trying to negotiate equilibria and/or trying to rely on secret handshakes.     

Inequality, cooperation, and growth: An experimental study

In a three player dynamic public goods experiment, social output today determines production possibilities tomorrow. In each period, players choose to sabotage, to cooperate, or to play best response. Sabotage harms social output and growth. Mutual cooperation maximises both. The property rights to social output are distributed unequally. Extent and skew of inequality are varied. We observe equilibrium play in most cases. There is also substantial cooperation, but little sabotage. Our exogenous variations of inequality are neutral to growth, neither negatively correlated to cooperation, nor positively correlated to sabotage. The neutrality result is shown to be sensitive to the dynamic nature of the employed game.     

Program equilibrium

In a computerized setting, players' strategies can be implemented by computer programs, to be executed on a shared computational devise. This situation becomes typical to new Internet economies, where agent technologies play a major role. This allows the definition of a program equilibrium. Following the fundamental ideas introduced by von Neumann in the 1940s (in parallel to his seminal contribution to game theory), a computer program can be used both as a set of instructions, as well as a file that can be read and compared with other files. We show that this idea implies that in a program equilibrium of the one-shot prisoners dilemma mutual cooperation is obtained. More generally, we show that the set of program equilibrium payoffs of a game coincides with the set of feasible and individually rational payoffs of it.     

Public randomization in the repeated prisoner’s dilemma game with local interaction

We consider a situation in which each agent observes only their neighbor’s actions and locally interacts with them by playing prisoner’s dilemma games. Introducing a public randomization, we construct an equilibrium which sustains cooperation and in which cooperation eventually resumes after any history.     

Coordination and learning in games with strategic substitutes and complements

This paper experimentally compares the impact of the presence of strategic substitutes (GSS) and complements (GSC) on players’ ability to successfully play equilibrium strategies. By exploiting a simple property of the ordering on strategy spaces, our design allows us to isolate these effects by avoiding other confounding factors that are present in more complex settings, such as market games. We find that the presence of strategic complementarities significantly improves the rate of Nash play, but that this effect is driven mainly by early rounds of play. This suggests that GSS may be more difficult to learn initially, but that given sufficient time, the theoretically supported globally stable equilibrium offers a good prediction in both settings. We also show that increasing the degree of substitutability or complementarity does not significantly improve the rate of Nash play in either setting, which builds on the findings of previous studies.     

On the impossibility of achieving no regrets in repeated games

Regret-minimizing strategies for repeated games have been receiving increasing attention in the literature. These are simple adaptive behavior rules that lead to no regrets and, if followed by all players, exhibit nice convergence properties: the average play converges to correlated equilibrium, or even to Nash equilibrium in certain classes of games. However, the no-regret property relies on a strong assumption that each player treats her opponents as unresponsive and fully ignores the opponents’ possible reactions to her actions. We show that if at least one player is slightly responsive, it is impossible to achieve no regrets, and convergence results for regret minimization with responsive opponents are unknown.     

Correlated equilibrium existence for infinite games with type-dependent strategies

Under study are games in which players receive private signals and then simultaneously choose actions from compact sets. Payoffs are measurable in signals and jointly continuous in actions. This paper gives a counter-example to the main step in Cotter?s [K. Cotter, Correlated equilibrium in games with type-dependent strategies, J. Econ. Theory 54 (1991) 48-69] argument for correlated equilibrium existence for this class of games, and supplies an alternative proof.     

Cooperation,imitation and partial rematching

We study a setting in which imitative players are matched into pairs to play a Prisonerʼs Dilemma game. A well-known result in such setting is that under random matching cooperation vanishes for any interior initial condition. The novelty of this paper is that we consider partial rematching: players that belong to a pair in which both parties cooperate repeat partner next period, while all other players are randomly matched into pairs. Under certain conditions, this rematching mechanism makes cooperation the unique outcome in the long run. Furthermore, we show that if imitation happens infrequently enough then cooperative behavior is always present in the population.     

AN EXPERIMENT WITH ULTIMATUM BARGAINING IN A CHANGING ENVIRONMENT*

We present experimental results on the ultimatum bargaining game which support an evolutionary explanation of subjects’ behaviour in the game. In these experiments subjects interacted with each other and also with virtual players, i.e. computer programs with pre‐specified strategies. Some of these virtual players were designed to play the equitable allocation, while others exhibited behaviour closer to the subgame‐perfect equilibrium, in which the proposer's share is much larger than that of the responder. We have observed significant differences in the behaviour of real subjects depending on the type of “mutants” (virtual players) that were present in their environment.