Repetition, Communication, and Coordination Failure

We examine repetition as an institution that affects coordination failure in a game with and without pre-play communication. We use probit regression with random effects to test hypotheses regarding the frequency and form of coordination failure in the presence of repeated play versus one-shot games. Our results indicate that repetition without pre-play communication results in a lower frequency of coordination failure relative to one-shot game outcomes. This result is reversed when pre-play communication is allowed. Our evidence also suggests that repeated play coordination failures tend to be suboptimal Nash equilibria, whereas one-shot game coordination failures are disequilibria regardless of the presence of pre-play communication.     

Individual characteristics and behavior in repeated games: an experimental study

Using a laboratory experiment, we investigate whether a variety of behaviors in repeated games are related to an array of individual characteristics that are popular in economics: risk attitude, time preference, trust, trustworthiness, altruism, strategic skills in one-shot matrix games, compliance with first-order stochastic dominance, ability to plan ahead, and gender. We do find some systematic relationships. A subject’s compliance with first-order stochastic dominance as well as, possibly, patience, gender, and altruism have some systematic effects on her behavior in repeated games. At the level of a pair of subjects who are playing a repeated game, each subject’s gender as well as, possibly, patience and ability to choose an available dominant strategy in a one-shot matrix game systematically affect the frequency of the cooperate–cooperate outcome. However, overall, the number of systematic relationships is surprisingly small.     

Stochastic uncoupled dynamics and Nash equilibrium

In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence of play (the period-by-period behavior as well as the long-run frequency) to Nash equilibria of the one-shot stage game, and present a number of possibility and impossibility results. Basically, we show that if in addition to random experimentation some recall, or memory, is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it suffices to recall the last two periods of play.     

Stochastic uncoupled dynamics and Nash equilibrium

In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence of play (the period-by-period behavior as well as the long-run frequency) to Nash equilibria of the one-shot stage game, and present a number of possibility and impossibility results. Basically, we show that if in addition to random experimentation some recall, or memory, is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it suffices to recall the last two periods of play.     

Testing static game theory with dynamic experiments: A case study of public goods

Game theory provides predictions of behavior in many one-shot games. On the other hand, most experimenters usually play repeated games with subjects, to provide experience. To avoid subjects rationally employing strategies that are appropriate for the repeated game, experimenters typically employ a “random strangers” design in which subjects are randomly paired with others in the session. There is some chance that subjects will meet in multiple rounds, but it is claimed that this chance is so small that subjects will behave as if they are in a one-shot environment. We present evidence from public goods experiments that this claim is not always true.     

Behavior in one-shot traveler’s dilemma games: model and experiments with advice

We report results of one-shot traveler’s dilemma game experiments to test the predictions of a model of introspection. The model describes a noisy out-of-equilibrium process by which players reach a decision of what to do in one-shot games. To test the robustness of the model and to compare it to other models of introspection without noise, we introduce non-binding advice. Advice has the effect of coordinating all players’ beliefs onto a common strategy. Experimentally, advice is implemented by asking subjects who participated in a repeated traveler’s dilemma game to recommend an action to subjects playing one-shot games with identical parameters. In contrast to observations, models based on best-response dynamics would predict lower claims than the advised. We show that our model’s predictions with and without advice are consistent with the data.      

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.     

Measuring Strategic Uncertainty in Coordination Games

This paper proposes a method to measure strategic uncertainty by eliciting certainty equivalents analogous to measuring risk attitudes in lotteries. We apply this method by conducting experiments on a class of one-shot coordination games with strategic complementarities and choices between simple lotteries and sure payoff alternatives, both framed in a similar way. Despite the multiplicity of equilibria in the coordination games, aggregate behaviour is fairly predictable. The pure or mixed Nash equilibria cannot describe subjects' behaviour. We present two global games with private information about monetary payoffs and about risk aversion. While previous literature treats the parameters of a global game as given, we estimate them and show that both models describe observed behaviour well. The global-game selection for vanishing noise of private signals offers a good recommendation for actual players, given the observed distribution of actions. We also deduce subjective beliefs and compare them with objective probabilities.     

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.     

On Sustaining Cooperation without Public Observations

This paper examines a dynamic game in which each player only observes a private and imperfect signal on the actions played. Our main result is that in a repeated prisoner's dilemma where defections are irreversible (at least for a long enough period of time), patient enough players may achieve almost efficient outcomes. Dealing with models of imperfect private monitoring is difficult because (i) continuation games are games of incomplete information, hence they do not have the same structure as the original game. In particular, continuation equilibria are correlated equilibria. (ii) Players are typically uncertain about their opponents' past observations and actions, and they use their entire own private history to learn about these actions. As a result equilibrium strategies are in general nontrivial and increasingly complex functions of past observations. We bypass these difficulties by looking at correlated equilibria of the original game and find correlated equilibria in which the decision problem faced by each player remains the same over time. Journal of Economic Literature Classification Numbers: C72.     

Monotone Games with Positive Spillovers

A monotone game comprises the infinitely repeated play of an n-person stage game, subject to the constraint that players' actions be monotonically nondecreasing over time. These games represent a variety of strategic situations in which players are able to make (partial) commitments. If the stage games have positive spillovers and satisfy certain other conditions, then the limit points of the subgame perfect equilibria are precisely the approachable action profiles. This characterization is applied to voluntary contribution games, market games, and coordination games. Journal of Economic Literature Classification Number: C7.     

Nearly compact and continuous normal form games: characterizations and equilibrium existence

Normal form games are nearly compact and continuous (NCC) if they can be understood as games played on strategy spaces that are dense subsets of the strategy spaces of larger compact games with jointly continuous payoffs. There are intrinsic algebraic, measure theoretic, functional analysis, and finite approximability characterizations of NCC games. NCC games have finitely additive equilibria, and all their finitely additive equilibria are equivalent to countably additive equilibria on metric compactifications. The equilibrium set of an NCC game depends upper hemicontinuously on the specification of the game and contains only the limits of approximate equilibria of approximate games.     

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.     

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.     

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.     

Learning to alternate

The Individual Evolutionary Learning (IEL) model explains human subjects’ behavior in a wide range of repeated games which have unique Nash equilibria. Using a variation of ‘better response’ strategies, IEL agents quickly learn to play Nash equilibrium strategies and their dynamic behavior is like that of humans subjects. In this paper we study whether IEL can also explain behavior in games with gains from coordination. We focus on the simplest such game: the 2 person repeated Battle of Sexes game. In laboratory experiments, two patterns of behavior often emerge: players either converge rapidly to one of the stage game Nash equilibria and stay there or learn to coordinate their actions and alternate between the two Nash equilibria every other round. We show that IEL explains this behavior if the human subjects are truly in the dark and do not know or believe they know their opponent’s payoffs. To explain the behavior when agents are not in the dark, we need to modify the basic IEL model and allow some agents to begin with a good idea about how to play. We show that if the proportion of inspired agents with good ideas is chosen judiciously, the behavior of IEL agents looks remarkably similar to that of human subjects in laboratory experiments.     

Subgame perfect reaction function equilibria in discounted duopoly supergames are trivial

This paper is concerned with infinitely repeated duopoly games with discounting. A question which has been open since Friedman's (Rev. Econ. Stud. 35 (1968), 257–272) reaction function article is settled for a general class of games. The question is wheter nontrivial reaction function equilibria can be subgame perfect. This question is answered in the negative. Such equilibria must be trivial in the sense of prescribing the stage game noncooperative equilibrium actions in every period, independent of prior history.     

Interconnected games and international environmental problems

The purpose of this paper is to introduce the concept of interconnected games and to show its relevance for modeling international environmental problems. It is argued that an interconnected game approach to international environmental problems may enhance cooperation and provide an alternative to the use of financial side payments to induce countries to cooperate. Two types of interconnected games are distinguished in this paper, i.e. direct sum games and tensor games. In the former all the constituting isolated games are games in strategic form and in the latter they are repeated games. In both cases the interconnected game can be interpreted as a multiple objective game, but only the setting where a trade-off is made for the vector-payoffs is considered. In addition to the formal definition of these types of interconnected games, some elementary results concerning Nash equilibria of such games are derived.Folmer and v. Mouche: Landbouwuniversiteit Wageningen, Postbus 8130, 6700 EW Wageningen, The Netherlands; Ragland: University of Colorado at Boulder, Boulder, Colorado 80309-0256, U.S.A. An earlier version of this paper was presented at the EAERE conference in Stockholm, June 1991. The authors appreciate comments made by conference participants and journal referees.     

Communication equilibrium payoffs in repeated games with imperfect monitoring

We characterize the set of communication equilibrium payoffs of any undiscounted repeated matrix-game with imperfect monitoring and complete information. For two-player games, a characterization is provided by Mertens, Sorin, and Zamir (Repeated games, Part A (1994) CORE DP 9420), mainly using Lehrer's (Math. Operations Res. (1992) 175) result for correlated equilibria. The main result of this paper is to extend this characterization to the n-player case. The proof of the characterization relies on an analogy with an auxiliary 2-player repeated game with incomplete information and imperfect monitoring. We use Kohlberg's (Int. J. Game Theory (1975) 7) result to construct explicitly a canonical communication device for each communication equilibrium payoff.     

Individually selecting among conventions - an evolutionary and experimental analysis

Conventions can be narrowly interpreted as coordinated ways of equilibrium play, telling all players which of possibly several equilibria to play or more broadly how to choose in a game without imposing the equilibrium property. Since coordination often takes place before learning about the game, one has to coordinate on a prescribing principle. For the subclass of 2×2-bimatrix games with two strict equilibria, we analyze the evolutionary stability of various such principles. In our experiment, we allow participants to coordinate on principles before playing various games. Based on between-subjects treatments, participants do so being completely (they know neither their role nor the game parameters), partially (they know either their role or the game parameters) ignorant, or with no veil of ignorance (they know their role and the game parameters).