Communication in repeated network games with imperfect monitoring

I consider repeated games with private monitoring played on a network. Each player has a set of neighbors with whom he interacts: a player's payoff depends on his own and his neighbors' actions only. Monitoring is private and imperfect: each player observes his stage payoff but not the actions of his neighbors. Players can communicate costlessly at each stage: communication can be public, private or a mixture of both. Payoffs are assumed to be sensitive to unilateral deviations. First, for any network, a folk theorem holds if some Joint Pairwise Identifiability condition regarding payoff functions is satisfied. Second, a necessary and sufficient condition on the network topology for a folk theorem to hold for all payoff functions is that no two players have the same set of neighbors not counting each other.     

Perfect communication equilibria in repeated games with imperfect monitoring

This paper introduces an equilibrium concept called perfect communication equilibrium for repeated games with imperfect private monitoring. This concept is a refinement of Myerson's [Myerson, R.B., 1982. Optimal coordination mechanisms in generalized principal agent problems, J. Math. Econ. 10, 67–81] communication equilibrium. A communication equilibrium is perfect if it induces a communication equilibrium of the continuation game, after every history of messages of the mediator. We provide a characterization of the set of corresponding equilibrium payoffs and derive a Folk Theorem for discounted repeated games with imperfect private monitoring.     

Perfect communication equilibria in repeated games with imperfect monitoring

This paper introduces an equilibrium concept called perfect communication equilibrium for repeated games with imperfect private monitoring. This concept is a refinement of Myerson's [Myerson, R.B., 1982. Optimal coordination mechanisms in generalized principal agent problems, J. Math. Econ. 10, 67–81] communication equilibrium. A communication equilibrium is perfect if it induces a communication equilibrium of the continuation game, after every history of messages of the mediator. We provide a characterization of the set of corresponding equilibrium payoffs and derive a Folk Theorem for discounted repeated games with imperfect private monitoring.     

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.     

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.     

Collusion through mediated communication in repeated games with imperfect private monitoring

Summary. This paper studies repeated games with imperfect private monitoring when there exists a third-party mediator who coordinates play by giving non-binding instructions to players on which action to take and by collecting their private information. The paper presents a Nash-threat folk theorem for a communication equilibrium based on such mediation when monitoring is jointly -perfect in the sense that every player is almost perfectly monitored collectively by other players.JEL Classification Numbers: C72, D82.I am very grateful to Mark Armstrong, V. Bhaskar, and Michihiro Kandori for helpful comments. Part of this research was conducted while I was visiting the University College London. Their hospitality is gratefully acknowledged.     

Infinitely repeated games with public monitoring and monetary transfers

This paper studies infinitely repeated games with imperfect public monitoring and the possibility of monetary transfers. It is shown that all public perfect equilibrium payoffs can be implemented with a simple class of stationary equilibria that use stick-and-carrot punishments. A fast algorithm is developed that exactly computes the set of pure strategies equilibrium payoffs for all discount factors.     

The folk theorem for irreducible stochastic games with imperfect public monitoring

This paper introduces stochastic games with imperfect public signals. It provides a sufficient condition for the folk theorem when the game is irreducible, thus generalizing the results of Dutta (1995) [5] and Fudenberg, Levine, and Maskin (1994) [9]. To do this, the paper extends the concept of self-generation (Abreu, Pearce, and Stacchetti, 1990 [1]) to “return generation,” which explicitly tracks actions and incentives until the next time the state returns to its current value, and asks that players not wish to deviate given the way their continuation payoffs from the time of this return depend on the public signals that have been observed.     

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.     

Communication in repeated games with costly monitoring

We study repeated games with discounting where perfect monitoring is possible, but costly. It is shown that if players can make public announcements, then every payoff vector which is an interior point in the set of feasible and individually rational payoffs can be implemented in a sequential equilibrium of the repeated game when the discount factor is high enough. Thus, efficiency can be approximated even when the cost of monitoring is high, provided that the discount factor is high enough.     

Coordination failure in repeated games with private monitoring

Players coordinate continuation play in repeated games with public monitoring. We investigate the robustness of such equilibrium behavior with respect to ex-ante small private-monitoring perturbations. We show that with full support of public signals, no perfect public equilibrium is robust if it induces a “regular” 2×2$2\times 2$ coordination game in the continuation play. This regularity condition is violated in all belief-free equilibria. Indeed, with an individual full rank condition, every interior belief-free equilibrium is robust. We also analyze block belief-free equilibria and point out that the notion of robustness is sensitive to whether we allow for uninterpretable signals.     

Social norms in repeated public good games

The paper explores the relationship between an individual’s preference for cooperation and the establishment of cooperative norms. Our aim is to provide an experimental test of the hypothesis (Fehr and Gachter, 2002) according to which the individuals’ willingness to punish defectors in social dilemma games favours the establishment of a common rule of behaviour leading to increased levels of cooperation. We conduct two experiments in which individuals decide both whether to participate in a common project and the institutional rules determining how the profits of the project were to be shared among the participants in the group. Our main question is to analyse whether high contributors react to free riding behaviour by adopting punitive sanctions. We also interview our subjects in order to elicit their opinions on civic values and free riding behaviour. Our results partly contradict the initial hypothesis; however, with reference to the institutional choices, we find that individual ethics may play an important role.     

Efficiency results in N player games with imperfect private monitoring

We demonstrate that efficiency is achievable in a certain class of N player repeated games with private, almost perfect monitoring. Our equilibrium requires only one period memory and can be implemented by two state automata. Furthermore, we show that this efficiency result holds with any degree of accuracy of monitoring if private signals are hemiindependent. Whereas most existing research focuses on two player cases or only a special example of N player games, our results are applicable to a wide range of N player games of economic relevance, such as trading goods games and price-setting oligopolies.     

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.     

Estimating the causal effect of beliefs on contributions in repeated public good games

We use instrumental variables for estimating the causal effect of beliefs on contributions in repeated public good games. The effect is about half as large as suggested by ordinary least squares. Thus, we present evidence that beliefs have a causal effect on contributions, but also that beliefs are endogenous. We compare the causal, belief-based model of contributions to alternative models based on matching the previous contributions of others and responding to one’s deviation from the average in the previous round. The causal, belief-based model performs well, indicating that beliefs have a central role in determining contributions.     

On the limit perfect public equilibrium payoff set in repeated and stochastic games

This paper provides a dual characterization of the existing ones for the limit set of perfect public equilibrium payoffs in a class of finite stochastic games (in particular, repeated games) as the discount factor tends to one. As a first corollary, the folk theorems of Fudenberg et al. (1994), Kandori and Matsushima (1998) and Hörner et al. (2011) obtain. As a second corollary, it is shown that this limit set of payoffs is a convex polytope when attention is restricted to perfect public equilibria in pure strategies. This result fails for mixed strategies, even when attention is restricted to two-player repeated games.     

Collusive behaviour in finite repeated games with bonding

In finite repeated games, it is not possible to enforce collusive behaviour using deterrent strategies if the state game has a unique Nash equilibrium, because of the ‘unravelling’ of cooperative behaviour in the last period. This paper demonstrates that under certain conditions, some cooperation among the players can be maintained if they can post a bond which they must forfeit if they defect from the cooperative mode. We show that the incentives to cooperate increase as the period of interaction grows in that the size of the bonds required to deter defection become arbitrarily small as the number of periods in the game increases.