Folk theorem with communication

This paper proves a new folk theorem for repeated games with private monitoring and communication, extending the idea of delayed communication in Compte [O. Compte, Communication in repeated games with imperfect private monitoring, Econometrica 66 (1998) 597-626] to the case where private signals are correlated.The sufficient condition for the folk theorem is generically satisfied with more than two players, even when other well-known conditions are not. The folk theorem also applies to some two-players repeated games.     

Tacit collusion in repeated auctions

This paper considers the question of tacit collusion in repeated auctions with independent private values and with limited public monitoring. McAfee and McMillan show that the extent of collusion is tied to the availability of transfers. Monetary transfers allow cartels to extract full surplus. A folk theorem proved by Fudenberg et al. (Econometrica 62 (1994) 997-1039) shows that transfers of future payoffs are almost as good if players are patient and communicate before auctions. We ask how the scope of collusion is affected if players dispense with explicit communication and their monitoring is limited. Collusion better than bid rotation is still feasible, but full surplus cannot be extracted. This constraint becomes less severe with more players and large cartels can become asymptotically efficient even with very limited monitoring.     

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.     

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.     

A belief-based approach to the repeated prisoners' dilemma with asymmetric private monitoring

This paper extends the belief-based approach to the repeated prisoners' dilemma with asymmetric private monitoring. We first find that the previous belief-based techniques [T. Sekiguchi, Efficiency in repeated prisoners' dilemma with private monitoring, J. Econ. Theory 76 (1997) 345-361; V. Bhaskar, I. Obara, Belief-based equilibria in the repeated prisoners' dilemma with private monitoring, J. Econ. Theory 102 (2002) 40-69] cannot succeed when players' private monitoring technologies are sufficiently different. We then modify the previous belief-based approach by letting the player with smaller observation errors always randomize between cooperate and defect along the cooperative path of the play. We show that with vanishing observation errors, efficiency and a folk theorem can be approximated using our modified belief-based strategies.     

A continuum of commitment

We examine a generic three-stage game for two players with alternating moves, where the first player can choose the level of adjustment cost to be paid in the last period to modify the action she announced in the first period. In the resulting continuum of commitment options, convexifying the choice between first-mover and second-mover advantage in pure strategies, we characterize when an intermediate adjustment-cost level is chosen in equilibrium. We show that the wastefulness of the adjustment cost may be in the players’ best interest, improving both of their individual net payoffs over making any fraction of the adjustment cost a side-payment from the first to the second player.     

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.     

A limit characterization of belief-free equilibrium payoffs in repeated games

The present paper studies repeated games with private monitoring, and characterizes the set of belief-free equilibrium payoffs in the limit as the discount factor approaches one and the noise on private information vanishes. Contrary to the conjecture by Ely et al. [J.C. Ely, J. Hörner, W. Olszewski, Belief-free equilibria in repeated games, Econometrica 73 (2005) 377-415], the equilibrium payoff set is computed by the same formula, no matter how many players there are. As an application of this result, a version of the folk theorem is established for N-player prisoner's dilemma games.     

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.     

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.     

Cooperative interim contract and re-contract: Chandler's M-form firm

Summary. At an interim stage players possessing only their private information freely communicate with each other to coordinate their strategies. This results in a core strategy, which is interpreted as an equilibrium set of players' alternative type-contingent contract offers to their fellows. From this set of offers each player then chooses an optimal one and engages in some subsequent action, thus possibly revealing some private information to the others. Now with new information thus obtained from each other, the players play a new game to re-write their contract. In all of the optimization and gaming just described, Bayesian incentive compatibility plays a central role. These ideas are formulated within a model of a profit-center game with incomplete information which formally describes interaction of the asymmetrically informed profit-centers in Chandler's multidivisional firm. Received: May 17, 1996; revised version: January 14, 1997     

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.     

On Failing to Cooperate When Monitoring Is Private

I consider a repeated prisoners' dilemma where in each period, each player receives an imperfect private signal about his opponent's current action. I show that when players are patient enough, any equilibrium where players use trigger strategies (i.e., do not revert to cooperation once they have started defecting) yields players a value arbitrarily close to the mutual minimax. I also examine the robustness of the result to perturbations of the game. Journal of Economic Literature Classification Numbers: C72.     

Screening and Signaling in Communication*

In this paper, we analyze a model of repeated strategic communication between a sender and a receiver. The sender has private information on a policy‐relevant variable but does not know the receiver's motives. Together with the desire of the sender to please the receiver and the desire of the receiver to be pleased, the two‐sided incomplete information creates an interesting communication problem in which players can choose to use information strategically. We analyze various modes of communication, which allow for information transmission, and we compare them in terms of both the quality of the policy implemented and the welfare of the players.     

Informed trading and the ‘leakage’ of information

This paper examines the effect of ‘leakage’ of information, private information becoming available to uninformed traders at a later date, on information acquisition and revelation. Using a Shapley-Shubik market game framework it is shown that (a) if information acquisition by the informed traders is costless, this leads to faster revelation of information; (b) if information acquisition is costly, there may be no acquisition of information; (c) information leakage leads to a fall in value of information but does not affect the incentive for informed traders to sell the information.     

Repeated Games with Almost-Public Monitoring

In repeated games with imperfect public monitoring, players can use public signals to coordinate their behavior, and thus support cooperative outcomes, but with private monitoring, such coordination may no longer be possible. Even though grim trigger is a perfect public equilibrium (PPE) in games with public monitoring, it often fails to be an equilibrium in arbitrarily close games with private monitoring. If a PPE has players' behavior conditioned only on finite histories, then it induces an equilibrium in all close-by games with private monitoring. This implies a folk theorem for repeated games with almost-public almost-perfect monitoring. Journal of Economic Literature Classification Numbers: C72, C73.     

Malice and patience in Rubinstein bargaining

If two players playing a Rubinstein alternating offers game are highly malicious (getting a high utility from “malice” in every period when the other player does not obtain a share in a fixed pie), and highly patient, no equilibrium with an agreement exists and players choose perpetual disagreement. This does not change if the players are subjected to a known deadline after which the pie will be appropriated by outside agencies or disappear: perpetual disagreement is still the only outcome. If in addition players are required to pay endogenously determined fines if they fail to reach agreement, players with discount factors in a certain range do reach agreement, but only at the deadline. However, infinitely patient players would never reach agreement for any feasible level of one-time fines. The result contrasts with spiteful or envious preferences. Our results highlight a novel reason for failure to resolve property disputes.     

Gambling in contests

This paper presents a strategic model of risk-taking behavior in contests. Formally, we analyze an n-player winner-take-all contest in which each player decides when to stop a privately observed Brownian motion with drift. A player whose process reaches zero has to stop. The player with the highest stopping point wins. Unlike the explicit cost for a higher stopping time in a war of attrition, here, higher stopping times are riskier, because players can go bankrupt. We derive a closed-form solution of a Nash equilibrium outcome. In equilibrium, highest expected losses occur at an intermediate negative value of the drift.     

A Robust Folk Theorem for the Prisoner's Dilemma

We prove the folk theorem for the Prisoner's dilemma using strategies that are robust to private monitoring. From this follows a limit folk theorem: when players are patient and monitoring is sufficiently accurate, (but private and possibly independent) any feasible individually rational payoff can be obtained in sequential equilibrium. The strategies used can be implemented by finite (randomizing) automata. Journal of Economic Literature Classification Numbers: C72, C73, D82.     

INCOMPLETE INFORMATION GAMES WITH MULTIPLE PRIORS*

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player “updates” his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players’ views on uncertainty about the environment affect the strategic outcomes.