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.     

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.     

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.     

Moral Hazard and Private Monitoring

We clarify the role of mixed strategies and public randomization (sunspots) in sustaining near-efficient outcomes in repeated games with private monitoring. We study a finitely repeated game, where the stage game has multiple equilibria and show that mixed strategies can support partial cooperation, but cannot approximate full cooperation even if monitoring is “almost perfect.” Efficiency requires extensive form correlation, where strategies can condition upon a sunspot at the end of each period. For any finite number of repetitions, we approximate the best equilibrium payoff under perfect monitoring, assuming that monitoring is sufficiently accurate and sunspots are available. Journal of Economic Literature Classification Numbers: C73, D82.     

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.     

Repeated games with present-biased preferences

We study infinitely repeated games with perfect monitoring, where players have β-δ preferences. We compute the continuation payoff set using recursive techniques and then characterize equilibrium payoffs. We then explore the cost of the present-time bias, producing comparative statics. Unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set is not monotonic in β or δ. Finally, we show how the equilibrium payoff set is contained in that of a repeated game with smaller discount factor.     

Strategic Buyers and Privately Observed Prices

A model of finitely repeated price competition between two sellers with differentiated goods and a large buyer is analyzed. The set of pure strategy sequential equilibria is investigated under public and private monitoring. With private monitoring, i.e., when prices are not observable to the competing sellers, all sales are made by the better seller and the set of repeated game equilibrium payoffs coincides with the stage game subgame perfect equilibrium payoffs. This is in sharp contrast to the game with perfect monitoring where the folk theorem obtains. Journal of Economic Literature Classification Numbers: C72, D43.     

Disappearing private reputations in long-run relationships

For games of public reputation with uncertainty over types and imperfect public monitoring, Cripps et al. [Imperfect monitoring and impermanent reputations, Econometrica 72 (2004) 407-432] showed that an informed player facing short-lived uninformed opponents cannot maintain a permanent reputation for playing a strategy that is not part of an equilibrium of the game without uncertainty over types. This paper extends that result to games in which the uninformed player is long-lived and has private beliefs, so that the informed player's reputation is private. The rate at which reputations disappear is uniform across equilibria and reputations also disappear in sufficiently long discounted finitely repeated games.     

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.     

Reciprocity Game

This paper shows that reciprocity comes from the desire to cooperate in finitely repeated prisoner's dilemma game. Before playing the finitely repeated prisoner's dilemma game, players choose the reciprocity level and commit to it, and the reciprocity level is public information. There are T equilibria if the prisoner's dilemma game is repeated for T periods, and each equilibrium is associated with different levels of cooperation. Further, if players choose their reciprocity levels sequentially, then the most cooperative equilibrium will be the unique equilibrium. However, reciprocity does not matter for the one‐period game and the infinitely repeated game.     

How complex are networks playing repeated games?

Summary. This paper examines implications of complexity cost in implementing repeated game strategies through networks with finitely many classifiers. A network consists of individual classifiers that summarize the history of repeated play according to a weighted sum of the empirical frequency of the outcomes of the stage game, and a decision unit that chooses an action in each period based on the summaries of the classifiers. Each player maximizes his long run average payoff, while minimizing the complexity cost of implementing his strategy through a network, measured by its number of classifiers. We examine locally stable equilibria where the selected networks are robust against small perturbations. In any locally stable equilibrium, no player uses a network with more than a single classifier. Moreover, the set of locally stable equilibrium payoff vectors lies on two line segments in the payoff space of the stage game. Received: May 9, 1997; revised version: November 18, 1997     

Coordination cycles

Players repeatedly face a coordination problem in a dynamic global game. By choosing a risky action (invest) instead of waiting, players risk instantaneous losses as well as a loss of payoffs from future stages, in which they cannot participate if they go bankrupt. Thus, the total strategic risk associated with investment in a particular stage depends on the expected continuation payoff. High continuation payoff makes investment today more risky and therefore harder to coordinate on, which decreases today's payoff. Thus, expectation of successful coordination tomorrow undermines successful coordination today, which leads to fluctuations of equilibrium behavior even if the underlying economic fundamentals happen to be the same across the rounds. The dynamic game inherits the equilibrium uniqueness of the underlying static global game.     

Efficient collusion in repeated auctions with communication

This paper studies collusion in repeated auctions when bidders communicate prior to each stage auction. For independent and correlated private signals and general interdependent values, the paper identifies conditions under which an equilibrium collusion scheme is fully efficient in the sense that the bidders’ payoff is close to what they get when the object is allocated to the highest valuation bidder at the reserve price in every period.     

Belief-Based Equilibria in the Repeated Prisoners' Dilemma with Private Monitoring

We analyze repeated prisoners' dilemma games with imperfect private monitoring and construct mixed trigger strategy equilibria. Such strategies have a simple representation, where a player's action only depends upon her belief that her opponent(s) are continuing to cooperate. When monitoring is almost perfect, the symmetric efficient outcome can be approximated in any prisoners' dilemma game, while every individually rational feasible payoff can be approximated in a class of such games. The efficiency result extends when there are more than two players. It requires that monitoring be sufficiently accurate but does not require very low discounting when a public randomization device is available. Journal of Economic Literature Classification Numbers: C73, D82.     

The equilibrium of the bargaining game and core convergence theorem on an exchange economy with limited traders

Debreu and Scarf (1963), Hildenbrand (1974), Aumann (1964), Dierker (1975), Bewley (1973), and others have shown that the core of an exchange economy with infinitely many or finitely many traders converges. However, an exchange economy does not always consist of infinitely many or finitely many traders. This note provides proof of the core convergence theorem on an exchange economy with limited traders by a bargaining game methodology. The main contribution of this note is to innovate the equilibrium solution to the bargaining game in the exchange economy. In this note, the concept of common payoff is introduced; in the bargaining game of a coalition on its common payoff, all coalition members will get the same distribution, thus the distribution scheme of the cooperation surplus of the exchange economy is determined. This note shows that the bargaining game among the traders on the distribution of the cooperation surplus will make the pure exchange economy with limited traders converge to the Walrasian equilibrium, all the allocations other than the Walrasian equilibrium will be eliminated from the core of this economy.     

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.     

Payoff dominance and risk dominance in the observable delay game: a note

We examine whether the payoff dominant sequential-move (Stackelberg) outcome is realized when timing is endogenized. We adopt the observable delay game formulated by Hamilton and Slutsky [Games Econ Behav 2(1):29–46, 1990]. We find that if one sequential-move outcome is payoff dominant, either (i) the outcome both players prefer is the unique equilibrium; or (ii) two sequential-move outcomes are equilibria and the one both players prefer is risk dominant. In other words, no conflict between payoff dominance and risk dominance in the observable delay game exists, in contrast to other games such as (non pure) coordination games. We also find that even if one of two sequential-move outcomes is the unique equilibrium outcome in the observable delay game, it does not imply that the equilibrium outcome is payoff dominant to the other sequential-move outcome.      

Dynamic common agency

A general model of dynamic common agency with symmetric information is considered. The set of truthful Markov perfect equilibrium payoffs is characterized and the efficiency properties of the equilibria are established. A condition for the uniqueness of equilibrium payoffs is derived for the static and the dynamic game. The payoff is unique if and only if the payoff of each principal coincides with his marginal contribution to the social value of the game. The dynamic model is applied to a game of agenda setting.     

TIME CONSISTENCY IN ALTERNATING‐MOVE POLICY GAMES*

We examine alternating‐move policy games where the government and the private sector alternate their moves. In contrast with the standard policy game, the set of equilibrium payoffs of the present model is bounded away from the payoff under the one‐shot Nash equilibrium, called the Kydland–Prescott outcome, and the upper bound is close to the payoff under the optimal policy (called the Ramsey policy) if the government is sufficiently patient. In other words, the Kydland–Prescott outcome is not time consistent, while the Ramsey outcome could be approximated by a time‐consistent policy of the same game.     

Interlinkage and Generous Tit‐for‐Tat Strategy

We investigate an infinitely‐repeated prisoners' dilemma with imperfect monitoring and consider the possibility that the interlinkage of the players' distinct activities enhances implicit collusion. We show a necessary and sufficient condition for the existence of a generous tit‐for‐tat Nash equilibrium. Such an equilibrium, if it exists, is unique. This equilibrium achieves approximate efficiency when monitoring is almost perfect, where the discount factors are fixed.