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.     

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.     

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.     

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.     

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.     

An experimental analysis of ultimatum bargaining

There are many experimental studies of bargaining behavior, but suprisingly enough nearly no attempt has been made to investigate the so-called ultimatum bargaining behavior experimentally. The special property of ultimatum bargaining games is that on every stage of the bargaining process only one player has to decide and that before the last stage the set of outcomes is already restricted to only two results. To make the ultimatum aspect obvious we concentrated on situations with two players and two stages. In the ‘easy games’ a given amount c has to be distributed among the two players, whereas in the ‘complicated games’ the players have to allocate a bundle of black and white chips with different values for both players. We performed two main experiments for easy games as well as for complicated games. By a special experiment it was investigated how the demands of subjects as player 1 are related to their acceptance decisions as player 2.     

Risk-dominance and perfect foresight dynamics in N-player games

In perfect foresight dynamics, an action is linearly stable if expectation that people will always choose the action is self-fulfilling. A symmetric game is a PIM game if an opponent's particular action maximizes the incentive of an action, independently of the rest of the players. This class includes supermodular games, games with linear incentives and so forth. We show that, in PIM games, linear stability is equivalent to u-dominance, a generalization of risk-dominance, and that there is no path escaping a u-dominant equilibrium. Existing results on N-player coordination games, games with linear incentives and two-player games are obtained as corollaries.     

Cheap talk in games with incomplete information

The paper studies Bayesian games which are extended by adding pre-play communication. Let Γ be a Bayesian game with full support and with three or more players. The main result is that if players can send private messages to each other and make public announcements then every communication equilibrium outcome, q, that is rational (i.e., involves probabilities that are rational numbers) can be implemented in a sequential equilibrium of a cheap talk extension of Γ, provided that the following condition is satisfied: There exists a Bayesian Nash equilibrium s in Γ such that for each type t_i of each player i the expected payoff of t_i in q is larger than the expected payoff of t_i in s.     

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.     

Characterizing belief-free review-strategy equilibrium payoffs under conditional independence

This paper proposes and studies a tractable subset of Nash equilibria, belief-free review-strategy equilibria, in repeated games with private monitoring. The payoff set of this class of equilibria is characterized in the limit as the discount factor converges to one for games where players observe statistically independent signals. As an application, we develop a simple sufficient condition for the existence of asymptotically efficient equilibria, and establish a folk theorem for N-player prisoner?s dilemma. All these results are robust to a perturbation of the signal distribution, and hence remain true even under almost-independent 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.     

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.     

Trading with a common agent under complete information: A characterization of Nash equilibria

We analyze an abstract model of trading where N principals submit quantity-payment schedules that describe the contracts they offer to an agent, and the agent then chooses how much to trade with every principal. This represents a special class of common agency games with complete information. We study all the subgame perfect Nash equilibria of these games, not only truthful ones, providing a complete characterization of equilibrium payoffs. In particular, we show that the equilibrium that is Pareto-dominant for the principals is not truthful when there are more than two of them. We also provide a partial characterization of equilibrium strategies.     

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.     

Proper belief revision and equilibrium in dynamic games

We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies and (3) a player at information set h should not change his belief about an opponent's ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every normal form proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Introduction to Repeated Games with Private Monitoring

We present a brief overview of recent developments in discounted repeated games with (imperfect) private monitoring. The literature explores the possibility of cooperation in a long-term relationship, where each agent receives imperfect private information about the opponents' actions. Although this class of games admits a wide range of applications such as collusion under secret price-cutting, exchange of goods with uncertain quality, and observation errors, it has fairly complex mathematical structure due to the lack of common information shared by players. This is in sharp contrast to the well-explored case of repeated games under public information (with the celebrated Folk Theorems), and until recently little had been known about the private monitoring case. However, rapid developments in the past few years have revealed the possibility of cooperation under private monitoring for some class of games. Journal of Economic Literature Classification Numbers: C72, C73, D43, D82, L13, L41.     

Preemption in a real option game with a first mover advantage and player-specific uncertainty

In this paper a two-player real option game with a first-mover advantage is analyzed, where payoffs are driven by a player-specific stochastic state variable. It is shown that there exists an equilibrium which has qualitatively different properties from those in standard real option games driven by common stochastic shocks. The properties of the equilibrium are four-fold: (i) preemption does not necessarily occur, (ii) if preemption takes place, the rent-equalization property holds, (iii) for almost all sample paths it is clear ex-ante which player invests first, and (iv) it is possible that both players invest simultaneously, even if that is not optimal. It is argued from simulations that real option games with a common one-dimensional shock do not provide a good approximation for games with player-specific uncertainty, even if these are highly correlated.     

Stochastic games with information lag

Two-player zero-sum stochastic games with finite state and action spaces are known to have undiscounted values. We study such games under the assumption that one or both players observe the actions of their opponent after some time-dependent delay. We develop criteria for the rate of growth of the delay such that a player subject to such an information lag can still guarantee himself in the undiscounted game as much as he could have with perfect monitoring. We also demonstrate that the player in the Big Match with the absorbing action subject to information lags that grows too rapidly will not be able to guarantee as much as he could have in the game with perfect monitoring.