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.     

The Nash-threats folk theorem with communication and approximate common knowledge in two player games

We show that the use of communications to coordinate equilibria generates a Nash-threats folk theorem in two-player games with “almost public” information. The results generalize to the n-person case. However, the two-person case is more difficult because it is not possible to sustain equilibria by comparing the reports of different players, and using these “third parties” to effectively enforce contracts.     

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.     

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.     

Delay and information aggregation in stopping games with private information

We consider equilibrium timing decisions in a model with a large number of players and informational externalities. The players have private information about a common payoff parameter that determines the optimal time to invest. They learn from each other in real time by observing past investment decisions. We develop new methods of analysis for such large games, and we give a full characterization of symmetric equilibria. We show that the equilibrium statistical inferences are based on an exponential learning model. Although the beliefs converge to truth, learning takes place too late. Ex-ante welfare is strictly between that without observational learning and that with full information.     

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.     

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.     

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.     

Equilibrium selection in global games with strategic complementarities

We study games with strategic complementarities, arbitrary numbers of players and actions, and slightly noisy payoff signals. We prove limit uniqueness: as the signal noise vanishes, the game has a unique strategy profile that survives iterative dominance. This generalizes a result of Carlsson and van Damme (Econometrica 61 (1993) 989-1018) for two-player, two-action games. The surviving profile, however, may depend on fine details of the structure of the noise. We provide sufficient conditions on payoffs for there to be noise-independent selection.     

Learning in games with unstable equilibria

We propose a new concept for the analysis of games, the TASP, which gives a precise prediction about non-equilibrium play in games whose Nash equilibria are mixed and are unstable under fictitious play-like learning. We show that, when players learn using weighted stochastic fictitious play and so place greater weight on recent experience, the time average of play often converges in these “unstable” games, even while mixed strategies and beliefs continue to cycle. This time average, the TASP, is related to the cycle identified by Shapley [L.S. Shapley, Some topics in two person games, in: M. Dresher, et al. (Eds.), Advances in Game Theory, Princeton University Press, Princeton, 1964]. The TASP can be close to or quite distinct from Nash equilibrium.     

Long persuasion games

This paper characterizes geometrically the sets of all Nash and perfect Bayesian equilibrium payoffs achievable with unmediated communication in persuasion games, i.e., games with an informed expert and an uninformed decisionmaker in which the expert's information is certifiable. The first equilibrium characterization is provided for unilateral persuasion games, and the second for multistage, bilateral persuasion games. As in Aumann and Hart [R.J. Aumann, S. Hart, Long cheap talk, Econometrica 71 (6) (2003) 1619-1660], we use the concepts of diconvexification and dimartingale. A leading example illustrates both geometric characterizations and shows how the expert, whatever his type, can increase his equilibrium payoff compared to all equilibria of the unilateral persuasion game by delaying information certification.     

Noise-independent selection in multidimensional global games

This paper examines many-player many-action global games with multidimensional state parameters. It establishes that the notion of noise-independent selection introduced by Frankel, Morris and Pauzner [D. Frankel, S. Morris, A. Pauzner, Equilibrium selection in global games with strategic complementarities, J. Econ. Theory 108 (2003) 1–44] for one-dimensional global games is robust when the setting is extended to the one proposed by Carlsson and Van Damme [H. Carlsson, E. Van Damme, Global games and Equilibrium selection, Econometrica 61 (1993) 989–1018]. More precisely, our main result states that if an action profile of some complete information game is noise-independently selected in one-dimensional global games, then it is also noise-independently selected in all multidimensional global games.     

Competitive experimentation with private information: The survivor's curse

We study a winner-take-all R&D race between two firms that are privately informed about the arrival rate of an invention. Over time, each firm only observes whether the opponent left the race or not. The equilibrium displays a strong herding effect, that we call a ‘survivor's curse.’ Unlike in the case of symmetric information, the two firms may quit the race (nearly) simultaneously even when their costs and benefits for research differ significantly.     

Robustness of equilibria in anonymous local games

This paper studies the robustness of symmetric equilibria in anonymous local games to perturbations of prior beliefs. Two priors are strategically close on a class of games if players receive similar expected payoffs in equilibrium under the priors, for any game in that class. I show that if the structure of payoff interdependencies is sparse in a well-defined sense, the conditions for strategic proximity in anonymous local games are strictly weaker than the conditions for general Bayesian games of Kajii and Morris (1998) [11] when attention is restricted to symmetric equilibria. Hence, by exploiting the properties of anonymous local games, it is possible to obtain stronger robustness results for this class.     

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.     

Full revelation of information in Sender-Receiver games of persuasion

We provide necessary and sufficient conditions on both players' preferences and information that can be certified for a Sender-Receiver game to possess a separating equilibrium, as well as sufficient conditions for every equilibrium of such a game to be separating. Accordingly, we generalize Seidmann and Winter's [D.J. Seidmann, E. Winter, Strategic information transmission with verifiable messages, Econometrica 65 (1997) 163-170] results to games with partial provability.     

Perturbed communication games with honest senders and naive receivers

This paper studies communication games in which the sender is possibly honest (tells the truth) and the receiver is possibly naive (follows messages as if truthful). The characterization of message-monotone equilibria in the perturbed games explain several important aspects of strategic communication including sender exaggeration, receiver skepticism and message clustering. Surprisingly, the strategic receiver may respond to more aggressive claims with more moderate actions. In the limit as the probabilities of the non-strategic players approach zero, (i) the limit equilibrium corresponds to a most-informative equilibrium of the limit (Crawford-Sobel) game; (ii) only the top messages are sent.     

Evolution in games with randomly disturbed payoffs

We consider a simple model of stochastic evolution in population games. In our model, each agent occasionally receives opportunities to update his choice of strategy. When such an opportunity arises, the agent selects a strategy that is currently optimal, but only after his payoffs have been randomly perturbed. We prove that the resulting evolutionary process converges to approximate Nash equilibrium in both the medium run and the long run in three general classes of population games: stable games, potential games, and supermodular games. We conclude by contrasting the evolutionary process studied here with stochastic fictitious play.     

Fictitious play in 2×n games

It is known that every discrete-time fictitious play process approaches equilibrium in nondegenerate 2×2 games, and that every continuous-time fictitious play process approaches equilibrium in nondegenerate 2×2 and 2×3 games. It has also been conjectured that convergence to the set of equilibria holds generally for nondegenerate 2×n games. We give a simple geometric proof of this for the continuous-time process, and also extend the result to discrete-time fictitious play.     

Endogenous private information structures

We formally incorporate the option to gather information into a game and thus endogenize the information structure. We ask whether models with exogenous information structures are robust with respect to this endogenization. Any Nash equilibrium of the game with information acquisition induces a Nash equilibrium in the corresponding game with an exogenous structure. We provide sufficient conditions on the structure of the game for which this remains true when ‘Nash’ is replaced by ‘sequential’. We characterize the (sequential) Nash equilibria of games with exogenous information structures that can arise as a (sequential) Nash equilibrium of games with endogenous information acquisition.