Bayesian repeated games and reputation

The folk theorem characterizes the (subgame perfect) Nash equilibrium payo¤s of an undiscounted or discounted in…nitely repeated game with fully informed, patient players as the feasible individually rational payo¤s of the one-shot game. To which extent does the result still hold when every player privately knows his own payo¤s ? Under appropriate assumptions (private values and uniform punishments), the Nash equilibria of the Bayesian in…nitely repeated game without discounting are payo¤ equivalent to tractable, completely revealing, equilibria and can be achieved as interim cooperative solutions of the initial Bayesian game. This characterization does not apply to discounted games with su¢ciently patient players. In a class of public good games, the set of Nash equilibrium payo¤s of the undiscounted game can be empty, while limit (perfect Bayesian) Nash equilibrium payo¤s of the discounted game, as players become in…nitely patient, do exist. These equilibria share some features with the ones of multisided reputation models. Keywords: Bayesian game, incentive compatibility, individual rationality, in…nitely repeated game, private values, public good, reputation. JEL classi…cation: C73, C72, C71; D82; H41 Conversations with Gorkem Celik, Martin Cripps, Gaël Giraud, Johannes Hörner, Vincent Iehlé, Frédéric Koessler, Marcin Peski and Péter Vida are gratefully acknowledged. Bernard Lebrun made a number of precise and helpful comments on an earlier draft. CEREMADE and LEDa, Université Paris-Dauphine. Member of Institut Universitaire de France. E-mail: [email protected] LEDa, Université Paris-Dauphine. E-mail: [email protected]