Jérôme Renault

The Value of Markov Chain Games with Lack of Information on One Side

We consider a two-player zero-sum game, given by a Markov chain over a finite set of states and a family of matrix games indexed by states. The sequence of states follows the Markov chain. At the beginning of each stage, only Player 1 is informed of the current state, then the corresponding matrix game is played, and the actions chosen are observed by both players before proceeding to the next stage. We call such a game a Markov chain game with lack of information on one side. This model generalizes the model of Aumann and Maschler of zero-sum repeated games with lack of information on one side (which corresponds to the case where the transition matrix of the Markov chain is the identity matrix). We generalize the proof of Aumann and Maschler and, from the definition and the study of appropriate nonrevealing auxiliary games with infinitely many stages, show the existence of the uniform value. An important difference with Aumann and Maschlers model is that here the notions for Player 1 of using the information and revealing a relevant information are distinct.

A folk theorem for minority games

We study a particular case of repeated games with public signals. In the stage game an odd number of players have to choose simultaneously one of two rooms. The players who choose the less crowded room receive a reward of one euro (whence the name “minority game”). The players in the same room do not recognize each other, and between the stages only the current majority room is publicly announced. We show that in the infinitely repeated game any feasible payoff can be achieved as a uniform equilibrium payoff, and as an almost sure equilibrium payoff. In particular we construct an inefficient equilibrium where, with probability one, all players choose the same room at almost all stages. This equilibrium is sustained by punishment phases which use, in an unusual way, the pure actions that were played before the start of the punishment.

The Value of Repeated Games with an Informed Controller

We consider the general model of zero-sum repeated games (or stochastic games with signals), and assume that one of the players is fully informed and controls the transitions of the state variable. We prove the existence of the uniform value, generalizing several results of the literature. A preliminary existence result is obtained for a particular class of stochastic games played with pure strategies.

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 Lehrers (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 Kohlbergs (Int. J. Game Theory (1975) 7) result to construct explicitly a canonical communication device for each communication equilibrium payoff.

Dynamic sender–receiver games

We consider a dynamic version of sender-receiver games, where the sequence of states follows an irreducible Markov chain observed by the sender. Under mild assumptions, we provide a simple characterization of the limit set of equilibrium payoffs, as players become very patient. Under these assumptions, the limit set depends on the Markov chain only through its invariant measure. The (limit) equilibrium payoffs are the feasible payoffs that satisfy an individual rationality condition for the receiver, and an incentive compatibility condition for the sender.

The Value of Markov Chain Games with Incomplete Information on Both Sides

We consider zero-sum repeated games with incomplete information on both sides, where the states privately observed by each player follow independent Markov chains. It generalizes the model, introduced by Aumann and Maschler in the sixties and solved by Mertens and Zamir in the seventies, where the private states of the players were fixed. It also includes the model introduced in Renault [19] [Renault J (2006) The value of Markov chain games with lack of information on one side. Math. Oper. Res. 31(3):490–512.], of Markov chain repeated games with lack of information on one side, where only one player privately observes the sequence of states. We prove here that the limit value exists, and we obtain a characterization via the Mertens-Zamir system, where the “nonrevealing value function” plugged in the system is now defined as the limit value of an auxiliary “nonrevealing” dynamic game. This nonrevealing game is defined by restricting the players not to reveal any information on the limit behavior of their ...

Optimal Dynamic Information Provision

We study a dynamic model of information provision. A state of nature evolves according to a Markov chain. An advisor with commitment power decides how much information to provide to an uninformed decision maker, so as to influence his short-term decisions. We deal with a stylized class of situations, in which the decision maker has a risky action and a safe action, and the payoff to the advisor only depends on the action chosen by the decision maker. The greedy disclosure policy is the policy which, at each round, minimizes the amount of information being disclosed in that round, under the constraint that it maximizes the current payoff of the advisor. We prove that the greedy policy is optimal in many cases – but not always.

Dynamic sender receiver games

We consider a dynamic version of sender-receiver games, where the sequence of states follows an irreducible Markov chain observed by the sender. Under mild assumptions, we provide a simple characterization of the limit set of equilibrium payoffs, as players become very patient. Under these assumptions, the limit set depends on the Markov chain only through its invariant measure. The (limit) equilibrium payoffs are the feasible payoffs that satisfy an individual rationality condition for the receiver, and an incentive compatibility condition for the sender.

General Properties of Long-Run Supergames

Supergames are repeated games in which a fixed known finite one-shot game is repeated over and over. Information about the actions chosen at each stage is provided by a signalling technology. This paper studies the main properties that are valid over this whole class of games and both surveys known results and provides new ones.

Discounted and Finitely Repeated Minority Games with Public Signals

We consider a repeated game where at each stage players simultaneously choose one of two rooms. The players who choose the less crowded room are rewarded with one euro. The players in the same room do not recognize each other, and between the stages only the current majority room is publicly announced, hence the game has imperfect public monitoring. An undiscounted version of this game was considered by Renault et al. (2005), who proved a folk theorem. Here we consider a discounted version and a finitely repeated version of the game, and we strengthen our previous result by showing that the set of equilibrium payos Hausdor-converges to the feasible set as either the discount factor goes to one or the number of repetition goes to infinity. We show that the set of public equilibria for this game is strictly smaller than the set of private equilibria.