Tristan Tomala

Repeated proximity games

Abstract. We consider repeated games with complete information and imperfect monitoring, where each player is assigned a fixed subset of players and only observes the moves chosen by the players in this subset. This structure is naturally represented by a directed graph. We prove that a generalized folk theorem holds for any payoff function if and only if the graph is 2-connected, and then extend this result to the context of finitely repeated games.

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.

Secret Correlation in Repeated Games with Imperfect Monitoring

We characterize the maximum payoff that a team can guarantee against another in a class of repeated games with imperfect monitoring. Our result relies on the optimal tradeoff for the team between optimization of stage payoffs and generation of signals for future correlation.

Belief-Free Equilibria in Games with Incomplete Information: Characterization and Existence

We characterize belief-free equilibria in infinitely repeated games with incomplete information with N \ge 2 players and arbitrary information structures. This characterization involves a new type of individual rational constraint linking the lowest equilibrium payoffs across players. The characterization is tight: we define a set of payoffs that contains all the belief-free equilibrium payoffs; conversely, any point in the interior of this set is a belief-free equilibrium payoff vector when players are sufficiently patient. Further, we provide necessary conditions and sufficient conditions on the information structure for this set to be non-empty, both for the case of known-own payoffs, and for arbitrary payoffs.

Empirical Distributions of Beliefs Under Imperfect Observation

Let (xn)n be a process with values in a finite set X and law P, and let yn f(xn) be a function of the process. At stage n, the conditional distribution pn P(xnx1,,xn1), element of (X), is the belief that a perfect observer, who observes the process online, holds on its realization at stage n. A statistician observing the signals y1,,yn holds a belief enP(pnx1,,xn) () on the possible predictions of the perfect observer. Given X and f, we characterize the set of limits of expected empirical distributions of the process (en) when P ranges over all possible laws of (xn)n.

Perfect communication equilibria in repeated games with imperfect monitoring

This paper introduces an equilibrium concept called perfect communication equilibrium for repeated games with imperfect private monitoring. This concept is a refinement of Myersons [Myerson, R.B., 1982. Optimal coordination mechanisms in generalized principal agent problems, J. Math. Econ. 10, 67–81] communication equilibrium. A communication equilibrium is perfect if it induces a communication equilibrium of the continuation game, after every history of messages of the mediator. We provide a characterization of the set of corresponding equilibrium payoffs and derive a Folk Theorem for discounted repeated games with imperfect private monitoring.

Mechanism design and communication networks

This paper characterizes the communication networks (directed graphs) for which, in any environment (utilities and beliefs), every incentive compatible social choice function is implementable. We show that any incentive compatible social choice function is implementable on a given communication network, in all environments with either common independent beliefs and private values or a worst outcome, if and only if the network is strongly connected and weakly 2-connected. A network is strongly connected if for each player, there exists a directed path to the designer. It is weakly 2-connected if each player is either directly connected to the designer or indirectly connected to the designer through two disjoint paths not necessarily directed. We couple encryption techniques together with appropriate incentives to secure the transmission of each players private information to the designer.

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.

Communication, correlation and cheap-talk in games with public information

This paper studies extensive form games with perfect information and simultaneous moves, henceforth called games with public information. On this class, we prove that all communication equilibrium payoffs can be obtained without mediator by cheap-talk procedures. The result encompasses repeated games and stochastic games.

Information design for strategic coordination of autonomous devices with non-aligned utilities

In this paper, we investigate the coordination of autonomous devices with non-aligned utility functions. Both encoder and decoder are considered as players, that choose the encoding and the decoding in order to maximize their long-run utility functions. The topology of the point-to-point network under investigation, suggests that the decoder implements a strategy, knowing in advance the strategy of the encoder. We characterize the encoding and decoding functions that form an equilibrium, by using empirical coordination. The equilibrium solution is related to an auxiliary game in which both players choose some conditional distributions in order to maximize their expected utilities. This problem is closely related to the literature on “Information Design” in Game Theory. We also characterize the set of posterior distributions that are compatible with a rate-limited channel between the encoder and the decoder. Finally, we provide an example of non-aligned utility functions corresponding to parallel fading multiple access channels.