Sylvain Sorin

Cooperation and bounded recall

A two-person game has common interests if there is a single payoff pair z that strongly Pareto dominates all other payoff pairs. Suppose such a game is repeated many times, and that each player attaches a small but positive probability to the other playing some fixed strategy with bounded recall, rather than playing to maximize his payoff. Then the resulting supergame has an equilibrium in pure strategies, and the payoffs to all such equilibria are close to optimal (i.e., to z).

On Repeated Games with Complete Information

We consider N person repeated games with complete information and standard signalling. We first prove several properties of the sets of feasible payoffs and Nash equilibrium payoffs for the n-stage game and for the λ-discounted game. In the second part we determine the set of equilibrium payoffs for the Prisoners Dilemma corresponding to the critical value of the discount factor.

Stochastic Approximations and Differential Inclusions

The dynamical systems approach to stochastic approximation is generalized to the case where the mean differential equation is replaced by a differential inclusion. The limit set theorem of Benaim and Hirsch is extended to this situation. Internally chain transitive sets and attractors are studied in detail for set-valued dynamical systems. Applications to game theory are given, in particular to Blackwells approachability theorem and the convergence of fictitious play.

Stochastic Approximations and Differential Inclusions, Part II: Applications

We apply the theoretical results on “stochastic approximations and differential inclusions” developed in Benaim et al. [M. Benaim, J. Hofbauer, S. Sorin. 2005. Stochastic approximations and differential inclusions. SIAM J. Control Optim.44 328--348] to several adaptive processes used in game theory, including classical and generalized approachability, no-regret potential procedures (Hart and Mas-Colell [S. Hart, A. Mas-Colell. 2003. Regret-based continuous time dynamics. Games Econom. Behav.45 375--394]), and smooth fictitious play [D. Fudenberg, D. K. Levine. 1995. Consistency and cautious fictitious play. J. Econom. Dynam. Control19 1065--1089].

AN OPERATOR APPROACH TO ZERO-SUM REPEATED GAMES

We consider two person zero-sum stochastic games. The recursive formula for the valuesvλ (resp.vn) of the discounted (resp. finitely repeated) version can be written in terms of a single basic operator Φ(α,f) where α is the weight on the present payoff andf the future payoff. We give sufficient conditions in terms of Φ(α,f) and its derivative at 0 for limvn and limvλ to exist and to be equal.We apply these results to obtain such convergence properties for absorbing games with compact action spaces and incomplete information games.

A uniform Tauberian theorem in dynamic programming

We prove that, in dynamic programming framework, uniform convergence of vλ implies uniform convergence of vn and vice versa. Moreover, both have the same limit.

Time Average Replicator and Best-Reply Dynamics

Using an explicit representation in terms of the logit map, we show, in a unilateral framework, that the time average of the replicator dynamics is a perturbed solution of the best-reply dynamics.

Game-theoretic methods in general equilibrium analysis

List of Figures. List of Authors. Introduction J.-F. Mertens. A: The Core and the Bargaining Set. I. General Equilibrium and Cooperative Games: Basic Results E. Allen, S. Sorin. II. Core Convergence in Perfectly Competitive Economies R.M. Anderson. III. Economies with Atoms J.-F. Mertens. IV. Bargaining Sets R. Vohra. B: The Value. V. The Shapley Value R.J. Aumann. VI. Value of Games with a Continuum of Players A. Neyman. VII. The TU Value: the Non-Differentiable Case J.-F. Mertens. Addendum: The Shapley value of a perfectly competitive market may not exist F. Lefevre. VIII. The Harsanyi Value S. Hart. IX. Value Equivalence Theorems: the TU and NTU Cases S. Hart. X. Economic Applications of the Shapley Value R.J. Aumann. C: The Cooperative Approach to Large Markets and Games. XI. An Axiomatic Approach to the Equivalence Phenomenon P. Dubey, A. Neyman. XII. Large Games and Economies with Effective Small Groups M.H. Wooders. D: The Non-Cooperative Approach. XIII. Strategic Market Games: a Survey of Some Results P. Dubey. XIV. From Nash to Walras Equilibrium E. Allen, H. Polemarchakis. XV. Correlated and Communication Equilibria J.-F. Mertens. XVI. Notes on Correlated Equilibrium and Sunspot Equilibrium J. Peck. XVII. Implementation with Plain Conversation S. Sorin.

The LP formulation of finite zero-sum games with incomplete information

This paper gives the LP formulation for finite zero sum games with incomplete information using Bayesian mixed strategies. This formulation is then used to derive general properties for the value of such games, the well known concave-convex property but also the “piecewise bilinearity”. These properties may offer considerable help for computational purposes but also provide structural guidelines for the analysis of special classes of games with incomplete information.

A Continuous Time Approach for the Asymptotic Value in Two-Person Zero-Sum Repeated Games

We consider the asymptotic value of two person zero sum repeated games with general evaluations of the stream of stage payoffs. We show existence for incomplete information games, splitting games and absorbing games. The technique of proof consists in embedding the discrete repeated game into a continuous time one and to use viscosity solution tools.