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

International audience1 Introduction: Shapley 2 Extensions of the Shapley operator : general repeated games 3 Extensions of the Shapley operator : general evaluation 4 Asymptotic analysis: the main results 5 Asymptotic analysis - the discounted case: games with incomplete information 6 Asymptotic analysis - the continuous approach: games with incomplete information 7 Asymptotic analysis - the continuous approach: extension

Asymptotic Properties of Optimal Trajectories in Dynamic Programming

We prove in a dynamic programming framework that uniform convergence of the finite horizon values implies that asymptotically the average accumulated payoff is constant on optimal trajectories. We analyze and discuss several possible extensions to two-person games.Comment: 9 page

Explicit Formulas for Repeated Games with Absorbing States

Explicit formulas for the asymptotic value and the asymptotic minmax of finite discounted absorbing games are provided. New simple proofs for the existence of the limits when the players are more and more patient (i.e. when the discount factor goes zero) are given. Similar characterizations for stationary Nash equilibrium payoffs are obtained. The results may be extended to absorbing games with compact action sets and jointly continuous payoff functions.Repeated games, stochastic games, value, minmax, Nash equilibrium

Irreversible Games with Incomplete Information: The Asymptotic Value

Les jeux irréversibles sont des jeux stochastiques où une fois un état est quitté, il n'est plus jamais revisité. Cette classe contient les jeux absorbants. Cet article démontre l'existence et une caractérisation de la valeur asymptotique pour tout jeu irréversible fini à information incomplète des deux côtés. Cela généralise Mertens et Zamir 1971 pour les jeux répétés à information incomplète des deux côtés et Rosenberg 2000 pour les jeux absorbants à information incomplète d'un côté.Jeux stochastiques; jeux répétés; information incomplète; valeur asymptotique; principe de comparaison; inégalités variationelles

Advances in Zero-Sum Dynamic Games

International audienceThe survey presents recent results in the theory of two-person zero-sum repeated games and their connections with differential and continuous-time games. The emphasis is made on the following(1) A general model allows to deal simultaneously with stochastic and informational aspects.(2) All evaluations of the stage payoffs can be covered in the same framework (and not only the usual Cesàro and Abel means).(3) The model in discrete time can be seen and analyzed as a discretization of a continuous time game. Moreover, tools and ideas from repeated games are very fruitful for continuous time games and vice versa.(4) Numerous important conjectures have been answered (some in the negative).(5) New tools and original models have been proposed. As a consequence, the field (discrete versus continuous time, stochastic versus incomplete information models) has a much more unified structure, and research is extremely active