Existence of the uniform value in repeated games.

Summary In this thesis, we are interested in a general model of repeated two-player zero-sum games and in particular in the problem of the existence of uniform value. A repeated game has a uniform value if there exists a payoff that both players can guarantee, in all games starting today and sufficiently long, independently of the length of the game. In a first chapter, we study the cases of a single player, called a partially observable Markovian decision process, and of games where one player is perfectly informed and controls the transition. It is known that these games admit a uniform value. By introducing a new distance on the probabilities on the simplex of Rm, we show the existence of a stronger notion where players guarantee the same payout on any sufficiently long time interval and not only on those starting today. In the next two chapters, we show the existence of the uniform value in two particular cases of repeated games: commutative games in the dark, where the players do not observe the state but the state is independent of the order in which the actions are played, and games with a more informed controller, where one player is more informed than the other player and controls the evolution of the state. In the last chapter, we study the connection between the uniform convergence of values in n-step games and the asymptotic behavior of optimal strategies in these n-step games. For each n, we consider the guaranteed payoff during ln steps with 0 < l < 1 by the optimal strategies for n steps and the asymptotic behavior when n tends to infinity.