vieille-1 - ISRAEL JOURNAL OF MATHEMATICS 119 (2000), 55-91...

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ISRAEL JOURNAL OF MATHEMATICS 119 (2000), 55-91 TWO-PLAYER STOCHASTIC GAMES I: A REDUCTION BY NICOLAS VIEILLE Laboratoire d'Economdtrie, Ecole Polytechnique 1 rue Descartes, 75 005 Paris, France and Grape, Universitd Montesquieu Bordeaux J av. Duguit, 33608 Pessac, France e-mail: ABSTRACT This paper is the first step in the proof of existence of equilibrium payoffs for two-player stochastic games with finite state and action sets. It reduces the existence problem to the class of so-called positive absorbing recursive games. The existence problem for this class is solved in a subsequent paper. 1. Introduction This paper* is the first of two papers devoted to the proof of existence of equi- librium payoffs in two-player stochastic games. In this introduction, we briefly sketch an overview of the topic. Stochastic games are games played in stages, over a set S of states. In any stage, the players are fully informed of the past play, including the current state s, and choose actions from given sets A and B. The state of the game changes from one stage to the next one, as a (random) function of the current state and the actions selected by the players. In any stage n, the players receive a payoff, which also depends on the current state and the * This is a thoroughly revised version of a discussion paper which circulated under the same title [22]. I wish to thank Eilon Solan, Sylvain Sorin and an anonymous referee for many helpful comments. Received January 28, 1998 55
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56 N. VIEILLE Isr. J. Math. actions selected. The game never ends. We assume that the sets S, A and B are finite. This model was introduced by Shapley [17], who proved that, when payoffs are zero-sum, and the infinite stream of payoffs (gn)n_>l is evaluated according to a geometric average )~ )-]~ 1 (1 - n-1 = A) gn, the game has a value v~, and, for both players, stationary optimal strategies do exist. This was extended to the non zero-sum case by Fink [8]. Many results followed, relaxing the finiteness assumptions on S, A and B; see Mertens and Parthasarathy [13] for general conditions. Unlike the one-player case (Blackwell [5]), the optimal strategies vary with )~, even when A is arbitrarily close to 0. This dependency has been investigated by Bewley and Kohlberg [3, 2, 4]. Using the algebraic structure of the graph of the equilibrium correspondence, they proved that vx has a Puiseux expansion in a neighborhood of 0, and that similar properties hold for optimal strategies. Shortly after Shapley, the undiscounted evaluation was introduced by Gillette [9], in the zero-sum case. The type of requirement introduced by Gillette has been strengthened by Aumann and Maschler [1] in the framework of games with incomplete information. Assume the game is stopped after the n-th stage, and each player wishes to maximize the arithmetic average of the payoffs he received up to that stage. This defines a finite game F~, which has a value v~. Does limv~ exist ? If so, does there exist a single strategy which is optimal (or e-optimal) in every F,~, for n large enough ? If the answer is positive, then v = lim,~-~o~ v,~ is called the value of the game. In the same volume, Milnor and Shapley [15] and
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vieille-1 - ISRAEL JOURNAL OF MATHEMATICS 119 (2000), 55-91...

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