vieille-2 - ISRAEL JOURNAL OF MATHEMATICS 119 (2000),...

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ISRAEL JOURNAL OF MATHEMATICS 119 (2000), 93-126 TWO-PLAYER STOCHASTIC GAMES II: THE CASE OF RECURSIVE GAMES BY NICOLAS VIEILLE Laboratoire d'Economdtrie, Ecole Polytechnique 1 rue Descartes, 75 005 Paris, France and Grape, Universitd Montesquieu Bordeaux av. Duguit, 33608 Pessac, France e-mail: [email protected] ABSTRACT This paper contains the second step in the proof of existence of equilib- rium payoffs for two-player stochastic games. It deals with the case of positive absorbing recursive games This paper* complements [12]. We prove here the existence of equilibrium payoffs in two-player, absorbing positive recursive games. Recursive games are stochastic games in which the players receive a payoff equal to zero until an absorbing state is reached. Positive recursive games are recursive games in which the payoff to one of the players is positive in each absorbing state. Such a game is absorbing if the other player cannot prevent the play from reaching an absorbing state in finite time. Zero-sum recursive games were first introduced by Everett [2], who proved the existence of stationary e-optimal strategies. Flesch, Thuijsman and Vrieze [3] exhibited a two-player recursive game with no stationary e-equilibrium profile. Independently of our work, Solan [9] proved the existence of equilibrium payoffs * This is a thoroughly revised version of a discussion paper which circulated under a slightly different title [10]. I wish to thank Eilon Solan, Sylvain Sorin and an anonymous referee for many helpful comments. Received January 28, 1998 93
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94 N. VIEILLE Isr. J. Math. in two-player, positive recursive games with two non-absorbing states that have the absorbing property. The paper is organized as follows. Section 1 contains definitions and the state- ment of the main result. Sections 2 and 3 are devoted to examples. The first example is a variation on the example in [3]: it is a two-player positive absorbing recursive game with no stationary e-equilibrium profile. The second example is used to present the main features of the e-equilibrium profiles that we obtain. Section 4 provides a sufficient condition for the existence of equilibrium payoffs. In Section 6, we define a family of constrained games, indexed by c > 0, and analyze the asymptotics of this family, as 6 goes to zero. 1. Definitions and main result A two-player recursive game is given by (i) a finite set of states S partitioned into S* and S\S*; (ii) finite sets A and B of available actions; (iii) a transition function p: S\S* x A x B -+ A(S), where A(S) is the space of all probability distributions over S, and (iv) a payoff function g = (gl, g2): S* -+ R 2. The game is played as follows. As long as S* has not been reached, the players choose actions, and the state changes from stage to stage according to p. As soon as a state s* C S* is reached, the game stops and the players receive the payoff g(s*). The elements of S* are called absorbing states. It is convenient to formalize this as follows. The set of stages is the set N* of positive integers. The initial state sl is given. At stage n, the current state sn is announced to the players. Player 1 and player 2 choose an action an and
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vieille-2 - ISRAEL JOURNAL OF MATHEMATICS 119 (2000),...

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