By L. S. Shapley (auth.), Abraham Neyman, Sylvain Sorin (eds.)
This quantity relies on lectures given on the NATO complex learn Institute on "Stochastic video games and Applications," which happened at Stony Brook, long island, united states, July 1999. It provides the editors nice excitement to provide it at the get together of L.S. Shapley's 80th birthday, and at the 50th "birthday" of his seminal paper "Stochastic Games," with which this quantity opens. we want to thank NATO for the supply that made the Institute and this quantity attainable, and the heart for video game concept in Economics of the nation collage of recent York at Stony Brook for webhosting this occasion. We additionally desire to thank the Hebrew college of Jerusalem, Israel, for supplying carrying on with monetary help, with no which this undertaking may by no means were accomplished. specifically, we're thankful to our editorial assistant Mike Borns, whose paintings has been necessary. We additionally want to recognize the aid of the Ecole Poly tech nique, Paris, and the Israel technological know-how origin. March 2003 Abraham Neyman and Sylvain Sorin ix STOCHASTIC video games L.S. SHAPLEY collage of California at l. a. l. a., united states 1. advent In a stochastic online game the play proceeds through steps from place to place, based on transition percentages managed together by means of the 2 players.
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A, ,8) >. )P(a,,8))-lr(a,,8). Average rewards: 1 N lim - N Lpn(a,,8)r(a,,8) N--+oo + 1 n=O Q(a, ,8)r(a, ,8). J. 3). (a,;3) + (1- 8)"(a,;3). Some comments might be in order. = (i) That L (1- A)npn(a,;3) = (I - (1- A)P(a,;3))-l follows from the fact that n=o N (I - (1- A)P(a,;3))(L(l- A)npn(a,;3)) n=O N (L(l- A)npn(a,;3))(I - (1- A)P(a,;3)) n=O while limN-+=(l - A)N+l p N+1(a,;3) = a. N (ii) By definition Q(a,;3):= lim N~l L pn(a,;3), which is called the N-+= n=O Cesaro limit. For later use, it easily follows that P(a,;3)Q(a,;3) = Q(a,;3)P(a,;3) = Q(a, ;3).
1. ASYMPTOTIC STUDY The first approach leads to the "compact case": under natural assumptions on the action spaces and on the reward function the mixed strategy spaces will be compact for a topology for which the payoff function will be continuous. -m=l rm). In the finite case, this reduces to a game with finitely many pure strategies. ii) the A-discounted game rA(z) with initial state Z and payoff equal to the 30 SYLVAIN SORIN discounted sum of the rewards: In this setup the first task is to find conditions under which: - in the two-person zero-sum case the value will exist; it will be denoted respectively by vn(z) and v),(z); - in the I-player case, equilibria will exist; the corresponding sets of equilibrium payoffs will be denoted by En(z) and E),(z).
This question can be answered positively, again using the limit theory. A state is called (c:- ) easy for a player ifthe player can guarantee the value for this game (up to c:) using stationary strategies. Theorem 6 Let Smax be the subset of states for which ,",(, the average reward value, is maximal and let Smin be the subset of state for which '"'( is minimal. (i) The states Smax are c:-easy for player 2 and some of the states of Smax are easy for player 1. (ii) The states Smin are c:-easy for player 1 and some of the states of Smin are easy for player 2.