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Unformatted text preview: Jnos Flesch, Gijs Schoenmakers, Koos Vrieze Stochastic Games on a Product State Space: The Periodic Case RM/08/016 JEL code: C73 M aastricht research school of E conomics of TE chnology and OR ganizations Universiteit Maastricht Faculty of Economics and Business Administration P.O. Box 616 NL  6200 MD Maastricht phone : ++31 43 388 3830 fax : ++31 43 388 4873 Stochastic Games on a Product State Space: The Periodic Case JAnos Flesch, Gijs Schoenmakers, Koos Vrieze & June 10, 2008 Abstract We examine socalled productgames. These are nplayer stochatic games played on a product state space S 1 & & S n ; in which player i controls the transitions on S i . For the general nplayer case, we establish the existence of equilibria. In addition, for the case of twoplayer zerosum games of this type, we show that both players have stationaryoptimal strategies. In the analysis of productgames, interestingly, a central role is played by the periodic features of the transition structure. Flesch et al. [2008] showed the exis tence ofequilibria under the assumption that, for every player i , the transition structure on S i is aperiodic. In this article, we examine productgames with pe riodic transition structures. Even though a large part of the approach in Flesch et al. [2008] remains applicable, we encounter a number of tricky problems that we have to address. We provide illustrative examples to clarify the essence of the di/erence between the aperiodic and periodic cases. Keywords: Noncooperative Games, Stochastic Games, Periodic Markov Decision Problems, Equilibria. 1 Introduction Stochastic games and productgames. n nplayer stochastic game is given by (1) a set of players N = f 1 ;:::;n g ; (2) a nonempty and &nite set of states S , (3) for each state s 2 S; a nonempty and &nite set of actions A i s for each player i; (4) for each & ddresses: JAnos Flesch: Department of Quantitative Economics Gijs Schoenmakers Koos Vrieze: Department of Mathematics. University of Maastricht, P.O.Box 616, 6200 MD Maastricht, The Netherlands. 1 state s 2 S and each joint action a s 2 & i 2 N A i s , a payo/ r i s ( a s ) 2 R to each player i; (5) for each state s 2 S and each joint action a s 2 & i 2 N A i s , a transition probability distribution p sa s = ( p sa s ( t )) t 2 S : The game is to be played at stages in N in the following way. Play starts at stage 1 in an initial state, say in state s 1 2 S . In s 1 ; each player i 2 N has to choose an action a i 1 from his action set A i s 1 . These choices have to be made independently. The chosen joint action a 1 = ( a 1 1 ;:::;a n 1 ) induces an immediate payo/ r i s 1 ( a 1 ) to each player i . Next, play moves to a new state according to the transition probability distribution p s 1 a 1 , say to state s 2 2 S . At stage 2 ; a new action a i 2 2 A i s 2 has to be chosen by each player i in state s 2 . Then, given action combination a 2 = ( a 1 2 ;:::;a n 2 ) , player i receives payo/ r i s 2 (...
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 Spring '09
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