janos - Jnos Flesch, Gijs Schoenmakers, Koos Vrieze...

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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 so-called product-games. These are n-player 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 n-player case, we establish the existence of- equilibria. In addition, for the case of two-player zero-sum games of this type, we show that both players have stationary-optimal strategies. In the analysis of product-games, interestingly, a central role is played by the periodic features of the transition structure. Flesch et al. [2008] showed the exis- tence of-equilibria under the assumption that, for every player i , the transition structure on S i is aperiodic. In this article, we examine product-games 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 product-games. n n-player 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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janos - Jnos Flesch, Gijs Schoenmakers, Koos Vrieze...

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