Thuijsman-LNMB - A Survey on Optimality and Equilibria in...

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Unformatted text preview: A Survey on Optimality and Equilibria in Stochastic Games Frank Thuijsman Department of Mathematics, Maastricht University 1 Abstract In this paper we discuss the main existence results on optimality and equilibria in two-person stochastic games with finite state and action spaces. Several examples are provided to clarify the issues. 1 The Stochastic Game Model In this introductory section we give the necessary definitions and notations for the two-person case of the stochastic game model and we briefly present some basic results. In section 2 we discuss the main existence results for zero-sum stochastic games, while in section 3 we focus on general-sum stochastic games. In each section we discuss several examples to illustrate the most important phenomena. It all started with the fundamental paper by Von Neumann [1928] in which he proves the so called minimax theorem which says that for each finite matrix of reals A = [ a ij ] m n i =1 ,j =1 there exist probability vectors ¯ x = (¯ x 1 , ¯ x 2 ,..., ¯ x m ) and ¯ y = (¯ y 1 , ¯ y 2 ,..., ¯ y n ) such that for all x and y it holds that 2 xA ¯ y ≤ ¯ xA ¯ y ≤ ¯ xAy . In other words: max x min y xAy = min y max x xAy . This theorem can be interpreted to say that each matrix game has a value. A matrix game A is played as follows. Simultaneously, and independent from each other, player 1 chooses a row i and player 2 chooses a column j of A . Then player 2 has to pay the amount a ij to player 1. Each player is allowed to randomize over his available actions and we assume that player 1 wants to maximize his expected payoff, while player 2 wants to minimize the expected payoff to player 1. The minimax theorem tells us that, for each matrix A there is a unique amount val( A ), which player 1 can guarantee as his minimal expected payoff, while at the same time player 2 can guarantee that the expected payoff to player 1 will be at most this amount. Later Nash [1951] considered the n-person extension of matrix games, in the sense that all n players, simultaneously and independently choose actions that determine a payoff for each and every one of them. Nash [1951] showed that in such games there always exists at least one (Nash-)equilibrium: a tuple of strategies such that each player is playing a best reply against the joint strategy of his opponents. For the two-player case this boils down to a “bimatrix game” where players 1 and 2 receive a ij and b ij respectively in case their choices determine entry ( i,j ). The result of Nash says that there exist ¯ x and ¯ y such that for all x and y it holds that ¯ xA ¯ y ≥ xA ¯ y and ¯ xB ¯ y ≥ ¯ xBy , where A = [ a ij ] and B = [ b ij ] are finite matrices of the same size....
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Thuijsman-LNMB - A Survey on Optimality and Equilibria in...

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