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highltd minimax_thm

highltd minimax_thm - OR 4350 Game Theory Prof Bland Linear...

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OR 4350 Game Theory February 21, 2012 Prof. Bland Linear Programming and Zero-Sum Two-Person Games These notes (revised from notes of Prof Todd) show how linear programming can be used to prove von Neumann’s Minimax Theorem and to obtain the value and the maximin and minimax strategies for the players in a two-person zero-sum game with a finite number of pure strategies for each player. We suppose that an m × n matrix A gives the payoffs to player I (and hence - A gives II’s payoffs). We’ll illustrate with an example in which A = - 1 1 3 0 2 - 4 - 2 6 - 1 3 2 - 3 . Before we proceed to the LP formulation, a little notation. Suppose we are given an m × n matrix A . For each j = 1 , . . . , n denote by A j the jth column of A . For each i = 1 , . . . , m denote by A i the ith row of A . Let P = { p R I m : p 0 , p i = 1 } and let Q = { q R I n : q 0 , q j = 1 } . We will assume, for now, that each p P is a row vector and each q Q is a column vector, so that the product pAq is well-defined. (Mea culpa: toward the end of this handout, when we get to the matrix notation for an LP formulation of player I’s optimal mixed strategy, we will treat p as though it is a column vector.) LP formulation Put yourself in player I’s place. Each p P is a mixed strategy, a randomization over I’s m pure strategies. For any fixed p P and any 1 j n the inner product pA j is the expected value of I’s payoff if I uses the mixed strategy p and II uses the pure strategy j . So min j pA j is a lower bound on I’s expected payoff, if she uses mixed strategy p . (Note that for fixed p , min j pA j = min q Q pAq .) Therefore player I would like to find among all p P one that makes min j pA j
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