This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: OR 4350 Game Theory February 21, 2012 Prof. Bland Linear Programming and ZeroSum TwoPerson 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 twoperson zerosum 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 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 ≥ , ∑ p i = 1 } and let Q = { q ∈ R I n : q ≥ , ∑ 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 welldefined. (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...
View
Full
Document
This note was uploaded on 03/18/2012 for the course ORIE 4350 taught by Professor Shmoys during the Spring '08 term at Cornell.
 Spring '08
 SHMOYS

Click to edit the document details