Operations Research: Applications and Algorithms Chapter 14 Game Theory to accompany Operations Research: Applications and Algorithms 4th edition by Wayne L. Winston Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

14.1 Two-Person Zero-Sum and Constant-Sum Games: saddle Points Characteristics of Two-Person Zero-Sum games There are two players (called the row player and column player) The row player must choose 1 of m strategies. Simultaneously, the column player must choose 1 if n strategies. If the row player chooses her ith strategy and the column player chooses her jth strategy, then the row player receives a reward for aij and the column player loses an amount aij. Thus, we may think of the row player’s reward of aij as coming from the column player. Such a game is called a two-person zero-sum game, which can be represented by a reward matrix.

A zero-sum game is a game in which the payoffs for the players always adds up to zero is called a zero-sum game. Two-person zero-sum is played according to the following basic assumption: Each player chooses a strategy that enables him/her to do the best he/she can, given that his/her opponent knows the strategy he/she is following. A two-person zero-sum game has a saddle point if and only if Max (row minimum) = min (column maximum) all all rows columns

If a two-person zero-sum game has a saddle point The row player should choose any strategy (row) attaining the maximum on the right side of (1). The column player should choose any strategy (column) attaining the minimum on the right side of (1). An easy way to spot a saddle point is to observe that the reward for a saddle point must be the smallest number in its row and the largest number in its column. A saddle point can also be thought of as a equilibrium point in that neither player can benefit from a unilateral change in strategy.

A two-person constant-sum game is a two-player game in which, for any choice of both player’s strategies, the row player’s reward and the column player’s reward add up to a constant value c.