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.

Why Game Theory? In many situation, two or more decision makers simultaneously choose an action, and the action chosen by each player affects the rewards earned by the other player. Game theory is useful for making decision in cases where two or more decision makers have conflicting interests.

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 of n strategies. If the row player chooses his/her ith strategy and the column player chooses his/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

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.

Example Row Players’ Stregegy Column Player’s Stretegy Row Minimum 4 10 Row 2 2 3 1 Row 3 6 5 7 Column Maximum

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.

Example During the 8 to 9pm time slot, two networks are vying for an audience of 100 million viewers. Then networks must simultaneously announce the type of show they will air in that time slot. The possible choices for each network and the number of network 1 viewers (in millions) for each choice are shown in following table. For example, if both networks choose a western, the matrix indicates that 35 million people will watch network 1 and 100-35 = 65 million people will watch network 2. Thus, we have a two-person constant-sum game with c = 100 (million). Does this game have a saddle point? What is the value of the same to network 1?

Network 1’ Stregegy Network 2’s Stretegy Row Minimum Western Soap Opera Comedy 35 15 60 45 58 50 38 14 70 Column Maximum

Solution Network 1’ Stregegy Network 2’s Stretegy Row Minimum Western Soap Opera Comedy 35 15 60 45 58 50 38 14 70 Column Maximum