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MODULE

4

Game Theory

LEARNING OBJECTIVES

After completing this supplement, students will be able to:

1. Understand the principles of zero-sum, two-person

games.

2. Analyze pure strategy games and use dominance to

reduce the size of a game.

3. Solve mixed strategy games when there is no saddle

point.

SUPPLEMENT OUTLINE

M4.1

M4.2

M4.3

M4.4

M4.5

M4.6

Introduction

Language of Games

The Minimax Criterion

Pure Strategy Games

Mixed Strategy Games

Dominance

Summary • Glossary • Solved Problems • Self-Test • Discussion Questions and Problems • Bibliography Appendix M4.1: Game Theory with QM for Windows

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M4.1

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MODULE 4 • GAME THEORY

Introduction

In a zero-sum game, what is

gained by one player is lost by

the other.

M4.2

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As discussed in Chapter 1, competition can be an important decision-making factor. The strategies taken by other organizations or individuals can dramatically affect the outcome of our decisions. In the automobile industry, for example, the strategies of competitors to introduce certain models with certain features can dramatically affect the profitability of other carmakers. Today, business cannot make important decisions without considering what other organizations or individuals are doing or might do.

Game theory is one way to consider the impact of the strategies of others on our strategies and outcomes. A game is a contest involving two or more decision makers, each of whom wants to win. Game theory is the study of how optimal strategies are formulated in conflict. The study of game theory dates back to 1944, when John von Neumann and Oscar Morgenstern published their classic book, Theory of Games and Economic Behavior.1 Since then, game theory has been used by army generals to plan war strategies, by union negotiators and managers in collective bargaining, and by businesses of all types to determine the best strategies given a competitive business environment.

Game theory continues to be important today. In 1994, John Harsanui, John Nash, and Reinhard Selten jointly received the Nobel Prize in Economics from the Royal Swedish Academy of Sciences.2 In their classic work, these individuals developed the notion of noncooperative game theory. After the work of John von Neumann, Nash developed the concepts of the Nash equilibrium and the Nash bargaining problem, which are the cornerstones of modern game theory.

Game models are classified by the number of players, the sum of all payoffs, and the number of strategies employed. Due to the mathematical complexity of game theory, we limit the analysis in this module to games that are two person and zero sum. A two-person game is one in which only two parties can play—as in the case of a union and a company in a bargaining session. For simplicity, X and Y represent the two game players. Zero sum means that the sum of losses for one player must equal the sum of gains for the other player. Thus, if X wins 20 points or dollars, Y loses 20 points or dollars. With any zero-sum game, the sum of the gains for one player is always equal to the sum of the losses for the other player. When you sum the gains and losses for both players, the result is zero—hence the name zero-sum games.

Language of Games

To introduce the notation used in game theory, let us consider a simple game. Suppose there are only two lighting fixture stores, X and Y, in Urbana, Illinois. (This is called a duopoly.) The respective market shares have been stable up until now, but the situation may change. The daughter of the owner of store X has just completed her MBA and has developed two distinct advertising strategies, one using radio spots and the other newspaper ads. Upon hearing this, the owner of store Y also proceeds to prepare radio and newspaper ads. The 2 * 2 payoff matrix in Table M4.1...