Game Theory

4

To accompanyQuantitative Analysis for Management, Twelfth Edition, by Render, Stair, Hanna and HalePower Point slides created by Jeff Heyl Copyright ©2015 Pearson Education, Inc.

After completing this module, students will be able to:

LEARNING OBJECTIVES

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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.

M4.1 Introduction

M4.2 Language of Games

M4.4 Pure Strategy Games

M4.5 Mixed Strategy Games

M4.6 Dominance

MODULE OUTLINE

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Introduction• 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– Game models classified by number of players,

sum of all payoffs, and number of strategies employed

– Two-person and zero-sum games

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Language of Games

• Two lighting fixture stores– A duopoly– Advertising strategies change

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GAME PLAYER Y’s STRATEGIES

Y1

(Use radio)Y2

(Use newspaper)

GAME PLAYER X’s STRATEGIES

X1

(Use radio)3 5

X2

(Use newspaper)1 –2

Table M4.1 – Store X’s Payoff Matrix

Language of Games

• Game outcomes

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STORE X’s STRATEGY

STORE Y’s STRATEGY

OUTCOME (% CHANGE IN MARKET SHARE)

X1 (use radio) Y1 (use radio) X wins 3 and Y loses 3

X1 (use radio) Y2 (use newspaper) X wins 5 and Y loses 5

X2 (use newspaper) Y1 (use radio) X wins 1 and Y loses 1

X2 (use newspaper) Y2 (use newspaper) X loses 2 and Y wins 2

Dominance

• The principle of dominance can be used to reduce the size of games by eliminating strategies that would never be played

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Dominance

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• Reduce the size of this game

Y1 Y2

X1 4 3

X2 2 20

X3 1 1

• Can be reduced to

Y1 Y2

X1 4 3

X2 2 20

Dominance

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• Reduce the size of this game

Y1 Y2 Y3 Y4

X1 –5 4 6 –3

X2 –2 6 2 –20

• Can be reduced to

Y1 Y4

X1 –5 –3

X2 –2 –20

Mixed Strategy Games

• Players will play each strategy for a certain percentage of the time

• Called a mixed strategy game• Commonly solved using the expected gain or

loss approach• Each player plays a strategy a particular

percentage of the time so that the expected value of the game does not depend upon what the opponent does

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Mixed Strategy Games

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PLAYER Y’s STRATEGIES

Y1 Y2

PLAYER X’s STRATEGIES

X1 4 2

X2 1 10

Table M4.4 – Game Table for Mixed Strategy Game

4P + 2(1 – P) = 1P + 10(1 – P)

P = 8/11

1 – P = 1 – 8/11 = 3/11

For Player Y solve

Mixed Strategy Games

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PLAYER Y’s STRATEGIES

Y1 Y2

PLAYER X’s STRATEGIES

X1 4 2

X2 1 10

Table M4.4 – Game Table for Mixed Strategy Game

4Q + 1(1 – Q) = 2Q + 10(1 – Q)

Q = 9/11

1 – Q = 2/11

For Player X solve

Mixed Strategy Games

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Table M4.5 – Game Table for Mixed Strategy Game with Percentages (P, Q) Shown

Y1 Y2

P 1 – P Expected gain

X1 Q 4 2 4P + 2(1 – P)

X2 1 – Q 1 10 1P + 10(1 – P)

Expected gain 4Q + 1(1 – Q) 2Q + 10(1 – Q)