to accompany quantitative analysis for management,9e by render/stair/hanna m4-1 © 2006 by prentice...

To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna

M4-1 © 2006 by Prentice Hall, Inc.Upper Saddle River, NJ 07458

Module 4Module 4

Game TheoryGame Theory

Prepared by Lee Revere and John LargePrepared by Lee Revere and John Large



Learning ObjectivesLearning Objectives

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 the game.

3. Solve mixed strategy games

when there is no saddle point.



Module OutlineModule Outline

M4.1 Introduction

M4.2 Language of Games

M4.3 The Minimax Criterion

M4.4 Pure Strategy Games

M4.5 Mixed Strategy Games

M4.6 Dominance



Game theory is the study of how optimal strategies are formulated in conflict. Game theory has been effectively used for:

War strategies Union negotiators Competitive business strategies

IntroductionIntroduction



Game models are classified by the number of players, the sum of all payoffs, and the number of strategies employed.

A zero sum game implies that what is gained by one player is lost for the other.

IntroductionIntroduction(continued)(continued)



Language of GamesLanguage of Games

Consider a duopoly competitive business market in which one company is considering advertising in hopes of luring customers away from its competitor. The company is considering radio and/or newspaper advertisements.

Let’s use game theory to determine the best strategy.



Language of Games Language of Games (continued)(continued)

STORE X’s PAYOFFs

Y’s strategy 1

(use radio)

Y’s strategy 2

(use newspaper)

X’s strategy 1

(use radio)

3 5

X’s strategy 2

(use newspaper)

1 -2

Below is the payoff matrix (as a percent of change in market share) for Store X. A positive number means that X wins and Y loses, while a negative number implies Y wins and X loses.

Note: Although X is considering the advertisements (therefore the results favor X), Y must play the game.



Language of Games Language of Games (continued)(continued)

Store X’s Strategy

Stores Y’s Strategy

Outcome (% change in market share)

X1: Radio Y1: Radio X wins 3

Y loses 3

X1: Radio Y2: Newspaper

X wins 5

Y loses 5

X2: Newspaper

Y1: Radio X wins 1

Y loses 1

X2: Newspaper

Y2: Newspaper

X loses 2

Y wins 2

Note: Although X is considering the advertisements (therefore the results favor X), Y must play the game.



The Minimax CriterionThe Minimax Criterion

The minimax criterion is used in a two-person zero-sum game. Each person should choose the strategy that minimizes the maximum loss.

Note: This is identical to maximizing one’s minimum gains.



The Minimax Criterion The Minimax Criterion (continued)(continued)

The upper value of the game is equal to the minimum of the maximum values in the columns.

The lower value of the game is equal to the maximum of the minimum values in the rows.



STORE X’s PAYOFFs

Y1

(radio)

Y2

(newspaper)

Minimum

X1

(radio)

3 5 3

X2

(newspaper)

1 -2 2

Maximum 3 5


Lower Value of the Game: Maximum of the minimums

Upper Value of the Game: Minimum of the maximums



STORE X’s PAYOFFs

Y1

(radio)

Y2

(newspaper)

Minimum

X1

(radio)

3 5 3

X2

(newspaper)

1 -2 2

Maximum 3 5


Saddle point: Both upper and lower values are 3.

A saddle point condition exists if the upper and lower values are equal. This is called a pure strategy because both players will follow the same strategy.



STORE X’s PAYOFFs

Y1

(radio)

Y2

(newspaper)

Minimum

X1

(radio)

10 6 6

X2

(newspaper)

-12 2 -12

Maximum 10 6


Saddle point

Let’s look at a second example of a pure strategy game.

Lower value

Upper value



Mixed Strategy GameMixed Strategy Game

A mixed strategy game exists when there is no saddle point. Each player will then optimize their expected gain by determining the percent of time to use each strategy.

Note: The expected gain is determined using an approach very similar to the expected monetary value approach.



Mixed Strategy Games Mixed Strategy Games (continued)(continued)

Y1

(P)

Y2

(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)

Each player seeks to maximize his/her expected gain by altering the percent of time (P or Q) that he/she use each strategy.

Set these two equations equal to each other and solve for Q

Set

th

ese

two

equ

atio

ns

equ

al t

o ea

ch o

ther

an

d s

olve

for

P



Mixed Strategy Games Mixed Strategy Games (continued)(continued)

4P + 2(1-P) = 1P + 10(1-P)4P – 2P – 1P + 10P = 10 – 2P = 8/11 and 1-P = 3/11

Expected payoff: 1P + 10(1-P) = 1(8/11) + 10(3/11) = 3.46

4Q + 1(1-Q) = 2Q + 10(1-Q)4Q – 1Q – 2Q + 10Q = 10 – 1Q = 9/11 and 1-Q = 2/11

Expected payoff: 2Q + 10(1-Q) = 2(9/11) + 10(2/11)

= 3.46



DominanceDominance

Dominance is a principle that can be used to reduce the size of games by eliminating strategies that would never be played.

Note: A strategy can be eliminated if all its game’s outcomes are the same or worse than the corresponding outcomes of another strategy.



Dominance Dominance (continued)(continued)

Y1 Y2

X1 4 3

X2 2 20

X3 1 1

Y1 Y2

X1 4 3

X2 2 20

Initial game

X3 is a dominated strategy

Game after removal of dominated strategy



Dominance Dominance (continued)(continued)

Y1 Y2 Y3 Y4

X1 -5 4 6 -3

X2 -2 6 2 -20

Initial game

Game after removal of dominated strategies

Y1 Y4

X1 -5 -3

X2 -2 -20

to accompany quantitative analysis for management,9e by render/stair/hanna m4-1 © 2006 by prentice...

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