- Game theory

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

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

Game TheoryGame Theory

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



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



ExerciseExercise

• Player A has a $1 bill and $20 bill, and player B has a $5 bill and $10 bill. Each player will select a bill from the other player without knowing what bill the other player selected. If the total of the bills selected is odd player A gets both bills, but if the total is even, player B gets both bills.

• Develop the payoff table for this problem.

• Determine the value of the game.



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

Question1Question1

• What is the value of the following game and the strategies for A and B?

B1 B2

A1 19 20

A2 5 -4



Questions2Questions2

• Shoe town and fancy foot are both vying for more share of the market. If Shoe town does no ad, it will not lose any share of the market if Fancy Foot does nothing. It will lose 2% of market if Fancy Foot invests $10,000 in ad, and it will lose 5% of the market if Fancy Foot invests $20,000 in ad. On the other hand, if Shoe town invests $15,000 in ad, it will gain 3% of the market if Fancy Foot does nothing; it will gain 1% of the market if Fancy Foot invests $10,000 in ad; and it will lose 1% if Fancy Foot invests $20,000 in ad.

• Q: Develop a payoff table for this problem.



Question3Question3

• For the following 2-person, zero-sum game, are there any dominated strategies? If so, eliminate any dominated strategy and find the value of the game.

Y1 Y2 Y3

X1 4 5 10

X2 3 4 2

X3 8 6 9



Question4Question4

• Solve the following the following game:

Y1 Y2

X1 -5 -10

X2 12 8

X3 4 12

X4 -40 -5



Documents

- Game theory