Chapter01 Game Theory

8/12/2019 Chapter01 Game Theory

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2008 Prentice-Hall, Inc.

Chapter 1

Game Theory


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2009 Prentice-Hall, Inc. 122

In t roduct ion

Game: a contest involving two or more

decision makers, each of whom wants to win.

Game theory: the study of how optimal

strategies are formed in conflict

Games classified by:

Number of players

Sum of all payoffs

Number of strategies employed

Zero-sum game: the sum of the losses must

equal the sum of the gains


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Payoff Tab le

Y1

Use radio

Y2

Use newspaper

X1

Use radio

3 5

X2

Use newspaper

1 -2

Game player Ys strategies

Game

Player

Xs

Strategie

s

+ entry, s X wins and Y loses

- entry, Y wins and X loses


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Outcomes

Store Xs Strategy Store Ys Strategy Outcome % change in

market share)

X1Radio Y1Radio X wins 3

Y loses 3X1Radio Y2Newspaper X wins 5

Y loses 5

X2Newspaper Y1Radio X wins 1

Y loses 1X2Newspaper Y2Newspaper X loses 2

Y wins 2


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Minimax Criter ion

In a zero-sum game, each person canchoose the strategy that minimizes themaximum loss


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Minimax Criter ion

Y1 Y2 Minimum

X1 3 5 3

X2 1 -2 -2

Maximum 3 5

Minimum of maximums

Maximums ofminimums

Saddle point

Note: an equilibrium or saddle point exists if the uppervalue of the game is equal to the lower value of the game.

This is called the value of the game.

This is a pure strategy game


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Pure Strategy

Whenever a saddle point is present, thestrategy a player should follow willalways be the same, regardless of the

strategy of the other player.



Pure Strategy

Second Players (Y)

Strategies

Y1 Y2

First Players (X)

Strategies

X1 3 5

X2

1 -2

Ys purestrategy

Xs purestrategy



Pure Strategy

Minimax Criter ion

Player Ys

Strategies

Minimum Row

Number

Y1 Y2

Player Xs

strategies

X1 10 6 6

X2 -12 2 -12

Maximum Column

Number

10 6



Mixed Strategy Game

Y1P

Y21-P

Expected Gain

X1

Q

4 2 4P+2(1-P)

X2

1-Q

1 10 1P+10(1-p)

4Q+1(1-Q) 2Q+10(1-q)



Solv ing for P & Q

4P+2(1-P) = 1P+10(1-P)or: P = 8/11 and 1-p = 3/11

Expected payoff:

EPX=1P+10(1-P)

=1(8/11)+10(3/11) = 3.46

4Q+1(1-Q)=2Q+10(1-q)or: Q=9/11 and 1-Q = 2/11

Expected payoff:

EPY=3.46



Dominance

A strategy can be eliminated if all its

games outcomes are the same or

worse than the corresponding outcomes

of another strategy



Dominat ion

Y1 Y2

X1 4 3

X2 2 20

X3 1 1

Initial Game

Y1 Y2

X1 4 3

X2 2 20

X3 is a dominated strategy

Game after removal of dominated strategy


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Dominat ion

Y1 Y2 Y3 Y4

X1 -5 4 6 -3

X2 -2 6 2 -20

Initial Game

Y1 Y4

X1 -5 -3

X2 -2 -20

Game after dominated strategies are removed

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Chapter01 Game Theory