A Little Game Theory 1

A LITTLE GAME THEORY

Mike Bailey

MSIM 852

A Little Game Theory 2

BASICS

• Two or more competitors• Each chooses a strategy• Pay-off determined when all strategies known

• John Von Newmann and Oskar Morganstern, Theory of Games and Economic Behavior (1944) seen by many as the first publication of Operations

Research Linear Programming is introduced in a chapter

A Little Game Theory 3

TWO-PERSON ZERO-SUM GAME

• Most common form

• Two competitors, each will be rewarded

• Fixed reward total What one wins, the other loses

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PAY-OFF MATRIX

• Presented as reward for player A

x y z

1 80 40 75

2 70 35 30

B’s strategy

A’s strategy

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MAXIMIN (MINIMAX)

• A chooses the strategy where he gets the best payoff if B acts optimally Maximizes the minimum

x y z

1 80 40 75

2 70 35 30

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MAXIMIN (MINIMAX)

• A chooses the strategy where he gets the best payoff if B acts optimally Maximizes the minimum

x y z

1 80 40 75

2 70 35 30

Does not alwaysoccur

“Saddlepoint”

Value of the Game

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DOMINANCE

• y dominates x for player B

x y z

1 80 40 75

2 70 35 30

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DOMINANCE

• y dominates x for player B• ...then 1 dominates 2 for player A

x y z

1 80 40 75

2 70 35 30

A Little Game Theory 9

DOMINANCE

• y dominates x for player B• ...then 1 dominates 2 for player A• ......then y dominates z for player B

x y z

1 80 40 75

2 70 35 30

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DOMINANCE

• y dominates x for player B• ...then 1 dominates 2 for player A• ......then y dominates z for player B• .........done

x y z

1 80 40 75

2 70 35 30

Doesn’t always happen

Useful for bigtables

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MIXED STRATEGIES

w x y z

1 75 105 65 45

2 70 60 55 40

3 80 90 35 50

4 95 100 50 55

A Little Game Theory 12

MIXED STRATEGIES

w x y z

1 75 105 65 45

2 70 60 55 40

3 80 90 35 50

4 95 100 50 55

A Little Game Theory 13

MIXED STRATEGIES

w x y z

1 75 105 65 45

2 70 60 55 40

3 80 90 35 50

4 95 100 50 55

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MIXED STRATEGY

• A will choose strategy 1 with probability p V(y) = 65p + 50(1-p) V(z) = 45p + 55(1-p)

• What value of p makes A indifferent to B’s choice?

y z

1 65 45

4 50 55

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MIXED STRATEGY

• A will choose strategy 1 with probability p 65p + 50(1-p) = 45p + 55(1-p)

• p = 0.8• V = 53

• B will choose y with probability q 65q + 45(1-q) = 50q + 55(1-q)

• q = 0.6• V = 53

y z

1 65 45

4 50 55

A Little Game Theory 16

PRISONER’S DILEMMA

• Payoffs are jail sentences (for A, for B) in years

silent betray

silent 1/2, 1/2 10, free

betray free, 10 2,2

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PRISONER’S DILEMMA• Pareto Optimum

No move can make a player better off without harming another• Nash Equilibrium

No player can improve payoff unilaterally

silent betray

silent 1/2, 1/2 10, free

betray free, 10 2,2

http://en.wikipedia.org/wiki/Prisoner's_dilemma

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APPLICATIONS

• ASW (Hide and Seek)

• Arms Control

• Advertising Strategy

• Smuggling

• Making the All-Star Team

• Multiethnic Insurgency and Revolt

• Drug Testing (Wired, August 2006)

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ITERATED PD

• Set a strategy involving a sequence of choices and memory of the (choice, outcome)

• Random termination of the game• Noise in the game• Specified payoff matrix

The Iterated Prisoner's Dilemma Competition:Celebrating the 20th Anniversary

                                                                    

http://www.prisoners-dilemma.com/