A LITTLE GAME THEORY
Mike Bailey
MSIM 852
A Little Game Theory
1
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
2
TWO-PERSON
ZERO-SUM GAME
• Most common form
• Two competitors, each will be rewarded
• Fixed reward total
 What one wins, the other loses
A Little Game Theory
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PAY-OFF MATRIX
• Presented as reward
for player A
B’s strategy
A’s strategy
x
y
z
1 80
40
75
2 70
35
30
A Little Game Theory
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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
A Little Game Theory
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MAXIMIN (MINIMAX)
• A chooses the strategy where he gets the best
payoff if B acts optimally
Does not always
occur
 Maximizes the minimum
“Saddlepoint”
x
y
z
1 80
40
75
2 70
35
30
A Little Game Theory
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
A Little Game Theory
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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
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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
x
y
z
1 80
40
75
2 70
35
30
A Little Game Theory
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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
A Little Game Theory
Doesn’t always
happen
Useful for big
tables
10
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
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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
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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
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MIXED STRATEGY
• A will choose strategy 1
with probability p
y
z
1
65
45
4
50
55
 V(y) = 65p + 50(1-p)
 V(z) = 45p + 55(1-p)
• What value of p makes A
indifferent to B’s choice?
A Little Game Theory
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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
1
4
y
65
50
z
45
55
• B will choose y with
probability q
 65q + 45(1-q) = 50q + 55(1-q)
• q = 0.6
• V = 53
A Little Game Theory
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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
A Little Game Theory
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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
A Little Game Theory
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)
A Little Game Theory
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ITERATED PD
The Iterated Prisoner's Dilemma Competition:
Celebrating the 20th Anniversary
• 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
http://www.prisoners-dilemma.com/
A Little Game Theory
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