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3/3/2009
1
Mixed Strategies – Matching Pennies
±
We denote the strategies on the game bimatrix
Guildenstern
q1

q
Heads
Tails
±
Rosencrantz’s payoff to
H
: (1)q + (1)(1q) =
2q1
T
: (1)q + (1)(1q) =
12q
Rosencrant
zp
Heads
1,1
1 , 1
1p
Tails
1 , 1
1 , 1
Mixed Strategies – Matching Pennies
±
Rosencrantz’s
&
Guildenstern’s
best responses:
q
1
The best response
curves intersect at
the Nash
p
01
1/2
1/2
equilibrium
p=1/2, q=1/2
Mixed Strategies – Matching Pennies
±
Suppose R’s payoff to H increases
Guildenstern

q
Heads
Tails
±
Rosencrantz’s payoff to
H
: (2)q + (1)(1q) =
3q1
T
: (1)q + (1)(1q) =
12q
±
Now, 3q1 = 12q requires q=2/5
Rosencrant
Heads
2,1
1 , 1
1p
Tails
1 , 1
1 , 1
Mixed Strategies – Matching Pennies
±
Rosencrantz’s
&
Guildenstern’s
best responses:
q
1
The best response
curves intersect at
the Nash
p
1/2
1/2
the Nash
equilibrium
p=1/2, q=2/5
2/5
Mixed Strategies – Matching Pennies
±
Suppose R’s payoff to H increases
Rt
Guildenstern

q
Heads
Tails
±
Rosencrantz’s payoff to
H
: (2)q + (1)(1q) =
3q1
T
: (1)q + (1)(1q) =
12q
±
Now, 3q1 = 12q requires q=2/5
Rosencrant
Heads
1 , 1
1p
Tails
1 , 1
1 , 1
Mixed Strategies – Chicken
±
Since we allow for randomization over pure
strategies, so we can consider mixedstrategy
Nash equilibria even in nonzerosum games
±
Recall the game of Chicken…
±
We have two purestrategy NE
Daisy Duke
Swerve
Don’t Swerve
BillyJo Bob
Swerve
1 , 1
1 , 2
Don’t Swerve
2 , 1
0 , 0
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Daisy Duke
q1

q
Swerve
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 Spring '07
 Emre

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