–1–
Prof. William H. Sandholm
Department of Economics
University of Wisconsin
March 13, 2003
Midterm Solutions – Economics 713
1.
Suppose to the contrary that
τ
1
strictly dominates
σ
1
.
Then
u
1
(
τ
1
.
s
2
) >
u
1
(
σ
1
,
s
2
) for
all
s
2
∈
S
2
.
Therefore, since
G
is zero sum,
u
2
(
τ
1
,
s
2
) <
u
2
(
σ
1
,
s
2
) for all
s
2
∈
S
2
, and so
min
ρ
1
1
∈
∆
S
(
max
s
S
2
2
∈
u
2
(
ρ
1
,
s
2
))
≤
max
s
S
2
2
∈
u
2
(
τ
1
,
s
2
) <
max
s
S
2
2
∈
u
2
(
σ
1
,
s
2
). That is,
σ
1
does
not minmax player 2.
(Alternatively, one could prove this statement by appealing to the Minmax
Theorem.
The Minmax Theorem tells us that if
σ
1
is a minmax strategy for player 1,
then (
σ
1
,
σ
2
) is a Nash equilibrium, where
σ
2
is some minmax strategy for player 2.
Hence,
σ
1
is a best response to
σ
2
, and so cannot be a strictly dominated strategy.)
2. This is a bad idea.
While a player will never choose a dominated strategy in a
Nash equilibrium of a normal form game, it is quite possible for him to play a
dominated stage game action in a SPE of a repeated game.
The “grim trigger”
equilibrium of the repeated prisoner’s dilemma provides one example.
3. (i)
This condition says that the change in a player’s payoffs when he
unilaterally deviates is always identical to the change in potential.
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 Spring '08
 SANDHOLM
 Economics, Game Theory, player, Nash, Minmax Theorem, s2 S2

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