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Nash Equilibrium: Existence
Let
G
= (
N
;
S
1
,...,S
n
;
u
1
,...,u
n
) be a strategic form game with
N
=
{
1
,...,n
}
.
Theorem
G has a Nash Equilibrium if, for every
i
∈
N
,
(a)
S
i
is a nonempty, compact, convex subset of
R
m
for some integer
m
;
(b)
u
i
is quasiconcave on
S
i
for every
s

i
∈
S

i
and is continuous on
S
.
Suppose that
G
is a ﬁnite game (each
S
i
is ﬁnite). I denote its mixed extension by Δ
G
=
(Δ
S
1
,...,
Δ
S
n
;
u
1
,...,u
n
), where for each
i
∈
N
, Δ
S
i
is the set of probability distributions over
S
i
, and
u
i
is a vNM utility. I denote a typical element of Δ
S
i
by
σ
i
; the probability that player
i
plays ’pure’ strategy
b
s
i
∈
S
i
by
σ
i
(
b
s
i
); a mixed strategy proﬁle by
σ
= (
σ
1
,...,σ
n
), where
σ
i
∈
Δ
S
i
for all
i
∈
N
; and the set of feasible mixed strategy proﬁles for Δ
G
by Δ
S
. Note that the expected
utility for player
i
when proﬁle
σ
is played is given by
U
i
(
σ
) =
X
s
∈
S
u
i
(
s
)
σ
1
(
s
1
)
σ
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 Spring '10
 schlee
 Microeconomics

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