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Unformatted text preview: ume also that the payoﬀ functions
˜
(u1 , . . . , uI ) are diagonally strictly concave for x ∈ S . Then the game has a
unique pure strategy Nash equilibrium. 26 Game Theory: Lecture 6 Continuous Games Proof
Assume that there are two distinct pure strategy Nash equilibria.
Since for each i ∈ I , both xi∗ and xi must be an optimal solution for an
¯
optimization problem of the form (5), Theorem 4 implies the existence of
∗
∗
¯
¯
¯
nonnegative vectors λ∗ = [λ1 , . . . , λI ]T and λ = [λ1 , . . . , λI ]T such that
for all i ∈ I , we have �i ui (x ∗ ) + λi∗ �hi (xi∗ ) = 0, (7) λi∗ hi (xi∗ ) = 0, (8) ¯
�i ui (x ) + λi �hi (xi ) = 0,
¯
¯
¯ i hi (xi ) = 0.
λ
¯ (9) and
(10) 27 Game Theory: Lecture 6 Continuous Games Proof
Multiplying Eqs. (7) and (9) by (xi − xi∗ )T and (xi∗ −
xi )T respectively, and
¯
¯
adding over all i ∈ I , we obtain
0 = (
¯ − x ∗ )T �u (x ∗ ) + (x ∗ − x )T �u (x )
x
¯
¯
(11)
∗
∗T
∗
T∗
¯
+ ∑ λ �hi (x ) (xi − x ) + ∑ λi �hi (xi ) (x −
xi )
¯
¯
¯
i i i i ∈I > i i ∈I ¯
¯
¯
¯
∑ λi∗ �hi (xi∗ )T (xi − xi∗ ) + ∑ λi �hi (xi )T (xi∗ −
xi ), i ∈I i ∈I where to get the strict inequality, we used the assumption that the payoﬀ
functions are diagonally strictly concave for x ∈ S .
Since the hi are concave functions, we have
hi (xi∗ ) + �hi (xi∗ )T (xi − xi∗ ) ≥ hi (xi ).
¯
¯ 28 Game Theory: Lecture 6 Continuous Games Proof
Using the preceding together with λi∗ > 0, we obtain for all i ,
λi∗ �hi (xi∗ )T (xi − xi∗ )
¯ ≥ λi∗ (hi (xi ) − hi (xi∗ )) ¯
= λi∗ hi (xi )
¯
≥ 0, where to get the equality we used Eq. (8), and to get the last inequality, we
used the facts λi∗ > 0 and hi (xi ) ≥ 0.
¯
Similarly, we have ¯
λi �hi (xi )T (xi∗ −
xi ) ≥ 0.
¯
¯ Combi...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.
 Spring '10
 AsuOzdaglar

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