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Unformatted text preview: (
¯ − x ∗ )T �u (x ∗ ) + (x ∗ − x )T �u (x )
x
¯
¯
(10)
∗
∗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 ).
¯
¯ 10 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium 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. (7), 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.
¯
¯ Combining the preceding two relations with the relation in (10) yields a
contradiction, thus concluding the proof. 11 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Suﬃcient Condition for Diagonal Strict Concavity
Let U (x ) denote the Jacobian of �u (x ) [see Eq. (5)]. In particular, if the xi
are all 1dimensional, then U (x ) is given by
⎛ ∂2 u (x ) ∂2 u (x )
⎞
1
1
···
2
∂x1 ∂x2
⎜ ∂x1
⎟
⎜2
⎟
..
U (x ) = ⎜ ∂ u2 (x )
⎟.
.
⎝ ∂x2 ∂x1
⎠
.
.
.
Proposition
For all i ∈ I , assume that the strategy sets Si are given by Eq. (3), where hi is a
concave function. Assume that the symmetric matrix (U (x ) + U T (x )) is
negative deﬁnite for all x ∈ S , i.e., for all x ∈ S , we have
y T (U (x ) + U T (x ))y < 0, ∀ y �= 0. Then, the payoﬀ functions (u1 , . . . , uI ) are diagonally strictly concave for x ∈ S .
12 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Proof
Let x ∗ , x ∈ S . Consider the vector
¯
x ( λ ) = λx ∗ + (1 − λ )x ,
¯ for some λ ∈ [0, 1]. Since S is a convex set, x (λ) ∈ S .
Because U (x ) is the Jacobian of �u (x ), we have
d
�u (x (λ))
dλ =
or �1
0 dx (λ)
d (λ)
U (x (λ))(x ∗ − x ),
¯ = U (x (λ)) U (x (λ))(x ∗ − x )d λ = �u (x ∗ ) − �u (x ).
¯
¯ 13 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Proof
Multiplying the preceding by (x − x ∗ )T yields
¯ (
¯ − x ∗ )T �u (x ∗ ) + (x ∗ − x )T �u (x )
x
¯
¯
�1
1
=−
(x ∗ − x )T [U (x (λ)) + U T (x (λ))](x ∗ − x )d λ
¯
¯
20
> 0,
where to get the strict inequality we used the assumption that the
symmetric matrix (U (x ) + U T (x )) is negative deﬁnite for all x ∈ S . 14 Game Theory: Lecture 7 Supermodular Games Supermodular Games
Supermodular games are those characterized by strategic complementarities
Informally, this means that the marginal utility of increasing a player’s
strategy raises with increases in the other players’ strategies.
Implication ⇒ best response of a player is a nondecreasing f...
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 Spring '10
 AsuOzdaglar

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