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Unformatted text preview: if for any x , y ∈ S and any λ ∈ [0, 1], λx + (1 − λ)y ∈ S . convex set not a convex set 12 Game Theory: Lecture 5 Existence Results Weierstrass’s Theorem Theorem
(Weierstrass) Let A be a nonempty compact subset of a ﬁnite
dimensional Euclidean space and let f : A → R be a continuous function.
Then there exists an optimal solution to the optimization problem
minimize
subject to f (x )
x ∈ A. There exists no optimal that attains it 13 Game Theory: Lecture 5 Existence Results Kakutani’s Fixed Point Theorem Theorem
(Kakutani) Let A be a nonempty subset of a ﬁnite dimensional
Euclidean space. Let f : A � A be a correspondence, with
x ∈ A �→ f (x ) ⊆ A, satisfying the following conditions:
A is a compact and convex set. f (x ) is nonempty for all x ∈ A. f (x ) is a convexvalued correspondence: for all x ∈ A, f (x ) is a convex set.
f (x ) has a closed graph: that is, if {x n , y n } → {x , y } with
y n ∈ f (x n ), then y ∈ f (x ).
Then, f has a ﬁxed point, that is, there exists some x ∈ A, such that
x ∈ f (x ). 14 Game Theory: Lecture 5 Existence Results Kakutani’s Fixed Point Theorem—Graphical Illustration is not convexvalued does not have a
closed graph 15 Game Theory: Lecture 5 Existence Results Proof of Nash’s Theorem
The idea is to apply Kakutani’s theorem to the best response
correspondence B : Σ � Σ. We show that B (σ) satisﬁes the
conditions of Kakutani’s theorem.
1 Σ is compact, convex, and nonempty.
By deﬁnition
Σ= ∏ Σi i ∈I where each Σi = {x  ∑j xj = 1} is a simplex of dimension Si  − 1,
thus each Σi is closed and bounded, and thus compact. Their product
set is also compact.
2 B (σ) is nonempty.
By deﬁnition, Bi (σ−i ) = arg max ui (x , σ−i ) x ∈ Σi where Σi is nonempty and compact, and ui is linear in x . Hence, ui is
continuous, and by Weirstrass’s theorem B (σ ) is nonempty.
16 Game Theory: Lecture 5 Existence Results Proof (continued)
3. B (σ) is a convexvalued correspondence.
Equivalently, B (σ )...
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 Spring '10
 AsuOzdaglar

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