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Then there exists an optimal solution to the

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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 non-empty 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 non-empty for all x ∈ A. f (x ) is a convex-valued 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 convex-valued 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 non-empty. 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 non-empty. By definition, Bi (σ−i ) = arg max ui (x , σ−i ) x ∈ Σi where Σi is non-empty and compact, and ui is linear in x . Hence, ui is continuous, and by Weirstrass’s theorem B (σ ) is non-empty. 16 Game Theory: Lecture 5 Existence Results Proof (continued) 3. B (σ) is a convex-valued correspondence. Equivalently, B (σ )...
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