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Dene the set of best response correspondences as b s

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Unformatted text preview: strategy existence in infinite games. Theorem (Debreu, Glicksberg, Fan) Consider a strategic form game �I , (Si )i ∈I , (ui )i ∈I � such that for each i ∈ I 1 Si is compact and convex; 2 ui (si , s−i ) is continuous in s−i ; 3 ui (si , s−i ) is continuous and concave in si [in fact quasi-concavity suffices]. Then a pure strategy Nash equilibrium exists. 20 Game Theory: Lecture 5 Existence Results Definitions Suppose S is a convex set. Then a function f : S → R is concave if for any x , y ∈ S and any λ ∈ [0, 1], we have f ( λx + (1 − λ )y ) ≥ λf (x ) + (1 − λ )f (y ) . concave function not a concave function 21 Game Theory: Lecture 5 Existence Results Proof Now define the best response correspondence for player i , Bi : S−i � Si , � � Bi (s−i ) = si� ∈ Si | ui (si� , s−i ) ≥ ui (si , s−i ) for all si ∈ Si . Thus restriction to pure strategies. Define the set of best response correspondences as B (s ) = [Bi (s−i )]i ∈I . and B : S � S. 22 Game Theory: Lecture 5 Existence Results Proof (continued) We will again apply Kakutani’s theorem to the best response correspondence B : S � S by showing that B (s ) satisfies the conditions of Kakutani’s theorem. 1 S is compact, convex, and non-empty. By definition S= ∏ Si i ∈I since each Si is compact [convex, nonempty] and finite product of compact [convex, nonempty] sets is compact [convex, nonempty]. 2 B (s ) is non-empty. By definition, Bi (s−i ) = arg max ui (s , s−i ) s ∈ Si where Si is non-empty and compact, and ui is continuous in s by assumption. Then by Weirstrass’s theorem B (s ) is non-empty. 23 Game Theory: Lecture 5 Existence Results Proof (continued) 3. B (s ) is a convex-valued correspondence. This follows from the fact that ui (si , s−i ) is concave [or quasi-concave] in si . Suppose not, then there exists some i and some s−i ∈ S−i such that Bi (s−i ) ∈ arg maxs ∈Si ui (s , s−i ) is not convex. This implies that...
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