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MIT14_15JF09_lec11

# MIT14_15JF09_lec11 - 6.207/14.15 Networks Lecture 11...

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6.207/14.15: Networks Lecture 11: Introduction to Game Theory—3 Daron Acemoglu and Asu Ozdaglar MIT October 19, 2009 1

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Networks: Lecture 11 Introduction Outline Existence of Nash Equilibrium in Infinite Games Extensive Form and Dynamic Games Subgame Perfect Nash Equilibrium Applications Reading: Osborne, Chapters 5-6. 2
Networks: Lecture 11 Nash Equilibrium Existence of Equilibria for Infinite Games A similar theorem to Nash’s existence theorem applies for pure strategy existence in infinite games. Theorem (Debreu, Glicksberg, Fan) Consider an infinite strategic form game �I , ( S i ) i ∈I , ( u i ) i ∈I such that for each i ∈ I S i is compact and convex; u i ( s i , s i ) is continuous in s i ; u i ( s i , s i ) is continuous and concave in s i [in fact quasi-concavity suﬃces]. Then a pure strategy Nash equilibrium exists. 3

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concave function not a concave function Networks: Lecture 11 Nash Equilibrium 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 ( λ x + (1 λ ) y ) λ f ( x ) + (1 λ ) f ( y ) . 4
Networks: Lecture 11 Nash Equilibrium Proof Now define the best response correspondence for player i , B i : S i S i , B i ( s i ) = s i S i | u i ( s i , s i ) u i ( s i , s i ) for all s i S i . Thus restriction to pure strategies. Define the set of best response correspondences as B ( s ) = [ B i ( s i )] i ∈I . and B : S S . 5

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Networks: Lecture 11 Nash Equilibrium 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. S is compact, convex, and non-empty. By definition S = S i i ∈I since each S i is compact [convex, nonempty] and finite product of compact [convex, nonempty] sets is compact [convex, nonempty]. B ( s ) is non-empty. By definition, B i ( s i ) = arg max u i ( s , s i ) s S i where S i is non-empty and compact, and u i is continuous in s by assumption. Then by Weirstrass’s theorem B ( s ) is non-empty. 6
Networks: Lecture 11 Nash Equilibrium Proof (continued) 3. B ( s ) is a convex-valued correspondence. This follows from the fact that u i ( s i , s i ) is concave [or quasi-concave] in s i . Suppose not, then there exists some i and some s i S i such that B i ( s i ) arg max s S i u i ( s , s i ) is not convex. This implies that there exists s i , s i �� S i such that s i , s i �� B i ( s i ) and λ s i + (1 λ ) s i �� / B i ( s i ). In other words, λ u i ( s i , s i ) + (1 λ ) u i ( s i �� , s i ) > u i ( λ s i + (1 λ ) s i �� , s i ) . But this violates the concavity of u i ( s i , s i ) in s i [recall that for a concave function f ( λ x + (1 λ ) y ) λ f ( x ) + (1 λ ) f ( y )].

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