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Unformatted text preview: ORIE 435 2006 Nashs Existence of Equilibrium Theorem These notes sketch a proof of Nashs Theorem that shows that an equi librium must exist (in mixed strategies) for any nperson game. The proof uses Brouwers fixedpoint theorem, which is as follows: let X R I m be a closed, bounded, nonempty, convex set and let f : X X be a continuous function; then there exists some x X such that f ( x ) = x . Nashs Existence of Equilibrium Theorem. For any nperson game in which each player has a finite number of pure strategies, there exists a Nash equilibrium point in mixed strategies. Proof . Let P i be the set of all mixed strategies for player i , i = 1 , . . . , n ; that is, if player i has m i pure strategies, then P i = { p i = ( p i 1 , . . . , p i m i ) T : m i X s =1 p i s = 1 , p i s , s = 1 , . . . , m i } . We set X = P 1 P n ; that is, each point in X corresponds to a selection of a mixed strategy for each player. It is easy to check that X is closed, bounded, nonempty, and convex. Next we define the function f . First, we let E i ( p 1 , . . . , p n ) (the same as i ( p 1 , . . . , p n )) denote the payoff to player i when each player...
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