Unformatted text preview: is quasiconcave on A i ; B has a closed graph since each is continuous. Thus by Kakutani's theorem B has a fixed point; as we have noted any fixed point is a Nash equilibrium of the game. Note that this result asserts that a strategic game satisfying certain conditions has at least one Nash equilibrium; as we have seen, a game can have more than one equilibrium. (Results that we do not discuss identify conditions under which a game has a unique Nash equilibrium.) Note also that Proposition 20.3 does not apply to any game in which some player has finitely many actions, since such a game violates the condition that the set of actions of every player be convex. • Exercise 20.4 ( Symmetric games ) Consider a twoperson strategic game that satisfies the conditions of Proposition 20.3. Let N = {1, 2} and assume that the game is symmetric: A 1 = A 2 and if and only if for all and . Use Kakutani's...
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 Spring '10
 VINH
 Game Theory, compact convex subset

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