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A_Course_in_Game_Theory_-_Martin_J._Osborne 35

A_Course_in_Game_Theory_-_Martin_J._Osborne 35 - is...

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Page 20 • Lemma 20.1 (Kakutani's fixed point theorem) Let X be a compact convex subset of and let be a set-valued function for which ·for all the set f ( x ) is nonempty and convex ·the graph of f is closed (i.e. for all sequences { x n } and { y n } such that for all , and , we have . Then there exists such that . • Exercise 20.2 Show that each of the following four conditions is necessary for Kakutani's theorem. ( i ) X is compact. ( ii ) X is convex. ( iii ) f ( x ) is convex for each . ( iv ) f has a closed graph. Define a preference relation over A to be quasi-concave on A i if for every the set is convex. • Proposition 20.3 The strategic game has a Nash equilibrium if for all ·the set A i of actions of player i is a nonempty compact convex subset of a Euclidian spaceand the preference relation is continuous ·quasi-concave on A i . Proof . Define by (where B i is the best-response function of player i , defined in (15.1)). For every the set B i ( a-i ) is nonempty since is continuous and A i is compact, and is convex since
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Unformatted text preview: is quasi-concave 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 two-person 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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