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A_Course_in_Game_Theory_-_Martin_J._Osborne 37

A_Course_in_Game_Theory_-_Martin_J._Osborne 37 - pairs of...

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Page 22 In words, a maxminimizer for player i is an action that maximizes the payoff that player i can guarantee . A maxminimizer for player 1 solves the problem max x min y u 1 ( x, y ) and a maxminimizer for player 2 solves the problem max y min x = u 2 ( x, y ). In the sequel we assume for convenience that player 1's preference relation is represented by a payoff function u 1 and, without loss of generality, that u 2 = - u 1 . The following result shows that the maxminimization of player 2's payoff is equivalent to the minmaximization of player 1's payoff. • Lemma 22.1 Let be a strictly competitive strategic game. Then . Further , solves the problem if and only if it solves the problem . Proof. For any function f we have min z (- f ( z )) = - max z f ( z ) and arg min z (- f ( z )) = arg max z f ( z ). It follows that for every we have . Hence ; in addition is a solution of the problem if and only if it is a solution of the problem . The following result gives the connection between the Nash equilibria of a strictly competitive game and the set of
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Unformatted text preview: pairs of maxminimizers. • Proposition 22.2 Let be a strictly competitive strategic game . a. If ( x * , y * ) is a Nash equilibrium of G then x * is a maxminimizer for player 1 and y * is a maxminimizer for player 2 . b. If ( x * , y * ) is a Nash equilibrium of G then max x min y u 1 ( x, y ) = min y max x u 1 ( x, y ) = u 1 ( x * , y * ), and thus all Nash equilibria of G yield the same payoffs . c. If max x min y u 1 ( x, y ) = min y max x u 1 ( x, y ) (and thus, in particular, if G has a Nash equilibrium (see part b)), x * is a maxminimizer for player 1, and y * is a maxminimizer for player 2, then ( x * , y * ) is a Nash equilibrium of G . Proof. We first prove parts (a) and (b). Let ( x * , y * ) be a Nash equilibrium of G . Then for all or, since for all . Hence . Similarly,...
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