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,...
View
Full Document
 Spring '10
 VINH
 Equilibrium, Game Theory, Nash

Click to edit the document details