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Unformatted text preview: 2.142 Analogous to Exercise 2.129, there exists a simplex containing and a retrac- tion of onto , that is a continuous function : with ( x ) = x for every x . Then : is closed-valued (Exercise 2.106) and uhc (Exercise 2.103). By the argument in the proof, there exists a point x such that x ( x ). However, since ( x ) , we must have x and therefore ( x ) = x . This implies x ( x ). That is, x is a fixed point of . 2.143 = 1 2 ... is the Cartesian product of uhc, compact- and convex- valued correspondences. Therefore is also compact-valued and uhc (Exercise 2.112 and also convex-valued (Exercise 1.165). By Exercise 2.106, is closed. 2.144 Strict quasiconcavity ensures that the best response correspondence is in fact a function :...
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- Fall '10