By Jean-Pierre Aubin (Eds.)
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Additional info for Mathematical Methods of Game and Economic Theory
Taking y = 0, we deduce that 0 < E 6 (p, x). On the other hand, supycp(p, y ) = = 0 since it is bounded above (see Proposition 4). This is the required contradiction. 0 As a consequence, we have the following result. Proposition 8. Let L belong to 2(U,V) and let X be a convex subset of U containing 0. Then (28) L(X)* = L*-1(X#). Furthermore, ifL ( X) is a closed convex subset containing 0, we obtain that (29) 5 L(X)= [(L'-'(X+)]*. 34 MINIMIZATION PROBLEMS AND CONVEXITY [Ch. 4 Proof. e. that L'p belongs to X#, or that p belongs to L'-l(X*).
E. Lx = y). The inf-convolution can be regarded in the following way. e. choose strategies xi such that E=lx’ = y. The inf-convolution Uy=lh associateswith any y the smallest total loss occurred during the sharing. 3. e. and (Q): inf (LS(y)+gW) Y€Y (RCV)) : inf f ( x ) Lx-y for a convenient y. 14 MINIMIZATION PROBLEMS AND CONVEXITY [Ch. 2 holds and the two following statements are equivalent: (1 1) 1 (a) 2 minimizes f ( x ) + g ( L x ) on the set X = L-I(Y) (b) j = LX minimizes Lf ( y ) + g ( y ) on Y and X minimizes f ( x ) on the set L-l(jj).
E. (20) B(J4X)) = (X). Ch. 31 25 MIXED STRATEGIES AND CONVEXITY Thus (21) Xis “convex” if and only i f B maps M ( X )into X . f(Bm) = f ”(4. A function f : U (24) (m,f) = f’(m) - 1- 00, + -1 is “convex” i f and only if V m = ZAiS(xi) E M(U), f (pm) =sf ”(m)= A function f : U --c [- 00, + -[ - n C A’f(xi). i=1 is “concave” i f -f is convex. It is clear that the domain Dom f of a convex function is convex. Also, if X c U is convex and f :X R is convex, its extension fx : U -,1- -, + -1 is convex.