By W W. Rouse 1850-1925 Ball

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Assume on the contrary that there is an 60 > 0 for which the assertion does not hold. Let a(l) E 17 be arbitrary. Since E,o(a(l)) does not cover 17, there exists a vector a(2) E 17 for which a(2) E E,o(a(l)). The sets E,o(a(l)) and E'o(a(2») do not cover 17 either, thus there is a a(3) E 17 such that a(3) ¢ E,o(a(l)) u E'o(a(2»). Proceeding in this fashion we get an infinite sequence a(l), a(2), ... with the property v-I a(V)E17, a(v)E U E,o(a(i)), (v=2,3, .. ). •• of {a(V)} and a vector (j(0) = (ii\O), ...

Let L j denote the set of all mixed strategies of the ith player. Lj is in fact a . Iex In . IIllNj sImp 11'\\ , ('-1 l - , ... , n). If the players are allowed to apply mixed strategies (they pick strategies randomly according to the probability distribution given by Xj), then it seems reasonable to redefine the pay-off function as the expected value of the pay-offs Kj(u\"Il , ... , u~"n)), (i = 1, ... , n). Thus by playing the mixed strategies XI' ... , X" the new pay-ofT functions are X\"tl ...

F,,)= = bl bn til an J ... ,u,,)dFt(u t ),··· dFII(u n) (k=I, ... ,n) depend linearly on the distribution functions F j if tlie remaining ones F,(l i= j, 1=1, ... , n) are held fixed. Thus weak compactness of the strategy sets Lt, ... , Ln with respect to the functions Kt, ... , Kn is only to be shown in order to be able to apply Theorem 5. Our proof consists of three steps. (a) Let {F~)}~t be a sequence of elements from Lk • We first show that there exists a subsequence {FfJ)} and a distribution function F10 ) ELk such that at every point of continuity of F~O), F(iJ)(u)~ F10)(u).