Thus, a mixed strategy for I is a function /, defined for all finite sequences a = (ax, . . . , an) with ak e X, n = 0, 1, 2, . . . , with values in the set Pr of y-vectors p = (px, . . . , pr), pt ^ 0, S pi = 1: the ith coordinate of f(ax, .. .,an) specifies the probability of selecting i at move 4n + 1 when ax, . . ., an are the ^-points produced during the first 4n moves. A strategy g for II is similar, except that its values are in Ps. For a given pair /, g of strategies, the X-points produced are a sequence of random vectors xx, x2, . . . , such that the conditional distribution of xn+x given xx, . . ., xn is 2 fi(xx, . . ., xn) m^g^x^ . . ., xn), where i,i fit gj are the ith and jth coordinates of /, g. The problem to be considered in this paper is the following: To what extent can a given player control the limiting behavior of the random variables %n = ( % + ••• + xn)/n? For a given closed nonempty subset 5 of X, we shall denote by H(f,g) the probability that xn approaches 5 as n -> oo, i.e., the distance from the point xn to the set 5 approaches zero, where xx, x2, . . . is the sequence of random variables determined by /, g. We shall say that 5 is approachable by I with /* (II with g*) if H(f*,g) = 1 (H(f, g*) = 1) for all g(f), and shall say that S is approachable by I (II) if there is an f(g) such that S is approachable by I with / (II with g). We shall say that S is excludable by I with f if there is a closed T disjoint from 5 which is approachable by I with /. Excludability by II with g, excludability by I, and excludability by 11 are defined in the obvious way. It is clear that no S can be simultaneously approachable by I and excludable by II. The main result to be described below is that every convex 5 is
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