Games of chance with multiple objectives

We consider games of chance between two players: Player M can win only by amassing point totals in several categories before player N scores a prescribed total number n of points. Let M have k objectives, with m(i) points required in category i and probability q(i) of scoring a point in that category. We resolve certain special cases: (a) For all m(i) equal, the probabilities of M winning are ordered by majorization of the vectors (q(1),...,q(k) ). (b) For all q (i) equal, the probabilities of M winning are ordered by majorization of the vectors (m(1),...,m(k) ). (c) For all m(i) equal and all q(i) equal, the probability of M winning approaches 0 as n -> infinity or as k -> infinity . The results, which follow from inequalities of majorization and Schur convexity, are in accord with intuition.