Games of chance with multiple objectives

Abstract

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 → ∞ or as k → ∞. The results, which follow from inequalities of majorization and Schur convexity, are in accord with intuition.