On the duration of the problem of the points

We consider an r-player version of the famous problem of the points, which was the stimulus for the correspondence between Pascal and Fermat in the 17th century. At each play of a game, exactly one of the players wins a point, player i winning with probability p_i. The game ends the first time a player has accumulated his or her required number of points—this requirement being n_ifor player i. A reliability application would be to suppose that a system is subject to r different types of shocks and failure occurs the first time there have been n_itype i shocks for any i = 1, …, r. Our main result is to show that N, the total number of plays, is an increasing failure-rate random variable. In addition, we prove some Schur convexity results regarding P{N ≤ k) as a function of p (for n_i≡ n) and as a function of n (for p_i≡ 1/r).