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 seventeenth century. At each play of a game, exactly one of the players wins a point - player i winning with probability p sub i. The game ends the first time a player has accumulated his required number of points - this requirement being n sub i for player i. 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 Pn or k as a function of p for n sub i n and as a function of n for p sub i 1r. Author