A Turn-Based Game Related to the Last-Success-Problem

Abstract

There are n independent Bernoulli random variables with parameters $p_i$ that are observed sequentially. We consider the following sequential two-person zero-sum game. Two players, A and B, act in turns starting with player A. The game has n stages, at stage k, if $I_k = 1$, then the player having the turn can choose either to keep the turn or to pass it to the other player. If the $I_k = 0$, then the player with the turn is forced to keep it. The aim of the game is not to have the turn after the last stage: that is, the player having the turn at stage n wins if $I_n=1$ and, otherwise, he loses. We determine the optimal strategy for the player whose turn it is and establish the necessary and sufficient condition for player A to have a greater probability of winning than player B. We find that, in the case of n Bernoulli random variables with parameters 1 / n, the probability of player A winning is decreasing with n toward its limit $\frac{1}{2} -\frac{1}{2\,e^2}=0.4323323\ldots$. We also study the game when the parameters are the results of uniform random variables, $\mathbf {U}[0,1]$.