The Prisoner's Dilemma

Abstract

The following is an interesting problem in probability: The prisoner's dilemma: A prisoner was to be executed and he begged the king for mercy. The king decided to give the prisoner a chance. He gave the prisoner 50 white balls and 50 black balls, all identical in shape. The prisoner was supposed to distribute these balls into two identical bags in any way he liked and then pick one bag at random and draw one ball at random from that bag. His life would be spared if the ball drawn was a white ball. (There are several other problems under the same name. For instance, see [1].) The question now of course is to decide how the balls should be distributed so that the prisoner has the best (and the worst) chance to live. For a small number of balls, one can easily find the answer by computing the probabilities of all possible distributions. A solution to the above problemn can be found in [2]. For a large number of balls, a calculus approach is more desirable. Assume now that there are N white balls and N black balls, N> 1, to be distributed into two identical bags. Let w and b be the number of white and black balls in one of the bags, 0 < w, b < N. Consider the domains