The Devil’s Game

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Ms. C dies and goes to hell, where the devil offers a game of chance. If she plays today, she has a 1/2 chance of winning; if she plays tomorrow, the chance will be 2/3; and so on. If she wins, she can go to heaven, but if she loses she must stay in hell forever. When should she play?

The answer is not clear. If she waits a full year, her probability of winning will have risen to about 0.997268. At that point, waiting an additional day will improve her chances by only about 0.000007. But at stake is infinite joy, and 0.000007 multiplied by infinity is infinite. And the additional day spent waiting will contain (presumably) only a finite amount of torment. So it seems that the expected benefit from a further delay will always outweigh the cost.

“This logic might suggest that Ms. C should wait forever, but clearly such a strategy would be self-defeating,” wrote Edward J. Gracely in proposing this conundrum in Analysis in 1988. “Why should she stay forever in a place in order to increase her chances of leaving it? So the question remains: what should Ms. C do?”