Playing Games With Eternity: The Devil’s Offer

by Edward J. Gracely

Suppose Ms C dies and goes to hell, or to a place that seems like hell. The devil approaches and offers to play a game of chance. If she wins, she can go to heaven. If she loses, she will stay in hell forever; there is no second chance to play the game. If Ms C plays today, she has a 1/2 chance of winning. Tomorrow the probability will be 2/3. Then 3/4, 4/5, 5/6, etc., with no end to the series. Thus every passing day increases her chances of winning. At what point should she play the game?

The answer is not obvious; after any given number of days spent waiting, it will still be possible to improve her chances by waiting yet another day. And any increase in the probability of winning a game with infinite stakes has an infinite utility. For example, if she wait a year, her probability of winning the game would be approximately .997268; if she waits one more day, the probability would increase to .997275, a difference of only .000007. Yet, even .000007 multiplied by infinity is infinite.

On the other hand, it seems reasonable to suppose the cost of delaying for a day to be finite — a day’s more suffering in hell. So the infinite expected benefit from a delay will always exceed the cost.

This logic might suggest that Ms C should wait forever, but clearly such a strategy would be self defeating: 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?

© Edward J. Gracely June 1988