Playing Games with Eternity


A legacy post originally published on OCTOBER 11, 2009 at 11:50 AM
🔗 Playing Games with Eternity by Mike Almeida

Ed Gracely in Analysis (1988) presented an instructive problem about wagering on heaven and hell. It tells us something interesting about Pascal’s Wager. Suppose you find yourself in hell and Satan himself has pity on you. He offers you this game of chance. If you win, you go to heaven for all eternity. If you lose, you remain in hell for all eternity. Your chances today of winning are 1/2. If you wait one day, your chances of winning go to 2/3. If you wait yet one more day, the chances of winning rise to 3/4, then to 4/5, 5/6…, and so on, infinitely. If you wait a year to play, your chances go to about 99.7. When should you play?

Gracely stipulates that each extra day in hell is only finitely bad. He does not add (but we should) that eternity in hell is infinitely bad. Eternity in heaven is, of course, infinitely good. The bizarre feature of this game is that it looks like you should play as soon as possible (NB: not Gracely’s conclusion). Suppose addition and subtraction are well-behaved for infinities. If I play today, my expected payoff is +∞ + -∞ = 0. I have some small chance of landing in heaven, and some small chance of landing in hell. If I play tomorrow, my expected payoff is [+∞ + -∞] + -D = -D, where -D is the finite cost of waiting in hell one more day. My expected payoff is higher if I play today!

But that can’t be right. And I think this shows part of what is wrong with Pascalian reasoning. The Wager does not give us reason to believe today that God exists. The fact is that we face a series of Pascalian wagers, not just one. Day after day, we have an opportunity to believe that God exists. Assume, as Pascal does, that believing has an infinite positive expected payoff. Each day that you do not believe, you are free to enjoy worldly pleasures. So long as there is some small chance that you live until tomorrow, you should postpone believing until then. Your expected payoff is the same—it is infinitely high—and you have the additional finite payoff of one more day’s worth of worldy pleasures! But then there is no day on which you should (rationally) believe, since there is always some small chance that you live until tomorrow (or till the next minute, for that matter). That can’t be right, either.

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