Chris’s post generated such an interesting comments thread that I feel I have to hop on the bandwagon. The following is a theologically revised version of a puzzle that’s been doing the rounds the past decade or so.
You, a rational agent, are in Purgatory, for good it seems. Things could be worse – you’ve heard horror stories about hell, but they could be better – you hear great things about heaven. These two possibilities seem to be equally valuable, in opposite directions. You’d be indifferent between your current state of affairs and a gamble with a 50% chance of a day in heaven and a 50% chance of a day in hell. (Purgatory is a lot like earth, so this kind of gambling is highly encouraged.)
One day an angel appears with a nice offer. God will give you some time in heaven for good behaviour. But He decided to play a little game to figure out how much time you’ll get. He wrote down two numbers, x and 2x, on slips of paper, and dropped them into identical envelopes. You will get one of the envelopes, it’s your choice which, and the slip will be good for the number of days in heaven that is written on it. The angel doesn’t know which envelope is which, and he doesn’t know what x is, except that it’s over 10. (They don’t send angels down for smaller missions than that.)
So you pick an envelope, and are about to see how long you’ll get in heaven when…
The angel makes you an offer. If you’ll do a day in Hell to cover for a friend of his that got caught up in a little scandal, he’ll give you the other envelope. He argues that this must be a good deal, as follows. Let y be the number written on your slip. The other envelope has either 2y or y/2 written on it. Each is equally likely, so your expected number of days in heaven if you switch envelopes is 0.5 * 2y + 0.5 * y/2 = 1.25y. So the expected gain from swapping is 1.25y – y = y/4. Since we know y > 10, y/4 > 2.5, so this is more valuable to you than a day in Hell.
The reasoning starts to sound attractive, until you worry about what other offers the angel has in mind if you accept. So you ask for some time to think about it. “It’s purgatory,” says the angel, “take all the time you want.”