Via Juan at Philosophy617 (who doesn’t think much of the proffered solutions, and probably won’t like this one) I came back to this version of the two-envelope problem put forward by Brian over at Crooked Timber last year.
In this case, once you observe that Brian’s angel is giving you faulty theology, it’s easy to show that you should reject his mathematics, and his offer. At the end of the problem, the angel says “It’s purgatory,” says the angel, “take all the time you want.” But the whole point of Purgatory is that it’s finite – you purge off your sins one at a time until they’re all paid off. Since we now have a finite problem, the solution is straightforward.
Recall that there are two envelopes, with numbers x and 2x representing remission of time in Purgatory, and that x is greater than 10. If your total time in Purgatory is M, we can assume that a just God is not going to give you more remission than that, so 2x is less than M, and x is less than M/2.
The trick in the problem is the apparent symmetry between the envelopes. If you pick one envelope, getting y, switching envelopes gives you y/2 or 2y with equal probability, which seems like a good bet. So it looks as though the angel can apply a Hell pump to you, with repeated offers to switch, paying a day in Hell each time.
The trick in the angel’s offer in the is that it’s not true, for any given y, that switching gives you even chances of y/2 or 2y. Suppose for example, you draw y greater than M/2. Then it’s certain that you’ve got the 2x envelope and that switching would be bad. Conversely, if you draw, say, 15 days, it’s obvious that you’ve got the x envelope and that switching would be good. Unfortunately, you can’t peek then decide whether to switch. If you could, the angel’s offer would probably be a good one. Since you can’t, and given any fixed distribution for x over the range [10,M/2], it’s easy to check that the expected gain from switching is zero.
It’s easy to extend the argument to allow for the case of a Bayesian soul, with a prior distribution that will be updated once the envelope is opened (of course, it’s too late to anything by then). You can also allow for some kinds of non-Bayesians but not too many, since the angel’s argument implicitly relies on the sure thing principle.
It’s also possible, in at least some cases, to refute the angel’s argument even when time in Purgatory may be infinite. All that’s really needed is a given probability distribution for remission time x with a finite mean.
fn1. I didn’t think angels were gendered, but the example uses male pronouns, and I’ll follow suit.