“John complains”:http://www.crookedtimber.org/archives/001872.html that the version of the two-envelope paradox I give is not theologically accurate. I was trying to come up with a more theologically accurate one, but I couldn’t really. Still, the following is intended to be a little closer to theological reality.

BRIAN: Where am I?
ANGEL: Purgatory.
BRIAN: Ah, that makes sense. Hang on, does that mean I get to go to heaven one day?
ANGEL: Yep, eventually.
BRIAN: WooHoo! So how long’s the wait then? And can you turn up the heat a little, it’s kinda chilly here?
ANGEL: Well, that’s a hard question. No.
BRIAN: Why is it hard? Hasn’t The Big Guy worked out how long my penance will be?
ANGEL: Well, it turns out decisiveness isn’t one of the divine attributes. So He hasn’t yet decided.
BRIAN: That makes sense. If Bushie is our paradigm of decisiveness, it isn’t obviously a virtue.
ANGEL: Right. So he’s going to toss a coin a few times to figure out how long you stay here. If it lands heads the first time, you stay here 2 days, and we’re done. If it lands heads the second time, you stay here 4 days, and again we’re done. If it lands heads the third time, you stay here 8 days, and again we’re done. You get the picture.
BRIAN: And if it never lands heads?
ANGEL: Then you stay here a forever.
BRIAN: Hmmm, I didn’t realise I’d been that bad.
ANGEL: It was a judgment call whether you went up or down, so don’t complain too much.
BRIAN: OK then. How long will this coin tossing take?
ANGEL: It’s already done.
BRIAN: So it landed heads at least once then!
ANGEL: I wouldn’t infer that too quickly. The first toss takes a {1/2} second, the second toss a {1/4} second, the third 1/8 of a second etc, so infinitely many tosses don’t take that long.
BRIAN: I don’t like the sound of this, but what’s the verdict.
ANGEL: It’s in this envelope. Go on, open it up … WAIT! I almost forgot. Before you do that, I’m meant to offer you a deal. We can tear up that envelope, and we’ll rerun the coin flips, and this time I’ll take a day off whatever the sentence is.
BRIAN: So it would be 1 day here, or 3 days, or 7 days, or 15 days, etc.
ANGEL: You got it.
BRIAN: That sounds like a great deal. Is there a catch?
ST PETER: Funny you should ask. If you open that envelope and see it says 2 days, would you prefer to keep the envelope or take the deal?
BRIAN: Since the deal has an expected infinite sentence, I guess I’d keep the envelope.
ST PETER: And what if it says 4 days?
BRIAN: I guess, well, I guess I’d keep the envelope again.
ST PETER: And 8 days?
BRIAN: Keep the envelope.
ST PETER: See a pattern here?
BRIAN: Yeah, but the envelope was constructed by the same mechanism as the angel is using, but without the discount. How can it be better to keep the envelope?

Some quick commentary on the case.

I’ve deliberately left out of St Peter’s argument what happens if I open the first envelope and see an infinite stay in purgatory. That’s a messy case, but there’s a few things to note.

First, it isn’t obvious I should prefer the deal rather than be indifferent between the deal and keeping the envelope even in this case.

Second, it has probability zero.

Third, we can avoid even this case if we are prepared to allow something like a discontinuity. Change the case so ‘tails forever’ has the same effect as heads on the first flip, i.e. 2 days in purgatory on the first run, 1 day in purgatory on the second run. Now we have a strict conglomerability argument for keeping the original envelope. I don’t understand at all the concern about discontinuous sequences, but I didn’t include this in the original case because of those concerns.

There’s no ‘infinite swapping’ outcome here like in the original two envelope case. But I think it’s very odd that keeping the envelope, which from a neutral perspective appears to be dominated by taking the deal, can be argued against by just the same kind of reasoning in the two-envelope case.

Moreover, if we change the case so the angel doesn’t go on to do the flipping, but has a second envelope from God, we can then give a parallel argument that we really should take the deal.

Nothing in this case relies on there being an equal distribution over [10, M), which doesn’t make any sense to me. But that was never essential to the two envelope case, as John Broome pointed out in his 1995 _Analysis_ paper. Really the two envelope paradox is just a variant on the St Petersburg paradox, as illustrated here. And that paradox works with countably additive probability distributions.