Sunday, November 18, 2012

Oswald Mega, the always welcome, has two envelopes and some money. He divides the money into three equal stacks.

He puts one of the stacks in one envelope, and the rest of the money into the other. He seals the envelopes, takes them to the usual watering hole, describes what he has done, swaps them around behind his back a few times, and then he hands you one of them.

The envelope contains £1000.

"Would you like to swap envelopes?" he says.

1. Does this version work? I mean, you've seen the stacks and therefore have a reasonably well- behaved prior for the amounts in the envelopes; doesn't this stop the paradox working?

2. Ah, I did not mean to imply that he divides the money in front of you! Will change. Thanks.

3. Actually, even with this version -- and indeed with any real-world version -- I think you have a distinctly reasonable prior (he's not going to have more than a trillion dollars) which is surely enough to make the paradox go away.

1. So uniform prior over [0,trillion] and you should always swap?

2. No. If you have a uniform prior over [0,10^12] for the total number of dollars then, e.g., you should not swap if you get an envelope with more than 10^12/3 dollars in it, because then the other one can't be bigger.

(And of course you should always swap if you get an envelope with an odd number of pence in it :-). But we should really pretend that money is infinitely divisible for this paradox.)

3. Sorry, I shouldn't have said always. I mean what would you actually do in the above situation when he's given you £1000? I'm guessing that the answer is 'I would swap', and that still seems paradoxical to me, in a more fierce way than the Monty Hall problem.

I love the bit about the odd number of pence. I'd never thought of that!

4. At £1000, with a uniform prior on [0,10^12] for total number of dollars? Yes, I absolutely would swap then.

Same uniform prior but before I've seen the contents? I would be indifferent between swapping and not swapping: £1000-like cases where swapping is a win are (in expectation) balanced out by £5.10^11-like cases where swapping is a loss.

5. >At £1000, with a uniform prior on [0,10^12] for total number of dollars? Yes, I absolutely would swap then.

Me too! But remember that you pulled that prior out of thin air. What would you actually do if the only information you had was as above?

6. I'd pull a prior out of thin air, of course. Well, not exactly out of thin air; I have some idea about how wealth is distributed, and about psychology, and so forth.

To be honest, I probably wouldn't actually explicitly decide on a prior. I'd wait until I'd opened the envelope and then make a guess at whether it was more likely 1/3 or 2/3 of the sum.

I think that to make this work as a paradox, you need the amount of money to be unbounded, and probably also need its expectation to diverge. (Perhaps you actually need *your expected utility from getting the money* to diverge; I haven't thought it through.)

4. Incidentally, I know why you called him Oswald Mega, but "the always welcome" sounds like it's also a joke and I'm failing to get that one.

1. No good joke, just sounded like an appropriate epithet for a man who is always giving money away with a twist.

Like wily Odysseus, or Achilles, who is swift-footed even when sulking in his tent.

5. There's a psychological element to this problem - if the sum is too small or too large I would definitely swap. But if the sum was, say, enough to clear my mortgage, or repay the mafia boss who has threatened to break my legs if I don't pay back a debt I owe him, then I wouldn't.