A Stronger Two-Envelope Paradox

24 05 2009

Consider the standard two-envelope paradox – there are two envelopes in front of you, and all you know is that one of them has twice as much money as the other. It seems that you should be indifferent to which envelope you choose to take. However, once you’ve taken an envelope and opened it, you’ll see some amount x of money there, and you can reason that the other envelope either has 2x or x/2, which has an expected value of 1.5x 1.25x, so you’ll wish you had taken the other.

Of course, this reasoning doesn’t really work, because of course you always know more than just that one envelope has twice as much money as the other, especially if you know who’s giving you the money. (If the amount I see is large enough, I’ll be fairly certain that this one is the larger of the two envelopes, and therefore be happy that I chose it rather than the other.)

But it’s also well-known that the paradox can be fixed up and made more paradoxical again by imposing a probability distribution on the amounts of money in the two envelopes. Let’s say that one envelope has 2^n dollars in it, and the other has 2^{n+1} dollars, where n is chosen by counting the number of tries it takes a biased coin to come up heads, where the probability of tails on any given flip is 2/3. Now, if you see that your envelope has 2 dollars, then you know the other one has 4 dollars, so you’d rather switch than stay. And if you see that your envelope has any amount other than 2, then (after working out the math) the expected value of the other one will be (I believe) 5/4 of the amount in your current envelope, so again you’d rather switch than stay.

This is all fair enough – Dave Chalmers pointed out in his “The St. Petersburg Two-Envelope Problem” that there are other cases where A can be preferred to B conditional on the value of B, while B can be preferred to A conditional on the value of A, while unconditionally, one should be indifferent between A and B. This just means that we shouldn’t accept Savage’s “sure-thing principle”, which says that if there is a partition of possibilities, and you prefer A to B conditional on any element of the partition, then you should prefer A to B unconditionally. Of course, restricted versions of this principle hold, either when the partition is finite, or the payouts of the two actions are bounded, or one of the unconditional expected values is finite, or when the partition is fine enough that there is no uncertainty conditional on the partition (that is, when you’re talking about strict dominance rather than the sure-thing principle).

What I just noticed is that it’s trivial to come up with an example where we have the same pattern of conditional preferences, but there should be a strict unconditional preference for A over B. To see this, just consider this same example, where you know that the two envelopes are filled with the same pattern as above, but that 5% of the money has been taken out of envelope B. It seems clear that unconditionally one should prefer A to B, since it has the same probabilities of the same pre-tax amounts, and no tax. But once you know how much is in A, you should prefer B, because the 5% loss is smaller than the 25% difference in expected values. And of course, the previous reasoning shows why, once you know how much is in B, you should prefer A.

Violations of the sure-thing principle definitely feel weird, but I guess we just have to live with them if we’re going to allow decision theory to deal with infinitary cases.