The Two‐Envelope Paradox, Nonstandard Expected Utility, and the Intensionality of Probability

You are given a choice between two envelopes. You are told, reliably, that each envelope has some money in it-some whole number of dollars, say-and that one envelope contains twice as much money as the other. You don't know which has the higher amount and which has the lower. You choose one, but are given the opportunity to switch to the other. Here is an argument that it is rationally preferable to switch: Let x be the quantity of money in your chosen envelope. Then the quantity in the other is either 1/2x or 2x, and these possibilities are equally likely. So the expected utility of switching is 1/2(1/2x) + 1/2(2x) = 1.25x, whereas that for sticking is only x. So it is rationally preferable to switch. There is clearly something wrong with this argument. For one thing, it is obvious that neither choice is rationally preferable to the other: it's a tossup. For another, if you switched on the basis of this reasoning, then the same argument could immediately be given for switching back; and so on, indefinitely. For another, there is a parallel argument for the rational preferability of sticking, in terms of the quantity y in the other envelope. But the problem is to provide an adequate account of how the argument goes wrong. This is the two-envelope paradox. Although there is fairly extensive recent literature on this problem, none of it seems to me to get to the real heart of the matter.' In my view, the flaw in the paradoxical argument is considerably harder to diagnose than is usually believed, and an adequate diagnosis reveals important morals about both the nature of probability and the foundations of decision theory. I will offer my own account, in such a way that the morals of the paradox will unfold first and then will generate the diagnosis of how it goes wrong. Thereafter I will briefly pursue some theoretical issues for decision theory that arise in light of the paradox's lessons.