The two envelope paradox and the foundations of rational decision theory

Terry Horgan
University of Arizona
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. In an earlier paper Horgan 2000) I offered a diagnosis of the paradox. I argued that the flaw in the argument is considerably more subtle and interesting than is usually believed, and that an adequate diagnosis reveals important morals about both probability and the foundations of decision theory. One moral is that there is a kind of expected utility, not previously noticed as far as I know, that I call nonstandard expected utility. I proposed a general normative principle governing the proper application of nonstandard expected utility in rational decisionmaking. But this principle is inadequate in several respects, some of which I acknowledged in note added in press and some of which I have meanwhile discovered..
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