Eric Schwitzgebel and Josh Dever have a paper on the two-envelope paradox arguing that the paradox arises because of faulty reasoning involving variables. They note that if we impose a constraint on which variables can be used in decision-theoretic reasoning, the paradoxical reasoning is blocked. I won’t repeat the formal version of the constraint (from page 4 of the paper) in HTML. But the effect is that X is only a legitimate variable if “the expected value of X is the same conditional on each event in the partition.” The problem is then that the paradoxical reasoning essentially involves appeal to a variable that does not satisfy this constraint.
As an aside, this kind of response is not entirely uncommon in discussions of the two-envelope literature, so it’s worth taking seriously. And Schwitzgebel and Dever’s version of the response is by far the most careful and plausible I have seen. (And it’s probably the earliest such version, given their note in the paper that they discussed this with people in Berkeley in 1993. Given the history of the two-envelope discussion, where so much happens online etc, this kind of fact seems quite relevant to priority, if priority matters at all here.) But it still seems flawed.
Here’s the reason. It’s true that their constraint blocks the paradoxical reasoning. But getting a constraint with that property is dead easy. Just say that any decision-theoretic reasoning is invalid and you’ll do that. The hard part is finding a constraint that knocks out the two-envelope reasoning, but not any reasoning that we want, both intuitively and on reflection, to preserve. And I think Schwitzgebel and Dever’s constraint fails that test.
Consider the following example. God partitions the reals in [0, 1] into two unmeasurable sets, S1 and S2. He picks a real at random from [0, 1]. If it’s in S1, He puts $10 into a red envelope, if it’s in S2 He puts $20 into that red envelope. He then rolls two fair and independent dice. If they land double-six, he puts an amount into a blue envelope equal to the amount in the red envelope plus $5. Otherwise, he puts an amount into that blue envelope equal to $5 less than the amount in the red envelope. Got it? (It’s easier with tables, but tables are hard in blogs.)
You are not told which number He picked, or how the dice landed, but you are told all of the above. You are then given a choice of the red or blue envelopes. How should you choose?
I take it that it’s obvious you should pick the red envelope. After all, whatever is in it, you have a 35/36 chance of getting $5 less with blue, and only a 1/36 chance of getting more. So I say, pick red.
But Schwitzgebel and Dever can’t say that. For they say the above reasoning violates their constraints on which variables can be used. (Or, more precisely, that any formalised version of the above reasoning would do so.) As near as I can tell, the reasoning I just made is just as bad, by their lights, as the paradoxical two-envelope reasoning.
As I see it, they are now under an obligation. For it seems obvious that red is better than blue, so they should tell us what principle they do endorse that gets that conclusion. It can’t just be the principle Always maximise expected utility, since in this case neither picking red nor picking blue has a defined expected utility. And, although this might just be a failure of imagination on my part, I can’t see what else it might be.
While I’m in this combative mood, I should also note that this example casts some doubt on any attempt to resolve the two-envelope paradox by appeal to expected utility reasoning. For the two-envelope paradox rests on principles that are plausible in cases like this one, even when expected utility reasoning fails. I’ll be polite/lazy enough to not quote anyone who actually does try and solve the problem that way.