[AWoodside]

This problem is troubling me as well. It seems like it must be due to some sloppy definitional issue but I can't put my finger on what it is.

To try to get at the root of it consider a similar situation, but where there is no hint of paradox. I have two envelopes and tell you that envelope A has $10, and envelope B has a 50/50 chance of being either $5 or $20 dollars. In this case you would obviously just pick envelope B because it has higher expected value. In a sense this is exactly what you're doing, albeit more abstractly, to get to the paradox in the case outlined in the OP. You're saying "let my envelope have $x, so the other one has a 50/50 chance of being either .5x or 2x." It seems like you're invalidly breaking some type of symmetry here. The situation is that the envelopes are distributed either as [(x, 2x) or (2x, x)]. By using the reasoning above you're changing the distribution to [x, (0.5x) or (2x)], which I'm pretty sure is not equivalent to the original situation. Given that you've recieved no additional useful information by picking an envelope, or looking even looking inside of one, it seems like you can't do this and still have to speak only in terms of expected value and using the (x, 2x) or (2x, x) formulation. And, as somebody has mentioned, there is no paradox in this case as long as you calculate the EV correctly (by recognizing the fact that you always get at least the low value). Because now the EV of switching is

EV(switch) = x + 0.5(x) + 0.5(0) = 1.5x

which is the same expected value you had when initially picking the envelope (and still have currently).

Thinking about it this way makes it a little more clear to me, but I'm not anywhere near to an answer I find rigorous/satisfactory. I do follow and agree with the explanations that mention the issue of not being able to come up with an equal-probability distribution over an infinite set and hence in some since the conditional probability questions you're asking aren't meaningful... but my gut tells me there must be a simpler explanation. Of course, my gut's definitely been wrong many times before.