I am presented with two envelopes A and B, and told that one contains twice as much money as the other. I choose envelope A, and am offered the options of either keeping it or switching to B. What should I do? I reason: (1) For any amount $x, if I knew that A contained x, then the odds are even that B contains either $2x or $x/2, hence the expected amount in B would be $5x/4. So (2) for all x, if I knew that A contained $x, I would have an expected gain in switching to B. So (3) I should switch to B. But this seems clearly wrong, as my information about A and B is symmetrical—once I have B, by the same reasoning I should switch back to A. It is known that if there is a finite upper bound on x the reasoning is faulty, since (1) and (2) will not together be true. Things become trickier if this restriction is dropped.