In the paradox, you are invited to play a game in which you may select either of two sealed envelopes. You know that one of them contains twice as much money as the other, but you don't know which is which. After choosing, you are offered the chance to swap envelopes. The paradox arises because, although it seems obvious that one has no more reason to swap envelopes than to keep the chosen one, there is a seemingly unimpeachable argument from standard decision theory calculus to the conclusion that swapping envelopes is in your interest.

We explain why previous efforts to resolve the paradox are inadequate, and argue instead that the real flaw consists in a confusion as to which amount of moneyŃ(a) the amount in the selected envelope, or (b) the total amount in the gameŃcan legitimately be taken as "fixed" in performing the calculations of expected utility. Where (a) is appropriate, then the reasoning justifying a swap is correct and unparadoxical. Where (b) is appropriate, the seemingly unimpeachable reasoning is fallacious after all, as it relies crucially on an equivocation. In particular, the variable representing the amount in the selected envelope is illicitly assigned two distinct values within the same formula.