So, my ONLY guess as to how there can be a strategy is that, if the number you see is positive, you keep it, and if the number you see is negative, you switch.

So, my ONLY guess as to how there can be a strategy is that, if the number you see is positive, you keep it, and if the number you see is negative, you switch. My reasoning is this: 1. If the number you pick is positive, then there are technically (although arbitrarily small) more negative numbers that can be chosen for the other envelope. So you have a slightly larger than 50% chance of winning. 2. The inverse is true if you chose negative, however. That said - I think this would generally imply you should always switch, which is why I'm not convinced I'm right. Consider a purely positive set, for example. If my number shows up as "12" - then I know there are infinitely more numbers larger than it, but only 11 numbers lower than it. But this starts to feel like the original two envelopes paradox. -- tldr; I'm not sure of the answer, although I think if I saw the answer I would have an "Aha!" moment.