A wealthy eccentric places two envelopes in front of you. She tells you that both envelopes contain money, and that one contains twice as much as the other, but she does not tell you which is which. You are allowed to choose one envelope, and to keep all the money you find inside.

This may seem innocuous, but it generates an apparent paradox. Say that you choose envelope 1, and it contains \$100. In evaluating your decision, you reason that there is a 50% chance that envelope 2 contains \$200, and a 50% chance that it contains \$50. In retrospect, you reason, you should have taken envelope 2, as its expected value is \$125. If your sponsor offered you the chance to change your decision now, it seems that you should do so. Now, this reasoning is independent of the actual amount in envelope 1, and in fact can be carried out in advance of opening the envelope; it follows that whatever envelope 1 contains, it would be better to choose envelope 2. But the situation with respect to the two envelopes is symmetrical, so the same reasoning tells you that whatever envelope 2 contains, you would do better to choose envelope 1. This seems contradictory. What has gone wrong?

### My solution

There really is no paradox. The generative process where there is \$100 in the first envelope AND there is no information about the distribution of the money in the envelope does not exist, you cannot simulate it, nor make money from switching. We can look at ways to make this process sort of possible and see how the paradox changes.

a) Give up on asserting you got \$100 in first envelope: If we assume we can draw a number uniformly from 0 to infinity, and we get X dollars. then E(X) < 1.25 E(X) merely states that infinity < 1.25*infinity, which is not meaningful since the usual inequality rules is not that meaningful for infs. Here the paradox is in reasoning about inf as if it is a finite number. The paradox stems from assigning a finite value to an unbounded variable, and make it sound okay (“say it is \$100”), while it is in reality impossible to draw a sample from that distribution.

b) Give up on saying we have no information about the money in the envelope: Since it is actually impossible to draw a sample uniformly from 0 to infinity, it is more reasonable to specify a sampling process where the value of money can be generated. Say we first pick the bigger number B for 0 to M uniformly at random, then set S = B/2, and show the player S or B with 50% chance each. Now there is no paradox for a finite M. If we let M grow say M = \$100 000, and the player sees \$100 in the first envelop, then he should in fact switch to get \$125 in expectation. But he cannot always switch to make more money, since if he sees 0.6M = \$60000 in the first envelop, he should not switch. In this case, the sample tells the player something about what he should do. One can also sample from say, the exponential distribution (log uniform), and get similar effects. Again, no paradox.

So basically, the assumptions that you can both draw a sample, and that it tells you nothing about the other envelope, while reasonable sounding, does not hold for any real process. Therefore no paradox.