Abstract:

We observe that arguments in the literature of the two envelope problem that the problem is paradoxical are not supported by the probability distributions meant to illustrate the paradoxical nature. To correct this, we present a distribution that does support the usual arguments. Most of the interesting aspects of the two envelope problem are assumed to require distributions on an infinite space. Our next main contribution is to show that the same counterintuitive arguments can be reflected in finite versions of the problem; thus they do not inherently require reasoning about infinite values. We also show how to topologically represent the problem in general, and this contribution in particular, yielding insights for both the finite and infinite cases.

Two envelope problem setup (for those who don't know it):
You are presented with two envelopes. One contains twice as much money as the other. You open one, and it contains, e.g., one hundred dollars. You are offered the opportunity to keep the amount you see or to exchange it for the contents of the other envelope. Should you switch?