Title: How to Open Two Envelopes

Paul Syverson

Naval Research Laboratory

**
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?