Two Envelopes Problem

The two envelopes problem, also known as the exchange paradox, is a brain teaser, puzzle or paradox in logic, philosophy, probability, and recreational mathematics, of special interest in decision theory and for the Bayesian interpretation of probability theory. Historically, it arose as a variant of the necktie paradox.

A statement of the problem starts with:

Let us say you are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are offered the possibility to take the other envelope instead.

It is possible to give arguments that show that it will be to your advantage to swap envelopes by showing that your expected return on swapping exceeds the sum in your envelope. This leads to the logical absurdity that it is beneficial to continue to swap envelopes indefinitely.

A large number of different solutions have been proposed. The usual scenario is that one writer proposes a solution that solves the problem as stated, but then some other writer discovers that by altering the problem a little the paradox is brought back to life again. In this way a family of closely related formulations of the problem is created which are then discussed in the literature.

There is not yet any one proposed solution that is widely accepted as being the correct one. Despite this it is common for authors to claim that the solution to the problem is easy, even elementary. However, when investigating these elementary solutions they often differ from one author to the next. During the last two decades several new papers have been published every year.