The exchange paradox, also known as the two envelopes problem, is a puzzle in logic and probability which originated as a variation of the necktie paradox (Hausmann, 2001). The exchange paradox is presented by formulation of a hypothetical challenge, which presents two contradicting situations taking the following form:
Two identical sealed envelopes contain money, where the sum in one envelop is twice the sum in the other envelop. They are presented to two players for picking. Whatever envelop a player chooses, he owns the amount of money in it. Before looking into the amount of money in the envelop a player chooses, he is offered an option of swapping the envelopes.
The first player may reason as follows: If his envelope has $2x and with a 50% probability of choosing either envelope, the other envelope contains either ½($2x) or ½($4x). The expected value of the amount of money in the other envelope is ½($2x) + ½($4x), resulting to $3x. This amount is greater than his current envelope’s $2x. In such a scenario, he decides to swap the envelopes. The second player reasons identically and the swapping keeps on.
Some authors argue that the flawed calculation of the expected value is caused by the insufficient information about the amounts of money in the envelops. Before the players do the calculation, information on prior distribution of money in the envelope is necessary. For example, if a player knew that the amounts in the envelopes were $2 and $4, and saw the $2 envelope, he would know the other envelope had $4. He will then swop. This case does not present a paradox.
In as much as many authors have presented solutions to this paradox, none has been widely accepted to be definitive (Epstein, 2012).
Epstein, R. A. (2012). The Theory of Gambling and Statistical Logic. San Diego: Elsevier Science.
Hausmann, W. (2001). On the two envelope paradox. Friedberg: Fachhochschule Gießen- Friedberg.