A Simple Solution to the Two Envelope Paradox

In the classical formulation of the two-envelope paradox, you are presented with two envelopes containing money and allowed to choose one to keep. After you pick one of the envelopes, but before you open it, you’re told that one of the envelopes contains twice as much money as the other, and if you wish, you can now switch envelopes. What should you do?

Intuitively, this is a toss-up: you have a 50-50 chance of getting more or less money if you switch. However, for problems of this sort (which include the Neck Tie paradox), the Expected Value (EV) is the standard approach. Formalized by Blaise Pascal, Pierre de Fermat, and Jakob Bernoulli in the 1600s, the EV calculation adds up the probability-weighted values of all the possibilities, which results in the value you should expect to see on average.

Applying the EV calculation to the two-envelope problem, you reason that you hold an envelope with an unknown value, A. The other envelope has a 0.5 probability of either being A/2 or 2A, so the EV, the sum of the value weighted probabilities, is 0.5*A/2 + 0.5*2A = 5/4*A. Standard decision theory asserts that, statistically, if you have a choice, you are better off switching to an unknown with the higher expected value. The Monty Hall Problem is a good example of where switching from your initial pick to another is the statistically correct choice. Since you are holding an envelope containing an unknown amount A, and the other envelope has been calculated to have an expected value of 5/4*A, standard decision theory suggests that you should switch.

This implies you should always switch, which in turn implies no envelope should ever be kept—a contradiction under symmetry. Numerous formal resolutions exist, typically involving improper priors, misapplied conditionals, or a contradiction of reference in the expected value formulation.

A simple resolution of the paradox: the assumption that the other envelope is equally likely to contain A/2 or 2A is structurally impossible. The envelopes contain two unknown amounts that are in a 2:1 ratio; the values proposed by the expected value calculation require a 4:1 ratio. This expected value calculation is invalid because its underlying probabilistic structure is incoherent.

The Expected Value is a foundational statistical metric, along with the median and the mode, but that doesn’t guarantee it is a meaningful measure in every context. In the case of the two envelopes, our intuition is correct, and the “expected value” is a mirage.