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, the classic argument is 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: from the information provided, the assumption that the other envelope is equally likely to contain A/2 or 2A is correct. However, what the typical construction of the classic expected value calculation for the two-envelope paradox misses is that the proposed calculation, with the values 0.5 A/2 and 2*A each having probability 0.5 is ill-formed. In the expected value calculation, your can’t incorporate non-shared values from two different realities. There are two possible sets of values, either A:2A, or A:A/2, but only one of these sets can exist at any point in time. They are mutually exclusive; an expected value calculation that uses non-shared values from each of two mutually exclusive situations is invalid.

The set of possible values a random variable can hold is called its support (the set of values it can take on with a non-zero probability), the values used in its expected value calculation must be present in the support of the random variable. The classical expected value calculation is invalid because it assumes the other envelope’s amount can be either A/2 and 2A with 0.5 probability, but the support of the random variable includes only one of these values, as the two envelopes’ amounts differ by a factor of two (e.g., B and 2B), not four.”

A correct way to determine the expected value of the paradox is to consider the actual amount of money in the envelopes, let’s call them B and 2B. When you pick an envelope, the chance of you picking the one with B is 50%, and the chance of you picking the one with 2B is also 50%. The expected value of your initial choice is:

Expected Value of initial choice = (0.5×B) + (0.5×2B) Expected Value of initial choice = 1.5B

The expected value of switching would be the same, as you’re just swapping one random variable for another. The paradox dissolves when you recognize that the expected values of both choices are equal, and therefore, there is no advantage to switching.

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 ill-posed “expected value” is a mirage.