Two Envelopes Problem - Alternative Interpretation

Alternative Interpretation

The first solution above doesn't explain what's wrong if the player is allowed to open the first envelope before being offered the option to switch. In this case, A stands for the value which is seen then, throughout all subsequent calculations. The mathematical variable A stands for any particular amount he might see there (it is a mathematical variable, a generic possible value of a random variable). The reasoning appears to show that whatever amount he would see there, he would decide to switch. Hence, he does not need to look in the envelope at all: he knows that if he would look, and go through the calculations, they would tell him to switch, whatever he saw in the envelope.

In this case, at Steps 6, 7 and 8 of the reasoning, A is any fixed possible value of the amount of money in the first envelope.

Thus, the proposed "common resolution" above breaks down and another explanation is needed.

This interpretation of the two envelopes problem appears in the first publications in which the paradox was introduced, Gardner (1989) and Nalebuff (1989). It is common in the more mathematical literature on the problem.

The "common resolution" above depends on a particular interpretation of what the writer of the argument is trying to calculate: namely, it assumes he is after the (unconditional) expectation value of what's in Envelope B. In the mathematical literature on Two Envelopes Problem (and in particular, in the literature where it was first introduced to the world), another interpretation is more common, involving the conditional expectation value (conditional on what might be in Envelope A). In order to solve this and related interpretations or versions of the problem most authors utilize the Bayesian interpretation of probability.