Two Envelopes Problem
At the start you obviously have an expected value of 1/2*(2A) + 1/2*(A). Your expected value is 3/2*A. Where 2A and A are the amounts in the envelopes you can chose.
So you make a choice. The envelope you select has an unknown value of X, and you are given the option to switch envelopes. The other envelope contains either 2X or X/2 which has an expected value of 1/2*(2X) + 1/2*(X/2) = 5/4*X. Therein lies the problem with expected value theory. It makes no sense to switch envelopes despite the higher expected value.
Your choice of an envelope does not change the original expected value of 3/2*A. Applying a second expected value calculation to a problem statement in which the initial conditions have not changed is simply wrong i.e. selecting an envelope has no effect on the expected value of its contents or the expected value of the contents of the envelop which was not selected.
Gell-Mann and Peters are exploring expected value theory from the ground up. It is extremely fragile, and time based observations are challenging it.
A simulation was run in which 6 sets of 50 trials were performed. In three of the sets the person kept the initial envelope selected. In three of the sets the person switched envelopes. No benefit was derived from switching envelopes. In 50 trials the expected value would be 50 * 3/2 = 75 if the value of A is set to one unit.
