in non-technical language, what goes wrong (see necktie paradox) is that, in the scenario provided, the mathematics use relative values of a and b (that is, it assumes that one would gain more money if a is less than b than one would lose if the opposite were true). however, the two values of money are fixed (one envelope contains, say, $20 and the other $40). if the values of the envelopes are restated as "x" and 2"x", it's much easier to see that, if a were greater, one would lose "x" by switching and, if b were greater, one would gain "x" by switching. one does not actually gain a greater amount of money by switching because the total "t" of a and b (3"x") remains the same, and the difference "x" is fixed to "t/3".