To expand on this, the switching argument is actually correct if you happen to know that the envelopes contain either {x, 2x} or {2x, 4x} dollars (where x = 50 or whatever), and you observe that A=100. These are effectively hidden assumptions of the argument.
If you wanted to, you could explicitly model all possible values of the envelopes, and all possible observations of A – and you'd get the right answer again, including in the case above. You just can't define A in a path-dependent way, and then treat it as having the same value in all paths.
If you wanted to, you could explicitly model all possible values of the envelopes, and all possible observations of A – and you'd get the right answer again, including in the case above. You just can't define A in a path-dependent way, and then treat it as having the same value in all paths.