I believe december is making this more complex than it really is. The original problem states:
That is, it is inescapable that there is a 50% chance of a randomly-chosen envelope being the “twice as much” envelope.
The only point I can see that could possibly be considered debatable is whether the envelope given to you is chosen at random, or chosen in a non-random fashion by someone who knows which envelope contains what. Since the problem is clearly presented as a logical/mathmatical puzzle, I think random selection is clearly implied. Otherwise, we are left with the pointless task of attempting to psychoanalyze the “envelope giver”, who is in no way identified or discussed in the problem.
The original problem goes on to suggest that:
That is, the mathmatical probabilities are as expected. There is no hint of any attempt to turn this into a psychoanalytical exercise.
So I wasn’t begging the question, but rather restating the original terms of the problem we’re considering.
But if we’re going to change to the psychoanalytical question: Well, then if my mom gives me the envelope, I keep it, and if Bill Clinton gives me the envelope, I trade.
I don’t even want to think about the “Yes, but he’d probably suspect you’d trade, so he’d give you the “win” envelope, but then he knew you’d suspect that he would suspect you’d trade, so he’d give you the “lose” envelope, but he knew that you’d know that he’d know that you’d suspect, so…” question. That was handled sufficiently in “The Princess Bride”.
december also states that:
Statements that are provable mathmatically or logically don’t need to be “tested”. However, repeated trials are useful for reassuring us that we haven’t done the math wrong - they back up our deduction with induction. For example, we don’t need to draw and measure a bunch of triangles to prove that the Pythagorean Theorem is true, but doing so may make us feel better about the proof. In the case at hand, it would be a simple matter to run a few trials (or a few million on a computer) and see that envelope traders and envelope keepers end up with the same amount of money in the long run.
It’s worth pointing out that there are two different sorts of gambling odds: those that are purely mathmatical, and those that are judgement calls. Odds for a coin toss, a slot machine, a roulette wheel, a lottery, and our envelope problem are of the first sort. They are 100% predictable (assuming that no one is “cheating” - in effect, violating the assumptions of randomness upon which the odds are based). Odds for a horse race, or any athletic contest, or weather (“30% chance of rain”), or the “psychoanalyze the envelope giver” problem are of the second sort. They are very dependant on the analytical skills of the odds-maker, and different people could legitimately come up with different odds.
One last point: logic problems commonly bear a rather tortured relationship to reality (“People from tribe A always tell the truth, but those from tribe B always lie…”, “Given a straight line on an infinite, flat plane…”, etc.), as a way to explicitly limit the problem to questions of logic. The solutions are, of course, applicable to “real life” only if you keep in mind how well, or poorly, the assumptions of the problem fit the actual situations encountered. That doesn’t make them any less “correct” on their own terms.