The Two Envelopes Problem

This is one of my favorite problems, mainly because it's so simple and yet stumps so many people, especially the "smart" ones.

Look at what the Wiki page has to say about it.

"The two envelopes problem is a puzzle or paradox within the subjectivistic interpretation of probability theory; more specifically within Bayesian decision theory. This is still an open problem among the subjectivists as no consensus has been reached yet."

http://en.wikipedia.org/wiki/Two_envelopes_problem

Look at the complex mathematics applied on that page. It makes me laugh since the solution is so simple, but I digress.


Here's the "paradox".
(I'll word it my own way, but I won't change the essential nature of the problem).


You are seated at a table, and before you are two identical envelopes. You are told that each envelope has a card in it with a monetary amount written on it. One has a certain unknown amount and the other has twice that amount. You cannot see through the envelopes and you have no idea what amount of money is involved or which envelope has which card. You are to choose one of the two envelopes, and after doing so, you are free to swap one for the other and to keep swapping as long as you wish. After having settled on choosing one of the two envelopes, you may then, and only then, open up one of the envelopes, read the card and collect that amount of money from the person offering the game to you. However, there is catch (there's always a catch). You are obligated to (a) follow the goal, which is to obtain as much money as possible, and (b) you must follow an optimal strategy that will allow you to achieve this goal, and (c) then and only then, can you stop swapping envelopes and collect your money. Otherwise, you must continue to swap envelopes in pursuit of a higher amount.


Now, you select an envelope. Let's say that the one you selected is has 'n' amount written on it's card (that you still haven't seen). But before looking at the card, you reason that the other envelope that you didn't select has either 2n or one half n written on it's card. If you swap cards, you stand to lose one half n, or stand to gain 'n' amount, so you're obligated by the terms of the game to swap envelopes, but now THE SAME reasoning now applies to the envelope you just selected. The OTHER envelope not in your possession always has either half or twice the amount as your envelope, so again, you are obligated to swap envelopes, and yet again, the same reasoning applies so you must swap yet again...

The end result is that you must KEEP swapping and NEVER actually receive any money because by the rules of the game, you are never allowed to stop swapping and to open any envelopes.

What's wrong with this reasoning?
(Try reasoning it out for yourself before reading the following solution)
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Solution:

What's wrong is, as you may have guessed, how the problem is being considered. We're approaching the problem as if there are three possible amounts (.5n, n, 2n), when there are actually two possible amounts on the cards. Instead of calling the amount on the card selected "n", let's just say that one envelope has a card with 'x' amount, and the other has 2x. If you first selected 'x', then by swapping you stand to gain 'x' amount, (2x-x). If, on the other hand, you first selected 2x, then by swapping you stand to lose 'x' amount. (-2x+x = -x).

That's the answer. By swapping, you stand to either gain 'x' or lose 'x', so there is no obligation to swap envelopes because there is no advantage in doing so.