Exchange Paradox

[callydrias]

Saw this written up in Nature last month and thought it was interesting. I'm not sure where the hole is. Maybe you can spot it.

There are two envelopes each containing money. One of the envelopes contains twice as much as the other and you can not tell which is which. You choose an envelope at random (you truly have no information on which to base your choice) and are then given the option to exchange if you'd like.

Consider the expectation of exchanging. There is a 50% chance that the other envelope contains twice as much money as you have now. There is also a 50% chance that the other envelope has half as much. The expectation for exchanging then is 0.5(2x-x) + 0.5(x/2-x) = 0.25x. Therefore exchanging envelopes is +EV so you should exchange.

If given the option to exchange again (if you haven't opened any of the envelopes yet), you would exchange over and over, ad infinitum, using the same reasoning each time. Discuss.