Two envelopes problem

The Two Envelopes Problem is a puzzle in probability. Two brothers have an uncle who is wealthy beyond measure. The uncle decides to give each nephew an envelope of money. He has two identical envelopes. Into one he places a sum of money, and into the other, he places twice that amount. Then both envelopes are sealed, and shown to the brothers. The first brother chooses one envelope, and the second brother gets the other. (This is a random choice, because they both look identical.)

Each brother goes to his own room to opens his envelope. The first one sees that the amount of money in the envelope is y. He wonders whether he should swap envelopes with his brother, and attempts a naive (and incorrect) probabilistic analysis. If he doesn't swap envelopes, he will have a profit of y. Now there is 50% chance that he has the smaller envelope, and 50% chance he has the larger one. Therefore, the expected profit of swapping is 50% (y/2) + 50% (2y) = 1.25y. Therefore, it is rational for him to seek to swap envelopes with his brother. Perhaps surprisingly, his brother has done the same analysis, and is also keen to swap. How can each brother expect to be better off by swapping, especially when the original choice was random?

References

Williams, David, 2001. Weighing the Odds: a Course in Probability and Statistics. Cambridge University Press. ISBN 052100618X.