Two envelopes problem

The two envelopes problem, also known as the exchange paradox, is a brain teaser, puzzle, or paradox in logic, probability, and recreational mathematics. It is of special interest in decision theory, and for the Bayesian interpretation of probability theory. Historically, it arose as a variant of the necktie paradox.The problem typically is introduced by formulating a hypothetical challenge of the following type: Of two indistinguishable envelopes, each containing money, one contains twice as much as the other. The subject may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, the subject gets the chance to take the other envelope instead.What is the optimal rational strategy for maximising the amount of money to be gained?There is no point at all in switching envelopes as the situation is symmetric. However, the story now introduces the so-called switching argument that shows that it is more beneficial to switch. The problem is to show what is wrong with this argument.