The Two Envelope Paradox

You’re on a game show. You’re given a choice between two envelopes containing money, knowing that one of the envelopes contains twice as much as the other. You get to keep the contents of whichever envelope you choose.

Having chosen the envelope, you open it, and find that it contains $1000. Before the game ends, though, you get one chance to change your mind, to exchange your envelope for the other one. The two envelope paradox arises because no matter which envelope you chose in the first place, it always seems that swapping is the rational thing to do.

Suppose that you chose the envelope containing the least money. If you swap for the other envelope, then you’ll double your money to $2000. You could gain $1000 by swapping.

Suppose that you chose the envelope containing the most money. If you swap for the other envelope, then you’ll halve your money to $500. You could lose $500 by swapping.

If you decide to exchange envelopes, then, then you could gain twice as much as you could lose. Your chances of gaining are equal to your chances of losing. Exchanging envelopes is therefore the rational thing to do.

Had you chosen the other envelope, though, then you could have reasoned in precisely the same way. Whatever amount of money you had taken from the other envelope, you could have reasoned that by exchanging you had twice as much to gain as to lose, and that your chances of gaining and losing were equal, and so that you should choose to swap.

Whichever envelope you choose in the first place, then, you’re better off swapping it for the other one when you get the chance.