the two-envelope paradox

Suppose you're on a gameshow where you can choose either of two sealed envelopes, A or B, both containing money.

The host doesn't say how much money is in each, but he does let you know that one envelope contains twice as much as the other.

You pick envelope A, open it and see that it contains $100. The host then makes the following offer: you can either keep the $100, or you can trade it for whatever is in envelope B.

You might reason as follows: since one envelope has twice what the other one has, envelope B either has 200 dollars or 50 dollars, with equal probability. If you switch, then, you stand to either win $100 or to lose $50. Since you stand to win more than you stand to lose, you should switch.

But just before you tell the host you would like to switch, another thought might occur to you. If you had picked envelope B, you would have come to exactly the same conclusion. So if the above argument is valid, you should switch no matter which envelope you choose. But that can't be right.

What's wrong with your reasoning?

Some mathematicians have argued that the problem here has to do with assigning a probability measure on an infinite set (the natural numbers). But logician Raymond Smullyan pointed out that the paradox can be restated without involving probability. Smullyan's restatement is as follows (from Smullyan's Satan, Cantor, and Infinity):

We can prove two contradictory propositions:

"Proposition 1. The amount that you will gain, if you do gain, is greater than the amount you will lose, if you do lose.

"Proposition 2. The amounts are the same."

The proof of Proposition 1 is essentially the one already given: "Let n be the number of dollars in the envelope you are now holding. Then the other envelope has either 2n or n/2 dollars.

"...If you gain on the trade, you will gain n dollars, but if you lose on the trade, you will lose n/2 dollars. Since n is greater than n/2, then the amount you gain, if you do gain — which is n — is greater than the amount you will lose, if you do lose — which is n/2. This proves Proposition 1.

"Now for the proof of Proposition 2. Let d be the difference between the amounts in the two envelopes, or what is the same thing, let d be the lesser of the two amounts. If you gain on the trade, you will gain d dollars, and if you lose on the trade, you will lose d dollars. And so the amounts are the same after all... This proves Proposition 2."


©2000 Franz Kiekeben