The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. It is of special interest in decision theory and for the Bayesian interpretation of probability theory. It is a variant of an older problem known as the necktie paradox. The problem is typically introduced by formulating a hypothetical challenge like the following example:
Imagine you are given two identical envelopes, each containing money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. Should you switch? Since the situation is symmetric, it seems obvious that there is no point in switching envelopes. On the other hand, a simple calculation using expected values suggests the opposite conclusion, that it is always beneficial to swap envelopes, since the person stands to gain twice as much money if they switch, while the only risk is halving what they currently have.
Introduction
Problem A person is given two indistinguishable envelopes, each of which contains a sum of money. One envelope contains twice as much as the other. The person may pick one envelope and keep whatever amount it contains. They pick one envelope at random but before they open it they are given the chance to take the other envelope instead.
The switching argument Now suppose the person reasons as follows:
By this mistaken reasoning (explained below), the person should switch.
History of the paradox
The envelope paradox dates back at least to 1943, when Belgian mathematician Maurice Kraitchik proposed a puzzle in his book Recreational Mathematics concerning two men who meet and compare their fine neckties. Each of them knows what his own necktie is worth and agrees for the winner to give his necktie to the loser as consolation. Kraitchik also discusses a variant in which the two men compare the contents of their purses. He assumes that each purse is equally likely to contain 1 up to some large number x of pennies, the total number of pennies minted to date. The puzzle is also mentioned in a 1953 book on elementary mathematics and mathematical puzzles by the mathematician John Edensor Littlewood, who credited it to the physicist Erwin Schrödinger, where it concerns a pack of cards, each card has two numbers written on it, the player gets to see a random side of a random card, and the question is whether one should turn over the card. Littlewood's pack of cards is infinitely large and his paradox is a paradox of improper prior distributions. Martin Gardner first wrote about the necktie paradox in his January 1958 column of Mathematical Games. He later popularized Kraitchik's puzzle in his 1982 book Aha! Gotcha, in the form of a wallet game:
Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the other (in the case where the amounts are equal, nothing happens). One of the two men can reason: "I have the amount A in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll have in my possession at the end of the game will be more than 2A. Therefore the game is favourable to me." The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the mistake in the reasoning of each man?
Gardner confessed that though, like Kraitchik, he could give a sound analysis leading to the right answer (there is no point in switching), he could not clearly put his finger on what was wrong with the reasoning for switching, and Kraitchik did not give any help in this direction, either. In 1988 and 1989, Barry Nalebuff presented two different two-envelope problems, each with one envelope containing twice what is in the other, and each with computation of the expectation value 5A/4. The first paper just presents the two problems. The second discusses many solutions to both of them. The second of his two problems is nowadays the more common, and is presented in this article. According to this version, the two envelopes are filled first, then one is chosen at random and called Envelope A. Martin Gardner independently mentioned this same version in his 1989 book Penrose Tiles to Trapdoor Ciphers and the Return of Dr Matrix. Barry Nalebuff's asymmetric variant, often known as the Ali Baba problem, has one envelope filled first, called Envelope A, and given to Ali. Then a fair coin is tossed to decide whether Envelope B should contain half or twice that amount, and only then given to Baba. Broome in 1995 called a probability distribution 'paradoxical' if for any given first-envelope amount x, the expectation of the other envelope conditional on x is greater than x. The literature contains dozens of commentaries on the problem, much of which observes that a distribution of finite values can have an infinite expected value.
Multiplicity of proposed solutions There have been many solutions proposed, and commonly one writer proposes a solution to the problem as stated, after which another writer shows that altering the problem slightly revives the paradox. Such sequences of discussions have produced a family of closely related formulations of the problem, resulting in voluminous literature on the subject. No proposed solution is widely accepted as definitive. Despite this, it is common for authors to claim that the solution to the problem is easy, even elementary. Upon investigating these elementary solutions, however, they often differ from one author to the next.
Example resolution Below is one simple resolution among several that have been proposed. The switching argument is not valid as stated because the symbol A is used inconsistently. It denotes both (i) the unconditional random variable representing the amount in the chosen envelope and (ii) the amount conditioned on whether that envelope contains the smaller or the larger of the two values. These conditional expectations generally differ and cannot be combined in a single expected value calculation. The switching argument effectively writes
… excerpt ends here. Continue reading the full article.
