Two Envelopes Problem - History of The Paradox

History of The Paradox

The envelope paradox dates back at least to 1953, when Belgian mathematician Maurice Kraitchik proposed a puzzle in his book Recreational Mathematics concerning two equally rich men who meet and compare their beautiful neckties, presents from their wives, wondering which tie actually cost more money. It 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 Schroedinger. Martin Gardner 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?

In 1988 and 1989, Barry Nalebuff presented two different two-envelope problems, each with one envelope containing twice what's in the other, and each with computation of the expectation value 5A/4. The first paper just presents the two problems, the second paper discusses many solutions to both of them. The second of his two problems is the one which is nowadays the most common and which 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 coin is tossed to decide whether Envelope B should contain half or twice that amount, and only then given to Baba.

In the Ali-Baba problem, it is a priori clear that (even if they don't look in their envelopes) Ali should want to switch, while Baba should want to keep what he has been given. The Ali-Baba paradox comes about by imagining Baba working through the steps of the two-envelopes problem argument (second interpretation), and wrongly coming to the conclusion that he too wants to switch, just like Ali.