One Observation behind Two-Envelope Puzzles
In two famous and popular puzzles a participant is required to compare two numbers of which she is shown only one. In the first one there are two envelopes with money in them. The sum of money in one of the envelopes is twice as large as the other sum. An envelope is selected at random and handed to you. If the sum in this envelope is x, then the sum in the other one is (1/2)(2x) + (1/2)(0.5x) = 1.25x. Hence, you are better off switching to the other envelope no matter what sum you see, which is paradoxical. In the second puzzle two distinct numbers are written on two slips of paper. One of them is selected at random and you observe it. How can you guess, with probability greater than 1/2 of being correct, whether this number is the larger or the smaller? We show that there is one principle behind the two puzzles: The ranking of n random variables X1, ... , Xn cannot be independent of each of them, unless the ranking is fixed. Thus, unless there is nothing to be learned about the ranking, there must be at least one variable the observation of which conveys information about it.
|Date of creation:||08 Oct 2003|
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- Nalebuff, Barry, 1989. "The Other Person's Envelope Is Always Greener," Journal of Economic Perspectives, American Economic Association, vol. 3(1), pages 171-181, Winter.