The Two-Envelope Paradox: A Complete Analysis? David J. Chalmers Department of Philosophy
University of Arizona
Tucson, AZ 85721 chalmers@arizona.edu Doug Hofstadter introduced me to the two-envelope paradox in 1988. This paper corresponds to more or less the position I came up with then. I wrote this up in 1994 after a couple of papers on the subject appeared in Analysis . I never published it, partly because it came to seem to me that this treatment resolves only part of the paradox: it resolves the "numerical" paradox but not the "decision-theoretic" paradox. For a more recent treatment of the decision-theoretic paradox, see The St. Petersburg Two-Envelope Paradox A wealthy eccentric places two envelopes in front of you. She tells you that both envelopes contain money, and that one contains twice as much as the other, but she does not tell you which is which. You are allowed to choose one envelope, and to keep all the money you find inside. This may seem innocuous, but it generates an apparent paradox. Say that you choose envelope 1, and it contains $100. In evaluating your decision, you reason that there is a 50% chance that envelope 2 contains $200, and a 50% chance that it contains $50. In retrospect, you reason, you should have taken envelope 2, as its expected value is $125. If your sponsor offered you the chance to change your decision now, it seems that you should do so. Now, this reasoning is independent of the actual amount in envelope 1, and in fact can be carried out in advance of opening the envelope; it follows that whatever envelope 1 contains, it would be better to choose envelope 2. But the situation with respect to the two envelopes is symmetrical, so the same reasoning tells you that whatever envelope 2 contains, you would do better to choose envelope 1. This seems contradictory. What has gone wrong? | |
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