Abstract
The Two-Envelopes problem involves no questions of human nature or survival, nor any dubious assumptions such as human randomness. There are no issues of cooperation or defection, no dominance principles at stake.
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Notes
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It’s not unprecedented to hold contradictory conclusions when it comes to the infinite. Compare three Medieval reactions to the paradox that two infinite magnitudes are unequal if one is a proper part of the other, yet equal because both are infinite: (1) infinites do not exist (2) “equal” and “unequal” do not apply to infinites (3) infinities can be at once equal and unequal (Kretzmann et al. 1982 pp. 569–571).
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A similar tendency to disregard useful information is seen in the more elementary Monte Hall problem. One of three boxes contains a (fixed) prize and the player is randomly given one box. A moderator who knows the location of the prize reveals one of the other two boxes to be empty; it’s always possible to do this. The player is offered the choice to switch for the remaining unopened box. One time in three she already possesses the prize and loses it by switching but two times out of three, she has an empty box and wins the prize by switching. Nonetheless many who should have known better declared the information revealed by the moderator to be useless for improving play.
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This is most easily seen in a large sample. Let C(2i) be the number of cases of S = 2i in the sample. Irrespective of the original probabilities, it is favorable over this particular sample to switch on 2i+1 if and only if C(2i+1) > C(2i)/2. This inequality tends to be true for the large counts that accompany small values of i, since Pi+1 > Pi/2. But for the small counts that accompany the sparsely sampled highest values of i, it often happens that C(2i+1) < C(2i)/2, making it unfavorable to switch on 2i+1.
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Meacham and Weisberg (2003) write as though expected value calculations and repeated trials are opposed in this problem: “it is misleading to speak of the expected utility (EU) of repeated trials, since in the peeking case the question is whether or not one should swap in a particular case, given that one has seen a particular amount in envelope A. … one’s decision regarding whether or not to swap in the peeking case should be determined by the EU of swapping for a particular value of A not on whether the EU of swapping is better ‘on average’ over repeated trials.” This misses the possibility of using repeated trials to test not only SW but selective strategies that depend on the particular amount A. With an exact model as in the two-envelope problem, expected value calculations and repeated trials ought to agree to within sampling error.
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Eckhardt, W. (2013). The Two-Envelopes Problem. In: Paradoxes in Probability Theory. SpringerBriefs in Philosophy. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5140-8_8
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