Two envelopes problem: Difference between revisions

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Byeong-Uk Yi, on the other hand, argues that comparing the amount you would gain if you would gain by switching with the amount you would lose if you would lose by switching is a meaningless exercise from the outset.<ref>{{cite journal |author=Byeong-Uk Yi |year=2009 |title=The Two-envelope Paradox With No Probability |url=http://philosophy.utoronto.ca/people/linked-documents-people/c%20two%20envelope%20with%20no%20probability.pdf |url-status=dead |archive-url=https://web.archive.org/web/20110929034017/http://philosophy.utoronto.ca/people/linked-documents-people/c%20two%20envelope%20with%20no%20probability.pdf |archive-date=2011-09-29 }}</ref> According to his analysis, all three implications (switch, indifferent, do not switch) are incorrect. He analyses Smullyan's arguments in detail, showing that intermediate steps are being taken, and pinpointing exactly where an incorrect inference is made according to his formalization of counterfactual inference. An important difference with Chase's analysis is that he does not take account of the part of the story where we are told that the envelope called envelope A is decided completely at random. Thus, Chase puts probability back into the problem description in order to conclude that arguments 1 and 3 are incorrect, argument 2 is correct, while Yi keeps "two envelope problem without probability" completely free of probability and comes to the conclusion that there are no reasons to prefer any action. This corresponds to the view of Albers et al., that without a probability ingredient, there is no way to argue that one action is better than another, anyway.

Bliss argues that the source of the paradox is that when one mistakenly believes in the possibility of a larger payoff that does not, in actuality, exist, one is mistaken by a larger margin than when one believes in the possibility of a smaller payoff that does not actually exist.<ref>{{cite journalarxiv |author=Bliss |year=2012 |title=A Concise Resolution to the Two Envelope Paradox |arxiv=1202.4669|bibcode=2012arXiv1202.4669B }}</ref> If, for example, the envelopes contained $5.00 and $10.00 respectively, a player who opened the $10.00 envelope would expect the possibility of a $20.00 payout that simply does not exist. Were that player to open the $5.00 envelope instead, he would believe in the possibility of a $2.50 payout, which constitutes a smaller deviation from the true value; this results in the paradoxical discrepancy.

Albers, Kooi, and Schaafsma consider that without adding probability (or other) ingredients to the problem,<ref name=":4" /> Smullyan's arguments do not give any reason to swap or not to swap, in any case. Thus, there is no paradox. This dismissive attitude is common among writers from probability and economics: Smullyan's paradox arises precisely because he takes no account whatever of probability or utility.