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Step 7 states that the expected value in B = 1/2(2A + A/2).
It is pointed out that the 'A' in the first part of the formula is the expected value, given that envelope A contains less than envelope B, but the 'A', in the second part of the formula is the expected value in A, given that envelope A contains more than envelope B. The flaw in the argument is that the same symbol is used with two different meanings in both parts of the same calculation but is assumed to have the same value in both cases. This line of argument is introduced by McGrew, Shier and Silverstein (1997).<ref>{{cite journal |last1=McGrew |first1=Timothy |last2=Shier |first2=David |last3=Silverstein |first3=Harry |title=The Two-Envelope Problem Resolved |journal=Analysis |date=1997 |volume=57 |issue=1 |pages=28–33 |doi=10.1093/analys/57.1.28 |url=https://academic.oup.com/analysis/article-abstract/57/1/28/139339|url-access=subscription }}</ref>
A correct calculation would be:
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=== Nalebuff asymmetric variant ===
The mechanism by which the amounts of the two envelopes are determined is crucial for the decision of the player to switch her envelope.<ref name="Tsikogiannopoulos"/><ref>{{citation |last1=Priest |first1=Graham |last2=Restall |first2= Greg |year=2007 |title=Envelopes and Indifference |url= http://consequently.org/papers/envelopes.pdf |journal= Dialogues, Logics and Other Strange Things |publisher=College Publications |pages=135–140}}</ref> Suppose that the amounts in the two envelopes A and B were not determined by first fixing the contents of two envelopes E1 and E2, and then naming them A and B at random (for instance, by the toss of a fair coin<ref name=":0">{{Cite journal|last1=Nickerson|first1=Raymond S.|last2=Falk|first2=Ruma|date=2006-05-01|title=The exchange paradox: Probabilistic and cognitive analysis of a psychological conundrum|url=https://doi.org/10.1080/13576500500200049|journal=Thinking & Reasoning|volume=12|issue=2|pages=181–213|doi=10.1080/13576500500200049|s2cid=143472998|issn=1354-6783|url-access=subscription}}</ref>). Instead, we start right at the beginning by putting some amount in envelope A and then fill B in a way which depends both on chance (the toss of a coin) and on what we put in A. Suppose that first of all the amount ''a'' in envelope A is fixed in some way or other, and then the amount in Envelope B is fixed, dependent on what is already in A, according to the outcome of a fair coin. If the coin fell Heads then 2''a'' is put in Envelope B, if the coin fell Tails then ''a''/2 is put in Envelope B. If the player was aware of this mechanism, and knows that she holds Envelope A, but do not know the outcome of the coin toss, and do not know ''a'', then the switching argument is correct and she is recommended to switch envelopes. This version of the problem was introduced by Nalebuff (1988) and is often called the Ali-Baba problem. Notice that there is no need to look in envelope A in order to decide whether or not to switch.
Many more variants of the problem have been introduced. Nickerson and [[Ruma Falk|Falk]] systematically survey a total of 8.<ref name=":0" />
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