Two envelopes problem: Difference between revisions

Revision as of 21:51, 8 October 2023

Revision as of 14:32, 9 January 2024

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==Multiplicity of proposed solutions==

There have been many solutions proposed, and commonly one writer proposes a solution to the problem as stated, after which another writer shows that altering the problem slightly revives the paradox. Such sequences of discussions have produced a family of closely related formulations of the problem, resulting in voluminous literature on the subject.<ref>A complete list of published and unpublished sources in chronological order can be found in the [[Talk:Two envelopes problem/Literature|talk page]].</ref>

No proposed solution is widely accepted as definitive.<ref>{{cite journal |first=Ned |last= Markosian |title= A Simple Solution to the Two Envelope Problem |journal=Logos & Episteme |year=2011 |volume=II |issue= 3 |pages = 347–57|doi= 10.5840/logos-episteme20112318 |doi-access= free }}</ref> Despite this, it is common for authors to claim that the solution to the problem is easy, even elementary.<ref>{{cite journal |first1=Mark D |last1= McDonnell |first2= Alex J |last2= Grant |first3= Ingmar | last3 = Land |first4=Badri N |last4=Vellambi |first5=Derek |last5=Abbott |first6=Ken |last6=Lever |title= Gain from the two-envelope problem via information asymmetry: on the suboptimality of randomized switching |journal = [[Proceedings of the Royal Society A]] |year=2011 | doi= 10.1098/rspa.2010.0541 |volume=467 |issue= 2134 |pages=2825–2851|bibcode= 2011RSPSA.467.2825M |doi-access=free }}</ref> Upon investigating these elementary solutions, however, they often differ from one author to the next.

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== Other simple resolutions ==

A widely-discussed way to resolve the paradox, both in popular literature and part of the academic literature, especially in philosophy, is to assume that the 'A' in step 7 is intended to be the [[expected value]] in envelope A and that we intended to write down a formula for the expected value in envelope B.

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}}</ref>

A correct calculation would be:

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The simple resolution above assumed that the person who invented the argument for switching was trying to calculate the expectation value of the amount in Envelope A, thinking of the two amounts in the envelopes as fixed (''x'' and 2''x''). The only uncertainty is which envelope has the smaller amount ''x''. However, many mathematicians and statisticians interpret the argument as an attempt to calculate the expected amount in Envelope B, given a real or hypothetical amount "A" in Envelope A. One does not need to look in the envelope to see how much is in there, in order to do the calculation. If the result of the calculation is an advice to switch envelopes, whatever amount might be in there, then it would appear that one should switch anyway, without looking. In this case, at Steps 6, 7 and 8 of the reasoning, "A" is any fixed possible value of the amount of money in the first envelope.

This interpretation of the two envelopes problem appears in the first publications in which the paradox was introduced in its present-day form, Gardner (1989) and Nalebuff (1988).<ref>{{Cite journal|last1=Nalebuff|first1=Barry|date=Spring 1988|title=Puzzles: Cider in Your Ear, Continuing Dilemma, The Last Shall Be First, and More|journal = Journal of Economic Perspectives|volume = 2|issue=2|pages=149–156|doi = 10.1257/jep.2.2.149 |doi-access=free}} and Gardner, Martin (1989) <i>'' Penrose Tiles to Trapdoor Ciphers: And the Return of Dr Matrix. </i> ''</ref>)

It is common in the more mathematical literature on the problem. It also applies to the modification of the problem (which seems to have started with Nalebuff) in which the owner of envelope A does actually look in his envelope before deciding whether or not to switch; though Nalebuff does also emphasize that there is no need to have the owner of envelope A look in his envelope. If he imagines looking in it, and if for any amount which he can imagine being in there, he has an argument to switch, then he will decide to switch anyway. Finally, this interpretation was also the core of earlier versions of the two envelopes problem (Littlewood's, Schrödinger's, and Kraitchik's switching paradoxes); see [[Two envelopes problem#History of the paradox|the concluding section, on history of TEP]].

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[[Martin Gardner]] popularized Kraitchik's puzzle in his 1982 book ''Aha! Gotcha'', in the form of a wallet game:

{{quoteblockquote|Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the other (in the case where the amounts are equal, nothing happens). One of the two men can reason: "I have the amount ''A'' in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll have in my possession at the end of the game will be more than 2''A''. Therefore the game is favourable to me." The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the mistake in the reasoning of each man?

| author = [[Martin Gardner]]

| source = ''Aha! Gotcha''