Talk:Two envelopes problem

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Notice This is the talk page for discussing editorial changes to the two envelopes problem article. Please place discussions concerning solutions to the two envelopes problem itself on the Arguments page. Questions or statements of this kind here will be moved to that page without notice.
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[edit] Merge with Necktie Paradox

I propose that Necktie Paradox is merged into this page, perhaps as a short paragraph or section of its own. It seems to fundamentally be the same problem. Thoughts? Andeggs 22:29, 22 December 2006 (UTC)

In a way that paradox is already mentioned here in the history section in the form of a wallet game. Maurice Kraitchik wrote about the necktie paradox already 1943, please see the text by Caspar Albers in the bibliography for a reference and citation. So I do agree we have a historical connection here, but I also think that the Necktie Paradox deserves an article of its own. Many of the ideas developed around the two envelopes problem isn't directly applicable to the Necktie Paradox. So historically it's the same problem, but now they are separate problems. However, this historical connection could be stressed more in both articles. iNic 00:28, 23 December 2006 (UTC)
Do not merge. Although essentially the same problem, they are expressed very differently and have different existences in the literature etc. Cross-refer but keep separate. Snalwibma 14:01, 4 January 2007 (UTC)
Do not merge. I agree with Snalwibma.--Pokipsy76 12:08, 30 January 2007 (UTC)
Merge The "history section (Two-envelope paradox#History of the paradox seems to be exactly this problem. —ScouterSig 19:12, 26 March 2007 (UTC)
Do not merge I agree that the history of the two-envelope paradox (the wallet switch) is the same as the necktie paradox, but as noted therein, the envelope problem differs in the presented relationship between values. The envelope problem is very much different to consider than Necktie. I fully agree with iNic above that perhaps the solution is further stressing of the common historic root.130.113.110.75 06:01, 12 April 2007 (UTC)

Also, this page is very similar to Exchange paradox—Preceding unsigned comment added by 64.81.53.69 (talk) 05:00, 11 April 2007

I definitely agree with merging with the Exchange paradox. I support further discussion on merging the necktie paradox, as they do seem similar, and the necktie paradox lacks a lot of the analysis of the other two articles, so would would suit being mentioned as a variation of the problem. Jamesdlow (talk) 04:54, 11 April 2008 (UTC)

I think it would be a bad idea to merge with the Exchange paradox. As the discussion page on that article mentions, that page seems more like an article on the paradox, while this page is more like a puzzle giving versions of the paradox and solutions. I also think this article is not in very good shape; citations are thin and the ones I checked don't actually say what they are cited for. Warren Dew (talk) 02:37, 3 May 2008 (UTC)

[edit] Formatting of References

I suggest to convert the formatting of the references ("Published papers") to an automated style instead of manually numbering the references. E.g., if I wanted to add a paper of Schwitzgebel and Dever, I would format it as follows: <ref name="Schwitzgebel_and_Denver_02">Eric Schwitzgebel and Josh Dever. ''The Two Envelope Paradox and Using Variables Within the Expectation Formula.'' 2006-07-07. ([http://www.faculty.ucr.edu/~eschwitz/SchwitzPapers/TwoEnvelope060707.pdf])</ref> at the position where it is cited and add a section with the following text

== References ==
<!-- ----------------------------------------------------------
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 discussion of different citation methods and how to generate 
 footnotes using the <ref>, </ref> and  <references /> tags.
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<references />

at the end of the article. --M.T. 22:49, 25 December 2006 (UTC)

I considered all different solutions for adding references available, with their pros and cons, when I finally settled for this one. The main advantages are that we can have the reference list in chronological order and don't have to have a reference to every single paper in the text. Your proposed system for references lacks these freedoms. iNic 05:57, 29 December 2006 (UTC)
The particular PDF you want to add is a more popular version of their original text already in the list. As such, I don't think it merits a new entry in the list, and it's in fact very easy to find via the extra link "A Simple Version of Our Explanation" already present (next to the link to their main paper from 2004). iNic 05:57, 29 December 2006 (UTC)
I would say that the paper that I suggested to add is very different from the one already listed, and it's much easier to understand.
What do you mean with "chronological order"? Does it mean that new papers should always be added at the end? --M.T. 14:21, 31 December 2006 (UTC)
The most common suggestions here recently at the talk page has been to either make this article more encyclopedic in style, even if it makes it more difficult to understand, or to make it more popular so that it's accessible to more people. One solution to this dilemma suggested itself when I read the Monty Hall paradox article recently. It has a separate version in Simple English! We could do the same for this article. No doubt, a Simple English version will be far more difficult to write than this one. But it might be worth a try, don't you think? iNic 04:25, 3 January 2007 (UTC)
Yes, the papers are added "in order of appearance." It makes it easy to add new ones at the end as they pop up. The list is already quite long and confusing, and if we add different versions of the same basic ideas from all authors as separate entries we only add to length and confusion, not to content. iNic 04:25, 3 January 2007 (UTC)

[edit] Equation source

{1 \over 2} 2A + {1 \over 2} {A \over 2} = {5 \over 4}A

Does anyone have the exact source were this equation first comes from? I can't see how anyone could take this equation seriously. "In the first term A is the smaller amount while in the second term A is the larger amount." This is the problem summed up perfectly, why it is listed as a 'proposed solution', it IS the solution.--Dacium 04:58, 17 April 2007 (UTC)

If you read on in the article you will find a short history of the paradox. I reverted your addition of a "correct equation" because it belongs to a different discussion. This article is about what is wrong about the switching argument (i.e., how to solve the paradox), not about different ways to reason instead (or different ways to act if put in this situation). However, I welcome a separate article about the decision problem where we can gather all these ideas (all "correct calculations" and strategies). (Please have a look at the archived discussions for this talk page as this question has been brought up before.) iNic 01:16, 18 April 2007 (UTC)

[edit] solution

I'm confused, is there a solution? In the intro, it says, "This is still an open problem among the subjectivists as no consensus has been reached yet." But the article seems to suggest there's a solution (if interpreted another way?). Could someone clarify this for me and amend the article if necessary? 129.120.94.151 22:37, 19 April 2007 (UTC)

I agree, it was confusing. I deleted recent edits that made the article contradictory in this respect. Thank you for pointing this out. iNic 13:13, 20 April 2007 (UTC)

[edit] crazy

I think this discussion is crazy. Just look at the french page on this topic (http://fr.wikipedia.org/wiki/Paradoxe_des_deux_ch%C3%A8ques), if you want the situation properly discussed.—Preceding unsigned comment added by 129.194.8.73 (talk) 13:26, 15 August 2007

[edit] This is a solved problem, not an open problem

The intro says "This is still an open problem among the subjectivists as no consensus has been reached yet." but that is not correct. There is a clear consensus about the solution in peer reviewed publications. Just have a look at Samet et al., 2004. Yes, this problem is strongly debated by the public and many solutions are proposed and put online but public debate does not mean there is no scientific consensus. It would be the same to call the Monty Hall problem an open problem. There is a clear solution to the Monty Hall problem with scientific consensus no matter what the public thinks or feels.

The solution is independent of Bayesian or frequency interpretation and can be captured in a single phrase:

"The Two Envelopes problem uses and impossible probability distribution."

The use of an impossible probability distribution is the core of most if not all statistical paradoxes. Just like "proofs" that show 1=2 sneak in a division by zero or set the square root of -1 equal to 1 or -1 ( which non-mathematicians easily accept ) so do statistical paradoxes sneak in an impossible probability distribution. The Two Envelopes problem uses the impossible distribution of an infinite set with every element having the same probability. "Any amount of money ( with no upper limit ) with every amount of money as likely". For a non-mathematician the probability distribution of the Two Envelope problem may sound completely reasonable but is mathematically utter nonsense just like dividing by zero.

For every probability distribution the sum of all the probabilities of every possible outcome has to be 1. If you don't have this statistics completely breaks down. You get infinite means and impossibilities as clearly shown in the Two Envelope problem.

If you change the impossible probability in the Two Envelope problem into a real probability by putting an upper limit to the money or by not giving the every amount of money the same probability, the paradox disappears.

There is a clear scientific consensus about this and any professional mathematician or statistician will confirm this. This is basic knowledge.

The article should clearly state this and use it as main point. Obviously this should mean a complete rewrite of the article.

I will not rewrite the article because I don't like to be sucked into an edit war. I just wanted to make it clear that the article is missing the main point. I hope in the end reason will prevail. Michael Korntheuer 17:37, 15 February 2008

If you read half way down the article you will learn that it's not the case that the paradox disappear for every proper prior distribution. If you read even further down you will discover that we actually don't even need a prior distribution to get the paradox, as we don't have to invoke the probability concept at all. iNic (talk) 22:35, 15 February 2008 (UTC)
True about a distribution summing to 1, although there are still infinities involved, specifically infinite expected values. False about not needing to invoke the probabity concept at all. In particular, how do you make the leap from "Let the amount in the envelope you chose be A. Then by swapping, if you gain you gain A but if you lose you lose A/2. So the amount you might gain is strictly greater than the amount you might lose" to "you should swap" without looking at probabilities? To decide to switch inherently involves an assessment of probabilities. To say otherwise is to say you'd be happy to play roulette as long as you can't see the wheel and are thus ignorant of the probabilities involved. Warren Dew (talk) 01:37, 3 May 2008 (UTC)