Talk:Two envelopes problem/Archive 8

This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

I have added this subject because user INic thinks that the addition of 28/5/2014 in the Alternative Interpratations section must not be published and he keeps deleting it. This addition is published in a well established mathematical journal (see here: Tsikogiannopoulos, Panagiotis (2014). "Variations on the Two Envelopes Problem". Hellenic Mathematical Society, Mathematical Journal (77–78): 3–25.) so it is not an original research and generaly it follows the rules of WP. In my opinion it is very useful to the readers of this article because it gives them an alternative explanation of how the paradox is resolved.
I welcome user INic to justify his action of deleting this explanation.

Are you the author of this paper? iNic (talk) 11:19, 7 October 2014 (UTC)

How is this relevant to your actions? Also I asked you nicely not to revert again so that we can talk about this but you reverted yet again. It seems that you are not interested for the good of wikipedia but only to "win" this warring. 89.31.176.13 (talk) 05:02, 8 October 2014 (UTC)

Please both stop edit warring and let us discuss the matter here.

On the face of it we have a reliable source giving a solution (of course despite my comment above it is not clear what the problem is but that is another matter). iNic makes the point that this is a primary source which he says is not cited by other secondary sources. Could we not agree something based on the above source that could be added to the article in appropriate section? Martin Hogbin (talk) 12:32, 8 October 2014 (UTC)

We already have a noticeboard for OR and primary sources which is the talk page called the Arguments page and I have already placed Mr IP's contribution there for everyone to see and read. So his executive summary of his article isn't deleted from Wikipedia at all. To have a talk page as a noticeboard for people that have problems getting their ideas read by others is OK and not against the Wikipedia rules. But to turn a Wikipedia article into a noticeboard for primary sources of original research is clearly against Wikipedia rules. In addition, all kind of self promotion is against the rules. To delete this from the main article is a no-brainer. iNic (talk) 15:19, 8 October 2014 (UTC)

iNic, this is the page for discussing changes to the article, please can you explain why you think something about this solution should not be included in the article, maybe a shortened version.

I already explained this. Please let me know what part of my explanation you didn't understand. iNic (talk) 22:51, 8 October 2014 (UTC)

I am not suggesting that we keep the whole post as it is but could we not fit something in somewhere? Martin Hogbin (talk) 16:39, 9 October 2014 (UTC)

Sure in theory we could have a section where we list minority views on how to solve this problem, in which this solution can be listed. There are a lot of minority views. But there are also whole classes of solutions very similar to each other from prominent sources that aren't even mentioned yet in the article, and to include a brief section about these has higher priority in my view. iNic (talk) 11:51, 10 October 2014 (UTC)

Do you know that 89.31.176.13 is the author of the cited reference? If you do not know this for sure, please do not make accusations about other editors.

(Redacted)iNic (talk) 01:08, 9 October 2014 (UTC)

You must not make accusations like this based on your guesses and you must not give out personal information on WP users. Martin Hogbin (talk) 14:52, 9 October 2014 (UTC)

I have not given out any personal information at all. At least not more than mr anonymous already did himself. iNic (talk) 11:51, 10 October 2014 (UTC)

89.31.176.13, if you are the author of the source, edit warring to get it into WP is a conflict of interest. Better to state here that you are the author and ask independent editors to discuss its inclusion. Also, I suggest that you consider registering as an editor, you still remain anonymous if you wish and your IP address will not be shown here. Registering does give your edits some greater credibility because others can see your past contributions to WP.Martin Hogbin (talk) 16:50, 8 October 2014 (UTC)

I am former user 89.31.176.13. I followed your advice and created an account. I sign this post with my new username. I didn't know that it makes a difference if I am the author or not of a source when I cite to it, but if it does, then I tell you that I am not the author of the source. The two anonymous comments in this thread are not posted by me. Please use ~~~~ to sign your comments to inform the rest of us who posts what (I just learned it myself!). Caramella1 (talk) 06:37, 9 October 2014 (UTC)

iNic, you made a harassment against me by intentionally revealing my work organisation, according to WP rules found here: http://en.wikipedia.org/wiki/Wikipedia:Harassment#Posting_of_personal_information. I demand that you revert it promptly, followed by a request for oversight to delete that edit from Wikipedia permanently. You also have been warned in your user/talk page. Caramella1 (talk) 06:37, 9 October 2014 (UTC)

I didn't reveal your work organisation at all. How on Earth would I know where you work? You revealed now yourself that the ISP you use is your workplace, not me. So your accusations against me are false and I would like you to apologize for that. If you want to delete your ISP from Wikipedia permanently you have to permanently delete all occurrences of your IP address from Wikipedia, and I can't help you with that unfortunately. I would love be able to help you out with that because in the process all your contributions to WP would be deleted as well. iNic (talk) 11:51, 10 October 2014 (UTC)

The issue has now been resolved. Martin Hogbin (talk) 18:10, 10 October 2014 (UTC)

How has it been resolved? Were you involved in these false accusations as well? iNic (talk) 07:10, 11 October 2014 (UTC)

The personal information has, at my request, been removed by an admin. ::The personal information has, at my request, been removed by an admin. Can we get on with discussing content please. Martin Hogbin (talk) 08:31, 11 October 2014 (UTC)

So you still accuse me for having revealed personal information? iNic (talk) 17:41, 11 October 2014 (UTC)

The purpose of this page is to discuss article content. One editor wants to include something in the article and you do not. You have been edit warring over this for days. best to discuss the issue here. Martin Hogbin (talk) 08:48, 12 October 2014 (UTC)

It's not about what I want or not, it's about following the Wikipedia guidelines for what type of content a Wikipedia article should contain. As I have already explained on this page, WP should not be a noticeboard for OR or minority views, published or not. So I know what the purpose of this talk page is, and I apparently use it exactly for that. But I'm still waiting for any good argument from you guys. Mr anonymous haven't given a single argument for why this particular theory without followers should be mentioned. Nor have you. Instead, you and mr anonymous have used this page for throwing false accusations of harassment against me. An apology from both of you would be in order. iNic (talk) 17:39, 12 October 2014 (UTC)

Let us discuss article content

INic and Caramella1, rather than continuing with your edit war let us discuss the issue here. On the face of it the addition by Caramella is a well sourced explanation but I agree with iNic that just adding one of many published resolutions in full detail is not appropriate. So, let us talk. Here are some questions for you both:

First of all I believe that it is not yet clear what is the right method to resolve the alternative interpretation, so every unique resolution for the alternative interpretation published in a refereed journal or book should be published in WP main article also. That way the readers will be better informed about the various proposed resolutions and the process will help them decide which one seems more logical to them. iNic's practice of deleting whatever he doesn't like is against Knowledge and against the right of the readers to be informed about the matter.

The new paper has been refereed by 3 mathematicians of the Hellenic Mathematical Society and if they think that it is interesting and not mathematically flowed then it should have a place in the main article. The full text is 23 pages long but I selected only an interesting part of it that I think suites well in the alternative interpretation proposed resolutions. Caramella1 (talk) 05:30, 13 October 2014 (UTC)

Caramella1, the best practice is given in

WP:BRD. You were right to add the material. iNic, or anyone else was entitled to revert it. The next step should be to discuss. You have now started that process here. Martin Hogbin (talk

) 09:14, 13 October 2014 (UTC)

Caramella, as a new editor please try to read and understand the rules of Wikipedia. The rules Martin hints you should read above and you also need to read

WP:PA. iNic (talk

) 13:45, 13 October 2014 (UTC)

To repeat: this is not my categories but the categories that emerge for anyone that reads the sources. I'm still waiting for the alternative categories you claim you have Martin. If you don't have any alternative grouping of the sources to suggest why do you constantly bring up the grouping as one of your issues? I don't get it. iNic (talk) 13:45, 13 October 2014 (UTC)

Is it OR or not?

INic, you have said before and you imply it now that the text I added is original research. So let's check what is original research according to WP "No original research".: "The phrase "original research" (OR) is used on Wikipedia to refer to material—such as facts, allegations, and ideas—for which no reliable, published sources exist". Now let's check what counts as a reliable published source according to WP"What counts as a reliable source".: "If available, academic and peer-reviewed publications are usually the most reliable sources, such as in history, medicine, and science. Editors may also use material from reliable non-academic sources, particularly if it appears in respected mainstream publications. Other reliable sources include, university-level textbooks, books published by respected publishing houses, magazines, journals, mainstream newspapers". Well, the text I have added is published in an academic peer-reviewed journal, refereed by 3 mathematicians, that is one of the most reliable sources according to WP, so it is NOT original research. Caramella1 (talk) 05:09, 14 October 2014 (UTC)

To repeat: This is what Jimbo Wales have said about what content belongs in Wikipedia:

The inclusion of a view that is held only by a tiny minority may constitute original research. Jimbo Wales has said of this:

If your viewpoint is in the majority, then it should be easy to substantiate it with reference to commonly accepted reference texts;

If your viewpoint is held by a significant minority, then it should be easy to name prominent adherents;

If your viewpoint is held by an extremely small minority, then — whether it's true or not, whether you can prove it or not — it doesn't belong in Wikipedia, except perhaps in some ancillary article. Wikipedia is not the place for original research.

It is the last case that applies to this paper in this case. If it is published or not doesn't matter. Extreme minority views doesn't belong in Wikipedia. Period. Try to make this theory known somewhere else first and come back in a couple of years. Then the situation might be different and the time is ripe for an inclusion. But right now? No. iNic (talk) 14:25, 14 October 2014 (UTC)

Martin, what is your opinion? Is the text I added OR or not? Caramella1 (talk) 18:15, 14 October 2014 (UTC)

It is obviously not OR as it is published in a peer reviewed journal. That does not mean, however that I support adding the text, at least in its current form. Maybe a brief reference would be in order. Martin Hogbin (talk) 07:49, 15 October 2014 (UTC)

I am afraid that a brief reference will not be helpful for the readers to understand the idea. However a solution must be found to this edit war. So I propose to you Martin to make the changes to the addition that you would be satisfied with and discuss it again. Agreed? Caramella1 (talk) 15:09, 15 October 2014 (UTC)

There are many ideas published which like this one are held only by an extremely small minority. Usually only supported by the author himself and his relatives. So if we include this in the article we need to include all the other extremely small minority views as well. This is perfectly doable as list for example where we have one or two sentences describing each of the extreme minority views. But before we include such a list I think it is good if we mention the ideas for a solution that are currently missing which have much more support in the sources. iNic (talk) 11:17, 15 October 2014 (UTC)

I agree with iNic that it isn't appropriate to feature the Tsikogiannopoulos reference in this article. That paper is just a single primary source that does not seem to have been cited by anyone (so it is not notable), and it is not the original source for any of the ideas discussed in the article. Featuring the Tsikogiannopoulos paper in this article would be giving it undue weight. (I might be inclined to argue for an exception if the paper contained some novel and valid insights on the subject of the article, but it doesn't, so I'm not.)Perswapish (talk) 01:55, 16 October 2014 (UTC)

There are currently four different views on how to organize this page. Martin Hogbin] thinks that the "Common resolution" section should be deleted from the article entirely, and if that isn't possible it should at least be placed at the end of the article. Gerhardvalentin on the contrary want to expand the "Common resolution" section making it more clear and include more versions of that interpretation of the problem. Caramella1 doesn't really care about the article as such as long as his favorite paper by Tsikogiannopoulos published this year is mentioned at a prominent place in the article. Then we have Perswapish and myself who thinks the overall structure of the current article is OK as it is. My view is that we can discuss and change the overall structure of the page and in what order different views should be presented, if we reach a consensus about a new and better structure of the page. Of course the content of each section can be discussed and changed as well. But to delete whole sections representing many authors, or, which is the opposite extreme, to highlight a single randomly chosen author and his recently published idea are both against the notion that Wikipedia should be a reliable source of information. iNic (talk) 10:21, 17 October 2014 (UTC)

I am not saying that, 'the "Common resolution" section should be deleted from the article entirely, and if that isn't possible it should at least be placed at the end of the article'. I believe that this non-mathematical section is very badly written, to the degree that it is meaningless, and that it should therefore be rewritten from scratch. Its position in the article should be based on the weight of sources supporting it. Martin Hogbin (talk) 11:45, 17 October 2014 (UTC)

I'm glad to hear that you no longer want to delete this section entirely. This is good news. I'm a little surprised by your assessment that this section is non-mathematical. My view is that it is more mathematical than what it need to be. The idea is after all very simple and can be explained almost without any math at all. But being the case that this section was written by a professional mathematician this is what we have now. I recently introduced collapsed sections for the more mathematical parts as these are not needed to undertand the main idea or read the article as a whole. I'm curious to learn in what way you think this section is so badly written, even to a degree that it is meaningless. And I'm even more curious about how you would like to rewrite this section from scratch. Regarding the ordering and why we have this section first is in my view only for pedagogical reasons. Solutions in this category have the benefit of not having to change the initial definition of the problem and can tackle the problem as it is. And not only is this idea the simplest and most non-technical to explain, it also evolves naturally into the next version of the problem that is more complicated both mathematically and philosophically. The last one (Smullyan's) is easy to state but the solutions here are perhaps the most technical ones and hence the hardest and to explain. It is therefore natural to put this as the last version. iNic (talk) 18:01, 17 October 2014 (UTC)

I think there are two separate issues here. One is just the usual issue of whether the article can be improved and clarified. I think everyone agrees that (like every other Wikipedia article) this article is not perfect and could be improved. But this generic and commendable striving for improvements isn't what has prompted the recent unproductive insert/delete cycle of edits. The recent problematic editing has been over the inclusion of the Tsikogiannopoulos paper.

My opinion is that this particular primary source, published just last year on a subject that has been written about extensively for many years, has not emerged as a notable or significant contribution to the literature. It is just one among a very large number of isolated writings on this subject, and has not been cited anywhere or received any attention (other than the vigorous attempt of one editor to have it featured prominently in this Wikipedia article). We would be giving it undue weight to feature this particular paper in the article.

Now, if I believed that the Tsikogiannopoulos paper contained some unique insight, not found in prior sources, I might set aside the undue weight concerns, and argue in favor of featuring it here. But I don't believe that's the case. The opinions of various editors may differ about this, depending partly on their levels of familiarity with the prior literature. Without getting into a detailed comparative discussion of the subject matter itself, which is not fit for a Discussion page, all I can say is that, in my opinion, the Tsikogiannopoulos reference does not warrant being featured prominently in the article.Perswapish (talk) 14:07, 17 October 2014 (UTC)

In your opinion, is the paper worth a mention of any kind? — Preceding unsigned comment added by Martin Hogbin (talk • contribs) 16:08, 17 October 2014

In my opinion, that paper isn't worth mentioning. The "resolution" it describes is not new, so even if we decided to add that "resolution" to the article, we would cite other (earlier) references for it. But I'm even doubtful of the value of doing this, because it isn't a valid "resolution", which is why it hasn't gotten much attention in the literature. I know that, as Wikipedia editors, we aren't really entitled to pass judgment, but I think we can base our editorial decision on the lack of attention that a primary source has received. If we want to make this article completely exhaustive, presenting every single idea on the subject that has ever been published, then we would get around to including this idea, but it wouldn't be near the top of my priority list.Perswapish (talk) 17:40, 18 October 2014 (UTC)

My view is in total agreement with Perswapish. I think the next natural improvement of the article would be to include a short explanation of the approach found in for example Clark & Shackel and Norton. This idea is quite common in the literature but not yet covered at all in the article. iNic (talk) 14:45, 19 October 2014 (UTC)

Unfortunately Tsikogiannopoulos' assumption "both players have no prior beliefs for the total amount of money they are going to share, so Events 1 and 2 must be considered to have equal probabilities to occur" leads to new paradoxes if one applies it rigorously to any amount of money which might be found in the envelope. This approach does not get to the heart of the paradox, it is a kind of diversionary tactic - a smoke screen. Obscures the issues. Anyone who wants to know the full, true story of how to resolve TEP ought to read http://www.math.leidenuniv.nl/~gill/tep.pdf but since I wrote that paper, and did not quite complete or publish it yet (though a number of editors of journals begged me to submit it to them), I am disqualified from doing any editing here.

It would be nice to actually see Tsikogiannopoulos famous paper ... I'm afraid my university library does not subscribe to the journal of the Hellenic mathematical society. One may have concerns about the quality of the journal. Richard Gill (talk) 11:04, 28 October 2014 (UTC)

Thanks for your comment. If you get a chance please respond in the last section below concerning the notability issue. Tkuvho (talk) 11:07, 28 October 2014 (UTC)

Thanks, I did. I do not think that the article is an unredeemable mess. I think it is rather good. The *problem* is a somewhat unredeemable mess, and consequently the literature on the problem is enormous and messy too. I explain why in my masterpiece http://www.math.leidenuniv.nl/~gill/tep.pdf. By the way, though "self-published" it has been extremely carefully scrutinised by a number of my colleagues around the world. But of course, that is what I would say, isn't it? Richard Gill (talk) 11:41, 28 October 2014 (UTC)

I have fully-protected the page - it is in no way an endorsement of the current version. Please continue to seek consensus for the article here, and if that is impossible follow normal

Since it was proved here and here that user INic applied unfair tactics to support his opinion, it is now clear that user INic was the only one who opposed with the explanation I added to the main article. So I believe that my last edit on the main article was justified. Also, editor Gerhardvalentin might consider making some useful changes to the main article also which he discusses in this talk page.

So GiantSnowman, I request to unprotect the main page and let it evolve with the contributions of good-faith editors. Caramella1 (talk) 06:25, 24 October 2014 (UTC)

Hear,hear. Indefinite full protection is not needed to end an edit war between two users. Better to tell the users to both stop warring. There are many other things that could be changed. Martin Hogbin (talk) 14:50, 24 October 2014 (UTC)

Caramella has started edit warring even of this talk page now why there is little hope he will not conduct that behaviour again when the article page is not protected anymore. iNic (talk) 14:25, 25 October 2014 (UTC)

GiantSnowman, all editors except INic agree that the main page is an "incomprehensible mess". I request again to unprotect it and let us make it better. Caramella1 (talk) 18:09, 25 October 2014 (UTC)

First, the editors should search agreement as to the main changes. Gerhardvalentin (talk) 21:11, 25 October 2014 (UTC)

I also think the page is *not* an "incomprehensible mess". The subject (the literature on the subject) is an *almost* incomprehensible mess. The page does a very good job at surveying this huge mess. It is encyclopaedic, in a good sense. Richard Gill (talk) 09:07, 29 October 2014 (UTC)

Giant Snowman has kindly un-fully-protected the page. Let constructive editing recommence. Richard Gill (talk) 10:47, 3 November 2014 (UTC)

This edit request to Two envelopes problem has been answered. Set the |answered= or |ans= parameter to no to reactivate your request.

The image of two envelopes (File:Envelop.jpg) has no caption, so I propose either adding an appropriate caption, or changing thumb to 200px (or similar).

Anon126 (notify me of responses! / talk / contribs) 22:13, 28 October 2014 (UTC)

Not done: The page's protection level and/or your

user rights have changed since this request was placed. You should now be able to edit the page yourself. If you still seem to be unable to, please reopen the request with further details. --Redrose64 (talk

) 10:56, 3 November 2014 (UTC)

• Wikipedia:Do not include the full text of lengthy primary sources. Snowman (talk) 18:16, 7 November 2014 (UTC)
• WP:QUOTEFARM. I am not sure if the paper that is being discussed has a suitable Creative Commons license or not, but if it has not then consideration must be give to its copyright and the possibility of fair use of a quote from the source in the Wiki article. Snowman (talk
  ) 18:16, 7 November 2014 (UTC)
• Wikipedia:Quotations. Quotes that are used are attributed to the author with the attribution very near to the quote. I note that there appears to be some quotes already in the article, and the article does not say where they come from. If they are old quotes, then they might be out of copyright, and hence a fair use rationale would not be needed. Snowman (talk) 18:16, 7 November 2014 (UTC)
• Another pitfall that does not apply here, is to be aware of the apparent appearance of plagiarism of material that is copied from somewhere else without the proper attribution, copyrighted or not. Snowman (talk) 18:16, 7 November 2014 (UTC)

As we all know, Wikipedia articles should be preferably based on tertiary or if necessary secondary sources, not on primary sources

Your paper is not the only paper that starts off with an overview of the current state of affairs, reviewing other published ideas before going into the main part of the paper which is the paper's new original ideas. This is not the kind of "secondary source" I had in mind. I agree that it's better than nothing and these kind of introductions shouldn't be discarded. But I don't view them as neutral surveys of the literature because they are all after all written by persons having their own original idea about the problem at hand. Both the selective selection of papers and how they are presented are likely to be tainted by the author's own views. An overview of all the papers, written by someone who is genuinely neutral to the different ideas isn't written yet. And an overview like that maybe never will be written either. iNic (talk) 20:01, 5 November 2014 (UTC)

I am not aware of any other comprehensive review paper, iNic. Please tell us which papers you have in mind. Richard Gill (talk) 19:41, 7 November 2014 (UTC)

I think the article should be structured to reveal a tree of puzzle variants, just like the sequels, prequels, and bifurcations of a movie franchise. This is possible because of the Anna Karenina principle: who's to way what is *the* error in a sloppy logical argument where not everything is carefully defined and inadequate notation makes it impossible to know what was going on in the writer's muddled mind? Remember that TEP started life as a brain-teaser invented by professional mathematicians in order to entertain and confuse amateurs. It has a long history going back to Littlewood and Schrödinger.

I would suggest that the article starts with a clear snappy *verbal* solution. For this purpose, simply transpose the Necktie paradox verbal resolution to Two Envelopes.

Then the article should go into depth (and make use of consistent and clean mathematical notation). Here one one can track the bifurcations and "resurrections" of the paradox. Again and again, the paradox is "resolved" but then a new variation is invented which shows that the story is a bit more complicated (i.e., is even more fun) than first appeared.

After that say that for a more detailed analysis some mathematical notation and concepts are required. Introduce them. Note the *two* uncertainties in the problem: what are the actual amounts x and y; y = 2x, x >0, in the two envelopes; and which envelope contains the smaller of the two. Then do the E(B | A = a) calculation and explain what might be going on wrong here. Namely: the author perhaps is confusing the probability that envelope A contains the smaller amount, with the conditional probability that it contains the smaller amount given that it actually contains "a" dollars (whatever "a" might be).

Here point out that according to some of the literature, the author is actually doing the E(B) calculation. But then he also derails, mixing up conditional and unconditional probabilities and/or expectations, but now in exactly the opposite way to what he did if he was doing the E(B | A = a) calculation. Giving different things the same name is called "equivocation" in the philosophy literature. From this point one can also "back-step" as it were, to a kind of pre-quel to TEP namely Smulyan's purely logical paradox.

On the other hand, perhaps the author did seriously intend us to believe that whatever "a" might be, that conditional probability Prob(A < B | A = a) really is always 1/2 (ie whatever "a" might be). Well that implies that an infinite number of amounts can possibly be in the envelope (double as many times as you like, or halve as many times as you like, any amount which you do consider plausible) and all of them equally likely. This is bringing you up against paradoxes of infinity. It becomes infinitely more likely that envelope A contains any amount *outside* of any finite interval, than any amount *inside* that interval. So the resolution would be that this assumption is "illegal" (takes you outside of conventional probability calculus).

Well the writer can retort, I don't mean *exactly* equal to a half and I don't mean *all* values "a*. I mean close to half, for many values "a".

Well then the analyst responds, still if you believe this chance is close to 1/2 for a whole lot of possible values of "a" then your distribution of possible values spreads out over a very wide range spanning many orders of magnitude, the logarithm of the amount is close to uniformly distributed over a large range. This means that it has a "close to infinite" expectation value and in that case, expectation values are not a good guide to decision making. As Keynes said, in the long run we are all dead. The expectation value is the long run average. If an expectation value is infinite then the short run average is never anywhere close to the long run average.

Then there are also the variants where we get further into math and decision theory including those variants where envelope A actually gets opened. Richard Gill (talk) 09:02, 4 November 2014 (UTC)

This is pretty much the structure the article already has today. Except that the "Common resolution" section is missing. Where do you want to put it? It should not go first, you have stated that. But where exactly do you want to put it? iNic (talk) 19:42, 5 November 2014 (UTC)

It is not missing. What you call "the common resolution", I call "the E(B) calculation" Richard Gill (talk) 19:55, 7 November 2014 (UTC)

Richard, I agree that this is a puzzle for the interested amateur. For that reason I think we should start with whatever we agree is the standard/original/canonical version of the problem with no complicating extensions. As was, no doubt intended by the originators of the problem, the player should be assumed risk neutral and the possible money in the envelopes should not include alien currencies, complex numbers, or infinities. These variations should be covered later after the resolution of the basic problem has been given.

iNic, I have never seen any evidence that the current 'common resolution' is common or even a resolution. The first resolution should be the one given in most mathematical sources on the subject. Martin Hogbin (talk) 10:22, 6 November 2014 (UTC)

Martin, to repeat, I didn't come up with the name "common resolution" here. Someone else did. Check the edit history. I'm just referring to what that section is currently called in the article. Got it now? You have all the rights in the world not to like or understand anything written here or anywhere else actually. But it's not a Wikipedia policy to include only theories or ideas that one particular Wikipedia editor likes or understand. I don't like any of the proposed solutions so if it was up to me this page would be blank. iNic (talk) 11:40, 6 November 2014 (UTC)

There is a solution supported by the majority of mathematical sources and that is what should come first. Richard agrees with me. Martin Hogbin (talk) 12:32, 6 November 2014 (UTC)

I really wish the situation was that easy and clearcut, but it isn't. It's impossible to find a single solution that a majority of sources agree is the correct solution. And I would really like to see the 150+ sources categorised into the two categories "mathematics" and "philosophy." Please go ahead and do that. After you have done that please indicate which of the mathematical sources that support the solution you now have in mind. After you have accomplished that you are free to put that solution first in the article. In fact the remaining ideas can be treated as being merely minority views in the article. iNic (talk) 14:39, 6 November 2014 (UTC)

Wikipedia doesn't have to take account of every fart that has ever escaped someone, with some connection to TEP. My structure encompasses in a logical way a large part of the literature. I think of the different approaches as arranged on a tree, a kind of family tree. I pick a suitable place to start and work outward from there. Of course anyone can come up with other ways to organise this mass of writings. If someone has a better idea, let them tell it here.

Many of the 150+ publications so far on TEP are trivial variations or even repeats of one another; a lot of it is intensively boring. iNic can tell us if some extremely notable solutions are missing from my overview. So I think that the situation *is* easy. I do not claim that there is a single solution that a majority of sources agree is the correct solution. Nobody claims that. The article should not claim that. This doesn't mean that it is hard to find a reasonable way to structure the article. I've told you how I would do it. Does anyone have anything better? Richard Gill (talk) 20:05, 7 November 2014 (UTC)

The Introduction does not introduce the Problem. (did not! - I have changed it Richard Gill (talk) 09:26, 8 November 2014 (UTC))

Notice, the body of the article is built around a particular formulation of the problem, in a section called "Problem". In particular, it is stated explicitly that you do *not* look in the envelope to see how much is in there. The body of the wikipedia article has to start with discussing solutions to the problem as formulated in the section "Problem" https://en.wikipedia.org/wiki/Two_envelopes_problem#Problem. And *later* still one can discuss variations of the original problem, and solutions to the variations. (By "solutions" I mean "solutions which have been presented by reliable sources". The words "reliable sources" to be understood according to wikipedia policy. We start with a canonical formulation of the problem. We continue with some "solutions" to that problem. Then we continue with some variant problems and their attendant "solutions". I put the word "solution" in quotes because we don't have to agree with these solutions, they may be total nonsense, but if notable and reliable sources have offered them, then we are obliged to report them here. Richard Gill (talk) 14:20, 8 November 2014 (UTC)

Right now, the article starts by confusing the reader with the Introduction: https://en.wikipedia.org/wiki/Two_envelopes_problem#Introduction . The section "Introduction" does not explain that the point is not to decide whether or not to switch envelopes, but to explain what is wrong with the argument for switching. Richard Gill (talk) 09:21, 8 November 2014 (UTC)

I have expanded the Introduction in accordance with the remarks I just made here. Richard Gill (talk) 09:25, 8 November 2014 (UTC)

I also took out some of the more nonsense remarks in the first and the second solution. Richard Gill (talk) 10:26, 8 November 2014 (UTC)

I did not do any *restructuring* of the article. Personally I do not think it is necessary. And I don't want to push my proposed restructuring, without consensus, because I don't want to be accused of editing the article to promote OR or my POV. Richard Gill (talk) 10:28, 8 November 2014 (UTC)

In fact my favourite structure is close to the present one. I would simply reverse the order of the first two "solutions". Richard Gill (talk) 14:21, 8 November 2014 (UTC)

In many mathematical resolutions of the problem it is assumed that the proposer of the argument for switching is suggesting that we should calculate the average sum (expectation) in the second envelope given a specific sum in the player's original envelope. The fault in the proposed calculation is that it is assumed that the probability of finding twice the original sum in the second envelope is always 1/2, regardless of what the player might have in his original envelope. In the case that there is an an upper limit on the possible sums that might be in the envelopes it is clearly not always true that the probability the other envelope contains twice the sum in the original envelope whatever sum is in the original envelope. For example, if the player's original envelope were to have more than half the maximum permitted sum then it is impossible for the other envelope to contain twice the original sum.

If there is no limit on what might be in the envelopes then the expectation in both envelopes is infinite and swapping is pointless. Martin Hogbin (talk) 16:15, 11 November 2014 (UTC)

Your last statement is not true, Martin. There can be a finite expectation yet no limit to the amount.

That is not what I understood from what you said above. If we stick to the version in which there is twice the sum in one envelope as the other, no limit means infinite expectation. Other variants can come later. Martin Hogbin (talk) 16:45, 11 November 2014 (UTC)

There is too much OR going on on this talk page and not enough reporting of what is written in reliable sources ... Richard Gill (talk) 16:32, 11 November 2014 (UTC)

What exactly is OR? My proposal just a verbal description of the, well-sourced, E(B|A=a) resolution. Martin Hogbin (talk) 16:45, 11 November 2014 (UTC)

You said that if there is no limit then the expectation is infinite and swapping is pointless. This is not true. There can be no limit, yet finite expectation. On the other hand, swapping (without looking) is always pointless.

I also said, 'If we stick to the version in which there is twice the sum in one envelope as the other, no limit means infinite expectation'. Is that not correct?

That is not correct. That's what I keep saying. Richard Gill (talk) 17:40, 12 November 2014 (UTC)

I suggest that nobody writes any text proposals without adding literature references which justify his or her proposal. Richard Gill (talk) 16:48, 11 November 2014 (UTC)

There is no need for that; this is the talk page. Of course we must have references when we add text to the article. Martin Hogbin (talk) 17:14, 11 November 2014 (UTC)

In the canonical/standard version of the two envelopes problem:

1. There are two envelopes.
2. One contains twice the sum of money (usually dollars) that is in the other.
3. The player does not look in his envelope.
4. The player's original envelope choice is random (from two).
5. An apparently simple, but flawed, line of reasoning is propsed for swapping.
6. We must find the error in the proposed argument.

Solving this problem should be first solution in the article. Attempts to complicate things with different ratios between the envelopes, alien currencies, imaginary quantities, risk averse players, the Ali Baba problem should be covered later. Martin Hogbin (talk) 17:30, 11 November 2014 (UTC)

I agree and in my view this is the approach the article currently tries to follow. A simple idea for a solution of the problem as stated is presented. This solution is maybe too simple as it is easy to show that the solution breaks down. This leads us to the different probabilistic versions of the problem which are more complicated. At the end we consider Smullyan's version which is the easiest to state but perhaps the hardest to solve. So having the simplest ideas first is a good principle in my view too and I'm glad you agree. Having the solution we think is "most true" or even "true" first is not a good approach and will take us nowhere in the end because we will never agree about that anyway. iNic (talk) 18:05, 11 November 2014 (UTC)

I agree with one proviso: there are *two* common "first solutions" of this "canonical problem". From the point of view of a mathematician, these *two* first solutions corresponds to two different interpretations of step 7: are we trying to calculate E(B) or E(B | A = a)? I would say (point of view of a mathematician) that in both interpretations, we try to calculate the target expectation by splitting over the two events A < B and A > B. In the present first presented solution (we are after E(B)) the argument should be using the formula E(B) = E(B | A < B) Prob(A < B) + E(B | A > B) Prob(A > B) = E(A / 2 | A < B) 1/2 + E(2 A | A > B) 1/2 = E(A | A < B) / 4 + E(A | A > B). But now the author assumes that E(A | A < B) = E(A) and E(A | A > B) = E(A). But this can never be true. Given envelope A has the larger of the two amounts, its mean value is larger than E(A) (or both are infinite).

According to the second interpretation we should be using the formula E(B | A = a) = E(B | A < B, A = a) Prob(A < B | A = a) + E(B | A > B, A = a) Prob(A > B | A = a) = 2 a Prob(A < B | A = a) + a / 2 Prob(A > B | A = a). Now he takes both those conditional probabilities to be equal to 1/2. But it is not possible (with a proper prior) that those two probabilities are both equal to 1/2 for all possible values of a. With an improper prior we head into disaster, anyway.

My simple example (which has been moved to "Arguments" page) was intended to illustrate this point concerning the second interpretation. If the amount 32 would be a possible value of "a", then the assumption that Envelope B is equally likely to contain 16 as 64 given A contains 32, is mathematically equivalent to the prior belief that the smaller of the two amounts is equally likely to be 16 as to be 32. Laplace's principle of insufficient reason. Now repeat this, going up and down in by a factor 2 each time. The basic probability assumption in the argument, is equivalent to the assumption that a priori, amounts 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 are equally likely. Now you can either say: we have to stop somewhere (e.g. here) and then you see the argument breaks down. Or you can say: we have to go on indefinitely. But then the smaller amount of money in the two envelopes is infinitely more likely to be outside of any finite range, as inside, which is ludicrous. I propose that this goes in the article. Richard Gill (talk) 10:33, 12 November 2014 (UTC)

Note: these considerations show that the conditional expectation interpretation of the two envelopes problem is mathematically interesting and subtle. And this is how the problem started (Littlewood, Schrödinger) ... a paradox concerning improper prior probability distributions, which is still a controversial topic today. Sometimes it seems to work and sometimes it leads to disaster. Richard Gill (talk) 17:57, 12 November 2014 (UTC)

I think we should let persons that understands a particular solution type write the text explaining that solution type. iNic (talk) 12:49, 12 November 2014 (UTC)

I am mystified that there are people here who want to edit the article but who don't understand both solution types. Sure, they may prefer one type to the other, but how can they contribute to a decent article on (the elementary approaches to canonical) TEP when more than half of the literature (on the elementary approaches to canonical TEP) is inaccessible to them? We are not talking about rocket science here. Richard Gill (talk) 17:44, 12 November 2014 (UTC)

I do not think there are many such people. Personally, I understand the E(B) solution but do not like it. I accept that we must have solutions of that type in the article because it is in the literature. I do question one thing though. How many of those who proposed this type of solution clearly stated that that is what they were doing. Sure, they were calculating E(B) but how many said that this was what they were doing, why they were doing it, and how this related to the proposed argument for swapping? In other words, have these philosopher's solutions been correctly classified as E(B) type.

There is a general consistency amongst sources giving E(B|A=a) solutions and many of them explicitly state that they think the proposed line of reasoning for switching is a mangled calculation of E((B|A=a). This means that we should be able to summarise those solutions here easily.

The problem with what we have called the E(B) solutions is that they mainly consist of words, thus it is very hard to summarise many different sources. Martin Hogbin (talk) 18:52, 12 November 2014 (UTC)

One of the main exponents of the philosopher's approach is Schwitzgebel and Dever. Tsikogiannopoulos (who was not aware of the mathematician's solution) thinks it is fine. They say as clearly as can be said in words, that they are calculating E(B) = E(B | B >A) Prob(B > A) + E(B | B <A) Prob(B < A) and they rewrite this sometimes as 2 E(A ) 1/2 + 1/2 E(A) 1/2 and sometimes without the "E". In other words, sometimes they follow what was written in the switching argument (the writer dropped the "E") but other times they correct that (obviously) wrong notation. Their explanation for why the result is not 1.25 E(A) is that E(A | B > A) and E(A | B < A) must obviously be respectively larger and smaller than E(A). When you know A is larger than B, your expected value of it should increase, and vice-versa. And the writer of the argument should have written E(...) on the left hand side. Tsikogiannopoulos says the same. He also says that the Schwitzgebel and Dever solution has been given by many authors. So we can refer to S & D and to T. What is written in our article should conform to the explanations in those two articles. Note: the paper by T exists in English translation on arXiv.org. It's worth reading, for its analysis of very many *variations* of TEP. T agrees with me that he missed the alternative explanation of standard TEP. He agrees that it is more subtle. Richard Gill (talk) 06:25, 13 November 2014 (UTC)

I think that you misinterpret Tsikogiannopoulos. Nowhere in his paper mentions anything about "When you know A is larger than B...". In fact he doesn't use conditional probabilities at all. And of course he does analyzes the variant where the player bases his expected return on the amount (s)he owns, whether this is known or unknown. Caramella1 (talk) 07:01, 13 November 2014 (UTC)

Richard I have a comment on your paper at Talk:Two_envelopes_problem/Arguments#The_philosopher.27s_solution.

I do not refer to Tsikogiannopoulos's paper. I refer to private communication (email) with him, 12 November 2014. "Yes, you maybe right about that. I meant that when the envelopes are closed one could think that the amounts are X and 2X and the resolve of the paradox is quite easy. But one could also think that he has the amount a in his own envelope without knowing what this is and the resolve is trickier." Richard Gill (talk) 07:28, 13 November 2014 (UTC)

Amusingly, Falk (2008) takes the unconditional expectation interpretation, Falk and Nickerson (2009) the conditional expectation interpretation. Each paper is written as if it is the only reasonable interpretation. I think that the conditional expectation interpretation, which indeed has a trickier resolution, is more interesting and more likely to have been the writer's intention, when one takes account of the history of the problem. Richard Gill (talk) 07:30, 13 November 2014 (UTC)

Yes I noticed this as well. I think it shows how difficult it can be to make up one's own mind about how to best solve this problem. iNic (talk) 12:30, 13 November 2014 (UTC)

It also shows that almost all articles written on TEP have the purpose to promote one particular point of view in order to make one useful point for one particular target audience; for instance, in Falk's case, she writes for people who care about elementary statistics education for non mathematicians. Neither of her papers is intended to survey the whole field. Neither is intended, for that matter, to be "the final word" on TEP. Each of those two papers has the purpose to make a particular educational point. Richard Gill (talk) 15:39, 13 November 2014 (UTC)

It has been the curse of this page from the very beginning that editors only familiar with or understanding one side of the story start to edit. And instead of contributing to the side they know and understand their main focus has been to delete the side of the story they do not understand. Because Wikipedia can't contain anything that is "false," right? This is not so productive. So if we instead start to only edit those sections where we do in fact have read the sources AND understood them this page will become so much better.

INic, remember what you just said when I will edit the side of the story that I understand. Caramella1 (talk) 06:24, 13 November 2014 (UTC)

I understand both sides. :-) Richard Gill (talk) 06:30, 13 November 2014 (UTC)

Richard, there are more than two sides or two opinions on how to solve this problem. There are many ideas out there and trying to understand them all is a big task. I'm currently reading a paper by Ishikawa and I'm only understanding it superficially. I would not be able to write a summary of that paper for example. So admitting that we don't understand all the papers written about this problem is a good first step for every editor on this page to take. If you think you understand a paper but you at the same time think the author of that paper is more or less insane, well chances are that you didn't understand the paper after all. iNic (talk) 11:41, 13 November 2014 (UTC)

Of course you are right, iNic Richard Gill (talk) 15:35, 13 November 2014 (UTC)

I propose that what is now called the "Common resolution" is renamed to "Logical resolutions." I also propose that what is now called "Alternative interpretation" be renamed to "Probabilistic interpretations." The last section I think we can rename to "Smullyan's non-probabilistic variant." With better and less anonymous section headings it will be easier to read and remember what is written on the page. iNic (talk) 14:43, 11 November 2014 (UTC)

Good plan! Richard Gill (talk) 15:12, 11 November 2014 (UTC)

I agree except that the 'Probabilistic interpretations' should be called 'Mathematical interpretations' and come first.

I think "Mathematical interpretations" makes the name too general. The best name would be "Bayesian interpretations" but that name has the disadvantage of sounding too technical for the casual reader. iNic (talk) 17:22, 11 November 2014 (UTC)

I changed my mind and I accepted your proposal "Mathematical interpretations" because we can add also other mathematical solutions under this heading which isn't probabilistic in nature. iNic (talk) 18:46, 11 November 2014 (UTC)

There are two problems though with what you call the 'Logical resolutions'. The first is that the simple version in this article is so simple as to be meaningless and Richard's (in my opinion rather generous) mathematical interpretation is too complicated as it is. I cannot rewrite the simple version because it makes no sense to me and fails if you look in your envelope. Anyone else is welcome to try to improve it.

This is why I invited Gerhard to rewrite this section because he understands these types of solutions very well. I agree that mathematical formalism should be kept at a minimum. Most of the ideas if not all can be explained without formulas. If formulas can be avoided we should avoid them. iNic (talk) 17:22, 11 November 2014 (UTC)

But Gerhard's explanation is not in accord with Richard's main categories of solution namely the attempted calculation of E(B|A=a) or the philosopher's solutions, which Richard says are an attempt to calculate E(B) = E(A) = the average sum in the two envelopes. It is also not in line with a intuitive understanding of what the proposed calculation states. A is the sum in the chosen envelope. Intuitively this is a fixed (but unknown) sum. Intuitively it cannot be two things it is one thing, and you hold that in your hand. It lives in a world between natural intuition, which can easily lead you astray, and proper mathematical anlysis which resolves the paradox. Martin Hogbin (talk) 17:59, 11 November 2014 (UTC)

Can anyone tell me what Gerhard's explanation is? Concisely, please. Or preferably: which reliable source does he claim to be paraphrasing? If he has a "simple solution" (of canonical TEP) it is either the philosopher's or the mathematician's, since there aren't any other. Either can equally well be expressed in words as in mathematical formulas. Richard Gill (talk) 17:24, 12 November 2014 (UTC)

Gerhard has his own view on this (as everyone else, right?) and you can read about it here. But of course Gerhard can't put his own theory in the main article, I'm not proposing that. My suggestion is merely that he can edit the current text based on published sources because I think he will understand those sources the best of us. iNic (talk) 17:41, 12 November 2014 (UTC)

I hope he will use plain English (short sentences, short words) and not his rather personal and special vocabulary and high complexity sentences. I hope he will distinguish *his* somewhat original and very lengthy solution from the solutions given by Falk, Schwitzgebel and Dever, Tsikogianopoulos. Richard Gill (talk) 16:04, 13 November 2014 (UTC)

Anyway, we *must* distinguish between the actual but unknown amount a in the envelope which we hold in our hand, and our beliefs concerning its possible values. Mathematicians have a convenient notation which emphasizes this distinction. A random variable A with a probability distribution (representing a priori beliefs) Prob(A = a), and a possible value a. And we must distinguish arbitrary possible values "a" and the actual fixed but unknown value "a". Richard Gill (talk) 10:48, 12 November 2014 (UTC)

No, all we *must* do is to report what the sources say. iNic (talk) 16:59, 12 November 2014 (UTC)

I have Panagiotis Tsikogiannopoulos' paper. One of my Greek students rapidly got hold of a very nice pdf, even though the journal is not available electronically. Tsikogiannopoulos is a physicist, working at Attica Bank. His email address is printed in the paper. The paper appeared in the Hellenic Mathematical Society (Greek language) journal "Mathematical Review", issues 77-78, 2012, pages 3 - 25, title: "Variations of the problem of exchanging envelopes". I have put it in my TEP dropbox, if anyone wants to see it ... . The author told me he will be uploading an English translation to arXiv soon.

Caramella1 could have saved everybody a lot of time by giving us this basic information. But I conclude that Caramella1 is most likely not Tsikogiannopoulos, and hence all those accusations of own research, fictitious paper, etc were false. Tsikogiannopoulos would at least have known the name of the journal in which he published!

Abstract of the paper: "There are many papers written on the Two Envelopes Problem that usually study some of its variations. In this paper we will study and compare the most significant variations of the problem. We will see the correct decisions for each player and we will show the mathematics that support them. We will point out some common mistakes in these calculations and we will explain why they are incorrect. Whenever an amount of money is revealed to the players in some variation, we will make our calculations, if possible, based on the revealed amount, something that is not accomplished in other papers."

References:

1. Dixit, A. & Nalebuff, B. (1993). Thinking Strategically: Competitive Edge in Business, Politics and Everyday Life. Norton paperback.

2. Nalebuff, B. (1989). The Other Person's Envelope is Always Greener. Journal of Economic Perspectives Vol.3, 1, 171-181.

3. Priest, G. & Restall, G. (2007). Envelopes and Indifference.

4. Schwitzgebel, E. & Dever, J. (2008). The Two Envelope Paradox and Using Variables Within the Expectation Formula. Sorites, 20, 135-140.

5. Bruss, F.T. (1996). The fallacy of the two envelopes problem. The Mathematical Scientist, 21, 112-119. Richard Gill (talk) 20:24, 7 November 2014 (UTC)

Notice that Tsikogiannopoulos seems strongly inspired by economist and game theorist Nalebuff. Who was the originator of the variant in which you do start by looking in your envelope. His paper is about a derivative TEP, not about *the* TEP as we define it in the article. His literature list is rather short. It seems to me that he is re-hashing just one point of view on TEP. Richard Gill (talk) 09:28, 8 November 2014 (UTC)

Tsikogiannopoulos has now posted an English translation of his paper on arXiv. http://arxiv.org/pdf/1411.2823.pdf Richard Gill (talk) 15:50, 12 November 2014 (UTC)

His paper starts classifying variants of TEP. Then he starts with canonical TEP. He says about it: "This variation has been sufficiently analyzed and all papers agree with each other. See for example Eric Schwitzgebel and Josh Dever".

He is wrong here. He is missing about half of the literature, because he doesn't realise that you can try to calculate E(B | A = a) for any value of a, you don't have to look in Envelope A first and then only do the calculation for the amount a which you see there. The paradoxical advice to switch arises precisely because it appears that you would be advised to switch whatever amount "a" you saw, hence no need to look. Just switch. Richard Gill (talk) 17:35, 12 November 2014 (UTC)

I raised this issue with the author. He says "Yes, you may be right about that. I meant that when the envelopes are closed one could think that the amounts are X and 2X and the resolution of the paradox is quite easy. But one could also think that he has the amount a in his own envelope without knowing what this is and the resolution is trickier." Richard Gill (talk) 18:12, 12 November 2014 (UTC)

Tsikogiannopoulos runs two (Greek) web-sites devoted to brain teasers. http://grifoi.org/ and http://pantsik.blogspot.nl/ Richard Gill (talk) 06:43, 14 November 2014 (UTC)

Suppose that the smaller of the two amounts is known (a priori) to be equally likely to be 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 (ten possibilities). If the amount in Envelope A is 1, one should obviously switch (and get 2). If the amount is 1024, one obviously should not switch (because then you would end up with 512). In all other cases, on average, you increase the amount "a" by the famous factor 5/4.

So switching is almost always favourable, but the single unfavourable case (which only has probability 1 in 20) would cause an enormous loss. The gains in all the other cases are balanced by the loss in the unfavourable extreme case. Richard Gill (talk) 16:48, 11 November 2014 (UTC)

What is this an example of? It does not show where the error in the proposed line of reasoning lies, just that you should not swap. Martin Hogbin (talk) 17:17, 11 November 2014 (UTC)

It is an example of a well-defined problem with a correct solution. That is such a pleasant contrast to most of the article and almost all of this talk page. Maproom (talk) 19:24, 11 November 2014 (UTC)

What problem? Martin Hogbin (talk) 19:55, 11 November 2014 (UTC)

The problem with which Richard Gill started this section. Maproom (talk) 20:05, 11 November 2014 (UTC)

Richard has not mentioned a problem he has just stated the obvious. The problem is to find the flaw in the stated line of reasoning. The flaw is exactly as I have said above, that the probabilty of finding double in the other envelope is not equal to 1/2 for every possible value in the initial envelope, as Richard, for some reason has confirmed above.

I just do not understand what this is meant to show. It is just what I said above, if the possible sums are bounded, the probabilty of 1/2 used in the proposed line of reasoning is not correct. It is what Richard says in his paper. Martin Hogbin (talk) 20:58, 11 November 2014 (UTC)

The example is supposed to show what happens when we take the Switching Argument seriously, when we interpret Step 7 as a computation of E(B | A = a).

Start e.g. by considering the case a = 32. Step 6 of the argument says that Prob(B = 64 | A = 32) = Prob(B = 16 | A = 32) = 1/2. This is true if and only if, a priori, Prob(X = 16) = Prob(X = 32). I am using standard notation with: smaller amount is X > 0, larger amount is Y = 2 X. I treat them as random variables. The probability distribution of X represents our a priori beliefs concerning the smaller of the two amokunts.

Now repeat with a = 16. Step 6 of the argument is correct if and only if, a priori, Prob(X = 8) = Prob(X = 16). Repeat again going downwards in powers of 2, and going up in powers of 2. The Switching Argument is assuming that if x is any amount possible, then all amounts 2^n x, with n = ... -2, -1, -, 1, 2, ... are equally likely.

We can now see what is wrong with the argument. Perhaps the author has an improper prior distribution in mind - in fact, it seems that he is assuming that the logarithm of the smaller amount x is uniformly distributed between - infinity and + infinity. Now the expected value of the amounts in the envelope is not defined, conventional probability calculus breaks down. Perhaps he has in mind that this distribution is correct over a large range but, for instance, breaks down at two endpoints. For instance, maybe the smaller amount cannot be smaller than 1 or larger than 512. Then the switching argument is correct for a = 2, 4, ... , 512 but incorrect for a = 1 and for a = 1024. Richard Gill (talk) 10:18, 12 November 2014 (UTC)

Yes, I know that. Why are you explaining it to me it is just what I said, in slightly more general terms, in my suggested text above. I see not reason why we should not give an example like yours as well though. Martin Hogbin (talk) 10:23, 12 November 2014 (UTC)

Good. By the way there is nothing new (no OR) here. The fact that the assumption of equal probabilities 1/2 is true *if and only if* the prior is uniform (on a logarithmic scale) is well known in the mathematical literature. Though there is some confusion with some authors believing in uniform priors in the original scale. That is a common mistake.

An important point is that if we take the argument seriously we can *derive* the implicitly assumed prior distribution of x. Uniform prior (on logarithmic scale) is a necessary and sufficient condition for the correctness of the computation in step 7 for all a. Richard Gill (talk) 10:40, 12 November 2014 (UTC)

Can you just confirm one thing as well please. In the standard version, where one envelope contains double the amount of the other, if the possible sums are not bounded then the expectation of both envelopes is infinite, with no exceptions. I am sure one of your sources says this and I can try and find it if you insist.

No, this is not true!!! I keep saying it is not true. It is the POV of Martin Hogbin, and it is wrong. Richard Gill (talk) 11:05, 12 November 2014 (UTC)

In the standard version, it is possible that E(B | A = a) > a for all a. But if that is true, it automatically follows that E(A) = E(B) = infty. This is well known and in many of the papers. Richard Gill (talk) 11:36, 12 November 2014 (UTC)

Richard, it is not a POV it was a question. All I am trying to do is to write the agreed facts in simple language. Your earlier answers were not clear to me, that is all. Which paper discusses this best? Martin Hogbin (talk) 16:50, 12 November 2014 (UTC)

Thanks for your answer though, For some reason I got being bounded and E(B | A = a) > a for all a muddled up in my head. Martin Hogbin (talk) 23:43, 12 November 2014 (UTC)

Which paper discusses this best? My paper, obviously. Otherwise I would not have written it. I think that my paper is the only one which explicitly compares both basic approaches (as well as a heap of other ones). Richard Gill (talk) 07:34, 13 November 2014 (UTC)

I appreciate that there is a long history of this problem starting with the two neckties in which the one object is not necessarily worth twice the other and we should have the history of the problem in the article, including the ways it has been modified to get round various resolutions, as described in your paper, but I am trying to give a simple E(B|A=a) resolution for the problem described in this article. Martin Hogbin (talk) 11:01, 12 November 2014 (UTC)

Sure. My paper starts with two simple resolutions of the problem described in this article, corresponding to the two main interpretations of steps 6, 7. Either way, the guy who made the argument is mixing up conditional and unconditional probabilities and/or expectations. That's the simplest possible resolution I can give. The verbal solution of the two neckties problem is the same. Richard Gill (talk) 11:08, 12 November 2014 (UTC)

I have restored two sections concerning the article.

Another important point is that Gerhard apparently believes that the 5/4 calculation is always wrong ... but in fact, it can be almost always correct! Richard Gill (talk) 10:42, 12 November 2014 (UTC)

Yes I was trying to explain that to Gerhard. I hope to continue on the arguments page. Martin Hogbin (talk) 16:50, 12 November 2014 (UTC)

Yes, and this is an important distinction between the two main types of solution ideas found in the sources. So it's not only Gerhard who "apparently believes" this. iNic (talk) 16:30, 12 November 2014 (UTC)

Obviously, the computation that leads to E(B) = 5/4 E(A) is wrong. The computation that leads to E(B | A = a) = 5/4 a can be entirely correct for a lot of values of a. Hopefully everyone is able to appreciate that. Ruma Falk has two papers on TEP and she takes a different interpretation in each. Gerhard is a fan of Falk so hopefully he can understand her other paper too and see that both interpretations are legitimate (thanks to the Anna Karenina principle). Richard Gill (talk) 17:18, 12 November 2014 (UTC)

Sorry Ruma Falk has three papers. The first one, Nickerson and Falk (2006) is an overview of many variants and many solutions, as well as giving some history. Richard Gill (talk) 16:27, 14 November 2014 (UTC)