Talk:Two envelopes problem/Archive 4

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Lead thoughts

[Afterword. These thoughts concern how to approach a family or several families of articles on problems, puzzles, paradoxes, etc. For understanding the issues, it may be useful to consider writing articles on "problems" in the sense of exercises that are set for students to write up. If not, why not? These thoughts accompany my reply to Richard Gill and the community a couple hours ago at Wikipedia talk:WikiProject Statistics#Two envelopes problem, two children problem. Then I supposed that I was writing at WikiProject Mathematics, which may not matter much.]

The lead sentence is unchanged since last spring or earlier: "The two envelopes problem is a puzzle or paradox for the subjectivistic interpretation of probability theory; more specifically within subjectivistic Bayesian decision theory."

Does anyone else see the wisdom of nearly empty boilerplate such as "The two envelopes problem is a puzzle in probability"? (If it were named a puzzle, I would call it a problem.) The very short lead section goes on to say something about it's current status. I don't know whether that is true but it is appropriate. In some words the lead should also convey that the problem is recent or current in pedagogy, illustration, or exposition.

Compare what the equally short lead paragraphs undertake at Bertrand paradox (probability) and Bertrand box paradox. What should be done where there is no original author? At the same time, should those leads note the status for Bertrand? Eg, did he pose and also claim to resolve?

These three leads locate their puzzles in "subjectivistic", "classical", and "elementary" probability. Does that follow a style guide for some WikiProject, or the systematic work of some editor? (The article doesn't now support the restriction to "subjectivistic" probability, nor the more specific one. And the lead paragraph goes on to indicate some relation with "frequentist" probability, unclear whether it poses any problem for frequentists.)

Regarding "puzzle or paradox", both keywords are linked to articles. Is some technical distinction intended? (I doubt it. I guess this one is called a puzzle or paradox for the poor reason that it is named a "problem".) Perhaps in lead paragraphs we may write all problems/puzzles/paradoxes as puzzles, say. I feel sure that for many articles in this family both the nature --if puzzles, paradoxes, etc are technically distinct-- and the scope of application must be at issue, open questions. The nature and scope need to be explained or argued in the articles.--P64 (talk) 19:14, 3 May 2011 (UTC)

I think your point about puzzle/paradox/problem is a good one, what do you suggest? Martin Hogbin (talk) 14:07, 5 May 2011 (UTC)

(edit conflict) Ha ha, Martin Hogbin, pleased to meet you. I did think you meant my "pouts" and I still suspect that.

Having skimmed the List of paradoxes, I would use "paradox" liberally in naming articles and in giving boldface alternatives. Offhand, I would not describe them as puzzles or paradoxes unless meaning that in some technical sense. Alternatively, Monty Hall problem says that it "is a probability puzzle" with a link only to probability, not to puzzle. I need to study further before talking more than "Offhand".

I'll read that article (http://www.pitt.edu/~jdnorton/papers/Exchange_paradox.pdf). It gives one interpretation parallel to Hilbert's paradox of the Grand Hotel. I wonder whether it's possible in User space to begin some valuable grouping beside the assignment to disciplines in the List of paradoxes.

Meanwhile I have discovered use of the {{Puzzles}} template. How do I find the template itself? or directly find its sponsoring project or author? --P64 (talk) 17:41, 5 May 2011 (UTC)

On the latter only: No project claims Template: Puzzles, whose discussion page is empty. It was created as Template: VG Puzzle for 'video game' and promptly moved. I have asked its author about any discussion of puzzles coverage in general.

P.S. Only the Lists and Philosophy projects claim List of paradoxes.--P64 (talk) 18:35, 7 May 2011 (UTC)

I think the problem is that the distinction between these concepts is not that clearly defined.

In the Monty Hall problem, the original statement by Selvin and the question by Whitaker both seem to be puzzles set to test the ingenuity of the reader. It is also, according to the most common definition of the word, a paradox in that the correct answer appears wrong to most people. 'Problem', to me has some suggestion of 'open problem', which the MHP is not. Martin Hogbin (talk) 22:05, 5 May 2011 (UTC)

The lead was a bit longer in the past but after a series of attacks from an angry editor this is what survived. I think the intro can be a little longer than what it is, that would not hurt at all. Regarding the problem/puzzle/paradox issue, these words are mentioned carefully in a way so that none of them is stressed more than any other. This has two reasons: The first is of course due to the fact that it is sometimes referred to as a problem, sometimes as a puzzle and sometimes as a paradox in the literature. The second reason is that some editors get upset if it's only referred to as a paradox because they have a strong opinion that it is not a paradox, causing endless discussions of what a "paradox" really is. Still others get upset if it's not called a paradox because they think it's a paradox. So in order to make as few people as possible upset all three words are used interchangeably. iNic (talk) 22:18, 5 May 2011 (UTC)

The word paradox is sometimes used in its technical meaning of an apparent contradiction in reasoning. Paradoxes, by defintion, always have resolutions: one shows that one of the steps in the reasoning is false (or one of the assumptions is never satisfied). In ordinary language the two envelope problem clearly is both puzzling, and it is a problematic problem. And within mathematics I think one is entitled to call it a paradox, or even a family of paradoxes. Richard Gill (talk) 13:10, 8 May 2011 (UTC)

The word problem is also used in maths education for a classic mind-bending exercise which is given again and again to every new generation of students. A good problem remains a challenge to every new generation of students. There is no implication that the problem is still unsolved. Then we would say unsolved problem and someone would offer a prize of one million Euro's for the first correct solution. Richard Gill (talk) 13:13, 8 May 2011 (UTC)

Popularly, a puzzle tends to have a solution in the sense that everyone agrees with the solution once they see it. I think this means that the two envelopes problem, and the Monty Hall problem, are not puzzles, even though they will always puzzle many people. Since people with different points of view will actually see completely different solutions, they sure are problems, if not paradoxes. (A paradox need not have a unique resolution - there can be different ways to resolve a paradox.) Richard Gill (talk) 13:27, 8 May 2011 (UTC)

Thanks.

FWIW, this article is in Categories that are new to me: Category: Probability theory paradoxes; Category: Decision theory paradoxes; Category: Named probability problems. Above I should have mentioned Categories along with the Puzzles template and the List of paradoxes.--P64 (talk) 17:23, 9 May 2011 (UTC)

Proposed structure for the article

I drafted a number of small sections containing alternately a resolution of the paradox followed by a modification of the original problem which brings the paradox to life again. From a few hours reading and Googling "two envelope problem" I can assure you that everything which I wrote there is contained in utterly reliable sources. I'm sorry that I did not yet add specific citations, whether section by section or sentence by sentence. I prefer to draft prose first using mathematical truth and an overall plan as editorial guide, and add the references later, as needed, so as to disturb the coherence of the complete story as little as possible. And maybe some fellow editors know the literature better than me (I still have to get hold of the papers of Falk and of McDonnald and Abbot, which are quite recent and quite authoritative, it seems).

How do people like this way of structuring the material?

The last section containing the details of the beautiful example which still confounds some mathematical economists (the ones who want to allow infinite expected utility in their theory) needed minor rewording, since it was already mentioned in my build-up / overview from paradox - resolution - new paradox - new resolution ... Richard Gill (talk) 17:45, 9 May 2011 (UTC)

You forgot to tag your sections. iNic (talk) 22:40, 9 May 2011 (UTC)

Sorry. To err is human, to forgive is divine. Anyway they are not "my" sections anymore but, maybe, "our" sections. I like "your" Smulyian addition. Richard Gill (talk) 05:01, 10 May 2011 (UTC)

PS can you add literature references to the articles of Smulyian and to Chase? Richard Gill (talk) 06:05, 10 May 2011 (UTC)

James Chase. The non-probabilistic two envelope paradox. Analysis (2002) 62(2): 157-160. Another article you have to pay for to be able to read. Anyone have a pdf for me? Richard Gill (talk) 06:14, 10 May 2011 (UTC)

I think the article is making good progress. It exhibits a branching family of related puzzles, problems and paradoxes, interesting for different reasons in different fields (philosophy, mathematics, probability, economics, psychology, mathematics education, statistics). Clearly there is a big task ahead to give good literature references wherever what is written is technical or not immediately obvious. I am still hampered by lack of access to some of the key publications and my time the coming weeks is severely limited.

The various lists of publications and links at the bottom of the article is in poor shape. There are strange duplications and important omissions. Richard Gill (talk) 10:27, 10 May 2011 (UTC)

Exchange paradox

I do not see the case for covering the so-called Exchange paradox separately. Merge with this article has been proposed four years and one year ago, without consensus. Now I suggest replacement by "Redirect" to this article --after checking its content for independent value, so maybe I should say Merge effected here. See Talk: Exchange paradox.

The first four sources identified at Exchange paradox (footnotes 1-4) are NOT identified here in the dozens of Notes and References, Further reading, and External links. For your reference I have improved those four citations in my new sandbox. --P64 (talk) 16:21, 10 May 2011 (UTC)

Please just redirect that page to this page. I tried to do it in 2007 when that page was new but the creator of the page refused to accept that. There should not be more than one WP article about this problem. iNic (talk) 22:39, 10 May 2011 (UTC)

Later yesterday I posted also at Talk: Necktie paradox, noting the short life expectancy for Exchange paradox as a separate article and wondering about the future of Necktie paradox, which is a cousin rather than a sibling. One distinct Necktie reference is listed in the Sources section below.--P64 (talk) 13:33, 11 May 2011 (UTC)

Three level paradox

Actually there are a number of levels to the paradox.

The argument which leads to infinitely often switching says that because we know nothing about the amount of money in the first envelope, whatever it may be, the other envelope is equally likely to contain half or double that amount. I wrote out the elementary argument, which iNic deleted because he didn't care to look for it in the published literature, that no probability distribution of the smaller amount of money can have the property that given the amount in a randomly chosen envelope, the other envelope is equally likely to have half of double the amount.

Firstly, by a very simple argument, such a probability distribution would have to give probability to arbitrarily large amounts of money. For pragmatists this is enough to disqualify the argument. There must be an upper limit to the amount of money in the envelopes.

Secondly, by a slightly more complicated argument (using the definition of conditional probability and one line of algebra) it follows that the probability distribution of the smaller amount of money must be improper. For many applied mathematicians this is also enough to disqualify the argument. Whether one is a subjectivist or a frequentist, all probability distributions are proper (have total mass 1). At least: conventionally this is so, in mainstream probability and statistics. (There is a school of decision theory where we do not demand the countable additivity assumption. And again in decision theory, if you want to have nice theorems, e.g. that the class of admissable decision rules is equal to the class of Bayes rules, you have to add Bayes rules based on improper priors).

However this is not satisfactory to all. There are many occasions where the use of improper priors as a way to approximate total ignorance leads to reasonable results in decision theory and in statistics (though there are also dangers and other paradoxes involved). More seriously, a slight modification to the original problem brings us right back to the original paradox without use of an improper prior (though we still need distributions with no finite upper limit).

Let's look for examples and show how easy they are to generate. Let's drop the requirement that one of the envelope contains twice the amount of money as the other. Let's suppose we just have two envelopes within which are two pieces of paper on which are written two different, positive, numbers. Call the lower amount A, the larger amount A+B. So A and B are also positive and I think of them as random variables, whether from a subjectivist or a frequentist viewpoint makes no difference to the mathematics.

We choose an envelope at random. Call the number in that envelope X and the other Y. Is it possible that $E(Y|X=x)>x$ for all x? Well, if $E(Y|X)>X$ it follows that $E(Y)>E(X)$ or both are infinite. But by symmetry (the law of $(X,Y)$ is equal to the law of $(Y,X)$) if must be the case that $E(Y)=E(X)$. Hence we must have $E(Y)=E(X)=\infty$.

So to get our paradox we do need to have infinite expectation values. Let's first just show that this is easy to arrange - I mean, in combination with $E(Y|X=x)>x$ for all x. Let the smaller number A have a continuous distribution with positive density on the whole real line, and let B, the difference between the larger and the smaller, also have a continuous distribution with positive density on the whole real line. Let's see what happens if we take A and B to be independent, with $E(B)=\infty$. A simple calculation shows directly that in this case, $E(Y|X=x)=\infty >x$ for all x.

So there exist examples a-plenty once we drop the assumption that the two numbers differ by a factor 2. How can we resolve the paradox?

Again some pragmatists will be happy just to see that the paradox requires not only unbounded random variables but even infinite expectation values.

But I would say to the pragamatists that though you might argue that such distributions can't occur exactly in nature, they do occur by a good approximation all over nature - just read Mandelbrot's book on fractals. Moverover as a matter of mathematical convenience it would be very unpleasant if we were forbidden to ever work with probability distributions on unbounded domains. For instance what about the standard normal distribution? And infinite expectations are not weird at all - what about the mean of the absolute value of the ratio of two independent standard normal variables? Not a particularly exotic thing. The absolute t-distribution on one degree of freedom.

Fortunately there are several ways to show why also idealists (non-pragmatists) need not be upset. In particular, we do need not switch envelopes indefinitely!

By symmetry, $E(X)=E(Y)=\infty$. So when we switch envelopes we are switching an envelope with a finite amount of money for an envelope with a finite amount of money, whose expectation value is infinite, given the number in the first envelope. But the number in the first envelope also has an infinite expectation value and in fact they two have the same marginal distribution (both with infinite expectation) and we are simply exchanging infinity for infinity. We can do that infinitely many times if we like, nothing changes. Of course an finite number is less than infinity. It's not surprising and it's not a reason to switch envelopes.

Suppose we actually looked at the number in the first envelope, suppose it was x. Should we still switch? The fact that the conditional expectation of the contents of the second envelope is infinite actually only tells us, that if we were offered many, many pairs of envelopes, and we restricted attention to those occasions on which the first envelope contained the number x, the average of the contents of the second envelope would converge to infinity as we were given more and more pairs of envelopes. We are comparing the actual contents of one envelope with the average of the contents of many. The comparison is not very interesting, if we are only playing the game once. (As Keynes famously said in his Bayesian critique of frequentist statistics, "in the long run we are all dead"). Now, the larger x is, the smaller the chance (since we are not allowing improper distributions) that the contents of the second envelope will exceed it. If we were allowed to take home the average of many second envelopes, the larger the first envelope the larger the number of "second envelopes" we would want to average, till we have a decent chance of doing better on exchanging.

So, on the one hand, infinitely switching is harmless, since each time we switch we are equally well off, if we don't look in the first envelope first. On the other hand, if we do look in the first envelope and see x, and we're interested in the possibility of getting a bigger amount of money in the other, we shouldn't switch if x is too large.

This is where the randomized solution of the variant problem comes in. If we are told nothing at all, and only use a deterministic strategy, there is no way we can decide to switch or stay on seeing the number X=x in the first envelope that guarantees us a bigger chance than 1/2 of ending up with the larger number. For any strategy we can think of, the person who offers us the envelopes can choose the numbers in the two envelopes in such a way that our strategy causes us to have a bigger chance of ending with the smaller number. However if we are allowed to use a randomized strategy then we can get the bigger number with probability bigger than 1/2, by using the random probe method. Choose a random number with positive density on the whol real line and compare it to x. When our probe lies between the two numbers in the two envelopes we'll be led to the good envelope. When it lies above both, or below both, we'll end up either with the second envelope, or the first envelope. But given the two numbers in the two envelopes, it is equally likely that the first is the smallest, as that the second is the smallest.

I hope this is all written down somewhere in the literature. (I believe it is all correct). But if not, I am happy to oblige, and get it peer reviewed so as to check the correctness, and then others may cite, if they find it useful and interesting. Richard Gill (talk)

PS if someone wants to move this material to the "Arguments" page I hope they will just go ahead and do so. I assume that the self-appointend guardian of these pages, @iNic, already moved my elegant analysis of the "equally likely double or half" assumption there too (others may well want to read it, even if he doesn't). Richard Gill (talk) 12:44, 8 May 2011 (UTC)

Sorry iNic, you did move it to the Arguements page, that's fine.

I am gathering my "original" contributions to the two envelope problem on my personal talk page, [1]. BTW I do not believe for a moment that my analyses are "original". This is not rocket science. They do not represent a personal Point of View either. Just routine analysis of the problem which any mathematician should be able to do after following Probability 101. Richard Gill (talk) 13:06, 8 May 2011 (UTC)

After writing this all up (partly reacting to comments by other editors here) I Googled "two envelopes problem" and read the first 20 hits which were mainly to semi-popular articles by professional mathematicians. They made a lot of sense, they were coherent, there was no controversy, no mutual contradictions, or unsolved problems. I did not discover anything that was not covered by my notes. Everything I have "invented" myself the last couple of days could be found in the literature. Of course: everything interesting which can be said about the two envelopes problem is common knowledge among the professionals, it's part of our folk-lore.

From the point of view of probability theory (whether interpreted in a Bayesian or frequentist way) there is absolutely nothing "unsolved" about the problem. The facts are known, they are not disputed.

It's a fact that different people like to "resolve" the paradox in different ways (some more mathematical, some more pragmatic). The chain of reasoning leading to the "switching for ever" conclusion is incorrect. You can resolve the paradox by pointing out the error in the reasoning. You can stop right there. Or, you can create a new paradox by altering the problem, and avoiding the earlier error, and coming up with a new argument which appears to lead to a similar crazy answer. You can fix the new paradox (either by pointing out a hole in the argument or by showing that the answer is not as crazy as it appears). And so on... So there is a branching family tree of related two envelope problems and different people might like to follow different paths through the tree before they are satisfied that there is only, at the end of the day, a paradox - an apparent contradiction, not a real one.

In this sense the two envelopes problem is never solved since it always remains a matter of personal taste which path to take through the branching family tree of paradoxes and resolutions and new paradoxes. So there will be never be a consensus how the problem should be solved. But this does not make it an unsolved problem, as is claimed in the first line of the article!

In the field of decision theory and mathematical economics there is debate about the meaningfulness of infinite expectations. The orthodox point of view is that we are interested in utility, not money; the utility of money is not linear. At some point having even more money gives us no further utility. Utility is always bounded. The switching paradox can only arise if we allow, not only for infinite utility, but even for infinite expected utility. Hence the paradox is simply not interesting, not relevant, to decision and utility theory.

I noticed a couple of very woolly articles by philosophers who clearly were not up to the mathematics and hence stayed in the "going round in circles" stage when one is not able to analyse the problem using appropriate tools and language.

Gill, can you please reveal which articles you think are the "very woolly" ones? What if others think that they are the good ones while dislike some of the ones you like? Please read (again) the NPOV policy of WP. iNic (talk) 22:37, 9 May 2011 (UTC)

Sorry iNic, I didn't notice your question, and now I have forgotten them. You may be sure these were not major articles in top philosophy journals. They were articles on blogs and newsletters. As you know, there must be thousands of such. Richard Gill (talk) 18:09, 6 July 2011 (UTC)

You said: what if others think they are good ones but dislike the ones I like? That would be great. Then they can add to the page and we can discuss why they are good, despite my inital impression. They can explain what "my sources" are overlooking. Collaborative editing. Good faith. We all learn from one another. We need experts from philosophy who can make sensible summaries of the philosophy literature, just as we need experts from mathematics who can make sensible summaries of the mathematical literature. Both can and should (constructively) criticize one another. Laypersons can criticize and improve both. By the way, the Wikipedia guidelines on verifiability say that the first thing to do when you come across unreferenced material is to add references yourself, not to delete. The second thing to do is to flag. The third is to delete. Richard Gill (talk) 07:16, 7 July 2011 (UTC)

It was nice to find out that the random probe method was actually invented by Thomas M. Cover, the famous information theorist, quite a few years before the two Australians wrote it up in the Proceedings of the Royal Society. He told it to them and they didn't believe it. But after 10 years it finally sank in, and then they wrote and published their paper and became briefly famous. In the meantime Tom had told his solution to many other people (I heard it along the grape-vine, very soon after Cover invented it, quite independent of the Australians). Pity I didn't write it up for a prestigious journal right then, I could have been as famous as they were. Richard Gill (talk) 09:45, 9 May 2011 (UTC)

Excellent job. This article has been a total mess for way too long. Tomixdf (talk) 11:47, 9 May 2011 (UTC)

Sources

By "Sources" I mean all of those identified in the sections now called "Notes and references", "Further reading", and "External links" AND those which are missing (per Richard Gill above, 10:27).

See my new sandbox for improved citations of the four distinct sources given for the "Exchange paradox", whose article should be deleted eventually (per P64 above, 16:21).

Here I have separately put the Further reading and External links in alphabetical order by surname --while tweaking a few entries. Along the way I deleted the one dead link to a presumably unpublished paper,

• http://www.public.iastate.edu/~dfaden/philosophy/two_envelopes/resolved.pdf —inactive

Unfortunately the Internet Archive missed it (~dfaden/philosophy/). <=That link will be valuable for anyone who needs introduction to the Internet Archive. --P64 (talk) 17:28, 10 May 2011 (UTC)

Necktie paradox closes by calling that "a rephrasing of the simplest case of the two envelopes problem". That article gives one citation,

• Aaron C. Brown. "Neckties, wallets, and money for nothing". Journal of Recreational Mathematics 27.2 (1995): 116–22.

That was evidently identified by its title and abstract, probably not used as a source. --P64 (talk) 18:21, 10 May 2011 (UTC)

Richard Gill, it may be useful to maintain an explicit wish list for copies of sources as one section of this Talk.--P64 (talk) 18:28, 10 May 2011 (UTC)

The list of sources on the page were very long some years ago (however far from a complete list). Too long some editors thought (fought over? -RDG Pls check the version history of the page /iNic) and deleted most of the sources. I propose that we this time create and maintain a separate page containing a complete list of all sources. (I propose further that we list the sources at that page in chronological order as it has several benefits: 1. It will be easy for readers new to the subject to follow the development of ideas. 2. The list will be very easy to maintain and keep up to date for editors as new published sources simply are added at the end. 3. Persons familiar to the subject can with a glance at that page see if something new has been published that they haven't read yet.) Sources on the main page should be restricted to the sources actually used as references in tha article. iNic (talk) 23:10, 10 May 2011 (UTC)

I have a dropbox folder (www.dropbox.com) of pdf's of the important papers which are not freely available on internet. I'll gladly invite active editors to share it. Richard Gill (talk) 08:17, 12 May 2011 (UTC)

I am not so sure now about the sense of creating a complete list of "all" sources. There are so many unpublished manuscripts. If articles never got published it might be because they're not much good. But if someone else is prepared to create and maintain this list, I'll certainly be glad of its existence. @iNic, you say that sources on the main page should be restricted to sources actually used as references in the article. I think that is a wise suggestion. Are suggesting deleting the sections "further reading" and "external links"? Richard Gill (talk) 18:48, 6 July 2011 (UTC)

Non probabilistic version

I have completely rewritten the section on the Smullyan variant of the problem. I did this, by the way, after collecting and studying a large amount of literature and communication with experts from logic ... I selected just a couple of sources for mention in the text. I tried to put the message(s) of this very technical literature across in a non-technical way. Did I succeed? If not, please improve! (I have pdf's of all these many sources, if anyone wants to see them). Richard Gill (talk) 09:22, 23 June 2011 (UTC)

Please list here all printed sources you have read regarding this variant. iNic (talk) 00:19, 25 June 2011 (UTC)

That will take a while. In the meantime, anyone is welcome to join my Dropbox.

There are at least three completely different two envelope problems. With and without opening the first envelope, with and without probability. They all have numerous very easy solutions, at least, easy in their own terms. And they have different easy solutions for readers with different backgrounds. Probabilists, economists, logicians all see different issues. A layperson can "defuse" the paradox (es) by common sense, but is not equipped to even see what the different academics are on about. Should the layman care? Probably not. Should the academic professional care that the layman doesn't care? Probably not. If the academics failed in the past to communicate their concerns to the laypersons, we can hardly expect wikipedia editors to be able to do it now. The literature is vast and complex. A big challenge to an encyopediist who has to write both for laypersons and for *all* the academic communities. Solution: collaborative editing. It requires that *all *participants recogniise the "relativity" and hence incompleteness of their own point if view Richard Gill (talk) 09:50, 27 June 2011 (UTC)

By the way, when studying the Smullyan version "without probability" it is important to read Smullyan himself. Some of the authors above rephrase his words ever so slightly, in a way which actually makes a difference to a philosopher-logician analyzing ever single word with care and attention. Richard Gill (talk) 08:06, 28 June 2011 (UTC)

Completed reorganization of material

I moved two "orphaned" sections at the bottom of the article to better locations, I think; now they are subsections of sections on the same material. This concerned James Chase's example of a proper distribution for which the expectation paradox holds, and the material on infinite expected utility in the foundations of mathematical economics. I also moved some references from section titles to the body of the text, corrected a mathematical derivation within the Chase example, and added the proof that the equal conditional probabilities implies an improper probability distribution.

Concerning this proof - I'm looking forward to iNic's reaction [What reaction? iNic (talk) 11:14, 8 July 2011 (UTC)] - Blachman, Christensen and Utts (BCU) deal separately with discrete dstributions and continuous distributions, and, writing for probabilists and statisticians in a brief "letter to the editor" correcting a big mistake in CU's earlier paper -, give very few details. My analysis can be seen as a synthesis - it combines their continuous and discrete "solutions" in one - but I am not using this synthesis to promote any new opinions or to promote Own Research (no Conflict of Interest). My only aim is to explain BCU's results as simply as possible so that the wikipedia article is as accessible as possible to the largest number of readers. The alternative would be a larger list of partial results, citing more articles, but all with the same message, no contradictions. I think it would not serve the reader as well. The message is that Step 6 requires an improper distribution and, if continuous, it has to look like the continuous density 1/x on the whole line, which integrates to infinity both at zero and at infinity. Or, if discrete, like the uniform distribution on all (negative and positive) powers of 2. In both cases we have a uniform distribution over the logarithm of the amount of money, from arbitrarily small to arbitrarily large. Discrete uniform in one case, continuous uniform in the other. The key equation 2p(n)=p(n)+p(n-1) and how to solve it can be found in their article and in many others too, both in proofs of the continuous case and the discrete case. Also the idea of splitting up the whole line in powers of 2 can be found all over the place. The point is that halving or doubling always takes you from one interval to one of its neighbors. We might as well "round down" the two amounts of money by replacing them with powers of 2. Write them in binary and keep only the leading 1, replace all succeeding binary digits by zero's.

The general theorem would be as follows (but this is OR!): change to currency units such that there's positive probability the smaller amount is between 1 and 2 monetary units. Now "copy" the distribution within this interval, which could be anything whatsoever, also to the interval from 2 to 4 (you have to stretch it by a factor 2), and to the interval from 4 to 8, ... And to that from 1/2 to 1 (compressing by factors of 2).... Now glue all these probability distributions together, infinitely many of them, giving each one of them equal weight. This is what you believe about the smaller amount of money, if you truly believe that the other envelope is equally likely the smaller or the larger, independently of what's in yours! Clearly ludicrous.

I think the body of the article is much cleaner now and gives a fair picture of the literature. The next thing to do, I think, is to make sure that all useful and/or notable references are actually cited so that separate sections on "other reading" and "external links" can be deleted. It would be useful however to collect a list of all the references which we have made use of, even if not citing or recommending them, on a separate page, as iNic proposed. Maybe on the "Arguments" page?

Also it's time to merge exchange paradox with this page. Richard Gill (talk) 09:44, 7 July 2011 (UTC)

Multiple issues tag

I have removed this tag as it seems no longer applicable. If anyone has any issues perhaps they could say here what they are. Martin Hogbin (talk) 08:55, 7 July 2011 (UTC)

Great! Good to see you back here, Martin. Richard Gill (talk) 09:20, 7 July 2011 (UTC)

This tag was added by editors not because the article originally had a lot of issues, but because they didn't like the content. So I'm very happy it's removed now! iNic (talk) 11:21, 8 July 2011 (UTC)