Talk:Two envelopes problem

First resolution

I have just noticed that a first resolution has been added This seems to me to be complete nonsense. For example it says:

Each of these steps treats A as a random variable, assigning a different value to it in each possible case. However, step 7 continues to use A as if it is a fixed variable

This makes no sense, and what is a fixed variable? Martin Hogbin (talk) 19:49, 26 January 2012 (UTC)

I did not write it, and do not like it much, but many sources present such an explanation. A mathematical variable stands for any value you like (within some perhaps only implicit domain). If it is mentioned several times it might be relevant to know whether the writer is intending the same value to be substituted at all instances, or not. I think this is what is intended by "fixed".

Language of random variables offers more opportunity for confusion. In mathematics the usual formalization is actually to represent a random variable by a function. The argument of the function is the particular instance. The value of the function is the value which the random variable has in that particular instance. One can represent the value of a random variable with a (mathematical) variable; standard notation is capital letters for random variables, lower case for a mathematical variable representing a possible value.

Amateurs only have a rough idea of mathematical language. Philosophers and logicians have their own language. They also talk about variables, are vague about random variables. One must distinguish free and bound variables. It's like "scope" in computer languages. And one must take account of the so-called quantifiers and their scope: "for all" or "there exists".

Perhaps we should check the cited article, Bruss (1996) I think. Maybe Falk's latest paper is better. Richard Gill (talk) 08:24, 27 January 2012 (UTC)

I cannot believe that any reliable source uses the language in the article but I will check. It is just plain wrong.

I think you agree that I have covered this kind of confusion on my two envelopes page. If A is meant to have a fixed value then it is a constant and the resolution does not need to wait until step 7; step 1 is wrong as a constant cannot take on more than one value.

If A is meant to be a random variable then the resolution depends what we think the writer us trying to calculate. If it is the unconditional expectation in the other envelope then step 7 is wrong, if it is the conditional expectation then step 6 cannot be justified as it is not valid for every possible value that might be in the original envelope (for all finite envelope spaces). Martin Hogbin (talk) 09:44, 27 January 2012 (UTC)

I think the "resolution" is the author's way of saying what you say, in the case A is a random variable and an unconditional expectation is being taken. Unfortunately the writer is a layperson or philosopher who works with probabilistic concepts at a purely intuitive level and without using (modern) standard probability language. The author of this solution moreover thinks that the writer of the TEP argument is similarly simplistic.

It's the blind trying to lead the seeing, thinking they are also blind, by ignorance of anything ese than blindness. Richard Gill (talk) 16:01, 27 January 2012 (UTC)

PS This phenomenon is the curse of TEP, as it was of MHP. How can the seeing explain what they see to the blind? Why should the blind believe them? That's why the literature is such a mess. Richard Gill (talk) 16:05, 27 January 2012 (UTC)

So do you agree that the first resolution in the article should go? Martin Hogbin (talk) 09:24, 31 January 2012 (UTC)

I think it should be completely rewritten, staying as close as possible to one of the reliable sources, and referring to which source it is using, too. Richard Gill (talk) 17:08, 31 January 2012 (UTC)

Please go ahead and rewrite, but remember to keep it short. iNic (talk) 17:16, 31 January 2012 (UTC)

Please do not delete the first resolution. iNic (talk) 09:38, 31 January 2012 (UTC)

iNic, there is no consensus for what you have written, see Richard's comments below.Martin Hogbin (talk) 15:50, 31 January 2012 (UTC)

For me, the first resolution, as presently in the article, is unintelligible (though I know where it is coming from). I hope someone will rewrite it so that it makes a bit more sense. I would recommend whoever does that to follow a particular well-chosen reliable source as closely as possible, and moreover to say which source they are copying the resolution from.

In my opinion this *kind* of resolution is just about as bad as the original argument. What is random, what is not? If things are supposed to be random what probability distributions are we using, where do they come from? Are we calculating a conditional or unconditional expectation? Does probability only refer to the labelling of the two envelopes, or also to the contents? If you don't say what assumptions you are making we cannot judge whether your arguments are correct or not.

I admit to having a certain bias: I believe that formal probability calculus was invented in order to help us avoid stupid mistakes when doing probabilistic reasoning, and that TEP is an example of the kind of stupid mistakes in question. Yet a whole load of writers want to solve TEP completely verbally and without reference to any specified probability framework.

People who never learnt any probability will prefer a quick verbal "solution" to a solution after translation into mathematics. But their "solution" will never satisfy people who do know about probability.

On the other hand, once we translate the problem into probability language, we discover (of course) that there exist many possible decent translations, and each one can have a different resolution. And every version does have a resolution, in the sense that one can determine where the logic went astray. Richard Gill (talk) 13:15, 31 January 2012 (UTC)

Richard, you say In my opinion this *kind* of resolution is just about as bad as the original argument. What is random, what is not? I agree completely. We are supposed to be writing an encyclopedia to inform the general public and experts alike. This 'resolution' is suitable for neither.

It is superficially like Falk and Nickerson's answer to their version 4 but there the authors are not trying to resolve the paradoxical line of reasoning given at the start of this article but they are showing in which cases either player should swap. They give a perfectly good reason why neither player should swap in version 4, but this is something we knew at the start and that is quite obvious by symmetry.

The main problem with the proposed resolution, if you can make any sense of it at all, is that it is far from clear why it would not apply equally to F&N's version 1, where the argument for Ali to swap is perfectly valid. Martin Hogbin (talk) 15:50, 31 January 2012 (UTC)

It is totally irrelevant if you like the proposed resolution or not. It is likewise totally irrelevant if you think the proposed solution is correct or not. It is even totally irrelevant if the proposed solution is actually true or false. What matters is if the solution is represented in "reliable sources," which in this context means if it has been proposed in papers published in peer reviewed journals. And it has. Many times. iNic (talk) 17:16, 31 January 2012 (UTC)

Can you show me where. I cannot find it in any reliable source. In any case, we should use a reliable secondary source to decide which resolutions we use. Martin Hogbin (talk) 17:59, 31 January 2012 (UTC)

Which sources have you read? Which secondary sources have you found? iNic (talk) 02:22, 1 February 2012 (UTC)

Bearing in mind that you are the only one who wants to include an unintelligible resolution it is up to you to show that it is supported by sources. Please do this or I will delete the section. Martin Hogbin (talk) 09:21, 1 February 2012 (UTC)

I'm not the only one who has read the sources. What's the problem with the sources referred to now? This is also the usual way to solve the Necktie problem which is the forerunner to TEP. So it very much deserves to be included here and also be mentioned as the "first" resolution. You are not entitled to delete everything at Wikipedia that you don't understand. That would be devastating. If you do that again I will report you for vandalism. iNic (talk) 09:46, 1 February 2012 (UTC)

This is not the usual way to solve the necktie problem. The way to solve the necktie problem is to notice that the expected price of the other's necktie is different when it is larger, from when it is smaller. If the probability distribution of X, Y is symmetric under exchange, then E(Y | Y > X) > E(Y | Y < X). All these problems are solved by using careful probability notation and easy facts from probability theory. The "equivocation" is in thinking that the amount in envelope A has the same probability distribution when it is the smaller of the two, than when it is the larger of the two. There is no equivocation in using the same symbol to denote the amount in envelope A in two complementary situations. Rawling doesn't know what he's talking about. Richard Gill (talk) 20:49, 7 February 2012 (UTC)

Be my guest. Deleting unsupported material for which there is no consensus in not vandalism. Please show me which source supports this claimed resolution. Martin Hogbin (talk) 21:16, 1 February 2012 (UTC)

Second revert now. You have to tell us what is wrong with the references in the section you keep deleting. Are they wrong? You need more references? iNic (talk) 12:04, 3 February 2012 (UTC)

What is wrong is that the references do not say what you have written. All I have asked you to do is to show me where in the cited references I can find your resolution. If you cannot do this I will delete it. Martin Hogbin (talk) 16:26, 3 February 2012 (UTC)

If you think it's badly written then please say so and we work out a better wording. Badly written is not a case for deletion. Our job as editors in this case is to summarize the common features of basically the same idea that has popped up in the literature many times. Can we agree that Falk (2008) and Rawling (1994) are good representatives for holding and explaining this view? If you want to include more sources please tell me which you have in mind. Let's only take the sources you think are the best. iNic (talk) 17:12, 3 February 2012 (UTC)

Tell me where in the above papers your proposed resolution is presented. What we have now is your OR. Martin Hogbin (talk) 09:57, 4 February 2012 (UTC)

Please read the whole papers! If we want to quote something I suggest something from the first paragraph p 87 in Falk or the second paragraph p 100 in Rawling. What do you suggest? iNic (talk) 11:04, 4 February 2012 (UTC)

The Falk paragraph you cite starts: The assertion in no. 6 (based on the notation of no. 1) can be paraphrased ‘whatever the amount in my envelope, the other envelope contains twice that amount with probability 1/2 and half that amount with probability 1/2’. The expected value of the money in the other envelope (no. 7) results from this statement and leads to the paradox. The fault is in the word whatever, which is equivalent to ‘for every A’. This is essentially the second resolution shown in the article and bears no relation to what you have written. Martin Hogbin (talk) 17:11, 4 February 2012 (UTC)

Rowling says: The first occurrence of 'contents of envelope' denotes $8, whilst the second denotes $2; and this is to commit the cardinal sin of algebraic equivocation - two occurrences of the same denoting term in the same equation must denote the same object., It is a natural language, and therefore rather vague, statement, your contribution. Martin Hogbin (talk) 17:15, 4 February 2012 (UTC)

The rest of the Falk paragraph you cite above reveals that it's not the second resolution Falk has in mind here: This is wrong because the other envelope contains twice my amount only if mine is the smaller amount; conversely, it contains half my amount only if mine is the larger amount. Hence each of the two terms in the formula in no. 7 applies to another value, yet both are denoted by A. In Rawling’s (1994) words, in doing so one commits the ‘cardinal sin of algebraic equivocation’ (p. 100). I think the current article has a fair account of their common idea: In the first term A is the smaller amount while in the second term A is the larger amount. To mix different instances of a variable or parameter in the same formula like this shouldn't be legitimate, so step 7 is thus the proposed cause of the paradox. We should include Rawling (1994) as a reference here, and maybe also a quote from his paper, What do you want to add or change? iNic (talk) 23:47, 4 February 2012 (UTC)

I think the problem with the first resolution (which historically, is not first at all) is that it doesn't make any sense at all, whether in Rawlings' or in Falk's words. The following is a true statement: E(B|A=a)=2a Pr(B > A |A=a) + a/2 Pr(B < A|A=a). Notice the terms 2a and a/2? It looks as though we are using a to denote both the amount in Envelope A when it has the smaller amount, as when it has the large amount! Yet this is not a sin at all, in this case.

The first resolution is a typical resolution by amateurs who don't know what they are talking about and whose solution exhibits even less mastery of probabilistic reasoning than the supposed writer of the original paradox.

If we do suppose that the writer is computing an unconditional expectation rather than the much more plausible conditional expectation, then the correct formula would have been E(B) = 1/2 E(2A | B > A) + 1/2 E (A/2 | B < A). The mistake is not using the same symbol twice with different meanings, but forgetting that he still needs to compute E(A | B > A) and E(A | B < A). Just replacing both expressions with A is doubly imbecilic. Especially since it is pretty obvious that A is on average smaller when you are told it is bigger than B, than when you are told it is larger. Richard Gill (talk) 14:10, 5 February 2012 (UTC)

This is the real problem with the Two Envelopes Problem page: about half of the sources are incompetent to write on the topic. So the wikipedia editor is forced to reproduce a load of nonsense. Richard Gill (talk) 14:12, 5 February 2012 (UTC)

As Martin mentioned, the fact that in the Ali Baba problem the same "resolution" appears to show that Ali shouldn't switch, is further proof that it is nonsense. Richard Gill (talk) 14:14, 5 February 2012 (UTC)

We simply can't split the sources into a "reliable" group and an "unreliable" group. We as editors would forever disagree about how to properly make such a split. But we are not here to directly or indirectly give a report on our own personal opinions. Richard, please keep in mind what you have said yourself elsewhere on these talk pages: "Any wikipedia editor's personal opinion is not relevant. We just have to get a decent overview of the actual literature. Some of us editors will be happier reading the academic philosophical literature, some of us will be happier reading the academic mathematical literature, some of us will be happier reading the popular literature. We have to trust one another ("good faith") and all of us have to develop some global understanding of what has been done in the fields where we are less competent." Our job is to as objectively and popular as possible present the most common ideas that have been published. Even if you and Martin hate the "first" idea and think it's rubbish, it's totally wrong to pretend that it doesn't exist in the published literature and simply delete it. Many many of the sources mentions this idea, even those that think the real solution is somewhere else.

Regarding the labeling "Second variant" and "Third variant" I agree that it's not optimal. It can give the false impression of an intended chronology. In the early days of this page the second variant was called "A harder problem", the third variant "An even harder problem" and the non probabilistic variant "The hardest problem". I thought that was quite good and made the subject more intriguing and the page more fun to read. Unfortunately some other editors objected to this after some years. They were at that time replaced with the more neutral but also more dull labels "second," "third," and "non probabilistic." iNic (talk) 00:25, 6 February 2012 (UTC)

I agree that *we* can't split sources into a reliable group and an unreliable group. It seems to me that I can better spend my time at the moment completing my own paper (actually doing a total rewrite) and hope that some time in the future it might be useful for wikipedia editors. While doing that I'll reread the philosophy literature and who knows maybe find a better version of the first solution.

But anyway, regarding the wikipedia article, I would prefer to put the second resolution first and save the first resolution for later in the article. The only papers which fairly neutrally survey a whole range of solutions are those of Nalebuff (1989), and of Nickerson and Falk (2006). Both of them focus on the second resolution, or rather, the second interpretation. Moreover, Nalebuff (next to Martin Gardner) is where it all started, and he knows the prehistory too.

What is presently the first resolution is certainly lighter reading but since it is evidently wrong on at two counts (1: there is nothing necessarily wrong with having the same symbol denote two different things in the same expression cf. my example, 2: it tells us that the argument that Ali should switch in the Al Baba problem is wrong, yet in that problem, Ali should switch) and rather inadequate on a third (3: it misses the more important mixup, namely between values of random variables and expectation values) I think we are foolish to highlight it. I see it more as an intermediate step to the two envelopes problem without probability. Richard Gill (talk) 17:43, 7 February 2012 (UTC)

I look forward to read your rewritten paper! (I'm writing my own now as well.)

To hide away the "first" solution in the article because you and Martin doesn't like it is not a very clever idea. This first idea is a very common idea throughout the history of TEP. The Spanish Inquisition could ban an idea they found intellectually offensive and pretend it never had existed. Today, this is generally regarded as a somewhat old fashioned way to handle views you can't accept. iNic (talk) 03:36, 8 February 2012 (UTC)

I have moved the disputed section here for discussion. Martin Hogbin (talk) 18:10, 6 February 2012 (UTC)

I have reverted your vandalism a third time now. iNic (talk) 00:31, 7 February 2012 (UTC)

I think the Rawling paper is incomprehensible. In fact, as far as I am concerned, it is mostly nonsense. It is a research paper in philosophy presenting what is claimed to be original research. It is therefore a so-called primary source. Wikipedia articles should primarily use as references tertiary sources. Standard textbooks, neutral review articles. Accepted knowledge.

I don't yet have the Bruss paper. It seems to me that only the latest Falk paper is a reasonable source for this solution. But then we should copy her solution, not change it so as to be unrecognisable. Richard Gill (talk) 20:33, 7 February 2012 (UTC)

We have to face it that there are no well written, unbiased and complete history or overview about this problem. In some rare cases the first pages in a research paper can be devoted to a short summary of the history. But these texts are never neutral or complete. There is no "accepted knowledge" regarding TEP in the form of an "accepted solution". The only accepted knowledge about TEP is that the discussion about how to properly solve it is still ongoing. iNic (talk) 03:36, 8 February 2012 (UTC)

It would be nice if the solution bore some similarity to the solution in Falk's paper. As it is it is iNic's home brew. Perhaps we should have a 'philosophical solutions' section.Martin Hogbin (talk) 20:52, 7 February 2012 (UTC)

OK so you will stop deleting the first section completely from now on? Great news in that case! Now when you apparently can read one paper how would you like to improve the home brew? By the way, I hate to make you disappointed but this section is not written by me at all. I copied it from an earlier version of this page.

It's pointless to have a "philosophy section" as everything is philosophy. The entire page would be included in such a section. iNic (talk) 03:36, 8 February 2012 (UTC)

It's your personal point of view, iNic, that everything (or everything in TEP?) is philosophy. I think that the list of sources themselves (especially when you check the academic status of the journals) show that TEP is not a notable topic in philosophy at all, but do show that it is a notable topic in mathematics education and communication (and specifically in education on probability and statistics) and in mathematical recreation. That is moreover where it started. That is where the big overview papers are written by well known authorities which survey many solutions and aspects of the problem and make some synthesis of everything which has gone before. By the way, the Stanford online philosophy encyclopedia does not have an article on two envelopes paradox.

Two envelopes problem is first and foremost a trick question in probability theory. The first solution which the article should give is the solution now called variant 2. It is not a variant, it is the original. The interpretation of the problem is the original interpretation, the solution is the original solution. That same solution has been confirmed by writer after writer. Too bad that the philosophers were not competent enough in mathematics (elementary probability theory) to understand it. Richard Gill (talk) 18:31, 8 February 2012 (UTC)

This is your own personal point of view, Richard. Which is obviously wrong. Too bad that you are not competent enough in philosophy to understand it. iNic (talk) 10:52, 10 February 2012 (UTC)

Proposed resolution

The most common way to explain the paradox is to observe that A isn't a constant in the expected value calculation, step 7 above. In the first term A is the smaller amount while in the second term A is the larger amount. To mix different instances of a variable or parameter in the same formula like this shouldn't be legitimate, so step 7 is thus the proposed cause of the paradox. For example, if we denote the lower of the two amounts by C we can write the expected value calculation as

${1 \over 2} C + {1 \over 2} 2C = {3 \over 2}C$

Here C is a constant throughout the calculation and we learn that 1.5C is the average expected value in either of the envelopes. So according to this new calculation there is no contradiction between the decisions to keep or to swap, and hence no need to swap indefinitely.

The preceding resolution was first noted by Bruss in 1996, and later explored, together with many other resolutions, in an exhaustive paper by Nickerson and Falk in 2006. A concise exposition is given by Falk in 2009. It is especially popular in the philosophy literature.

However, this resolution depends on a particular interpretation of what the writer of the argument is trying to calculate: namely, it assumes he is after the (unconditional) expectation value of what's in Envelope B. In the mathematical literature on Two Envelopes Problem, another interpretation is more common, involving the conditional expectation value (conditional on what might be in Envelope A), to which we now turn.

Not in any source

iNic this is not a question of one source vs another. You have not managed to show where the above resolution is presented in any source.

I for sure have. This "discussion" is extremely silly. Do you need new glasses or what? iNic (talk) 00:31, 7 February 2012 (UTC)

You might also like to tell us what sort of quantity 'C' is above. Is it a constant, a random variable or what? Martin Hogbin (talk) 18:06, 6 February 2012 (UTC)

I think Richard can put you into contact with some of the authors of these papers if you need help reading. iNic (talk) 00:31, 7 February 2012 (UTC)

iNic better, and more persuasive, to stick to the subject than to attack the writer. All you have to do is show me where the proposed resolution is given in a reliable source. At the moment you are giving your interpretation.

By the way, I would be happy to play the Broome game with you on the arguments page.Martin Hogbin (talk) 10:25, 7 February 2012 (UTC)

We can't quote all text from all writers that explains a certain idea. As a Wikipedia editor you have to do some creative work when summarizing a lot of text from many sources into a clear and succinct presentation of the idea in question. I have asked you how you would summarize this idea as outlined in two simple sources, but you refuse to answer. Instead all you do is to repeatedly delete the whole section. That is not constructive at all, just a silly game vandals play. I'm not interested in playing any other games with you. iNic (talk) 14:38, 7 February 2012 (UTC)

Renamed sections

In true wikipedia spirit ("be bold") I have changed the names of the sections and edited the text in such a way that though the first variant still comes first, only indisputable statements are made about it, and a critical reader is enticed to continue. I say that it is a common resolution, and that the resolution only says that you must not use the same symbol to denote different things. But I don't say that wikipedia says that this is a sin. As a wikipedia editor, I merely report a statement made in what appear to be reliable sources by wikipedia criteria. It does not thereby become the truth. I think that now, the first variant is reported in a neutral way; peole who are happy with it can stop reading (they'ld be happy with anything!), people who are not happy can read on. The second variant is now called "an alternative interpretation". In fact, it's the original interpretation, but OK, TEP is a part of living culture, hence evolves and mutates and branches, it is not for us wikipedia editors to claim that a particular version is the true version. Richard Gill (talk) 19:57, 8 February 2012 (UTC)

Well done. Martin Hogbin (talk) 13:19, 10 February 2012 (UTC)

Very well said Richard! And if your slight modifications of the text in the first resolution will make Martin so happy that he will stop deleting it I'm more than happy. Also, I notice that you have come to the same conclusion as me, that even if the resolution mentioned first is not historically the first one it is easiest to outline the ideas around this paradox when putting the different variants/interpretations in this order. iNic (talk) 16:39, 10 February 2012 (UTC)

Already in my draft paper I have the two resolutions in this order. Deliberately, in order to start off with what will be a surprise for my readers, who never realized that such an alternative interpretation existed. And to show that my "unified solution", mathematically at least, gives two complementary results which each take care of one of the two interpretations, reproducing (mathematically) the common resolutions to the two interpretations. Our discussions have been immensely valuable in forming what I think is mathematically a new (though modest) synthesis. Richard Gill (talk) 11:11, 11 February 2012 (UTC)

[I have moved the following discussion to the arguments page]

I added some references on the "sources" (Literature) page. In particular it seems that Marilyn Vos Savant has written on TEP, a couple of years after Nalebuff and Gardner. I would like to see her solution.

I noticed that chronologically, "TEP without probability" was also very early. I suspect that this purely logic/semantic version actually led to the philosophy interest and fuelled the bifurcation. Note that it specifically targets the "equivocation" issue. Richard Gill (talk) 11:11, 11 February 2012 (UTC)

Yes, this is why trying to outline the ideas in strict chronological order is not a very good idea. It would lead to an article that would be very hard to comprehend for the intended general reader. iNic (talk) 08:40, 12 February 2012 (UTC)

Remove Section "Extensions"?

I think the article is looking pretty good now, especially for academic readers. I think there is just one big improvement which could be made: delete the material in the section "Extensions" taken from the recent paper by McDonnell and Abbott. I don't find this material particularly notable, and the kind of analysis which these authors have done has been done earlier in many other papers in the more statistical literature. I think it merely deserves a reference as being a recent contribution, on the lines of: "if one looks in the envelope, and if one has a prior distribution over the possible values in the two envelopes, it is possible to compute optimal decision rules, typically of the form "change envelopes if the amount in A is smaller than some critical amount". Richard Gill (talk) 16:43, 12 February 2012 (UTC)

In general the article need to be shorter and easier to read. iNic (talk) 03:31, 13 February 2012 (UTC)

I completely rewrote the section on Extensions and did a little rewriting of the section on Randomized Solutions. I need to add references to the mathematical claims here. Everything I say can be found in various papers in the more statistical/mathematical literature on two envelopes problem. I think all I wrote is "notable" enough to be written up. One test of "notability" is that people do keep independently rediscovering all these results. The counter-intuitive difference between the results for discrete and continuous distributions keeps confusing authors, too. Richard Gill (talk) 18:00, 13 February 2012 (UTC)

A solution which ANYONE can understand

Here is a solution which ANYONE can understand - certainly easier and more acceptable than any of the solutions proposed here: OK - two envelopes, A and B, one of which contains x dollars and the other contains 2x dollars. You get to pick either envelope and keep its contents, or exchange it and keep the contents of the 2nd envelope. This would be similar to being given $1.5x dollars, keeping $x dollars, and flipping a coin double or nothing for the other $.5X. Whatever happens, you get to keep $x, and have a 50% probability of winning the other $x. Now, you don't need to actually do this, but it may help to think that you actually can do it. Without knowing how much x is, take $x from both envelopes, and put $x in your pocket - after all, you are guaranteed that $x. Well now one of the envelopes is empty while the other contains $x, and there is absolutely nothing which would suggest which is which. Pick one - there is a 50% change you pick the wining envelope and a 50% change you get the emply envelope. Flip of a coin - double of nothing for $x/2 QED.

OK, how about the case where you know that your envelope contains $10,000 and that the other envelope contains either $5000 or $20000. That's another probem entirely, which I will treat subsequently. Note that knowing one envelope contains $10000 requires nothing special - the x in the pocket still works, but we don't know whether x=$5000 or x=$10000. No matter.

It is quite obvious to most people that you should not swap and there are many simple ways to show this. The problem is to find the flaw in the proposed line of reasoning given in the article, which, at first sight, appears to be quite reasonable. This is not so simple. Many of the sources on the subject tend not to make clear what claimed paradox they are addressing. Martin Hogbin (talk) 09:37, 13 February 2012 (UTC)

Introduction to Solutions

I corrected and expanded slightly Martin's new "Introduction" so that it now refers explicitly to both of the first two resolutions, and also explicitly places his "easy" solution within the second interpretation. I also tried more systematically to distinguish "Envelope A" from "the first envelope". I think it does now serve as an honest introduction to the whole article, including a simple solution of a common interpretation of the problem which will satisfy many readers. At the same time it hints at the many possible mathematical subtelties which can turn up if one goes into the problem in depth.

I disagree that the TEP paradox is "just" a reflection of a deep failure of usual Bayesian decision theory (vN-M). I am not aware of any notable publications which put forward this point of view, either. I agree that there are a lot of problems with vN-M, which is neither prescriptive nor descriptive. A beautiful mathematical construction which unfortunately does not describe reality very well, and does not even describe how we would like reality to be, let alone how it actually is. It certaily would be interesting to review TEP within the context of criticism of vN-M, and to investigate whether any of the available alternatives to vN-M allow for a more satisfactory resolution of appropriate variants/interpretations of TEP. I understand that this is going to be iNic's mega-opus OR. Richard Gill (talk) 11:26, 15 February 2012 (UTC)

Correct, this is totally OR on my part so I will not discuss it here. iNic (talk) 13:53, 15 February 2012 (UTC)

The introduction text now include this paragraph:

It can be envisaged, however, that the sums in the two envelopes are not limited. The requires a more careful mathematical analysis, and also uncovers other possible interpretations of the problem. If, for example, the smaller of the two sums of money is considered to be equally likely to be one of infinitely many positive integers, thus without upper limit, it means that the probability that it will be any given number is always zero. This absurd situation is known as an improper prior and this is generally considered to resolve the paradox in this case.

This is not correct as a definition of an improper prior as any distribution over the real numbers has this property. Anyone that knows anything about mathematics will stop reading here. iNic (talk) 15:03, 15 February 2012 (UTC) 

This is not intended to be a definition of an improper prior but a particular example of one. I will make that clearer. Martin Hogbin (talk) 23:38, 15 February 2012 (UTC)

You apparently don't understand what's wrong with what you have written. You still claim that a property is absurd which obviously isn't absurd. That is absurd. Moreover, I thought Richard said that he would add all versions you "forgot" into this absurd mini-version of the page. But I can't find the non-probabilistic versions of TEP here. Was it maybe forgotten once again purely by accident? Wy am I not surprised? iNic (talk) 07:51, 16 February 2012 (UTC)

What is absurd and what is not is clearly a matter of opinion. Many people, as demonstrated by the literature on the subject, find the idea of an improper prior sufficiently absurd that they consider it to constitute a resolution of the paradox.

As I have explained before this introduction is intended to provide an understandable introduction to the most generally accepted interpretations and resolutions of the paradox, as typified by Nalabuff's review of the subject. A summary of the article as a whole belongs in the lead. Martin Hogbin (talk) 09:25, 16 February 2012 (UTC)

No no. To repeat: you say that it's absurd that any given possible number for a distribution has probability zero. If that is absurd then the normal distribution, for example, is absurd as well. But it's not absurd. It's your statement that is absurd. (There are more absurdities but let's start here.) iNic (talk) 10:38, 16 February 2012 (UTC)

Are you saying that the uniform distribution on an infinite interval is not an improper prior? See [1] Martin Hogbin (talk) 11:16, 16 February 2012 (UTC)

The passage in question does not define "improper distribution", it merely gives an example. The example concerns probability distributions on the integers, to make it easier to grasp by laymen (and closer to the spirit of TEP - money is not continuous, but discrete). I think that what is written now should satisfy both laymen and experts. Richard Gill (talk) 15:37, 16 February 2012 (UTC)

Of course it's not a definition of an improper distribution. You and I know that. Only problem is, how will anyone that have no clue what an improper distribution is, and reads the intro, know that? A WP article can't write crap and simply assume that all readers already know the subject anyway so that it's crap won't do any harm. In addition, there is no discussion whatsoever why Bayesian concepts has been introduced in this context by some authors. They are just introduced out of the blue. Why? Because according to Martin he is only "stating the obvious." But what if Bayesian thinking is not obvious to everyone, which, by the way, is the case? This intro might work fine as Martin's personal note, but not as anything belonging to an encyclopedia. iNic (talk) 22:10, 16 February 2012 (UTC)

A common resolution

Although Richard has now changed this section so that it makes some kind of mathematical sense it is clearly not now a resolution of the paradox, which requires us to find the flaw in the proposed line of reasoning. The 'resolution' now simply shows that there is no point in swapping, which is something that everybody finds obvious at the start, indeed, without this natural assumption there would be no paradox. That is the problem with the 'common resolution, it can either be a mathematically sound statement which fails to address the problem at hand or it can be a confused and non-mathematical rambling which is worse than the paradox itself. Martin Hogbin (talk) 09:32, 16 February 2012 (UTC)

This solution claims that step 7 in the line of reasoning is wrong. It doesn't claim "simply that there is no point in swapping." You have used this pointless argument yourself in our discussions on these talk pages but the defenders of this view hasn't. Please show me a single source for your claim. It is obvious that you haven't understood this solution at all and that you have created your very own interpretation of it, which might very well be absurd. iNic (talk) 10:46, 16 February 2012 (UTC) 

Yes, you are quite right, there are two unrelated attempts at a resolution in this section. The first is as shown below. Note carefully Richards wording which I have put in bold.

'A common way to resolve the paradox, both in popular literature and in the academic literature in philosophy, is to observe that A stands for different things at different places in the expected value calculation, step 7 above. In the first term A is the smaller amount while in the second term A is the larger amount. To mix different instances of a variable in the same formula like this is said to be illegitimate, so step 7 is incorrect, and this is the cause of the paradox.'

This is Richard's way of trying to express the vague notion about using one thing being used to be be two different things. It is there because such vague thoughts often appear in some literature. If you can find some clear explanation of exactly what is being claimed by these authors then please replace the paragraph with that. You need to start with defining exactly what kind of quantity 'A' is supposed to be.

The rest of the section merely states the obvious. Martin Hogbin (talk) 11:07, 16 February 2012 (UTC)

The reason I inserted "said to be" is because we're reporting the claim of the typical writer who supports this solution. We wikipedia editors are not endorsing any particular solution, just reporting what's in the literature, and as far as possible trying to do that so that our readers will understand it. I don't find these resolutions satisfactory at all. But they are common.

I also made an attempt to convert this vague solution into a careful mathematical analysis, which at least mathematicians hopefully would find meaningful. In the mathematical analysis I adopt what seems to be the interpretation of these solutions, namely that we are after an unconditional expectation, and I explain where the writer then derails, under this interpretation.

Personally I don't like this solution at all - I find the interpretation far fetched, I find the vague resolution unsatisfactory. Moreover it misses one of the errors which the writer is making, according to this interpretation. He is also confusing random variables and their expectation values. So the mathematically precise version of this resolution is just as complicated as that of the second resolution.

I hope our readers will do their best to understand the second interpretation and the standard resolution which goes with it.

Finally we must leave our readers to judge for themselves. We are merely reporters, not judges. Richard Gill (talk) 15:44, 16 February 2012 (UTC)

I agree 100% that we are only reporters of opinions and if we agree or not, think the ideas are silly or not, worthless or not, brilliant or not, pointless or not, complicated or not, ridiculous or not, far fetched or not, satisfactory or not, vague or not, meaningful or not, erroneous or not, stupendous or not or even dangerous or not, should not make a damn difference. And we should not "hope our readers will do their best to understand the second interpretation and the standard resolution" because there is no "standard resolution." If you have this agenda you should back off from WP and not edit a single page from now on. The ideas you happen to like doesn't automatically transform into any "standard resolution." That is just BS. iNic (talk) 16:17, 16 February 2012 (UTC)

By the way, I can tell you the real difference between a philosopher and a mathematician. A philosopher can grasp more than one idea at the same time. iNic (talk) 16:17, 16 February 2012 (UTC)

Tools required to be a mathematician: pencils, paper, waste paper basket. Tools required to be a philosopher: pencils, paper. Martin Hogbin (talk) 17:16, 16 February 2012 (UTC)

I think I will do the same as you Richard and leave this page entirely and let the vandals take over. I have done that before and the result was a disaster. Then you came along and I started to believe in making a fun, short, readable and yet accurate page about TEP again. But now I have lost hope in this project once again. It's kind of sad but I will leave the corpse to the hyenas again. iNic (talk) 22:24, 16 February 2012 (UTC)

Your offensive tone to other editors will not be missed. Martin Hogbin (talk) 23:03, 16 February 2012 (UTC)

Yes leaving this page will leave both you and me happier. It's a true win-win. Who said that two opponents can't both win when playing a game? iNic (talk) 00:24, 17 February 2012 (UTC)

iNic, when I said hat I hoped the readers would come to their own conclusions, I did not mean they should come to my conclusion. Maybe the neutrally and clearly presented, and well sourced material, will lead them finally to your opinion, or to Martin's, or to mine, or to yet another, I think the article, though not perfect, does honestly survey what's in the literature. The "vandalism" which previously plagued the article was that editors who thought they understood TEP wrote up their own solution. I think the amount of vandalism will go down strikingly in the future; we have set some "academic standards" which will encourage serious editing.

Maybe the properties "fun, short readable" are mutually inconsistent. Or more precisely, the resulting article would strongly depend on the point of view of the writer. Wikipedia is not there for that purpose. We give references to articles which some readers will find short, fun, readable. The lead of the article should be short, fun and readable. I think that present lead and intro are pretty good.Richard Gill (talk) 06:56, 17 February 2012 (UTC)

The current article is already strongly dependent on the point of view of the writer(s). For example, the logical version isn't mentioned in the new fabulous introduction at all, despite it's one of the very first versions. Why? Because of biased editors. When the logical version is presented in the article the first comment about it is from two probabilists who knows nothing at all about logic(!). Then their silly idea is presented that the problem can be dismissed because the non-probabilistic version doesn't include probability(!). This is a very biased way to present, or rather suppress, a topic in WP. It is moreover in perfect accordance with your own personal view of this topic. Wikipedia is not here for your purpose. iNic (talk) 15:02, 17 February 2012 (UTC)

(1) Smullyan's version is not mentioned explicitly in the introduction because of its relative lack of notability. (2) The present discussion of it presents in an unbiased way the points of view of Albers et al., Chase, and Li, and moreover points out that they are consistent with one another. In particular the logician Li agrees 100% with the mathematicians Albers et al.: without any probability there are no reasons to switch or not to switch; both of Smulyan's arguments are wrong. Chase adds probability and shows that there now is a good reason why switching is pointless: one of Smulyan's arguments is correct, the other is incorrect. (3) As far as the article reflects any editors' editorial (rather than substantive) opinions, it seems to me to be a decent compromise between those of the three currently active editors. Richard Gill (talk) 11:05, 18 February 2012 (UTC)

iNic, you say, 'the logical version isn't mentioned in the new fabulous introduction at all'. One reason for this is that this is not the version described at the top of the article. One thing you must agree on is that different versions of the problem can have different resolutions. The is absolutely no point in carefully describing one version of the paradox (as we currently do) and then giving the resolution to another version.

Why not add a new section at the end called 'Variations of the problem' or similar in which you describe the less notable variants of the paradox and their resolutions? Martin Hogbin (talk) 12:54, 18 February 2012 (UTC)

It's obvious that the intro violates the NPOV policy, from the first sentence to the last. You are both of you so totally into your own way of thinking about this problem that you are unable to write a neutral and unbiased introduction. Richard, who has decided that the logical version has a "lack of notability"? No one has. That is just nonsense. You are simply violating one of the fundamental rules of WP wich is neutrality. And Martin, you think that all versions except the logical is explicitly mentioned at the top of the article? This is obviously not true. Your argument to justify a biased intro is equally much nonsense as Richard's argument. (Interesting to note that while you have a silent agreement that the logical version should be abandoned from the intro, when forced to speak about it you totally disagree on why it should be abandoned...)

Martin, different writers differ when it comes to how many distinct versions they think there are, and which versions they think are only different wordings of one and the same version. This is the problem with both of you. You are both convinced that you know the real resolution(s) to this problem(s) and how many different versions there really are and which versions are less important and thus can be dismissed. Then in the name of enlightening the masses of the correct interpretations and their solutions you bias the article to fit your "true" views. That is POV editing, resulting in a POV article.

Let's take the first sentence of the intro as an illustration: "The ways in which the paradox can be resolved depend to a large degree on the assumptions that are made about the things that are not made clear in the setup and the proposed argument for switching." This is simply not true. It's not the case that all authors of papers complain over lacking information in the setup or history of events leading to the situation. The writers with a Bayesian approach typically do that, but not all solutions are written from a Bayesian standpoint, or even can be written from a Bayesian standpoint. So even if you are both convinced Bayesians you need to be able to take off your Bayesian glasses, just for a little while, and view your personal standpoint as one among others when editing this page. If you can't do that you are simply not suited to edit this page. Right now you are both too much engaged in the battle to be able to be neutral reporters of the battle. iNic (talk) 11:22, 19 February 2012 (UTC)

What exactly is the POV that you believe is missing? Martin Hogbin (talk) 11:34, 19 February 2012 (UTC)

The missing POV is the NPOV. iNic (talk) 11:52, 19 February 2012 (UTC)

You are not being particularly helpful. You clearly believe that one POV is being pushed at the expense of one or more others. What is the POV you believe is being excluded and where can I find a reference to it? Martin Hogbin (talk) 12:17, 19 February 2012 (UTC)

As I have already said above, not all writers blame on a not well enough specified set up of the problem as stated. This is however stated as a fact without any reservations in the very first sentence of your intro. The whole intro also takes the Bayesian view for granted. Without even mentioning the Bayesian view with one word. Instead exclusively Bayesian concepts like priors , or rather improper priors, are blamed at just like that. No explanation of context. No explanation of what a prior is to start with and why some authors talk about these for some of the versions of TEP. All we get is a silly and wrong reason why a thing called "prior" should be called "improper." Extremely unhelpful and biased text. iNic (talk) 13:11, 19 February 2012 (UTC)

Please stop complaining and tell me what it is you would like to see.Martin Hogbin (talk) 18:11, 19 February 2012 (UTC)

The intro would have to be rewritten entirely to live up to NPOV standards. But what's the point having an intro in the first place? Better to make the article itself easier to read, then we can skip having intros. iNic (talk) 19:01, 19 February 2012 (UTC)

Are you suggesting that it is possible to give a clear resolution of a paradox without knowing either the setup or what the claimed paradox is? Martin Hogbin (talk) 11:39, 19 February 2012 (UTC)

Please read the sources. This is the Talk page, not the Arguments page, so I'm not suggesting anything about the problem. This is precisely the problem with you and Richard. You treat this Talk page as if it's the Arguments page (I've moved tons of your contributions here to the Arg page) and you treat the article as if it's the Talk page. iNic (talk) 11:56, 19 February 2012 (UTC)

Is there a particular source you would like me to read? The article as it is represents the sources, in particular the closest that we have to a general review, the Nalebuff paper. Martin Hogbin (talk) 12:08, 19 February 2012 (UTC)

Nalebuff writing in 1989 can't possibly give an overview of all the papers written after 1989. Since then the number of papers has exploded. The sources page lists 111 different papers/contributions after 1989 and just a handful before 1989. iNic (talk) 13:11, 19 February 2012 (UTC) 

If you are not going to say what view you think is not properly represented here and what sources propose this view, please stop putting POV tags on the article. Martin Hogbin (talk) 18:11, 19 February 2012 (UTC)

I have said that. The logical view is never mentioned and Bayesianism introduced as a fact when it's not a fact, only a point of view some have in some contexts. iNic (talk) 18:56, 19 February 2012 (UTC)

The logical view clearly does not address the paradox described at the start of the article. Why do you not do as I suggest and add a section on this variant and its resolution?

There is nothing specifically Bayesian about the section in question except that it mentions the term improper prior. The resolutions referred to apply equally to the frequentist interpretation of probability as at least one source says. I will try to find a reference and add it. Martin Hogbin (talk) 20:36, 19 February 2012 (UTC)

If you classify the papers written since Nalebuff's paper you will find that most of them repeat material that is already in his paper. Several use it to invent new paradoxes inspired by the original (cf. the Broome variants, already in Nalebuff) or go more deeply into some technical aspects, or wander off on some tangent seeing connections to other problems (e.g., Saint Petersburg paradox, already in Nalebuff). The more mathematical literature almost exclusively takes the point of view that the TEP argument is about a computation of the conditional expectation value of the contents of Envelope B given some hypothesized amount in Envelope A. It takes for granted that the reader has some basic understanding either of the Bayesian point of view, or of the frequentist point of view, in both cases seeing the pair of amounts in the two envelopes a priori as random variables.

The philosophical literature almost exclusively takes the point of view that TEP is about a computation of the unconditional expectation value of the contents of Envelope B. It is also highly repetitive. Many of the papers are very long and difficult to read and aimed at an academic audience in philosophy interested in particular technical issues in philosophy (problems of how to name things).

Smullyan's little paradox is a breath of fresh air in the philosophy literature. He separates the issues of how to name things from the issues of probability calculation. There are several resolutions of his paradox since there are several ways to interpret his problem statement, but they are all discussed in the wikipedia article and there is no mutual contradiction between them. The literature on his TEP is rather small.

My opinion is therefore that the present introduction to the paper represents the literature in a rather fair way, and in a way which makes it maximally accessible to the lay person. Specialists will also find in the article a discussion of all the notable specialist (academic) issues. The field is complex and messy, many of the publications are not very high quality, so it is quite a challenge to give a comprehensive overview. But that is what we have done. Richard Gill (talk) 09:27, 20 February 2012 (UTC)

There are three main categories of solutions to TEP in the literature. One of them, the one you call the philosophers' solution, is satisfied with how the problem is stated. No information is lacking but no information is superflous either. This is the one currently called the "A common solution" in the article. Then there is the class of solutions which I call Bayesian and you call the mathematicians' solution. In their view the problem as stated doesn't contain sufficient information; the problem as stated is underdetermined. So to be able to solve the problem they first have to fill in the missing parts. This can be done in different ways leading to quite different types of arguments and solutions. Here we find the bounded prior game, the unbounded Broome type prior game, my bounded ticket game and so on. Then the third category of solutions we could call the logicians' solution. Their view is that the problem as stated contain superflous information. According to this view we can reduce the problem to a logical problem without any probabilities. Your intro is not NPOV because already in the first sentence it states the middle POV that the problem id underdetermined.

Why don't we call these three main interpretations "The philosophers solution", "The mathematicians solutions" and "The logicians solutions" in the article? I think that would be an enhancement. iNic (talk) 10:06, 20 February 2012 (UTC)

An interesting argument but let me ask whether this particular analysis of the literature is your own or if you have source which analyses the literature in this way. Martin Hogbin (talk) 10:42, 20 February 2012 (UTC)

There are no neutral surveys of the literature yet, probably because this is still an ongoing debate. Richard agrees with me on this point. How many of the sources have you read? Only Nalebuff? Maybe our different views of TEP as a subject is due to different views of the TEP history. To you and Richard TEP as a field of research essentially ended in 1989 with the Nalebuff paper. In my view it on the contrary took off for real in 1989 with the Nalebuff paper. I think that just a glance at the sources page supports my view. iNic (talk) 12:09, 20 February 2012 (UTC)

I disagree, iNic, that the mathematicians think the mathematician's version of TEP is incomplete. Whatever prior distribution is put over the possible amounts of money in the two envelopes, as long as it is proper, the TEP argument deriving the conditional expectation of B given A=a is incorrect and one can clearly state where it fails. It simply can't be the case that B is equally likely a/2 as 2a for all possible values of a. For ordinary folk, the example where we assume an upper bound to the sums of money makes this clear enough.

Improper priors are dealt with by remarking that they are improper. Littlewood uses the word "monstrous". Broome's paradox is a new paradox. Interesting for mathematicians, not too interesting for ordinary folk, I suspect.

Regarding Smulyan's paradox, I am not aware of any logician who thinks that Smulyan's problem is "the only true" TEP problem because it is somehow more pure to leave out the probability ingredients altogether. It is simply considered a new paradox.

It is obvious that the usual (standard) statement of basic TEP (as presented in the article) is incomplete, and needs extra information before it can be resolved: the philosophers and the mathematicians fill in key missing steps in different ways: the missing information being what is thought to be random and what is thought to be fixed, hence also implicitly whether a conditional or unconditional expectation is being taken.

Both common interpretations have simple resolutions if one is familiar with elementary probability calculus. However it is all rather heavy going for those who do not have any familiarity with elementary probability concepts. Richard Gill (talk) 17:41, 20 February 2012 (UTC)

The problem with the simple solutions given by some philospohers and others is that they are mathematical nonsense, just as bad as the problem itself. Unfortunately, there do not seem to be any reliable sources which point this out, maybe someone needs to write one.

The logicians solution is, obviously, to a different variant of the problem, which could be added at the end of the article. Martin Hogbin (talk) 10:04, 21 February 2012 (UTC)

The logician's problem and two logician's solutions are already in the article "non probabilistic variant". iNic was complaining that it is not mentioned in the introduction. I think it is not notable enough for this. Only three or four articles out of the hundred, I think. It is the case however that it focusses on the confusion of wording ("equivocation") which the philosophers concentrate on, in their interpretation of TEP. In that sense it is, In my opinion, a better paradox than the philosopher's version. Richard Gill (talk) 12:48, 21 February 2012 (UTC)

Perhaps we should just change the name of the introduction to, 'Introduction to the solutions using probability theory' or the like. It was never intended to be a summary of the whole subject, that is the purpose of the lead. Martin Hogbin (talk) 17:42, 21 February 2012 (UTC)