Talk:Two envelopes problem/Archive 6

Falk (2008), Falk and Nickerson (2009), Nickerson and Falk (2005)
I have also been in email contact with Ruma Falk and have read the three papers on TEP of which she is author or co-author. They are all excellent. Note that they are written for different audiences. Nickerson and Falk (2005) is perhaps the first paper on TEP which seriously surveys all the solutions known to the authors at that time. It explicitly points out that the different solutions make different assumptions as to context and intention of the writer. It pays attention to the Bayesian context with and without improper priors. Explains that improper priors give infinite expectation values and hence unreliable advice as to decision or action. It's all there. All three are now in my dropbox, if anyone else is interested to read them. I think that Nickerson and Falk (2005) is obligatory reading for any wikipedia editor of TEP. It's the only paper with a complete and neutral survey of as many solutions as possible. Richard Gill (talk) 13:42, 4 November 2011 (UTC)

The error in the switching argument
Please help to check which sources say the following, and help to improve the article accordingly:

The basic setup  does not accent in any way the fact that there is a huge difference  between having chosen the envelope "filled at first with any determining amount of money", i.e. containing the "original amount" of X  (let's call that amount "Xoriginal", and let us denote that amount with the value of "1") with probability of 2/4, or having chosen the envelope that was filled thereafter with the dependent amount, say "the second envelope", containing dependently either 2X ("Xdouble", with the denoted value of "2") with probability of 1/4, or with equal probability of 1/4, containing only X/2 ("Xhalf", with the denoted value of "0,5").

For only in the case that (with probability of 1/2) you should have chosen the envelope with the determining amount of "Xoriginal", then with probability of 1/2 the other envelope will contain dependently "Xdouble" (2X) or will contain with equal probability of 1/2 only  "Xhalf" (X/2). And note that only in that case that, by chance, in 1/2 of cases you should have chosen the envelope containing the determining "Xoriginal" amount, all the further conclusions of the TEP, including "5A/4") do apply, otherwise not.

But fully neglecting that draconic precondition, "the basic setup" is based on this last variant only. Without saying so. The basic setup and all further conclusions do fully neglect the scenarios of the two other possible constellations with quite other assigned prospects, having equal probability of 1/4 each: That you could have chosen the dependent "second envelope" of 2X ("Xdouble"), having quite other prospects assigned, or that you could have chosen the dependent "second envelope" of X/2, again having quite other assigned prospects, contradicting all further conclusions of the TEP.

The value of 5A/4 is only correct if the candidate, by chance, should have chosen the determining envelope "filled at first with any amount of money". Then the dependent envelope B (its "slave") can be expected to contain 5A/4.

But if the candidate should have chosen the dependent envelope filled thereafter only, remunerated either with considerable 2X or with X/2 only, then in that case the expected determining value of the other envelope can never be 5A/4 but will forever be 4A/5 only. That's a given fact:  E(B) = (2/2+1/2*2)/2,5 = 4/5 A.

And – without knowledge which one of those both envelopes had been chosen, the determining first one or the dependent second one – the expected value of the other envelope is exactly 1A:

E(B) = ((2*1+1/2)+(2/2)+(0,5*2))/4,5 = 1A.

Which sources do show that coerciveness? It could help to better articulate the inevitable facts. Who can / will help? Gerhardvalentin (talk) 18:56, 23 November 2011 (UTC)


 * The standard TEP story has two envelopes filled with two positive amounts of money, one twice the other. Then the player picks one of the two envelopes completely at random. How the envelopes are filled is not part of the story. Some authors, by way of example, discuss particular ways in which the host could have chosen the two amounts of money. They show that for their particular story, one or more of the steps of the argument is actually wrong. But such examples do not solve the paradox. For one thing, for many authors the important thing is not how the host filled the envelopes, but what the player believes about the amounts of money finally in the two envelopes. For another thing, showing that there is an inconsistency in the TEP argument in one particular case does not prove that the argument is incorrect in all cases. It might help you to discover what is the mistake in general, but it might not. Anyway, the general mathematical principles behind all knowns solutions is given in Samet, Samet, and Schmeidler and later improved by Gill. Richard Gill (talk) 00:05, 24 November 2011 (UTC)


 * Thank you.  Am no prob. champ, nor do I know all the sources, but you helped me to see that the 5A/4  result can never be based on any symmetric distribution, because in that case of symmetric distribution the only valid result E(B) = ((2*1+1/2)+(2/2)+(0,5*2))/4,5 = 1A.

(but the "basic setup" of the TEP ignores that fact and doesn't even mention this "a priori given draconic restriction".) Whereas in case you know for sure that envelope A is of dependent type "β" then E(B) = (2/2 + 1/2 * 2) / 2,5  = 4A/5 only.
 * And you helped me to see that one thing is quite obvious: The TEP story and any game is based on two quite different types of envelopes. Type "α" had been filled first with any certain determining amount "X" (let us say that this amount is "1" e.g., and only "thereafter" another envelope of type "β" has been filled either with dependent "2X" or with "X/2". And the 5A/4 result is only valid in case you know for sure that the determining envelope A is of type "α", otherwise never: 1/2 * 2A + 1/2 * A/2 =5A/4


 * And as long as you do not know for sure whether you hold a determining "α-envelope" or a dependent "β-envelope", as this is not "part of the TEP story", then only E(B) = ((2*1 + 1/2)+(2/2) + (0,5*2))/4,5 = 1A can be correct. That's it.


 * Result of switching to envelope B:  5/4 A, 4/5 A and 1A respectively (millionfold verified): Only if envelope A is of type "α" with determining amount, then E(B) = 1/2 * 2A + 1/2 * A/2 = 5/4 A (as the article – without destinguishing - incorrectly claims for all types of envelopes) Only if envelope A is of type "β" with dependent amount, then E(B) = (2/2 + 1/2 * 2) / 2,5  = 4/5 A If type of envelope A ("α" or "β") is unknown:    E(B) = ((2*1+1/2)+(2/2)+(0,5*2))/4,5 = 1A


 * Is that correct distinction mentioned anywhere in the sources?  Regards, Gerhardvalentin (talk) 01:52, 24 November 2011 (UTC)


 * This distinction is mentioned in many papers. You should read the sources if you want to work on this article. If you like you are welcome to join my dropbox folder which contains all the important papers and many less important. There is a problem with your notation. You relate E(B) to A. But in many formulations A is also random. If you want to *do* probability calculus, please *learn* probability calculus. There are excellent free eBooks easily available. Many of the sources point out that the error in the TEP argument is the error of not distinguishing between random variables and possible values thereof. If you do not make this distinction yourself you cannot contribute to writing an article which correctly summarizes what is in the sources! Secondly, your conclusions are not correct, because in the case that we use probability also to represent prior ignorance of, say, the smaller amount of money x in the two envelopes, then it is the case that if we a priori are so collosally ignorant concerning x that we use the improper prior distribution which makes log(x) uniformly distributed on the whole real line, then it is true that E(B|A=a)=5a/4 for all a. If that interpretation was the intention of the writer, then we must resolve the paradox in another way. Richard Gill (talk) 18:47, 24 November 2011 (UTC)
 * Yes, thank you Richard. I meant to ask whether – for a better understanding of the very  "nature of this paradox"  for grandma and grandson – any sources do clearly accent that "E(B)=5A/4" is fully valid,  but fully valid only if you know for sure that envelope A had FIRST already been filled with any arbitrary determining amount and only afterwards it was decided whether the dependent envelope B was filled with dependent half or dependent duplicate of the already fixed determining amount of envelope A. Because only then, although the already "prefixed" determinng amount in envelope A  will equally likely be double or half of the "dependent" amount in envelope B,  "5A/4"  for envelope B is perfectly correct then, but never otherwise. Because "E(B)=5A/4" does not tolerate any prefixed amount in envelope B with some "dependent" content in envelope A. This given restriction should be made obvious and may never be kept secret, otherwise creating an apparent paradox. This given axiomatic precondition not explicitly being "part of the story",  E=(B)=5A/4 should never unrestrained and nonrestrictive be laconically quoted. Do any sources clearly accent this basic fact, just to help grandma and grandson for a better understanding of the very nature of this just assumed "mystic paradox"? Gerhardvalentin (talk) 11:58, 25 November 2011 (UTC)


 * The present text of the article says explicitly: there are two envelopes, you pick one completely at random, and that is Envelope A, the other is B. Now, before you even look in A, you consider whether or not you'ld like to switch for B. There is no point at all thinking about the wrong scenario, except to underline the point that it is wrong. Sure, there are plenty of sources which compare the true TEP scenario to the wrong scenario. But another point: why do write E(B)=5A/4? Is A supposed to be random or fixed? Please think carefully about your notation! You have got to get this straight!
 * Richard, but it's the article that misleadingly states: 7. So the expected value of the money in the other envelope is $${1 \over 2} (2A) + {1 \over 2} \left({A \over 2}\right) = {5 \over 4}A$$ 8. This is greater than A, so I gain on average by swapping. It's the article that misleadingly says that you can expect B to be  5A/4  or  5/4 A. – It's the article that incorrectly says so.  Gerhardvalentin (talk) 12:55, 27 November 2011 (UTC)


 * Yes, the reasoning would be easier to follow and easier to see where it went wrong if a good notation was being used. I think that the writer was trying to compute E(B|A=a). He gets the answer 5a/4 by supposing that, given A=a, the two possible values of B are 2a and a/2, and the two conditional probabilities thereof are 1/2, 1/2. And that's where he is wrong. Richard Gill (talk) 14:11, 29 November 2011 (UTC)


 * Next point: it is not only in the special (wrong) scenario you just described that E(B | A=a )=5a/4 can be true for some values a. It can be true in the proper TEP scenario for almost all values a. The statement E(B | A=a )=5a/4 is true if and only if, given that A=a, envelope B is equally likely to contain 2a or a/2. For instance, the chess-board example: the host chooses one of the 64 squares of the chess board completely at random. The squares were pre-numbered 0 to 63. Given that he chose square r, he puts amounts of money 2 to the power r and 2 to the power r+1 into the two envelopes, shuffles them; you pick one and call it Envelope A. Unless there is 2 to the zero, or 2 to the 64, in your envelope, it is equally likely that Envelope B contains half or double the amount. E(B | A=a )=5a/4 for almost all values of a. With probability 63/64 therefore, E(B|A)=5A/4. Now consider bigger and bigger chess-boards. As they get bigger and bigger, the probability that E(B|A) is not equal to 5A/4 goes to zero. The precursors of two envelopes (Schrödinger and Littlewood's two-sided cards problems, Kraitchik two neckties) were thinking of this limiting situation. Another example. The host tosses a biased coin with probability of heads equals 1/3 till the first time he gets heads. Call the number of tails he saw before he got heads r. He puts two to the r in one envelope and 2 to the r+1 in the other, shuffles them, and you choose one and call it Envelope A. Whatever amount a is in your envelope, it's the case that E(B|A=a)>a. Do the calculations yourself, please! You need to learn some elementary probability and the good way to learn it is by carefully working through some simple examples. Build up your probabilitistic intuition, practice using a good notation which distinguishes random variables and possible values of random variables. Install the statistical language R to your computer (www.R-project.org) and do some simple simulation experiments. Learn! Only after learning, can you teach. Wikipedia editors have to be good teachers, they need to be on top of their material or they'll only confuse their pupils. That would be like the blind leading the blind. I already wrote some R scripts for playing with various scenarios. Let me know if anyone wants them. Richard Gill (talk) 11:57, 27 November 2011 (UTC)


 * Thank you, but my point is what the article says: I showed above that it is fully right that you can expect the dependent envelope B on average to contain 5/4 of the determining amount in envelope A. But regard this draconic restriction, that the TEP-article hides and reprobately keeps secret: This is only true if you know for sure that the "determining" envelope A, independently from the empty envelope B, has been filled with any random amount first, and only afterwards it was decided to fill the "dependent" envelope B with the dependent double amount of "determining amount in A", or with equal probability with dependent half that "determining amount of A". (added): This will apply only in one HALF of cases, never in the second half. And to get efficient for any analysis you definitely have to KNOW FOR SURE that it applies in the special case, otherwise you may never use it in your theorem. If it is unknown it cannot be considered in any way. As this restriction is nowhere mentioned in the TEP article, and you never can nor will know which envelope is the substantially "determining" one and which is the "dependent" one in the TEP,  A or B, this theorem leading to 5/4 A may never be laconically considered for the TEP. If you have chosen an envelope with "dependent amount", you forever have to expect to get only 4/5 A by swapping to B (twice if A is the dependent half amount of envelope B, half if A is the dependent double amount of envelope B). And as I said above - in lack of knowledge - you only can expect B to contain ((2*1+1/2)+(2/2)+(0,5*2))/4,5 = 1A. 5/4 A,  4/5 A  and  1A.  All of this is millionfold verified. If you put the cards on the table, then there is no paradox at all. No more need for infinity. The article should be accessible for grandma and grandson also. With open cards on the table, as per the sources (Ruma Falk, e.g.). Regards, Gerhardvalentin (talk) 13:22, 27 November 2011 (UTC)

What the article says
Gerhard, read the article! It says


 * There are two indistinguishable envelopes,


 * each of which contains a positive sum of money,


 * one envelope contains twice as much as the other.


 * You pick one envelope at random.


 * You denote by A the amount in your selected envelope.

In short: Envelope A is a randomly chosen envelope from an indistinguishable pair, Envelope B is the other.

The introduction to the article, where standard TEP is described (the problem which most of the sources, and in particular most of the popular sources, write about), doesn't say anything about how the envelopes are filled. It doesn't have to. You don't know how.


 * That's right, Richard. Not knowing how, and not having to know it. But alone your experience teaches you that there must have been a certain history, and "if" the window glass has been crushed from outside, makes other conditions for the insurance company as if it had been crushed from within the flat. There are quite other conditions for both variants. And as  "5/4 A"  will forever be valid in only one special HALF of cases, and never in the second half, and only if you already know for sure, just from the outset, that your envelope "A" is the determining envelope resp. the determining amount, but never never ever otherwise, you have to admit that - not knowing this significant condition indeed to be given for sure - makes  "5/4 A FORFEVER and EVER !!! ? ! ? !"  a really ridiculous farce, makes "5/4 A"  an unforgivable error and a gross fallacy, please ask the insurance company to get their expertise. Gerhardvalentin (talk) 17:52, 29 November 2011 (UTC)

You don't know the two amounts, you don't even get to look in your own envelope. That's the whole point. Yes you convince yourself to switch. And then you could convince yourself to switch back. It's a paradox because the conclusion is obviously wrong. The problem is to show where the apparently logical reasoning goes astray. There are alternative solutions depending on what you imagine the writer was trying to do and what he was assuming. We don't know either. Over the historic development of the paradox, and as it spread to different cultures (from mathematical recreations to economists, statisticians, philosophers), people tended to have different ideas what the writer was up to. That is why the article is hard to write: because there is not one solution, but many, and they are different. The problem is how to organise the different solutions. My own original research is an attempt to find synthesis. But wikipedia editors are not allowed to make new syntheses so I am not allowed to edit the article. But I offer my findings for discussion and I am likely getting them published in a good peer-reviewed journal soon. The results are still getting better and better, I think, thanks to discussions here and elsewhere. Sure, some sources are confused, and very many sources are bloody confusing. One source of confusion seems to be with Nalebuff (1988, 1989) who introduced a new version of TEP in which we know that envelope A is filled with x>0, amount unknown, by the host, then the host tosses a coin to put x/2 or 2x in envelope B. The player is given Envelope A and asked if he would like to switch. Nalebuff then gives a correct reasoning with the host and owner, Ali, of Envelope A would like to switch, followed by the paradoxical (wrong) TEP-like reasoning why the owner, Baba, of Envelope B would also like to switch. At the end of his problem description he also mentions the original or standard version - the version described in the wikipedia article, and from which he invented his new problem. At the time Nalebuff wrote his article the story was being discussed by statisticians and mathematical economists (he heard it from colleagues). Gardner also heard about it at the same time, and put it into his 1989 book "Penrose tiles to trapdoor ciphers", p.147-148. Gardner had previously (1982) discussed the Kraitchik 1942, 1953 two envelopes version, converting it to two envelopes, in his 1982 book "Aha, Gotcha". Gardner (1989) discusses our basic, "proper" TEP, and most of the people who wrote in to Nalebuff, whose solutions Nalebuff discusses in his second paper, also discussed our basic, proper TEP. The TEP described in Wikipedia, the version described by Falk, who even quotes from Wikipedia, and the version most discussed in the literature. Please let's focus on the basic TEP. Not on a variant. Of course the difference with the variant is interesting. Nalebuff's new Ali and Baba TEP problem is different. Your grandmother and grandchild will understand that Ali should want to switch and Baba should want not to switch. In the original TEP problem they understand that there is no point in anyone switching. But the problem is not to realise what the right answer is, it is to understand where the reasoning which leads to an absurd answer goes wrong. Richard Gill (talk) 07:32, 29 November 2011 (UTC)


 * Thank you Richard for your incredible patience. Yes, the real "shame" is that "the given actual content of envelope A" and "All possible contents of envelope A" are confusingly mixed together. Let me repeat again: Only items 1–3 of the 12 claims of the article (1-12) do apply in every case. But in one full halve of cases, forever when "A" inevitably is to be the dependent amount, all steps 4–12 are completely inconsistent with the given facts, no matter whether you know the actual content of envelope A, or not. Yes, everything seems to be consistent, but this consistency does not exist: step 4 does never apply in 1/4 of cases, and step 5 does never apply in the remaining 1/4 of cases. Once more: Step 4 is completely incorrect in 1/4 of cases, whether you know the given actual content of envelope A, or not. And step 5 is incorrect in another 1/4 of cases, whether you know the given actual content of envelope A, or not. So steps 4–12 are false allegations like "The Emperor's new clothes", if you just have a look there. Why is it so hard to name those false allegations, that might apply in exactly 1/2 of cases only, but never in the remaining 50 % of cases, where either item 4 is completely incorrect in 1/4 of cases, or item 5 is completely incorrect in the other 1/4 of cases. So you are required to add to step 4: "Incorrect in 1/4 of cases", and to add to step 5: "Incorrect in another 1/4 of cases". And, before step 6, to add "As a consequence, the rest is nonsense in 1/2  of cases". Just have a look to the four equal likely scenarios. It should be possible to object items 4 and 5, and as consequence the rest of 6 to 12. Regards, Gerhardvalentin (talk) 14:39, 29 November 2011 (UTC)


 * You say "four equally likely cases". You are still assuming things we don't know. You are assuming first the host picks an amount, then he halves or double it, then randomly one of the two envelopes becomes envelope A. So you are doing probability with just two independent coin tosses in the whole story. But other people do the probability differently. Very many people suppose the host first picks a number x, then puts x and 2x in two envelopes. One of the envelopes is picked at random and called Envelope A. Some people "do the probability" only using the final random pick of an envelope. Many other people also use probability to describe their uncertainty as to what x might be. I'm not saying your analysis is wrong. I'm just saying it's relative to particular assumptions about what is fixed and what is random, and what the randomness is. Your choice is not common on the literature! From a mathematical point of view, my unified solution is a good solution because it applies to every single interpretation I know. Unfortunately however it is mathematical so your grandmother or little grandson won't understand it. This is the problem with writing the article. It must start with easy solutions which are however restricted to a particular interpretation. And the particular interpretation must be made absolutely clear each time. Unfortunately many of the sources are not very explicit about which interpretation they are taking, so the editors of wikipedia have to do some "reconstruction". The problem is, in fact, that there hardly exists an authoritative source which discusses all interpretations. Possibly the only one so far is the Nickerson and Falk paper. And my up-coming paper. Richard Gill (talk) 08:16, 30 November 2011 (UTC)


 * Yes Richard, you are correct again, thank you – I steadily just am trying ... and I didn't "see" those aspects before. But what you said shows clearly the direction, where to search. Thank you once more. Kind regards, Gerhardvalentin (talk) 12:40, 30 November 2011 (UTC)

Falk and Konold (1992)
Just found a really great reliable source: Ruma Falk and Clifford Konold (1992). With the most succinct solution I have yet seen. I added the reference to the sources page and put a pdf in my Dropbox.(Ruma Falk sent me a photocopy - that's how I know about this one). Richard Gill (talk) 10:05, 30 November 2011 (UTC)
 * Thanks for the paper. Never read that before. iNic (talk) 13:53, 6 December 2011 (UTC)


 * Do you have the two papers by Zabell? Could you put them in the dropbox? Richard Gill (talk) 22:23, 12 December 2011 (UTC)


 * I just got the Zabell book Symmetry and its discontents which has the 1988 paper with the same name as its first chapter. Somewhat disappointingly this paper only states the TEP puzzle as a final illustration, a nice little challenge for the reader(!), without any real hint on his own solution to the problem. However, he ends with the comment "I will resist the temptation to explain what I take to be the paradox, other than noting that all hinges on [the player] A's apparently harmless symmetry assumption that it is equally likely that B holds the envelope with the greater or the lesser amount." iNic (talk) 00:20, 30 December 2011 (UTC)

PS my own "unified solution", latest version, is now on the TEP Talk/Arguments page, and on my own Talk page. I have been showing it to experts in the field. Richard Gill (talk) 09:31, 5 December 2011 (UTC)


 * Wow, so there are professional TEP experts now? How cool! iNic (talk) 13:53, 6 December 2011 (UTC)


 * There are probabilists who specialize in exchangeability (symmetry, under exchange). TEP is also called the exchange paradox and all of its variants (except Nalebuff's Ali-Baba problem) are about a joint probability distribution of two random variables which is unaltered under exchange of the two variables. Symmetry. Richard Gill (talk) 22:19, 12 December 2011 (UTC)

Chase
In the section called "Non-probabilistic variant" on Smullyan's presentation is the following:

"(Actually Smullyan only mentioned arguments 1 and 2; argument 3 was added later, by James Chase, who was the first to publish a solution of the paradox, see below)."

The claim that Chase's article contains a solution is not a matter of consensus. Furthermore, if the article does contain a solution, whether it is the first published solution is not a matter of consensus. I have therefore removed the phrase ", who was the first to publish a solution of the paradox".

It is also worth noting that "argument 3" is identical with "argument 1", except that the roles of the two envelopes are reversed. "argument 3" is therefore superfluous. ---Dagme (talk) 20:32, 26 December 2011 (UTC)


 * Good points! I deleted the third redundant argument and also the uninteresting fact that Chase was the one who added this argument. I also changed the headings from 'resolution' to 'proposed resolution' to remove the false impression this article otherwise conveys that the presented proposed resolutions are uncontroversial. iNic (talk) 01:01, 27 December 2011 (UTC)


 * Indeed argument 3 is identical to argument 1 with the roles of the two envelopes reversed. That does not make it superfluous, since the conclusion of argument 3 is opposite to the conclusion of argument 1. This observation supports Chase's reasoning that argument 1 must be wrong. Arguments 1 and 3 both make it favourable to go for one of the two envelopes, but a different one each time. Argument 2 makes it neutral. It is a fact that Chase was the first to publish an attempted resolution of Smulyan's paradox. As far as I know, no-one disagrees strongly with his analysis; some have tried to improve or refine it. The only author who really disagrees is Byeong-Uk Yi. The fact that Li has a different proposed resolution doesn't change the fact that Chase was first to publish a proposed resolution. One reason there is not complete concensus about the resolution is because there is no concensus among philosophers on the proper logic of counterfactual reasoning. To each different formal logic of counterfactual reasoning, there would have to be a different formal resolution to the paradox. Another reason for apparent lack of concensus is that there is disagreement about the precise statement of the problem. For Chase, it is given as part of the problem that Envelopes A and B are defined by choosing one envelope at random. Because this ingredient is assumed, we know that we should be indifferent as to which envelope is chosen. Thus we know that arguments 1 and 3 must both be false. Argument 2 is argued to be correct because it is based on the most recent fork in history: which of the two envelopes is taken to be Envelope A. At that point in history, the contents of the pair of envelopes is fixed. Chase uses a particular approach to counterfactual reasoning which is certainly popular among philosophers and moreover easy to explain to non-specialists As far as I know there is only one publication disagreeing completely with Chase's analysis: that is Byeong-Uk Yi (whose paper is not actually finished yet, it is "work in progress"). There are however two big differences between Yi's and Chase's background assumptions. One: Yi does not take as part of the problem statement that Envelope A was defined by picking one of the two envelopes completely at random. Two: Yi works with his own, different, logic of counterfactual reasoning. According to his framework, all three arguments are wrong since the statements they aim to derive are all completely meaningless. His solution, you could say, is that there is no solution. In conclusion, I think the idea that Smullyan's paradox is still unresolved or that there is controversy about how to solve it, nonsense. Richard Gill (talk) 11:44, 1 January 2012 (UTC)

3 Contexts, 2 Aims gives 6 problems, 6 solutions
Our task is not to come up with new solutions of the TEP paradox but to survey the existing proposed solutions, of which there are many. In my humble opinion the multiplicity of solutions comes from the multiplicity of ways to imagine a context within which the argument is placed, and the actual aim of the writer of the argument. I can see 6=3*2 main possibilities. Most of them can be found in the literature. There are 3 different implicitly understood contexts, or probability models. Context 1: the amounts of money in the two envelopes are fixed, say 2 and 4 Euros, and the only randomness is in the random allocation of one of these two amounts to Envelope A. Context 2: the amounts of money in the two envelopes are initially unknown, and our uncertainty about them is described by a (proper) prior probability distribution of, say, possible values of the smaller amount of money. For instance, it could be 1, 2, 3, ... up to 100 Euros, each with equal probablity 1/100. Context 3: we have no information at all about the possible amounts of money in the smaller envelope, so we use the standard non-informative Bayesian prior according to which the logarithm of the amount is uniformly distributed between -infinity and +infinity. There are two different implicit aims of the writer. Aim 1: to compute the unconditional expectation of the amount of money in Envelope B. Aim 2: to compute the conditional expectation of the amount of money in Envelope B, given any particular amount, say a, which might be imagined to be in Envelope A. Note that this second aim does not depend on actually looking in Envelope A. In any of the three contexts we can imagine being informed the amount in Envelope A and thereupon doing the calculation. If the result of the calculation is that we would want to switch envelopes whatever the imagined a might be, then the calculation is useful even when we can't look in the envelope: it tells us to switch, anyway. From studying the origins of TEP (two-necktie problem, two-sided cards problem) and knowing the authors of the papers which introduced all these problems, I think we can deduce that the writer is a sophisticated mathematician who is deliberately trying to trick the unwary not-mathematicially-sophisticated reader. This implies, for me, that the intended context of the problem is Context 2 (proper Bayesian) or perhaps even Context 3 (improper Bayesian); and that the aim of the writer was to compute the conditional expected amount in Envelope B given any particular imagined amount a in Envelope A. He uses the ambiguity of ordinary language to seduce us to take the conditional probabilities that the second envelope contains twice or half what's in the first, given any particular amount in the first as 50/50, whatever that amount (in the first envelope) might be. It's a mathematical theorem that this last fact is impossible, at least, with a proper prior distribution over the amounts possible. We are easily seduced, since these conditional probabilities are in fact entirely correct, had we done formal calculations using the traditional improper prior which is conventionally often used to describe total ignorance. However the philosophical literature is written by people who are often complete amateurs in probability theory, and they cannot even imagine such a sophisticated context as Context 2 or Context 3, nor such a sophisticated aim as Aim 2. This literature focusses on Context 1 and Aim 1. Personally I find this unsatisfactory since the more sophisticated context makes the paradox much more interesting; the "mistake" in the argument is much more subtle. On the other hand, if we assume Context 1 and Aim 1, then the writer of the TEP argument is not just making one stupid mistake but several. He is not only confusing random variables and possible values they could take, but he is also confusing expectation values and random variables. The economic and decision theory literature tends to take the more sophisticated context and aim. Richard Gill (talk) 18:13, 1 January 2012 (UTC)


 * Sure I agree with most of what you say. (But you still don't even mention the non probabilistic versions of TEP when listing all the versions of TEP you think there are. Why you always omit them is still a mystery to me. It might be some kind of psychological blind spot that I don't understand.) I think the article now clearly states the fact that the problem has different interpretations and exist in different versions with different wordings. Your tree main versions are there for sure. Your Aim 1 and Aim 2 is relevant in the first version of TEP (when we don't look in an envelope) but I can't see the relevance of Aim 1 in version 2 and 3 where Aim 2 is stated explicitly. Which authors made such interpretations? Please remind me. I can't remember I read any accounts of that sort. If these interpretations really do exist in the literature we should represent them the article as well. However, the number of different proposed solutions are not only caused by the number of different versions and interpretations. There are genuine disagreements on how to solve all the versions. As well as fundamental disagreements on how many different versions there really are. This fact is still not explained with enough clarity in the article I think. iNic (talk) 13:02, 2 January 2012 (UTC)


 * I am only talking about the standard TEP problem as introduced in the article: the one with the 5/4 calculation, and where we do not look in Envelope A. As far as I can see the philosophical literature tends to choose as context either Context 1 or Context 2, and as aim always Aim 1. In my opinion the philosophical literature is wordy and fuzzy but there is broad agreement how "their" paradox is resolved. The statistical and the mathematical economics literature tend to go for Context 2 or Context 3, and as aim Aim 2. I don't see any disagreement concerning how it can be resolved. Regarding the educationalist and psychology literature, Ruma Falk seems to have moved from Aim 2 to Aim 1 over the years in her published papers, but in recent correspondence with me she seems to go back to her earliest position. Her first paper (Falk and Konold, 1992) also has explicitly the right improper prior (uniform prior on the logarithm of the amount), whereas later papers make the common mistake to think the prior being used is the wrong one (uniform prior on the amount itself). I don't mention Smulyan's TEP without probability as I consider it has been adequately resolved and in any case much less interesting. When we allow the player to look in Envelope A, the problem changes completely. There is no paradox any more. Depending on your prior, you will switch for some values of a, and not for others. Or if you don't want to make your decision by optimizing the expected gain relative to a given prior, you can use the trick of comparing the observed value of a with a random probe. These are interesting problems to look at, but there is nothing paradoxical any more. I plan to go through all the papers I know and classify them according to my scheme. Do you have the papers by Zabell? If so could you put them in the dropbox? My present impression is that authors who agree on the same context and aim, tend to agree as to resolution. Please tell us if you know any striking disagreements. Richard Gill (talk) 16:08, 2 January 2012 (UTC)


 * I have one of the Zabell papers, but only in print and not as a pdf file. I could scan it and put it in the dropbox if you want. I will go through all the papers as well and classify all different views. In some cases it was years since I read the papers so I need to refresh my memory. iNic (talk) 01:25, 4 January 2012 (UTC)


 * Yes please! I also need to sit down and rewrite my own paper, starting completely fresh, and reread the important papers. Richard Gill (talk) 09:51, 5 January 2012 (UTC)

Moved "sources" to talk page
I moved the "sources" page to http://en.wikipedia.org/wiki/Talk:Two_envelopes_problem/sources, and tried to delete http://en.wikipedia.org/wiki/Two_envelopes_problem/sources. The reason for this is because the "sources" page is very very useful to editors of TEP, but it has had a notice on it that it is going to be deleted for more than half a year now. It would be a pity if it were lost. Richard Gill (talk) 16:34, 2 January 2012 (UTC)


 * Oops. Now an editor called Fastily has deleted both pages! See User talk:Fastily. Help! Richard Gill (talk) 13:01, 3 January 2012 (UTC)

I think we lost it. iNic (talk) 05:11, 4 January 2012 (UTC)


 * I think we have it back, see /Literature. Richard Gill (talk) 15:31, 4 January 2012 (UTC)

Great thanks. iNic (talk) 16:19, 4 January 2012 (UTC)


 * I see that someone has added links on the main article page to the two TEP talk subpages Literature, and Arguments. I think this is illegal according to standard wikipedia rules. Subpages of main pages are not allowed, and links to talk pages are not allowed. Richard Gill (talk) 14:11, 19 January 2012 (UTC)

But you created the subpage for the sources and moved the sources there. Now you say that "subpages of main pages are not allowed." So now the page must be deleted? Why not then delete it in the first place? I don't get it. And if we are not allowed to link to it how on earth will people find it? iNic (talk) 15:37, 20 January 2012 (UTC)


 * To be precise: wikipedia *articles* (pages in what is called the main or article or default namespace or just the namespace) are not allowed to have subpages, and not allowed to link to wikipedia pages which are not in main namespace. On the other hand, Talk pages (both User and Article) can have subpages and can refer to other talk pages or talk sub-pages (pages in Article Talk namespace, User Talk namespace). I discovered these rules when trying to find out why the "sources" page had been deleted. And I put links from the main talk page to the "Literature" subpage alongside of that to "Arguments" subpage. So interested *editors* can find this material. For ordinary *readers*, the article has to be self-contained. Richard Gill (talk) 15:17, 21 January 2012 (UTC)


 * PS See WP:SP and WP:WORP - the guidelines on subpages and workpages. Richard Gill (talk) 15:42, 21 January 2012 (UTC)

I read the guidelines regarding this now. Pity that subpages to articles aren't allowed. I think we should ask an admin for advice here. There is no point having a page for all sources if we're not allowed to link to it from the main page. If it's formally a Talk page or not doesn't really matter. What do you think? iNic (talk) 05:13, 22 January 2012 (UTC)


 * I think that the big list of references is mainly a resource for *editors*. A splendid resource, in fact! For *readers* who are looking for a big literature list, we should refer to one of the more recent papers with a more or less complete reference list, at least, regarding material which has been properly published. There is a lot of stuff on "our" literature list which doesn't meet the wikipedia criteria of a "reliable source". Self published, un-refereed, not-independently-edited.Richard Gill (talk) 19:53, 22 January 2012 (UTC)

Second variant does not require Envelope A to be opened
The second variant starts by supposing that Envelope A is opened, so that the amount A can be considered a fixed known quantity. However that is not the usual interpretation of this way of reading TEP. One just imagines opening Envelope A. One argues that for any amount which might be found there, one would prefer to switch. Hence one should switch anyway. No need to take a look.

I have added this interpretation, and also added a reference to support it. In fact, this way of reading the TEP argument is the original way in which Nalebuff (1989) intended the paradox to be understood. In the same paper, he also gives the resolution which the article attributed to a later paper (Christensen and Utts). The comprehensive paper of Nickerson and Falk also pays a lot of attention to this solution.

A few later authors seem to think that this second resolution only applies to the situation when Envelope A is opened, but others are quite explicit that this interpretation is not necessary. This is a point where non mathematically trained authors sometimes betray lack of understanding of the more mathematical parts of the literature. If someone thinks that this is a point of controversy then they can add some more references, and I will add some more counter-references. For instance, Ruma Falk has held different points of view at different times. Richard Gill (talk) 14:11, 19 January 2012 (UTC)


 * I also edited the third variant (Broome example and similar) to show that the reasoning does not demand that Envelope A is opened. I note that an example like this is already contained in Nalebuff's 1989 paper. It seems to me that actually, very little has been added to the literature on TEP since Nalebuff (1989) except for the "noise" generated (imho) by the philosophers, who did not understand enough probability to understand Nalebuff's work. Richard Gill (talk) 13:34, 20 January 2012 (UTC)

You deleted the explanation for why looking in the first envelope still lead to a contradiction. Why did you do that? iNic (talk) 15:20, 20 January 2012 (UTC)


 * I thought it was a minor point and distracted from the fact that we have a full-blown paradox when we don't look in the envelope at all. This variant is *not* about the situation when you look in Envelope A, first. It is about the situation when you *imagine* looking in Envelope A. I realise that in the literature there is some confusion about this issue. It's clear to me that the writers in the philosophy literature simply did not understand the points and arguments made in the mathematical literature. There is quite a big problem in the literature with writers reinventing the wheel, over and over again. As far as I can see Nalebuff (1989) covers just about all interesting variants of standard probabilistic TEP up to and including Broome-type examples. But he writes for mathematical economists who are very familiar with probability calculus and not scared of mathematical formulas. Richard Gill (talk) 10:56, 21 January 2012 (UTC)


 * PS, personally, I don't think that the case in question should be called the "second variant" as if it were a new paradox, and a different paradox, from the "first variant". In fact, the so-called "second variant" is actually the original two envelopes problem, Nalebuff's 1989 paper contains the resolution. What in the article is presently presented as the original paradox and the standard resolution of the paradox is actually a new variant. A lot of the writers following Nalebuff did not read or understand his paper, and invented a new TEP paradox by supposing that the writer is trying to compute the unconditional expectation value of B. That is of course a less sophisticated understanding of the context. On the other hand, it requires us to believe that the writer is making two cardinal errors, not one; he is confusing expectation values with actual values, as well as using the same symbol to stand for two different things. He's not only using the same symbol for two different things "at the same level" so to speak, he is also using the same symbol for concepts "at different levels": an unknown amount, and the expected value thereof. TEP was invented by mathematically sophisticated writers who wanted to tease amateurs. As you yourself said I think, their joke has indeed misfired. Richard Gill (talk) 11:04, 21 January 2012 (UTC)

But we must keep in mind that we also have to write for the general public here. In lack of a good popular account and overview of the subject we have to try our best to write this popular overview ourselves. The first variant is therefore important not to omit. It is also easier to explain that it's a new situation if we are allowed to actually look in the first envelope and why that leads to a paradox as well. That there should be a new variant just by imagining that one envelope is opened is too esoteric for most people. Some years ago I read a blog where they discussed TEP and there one guy said that he found the twin construct in the Wikipedia article helpful for understanding this version of TEP. iNic (talk) 05:33, 22 January 2012 (UTC)


 * I agree with most of what you say, iNic. In particular, one can create two different families of TEP paradoxes depending on whether or not the player looks in Envelope A before being asked whether or not to switch. But at the same time, within the family of paradoxes where the player does not look, there are two main interpretations of what the argument is trying to do. Hence two main diagnoses of where it breaks down. I agree that for many readers the notion that you can imagine opening the envelope, and then imagine what your decision would be for any possible amount you found there, will initially seem esoteric. Apparently this is exactly what happened when TEP made the step from the economics and mathematics literature to the philosophy literature: it mutated, one could say, from what presently is called Second Variant, to what presently is called First Variant. HIstorically these appeared in the opposite order, and they were two different interpretations of what the writer was trying to do, within the same variant, the same general set-up. One can see the effect of this mutation in the papers of Ruma Falk: over the years, perhaps under the influence of different co-authors, perhaps under the influence of classes of new students, her own favourite resolution shifts from Variant Two to Variant One. It's our job to explain to the ordinary reader why Variant 2, but with Envelope A still closed, is less esoteric than first appears. Probably you, iNic, can do that better than me. I also agree that we need both interpretations alongside one another, and there is nothing wrong with putting the one which is easier for ordinary readers to understand, first. In mathematical terms the difference is whether we are trying to compute an unconditional expectation or a conditional expectation. In both cases we are doing this by splitting over the two possibilities that Envelope A contains the smaller or the larger amount. In the interpretation that the writer is doing the unconditional expectation, he gets the probabilities in the two cases right, but the values in the two cases wrong. In the interpretation that the writer is doing the conditional expectation, it is exactly the other way round. Richard Gill (talk) 09:26, 22 January 2012 (UTC)


 * BTW there seem to be only two good overviews of the subject: Nalebuff (1989), and Nickerson and Falk (2006). Neither of them "popular". As far as I know, every "popular" account focusses on just one interpretation, just one variant, and presents the author's favourite solution. Both Nalebuff (1989) and Nickerson and Falk (2006) emphasize that there are many different ways to interpret the problem. See especially Nickerson and Falk's summary of their main points, on page 184, end of the section "Overview". Richard Gill (talk) 10:33, 22 January 2012 (UTC)

My suggestion is that we present the second variant as the one where we look into the first envelope. The paradoxical nature of this version can be explained in two ways. The first is via the twin construct. The second is to infer that as the reasoning is the same whatever we see in the first envelope we don't have to look in the envelope, only imagine that we do it, and conclude that this variant is paradoxical for exactly the ame reason as the first variant. In this way we will cover the case when we only imagine that we look, as an intermediate step in the reduction of the version to the first one. iNic (talk) 02:33, 23 January 2012 (UTC)


 * That's a possible way to proceed. The twin construct works to show that all (symmetric) variants are equally paradoxical. You imagine your twin, who is the owner of Envelope B, doing just the same as you, and also coming to the conclusion that they want to switch. So the owners of both A and B want to switch, whether or not they look in their envelopes and (if they do look) whatever they both see. Both think their situation is improving. But they can't both be right. Nickerson and Falk (2006) present the paradox in this way, not as leading to the wish on the part of one player to go on switching infinitely often. Also Nalebuff (1989) presents it this way: there are two players, how can it be that both of them want to switch? This way of thinking about what is paradoxical about the TEP argument makes the question whether or not we are actually looking in the envelope irrelevant. NB Nalebuff concentrates on the situation where the envelopes are opened but he also mentions that according to the conclusion of the argument, there is no need to look at all. Gardner (1989) considers the envelopes (for him, they are boxes) closed but is clearly conditioning on the amount x which might be in the player's envelope (box). NB I don't think there is anything esoteric in remarking that the conclusion to switch did not depend on what was seen in the envelope, hence also can be used by someone who does not look. We are imagining the whole thing, anyway! Richard Gill (talk) 12:06, 23 January 2012 (UTC)


 * PS to iNic, do go ahead and put the twin argument back in Variant 2 if you think it is important. As long as Variant 2 also leads to Variant 2-b: the TEP argument is an argument about conditional probabilities given A=a, whatever a might be, and without actually finding out what a is in the case at hand. Richard Gill (talk) 12:25, 23 January 2012 (UTC)

Question from an anon.
I removed this question from the article:

{Question to author: How do the men know that the other person has either half or double. That is not given in a normal situation. If the two possibilities were 0.7 and 1.1, each would stand to lose by switching :-)}

The simple answer is that that information is given to us as part of the setup. But already the conjurer has started his deception. Martin Hogbin (talk) 11:38, 20 January 2012 (UTC)


 * It's important to make the set-up as clear as possible. First two envelopes are filled, one with twice what is in the other. The envelopes are closed, from the outside they are indistinguishable. Then one envelope is picked at random and named Envelope A. For a probabilist, this has an unambiguous mathematical description. The contents of the two envelopes are determined by the smaller of the two amounts. So first we create X > 0, then we define Y =2 X, then we toss a fair coin, independently of the actual values of X and Y, and define A=X, B=Y if the outcome is "heads", A=Y, B=X if the outcome is "tails". This mathematical description covers both a frequentist model in which X is the outcome of a truly random process, whether known or not, and a subjectivist model, in which the probability distribution of X is merely a description of our personal prior knowledge (guesses, beliefs) as to what the smaller amount might be. It also covers the situation where X is actually fixed, say, at the value 2. That's because a "constant" is a special case of a "random variable". This the advantage of an abstract mathematical description: the same abstract mathematical framework can apply to very different real world situations, situations which an amateur might tend to think of as being completely different. Richard Gill (talk) 13:46, 20 January 2012 (UTC)

Article gives wrong impression about status of TEP
The article starts with the statement "It is quite common for authors to claim that the solution to the problem is easy, even elementary. However, when investigating these elementary solutions they are not the same from one author to the next. Currently, at least a couple of new papers are published every year."

This seems to me to be very misleading. New papers will be published on TEP for ever, because of its status as popular brain-teaser, popular example in elementary probability classes, etc. So new writers will keep looking at the problem and write their own opinion in their own words. Many won't study the whole literature on the problem. However, looking at the whole literature, though many authors claim that earlier solutions were false and theirs is the first correct solution, it seems to me that there are only a couple of resolutions to the standard TEP paradox, which keep being reinvented. And each different resolution corresponds to a different imagined context to the problem. Then some people like to resolve such a paradox by demonstrating an internal, logical error in the argument, others like to resolve such a paradox by demonstrating an external, modelling error. Of course there have been some inventions of new paradoxes, e.g. Smulyan's no probability version; and like Cover's randomized solution without use of prior probability distributions, for the new variant in which you open your envelope, right away. These are different, but related paradoxes; it can be debated whether they even deserve the name "paradox". One is a nice example of the ambiguity of ordinary language, the other is a cute trick. Richard Gill (talk) 11:15, 21 January 2012 (UTC)


 * I think the article should make clear, especially for the general reader, that the TEP was never intended to be a serious mathematical problem. It is more like the missing pound problem:


 * Three young chaps go into a restaurant. They buy a meal for £30, so they pay £10 each and then leave. The manager then realises they have been over-charged by £5, so sends the waiter after them with the money. The waiter keeps £2 as a tip, and gives each chap £1 change. That means, they each paid £9. 3 x £9=£27 The waiter's tip is £2, making £29, so where has the other £1 gone? 


 * We could argue forever about what the author of the above problem intended the calculation to be, and exactly where the error lies, but the fact is that the author probably did not have any proper calculation in mind at all. They give an intentionally befuddled calculation with the intention of generating an non-existent paradox.  Once the problem is put on a proper and clear mathematical footing the paradox disappears.


 * The TEP is not so different. It is never clear exactly what the author is claiming but the hope is that the reader will interpret it in some paradoxical way. Martin Hogbin (talk) 14:55, 22 January 2012 (UTC)


 * Sure, the originators of TEP were mathematicians who wanted to create a simple brain-teaser which would trick amateurs into a nonsensical deduction, and thereby get the amateurs thinking more carefully. I am pretty sure that the originators were aiming to show that using uniform distributions to stand for total ignorance can easily generate nonsense when the number of possibilities is infinite. The idea is that if we know nothing about how the envelopes were filled, and if we imagined looking in Envelope A and seeing the amount a there, we would find it equally likely that the other envelope contains 2a or a/2, since a priori the "host" is equally likely to have put a/2 and a into the two two envelopes, as that he put a and 2a into the two envelopes. This presumption leads to a nonsensical conclusion. Hence the assumption must be wrong. And on further thought, it is indeed impossible to consistently believe that the assumption is exactly true whatever a might be. It leads to an improper prior. Please note that the inventors of TEP were mathematically sophisticated, and in particular, were very familiar with probability calculus and with the impropriety of improper prior distributions (I refer to Schrodinger, Littlewood, Kraitchik, and Nalebuff). Martin Gardner was not a professional mathematician and never understood either the two necktie paradox nor the two envelopes problem. But what must the wikipedia article cover? It must cover the notable literature on TEP. Part of this literature is quite mathematically sophisticated, part is very unsophisticated (to say it politely). The literature consists of popular material, educationalist and psychological literature, philosophical literature, mathematical economics and mathematical statistics. There does not exist an authoritative survey. The closest we have are the articles of Nalebuff (1989) and Nickerson and Falk (2006). (Is there also some kind of survey article in the philosophical literature?). Like it or not, we have to survey what is out there. I think that as well as thinking about the problem themselves, editors of the article ought to do their best to read and understand the two just mentioned articles. Then we should try to write an article which summarizes the main ideas in those two articles. The idea of collaborative editing is that experts and amateurs should collaborate to produce an article which is accessible to amateurs and at the same time does justice to the expert point of view. Richard Gill (talk) 19:44, 22 January 2012 (UTC)

And in the process we must not forget to always have WP:NPOV before our eyes. Our own views within the subject must never ever in any way affect the manner in which the article is written. For example, if we think that some of the TEP variants are vastly more important than the others. Or if we believe that some of the proposed solutions are 100% correct, while all others flawed. iNic (talk) 03:09, 23 January 2012 (UTC)


 * Sure. But we can report that reliable source X thinks that the results of reliable source Y are flawed. The article needs to be written with constant references to sources. Richard Gill (talk) 12:20, 23 January 2012 (UTC)

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 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)