Talk:Two envelopes problem/Archive 3

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Archived Harder Problem

The solution above doesn’t explain what’s wrong if the player is allowed to open the first envelope before being offered the option to switch. In this case, A (in step 7 of the expected value calculation) is indeed a constant. Hence, the proposed solution in the first case breaks down and another explanation is needed.

reworded "constant" to "consistent" in first proposed solution.

Opening the envelope to check the amounts does not fix the inconsistent use of the variable A in step 7 of the original problem.128.113.65.168 (talk) 17:55, 8 October 2008 (UTC)

Archived Proposed solution [to the harder problem]

Once the player has looked in the envelope, new information is available—namely, the value A. The subjective probability changes with new information, so the assessment of the probability that A is the smaller and larger sum changes. Therefore step 2 above isn’t always true and is thus the proposed cause of this paradox. [This paragraph is not true given that the problem with step 7 was inconsistent use of variable A, and not the requirement that the amount in your opened envelope is a constant. Invoking "subjective probability" seems to be incorrect here, in that it doesn't invoke Bayes's forumula in order to recalculate any probablities based on the observation made. 128.113.65.168 (talk) 18:40, 8 October 2008 (UTC)]

Step 2 can be justified, however, if a prior distribution can be found such that every pair of possible amounts {X, 2X} is equally likely, where X = 2ⁿA, n = 0, ±1, ±2,.... But as this set is unbounded (i.e., genuinely infinite) a uniform probability distribution over all values in this set cannot be made. In other words, if each pair of envelopes had a non-zero constant probability, all probabilities would add up to more than 1. So some values of A must be more likely than others. However, it is unknown which values are more likely than others; that is, the prior distribution is unknown. [The argument above is: S is a set with an infinite number of members with non-zero probability, it follows that the sum of the probablity of all members must be greater than 1. This is not true because the elements may then be assigned infinitessimal probabilities. The rest of the paragraph makes no sense because it proposes increasing probablities of certain members when the problem was that the total probablity should be less than 1. I am deleting the "harder problem", and while preserving the second paragraph of "the harder problem" to "twin argument". .128.113.65.168 (talk) 18:40, 8 October 2008 (UTC)]

Other comments

isn't harder. If you say to me "I'm going to flip a coin. If it's heads, I'll give you $20 dollars, tails you give me $10" I will say OK, let's go. Then if you ask me again, I'll keep at it until either you're bankrupt or we've flipped one-hundred tails in a row, in which case, $1000 is a small price to pay to be able to say that I paid a thousand dollars to watch a man flip 100 tails in a row. I'd probably drop $100 to tattoo those words onto my hands so everytime I flipped a coin I got to remember that ridiculous event. A simpler way to see this is to suggest I put ten dollars in your hat, you put twenty dollars in your hat, we flip a coin, and the winner gets all the money in the hat. I will walk away wearing your hat because it's a stupid game.

Point being, there is no contradiction when two people have opened their envelopes, looked inside, and decided to switch. They both see that they have a 50% chance of losing half their money and a 50% chance of doubling it. The potential gain outweighs the potential loss. This is a simple gamble and I'd take it every time. If the problem was "One envelope has an extra $20 bill in it" it would not be any better to switch, since you might lose twenty or gain twenty. —Preceding unsigned comment added by 129.97.194.132 (talk) 17:12, 25 September 2008 (UTC)

If you are offered an envelope and you're being told the other envelope contains either half or double the amount than yours, would you swap ? I'd say, based on this information, you should not make a decision at all. If you'd also been told that the amount in the other envelope was calculated from the amount in yours, I would swap. If, on the other hand, you'd een told that the amount in your envelope was calculated from the amount in the other envelope, I would definately keep my envelope. The 2-envelope problem puts you in the situation you don't know which amount was calculated on which, and both possibilites are equally likely to have occured, so you should not make a decision whether to stay or swap. —Preceding unsigned comment added by 193.29.5.6 (talk) 11:27, 1 April 2010 (UTC)

So-called open problem

Currently this problem is called an "open problem" without even providing a reference for that statement. This paradox has a trivial solution according to statisticians (see for example, http://www.maa.org/devlin/devlin_0708_04.html). In philosophy, it still seems to be a matter of some discussion. Rather than simply stating "open problem", we should dicuss who considers it an open problem (some philosophers) and who does not (statisticians), and provide references for both sides. Simply calling it an "open problem" is grossly misleading and incorrect. Tomixdf (talk) 08:16, 12 October 2009 (UTC)

The page you are linking refers only to the first version of the paradox, not on the second and harder one, indeed the conclusion of the author is:

To summarize: the paradox arises because you use the prior probabilities to calculate the expected gain rather than the posterior probabilities. As we have seen, it is not possible to choose a prior distribution which results in a posterior distribution for which the original argument holds; there simply are no circumstances in which it would be valid to always use probabilities of 0.5.

so apparently he is not considering the harder problem when a probability distribustion is given and the paradox still holds.--Pokipsy76 (talk) 17:27, 17 May 2010 (UTC)

I suggest we put in Devlin's nice explanation for the statistics side (case closed for statistics), with references. Then we can have a section on why this is still considered an open problem in philosophy, again with references. Tomixdf (talk) 09:08, 12 October 2009 (UTC)

Devlin's article is already represented in the article as an external link which is appropriate for that kind of text. What makes you think his solution is without opponents? The irony here is that you start out to claim, as if it were a fact, that there is no controversy here, only to reveal that you think that Devlin's very short explanation from 2004 ended all controversies. This is simply not true. Unfortunately we can't rearrange the article according to your personal opinions, however strong your feelings might be. iNic (talk) 22:19, 12 October 2009 (UTC)

We need to find a consensus as this is turning into an edit war. You can't simply keep on calling this an "open problem" without providing a decent reference. Quantum gravity is an open problem: every agrees that it is not solved. The enveloppe paradox is NOT widely accepted as an open problem. For statisticians, it simply arises from not applying the rules of the Bayesian calculus corectly. We can certainly mention that _some_ still see it as an open problem. But the way in which it is presented now is totally misleading. Tomixdf (talk) 07:59, 13 October 2009 (UTC)

Calling frequentist statistics "more technical" than Bayesian statistics illustrates the extremely low quality of this article quite nicely. What on earth is meant by that? No wonder the article is flagged as problematic. Tomixdf (talk) 08:04, 13 October 2009 (UTC)

I read through the reference that was provided for the "Note". This reference in no way mentions that "frequentist statistics is more technical", or any other of the nonsense statements in the Note. Tomixdf (talk) 18:40, 13 October 2009 (UTC)

Please try to calm down. Simply deleting sections because you don't like the content isn't called edit warring, it's called vandalism. Your "arguments" for your vandalism doesn't make sense. Try to improve the article instead of destroying it. It can be greatly improved by anyone that cares to read a substantial part of the articles (and not just one or two!). When you have studied the subject you are welcome back in helping to improve the article. iNic (talk) 22:41, 13 October 2009 (UTC)

Let's stick to the topic and to arguments, please. (a) Where in the reference that you provide does it say that "frequentist statistics is more technical than Bayesian statistics"? (b) Where is the reference that this is considered to be an open problem in statistics? One can find many articles that attack the theory of evolution. That does not mean it is an "open problem in science". (c) There is a controversy out there - I do not dispute that at all. It just needs to be described in a correct way. For example: this is a solved problem according to A and B, but C and D dispute this, because of this and this reason. Again, there is no agreement that this is an open problem, so it should not be stated as an undisputed fact. (d) If this is an article on Bayesian decision theory, then why is there no example of the application of the Bayesian calculus? The (again unreferenced) "proposed solution" is clearly wrong in that respect - it does not result from any Bayesian reasoning. Tomixdf (talk) 07:14, 14 October 2009 (UTC)

This article IS disputed. We are disputing it right now, and there are numerous complaints about the article in the talk page. So removing the "disputed" box is unreasonable. Tomixdf (talk) 07:17, 14 October 2009 (UTC)

But please go ahead and write your own account of this problem and it's true solution, according to you! There are lots of space on the internet. There is space for both of us, believe me. One other editor did exactly that in the past because he thought that this article didn't show the "true" solution. He had studied three published articles and seen the light. You find his article here Exchange paradox. It will be very interesting to read your account where Devlin is the Darwin of the two envelopes problem, and the rest of us are just irrational dumbfucks. If this trend continues we will in the end have a bouquet of articles at Wikipedia all claiming that this problem is solved and not an open problem. But of course, they will not claim that the same solution is the true solution... Retorical question: who are then the real dumbfucks? iNic (talk) 19:31, 14 October 2009 (UTC)

iNic, you are now edit-warring. You've made three reverts in 24 hours, and you are using reverts as a substitute for answering the valid points raised above by Tomixdf. You're also calling our edits "vandalism" which misrepresents what you are doing. If you really think we are vandalising the article, report us to the admins and see what happens. Removing a paragraph which is poorly referenced and contains multiple absurdities is necessary to stop the article being misleading. Are you going to engage in constructive discussion or are we going to involve the edit-warring noticeboard? MartinPoulter (talk) 22:21, 14 October 2009 (UTC)

Within Bayesian statistics, this is a solved problem, which is even used in teaching Bayesian statistics (see for example Teaching statistics, Vol. 31:2, 2009, pg. 39-41 for a recent reference with the solution, and many others). The solution for the problem as formulated on Wikipedia has even been discussed in a peer reviewed publication (Teaching statistics, Vol. 30:3, 2008, pg. 86-88). Nonetheless, it seems that this problem is still very much discussed in publications on the philosophy of probability (ie. by philosophers). The article should simply reflect this situation: to Bayesian statisticians this is a solved problem with a trivial solution, but philosophers are still discussing for this/that reason. An interesting story, in fact, which can be resolved without edit war IMO. Tomixdf (talk) 06:51, 15 October 2009 (UTC)

So there are two articles on Wikipedia about the same problem? In one it is called an open problem in Bayesian statistics (without providing a reference), and in the other it isn't (with references)? What a mess - this needs to be fixed. On first glance, the other article is fine. It's the explanation that is found in Devlin and many other references, and it involves using the prior distribution of the amount in the envelopes (which you need to solve the probem). Tomixdf (talk) 07:44, 15 October 2009 (UTC)

Aha there is finally a solution that conclusively solves the problem? Wow that's really great news! Why didn't you tell me from the beginning? And why don't you add this final solution to the article??? At the same time you of course need to explain why all other suggestions for a solution are wrong. Also don't forget to mention who finally solved the problem first. I will applaud this kind of enhancement of the article, any day! iNic (talk) 23:14, 15 October 2009 (UTC)

But until you do this kind of total rewriting of the article I will consider all partial deletions of the article as it stands now as vandalism. The reason is that the article doesn't make sense if it tries to hide the fact that the problem is open when at the same time several different solutions are displayed in the article. Everyone that can read will be able to see that an article that says that a problem is solved while it displays several contradictory solutions is incoherent. iNic (talk) 23:14, 15 October 2009 (UTC)

As there already is a wikipedia article of your liking partially covering this subject (the one by Xbert) I suggest that you start out editing that article to include your new (or old?) groundbreaking news. In this way we will have one article claiming the problem is solved and another one (this one) where all solutions are welcome, which will of course include your favorite solution. iNic (talk) 23:14, 15 October 2009 (UTC)

I will again request that you adopt a constructive attitude, stick to arguments, and avoid name-calling (calling people "dumbfucks" does not belong in a discussion on a Wikipedia talk page), threats and sarcasm. Often an article gets better when several editors with different initial opinions try to reach consensus - I suggest we do that. (a) I did not say the problem is solved: I said the problem is considered to be solved within the Bayesian statistics community (which it is). I also provided several references to back that up (see above), and can provide a ton more. (b) What are your _arguments_ for not including this? (c) Are you aware that in Wikipedia, having two articles on the same subject which contradict each other is not acceptable? I'm not interested in more threats or sarcasm, but expect arguments this time as answer to (a), (b) and (c). Thanks. Tomixdf (talk) 05:48, 16 October 2009 (UTC)

This article is clearly strongly disputed (see above discussion, and I'm not the only one taking part). I've added the disputed box again. I don't understand why you keep removing it: it simply flags that there is a discussion going on. We can remove it when we have reached consensus. Tomixdf (talk) 05:53, 16 October 2009 (UTC)

(a) It is obvious from the article itself that the problem isn't solved. Everyone that can read can see that. For everyone that bothers to read the references it's even more obvious. The article would be too long if all ideas would be represented. Your claim that the problem is already solved "within the Bayesian statistics community" is of course unreferenced and absurd. All suggestions so far has has come from "within the Bayesian statistics community," (except the ones that try to solve the non-probabilistic variant). The articles you refer to doesn't support your claim at all, on the contrary. The second sentence in Falk 2008 that you refer to states that the problem "has not yet been entirely settled." So for those that can't infer that conclusion themselves can at least take Falk's word for it. (b) Sure I will include these references, no problem. (Falk and Nickerson's papers have in fact been included in the list on this page before as 'forthcoming.') (c) Sure I agree here too. There shouldn't be two articles covering the same subject, let alone a bouquet of articles. If your read the talk pages of Xberts article you will see that I've tried to merge the articles before. But sometimes I just give up when confronted with compact stupidity. I'm sorry for my strong language in this context. iNic (talk) 03:15, 19 October 2009 (UTC)

I note that INic is refusing to debate the specific points of the disputed text but is using claims of vandalism (which can't be backed up) as a substitute for proper procedure. The claim (twice) that Bayesianism is "less technical" than another interpretation is unsupported (and absurd) original research, and INic shows no interest in actually the defending the disputed text. MartinPoulter (talk) 15:42, 17 October 2009 (UTC)

It's always vandalism when someone deletes an entire section aimed at placing the the subject of an article into the correct context. It's definitely not "proper procedure." The readers that want to read more about Bayesianism, frequentism and their similarities and differences will follow those links to learn more. I suggest that you do the same. iNic (talk) 03:15, 19 October 2009 (UTC)

iNic: That is not a correct definition of vandalism. Please read WP:VAND (In particular, the second paragraph: "edits/reverts over a content dispute are never vandalism") -- Foogod (talk) 00:38, 27 October 2010 (UTC)

Consensus for progress

Just to spell it out, here is the argument for removing the edit-warred "Note".

• "Because the subjectivistic interpretation of probability is closer to the layman's conception of probability, this paradox is understood by almost everybody." -two unreferenced, dubious factual statements.
• "(This follows from the fact that Bayesianism is a project that tries to mathematically capture the lay man's conception of probability, without running into paradoxes.)" -unreferenced and obviously false "fact".
• "However, for a working statistician or probability theorist endorsing the more technical frequency interpretation of probability this puzzle isn't a problem, as the puzzle can't even be properly stated when imposing those more technical restrictions." -two descriptions of frequentism as "more technical" are unreferenced and absurd.
• <ref>Priest and Restall, ''Envelopes and Indifference'' [http://consequently.org/papers/envelopes.pdf PDF], February 2003</ref> -Self-published source rather than reliable source. Source doesn't back up the statements made in the note.

Since there are two editors setting out arguments in terms of policy why the note should be removed and one editor who is edit-warring rather than addressing these obvious problems, we have consensus for a change. MartinPoulter (talk) 14:42, 18 October 2009 (UTC)

Please see my comment above. iNic (talk) 03:15, 19 October 2009 (UTC)

...which doesn't even address the points I've raised. You do realise that Wikipedia has a no original research policy, and that this policy is not optional? MartinPoulter (talk) 12:15, 19 October 2009 (UTC)

Agreed. The "note" is OR and unreferenced: it has to go. Tomixdf (talk) 06:24, 20 October 2009 (UTC)

Generalizing the fallacity

Should we dedicate a page for the fallacities we can intuively make when calculating the expected value and support it with a variety of examples, like this paradox ? Are the fallacies even clear ? —Preceding unsigned comment added by 94.225.129.117 (talk) 07:48, 3 April 2010 (UTC)

This would be a great idea. A nice example to think about is the following:

Suppose God gives you some money and gives you the choice to keep or switch it. He also tells you either A or B: (assume the prior distribution of any money God has is linear and a coin has 50% chance on tail / head)

A. If you decide to switch, I'll toss a coin. On tails, I'll double the amount, on head, I'll half it. B. If you decide to switch, You'll get the initial amount before I tossed a coin (tails -> it doubled, head -> it halved).

What would you decide in situation A, respectively B knowing your goal is to get the most money out of God.

Note: the two envelope paradox combines both situations. —Preceding unsigned comment added by 94.225.56.12 (talk) 21:28, 5 January 2011 (UTC)

Calling all Probability Theorists: This article needs an edit

Not a probability expert, but from reading the article, talk page, and links, it does seem like there's a concensus solution. Though it doesn't come through very well in the article.

Putting all the peices together, looks like step seven (in the first statement of the paradox) is the incorrect step, because it assumes a uniform distribution (of, say, the amount of money in the lesser envelope). I don't think "impossible probability distribution" is the right phrase (quoting one of the above comments), but in any case, the distribution is not specified in the problem setup, so it's a mistake to assume - in step seven - that it's uniform. For example, what if you were told in advance that the probability distribution is really trivial - like, the amount in the lesser envelope has a 100% chance of being $10. Given this information, step seven would be obviously wrong. Well, you're not told that the lesser envelope has a 100% chance of being $10 - but that doesn't make step seven any more reasonable - since you aren't told anything at all. The fallacy is to assume a uniform distribution when no distribution is specified.

This reasoning is hinted at in the article - for example, it shows that a uniform distribution would have divergent expected values, and that things come out better when you use different variables (for which you aren't assuming a distribution). Hopefully a subject-matter expert will come along and simplify things - it's too much of a re-write for a non-mathematician to attempt.

I'm not suggesting that all the flavor from the article be removed - clearly is has become almost a cultural phenomenon which is itself deserving of some attention - and maybe there are some philosophical implications as well - but it would be improved if the concensus solution were clearly stated somewhere in there.

206.124.141.187 (talk) 11:06, 12 April 2010 (UTC)

By the way, has anyone else noticed the silliness of this statement in the "non-probabilistic" section?

By swapping, the player may gain A (in the case when A = $10) or lose A/2 (in the case when A = $20). So the potential gain is strictly greater than the potential loss.

If I may gain A when A is $10, or lose A/2 when A is $20... turns out that's $10 either way. I think this version of the paradox is self-solving.  :)

206.124.141.187 (talk) 11:25, 12 April 2010 (UTC)

I agree, this article needs a serious edit. The given solutions do not make clear that the paradox is related to a priori distribution assumptions. It also needs some sources. Martin Hogbin (talk) 22:30, 16 May 2010 (UTC)

For an intuitive resolution does anyone think this argument is along the right lines:

Before we can consider whether to swap or not we should consider what is the average sum we should expect to find in our initial choice. Since there is no limit set in the question to the sum that might be placed in the envelope the average sum we should expect is infinite. Herein lies the resolution of the problem. The other envelope therefore might contain infinity/2 or infinity*2 both of which are infinity. Dealing with infinity is like dividing by zero. You can get any answer you want. Martin Hogbin (talk) 22:41, 16 May 2010 (UTC)

Much better now

Well done, Dilaudid. The article is much better after your recent changes. Martin Hogbin (talk) 10:27, 27 May 2010 (UTC)

I think that Clark and Shackel's argument is bogus. Finding an example with infinite sets where there is no paradox does not mean that infinite sets cannot be responsible for this paradox. It is just as easy to use division by zero to show that 2=2 as it is to use division be zero to show that 2=3. Martin Hogbin (talk) 10:33, 27 May 2010 (UTC)

Disruptide edits by iNic again

We're having trouble with iNic again. This time he's using a reference that clearly states that the problem is solved to claim the exact opposite. I've got the article in front of me, and it's MILES away from stating that it is an "open problem". I've reverted using essentially verbatim quotes from the article, but I'm sure that won't be much help. Opinions welcome. I'd welcome a consensus but past experiences indicate that this user does not listen to reason and is abusive (calling people "dumbfucks", stupid etc.). Tomixdf (talk) 10:37, 9 June 2010 (UTC)

I do not think it is fair to call any edits to this article disruptive considering its present state. It is not even clear what the exact problem is. WE need to find more sources, preferably on-line ones that can be easily checked out. Martin Hogbin (talk) 12:55, 9 June 2010 (UTC)

It's disruptive to willfully attribute statements to references that do not contain them, in order to push a POV, while refusing to find a consensus. IMO it's easy to find a consensus. Something like "In statistics, this is widely considered a trivially solved problem [references from stat journals here]. In philosophy, some variants of the paradox still cause debate [references from philosophy journals here]." Also, in WP, a decent peer reviewed article that is accessible online from any university library (such as the ref I provided) is just about as good as it can get. Tomixdf (talk) 13:57, 9 June 2010 (UTC).

I agree with Martin Hogbin; considering the present bad state of the article any (sane) edit will be to the better. I will have more time for Wikipedia now and will try to restore this article. I say "try" because I have the unpleasant feeling that tomixdf will revert almost every edit I will make. User Tomixdf has his own peculiar view of the state of this problem. He claims, without any proof or reference, that this problem is solved "among Bayesian statisticians" but still unsolved "among philosophers." How a problem can be both solved and unsolved at the same time is of course very hard to understand for anyone that adhere to ordinary logic. Who supposedly solved the problem and when it was solved he has failed to explain. This is not a bit surprising of course because it is still unsolved. But this basic, simple and obvious fact (that the problem is still unsolved) can't be stated explicitly in the article as long as Mr tomixdf is in control of the article. In lack of valid arguments he even resort to ad hominem attacks to keep the article in a bad shape. The paper he refers to above is a short article by Ruma Falk which first two sentences read "This probability paradox is one of the most widely discussed problems in recent years. It has not yet been entirely settled, and many find it still disturbing notwithstanding the many insights gained in studies that have analysed the problem from diverse angles" And yet, according to tomixdf, this paper "clearly states that the problem is solved"(!) I hate to get personal, but either I can't read or he can't read. iNic (talk) 01:30, 20 June 2010 (UTC)

There are 1000s of creationists who do not accept the theory of evolution; that does not mean that evolution theory should be presented as an "open problem in science" in Wikipedia. Also, you know very well that Falk's article calls this a solved problem with a trivial solution; he simply mentions that there are many subtle VARIANTS of the problem, which leads to "futile controversies". I simply cannot understand why you keep misrepresenting the sources. For the final time, I request that you adopt a constructive attitude. Tomixdf (talk) 08:51, 20 June 2010 (UTC)

Your "creationist analogy" is totally unfounded and absurd as an analogy in this case. This analogy showcases your personal view on the subject but you have never been able to substantiate this view in any way. I have long ago asked you who, if this analogy were correct, would be the "Darwin" of the Two Envelopes problem. I never got an answer. It would also be interesting to know who you think plays the role of "creationists" in this case. I never got an answer to that either. Please keep in mind that this is your silly analogy, not mine. Last time I asked you about this you complained to the administrators that my tone in my questions was too ironic for your taste. But if you push a POV that is silly, you must be prepared to get questions about it that might have to be on the same level of silliness. iNic (talk) 02:13, 21 June 2010 (UTC)

And no, Falk doesn't call this a solved problem at all. On the contrary she correctly states that this problem "has not been entirely settled" as you can read in the second sentence of his article (as quoted in full above for your convenience). It is easy to find other papers stating the same thing, for example in the recently added paper in the Further reading section by McDonnell and Abbott from 2009. The first two sentences in their paper reads "The two-envelope problem has a long history, and it is sometimes called the exchange paradox (Zabell 1988) or the two-box paradox (Agnew 2004). It is a difficult, yet, important problem in probability theory that has intrigued mathematicians for decades and has evaded consensus on how it should be treated." iNic (talk) 02:13, 21 June 2010 (UTC)

And no again, Falk doesn't at all say that all other variants of the problem that differs from her own only lead to futile controversies. What she says is instead this "There are many variations of this elusive problem in which subtle changes in assumptions, or details of the underlying experiment, call for different solutions. Those apparently minor distinctions have sometimes been overlooked by readers and resulted in futile controversies. To avoid ambiguity, I start by referring to Wikipedia’s puzzle entitled ‘Two Envelopes Problem’." So to be sure to avoid all potential ambiguities in her paper she uses the clear statement of the problem that she could find, at that time, in this Wikipedia article. iNic (talk) 02:13, 21 June 2010 (UTC)

I leave your personal attacks on me without comment. iNic (talk) 02:13, 21 June 2010 (UTC)

I have never reported you to the administrators, even after you started calling your oponents in the discussion "dumbfucks" and "stupid", but I certainly will if you start butchering the article again. The "creatonist analogy" is right on spot: even though there is a trivial solution in Bayesian probability theory (see Falk: "I try to dispell the doubts that sometimes linger despite the sound arguments in the above sources"), the problem continues to raise discussion from outsiders. Within the Bayesian community, the basic form of the paradox is a solved, trivial problem. And once again, I think it is fine to point out that there is a discussion about certain formulations of the paradox in certain communities. So again, there is a simple consensus possible, IMO. Tomixdf (talk) 08:59, 21 June 2010 (UTC)

Do you know that POV-pushing is not at all accepted at Wikipedia? I already know that you think that your silly analogy is "spot on," so I didn't ask for yet another statement of that kind. What I asked for was the sources you have for this claim. If the only source you have is your own opinion we need to move on. (If you want to promote your own interpretation of this problem you are welcome do so on the arguments page. The talk page is devoted to serious editorial discussions only. Please respect that.) iNic (talk) 22:52, 21 June 2010 (UTC)

My issues with the current statement of the problem and solution

The player is presented with two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other (say $1 and $2).

There are two possible interpretations of this: (1) the player is told what sums the envelopes contain, for example $1 and $2, or (2) the sums of money can be anything, with $1 and $2 in the statement of the problem being just an example of sums of money with one being twice the other.

The given "solution" -- that A and B both have expected value $1.5 -- is of course only consistent with interpretation (1). I have to say that when I've encountered the problem before (e.g. in Smullyan's Satan, Cantor and Infinity) interpretation (2) has been explicit. Indeed, interpretation (2) is much more troubling, since you can't simply say "by symmetry, the conclusion is obvious" -- if you don't know the amounts, there is still a paradox about whether or not to switch after you've opened one envelope but in that case there is no symmetry. So surely the presentation ought to focus on interpretation (2) -- otherwise the reader who notices the two interpretations may go away saying, "Well, (1) has been solved, but what about (2)?"

"The error is at the sixth step above. It imitates a calculation of expected value, but is mathematically nonsense. Demonstrably, the result 1.25 A is not a value but a random variable. Any rigorous investigation of the game with conditional probability concludes that A and B both have expected value $1.5."

"Demonstrably" is an infuriating word. It here seems to mean "There is a proof of this, but this Wikipedia page is too small to contain it." What is the proof?91.105.61.167 (talk) 21:41, 15 June 2010 (UTC)

Exactly right. The real problem is that this article is not based on what is said in reliable sources but is just one editors opinion on the subject. We need to find out what the scientific literature says about this problem. Martin Hogbin (talk) 23:02, 15 June 2010 (UTC)

This article it has been in free fall for quite some time. A lot of bad, unsourced edits from anonymous editors has been added and some good sourced sections have been removed. I will try to restore the article as it once were and start to improve it from there. That is, if user tomixdf doesn't revert every change I do... iNic (talk) 01:49, 20 June 2010 (UTC)

That previous version claimed absurdities such as "the paradox cannot be stated in frequentist statistics" and "frequentist statistics is more technical than Bayesian statistics". This was all backed up with "references" that did not back up these statements at all. I was not the only editor who judged the article an absolute mess (see discussion above). Tomixdf (talk) 11:55, 20 June 2010 (UTC)

Can you provide some references for your statement that, 'Within the Bayesian community, the basic form of the paradox is a solved, trivial problem'. Surely we need to start by reading and understanding these references then incorporating what they say into the article. I would be happy to help with this. If there is a significant minority view, or if another solution holds in some circumstances we should say this too. Martin Hogbin (talk) 11:06, 21 June 2010 (UTC)

I agree. Apart from the article that I already provided in the introduction of the article, this article seems to give a good overview of the situation. It states: "The original version of the two envelope paradox is not all that paradoxical", but also "there are strengthened versions of the two envelope paradox where the familiar reasoning seems to hold, but it still does not make any sense to prefer the other envelope." It is these strengthened versions of the paradox (involving infinite expected utilities) that still elicit discussion. Tomixdf (talk) 13:25, 21 June 2010 (UTC)

Do we have any papers on the subject from peer-reviewed mathematical or statistical journals? We also need sources stating the different versions of the problem. Martin Hogbin (talk) 23:07, 21 June 2010 (UTC)

The Falk paper (which is now being abused by iNic to call ALL versions of the paradoc "unsolved") is a peer-reviewed paper in a statistical journal. The other reference I provided (which is also peer reviewed, BTW) contains many other peer reviewed references. It's very sad that the article is turning to bogus again. Tomixdf (talk) 17:59, 22 June 2010 (UTC)

I have some statistical background - I think it's pretty clear that the exchange paradox (or two envelope problem, if you prefer) is resolved. This link has a nice summary of the literature: [1]. "I think it fair to say that most mathematicians would consider the two-envelope paradox to be resolved for all practical purposes. But there are some loose ends that continue to trouble

some thinkers (philosophers particularly)." I appreciate that this may be an unsolved philosophical problem (is there any other kind?) but it is not an unsolved statistical problem. If it were, statistics would be in a very bad state. --Dilaudid (talk) 15:15, 23 July 2010 (UTC)

This is an unsolved problem in decision theory, not in statistics. No one ever claimed that it's an unsolved problem in statistics. This article has been in a bad state for a long time due to anonymous attacks but at least this part is restored now. I'm sorry if the article confused you before. I will restore the whole article as soon as I find time. iNic (talk) 00:03, 24 July 2010 (UTC)

I do get confused a lot :) I'm glad that you agree that this is a solved problem in statistics. If it's an unsolved problem in decision theory - is it worth adding a short note to the article explaining what the decision theory paradox is? I'm not 100% clear what decision theory is. Dilaudid (talk) 21:08, 24 July 2010 (UTC)

It's not the case that we have two problems here, one in "statistics" and one in "decision theory" where the first one is solved and the second unsolved. Sorry if my wording made you think that. The problem itself is safely within decision theory -- and only within decision theory. It is not a statistical problem at all. So that this problem isn't an unsolved problem in statistics DOES NOT mean that it is a "solved problem in statistics." I repeat: it's not a problem in the domain of statistics at all. Therefore it's neither solved nor unsolved in statistics. It's a non-statistical problem. This is made clear in the very first sentence of the article where you also find a link to decision theory. "The two envelopes problem is a puzzle or paradox within the subjectivistic interpretation of probability theory; more specifically within Bayesian decision theory." I don't know how to state this in a more clear manner. If you have an idea how to improve it please let me know. iNic (talk) 15:42, 25 July 2010 (UTC)

Two Envelopes "Paradox" is Resolved - New sources

Hi Folks, I think this "unsolved paradox" is actually solved. In order to get this information out to as many contributors as possible I thought I'd make a new section. Check this source out, and we can process it into the article. [2] --Dilaudid (talk) 15:25, 23 July 2010 (UTC)

Where was this source published? Martin Hogbin (talk) 00:31, 26 July 2010 (UTC)

No this is not a solved problem and your new source doesn't disprove all other solutions, let alone is there a general consensus that Federico O’Reilly finally resolved all problems in this paper. To claim that is just silly. iNic (talk) 00:12, 24 July 2010 (UTC)

iNic - I take your point (section above) that you consider this an unsolved paradox in decision theory, hopefully we can get some clarification on this. Since we both agree that this is a solved problem in statistics (indeed a fallacy, as the "Teaching Statistics" source maintains) - so can anyone who regards this as an unsolved statistical paradox please chat here? If everyone agrees this is a fallacy, we can start to move towards de-mystifying this article. Dilaudid (talk) 21:14, 24 July 2010 (UTC)

No I never said that this is a "solved problem in statistics." And yes, everyone agrees that the reasoning contains at least one fallacy. Where different writers disagree are where the fallacy is and what kind of fallacy it is. If you want to improve the article please read at least ten of the published papers to get an overview of the debate. This discussion page doesn't reflect the academic debate about this problem at all. iNic (talk) 16:00, 25 July 2010 (UTC)

Are any of the published papers freely available online? Martin Hogbin (talk) 13:23, 26 July 2010 (UTC)

Some are available online for free, others are available online but not for free, and still others are available for free but only at the library. iNic (talk) 14:21, 26 July 2010 (UTC)

I have looked at the few papers available online and I have to agree with you that there is no universally agreed solution to this problem, there is hardly an agreed problem. The solution given in this article is actually part of the paradox according to one paper. Martin Hogbin (talk) 17:36, 27 July 2010 (UTC)

Exactly. The simple form of the paradox has a simple solution; nobody disputes that. But there are many intricate variants (typically involving infinity somehow, or ambiguously formulated) that can most definitely be regarded as "unsolved", or at least "controversial". You can find a very good discussion of this in Falk (Teaching statistics, vol. 30, 2008), but there are numerous other references. So the current Wikipedia article gives a completely false view: it states that ALL versions of the paradox are unsolved, which is totally incorrect. On top of that, it limits the paradox to the "subjective interpretation of probability", which is equally incorrect. Note that none of these statements are backed up by the two references provided. Tomixdf (talk) 08:52, 28 July 2010 (UTC)

You will see that I have deleted the 'Solution' section given because it was completely unsourced. It seemed to be the OR of one particular editor. Solutions of that type are presented by some sources but are discredited by others. We need to give a balanced view of the situation in the article showing the generally accepted academic consensus on the subject if there is one. Martin Hogbin (talk) 09:13, 28 July 2010 (UTC)

That is fine - I was talking about the introduction. I hope you at least agree that the current introduction is not acceptable? Why don't we put the Falk discussion in, as an example of a solution for one version of the paradox? Tomixdf (talk) 09:26, 28 July 2010 (UTC)

Why not? I am new to this problem but it is clear that there is not one simple, clear, universally accepted solution. Martin Hogbin (talk) 12:12, 28 July 2010 (UTC)

Martin - continuing from your line above, there seem to be at least 3 separate paradoxes that have come out of this. Two are trivial, one appears to be unresolved. 1) Smullyan's "logical paradox", a trivial fallacy, see Albers' "Trying to Resolve the Two-Envelope Problem" [3]. Smullyan uses the same word to refer to two separate things. This is the same as the random variable argument that some have tried to add to the (wiki) article. 2) A second fallacy - the failure to incorporate the information about the amount in the envelope into a posterior/conditional probability (see Albers, p.90, top). This is also trivial, and is the solution mentioned in the current article. 3) A decision theory problem, [4]. Dietrich and List are using the paradox as a highly unrealistic test case for decision theory, to see if decision theory can be applied to infinite expectations. This loses any basis in the real world, since real world expectations of quantities of money cannot be infinite, but it appears to be an interesting problem. So I would regard the first two paradoxes as "explained" or "resolved" and the third as open (this is the point that iNic was trying to make).

I might be wrong, but I think that the difference between 1 and 2 is that before you look at the amount in the envelope, 1 applies - X is a random variable. After you look at the amount in the envelope, X needs to be treated as evidence - it's a definite amount of money.

A few more things - the word "solved" was an unfortunate word for me to use, the paradox may be resolved/explained without the problem (of whether to swap or not) being solved. In answer to Martin's question - I don't think O'Reilly was published anywhere. Is it worth grouping the sources by which paradox they are referring to? Also Tomixdf/Martin - do you have a free source for Falk? I'd like to read the paper. Cheers all. Dilaudid (talk) 17:37, 31 July 2010 (UTC)

I agree with several things that you say. As with all problems of this nature it is important to decide exactly what the question is. There seem to be several versions of the problem, some where you look inside the envelope before deciding and others where you do not. The question also needs to be quite clear on what sums could possibly be in the envelopes.

Once all this is settled we need to see which situations are indeed paradoxical in that two different lines of reasoning give two different answers. In some formulations it could be correct that you should swap.

What notable formulations are there and is one regarded as the 'classic' formulation? Martin Hogbin (talk) 18:19, 31 July 2010 (UTC)

lack of explanation

there is not enough substance in the article to denote there even being a problem, example: why baysian probability even kicks in with theese few outcomes and so few data, esp. because as any middle school algebra student can tell you. this word problem can only result, by its own definition in two outcomes, x or 2x. any math that cannot stand up to its own definition is suspect to say the least.--67.170.10.189 (talk) 01:56, 29 August 2010 (UTC)

I'd like to disagree. This is a well known problem and this article is both comprehensive and concise, and more details can be found via the lists of further reading and external links. E.g. see my contribution to the latter for one view of how a closely related problem remains even when a trivial resolution is realised, which was in last May's edition of The Reasoner. Username12321 (talk) 09:19, 3 September 2010 (UTC)

I agree there clearly is some kind of paradox here, depending on exactly what the question is. This article ideally needs to state exactly what problem formulations are being considered, what paradoxes there are for each, and how each paradox is resolved, all based on reliable sources of course. Not easy. Martin Hogbin (talk) 10:01, 27 October 2010 (UTC)

Some organisation

Reading the cited and other sources it would seem that two important distinctions are made in the formulation of this problem. One distinction is whether or not the player opens his envelope. Case where the envelope is opened are generally easier to understand and explain and they are sometimes used as a basis for deductions about cases where the envelope is not opened.

The other important distinction is in the range or distribution of sums that might be in the envelopes. For cases where there is a simple finite distribution of possible sums (pretty well any realistic formulation) it seems to be generally agreed that the problem is easily solved. It is in the case of infinite distributions that there appears to be no general agreement.

I would like to rewrite this page to show the various possibilities and to help make clear where there is agreement between sources and where there is not. Is anyone interested in joining me? Martin Hogbin (talk) 12:20, 1 January 2011 (UTC)

Sure I'm interested! There are a lot of interesting ideas around this paradox that easily can be stated in the article, so a lot of improvements are for sure possible. The article had the kind of structure you talk about a couple of years ago with different sections for different versions/interpretations/formulations of the paradox. Another thing that I think would be an improvement is to create a separate page with links to further reading. That page could be the 'complete' reference guide for anyone that want to learn more about this. iNic (talk) 16:01, 7 January 2011 (UTC)

Merge with Necktie Paradox

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

In a way that paradox is already mentioned here in the history section in the form of a wallet game. Maurice Kraitchik wrote about the necktie paradox already 1943, please see the text by Caspar Albers in the bibliography for a reference and citation. So I do agree we have a historical connection here, but I also think that the Necktie Paradox deserves an article of its own. Many of the ideas developed around the two envelopes problem isn't directly applicable to the Necktie Paradox. So historically it's the same problem, but now they are separate problems. However, this historical connection could be stressed more in both articles. iNic 00:28, 23 December 2006 (UTC)

Do not merge. Although essentially the same problem, they are expressed very differently and have different existences in the literature etc. Cross-refer but keep separate. Snalwibma 14:01, 4 January 2007 (UTC)

Do not merge. I agree with Snalwibma.--Pokipsy76 12:08, 30 January 2007 (UTC)

Merge The "history section (Two-envelope paradox#History of the paradox seems to be exactly this problem. —ScouterSig 19:12, 26 March 2007 (UTC)

Do not merge I agree that the history of the two-envelope paradox (the wallet switch) is the same as the necktie paradox, but as noted therein, the envelope problem differs in the presented relationship between values. The envelope problem is very much different to consider than Necktie. I fully agree with iNic above that perhaps the solution is further stressing of the common historic root.130.113.110.75 06:01, 12 April 2007 (UTC)

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

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

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

The envelope is slightly different then the nectie, but the neckties is 100% same as the wallet. Cheaper guy gets the booty! 76.112.206.81 (talk) 07:13, 30 September 2008 (UTC)

Do not merge. The envelope model is superior to the neck-tie device - in the neck-tie conundrum it is an artificial construct to say that a man could see it is a 50/50 chance that his neck-tie was cheapest (in reality a man would guess his tie was cheap/average/expensive) - but in the envelope model the statement that one envelope contains double the other can be considered absolute. Gomez2002 (talk) 13:53, 14 February 2011 (UTC)

Stop substantially editing this if you aren't an expert

I understand referencing sources for citation, but if you need to check the literature to understand how to unravel this 'paradox,' you're not qualified to edit this page. Please be humble and don't. You can argue that 'experts' disagree how to resolve it, but the reason we need experts writing technical articles is that not every PhD who publishes is good at what they (or especially others) do, and experts can sift the noise. This leaves the question of who the real experts are, which we don't need to answer. It's enough to say that if you aren't certain that you're an expert on this topic, you aren't, even if you've read a dozen papers about it.

Paradoxes arise when two different formulations of supposedly the same problem contain divergent, implicit assumptions. The paradox disappears when you identify the missing or altered assumptions in the flawed model. This has been done ad nauseum on these pages. The answers have been expounded comprehensively, but that work is either absent, or marginalized by unwarranted hedging and convolutions in the article, particularly 1) the last sentence of the intro, 2) the title 'Possible Solutions,' 3) the last sentence of that section, and 4) most of the 'A second problem' section. If you claim ultimate editorial authority over this article and you aren't sure of this, please pass the torch. I'm not embarrassed for you, but I cringe a little that, even as a work in progress, you default to favoring ambiguous quibbling over rigorous lucidity. This is not a perplexing philosophical debate, it's semantic obfuscation that has been exposed and corrected. Hand the reins off to workhorses who are at least clever enough to have solved the puzzle themselves. — Preceding unsigned comment added by Nimblecymbal (talk • contribs) 18:56, 26 March 2011 (UTC)

Rewrite (2011-04-08)

I realize I might be stepping into a hornets' nest, but I read this article the other day and found it to be incredibly confusing. The problem is not difficult if probabilities are correctly applied, and I added a more mathematical analysis to demonstrate the proper handling of the problem as well as to explain the obvious errors in the "infinite swapping" argument. I hope this rewrite is not offensive to anyone on this discussion page. Everyone seemed to be in agreement that a rewrite was necessary, and I thought I would take a go at it. If this was inappropriate, feel free to revert the changes.

Also, I'd like to chime in and agree that this problem is not "open" any more than the Monty Hall Problem is considered an open problem. I also don't see what Bayesian statistics has to do with this problem at all. The only Bayesian analysis I have seen includes prior and posterior probabilities that suffer from the same mis-application of probabilities that result in the infinite swapping argument. If we were playing the extended game where X is drawn from some probably distribution and given to us with probability p we could use a Bayesian analysis to determine the likelihood for different probability distributions or different values of p, but this is not how the discussion of subjective probabilities was framed.

Bodhi.root (talk) 20:44, 8 April 2011 (UTC)

Some thoughts

What is this article about

It is important to understand what this article is all about. It is not about getting the right answer to any given two envelopes problem, it is about giving different lines of reasoning that give different answers to the same question. Just giving one clear and correct line of reasoning is not what makes this a paradox.

Agree. Richard Gill (talk) 16:50, 2 May 2011 (UTC)

Different problems

There are several versions of the problem with at least two important distinctions. The first distinction concerns the possible sums of money that might be in the envelopes. This is not normally made clear but it is often assumed to be equally likely to be any whole (non negative) number of currency units. In an alternative formulation there is a range of possible values, say, up to 1000,000.

Possible amounts of money: this is not part of the "given". It is part of the resolution of the paradox. The argument which leads to the silly "switch for ever" conclusion depends on an assumption that the other envelope could equally well be twice as half the first one, *whatever* might be in the first. This assumption however implies that indefinitely large (and indefinitely small) amounts of money could be involved, in fact in such a way that each time you double the size of the amount of money, it it still "equally likely" to be in one of the envelopes. The resolution of the paradox is therefore "garbage in, garbage out". It might be approximately true for some range of values that whatever is in the one envelope, the other is roughly equally likely to be twice or half the first, but it can't be exactly true, whatever.

The other distinction is whether the player opens their envelope before deciding whether to swap or not.

That is a different problem, with a completely different solution, obviously "inspired" or related to the first. The solution is that you should compare what is in the envelope you opened with a "probe" guess; it has to be a very fine probe, in that it has some chance to be inbetween any two *different* amounts of money, whatever they might be. Obviously a mathematical abstraction but still not so crazy as it seems. Choose a random number between 0 and 1, for instance by tossing a lot of coins and writing down heads and tails as binary digits: 0.110101... stands for HHTHTH.... Now take minus the natural logarithm of this number. You end up with a random number Y between 0 and infinity. If the two amounts of money are X1 and X2, then Y will with probability one not be equal to X1 or X2, and with positive probability it will lie between them.

I have posed this problem to many people. No one believes that it is possible to end up with the larger amount of money with probability greater than 1/2. (The same happened to me the first time someone told it to me, 20 years ago). But everyone is finally convinced and amused that it can be done, and goes on to ask the question to new people.

There is no point in talking about solutions unless both these questions, and maybe others, are answered. We need to find out what the versions in sources are. Martin Hogbin (talk) 17:58, 1 May 2011 (UTC)

Agree. Richard Gill (talk) 16:50, 2 May 2011 (UTC)

I am a mathematician, and in the Real World I am the Real Person Richard D. Gill (mathematician). Consequently, I believe I know The Truth about several of these issues. However this is totally irrelevant to Wikipedia, of course. I have drafted a couple of supplementary sections which contain the solutions which I have, over the years, frequently discussed with colleagues - of the two main variants of the problem. I hope (a) the reasoning is clear, (b) these solutions have been published somewhere. If the latter is not the case, I will be happy to generate a "reliable source" for these arguments. However obviously I am not the one who should cite such publications on wikipedia.

If my contributions are controversial, someone could just move them either to this Talk page or to the Arguments page. Richard Gill (talk) 16:50, 2 May 2011 (UTC)

By the way, the present text is totally confused about whether the probabilities in question are to be interpreted in a frequentist or a subjectivist sense. It is also totally confused about unconditional and conditional probability. Of course it makes no difference to mathematics whether we are talking about subjectivist or frequentist probability (the calculus is the same), but the interpretation is different. So you cannot criticise a solution by saying it makes no sense from a frequentist point of view in which the amount of money in the two envelopes is fixed. Clearly the person who thought up the problem had a subjectivist notion of probability. Or they were thinking of many repetitions in which the amounts of money in the two envelopes vary from occasion to occasion according to some probability law.

You also cannot criticise a solution by saying that the amounts of money in the two envelopes are fixed and therefore can't be all the other possible values which they would have to be able to have, if for any amount of money, both half and double are also possible amounts. Obviously the person who posed the problem is either a Bayesian where the player has no idea what the two amounts are, but expresses his uncertainty thereover according to a probability distribution, or the person is a frequentist, thinking of repetition after repetition where the two amounts of money vary according to some probability law. My fingers are itching to delete all "nonsense" parts of the article. I think that what is left could be quite coherent and a whole lot clearer. But OK, let's discuss this first. And let's do some literature research to find Reliable Sources who propose sensible solutions. Richard Gill (talk) 17:16, 2 May 2011 (UTC)

All: Please only present ideas and reasoning from published sources. All unsourced ideas and OR will be removed or moved to the Arguments page. The ordinary Wikipedia rules are appliccable and valid even for this article. This article is in no way excluded form the ordinary Wikipedia rules. Violating the rules will never make the article better or clearer, on the contrary. For those new to Wikipedia please read the rules. iNic (talk) 19:34, 2 May 2011 (UTC)

Sure @iNic. Go ahead and move my contributions if they are unhelpful. I just want to point out that my day job is to write Reliable Sources in exactly this field. If any fellow editors find my comments useful, but cannot locate them (exactly) in the literature, I am happy to oblige. But the most important thing for now, for wikipedia, is to actually study the literature; I fully agree with that. I have not done it - I just frequently discuss these problems with colleagues in mathematics and in particular in probability and statistics, and I assign these problems to my students. It is (for me) elementary probability, elementary reasoning, common knowledge (folk-lore) so no great need (for a teacher of mathematics, in his daily work talking to colleagues and students) to go to the trouble of actually consulting the sources. Though it would be interesting. I realise that this is different for an editor of wikipedia. Anything that is challenged has to be reliably-source-able. Richard Gill (talk) 20:44, 2 May 2011 (UTC)

I just want to point out that Wikipedia is not a public notepad for academics, nor for anyone else. Your careless and arrogant attitude you display when you say "It is (for me) elementary probability, elementary reasoning, common knowledge (folk-lore) so no need (for a teacher of mathematics) to go to the trouble of actually consulting the sources" will not serve you well at all at Wikipedia. Please try to read and understand the rules before you edit an article. The only way for you to get your own ideas into a Wikipedia article is to write an article about this problem and get it published in a peer reviewed (and relevant) journal. After that, someone else has to refer to your article and write a short summary of your ideas. (However, I doubt that you will get an article published in this area as your ideas are not very original. Your basic idea has been published many times before and it has also been shown that this solution is insufficient to solve the paradox. But Good Luck anyway!) iNic (talk) 01:39, 3 May 2011 (UTC)

Richard Gill's response to apparent failure of "Good Faith" assumption by iNic

How come you think I *want* to get *my own* ideas into wikipedia? How can you accuse me of being careless and arrogant in my daily work teaching my students how to do mathematics? Because I'm a professional in this area I often sound arrogant when I open my mouth. It doesn't mean I *am* arrogant. I'm just trying to give you info, not trying to put you down. Do you have an inferiority complex? I assure you, I do not have a superiority complex. Could you please try to understand the matheamtical/logical arguments I gave you and if you succeed, and if you find them useful, then let's work together to make them accessible to the general public. If not they can be consigned to oblivion. Those arguments surely must be in the published literature and I am going to help look for them. Possibly they need some explanation (translation from academic to popular language). Just need to log in to my university workstation so I can get the academic articles without having to pay tax to the publishing companies. Wikipedia is about collaborative editing. I have some specialist knowledge which might be useful for this article. You no doubt have a lot of communication skills, much more than me. Could you perhaps explain on my talk page, @iNic, how you can know that my solution is insufficient to solve the paradox? It seems that you too have access to The Truth. Or do you know reliable sources which I'm unaware of? BTW I was involved for two years on the infamous Monty Hall problem (MHP) wikipedia page, almost received a wikipedia ban for COI and OR, and was accused of arrogance and incivility ten to twenty times. I'm not proud of that, since I didn't mean to be uncivil. I'm sorry that I often appear arrogant but it didn't help when I tried not to be, so now I just try to be myself. I report what I see to be facts (and, for what they're worth, I also report my opinions - I don't demand agreement with the latter). The ideas which came up during the MHP wars - ideas which I learnt from fellow editors, which led me to change my own opinions several times - led to three or four papers by me and others in the academic literature. Nowadays I'm getting asked to review new research articles submitted to maths or philosophy journals on MHP, and I've corresponded with almost all the authors of major sources in the area. So don't be so sure new and publishable research can't come out of collaborative wikipedia editing on the two envelopes problem! I find the conflict between "popular" and "academic" solutions to puzzles like this fascinating, and a microcosm of the problems of communication between the ordinary and the academic worlds which lead to major miscarriages of justice whenever statistics is used in courts of law, for major public policy decisions, etc. Sometimes it hurts to be kind. It can be kind to be cruel. It can be better to be abrasive, so as to bring issues out into the open, rather than waste time going round in polite circles avoiding them. Why should I hide from you that I'm a mathematics professor? It doesn't mean that I demand you to believe everything I say. I hope it would be a good reason to take what I say seriously, as far as that domain is concerned. But it's the facts which count, the sources. Wikipedia editors have to use common sense and common logic and background knowledge in their work. In topics which lie partly within some scientific domain, and partly in the public domain, there are going to be problems defining "own research". Mathematicians who (co-)author articles on (partly) mathematical topics will automatically and uncontroversially perform what, for them, are routine logical deductions and standard derivations, even though the specific deduction or specific derivation is not written down somewhere in accessible published literature. It only needs to be reliably sourced if it is (likely to be) challenged. A scientist who is also a wikipedia editor can publish pedagogical papers containing material which is common knowledge in his field among the fellow professionals, but not written down explicitly enough for educational purposes for the outside world. I can do this for the two envelopes problem, if it would seem useful to the community of wikipedia editors interested in the problem. I did it for the Monty Hall problem. Mathematics consists of tautologies. Things which have to be true by definition. Regarding MHP, it seemed to me that the popular solutions contained valuable insights, and the exciting thing was to create a new synthesis out of apparent contradiction between popular and academic reasoning. Thesis, antithesis, synthesis. That's how science makes progress. It would be wonderful if the same would happen again regarding the two envelopes problem. Richard Gill (talk) 12:32, 3 May 2011 (UTC)

Gill, you explain your unhonest mindset best yourself here: [5] I will not treat you as a WP rookie from now on. iNic (talk) 00:54, 4 May 2011 (UTC)

I think I'm honest and open! You call me dishonest! I call that failure to assume Good Faith. What you see is what you get. I hide nothing. You are welcome not to like it. (By the way, you shouldn't take me so seriously. I do not take myself seriously, either. This is called tongue-in-cheek British humour. To be sure, it confuses a lot of people a lot of the time). Richard Gill (talk) 08:57, 4 May 2011 (UTC)

Please find yourself some other playground if you want to tell jokes. So far no one is laughing. iNic (talk) 12:50, 4 May 2011 (UTC)

I'm laughing.

How about you start being civil, assume good faith on my part, and try to be constructive? Is this *your* article, or something? You just accused me of dishonesty. For no reason whatever (except perhaps that you can't read plain English). Richard Gill (talk) 14:38, 4 May 2011 (UTC)

Can we start by determining exactly what is the definitive or maybe just the first exact version of the question. Is there any limit on the possible money and does the player open their envelope before deciding? Martin Hogbin (talk) 20:29, 2 May 2011 (UTC)

In my professional opinion there are two main problem versions current in the (professional) world of probability, statistics, decision theory, mathematics education, foundations of mathematics. Obviously not all of that literature is of interest to the general reader. But many people do come to wikipedia from these professional worlds (in particular, our students) so if there is some serious advanced maths literature on this subject it should, I think, also get a modest place in the wikipedia article.

Of course the popular literature can contain all kinds of variants, and both a lot of good sense, and a lot of nonsense. I am not familiar with it.

In the first version the envelope is not opened before deciding. There is no stated limit on the possible money. The paradox is solved by realising that no probability distribution on the amount of possible money (say, the smaller amount) is compatible with the "equally likely" assumption of the "silly (switch forever back and forth) solution". In particular, no probability distribution with a finite upper limit would work. But even if we allow smaller and smaller chances of larger and larger amounts of money, the definition of conditional probability and one line of algebra shows that the assumption "the other envelope can equally likely contain half or double the first envelope" leads to the conclusion that it is equally likely that the smaller amount of money is between 1 and 2, as between 2 and 4, as between 4 and 8 .... (and for that matter, equally likely between 1/2 and 1, and between 1/4 and 1/2, .....). So the apparently innocuous assumption "equally likely", if taken seriously and exactly (ie supposed to be true exactly, whatever x) leads to a contradiction with common sense, with reality, and with mathematics for that matter. Thus the argument leads to a nonsensical conclusion because the premises are logically self-contradictory. In other words, garbage in, garbage out.

In the second version the envelope may be opened before deciding. (And there's no restriction that the two amounts of money differ by a factor 2, they just have to be different and positive. They are just two numbers, not necessarily amounts of money; you win if you end with the larger number in your hands). Now it is quite amazingly possible to arrange that you end up with the envelope with the larger amount of money, with probability greater than 1/2. But only if you are prepared to use a randomized strategy. This is wonderful. The idea is that the guy who opens the envelope compares the amount in it with a random amount of money Y (which he conjured up in his head), which has a positive probability to be *between* the two amounts of money in the two envelopes. If both envelopes are smaller than Y this doesn't help. You switch but the amount you get could equally likely be the larger or the smaller (depending on which one you got first). If both envelopes are larger than Y this doesn't help. You don't switch but the amount you get could equally well be the smaller or the larger (depending which one you got first). If however your random "probe" amount of money Y lies between the two amounts in the envelopes, then whichever you open first, you will be led to the right choice.

I'd recommend that someone posts a question on the maths portal or the statistics portal, to find out if any wikipedia editors from those fields know of good references. Richard Gill (talk) 20:50, 2 May 2011 (UTC)

I put an item on the Wikiproject Statistics Talk page and the Wikiproject Mathematics Talk page, alerting people to two envelopes. Hope this leads to some expert contributions concerning the actual sources. Richard Gill (talk) 08:19, 3 May 2011 (UTC)

The philosophy of probability is also a field where people have been going round in circles for four hundred years or more. To some extent, because some philosophers are unable to read basic mathematics literature. To some extent, because philosophers make distinctions which to many of us are so subtle as to seem meaningless. So for philosophy, any basic probability paradox is *never* resolved, but is always reappraised with the rise of every new movement in philosophy. And it is re-solved by every new generation of philosophy students. Richard Gill (talk) 09:11, 3 May 2011 (UTC)

Proposal to delete two unsourced and unhelpful sections

I suggest we simply delete the two sections "Possible solutions" and "Mathematical analysis". As far as I can see these are totally unsourcable and simply represents an earlier editor's own research. It would be fine by me if they were convincing (and then we would probably be able to find sources for them, since this problem is almost old as the hills) but they're not (IMHO).

The new section (which I drafted) "Mathematical analysis II" needs sourcing, but that can be done. It also needs a whole lot of improvement, so that the main idea gets across to the non-technical reader, but that can be done too, I am sure.

Once this is done the main line of the article becomes transparent again. There are two different "two envelope" problems - the original paradox where no envelope is opened, and the later variant, where one of the envelopes is opened at random. The original paradox is resolved by showing that the assumptions used in the "silly solution" are untenable. You cannot rationally believe that the other envelope is twice or half as large as the first one, with equal probabilities, whatever the amount in the first. The argument for eternally switching usings a premise which actually is untenable. So: "garbage in, garbage out". Replacing the premise with anything realistic destroys the silly argument.

The variant where the player randomly opens one of the envelopes is quite different. It turns out that there is a rational strategy whereby the player can improve his chances of ending with the envelope with the bigger number, provided he is able to generate random numbers himself. He needs to have a kind of probe which, if he is lucky, will happen to fall between the two numbers in the two envelopes. It's not difficult to invent such a probe. You just need to be able to toss a coin a few times and you need to know binary fractions and logarithms.

Not everybody's cup of tea, but that's life. It's a mathematical paradox.

Then we need to find out what the philosophers have said, and what the economists and game theorists have said. Richard Gill (talk) 14:43, 3 May 2011 (UTC)

I deleted the two unsourced sections. As your newly written sections are as unsourced, and hence unhelpful, as the now deleted sections, they will be deleted next. iNic (talk) 23:56, 3 May 2011 (UTC)

You think they are equally unhelpful? I think they make a load more sense, and I assure you that this is how people in my field solve these problems, so I am pretty sure we will be able to support that by appropriate literature citations. And if all else fails I'm happy to write the literature myself.

So: Sure. Anyone can delete them. Or improve them. Or study the literature and add appropriate citations. I plan to do the latter, but Rome was not built in a day, and I don't have easy access to my university electronic journal library at the moment, so this will take some time. Richard Gill (talk) 05:55, 4 May 2011 (UTC)

OK when you have proved that you can read you are welcome back! In the meantime all unsourced OR will be deleted. This is a very simple and basic rule at WP and if you don't understand this extremely simple rule I honestly think you shouldn't edit anything at Wikipedia. Please just delete your account. I can help you out with the technical details. iNic (talk) 13:00, 4 May 2011 (UTC)

Dear iNic, maybe you could also take some reading lessons -- you systematically misinterpret everything I wrote. And do you own this article, or something? Is it your job to advise me how to delete my account or to read me the Wikipedia law? I know the policies and I support them fullheartedly. Richard Gill (talk) 14:23, 4 May 2011 (UTC)

iNic, why all the fuss about Richard Gill's contribution. It has been written by someone who is an expert in this field, which is something that this page needs. It is not yet sourced but Richard says that he intends to do this, in the meantime the section is appropriately tagged. I do not think that his contribution is perfect so we should work together to improve it. Martin Hogbin (talk) 17:02, 4 May 2011 (UTC)

Gill is no expert in this field at all. Gill is a mathematician and not a philosopher of mathematics. If he studies and understands the published literature he can in the future for sure contribute to this page as a layman in philosophy. But he openly confess that he up to this point hasn't read a single published paper about the problem, so right now he isn't even an informed layperson in this field. However, in his own eyes he's such an authority and natural talent that he doesn't need to read any of the published papers to know what is written in them! On top of this he openly admits that he on purpose violates basic WP rules. Let me quote him: "I just wrote up two mainstream solutions to two main variants of the two envelopes problem, both "out of my head", ie without reliable sources. (Very evil of course, very un-wikipedian)" and also about this page "... two of the sections on the page itself were hurriedly and entirely written purely as uncited "own research" by me." He thus knows the WP rules very well but choose to systematically violate them anyway. Now, in his latest comments here, he tries to convince us that he knows the WP policies and that he "support them fullheartedly" which, of course, is a blatant lie. iNic (talk) 23:56, 4 May 2011 (UTC)

This unseemly argument does nothing to help improve this article. Richard has said he intends to add references to his addition. In the meantime can you be more specific in you criticism. Martin Hogbin (talk) 14:31, 5 May 2011 (UTC)

Sure, I can be more specific if you want and need that. Every editor of Wikipedia is supposed to know some basic rules when editing and collaborating for creating good articles. This external page summarizes the "ten commandments" of Wikipedia. Rule 2 is Learn the Five Pillars. On of the most basic of these five pillars is that Wikipedia is not a publisher of original thought or research. This must never ever happen and any attempt to do so must be deleted immediately. There are a LOT of other places outside Wikipedia, both online (blogs for example) and offline, where you can promote your own pet theory of anything whatsoever. However, Wikipedia is NOT a blog. Rule 8 is Share Your Expertise, but Don't Argue from Authority. Argument based on authority is simply not allowed at Wikipedia. This can be a minor shock for people that are used to do that in the real world when they realize that this is simply not allowed here. Here all arguments must have some substance. Your degree or position in the real world means absolutely nothing at all here. Your knowledge as an expert might do, but never your degree. A quote from the external page: "Your specialist knowledge should enable you to write in a neutral manner and produce reliable, independent sources to support each assertion you make. If you do not provide verification, your contributions will be rightly challenged irrespective of how many degrees you hold."

Hope this short summary helps. As soon as all editors are aware of and embrace these basic rules we can move on with the actual editing of the article at hand. As long as we disagree on the basic WP rules it's an hopeless task to try to improve the article. You can't play chess if you don't first agree on the chess rules to follow, can you? It's as simple as that. iNic (talk) 16:46, 5 May 2011 (UTC)

No it does not help at all. I was asking for some criticism of what Richard has written. What do you think is wrong with it? We all know and all accept with the rules of WP and I certainly agree that everything contentious should have a reference to a reliable source. However, Richard has said that he intends to do this and the article has been tagged to show that it needs doing. Surely we can allow a little time to do this. It is not easy to write cogent prose if you have to stop after every sentence to find a reference. Most people find it easier to write the whole thing first and then look for references afterwards. Martin Hogbin (talk) 17:31, 5 May 2011 (UTC)

What is wrong with it? Well, did I mention that OR is not allowed at Wikipedia? I think I just did. Gill has explicitly admitted he took 100% right out of his head. I can't think of a clearer case of OR. If you can, please let me know! To search for reliable sources after the OR is written is just silly. What is gained by that? Nothing. If Gill is anyway going to take the pains and read the published papers he has to do as everyone else and read them before he contribute to Wikipedia. We can't have one set of rules valid for Gill and another set or rules valid for everyone else. iNic (talk) 21:32, 5 May 2011 (UTC)

I meant what is wrong with his argument or logic?

Did I ever say it was wrong? Did I ever say it was true? I don't think so! Please read this basic WP policy iNic (talk) 22:49, 5 May 2011 (UTC)

You seem to misunderstand the meaning of references. It is not expected that every word of each article should be copied directly from a source; that is plagiarism. If somebody knows a fact (because the learned it from a reliable source) then they are fully entitled to put it in WP, in their own words, 'out of their head' so to speak. They can then try to look up a suitable source to verify their statement and add it to the article. That is exactly what Richard said he would do. OR is when you make something up on your own. Martin Hogbin (talk) 21:54, 5 May 2011 (UTC)

OK so how could Gill have learnt it from a reliable source if he never ever read a reliable source? This is the spooky part I never will understand. Or do you claim that Gill lied when he said that he hasn't studied the literature at all? Why would he lie about that? Maybe he walks in the sleep every night and walk into his library and take out the books and papers he never read during the day (by accident containing all published envelope puzzle papers) and in this way subconsciously get access to the literature that he during the day can just pop out of his head? Well, I find many stories about flying saucers much more likely. And I don't believe in flying saucers. Please get real. iNic (talk) 22:49, 5 May 2011 (UTC)

People at academic institutions spend a great deal of their time reading published papers and other sources of information. As Gill is an expert on probability I guess he tries to keep himself up to date on the many aspects of the subject by reading sources on probability, game theory, decision theory etc. He said that he was replying from a mainstream POV, which means the general consensus of expert opinion on the subject, on which he would, no doubt, try keep himself up to date.

Can we return to discussing the subject of article rather than other editors please. Martin Hogbin (talk) 23:19, 5 May 2011 (UTC)

The only problem with your reasoning is that it is very far from how Wikipedia works and what Wikipedia is. It is not the case that by simply being at a special place, or being a special type of human being, or by having some special type of friends, or by smoking a special type of pipe will give you a free ticket to write whatever you want at Wikipedia. Not even the Pope is allowed to freely edit the article about God! It's not the case that Wikipedia tries to ruin the party for all the wonderful persons that think they are special. Not at all. They are still very special. Really! Only not at Wikipedia. iNic (talk) 23:41, 5 May 2011 (UTC)

I do not think I am special. We are all different. We all bring different resources to wikipedia. Where we share them. There are more ways of being well-informed on a subject than reading published papers. I have been talking about the two-envelopes problems with students and colleagues for forty years by now. I use variants of the problem as an exam question. I do not publish in the philosophy and/or foundations of mathematics/probability/statistics but I have been reading this literature and talking to experts, going to conferences, for 40 years too. This does not mean that I am a special person with special authority on wikipedia. No. But it might mean that other editors could find some of what I say useful. Let's collaboratively edit. Let's combine our different skills. It will help to collaboratively understand the Mathematical Truth(s) of the Two Envelope Problem(s). There is not much point in writing a lot of rubbish into the wikipedia article, just because some people have published rubbish in so-called academic journals. Other people have refuted the rubbish. Some non-rubbish is more interesting to a broad audience than other. Let's try to use some good taste, balance, get an overview of our field. A wikipedia editor is not just a scribe who copies what others have written. Wikipedia not only wants to be an encyclopedia, it wants to be a good encyclopedia, a useful encyclopedia. That requires a lot of skills at the meta-level. Editing requires sifting, selecting, ordering. Richard Gill (talk) 13:47, 8 May 2011 (UTC)

It seems that this page is broken. I tried some time ago to improve the readability and added a referenced section that explained the logical fallacy with the unopened envelope problem. Perhaps my contribution was not perfect, but it was replaced over time by other editors by an unreferenced segment, which then was deleted and replaced by a self-declared math "expert" with a long, hard to read and unreferenced section that does not address the logical fallacy / unclear definition of the problem setup (point 6) or the difference between unopened and opened cases. I do not think anyone disputes that the player in the game that can decide to switch can win over time if he/she can open an envelope, and the opposite player will lose over time. This can be stated much more clearly than what is done now. If the envelopes are not opened, there is no way to win and there is no problem outside that created by a very clearly definable logical fallacy. It is impossible to read the article and come out with these simple facts.

In cases where the amounts are uniformly distributed, it is beneficial to switch all envelopes, except the one that is the largest in the game. If the priors are set like in the "second problem", it is beneficial to switch all opened envelopes; this is due to strange behavior of infinity. —Preceding unsigned comment added by Yuzta (talk • contribs) 08:27, 7 May 2011 (UTC)

I agree that the article is in a bad state. Richard Gill added some stuff with the intention of adding references. Unfortunately iNic deleteted it before he could do.

Do we agree on the answer that:

• For all 'sensible' distributions of money in the envelope the probability of doubling your money is not independent of the value in your envelope, thus proposition 6 fails.
• Some distributions result in an improper prior, which means that it is a silly question.
• Some distributions have a proper prior but an infinite expectation, thus you can double your money by swapping, but only from infty to 2 * infty?

For any proper distribution of money in the envelopes the probability of doubling indeed depends on the value in the envelope, thus proposition 6 fails. This is Ruma Falk's solution. As she points out, the writer of the solution doesn't distinguish between a random variable and a possible value of a random variable, does not distinguish between unconditional and conditional probabilities.

The only way to get the probability of doubling independent of the value is by use of an improper prior distribution. Thus it is not surprising we get into a mess. However, improper priors are a useful tool. And anyway, it is possible to arrange that the expectation value of the other envelope, given what is in the first, is always greater than what is in the first, with a proper prior distribution, if cleverly chosen. So blaming the paradox on an improper prior is not really finding the root of the paradox.

We then find that the paradox can only exist if the expected amount of money in either (and then both) envelopes is infinity. So the switching strategy says that exchanging a finite amount of money for an amount which is infinite in expectation is a clever thing to do. This is not really paradoxical (and switching forever is not interesting) since the conclusion is that we have switched an envelope with expected infinite contents for another envelope whose contents are infinite in expectation, and indeed even have the same probability distribution. We do not improve ourselves by making this switch. From an economic point of view one can say that all utilities are finite and even bounded. Even if we allow mathematically the existence of arbitrarily large amounts of money in the world, they are not thereby arbitrarily hugely valuable to us. At some point there is no gain in having even more money. So from the economic point of view the paradox is resolved by saying that we are not interested in the expectation value of the amount of money in the envelopes at all, the only things we are interested in have finite expectations. From a mathematical point of view one can say that the paradox is pretty silly. And especially if you look in the envelope and see amount x. Whatever x may be, the expectation of the amount in the other envelope may be larger, but the larger x is, the chance that the other envelope contains even more gets smaller and smaller till it is negligeable. Since we are offered the game only once, we are just not interested in the average amount that we would get by switching every time, on many many many repetitions, on those occasions when the first envelope contains x.

It just isn't difficult, it isn't unsolved or controversial, and it has all been said before in the literature. Richard Gill (talk) 10:48, 9 May 2011 (UTC)

Intro

I rewrote the first lines of the article. iNic reverted. I re-reverted. Maybe we could discuss and some others might like to contribute?

Just in case iNic reverts again, here is my proposal:

The two envelopes problem is a puzzle or paradox for the subjectivistic interpretation of probability theory; more specifically within subjectivistic Bayesian decision theory. The problem continues to generate discussion, especially in the philosophy of probability and in popular literature. New variants of the problem also turn out to have surprising features, independent of whether the problem is considered from the point of view of frequentist probability or Bayesian probability.

Here's the original:

The two envelopes problem' is a puzzle or paradox within the subjectivistic interpretation of probability theory; more specifically within subjectivistic Bayesian decision theory. This is still an open problem among the subjectivists as no consensus has been reached yet.

The original lead sentence stated that the problem was unsolved. This is simply not true. In the professional literature it has been exhaustively analysed from many angles and is not subject of controversy. Of course newcomers will keep coming to the problem and re-inventing the wheel. And writers of pedagogical literature on mathematics education will write about the problem again, and present it to newcomers. This does not mean that it is an "open problem".

The paradox is equally interesting within frequentist probability, especially in its second variant.

It's clear that a non-mathematician might like to resolve the paradox in a completely different way from a mathematician. For the practical minded man-in-the-street it's enough to say that talking about indefinitely large amounts of money is stupid. There must be an upper limit. This is true and it's fine but it is not interesting for the professionals. In mathematics we use infinities all the time, we need to since otherwise we couldn't solve the problems (real problems, practical problems), so it's an issue for us to know when this is safe and when it is not safe. "Improper priors" are often harmless and simply a convenient approximation to something more realistic. When can they cause trouble, when are they safe? This is an important practical problem for statisticians solving real problems in the world.

I note that in the talk above, many others have affirmed my point of view. Richard Gill (talk) 06:10, 4 May 2011 (UTC)

Please do not remove sourced parts of the article and replace it with your own OR. POV pushing is never allowed at Wikipedia. And no, the talk page here is NOT the same as the published literature in this subject. Have you read the rules at WP at all? Or do you see the rules as just bad jokes? If you do others will start to see you as a bad joke. iNic (talk) 13:20, 4 May 2011 (UTC)

I did not remove sourced parts of the article, and what I replaced it with was not OR. The two references given in the original lead do not support the text that was written there. I strongly support the rules/policy of wikipedia. Hiding behind the rules is no excuse to write nonsense on wikipedia pages. I suspect that the author of those original sentences did not understand the articles he or she was citing.

And who said the talk page here is published literature? My day-job is to publish literature in this field, in peer-reviwed academic journals and other media. I am not POV pushing, I am reporting on the talk page (and writing into the article) the standard analysis and points of view on the two envelopes problem in the academic mathematics community. I'm looking for more references and I'm reading up on the history.

The rules of wikipedia are splendid, and I support them. The citations added to the old opening sentences of the article do not support the Point of View which was expressed in those sentences. I suspect that the author was not actually able to actually understand what is written in those papers. Richard Gill (talk) 14:29, 4 May 2011 (UTC)

OK, so you somehow know that these papers doesn't support the intro even as you haven't read them? Isn't that very strange? I find it almost a little spooky. Please rest assured that these papers, as well as all other published papers about this problem taken together, supports the wording in the intro. iNic (talk) 00:17, 5 May 2011 (UTC)

Yes, spooky isn't it! On Monty Hall Problem I was frequently accused of ESP. However on those occasions when I guessed what authors had actually meant, and we later asked the authors if I was right or wrong, I was always right. I don't believe it's spooky or ESP. It's just that I'm a professional in the same field as those authors, I write for the same audience and know the hidden assumptions, the unstated context. Richard Gill (talk) 13:02, 8 May 2011 (UTC)

I also can read papers written by others which refer to the two papers mentioned in the intro, in particular, other papers written by the same authors. I can read the abstracts of those papers. I can read the summary of those papers which others have made for wikipedia and deduce what the authors must have originally meant, so as to be consistent with other writings by themselves and others. Since the basic maths is all pretty elementary and un-controversial, this is not a difficult job. I do look forward to getting hold of Falk and of McDonnell and Abbott's papers. I hope someone has got a pdf and can email them to me. Richard Gill (talk) 13:33, 9 May 2011 (UTC)