Talk:Two envelopes problem

Constructive suggestions

Much of the debate here and inclarity in the article stems from the fact that there are two different ways to present the paradox, a mathematical one and a realistic one. The solution to the realistic one is easy, and it involves finite upper limits to the prize sums, individual utility functions and such. The mathematical paradox (where you cannot make such assumptions) is the actual paradox and much harder to crack. This distinction should be made more clear.

Another suggestion: Many of the references seem to be missing the point of the paradox entirely, or are otherwise irrelevant. They should be pruned away. Smite-Meister (talk) 22:42, 24 March 2009 (UTC)

the danger of concrete examples

The page as it currently exists uses in example "the St. Petersburg lottery has an infinite expected value" (sic) which falls prey to Richard Feynman's old trick of defeating mathematical conundrums by forcing expression in concrete terms. There is no lottery which has an infinite expected value in the real physical world. Lotteries have set values; one method of calculating expected value is to use the chance of winning as related to the set of known or possible payoff values. There are no valid methods of calculating expected value of a lottery ticket that can yield infinity, so the example as stated does nothing to improve understanding. Expecting infinite yield from any real-world monetary instrument is literally insane. —Preceding unsigned comment added by 204.27.178.252 (talk) 23:45, 25 November 2008 (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 (UTC)

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)

Formatting of References

I suggest to convert the formatting of the references ("Published papers") to an automated style instead of manually numbering the references. E.g., if I wanted to add a paper of Schwitzgebel and Dever, I would format it as follows: <ref name="Schwitzgebel_and_Denver_02">Eric Schwitzgebel and Josh Dever. ''The Two Envelope Paradox and Using Variables Within the Expectation Formula.'' 2006-07-07. ([http://www.faculty.ucr.edu/~eschwitz/SchwitzPapers/TwoEnvelope060707.pdf])</ref> at the position where it is cited and add a section with the following text

== References ==
<!-- ----------------------------------------------------------
 See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for a 
 discussion of different citation methods and how to generate 
 footnotes using the <ref>, </ref> and  <references /> tags.
----------------------------------------------------------- -->
<references />

at the end of the article. --M.T. 22:49, 25 December 2006 (UTC)

I considered all different solutions for adding references available, with their pros and cons, when I finally settled for this one. The main advantages are that we can have the reference list in chronological order and don't have to have a reference to every single paper in the text. Your proposed system for references lacks these freedoms. iNic 05:57, 29 December 2006 (UTC)

The particular PDF you want to add is a more popular version of their original text already in the list. As such, I don't think it merits a new entry in the list, and it's in fact very easy to find via the extra link "A Simple Version of Our Explanation" already present (next to the link to their main paper from 2004). iNic 05:57, 29 December 2006 (UTC)

I would say that the paper that I suggested to add is very different from the one already listed, and it's much easier to understand.

What do you mean with "chronological order"? Does it mean that new papers should always be added at the end? --M.T. 14:21, 31 December 2006 (UTC)

The most common suggestions here recently at the talk page has been to either make this article more encyclopedic in style, even if it makes it more difficult to understand, or to make it more popular so that it's accessible to more people. One solution to this dilemma suggested itself when I read the Monty Hall paradox article recently. It has a separate version in Simple English! We could do the same for this article. No doubt, a Simple English version will be far more difficult to write than this one. But it might be worth a try, don't you think? iNic 04:25, 3 January 2007 (UTC)

Yes, the papers are added "in order of appearance." It makes it easy to add new ones at the end as they pop up. The list is already quite long and confusing, and if we add different versions of the same basic ideas from all authors as separate entries we only add to length and confusion, not to content. iNic 04:25, 3 January 2007 (UTC)

Equation source

${1 \over 2}2A+{1 \over 2}{A \over 2}={5 \over 4}A$

Does anyone have the exact source were this equation first comes from? I can't see how anyone could take this equation seriously. "In the first term A is the smaller amount while in the second term A is the larger amount." This is the problem summed up perfectly, why it is listed as a 'proposed solution', it IS the solution.--Dacium 04:58, 17 April 2007 (UTC)

If you read on in the article you will find a short history of the paradox. I reverted your addition of a "correct equation" because it belongs to a different discussion. This article is about what is wrong about the switching argument (i.e., how to solve the paradox), not about different ways to reason instead (or different ways to act if put in this situation). However, I welcome a separate article about the decision problem where we can gather all these ideas (all "correct calculations" and strategies). (Please have a look at the archived discussions for this talk page as this question has been brought up before.) iNic 01:16, 18 April 2007 (UTC)

I don't get it. This is what is wrong about the switching argument. The A in 4. is smaller than the A in 5. Therefore the proposed equation makes no sense.--90.179.235.249 (talk) 17:58, 19 January 2010 (UTC)

solution

I'm confused, is there a solution? In the intro, it says, "This is still an open problem among the subjectivists as no consensus has been reached yet." But the article seems to suggest there's a solution (if interpreted another way?). Could someone clarify this for me and amend the article if necessary? 129.120.94.151 22:37, 19 April 2007 (UTC)

I agree, it was confusing. I deleted recent edits that made the article contradictory in this respect. Thank you for pointing this out. iNic 13:13, 20 April 2007 (UTC)

Why no game-terminating event?

Doesn't the paradox arise from purposefully leaving out a game-terminating event, i.e. a time limit, or some other contest event that causes the player to end up with zero dollars, and thus that either of the envelopes result in a positive outcome? Using a game-terminating event , the solution is obvious -- keep any envelope and minimize risk for a zero gain. However, without some sort of terminating event, of course the game must continue forever because there is no incentive to pick one or the other envelope since either represents a 50/50 chance at the highest gain. Fine Arts (talk) 18:02, 27 October 2008 (UTC)

If you only have 2 envelopes and you pick one and then take the other for any reason, deosn't the game end? Wouldn't the game end the moment you picked the second envelope? People don't hover indecisevely forever between these sorts of choices, even if they seem equal. Any number of factors outside of the little scenario will cause the game to end one way or another. The solution I propose is that, if the amount of money in the first envelope is an indespensible amount, say abitrarily 1 million dollars, you would take it where as half of that would be a huge loss and double that would be just gravy. The incentive to 'go on' is lessened. Where as, though if the amount in the first envelope is more trivial, say abitrarily, 4 dollars, double it is just as meaningless as half, so you might as well just press your luck.Zavion (talk) 16:34, 14 November 2008 (UTC)

crazy

I think this discussion is crazy. Just look at the french page on this topic (http://fr.wikipedia.org/wiki/Paradoxe_des_deux_ch%C3%A8ques), if you want the situation properly discussed.— Preceding unsigned comment added by 129.194.8.73 (talk) 13:26, 15 August 2007 (UTC)

---

I've looked at the French page but as far as I can see it doesn't really cover all the bases. The explanation they use does come up with the correct answer, however, that is not required. We already know the correct answer i.e. there is no reason to swap. That is why it is a paradox. The problem is, what is the logical error in the reasoning?

Now, the French page states the 50% term should be 50%*2. But this is right at the heart of the issue and can't just be changed because the answer comes out right. Let me expand...

In the English article the non-probabilistic variant section says

1. Let the amount in the envelope chosen by the player be A. By swapping, the player may gain A or lose A/2. So the potential gain is strictly greater than the potential loss.

2. Let the amounts in the envelopes be Y and 2Y. Now by swapping, the player may gain Y or lose Y. So the potential gain is equal to the potential loss.

Now, it is clear that 1. must be wrong and 2. must be correct. But at point does 1. go wrong?

As far as I can see (and I might be wrong here), the French article is doing the equivalent of saying "Well, this formula (relating to 1.) is wrong but if we use this other formula (relating to 2.) it comes out right." Very true. But what is the logical error in 1.?

Also, I think it specifically does not pass the test where we open our envelope and see that it contains $10. In this case, the paradoxical formula can still be (apparently) justified - gain $10 vs loose $5. Whereas the 50%*2 term now appears to make no sense at all. Fontwell (talk) 14:42, 5 September 2008 (UTC)

Actually, "1" is correct and "2" is wrong. "2" is wrong because "gain Y or lose Y" depends on two different selections of the first envelope--to gain Y by switching I must have chosen the Y envelope, and to lose Y by switching I must have chosen the 2Y envelope. These two things cannot simultaneously be true. "1" is correct whether A is the larger amount or the smaller amount. 76.234.120.116 (talk) 01:15, 21 September 2008 (UTC)

Erm, you seem to have fallen for the original paradox. Which just goes to show what a corker it is!

The overview is this:

The initial choice of envelope was arbitrary and had no way of choosing the better outcome. No new information was presented to us after the choice. If we subsequently swap envelopes it cannot put us in an advantageous position. We should all believe this to be correct.

Thus, any mathematical argument that shows an advantage in swapping is flawed. This is the paradox. Thus 1. must be wrong.

The explanation for why 2. is wrong is flawed. I think there is a confusion between 1. and 2. going on here.

The big difference in 2. is that all it says is that the two amounts of money are Y and 2Y. It doesn't specify which envelope you choose. So, if you don't know which is which, then swapping will either swap you from holding Y to holding 2Y (gain of Y) or swap you from holding 2Y to holding Y (loss of Y). Thus the possible gain matches the possible loss.

The comment:

"...gain Y or lose Y" depends on two different selections of the first envelope--to gain Y by switching I must have chosen the Y envelope, and to lose Y by switching I must have chosen the 2Y envelope. These two things cannot simultaneously be true."

This is all correct. However to say "2 is wrong because" of it is just a non-sequitur. All that is missing from that comment is to finish it off: So what we do in game theory (or in fact probabilities in general) is to say: "These two things cannot simultaneously be true" but we don't know which is actually true. So let us assign a probability (of truth) to each case so that we can determine our 'expected gain'.

In this case (2.) we have no reason to think that holding Y or 2Y is more probable. So we have two equally likely possibilities. One of them gains us Y while the other looses us Y. The overall 'expected gain' is the sum of these two. Strictly, Expected Gain = {Y x 0.5} + {-Y x 0.5} = 0

Fontwell (talk) 17:31, 23 September 2008 (UTC)

This paradox teaches us what expected value really is. Expected value is calculated from either the envelope with the least amount in it, or the other one. The fallacity many of us (including me at first) make is to calculate it from the envelope i pick or the other envelope. I do understand why many bayesians make this mistake, yet a bayesian solution is available. Let the amounts in the envelope a priori be x and 2x. you pick 50%x or 50%2x. By swapping you gain x or lose x. Expected value is zero. Now let us call the amount in the envelope we pick A and substitute the x'es in the above tree. We pick A, we could gain A or we could lose 1/2A. Seems like there is a expected gain. But, also substitue the prior. We now see that the prior could have been {A, 2A} or {1/2A, A}. Meaning we no longer have a prior, rather a subjunctive hyper-prior. This proves the fallacy. We compare leaves from both seperate trees. —Preceding unsigned comment added by 193.29.5.6 (talk) 06:27, 1 April 2010 (UTC)

Let me try to explain the fallacity in a number of steps:

1. I'm offered to play a game. I bet for instance €10, the host throws a coin (50% head, 50% tails). On head, I double up, on tails I half down. However, I'm allowed to withdraw before the host throws. Would I go for it or not ? It is clear I have an expected gain by playing the game (1/2 €+10 + 1/2 €-5 = €2.5).

2. I'm playing the game in a different situation. I'm offered €10. The host tells me he had drawn a coin prior to the current event. On head, he doubled his initial amount, on tails he halfed it. Anyway, the outcome is €10. I'm now offered to either go home with €10 or switch to the initial amount. Would I got for the switch ? When switching the expected value appears to be in my favour (1/2 €+10 + 1/2 €-5 = €2.5). Yet it is not. Expected value is calculated from prior to posterior, not the other way around. So the way I must think is: from my initial value I could have either doubled up or halved down to €10. Switching would mean I either lose my initial amount or gain half my initial amount. So I do not want to switch at all ! The problem we have with reasoning this way (the correct way), is that threat our €10 is if it were the prior situation, and the past events are the posteriors.

3. The 2 envelope paradox is a combination of both steps above. You pick an envelope. The amount inside of it is either the initial amount (step 1). In this case you want to switch. Or the amount inside of it is the result of the experiment of the other envelope (step 2). In this case you want to stay. Both scenarios are equally likely, so it should not matter wheter you switch or stay. —Preceding unsigned comment added by 94.225.129.117 (talk) 07:42, 3 April 2010 (UTC)

This is a solved problem, not an open problem

The intro says "This is still an open problem among the subjectivists as no consensus has been reached yet." but that is not correct. There is a clear consensus about the solution in peer reviewed publications. Just have a look at Samet et al., 2004. Yes, this problem is strongly debated by the public and many solutions are proposed and put online but public debate does not mean there is no scientific consensus. It would be the same to call the Monty Hall problem an open problem. There is a clear solution to the Monty Hall problem with scientific consensus no matter what the public thinks or feels.

The solution is independent of Bayesian or frequency interpretation and can be captured in a single phrase:

"The Two Envelopes problem uses an impossible probability distribution."

The use of an impossible probability distribution is the core of most if not all statistical paradoxes. Just like "proofs" that show 1=2 sneak in a division by zero or set the square root of -1 equal to 1 or -1 ( which non-mathematicians easily accept ) so do statistical paradoxes sneak in an impossible probability distribution. The Two Envelopes problem uses the impossible distribution of an infinite set with every element having the same probability. "Any amount of money ( with no upper limit ) with every amount of money as likely". For a non-mathematician the probability distribution of the Two Envelope problem may sound completely reasonable but is mathematically utter nonsense just like dividing by zero.

For every probability distribution the sum of all the probabilities of every possible outcome has to be 1. If you don't have this statistics completely breaks down. You get infinite means and impossibilities as clearly shown in the Two Envelope problem.

If you change the impossible probability in the Two Envelope problem into a real probability by putting an upper limit to the money or by not giving the every amount of money the same probability, the paradox disappears.

There is a clear scientific consensus about this and any professional mathematician or statistician will confirm this. This is basic knowledge.

The article should clearly state this and use it as main point. Obviously this should mean a complete rewrite of the article.

I will not rewrite the article because I don't like to be sucked into an edit war. I just wanted to make it clear that the article is missing the main point. I hope in the end reason will prevail. Michael Korntheuer 17:37, 15 February 2008

If you read half way down the article you will learn that it's not the case that the paradox disappear for every proper prior distribution. If you read even further down you will discover that we actually don't even need a prior distribution to get the paradox, as we don't have to invoke the probability concept at all. iNic (talk) 22:35, 15 February 2008 (UTC)

True about a distribution summing to 1, although there are still infinities involved, specifically infinite expected values. False about not needing to invoke the probabity concept at all. In particular, how do you make the leap from "Let the amount in the envelope you chose be A. Then by swapping, if you gain you gain A but if you lose you lose A/2. So the amount you might gain is strictly greater than the amount you might lose" to "you should swap" without looking at probabilities? To decide to switch inherently involves an assessment of probabilities. To say otherwise is to say you'd be happy to play roulette as long as you can't see the wheel and are thus ignorant of the probabilities involved. Warren Dew (talk) 01:37, 3 May 2008 (UTC)

The section Two envelopes problem#A non-probabilistic variant does describe a paradox that has nothing to do with probability. (Though I don't see it as a variant.) The solution of that paradox (IMO) is that the term potential gain is ambiguous and used in two different meanings, giving different results. Oded (talk) 18:12, 27 July 2008 (UTC)

Yes, the statement "if you gain you gain A but if you lose you lose A/2" is making exactly the mistake described in the first section! The "A" that you "gain" is the same actual numerical value as the "A/2" that you "lose".

The source of the apparent paradox in "an even harder problem" is similar to what Michael Korntheuer was driving at--the fact that there are situations where the "double" value of one possible set is the same as the "single" value of another set. That is, X^a = 2X^b. Since a and b have equal probability, there is no way to identify whether an envelope has X^a or 2X^b--even if we open the envelope. 192.91.147.35 (talk) 01:14, 11 September 2008 (UTC)

The problem vanishes once the problem is set up correctly : we have only two envelops, and their value is deterministic. Which one will be elected is not determined. The experiment as stated in the paradox is this : there are two envelops : A with value X and B with value 2X. The experimentor will chose A with probability 1/2, and B with probability 1/2. Once the choice has been made (say the choice was A), a fairy destroys the envelop B and replaces it with a new envelop C with value X/2 or 2X (1/2 probability each). This is clearly not the meant experiment. In the experiment, B stays there with its determined content. To modelize the experiment, you must not forget to take into account that both values are determined through the whole experiment. The fact that you don't know which envelop is A and which envelop is B makes the only probabilistic problem in the experiment. This is the only probabilistic variable in the experiment : no amount in the envelops is probabilistic. Subtenante (talk) 11:31, 17 December 2009 (UTC)

I slightly disagree with you there. The amount in envelope A is deterministic, the amount in envelope B is probalistic (50% A*2 50% A/2). The experiment starts at a point the probalistic event has taken place. At this point the only thing you should care about is to know which envelope is A, and which is B. You know that the amount in envelope B has a higher expected value than the amount in envelope A. Because you can't distinguish the envelopes, you cannot make a decision, other than to take a gamble. The fallacity people tend to make is to threat both envelopes as if they were envelope A, and the amount in the other envelope was determined from the one they have picked. That's why switching seems always the best decision.

Let's dig a bit deeper into the fallacious thinking process. When picking an envelope there are two outcomes possible. You pick envelope A, you pick envelope B. If you pick A, you may reason as follows: B = 50%A*2 + 50%A/2 > A. The amount in B is expected to be higher than A, so you want to switch. If you pick B, you reason the exact same way: A = 50%B*2 + 50%B/2 > B. The amount in A is expected to be higher than B, so you want to switch. But, you must not threat A - deterministic - as B - probalistic -. The above reasoning is fallacious. Instead reason as follows: B = 50%A*2 + 50%A/2. I got B, so switching to A would mean: 50%(-A) + 50%(+A/2) < 0. There is a potential loss in switching to A, so you should stay with B. The fact A can take several values doesn't even matter. Assume B = €10, A0 = €5, A1 = €20. Swithing to A gives 50%(-A0) + 50%(+A1/2) = €+2.5. Even though there seems an expected gain by switching to A, there isn't (because: 50%(-A) + 50%(+A/2) < 0). The fact we distributed A by the hyper-prior A0, A1 misleads us.

There is no benefit to switching

This is in the article at the moment:

"But in every actual single instant when an envelope is opened, the conclusion is justified: the player should switch! Not many authors have addressed this case explicitly trying to give a solution. Chalmers, for example, suggests that decision theory generally breaks down when confronted with games having a diverging expectation, and compares it with the situation generated by the classical St. Petersburg paradox."

This is wrong, there is no benefit in switching. It is proven by the fact that there does not exist any pair of envelopes for which switching makes sense. Since this result is based on having perfect knowledge about the system it has to be right. The calculation from the point of view of the person who opened one envelope does not assume perfect knowledge of the system (he does not know what is in the other envelope) so this calculation should be put in doubt.
This means switching is NOT beneficial and the above text in the article is misleading and should preferably be removed.
Another thing is that there is no need to rehash St. Petersburg paradox in my opinion, the link to it should suffice.
Enemyunknown (talk) 18:31, 14 September 2008 (UTC)

If you think this is wrong please write a paper about this and get it published in a peer reviewed magazine. When this is done someone else, not you, can add your opinions and ideas in the wikipedia article. What matters in wikipedia is verifiability, not truth. iNic (talk) 18:05, 15 September 2008 (UTC)

I'm not going to bother, but I'd like to see the source of that statement, on page it was only said "See John Norton, 1998, for a suggestion", i fixed it with the link from exchange paradox page. I tried to find the suggestion in the long pdf skimming through it but to no avail, can you maybe point me to the page on which it is stated? http://www.pitt.edu/~jdnorton/papers/Exchange_paradox.pdf

It should also be clearly stated if the advice to switch relates to the version of the paradox described immediately above it or to all versions.

What about suggestion about the lottery? I think inclusion of its description here is confusing as its a completely different paradox, the reader won't understand why its expected value is infinite without reading the full description anyway, so i think it would be better to just leave the link to it.Enemyunknown (talk) 10:26, 18 September 2008 (UTC)

If you notice the levels of the headings, they indicate that this text applies to the specific lottery situation stated. I have not checked sources, but it looks right to me. And the comments about the infinite expected value are highly relevant.

The point of the example is this, in my understanding, this:

The whole paradox has something to do with assumptions about what these envelopes may contain. If you are in a quiz program where the max prize is known to be $1 000 000 and your envelope contains $500 000 or less, you may consider switching, but if it contains $1 000 000, don't switch - you can't win more anyway; you will be sure to loose. If you are given the choice of envelopes in some other situation where you have clues that allow you to assign a priori probabilities to various pairs of amounts, you may have a similar situation where swithcing is favourable for small amounts but not for large amounts - and this may be perceived as solving the paradox: In real situations with real distributions (whether they are known or not), the IS no paradox; it's as fictitious as the Martingale system. But alas, if your a priori probabilities (wherever you've got them from) are as in the lottery given here, you should switch in all cases, and we do have a paradox, contrary to all common sense and reason! That is, until you realise that this "lottery" is NOT a real situation, as the expected value of the contents of the envelopes is infinite. Not the richest bank in the world could finance a lottery which truly had this distribution, and if you are TOLD that this is the distribution, you KNOW you've been lied to. E.g., you could obtain your own a priori distribution by modifying the given distribution: The probability of prizes larger than $1 000 000 000 is not small, as given by the lottery probabilities, but truly zero. (Don't believe me? Then make the cut at $1 000 000 000 000 000 000 instead!) Then, again, there is no paradox; only the baffling question about how to construct your a priori distribution when your information is incomplete.--Noe (talk) 11:40, 18 September 2008 (UTC)

There are lots of sources for this statement, as well as there are many authors that make the connection to the St Petersburg paradox, and they think this connection is central for a proper understanding this paradox. You are of course free to disagree personally that this is a valid connection, but as I said before: wikipedia should only reflect the discussion found in the literature. Of course, there are also authors that denies that these two paradoxes are connected, and this view is also mentioned in the article. iNic (talk) 14:29, 20 September 2008 (UTC)

The situation is symmetrical and there does not exist a single pair of envelopes for which switching gives benefit, the article many times this to say about the decision to switch: "This conclusion is just as clearly wrong as it was in the first and second cases."

However someone included this surprising paragraph:

"But in every actual single instant when an envelope is opened, the conclusion is justified: the player should switch! Not many authors have addressed this case explicitly trying to give a solution.

I looked into the reference and I couldn't find this suggestion, so I asked you for the source, since you failed to provide it I looked once again and this time I found that the author of the provided reference claims something directly opposite to what this paragraph says, here is the quote from the source:

"I consider a variant form of the paradox that avoids problems with improper probabilities and I argue that in it this expectations give no grounds for a decision to swap since that decision has to be based on summation on all the expectations. But this sum yields a non-convergent series that has no meaningful value."

http://www.pitt.edu/~jdnorton/papers/Exchange_paradox.pdf

I therefore remove this paragraph, do not put it in unless you can properly source it.

Enemyunknown (talk) 04:31, 21 September 2008 (UTC)

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)

Comments about the Wikipedia Article

The following discussion is related to the Wikipedia article as stated in January 2009. The following topics were split from a more comprehensive discussion that included a detailed discussion of the solution to the paradox as well. The basis for my including a discussion of the solution in my original discussion of the article was my belief that it was probably necessary to understand the solution in order to evaluate and edit the existing article. The solution portion of that original discussion is presently on a separate discussion section page identified as the “Arguments Page.” This page can be accessed from the link at the top of this page.

Questionable Title

The title of the Wikipedia article appears questionable with respect to the use of the word “problem,“ a more general term, rather than the word “paradox.” Presently the title of the Wikipedia article is “Two envelopes problem.” However, in another Wikipedia article titled “List of paradoxes,” the title that is linked to this article is stated as “Two-envelope paradox.” Interestingly, the opening sentence of the article states that the problem is a puzzle or paradox, so according to the article itself, it does not appear that classifying the problem as a paradox is in any way a contested issue.

I also performed a quick Internet search using Google. Search words “two envelope problem” and “two-envelope problem” provided 551 results, while search words “two envelope paradox” and “two-envelope paradox” provided 3870 results. Although popular usage may not be the true or ultimate test for being correct, it does indicate a popularity in usage of nearly eight to one.

It also appears that irrespective of the title containing the word problem or paradox, the title should probably begin with the article “the.” Accordingly, the title should probably be either “The Two-Envelope Problem” or the “The Two-Envelope Paradox.” I am also somewhat confused by this new trend to not capitalize the first letter of each major word in a title. Does anyone have an explanation for this?

General Comments on the Wikipedia Article

In general, I found parts of the article detailed and well done, but overall I found the article poorly organized, poorly worded, and not thoroughly or well explained. When I performed an Internet search of the paradox, I found a completely different wording to the more popularly stated version of the paradox and found at least three popular approaches to disproving the paradox. Interestingly, the popular version of the paradox included the provision that the amount contained in the first enveloped was known to the player, while none of the four versions mentioned the Wikipedia article included that provision. Also, though my research was rather limited, I did not come across any of the various specific versions of the paradox that are discussed in the Wikipedia article. The Wikipedia article also contains at least one comment claiming that there is some contention or dispute over the flaw or the validity of the flaw in the paradox. After reading several of the articles and papers posted on the Internet, I find that fact somewhat difficult to believe without further qualification or explanation.

Wording Used in the Article

My observation while reading the article was that there were instances where the meaning or thought was not clear. Because probability is not my field of expertise, I wasn’t sure whether this was true because the text was simply not worded well or whether the text was stated in terminology that was foreign to me. A good example is the first sentence of the section titled “An even harder problem.” The subject sentence is as follows: “The solution above does not rule out the possibility that there is some non-uniform prior distribution of sums in the envelopes to give the paradox force.” The main problem I found in understanding the sentence was in the meaning of the phrase “to give the paradox force.” At first, I suspected the meaning to be related to that which established a stronger or better-established paradox. However, two sentences later in the discussion the subject is regarding “a sensible strategy that guarantees a win.” It wasn’t until I began reading the following section on the “Proposed solution” that the discussion returned to the paradox. Then, even after reading the section on the solution, I was still unable comprehend the meaning of the subject sentence.

Accordingly, if this readability problem is related to technical terminology, then perhaps the terminology should be replaced by terminology more suited to the general public, the intended readers of an encyclopedia. Perhaps otherwise, the article should be reviewed and the general wording improved.

Four Versions of the Paradox

The present version of the Wikipedia article addresses four versions of the paradox. These are identified as “The problem,” “An even harder problem,” “A non-probabilistic variant,” and “History of the paradox.” I address each of these divisions individually in the following sections. I also included some discussion related of the respective solutions. I did this being fully aware of the notice to include solutions on the “Arguments Page.” My rationale for retaining these discussions was twofold. First, for the most part the existing solutions as provided in the article are problematic and require some discussion for the purpose of editing. Accordingly, these points could not be easily made without some reference to a common understanding of the solution. Second, I also did not wish to expend the time to edit and/or rewrite these sections.

Notes on the Section Titled “The problem”

The most elementary version of the paradox described in the Wikipedia article is stated in 12 enumerated statements. Although the 12 statements may or may not have some basis in the historically correct version of the paradox, this version is not only inconsistent with the popular versions, but appears to be extended beyond the basic version required for gaining a fundamental understanding of the paradox.

Regarding the discussion of the flaw in this version of the paradox, the flaw is explained mathematically in the section identified as “Proposed solution.” In brief, the flaw is attributed to a misrepresented and thereby incorrect calculation of the expected outcome stated in step 7 of the problem. However, I also found it possible to analyze the flaw by the path of reason. Such a solution may be more attractive to readers not having a strong math background.

First, if the player reasons that there are only two envelopes containing two amounts, and for the purpose of reasoning simply calls these amounts C and 2C, then the player can extend this reasoning as follows. If initially the player randomly chooses the envelope containing the smaller amount C, then A equals C by definition. If the player then chooses to swap that amount, amount A, for what could be perceived as being either ½A or 2A, the actual swap can only be for amount 2A, thereby providing an apparent gain of A. In terms of the C, however, the swap is equivalent to having swapped the initial amount C for amount 2C. Because the player initially selected the envelope containing amount C, the smaller amount, there is no such amount equal to ½A in this case.

By comparison, if initially the player randomly chooses the envelope containing the larger amount 2C, then A equals 2C by definition. If the player then chooses to swap that amount, again amount A, for what again could be perceived as being either ½A or 2A, the actual swap this time can only be for ½A, thereby providing an apparent loss of ½A. In terms of C, again, the swap is equivalent to having swapped the initial amount 2C for amount C. Similarly, because the player initially selected the envelope containing 2C, the larger amount, there is no such amount equal to 2A in this case. Accordingly, despite the truth to the statement that the player can gain A and only lose ½A, we can see that this statement is only true because A has twice the value in a loss than in a win. In terms of C, the player can only win or lose the same amount, amount C.

The source of confusion creating the paradox is in being drawn into false reasoning based on believing that because the amount contained in the initial envelope can be represented by A, that A must be a fixed amount. It is easy to be drawn into this trap because the player can visually see and hold the envelope or, at least, imagine doing so. In this context it is then difficult to revert back to the logic and reason that the outcome was predetermined at the time the initial envelope was assigned.

Also worthwhile noting is that statement 6 in what is called “The switching argument” (stated as, “thus, the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2”) is very open to misinterpretation. The truth of this statement is dependent on the precise definition of variable A. As stated previously, although it is true to state that (1) the probability of A being the smaller amount is one-half and (2) the probability of A being the larger amount is also one-half, once the value of A is known, these statements are no longer true for all specific values of A.

Notes on the Section Titled “An even harder problem”

A second version of the paradox is described in this section. This version of the paradox alters the frequency probability of the amount pairs that contain a common amount. For instance, in the primary version of the problem the frequency probability of a pair of envelopes containing $10 and $20 is the same as for a pair of envelopes containing $20 and $40. In this variation, the probability is determined by a mathematical formula that results in the pair containing the lower amount to have a frequency probability of 3/5, while the pair containing the higher amount to have a frequency probability of 2/5. Doing this causes some confusion. Although it might appear that this has an affect on the overall 50:50 win-loss probability of a swap, it only does so in certain conditions. Tracking this distinction appears to add another layer to the confusion already present in the original version of the paradox. My suggestion would be to include this topic under another title. In effect, this discussion is more related to frequency probability than to the Two Envelopes Problem.

Regarding the mathematical formula ( 3/5 * x/2 + 2/5 * 2x = 11/10 x ) that is used to calculate the expected outcome to be 11/10 x, the formula is incorrect for the same reason that the previous mathematical formula ( ½ * 2A + ½ *A/2 = 5/4 A ) was incorrect in the primary version of the problem. Interestingly, in the primary version of the problem the flaw in the formula was uncovered and was explained in terms of the expected amount contained in the two envelopes. The formula for that calculation was ½ * C + ½ * 2C = 3/2 C. For example, if C is equal to $10 and 2C is equal to $20, then the average or expected value is $15.

If we now apply this same reasoning to this version of the problem, the formula becomes 3/5 * C + 2/5 * 2C = 7/5 C. And, if again we use the same values for C and 2C, the average or expected amount contained in the envelopes is $14. Because the frequency distribution of $10 is stipulated to be 3/5 while the frequency of $20 is stipulated to be the remaining 2/5, the expected amount is correct. Accordingly if this reasoning is applicable and sufficient for the basic version of the problem as stated in the Wikipedia article, it should also be applicable and sufficient for the “harder” version The article simply does not address this issue and instead discusses reasoning along a completely different line. However, despite all of the discussion, it certainly appears that the solution should be identical to that of the primary version of the problem.

Regarding a solution to uncovering the flaw in this version of the paradox (if that is the objective), I also found it possible to analyze the flaw by using the same path of reason I used in the primary version. Accordingly, I provided that path of reasoning below.

First of all, to minimize confusion over the frequency of the pairs of amounts and the amounts themselves, let’s assign designations for everything. Accordingly, let’s call the pair containing the lower amount X and the pair containing the higher amount Y. In terms of the common member B, it follows that X represents B/2 and B, while Y represents B and 2B.

Next, suppose that the envelopes contain the pair designated as X. And, as before, if the player reasons that there are only two envelopes containing two amounts, and for the purpose of reasoning simply calls these amounts C and 2C, then the player can extend this reasoning as follows. If initially the player randomly chooses the envelope containing the smaller amount C, then A equals C by definition. If the player then chooses to swap that amount, amount A, for what could be perceived as being either ½A or 2A, the actual swap can only be for amount 2A, thereby providing an apparent gain of A. In terms of the C, however, the swap is equivalent to having swapped the initial amount C for amount 2C. Also worth noting is that in terms of B, C equals B/2 and 2C equals B. Therefore the swap was also equivalent to having swapped the initial amount B/2 for amount B.

By comparison, if initially the player randomly chooses the envelope containing the larger amount 2C, then A equals 2C by definition. If the player then chooses to swap that amount, again amount A, for what again could be perceived as being either ½A or 2A, the actual swap this time can only be for ½A, thereby providing an apparent loss of ½A. In terms of C, again, the swap is equivalent to having swapped the initial amount 2C for amount C. Again, worth noting is that in terms of B, C still equals B/2 and 2C still equals B. Therefore the swap was also equivalent to swapping the initial amount B for amount B/2. Accordingly, in terms of either B or C, the gain (B/2 or C) and loss (B/2 or C) is the same.

Next, we simply repeat the same reasoning for the pair of amounts designated as Y. Again we would expect to get the same matching gain and loss. However, now worth noting is that in terms of B, C equals B and 2C equals 2B. Accordingly, in terms of either B or C the gain (now B or C) and loss (B or C) is the same.

In this version of the problem the source of confusion creating the paradox extends one level deeper than in the classic version. Accordingly, not only is there the confusion of being drawn into false reasoning based on believing that because the amount contained in the initial envelope can be represented by A, that A must be a fixed amount, there is also the draw to confuse the probability of the frequency of the amount pairs with the probability of the frequency of the amounts within a pair.

Notes on the Section Titled “The non-probabilistic variant”

The non-probabilistic version is stated in two enumerated statements as follows: 1. Let the amount in the envelope chosen by the player be A. By swapping, the player may gain A or lose A/2. So the potential gain is strictly greater than the potential loss. 2. Let the amounts in the envelopes be Y and 2Y. Now by swapping, the player may gain Y or lose Y. So the potential gain is equal to the potential loss.

Although the Wikipedia article states that “so far, the only proposed solution of this variant is due to James Chase,” I find this confusing and a somewhat difficult to believe. This variation does not appear sufficiently different from the primary version for the same reasoning to not apply.

Regarding the truth of the two enumerated statements, statement 1 is false and statement 2 is correct. Accordingly, the part of statement 1 that is incorrect is the stated conclusion (“so the potential gain is strictly greater than the potential loss”). The flaw in the conclusion is the result of faulty reasoning between the previous statement (“by swapping, the player may gain A or lose A/2”) and the stated conclusion. The suggested reasoning is that because the player may gain twice the amount that he (or she) may lose, that there must be a potential gain. This conclusion is, in fact, incorrect, because it does not consider the offset created by the fact that swapping larger values of A will result in a greater number of losses than gains, and that swapping smaller values of A will have a greater number of gains than losses. Although neither of the enumerated statements specifically claims the probability for the second envelope to contain either ½A or 2A to be 50:50, it is implied. Accordingly, the implied 50:50 probability for the second envelope to contain either ½A or 2A, however, does also imply that each and every specific values of A must also have that same 50:50 probability. The details of this were discussed in greater detail previously.

Regarding the solution to the paradox, if the player initially chooses the envelope containing the amount Y, then A equals Y as defined in statement number 1. Accordingly, the player can only gain an amount Y in swapping for the remaining envelope containing the amount 2A. In terms of amount A, the player effectively swaps amount A for amount 2A. Because the player initially selected the envelope containing the amount Y, the smaller amount, there is no such amount equal to ½A. By comparison, if the player initially chooses the envelope containing the amount 2Y, then A equals 2Y as defined in statement 1. Accordingly, the player can only lose an amount Y in swapping for the remaining envelope containing the amount ½A. And, in terms of amount A, the player effectively swaps amount 2A for amount A. Similarly, because the player selected the envelope containing 2Y, the larger amount, there is no such amount equal to 2A. According, the player can only gain or lose the amount Y, and the reasoning is an exact parallel to the reasoning in the primary version of the paradox.

Regarding the paradox of the non-probabilistic version being difficult to resolve as claimed in the article, the above explanation does not indicate that to be true. Perhaps there is some misunderstanding or perhaps this issue requires further research.

Notes on the Section Titled “History of the paradox”

The historical version of the problem that is characterized by the narrative of two strangers wagering over the amount of money in their wallets appears to be another version of the Necktie Paradox rather than a version of the two-envelope problem. However, the historical version differs from the Necktie Paradox in that each man may or may not possess information regarding the contents of his own wallet that may alter his own advantage or disadvantage in the wager. Accordingly, if both men are oblivious to the contents of their wallet, the structure of the wager becomes identical to that of the Necktie Paradox. However, if either man is aware of the contents of his wallet, that knowledge can have a significant effect on the decision to partake or not partake in the wager.

For instance, if one of the men were carrying more than an expected amount of money in his wallet, he would be wise to not partake in the wager. Similarly, if one of the men were carrying less than an expected amount of money in his wallet (for example, no money), he would likely have a winning advantage. Also, because either man can have the ability to base the decision of whether to make the wager on his knowledge of the contents of his wallet, the problem provides no rationale for providing credibility to having a 50:50 or random probability for either man winning the wager. However, in the interest of focusing only on the paradox portion of the problem, it would probably be more prudent to play along with the intent of the narrative and not be picky about the unintentional flaws in the narrative.

Despite the history of how the two-envelope problem may have evolved, I disagree that the two-envelope problem is simply another version of the two-wallet problem. First, there is the problem of correlating the parenthetically stated “½” probability to the narrative as I mentioned above. Second, even though neither man knows the amount in the wallet of his rival, either man is not precluded from being fully aware of the amount in his own wallet. And, assuming that this amount is reasonable and thereby does not completely override the presumed 50:50 probability to win or lose, one man knowing the amount does affect the path of reasoning to be distinctly different from the two-envelope problem. Third, the two-envelope problem specifically addresses amounts that differ by a factor of two. Although this does not alter reasoning in the problem, it does simplify the discussion when attempting to represent the amounts as variables. And fourth, because the two-envelope problem involves amounts that are unknown and undisclosed, it allows for the inclusion of the argument that proposed a continuous and endless exchange. That same argument is more difficult to propose in the two-wallet problem because the narrative does not support the exchange of unopened wallets.

Accordingly, it appears out of perspective to relate the two-wallet problem to the two-envelope problem without mentioning the more closely related Necktie Paradox. This is similar to (though perhaps not as important as) misplacing a species in the evolutionary tree.

Bill Wolf (talk) 21:10, 21 February 2009 (UTC)

Suggestion to remove non-probabilistic version

It seems to me that the non-probabilistic version of the paradox does not make any sense, since there is no point in comparing potential rewards if no probabilities are attached to them. It's like asking, is it preferable to enter a lottery where the prize is $1,000 or one where it is $100,000? If you are given no more information then the question is meaningless. Or am I missing something? Soler97 (talk) 15:46, 17 July 2009 (UTC)

Yes you missed the point. The logical variant is not a decision problem. And even if you still don't think this version makes any sense, that is of course not a valid reason for removing the section from the encyclopedia. iNic (talk) 01:18, 2 October 2009 (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)

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)

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)

I propose adding a link in external references

I propose adding a link to http://www.opentradingsystem.com/quantNotes/Two_envelopes_.html

It contains a quick experimental resolution argument. Kaslanidi (talk) 20:34, 30 December 2009 (UTC)

I object, per WP:ELNO points 1, 4, and 11. - MrOllie (talk) 20:35, 30 December 2009 (UTC)

I second the objection. OhNoitsJamie ^Talk 20:44, 30 December 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)