Talk:Two envelopes problem/Archive 2

message: I don't like the introduction:
:The Two Envelopes Problem is a puzzle within the subjectivistic interpretation of probability theory. This is still an open problem among the subjectivists as no consensus has been reached yet.
:Because the subjectivistic interpretation of probability is closer to the layman's conception of probability this paradox is understood by almost everybody. 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 stated when imposing those more technical restrictions. Consequently, all published papers below with different ideas on a solution are written from the subjectivistic or Bayesian point of view.
I don't like it because it tells me the point of view I should have about the problem before expainig what the problem is. Comments about subjectivism should come later as a possible pont of view, not at the very beginning BEFORE everithing else!!
--Pokipsy76 16:22, 24 October 2005 (UTC)
replies: text: Good point! I moved the second paragraph to after the presentation of the puzzle and made it a comment. Is this good enough you think? If not, please feel free to improve it yourself.  INic 01:14, 25 October 2005 (UTC)
thread_title:  The introduction 

message: The explanation given doesn't make sense. Let's say I open the envelope and there is a $100. If I switch I expect to gain $125. Now that I switched I do the following calculation: I have 1/2 * 50 + 1/2 * 200 = $125. In the other envelope I see $100. So why would I switch back? The fact that I openned the envelope and found a sum greater than 0 added additional information about the expected value in the game. There is no paradox in this case.
-- Martin
:You are right, it should be made more clear in the article how the contradiction appears in this case. In essence it's the same contradiction as in the first case. The only difference is that here the explanation given for the first case doesn't work anymore. I'll try to improve the text. INic 13:28, 2 August 2006 (UTC)
thread_title:  A Harder Problem 

message: I have a problem with the 'An even harder problem'.
First of all, there are two major mistakes in the calculation of the expected gain:
1) the first 1/2 should be omitted
2) a factor (1 - q)^(n-1) is missing in the final expression.
so it should be
(2^n q(1 - q)^n - 2^(n-1)q(1 - q)^(n-1))/R 
= 2^(n-1)q(1 - q)^(n-1)(1 - 2q)/R 
which is still positive if q < 1/2
But most of all, it does not make sense to consider distributions with infinite mean. They don't have practical value and you can't make any conclusion based on calculations with these kind of distributions.
It is easy to prove that the expected gain of switching envelopes before opening them, should be zero for each 'real' (finite mean) distribution.
The expected amount in the two envelopes is the same by symmetry which means that the expected gain of switching (difference of the two) is zero, if the expected amount in the envelopes is finite.
This means that for every 'real' distribution that the mean of the gain is zero.
If there are amounts, with probability > 0, for which it is better to switch then there are other amounts, with probability > 0, for which it is better not to switch.
The only distribution for which the reasoning of the paradox holds, is the one for which the envelopes contain 0 with probability 1. But then, the gain of switching remains zero even if the amount is doubled in the other envelope.
To conclude, It is better to remove the section about 'An even harder problem'. Distributions with infinite mean really don't make any sense and it just confuses the reader.
Regards,
Stefaan
:I agree that the example presented here can be better. I will soon replace this with an example found in the literature. But I don't agree that the only admissible distributions are the ones with finite means. Cauchy distributions for example lacks both mean and variance but that doesn't make them 'unreal' or unuseful. INic 21:17, 30 July 2006 (UTC)
thread_title:  The 'an even harder problem' does not make sense 

message: I recently edited the intro so it is easily understood:
The two envelopes problem is the following:
*You are shown two indistinguishable envelopes and asked to select one, which you do.
*You are then told that each envelope contains money, and one contains twice as much as the other.
*You are given the option to swap the envelope you have chosen for the other one.
*What should you do? 
Some writers analyse this as a puzzle or paradox within the subjectivistic interpretation of probability theory; more specifically within Bayesian decision theory. This is still an open problem among the subjectivists as no consensus has been reached yet. 
However, this has now been removed so the definition is 
  a puzzle or paradox within the subjectivistic interpretation of probability theory; more specifically within Bayesian decision theory. This is still an open problem among the subjectivists as no consensus has been reached yet. 
IMHO it's far better to explain the problem BEFORE going into how it might be analysed. (I've tried it on lots of people - they have had good analyses without a whiff of Bayesianism!
:First of all the two envelopes problem is not the one you have written above ending with the question "What should you do?". Instead the very statement of the problem must include the reasoning leading to the contradiction. That is followed by the question where the flaw in the reasoning is. That is the problem. See also my answer in the section above with title "What's the problem?". INic 00:13, 31 August 2006 (UTC)
replies: text: Secondly, it is actually considered good wikipedia practice to have a short introduction to set the subject of the article into the right context before going into the meat of the matter. Please see WP:LEAD. INic 00:13, 31 August 2006 (UTC)

text: Thirdly, wikipedia articles should only be based on the contents of already published papers in reputable magazines. That you have friends that have solved this problem in an entirely different way doesn't matter I'm afraid. All published suggested solutions so far are Bayesian in character, and that is a fact that should be reflected here. When your friends have published their solutions we will rephrase this article accordingly. INic 00:13, 31 August 2006 (UTC)

text: The problem or puzzle is to solve the paradox or get rid of the contradiction. I haven't seen any disagreement here in the literature. The only disagreement here is about what to call it, not what it is. My personal opinion is irrelevant. INic 02:28, 1 September 2006 (UTC)

text: I can't find any clear distinction in the literature between what you call 'problem' versus 'paradox.' These words are used interchangeably. INic 02:28, 1 September 2006 (UTC)

text: No, as is explained in the comment of the lead section Bayesianism enter the stage already in the set-up of the problem. The only exception to this is the last, hardest, variant of the puzzle which isn't connected to probability at all. INic 02:28, 1 September 2006 (UTC)

text: This is clearly not the same problem. This is seen by the fact that no contradiction or paradox can arise based on this situation. And that this problem merits an non-Bayesian analysis and solution is thus totally beside the point. INic 02:28, 1 September 2006 (UTC)

text: Yes that is OK. INic 02:28, 1 September 2006 (UTC)
thread_title:  Recent Edits - lets' express the probelm clearly before going into Bayesiana -Do we need to START with Bayes? 

message: I suggest to replace the word "solution" in the paragraph titles by "discussion" because what we do is discussing the problem and it is not clear who is proposing the "pèroposed solutions" and who is suggesting that these arguments *are* actually solution.--Pokipsy76 13:32, 15 September 2006 (UTC)
replies: text: I agree that "discussion" is more fitting, especially for the later paragraphs. However, for the first "proposed solution" there isn't much disagreement that this is the correct solution. Hence to replace that with "discussion" would be incorrect as no discussion exists here really. "Proposed solution" I think is appropriate for covering all cases, from almost total agreement to total disagreement. Another option would be to rename them "solution," "proposed solution" and "discussion" respectively. However, some might argue that this introduces an unnecessary element of POV to the article. INic 01:09, 16 September 2006 (UTC)
thread_title:  "solution" -> "discussion" 

message: I was posed this paradox a while back (the "harder" version), and passed it on to a friend of mine. He then pointed me to this page. I'm no wiki expert (I only registered today) but feel that it is not readily comprehensible to non-specialist readers. Or someone as stupid as me.
My way to explain the paradox is to see that the 
values are chosen before you even see the envelopes. You having chosen the 
one with $512 in means that the value of the other envelope is already 
set. The chances of that envelope having a $1024 or $256 in it are not 
50/50, they depend upon the generosity of the person filling the envelopes.
Choosing the envelope with $512 in it was the 50/50 part of it, not whether the other 
one is higher or lower. That is why the maths doesn't work, it is a failure to interpret the question correctly.
I think that this agrees with the proposed solution, but is there any chance that someone could attempt to make the page a little more accessable to the layperson?  Situationist 00:48, 8 November 2006 (UTC)
replies: text: Math is not psychological or material, it is conceptual. Generosity does not play any part in this, as evidenced by the following two concepts.  1) The envelope filler MUST put two amounts of money, one in each envelope. 2) One MUST ALWAYS be twice as great as the other. Your example, in which you use real dollar amounts, can be contradicted by the assumption the filler is wealthy beyond belief. Or look a it this way: Have him put in $10 and $20. If you pick $10, you would see his as "generous". But if you pick $20 the first time, you would see him as stingy. Things like this are all a matter of opinion anyway. Try using variables when examining the problem, and imagine a computer filling the envelopes as well. It may help you understand it better. MateoCorazon 01:42, 12 November 2006 (UTC)
thread_title:  One stupid persons viewpoint 

message: Hi,
Congratulations on creating an informative article. One grips is that the Manual of Style says articles should where possible not address the reader directly using 'we' or 'you'. I think the tone of this article should reflect that using more passive statements and so on. Thanks Andeggs 22:19, 22 December 2006 (UTC)
replies: text: Personally, I like this direct style and don't think it becomes too unencyclopedic. The disadvantage of using passive statements everywhere is that the text can seem too academic and abstract for many readers. In addition, the common way to express this paradox in the (academic) literature is in this direct way. I don't think wikipedia should try to give a more academic impression of a topic than the academic articles that it's the summary of. iNic 00:50, 23 December 2006 (UTC)
replies: text: Actually having had a look at the article again and at the manuals of style I think its the use of 'you' that is more of a problem than the use of 'we'. As the main Manual of Style says: "Avoid second-person pronouns". I'm sure this is possible with a bit of rewording but I don't want to do it and then be reverted immediately. I think the point about the academic literature is a little bit weak because we are writing an encyclopedia but they are writing maths/philosophy books and articles - and these are necessarily different. Thanks Andeggs 12:03, 23 December 2006 (UTC)

text: Please go ahead and do the changes you find appropriate. English isn't my native language so at least I won't revert any language changes. Regarding the academic style I was thinking about another writer at the talk pages here requesting a summary of this article for non-specialists that he could understand. Even the guideline you refer to include skepticism about its own applicability: While opinions vary on how far this guideline should be taken in mathematics articles — an encyclopedic tone can make advanced mathematical topics more difficult to learn — authors should try to strike a balance between simply presenting facts and formulas, and relying too much on directing the reader or using clichés [...] iNic 12:37, 23 December 2006 (UTC)
thread_title:  Use of 'we' and 'you' 

message: 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)
thread_title:  the danger of concrete examples 

message: 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)
replies: text: 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)
thread_title:  Why no game-terminating event? 

message: 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)
thread_title:  Comments about the Wikipedia Article 

message: 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)
replies: text: 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)
thread_title:  Suggestion to remove non-probabilistic version 

message: 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)
replies: text: I object, per WP:ELNO points 1, 4, and 11. - MrOllie (talk) 20:35, 30 December 2009 (UTC)
replies: text:  I second the objection. OhNoitsJamie Talk 20:44, 30 December 2009 (UTC)
thread_title:  I propose adding a link in external references 

message: 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)
replies: text: Having worked through the "even harder problem" I think this is valid and quite interesting. I have tried to add points. Since the original two-envelope problem is now clearer, I don't think including the more complicated example will detract from it - is everyone else happy with this version? Dilaudid (talk) 21:24, 26 May 2010 (UTC)

text: Suppose that the envelopes contain the integer amounts {2n, 2n+1} with probability 2n/3n+1 where n = 0, 1, 2,… 
Why was this probability formula chosen, and what significance does it have for the problem?   96.245.11.102 (talk) 13:08, 26 November 2010 (UTC)
thread_title:  Constructive suggestions 

message: {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)
replies: text: 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)
replies: text: 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)
thread_title:  Equation source 

message: 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)
replies: text: 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)

text: It is not as simple as that.  The main problem is the lack of clarity in the problem definition.  Priest and Restall give an interesting discussion of some of the issues involved. But there is no universally accepted solution (or problem). Martin Hogbin (talk) 10:41, 26 December 2010 (UTC)
thread_title:  solution