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2025/10/25 18:54:07
First edition 2025/10/25

Maybe flawsome descriptions in the Eng Wiki article 'Two envelopes problem' (revision at 04:28, 14 August 2025)

Caution
I who am a Japanese wrote this page in English, but I am not so good at English.


In my perception, the English language Wikipedia article 'Two envelopes problem' (revision at 04:28, 14 August 2025) has some points I think better to be corrected. Such points have increased from the revision 21:39, 23 November 2014.
This page was written revising the page "Maybe flawsome sections of the English Wikipedia article 'Two envelopes problem' (revision at 21:12, 4 January 2021)".


☟ Important
I hope you at least read the following important paragraphs.

Contents

Composition of the revision at 04:28, 14 August 2025 of the article

Revision at 04:28, 14 August 2025 of the article has following composition.

Sections that I think probably better to be corrected

Top (The lead section)

In my eyes, the problem statement summary in this section is not appropriate.

In the part after the phrase "The problem is typically introduced by formulating a hypothetical challenge like the following example:", in my eyes, there are problems such as: As the consequence, I think it is preferable to express only the essence.

Example of the expression of the essence.
The essence of the problem is as follows:
There are two envelopes such that one contains twice amount of money as much as the other. You are randomly handed one of the two envelopes, and you are given the chance to switch envelopes. If you let x be the amount of the handed money, then the expected value of the amount of money in the other envelope will be 1.25x that is greater than x. Therefore, it is always preferable to switch envelopes, regardless of the value of x and regardless which envelope is handed.
This is a paradox.

Section "Introduction"

About a new section that gives examples of problem statement.

For the following reasons, I think it would be better to have a section that gives examples of problem statement. I think the problem statements in the following literatures are suitable as examples.

Section "Problem"

☟ Important
There is a circular reference in the section "Problem"

There is a circular reference among the paper Falk, Ruma (2008) and the English language Wikipedia article "Two envelopes problem". Therefore, instead of the paper by Falk, I recommend to refer to McGrew, T. J., Shier, D., & Silverstein, H. S. (1997) which has the following characteristics.

☟ Important
I think that it is desirable to make sure the opportunity to swap envelope is vary depend on the resolutions

To my eyes, it is better to refine the sentence "You pick one envelope at random but before you open it you are given the chance to take the other envelope instead" as follows.
You pick one envelope at random but before or after you open it you are given the chance to take the other envelope instead.
※ People who adopt the simple resolution such like those presented in the section "Example resolution" need the word "before". On the other hands people who present Bayesian resolution accept both words "before" and "after" while the early literatures of the Bayesian resolution used "after".

☟ Important
I think that it is desirable to make new paragraph "Paradoxes" and move the paradox described by lines 9-12 of the switching argument there.

The paradox described by lines 9-12 of the switching argument is unique to the switching argument before opening envelope. So, I think it must not be included in the common switching argument. And I think some famous paradoxes should be presented in a new paragraph "Paradoxes" with reference to the literature.

I think the new paragraph "Paradoxes" should explain the following:
Mathematical discussion and philosophical discussion:
  • Mathematician try to solve the problem by careful calculation of the probability.
  • Philosophers try to solve the problem by proving that it is impossible to use x/2 and 2x simultaneously in the expected value formula without contradiction.
Examples of paradoxes

I'm afraid that the claim " The puzzle is to find the flaw in the line of reasoning in the switching argument" is unsuitable

From the following, I think mathematicians and philospphers both did not try to find the flaw along whith the line of reasoning. However, the claim is very traditional (the first revision with this claim was at 22:13 on October 3, 2005).
Therefore, I don't think this claim needs to be removed. ☜ Important

I'm afraid that the claim "The puzzle is not solved by finding another way to calculate the probabilities that does not lead to a contradiction" is not suitable for this section.

This claim contradicts the section "Bayesian resolutions". So I think it is more desirable to write this in the section "Example resolution". ☜ Important

I think that it is desirable to create new paragraph "Wording variations of the problem statement".

I think it is desirable to show important variations of the wording of problem statement

Examples of important variations: Examples of less important variations:

Order of solutions

☟ Important
I think the section "Example resolution" should be placed after the section "Bayesian resolutions" with revised title, as the Bayesian resolution was historically discussed before the simple resolutions such like those presented in the section "Example resolution".

Section "History of the paradox"

I think the first sentence "The envelope paradox dates back at least to 1953, when …" is inaccurate

I think the first sentence "The envelope paradox dates back at least to 1953, when …" is inaccurate. This is because, although the envelope paradox and the "Necktie paradox" (presented in Kraitchik,M.(1943).) are very similar, there are essential differences between them.
For the same reason, I think some elements of the envelope paradox do not date back to the "Wallet game" shown in Gardner, M. (1982).

☟ Important
From the above, I think it is beｔter to change the explanation of the relation of the early paradoxes and the exchange paradox as follows.
The envelope paradox was preceded by the "Necktie paradox" (presented in Kraitchik,M.(1943).) and the "Wallet game" (presented in Gardner, M. (1982)) which are similar in nature.
But there are significant differences between the envelope paradox and the earlier paradoxes, so the envelope paradox cannot be seen as a variation of the earlier paradoxes.

Similarities between the earlier paradoxes and the envelope paradox
  • In each paradox, two players have same kind items which are assigned value.
    In the Necktie paradox, each player holds expensive necktie.
    In the Wallet game, each player holds wallet containing money.
    In the Envelope paradox, each player holds envelope containing money.
  • If a player expects the payoff of the game using the mean value of the holding value, the player always has an advantage. It means each player always has an advantage over the other player. This is a paradox.
Significant differences the envelope paradox has comparing the earlier paradoxes
  • The game setting process includes the step to determine a pair of amounts contained in the two envelopes and the step to decide which envelope to give to which player.
  • The ratio of the lesser amount and the greater amount is fixed as 1 to 2.
  • When thinking of one pair of amounts of money, we can discuss the "Simple resolution" such like those presented in the section "Example resolution" to the confusion over the rules of the game and the confusion over the meanings of letter symbol.
  • When calculating conditional expected value conditioned on the handed amount of money, the Base rate fallacy will cause a probability calculation error.
  • In the envelope paradox, it is easy to analyze the discharge fallacy which we often make when calculating the other amount mean value based on the handed amount mean value.

☟ Important
Referring Zabell, S. (1988) is missing.

In the section "History of the paradox", Nalebuff, Barry. (1988), Nalebuff, Barry.(1989) and Gardner, M. (1982) are introduced as early literatures. However, non-refered literatures Zabell, S. L.(1987), Zabell, S. (1988) and Zabell, S. L. (1988) are earlier than them for the following reasons: From the above, I think the following point should be emphasized
  • The above literatures by Zabel as the earliest references of the envelope paradox. ☜ Important
  • The literatures by Nalebuff spreaded the envelope paradox presented by Zabell, S. L.(1987). (Nalebuff used unique words "Ali" and "Baba" to call the game players)
  • The literatures by Nalebuff also originated the Nalebuff's asymmetric variant.

Section "Multiplicity of proposed solutions"

☟ Important
I think the title of the section "Multiplicity of proposed solutions" suggests that there is only one problem and multiple solutions for it. However, such an understanding is a big misunderstanding.
It is not the solutions that are diverse, but the interpretations. ☜ Important
I think it should be emphasized at the outset that there are multiple interpretations of the two envelopes problem as below.

Interpretation that the problem demands the solution based on a pair of amounts of money.
On this interpretation, the fallacy may be thinking of a uniqe amount in the selected envelope.
Interpretation that the problem demands the solution based on a unique amount of money in the selected envelope.
On this interpretation, the fallacy should be misculculation of probability.
Interpretation that the problem demands the solution based on all pairs of amounts of money
On this interpretation, the fallacy may be the discharge fallacy.

Section "Example resolution"

Simple resolutions by philosophers may not be so simple

I think this section "Example resolution" presents resolutions by philosophers who didn't accept mathematical resolution. And the discussed problems, paradoxes, and resolutions are different from those discussed by mathematicians.

※In the English Wikipedia article "Two Envelopes Problem," the solution presented in the "Example resolution" section has been called "Sinple resolution" since the revision around December 2018 to the revision at 03:39, 28 August 2022. So I often call such a resolution by philosophers "Simple resolution by philosophers"

Comparing philosophical and mathematical resolutions
  problem and paradox for philosophers problem and paradox for mathematicians
☟ Important
Interpretation of the expected value calculating formula 		(I'm not sure, but one possibility is as follows)

The formula calculates a conditional expected value based on a unique pair of amounts of money. 		The formula calculates a conditional expected value based on a unique amount of money in the selected envelope.
chance to switch envelopes In all literatures, the chance is given before opening the chosen envelope. In the almost literatures, the chance is given after opening the chosen envelope.
the fallacy of calculation of the expected amount of money in the other envelope fallacious use of x/2 and 2x in the calculation formula groundless probability 1/2
diagnoth of the expected value calculation formula method1 :
detection of non rigidity of the symbol x.

method2 :
detection of the fact that the amount of money in the other envelope is not determind acording the amount of money in the chosen envelope after choice. 		method 1:
exemplification of the case that the probability is not 1/2.

method 2 :
proof that the prbability distribution can not be proper if the probability is always 1/2.
fixing of the calculation formula Often no fix presented, but lessons presented to avoid the error. Almost mathematician derived correct calculation formula using a symbol of the probability density function.
cause of the fallacy expected value calculating formula itself probability illusion
(A few mathematician thought that a careless assumption of the prior distribution is the cause of the fallacy)
thinking about the opposite problem All philosophers ignored the case of chance of switching given after opening. Some mathematician thought that there is no paradox befor opening.
☟ Important
goals of the problem 		Making the paradox disappear along with the falasious calculation formula
※ I think that the formula that derives no paradox is not a goal but a by-product.
  1. confirmation that the probabilty is not always 1/2
  2. confirmation that there is no problem even if the expected value calculation results in an advantageous exchange
  3. fixing of the calculation formula using the correct probability
  4. confirmation of the law of total expectation using the corrected formula

And the philosophical literatures, unlike the mathematical literatures, are so complicated to me that I had a hard time understanding them.
As the consequence, the following may be desirable.

In the section "Example resolution", in my eyes it seems that the resolution is not explained easily understable.

In my understanding, the resolution by philosophers which this section intended to explain is a "Diagnosis" such as follows. I hope that the above "Diagnosis" becomes explained more straightforward.

In addition to the resolution presented in this section, I think it would be desirable to present the most popular simple solutions.

The simple resolution showed in the section "Proposed Solution 1" of the English Wikipedia article "Two envelopes problem" (revision at 22:05, 3 October 2005) is much simpler than the resolutions described in the "Simple resolution" section of the same article revision at 21:12, 4 January 2021.  And it seems to me that the former resolution is more common among people proposing simple resolution on the Internet.
Unlike the latter resolution, the former resolution discusses only the uniqueness of the value of the symbol denoting the amount of money in the handed envelope. And it does not mention how the fallacious calculating formula confuse the reader.

Considering the above, I think that it is desireble to divide the section to the following two parts.
And I think that the following common interpretations should be written before the above two parts.

I think that the phrase "any average A" should not be used.

I think there is a duplication among the following statement in this section and the section "Other simple resolutions", as it uses the phrase "any average A".
Whereas step 6 boldly claims "Thus the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2.", in the given situation, that claim can never be applicable to any A nor to any average A.

A same paper (Priest, G., & Restall, G. (2007)) has been referred to with different reference number in by locations.

I was confused because that paper has reference numbers ``8'', ``9'', and ``13'', but other literature with multiple references each have only one reference number. ☜ Important

Section "Other simple resolutions"

☟ Important
To my eye, the interpretation described in this section seems not so common

I have understood that the first sentence of this section "Other simple resolutions" claims that it is common way to assume the simbol asigned to the amount of money in the selcted envelope is intended to be "expected value". However, as the following reasons, I think that such an interpretation is not so common way. Therefore, I think it is better to replace the first sentence which starts with "A widely discussed way to resolve the paradox" with a more modest sentence like this:
Sometimes we try to make a formula to calculate the expected value of the amount in the other envelope using the expected value of the amount of money in the selected envelope rather than using the amount itself.
Let A, B denote the random variables of the amount of money in the selected envelope and the other envelope.
Then we will get :
  E(B) = (1/2) (2 E(A)) + (1/2) (E(A)/2) = (5/4) E(A) > E(A).
Because this formula and the formula in step 7 of the switching argument have the same structure, the relationship between them is sometimes discussed.

In my eyes, the title "Other simple resolutions" is not suitable to the content of the section.

I understood that the opinion of the section "Other simple resolutions" is as follows. And because of the following reasons, I think that the title "Other simple resolutions" of the section is not appropriate.

Reaons :
As the consequence, it would be desirable to change the title of the"Other simple resolutions" section.
For example:
"Other fallacies and resolutions in cases the chance to switch is given before opening"

Addition : The discharge fallacy associated with the two envelopes problem
The concept of "discharge fallacy" is explained in Jeffrey, R.(1995).
And I understood that the discharge fallacy is explained in the literature like below.
Let X and Y be the amount of money in the selected envelope and the other envelope.
Then by the law of total expectation , we get:
E(Y) = P(Y=2X) E(Y|Y=2X) + P(Y=(X/2)) E(Y|Y=(X/2)) =
(1/2)E(Y|Y=2X) + (1/2)E(Y|Y=(X/2)).
If correctly change conditional expectations, we will get:
E(Y) = (1/2)E(2X|Y=2X) + (1/2)E(X/2|Y=(X/2)).
However, by the discharge fallacy, we often get:
E(Y) = (1/2)E(2X) + (1/2)E(X/2).
(↑ conditions are incorrectly discharged)

However, I think that the above mechanism cannot explain the discharge fallacy that I myself experienced, and I hypothesized that the real mechanism is as follows.
Let X and Y be random variables which are assigned to the amount of money in the selected envelope and the other envelope respectively.
Then we get:
In the situation that X < Y, Y = 2X and E(Y) = 2E(X).
In the situation that X > Y, Y = (1/2)X and E(Y) = (1/2)E(X).
If we have combined them correctly, we will get:
E(Y) = (1/2)2E(X|Y=2X) + (1/2)(1/2)E(X|Y=(X/2)).
However, by the discharge fallacy, we often get:
E(Y) = (1/2)2E(X) + (1/2)(1/2)E(X).
(↑ conditions are incorrectly discharged)


In my eyes, it seems that Schwitzgebel, E., & Dever, J. (2008) is not accurately quoted in the section "Other simple resolutions"

In this section, it is claimed that:
A widely discussed way to resolve the paradox, both in popular literature and part of the academic literature, especially in philosophy, is to assume that the 'A' in step 7 is intended to be the expected value in envelope A and that we intended to write down a formula for the expected value in envelope B.
And the following correct calculation is presented with reference to Schwitzgebel, E., & Dever, J. (2008).
Expected value in B = 1/2((Expected value in B, given A is larger than B) + (Expected value in B, given A is smaller than B))
However, such a reference to Schwitzgebel, E., & Dever, J. (2008) contradicts to the followings.

In my eyes, the paragraph which contains "E(A|A<B) + (1/4)E(A|A>B)" has some flaws

I think the paragraph which starts with "Line 7 should have been worked out more carefully as follows:" has the following flaws.

I'm afraid that the explanation presented in this section "Other simple resolutions" is merely hypothetical.

Im my eyes, the explanation written in this section saids as same as the opinion in Jeffrey, R.(1995) which discusses the "discharge fallacy".  However, as far as I know, only the literatures written by Jeffrey discuss similar fallacy.  In addition, I think such an opinion does not explain my private experience.
So I think that it should be made clear that the explanation presented in this section "Other simple resolutions" is merely hypothetical.

☟ Important
It seems better not to have the paragraph that begins with "Tsikogiannopoulos presented a different way to do these calculations"

Reading the following discussion in an archived talk page, I found that many editors of this article "Two envelopes problem" had thought the opinion by Tsikogiannopoulos better not presented in this article.
The discussion titled "Request for comments" which had started at 11:30, 19 October 2014 , and had been NACed at 06:46, 6 December 2014.
(In April 2021, We can read this discussion in the page titled "Talk:Two envelopes problem/Archive 9 - Wikipedia")
I think it's thanks to the editor who NACed the discussion that we can still read it even now in June 2021. ☜ Important
I think editors who didn't accept the opinion have not yet accepted it.
And in my eyes, in subsequent discussions about the opinion, no editor agreed to present it in the article "Two envelopes problem". ☜ Important
And the opinion by Tsikogiannopoulos is still very minor in 2021.  On March 26, 2021, I searched Tsikogiannopoulos, P. (2014) using Google Scholar and found that it is cited by only one other paper.
And I myself have never seen a similar opinion on internet web pages, except for one blog page. ☜ Important

Most editors didn't seem to focus the meaning of the opinion.
So I think the editors of the article "Two envelopes problem" should discuss about the meaning of the opinion in order to clean up the article. ☜ Important

In the first place, the opinion should not be written in the section of its current location, as it deals with two pairs of amount of money.

Addition: Strange points of the opinion by Tsikogiannopoulos


Addition: I have tried to explain the meaning of "success factor"
(Added on April 30, 2021)

I think that the concept of "imbalance rate" helps to understand the meaning of "success factor".
Let A be the amount of money enveloped in the chosen envelope, and let SL and SG be the success factor of the lesser pair and the success factor of the greater pair respectively. Then,
  SL = (A/2 - A)/(3A/4) = (A/2 - 3A/4)/(3A/4) - (A - 3A4)/(3A/4) and
  SG = (2A - A)/(3A/2) = (2A - 3A/2)/(3A/2) - (A - 3A2)/(3A/2).
(A/2 - 3A/4)/(3A/4) and (A - 3A/4)/(3A/4) are understandable as the "imbalance rate", i.e. the rate of the difference of the amount from the average.  Similarly, (A - 3A/2)/(3A/2) and (2A - 3A/2)/(3A/2) are understandable as the "imbalance rate" too.
As mentioned above, the success factor is interpretable as the switching gain of the imbalance rate.

Addition: An example of very similar opinions

On November 8, 2004, an opinion which was very similar to the opinion by Tsikogiannopoulos had been written in a blog page. In the opinion, the expected value of the imbalance rates had been calculated instead of the success factors as follows.
Assume that the chosen envelope contains 10,000 JPY, then the expected value will be as follows.
Case with switching :  (-2500/7500) × 1/2 + (+5000/15000) × 1/2 = 0
Case without switching :  (+2500/7500) × 1/2 + (-5000/15000) × 1/2 = 0
On November 10, 2004, the author of the blog page himself withdrew the above opinion after reading the mathematical resolution using the Bayes' rule of probability.  ☜ Important

Section "Nalebuff asymmetric variant"

I think it would be better to refer to the Nalebuff asymmetric variant at an another section

In the section "Other simple resolutions", I think it is better not to refer to Nalebuff asymmetric variant presented by Nalebuff, Barry. (1988) for the following reasons.

If the Nalebuff asymmetric variant is cited in any way, it should be emphasized that Nalebuff, Barry.(1989) was a major inspiration to the simple resolution literatures.

I think it's important that a lot of popular literature on simple resolutions without probabilistic/statistical thinking referred to Nalebuff, Barry.(1989).
Examples:

Section "Bayesian resolutions"

It may be beter to clarify the caracterlistics of the mathematical resolution

I think the mathematical resolution has characteristics described in table "Comparing philosophical and mathematical resolutions.
And I think the most important characteristics is that the chance to switch is as frequently given after opening as before opening.
※ Mathematicians can think conditional expected value before opening but are not fond of doing so.

And I think the mathematical resolution is not so Bayesian.
Zabell, S. (1988) was published in a jornal of the Bayesian statistics. And the title of Christensen, R; Utts, J (1992) has word "Bayesian".
But in my eyes, these literatures did not use any concepts specific to Bayesian statistics other than "posterior probability" and "prior probability".
Even if the authors are Bayesian, I don't think their resolution is not so Bayesian.
However, I think it is not bad to use the words "Bayesian probability theory", "Bayes' theorem" or "Bayes' rule" in section titles.

Addition:
As far as I know, the following Wikipedia articles mathematically discuss, but the term "Bayes" is not used in the section titles.

language of Wikipedia title of the article about the two envelopes problem   revision   Section titles
beginning with
"Bayes" 		Words
beginning with
"Bayes"
German Umtauschparadoxon 16:55, 22. Aug. 2016‎ Nothing Nothing
Italian Paradosso delle due buste 15:12, 16 apr 2016‎ Nothing Nothing
Hebrew פרדוקס המעטפות 04:20, 1 במאי 2016‏ Nothing Nothing
Dutch Enveloppenparadox 13 feb 2014 18:33‎ Nothing "Bayes' rule"
Russian Задача о двух конвертах 05:17, 19 ноября 2016‎ Nothing Nothing
As the consequence, the following seems desirable.
Addition: Many mathematical literatures provide the opportunity for switching after opening envelope.
A mathematical article Zabell, S. (1988) is the most old literature of the two envelopes problem, and the chance to switch envelopes is given after opening the chosen envelope. And many famous literatures following this article presented the problem with same fashion.
And many mathematical literatures said that there is no paradox before opening the chosen envelope.
The English Wikipedia article "Two Envelopes problem" had the section "A harder problem" until the revision at 17:57, 8 October 2008, and in the section, the chance to switch was given after opening.

The explanation of the fallacy is merely hypothetical.

In my perception, this section explains that the cause of the fallacy is careless assumption of flat distribution.
But I think that such an explanation is merely one hypothesis by non psychological researchers, and that hypothesis has not been verified by cognitive psychological experiment.
So I think it needs to be clarified that the explanation is just one hypothesis.

For mathematicians, the correct calculation formula is one of their goals

I think that the corrected calculation formula should be presented as one of the goals of the problem. Detection of the probability error should be presented as introduction of the corrected calculation.
And I think that the way to calculate conditional expected value of the amount of money should be presented as basic knowledge, not as "further developments in connection with Bayesian probability theory". ☜ Important
Historically, Zabell, S. (1988) (The most old literature of the two envelopes problem) presented calculation formula of conditional probability as below.
P[X=y|Y1=y] = p(y)/{p(y)+p(y/2)}. (X is the lesser amount, Y1 is the amount contained in your envelope, and p is the continuous or descrete dencity function.)
(I'm afraid that the above formula is wrong for the continuous case)
 I think the German language wikipedia article "Umtauschparadoxon" (revision am 3. April 2020 um 09:31) is exemplary as it clearly presents each conditional expectation formula for the case of descrete distribution and the case of continuos distribution.

☟ Important
Nalebuff (1989) is referred incorrectly

I'm afraid that the sentence "It also applies to the modification of the problem (which seems to have started with Nalebuff) in which ……" is incorrect, as it is not a modification started with Nalebuff. In the earliest literatures of the two envelopes problem (example: Zabell, S. (1988)), the opportunity to swap is given after opening selected envelope.

In the first place, I'm afraid that Nalebuff, Barry.(1989) is not suitable to refere as a literature of Bayesian resolutions. Because it did not present the problem statement in full form.  So I think it is better to reffer it only as a litterature of Nalebuff asymmetric variant. Instead it, as an literature of so called Bayesian resolution on the case before opening envelope, I recommend Jackson, F., Menzies, P., & Oppy, G. (1994).

Section "Second mathematical variant"

It may be better to change the title

I don't think it's appropriate to number it like "Second mathematical variant". So I think the following titles may be better. However, at the revision of 14:46, 27 August 2004, the English language Wikipedia article "Envelope paradox" had a section titled "A second paradox" on this theme. Therefore, it cannot be said that the title "Second mathematical variant" is not traditional, and this problem might be not so important.

Section "Conditional switching"

☟ Important
I'm afraid that the important theme of the randomised switcing is not properly explained.

Randomized switching has been very important theme from the early days of the two envelopes problem.
However in my eyes, it is not properly explained - for example, the phrase"conditional switching problem" is not unfamiliar and the important technique of the randomized switching is not properly explained.
So I think the section "Conditional switching" should be replaced with the section "Randomized solutions" revision at 12:56, 28 December 2019 . ☜ Important

Addition:


Reference

Terms





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