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Return to the list of my pages written in English about the two envelopes problem
2025/09/13 15:59:07
First edition 2021/04/04

Maybe flawsome sections of the English Wikipedia article 'Two envelopes problem' (revision at 21:12, 4 January 2021)

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 21:12, 4 January 2021) has some points I think better to be corrected. Such points have increased from the revision 21:39, 23 November 2014.

☟ Important
I hope you at least read the following important paragraphs. (← Added on February 1, 2022)

Contents

Composition of the revision 21:12, 4 January 2021 of the article

Revision 21:12, 4 January 2021 of the article has following composition.

Sections that I think probably better to be corrected

The lead section

(Added on February 1, 2022)

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

In my eyes, in the part after the phrase "The problem typically is introduced by formulating a hypothetical challenge of the following type:", there are problems such as: (← Quoted sentence was revised on April 24, 2025) 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 "1 Introduction"

(Added on May 25, 2024)

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 "1.1 Problem"

(Added on July 25, 2021)

☟ Important
There is a circular reference in the section "1.1 Problem"

(Moved here on July 25, 2021. The title was revised on December 21, 2021)

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

(Added on May 25, 2024)

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 present the simple resolution need "before". On the other hands people who present Bayesian resolution accept both "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.

(Added on February 1, 2022)

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 very compelling line of reasoning above" is unsuitable

(Added on December 21, 2021)

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 "Thus, in particular, the puzzle is not solved by the very simple task of finding another way to calculate the probabilities that does not lead to a contradiction" is not suitable for this section.

(Added on May 25, 2024)

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

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

(Added on February 1, 2022)

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

(Added on April 15, 2024)

☟ Important
I think the section "Bayesian resolutions" should be placed before the section "Simple resolution"", as the Bayesian resolution was historically discussed before the simple resolutions.

Section "2 Multiplicity of proposed solutions"

(Added on April 15, 2024)

☟ 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. (← Revised on May 25, 2024)
It is not the solutions that are diverse, but the interpretations. ☜ Important
(↑ Added on May 9, 2024)
I think it should be emphasized at the outset that there are multiple interpretations of the two envelopes problem as below. (← Revised on May 25, 2024)

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 "3 Simple resolution"

I think that it is desirable to present the most popular simple resolution

(Added on September 13, 2025)

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 reolution" 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.

Simple resolutions by philosophers may not be so simple

I think this section "Simple 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.

Comparing philosophical and mathematical resolutions
  problem and paradox for philosophers problem and paradox for mathematicians
☟ Important
Interpretation of the expected value calculating formula
(Added on May 25, 2024) 		(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
(↑ Revised on April 5, 2021)
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.
(↑ Revised on May 25, 2024)
  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
(↑ Revised on May 25, 2024)

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. (Totally reivised on September 13, 2025)

In the section "Simple resolution", in my eyes it seems that the simple resolution by philosophers was not explained easily understable.

(Added on November 26, 2022)

In my understanding, the simple resolution by philosophers is a "Diagnosis" such as follows. And I think it is explained in the section "Simple resolution", but the explanation is not easily understandable.
I hope that the above "Diagnosis" becomes explained more straightforward.

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

(Added on November 26, 2022)

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.

(Added on May 25, 2024)

I was confused because that paper has reference numbers ``6'', ``7'', and ``10'', but other literature with multiple references each have only one reference number. ☜ Important

Section "4 Other simple resolutions"

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

(Added on June 08, 2024)
I have understood that the first sentence of this section "Other simple resolutions" claims that it is common way to interpret the simbol asigned to the amount of money in the selcted envelope as "expected value". (← Revised on June 28, 2025)  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 common 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.

(On November 26, 2022, this paragraph was written as the revised virsion of the paragraph 'In my eyes, the paragraph which contains "E(A|A<B) + (1/4)E(A|A>B)" is not suitable to the section "Other simple resolutions"')
I understood that the opinion of the section "Other simple resolutions" is as follows. (↑ Revised on May 1, 2025)

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

Reaons :   (Greatly revised on June 08, 2024)
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"

On June 08, 2024, the paragraph "discharge fallacy" was divided to two parts as below.

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. (← Revised on June 08, 2024)
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)
(↑ Added on June 08, 2024)
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. (← Revised on May1, 2025)
(This paragraph was added on June 08, 2024)
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)
(↑ Revised on May 1, 2025)

(On September 13, 2025, the paragraph "Addition 2 : Relationship of the discharge fallacy and so called "the simple resolutions" discussed by philosophers" was deleted)
(On June 08, 2024, the paragraph about the psychological similarity between the discharge fallacy and the confusion between average of rates and rate of averages was deleted)

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

(Added on April 7, 2021)

In this section, it is claimed that:
A common 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.
(On November 26, 2022, the paragraph "In my eyes, the paragraph which contains 'E(A|A<B) + (1/4)E(A|A>B)' is not suitable to the section '4 Other simple resolutions' " was deleted.)

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

(On November 26, 2022, the follwing was created using some contents of the removed paragraph "In my eyes, the paragraph which contains 'E(A|A<B) + (1/4)E(A|A>B)' is not suitable to the section '4 Other simple resolutions' ")
(On June 28, 2025, the following was totally revised.

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.

(Added on September 13, 2025)

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. (← Revised on April 10, 2021)
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")
(↑ Revised on April 9, 2021)
I think it's thanks to the editor who NACed the discussion that we can still read it even now in June 2021. ☜ Important (← Added on June 10, 2021)
I think editors who didn't accept the opinion have not yet accepted it. (← Added on April 6, 2021)
And in my eyes, in subsequent discussions about the opinion, no editor agreed to present it in the article "Two envelopes problem". ☜ Important (← Added on December 1, 2021)
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 (← Added on August 14, 2021)

Most editors didn't seem to focus the meaning of the opinion.
(Added on July 18, 2021)
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
(↑ Added on November 23, 2021)

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.
(↑ Added on April 24, 2021)

Addition: Strange points of the opinion by Tsikogiannopoulos
(Added on April 18, 2021)


↑ On December 14, 2021, one item relating to the confusion between average of rates and rate of averages was removed from the above list. On December 21, 2021, another low-value item was removed from the above list.

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
(Added on April 30, 2021)

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. (One zero was added on May 11, 2021)
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 "4.1 Nalebuff asymmetric variant"

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

(The paragraph "I think it may be better not to refer to Nalebuff, Barry. (1988) here" was totally revised on June 14, 2024 with new title)

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.

(Added on April 16, 2024, Revised on May 25, 2024)

I think it's important that a lot of popular literature on simple resolutions referred to Nalebuff, Barry.(1989).
Examples:

Section "5 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. (← Revised on May 25, 2024)
※ 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. (← Revised on November 11, 2021)
language of Wikipedia title of the article about the two envelopes problem   revision   Section titles
beginning with
"Bayes"
(Added on
November 11 ,2021) 		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. (← Revised on September 13, 2025)
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.

(This title was revised on September 13, 2025)

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. (← Revised on April 8, 2021)

☟ Important
Nalebuff (1989) is referred incorrectly

(Added on May 25, 2024)

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 "5.3 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.
(↑ Added on August 1, 2021)

Section "7 Conditional switching"

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

(This title was revised on September 13, 2025)

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. (← Added on September 13, 2025)
So I think the section "Conditional switching" should be replaced with the section "8 Randomized solutions" revision at 12:56, 28 December 2019 . ☜ Important

Addition: (← Added on April 15, 2024)

Section "8 History of the paradox" (← Added on June 28, 2025)

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" 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.


The photo paseted on the lead section

(On February 1, 2022, this title was changed)

It is not an error to fold bills in the photo titled "The puzzle concerns two envelopes containing money". But …

When I first saw it, the envelopes in the photo looked like the simplified version of NOSHI BUKURO rather than the formal version. (← Revised on June 23, 2021, July 25, 2021)
Formal NOSHI BUKURO is a kind of gift wrapping and it is used for congratulations party such as wedding receptions.
The NOSHI BUKURO in the photo looked like just a simplified version , but I was startled to see the folded bill. (← Revised on June 27, 2021, November 11, 2021)
However I can understand that it is not an error to fold bills in the simplified version NOSHI BUKURO which is used for informal celebrations.
And I have found there is a version of POCHI BUKURO (Envelopes for a small amount of money gifts) which imitates NOSHI BULURO and we can't put bills in it unless folding. (← Revised on May 25, 2024)
And over time, I've become accustomed to seeing the photo.

(↓ Added on June 18, 2021)
Recently (June, 2021), I found that POCHI BUKURO which imitates NOSHI BUKURO is sometimes used as OTOSHIDAMA BUKURO (Envelopes for New Year's money gifts). (← Revised on May 25, 2024)
And I found that the photo which was pasted on the lead section on September 18, 2020 is exactly of such a POCHI BUKURO.
That photo had been registered to Wikimedia on December 31, 2019 and paseted on the Japanese language Wikipedia article "お年玉" (OTOSHIDAMA) on the same day.
(↑ Revised on August 14, 2021, November 29, 2022, May 25, 2024)
I had misunderstood that the photo is of the simplified version NOSHI BUKURO. (← Added on July 02, 2021, May 25, 2024)

(On June 23, 2021, the text explaining my personal experience as a Japanese about OTOSHIDAMA BUKURO was deleted)

(↓ Added on June 23, 2021)
However, another problems about the photo still remain. (← Revised on June 27, 2021, July 25, 2021, August 14, 2021)
(↑ Revised on May 25, 2024)
 (↓ Added on July 02, 2021)
The former photo pasted at the revision 11:42, 27 August 2009 and removed at the revision 22:34, 18 January 2016 may be more suitable for the following reasons. (← Revised on August 14, 2021) However, in my eyes, this photo is very dreary. It may be better not to have a photo, as many other language Wikipedia articles about the two envelopes problem dont have. (← Added on August 14, 2021)

About Afterwards

(↑ Added on November 26, 2022)

The photo was removed at the revision 18:21, 2 August 2022.


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