Return to the list of my pages written in English about the two envelopes problem

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

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)

• Paragraph about the order of solutions → here  (← Added on April 15, 2024)
• Paragraph about the multiple solutions → here  (← Added on April 15, 2024)
• Paragraph about the opinion by Tsikogiannopoulos → here
• Paragraph about the theme of randomized switching → here

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

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

• 1 Introduction
  • 1.1 Problem
• 2 Multiplicity of proposed solutions
• 3 Simple resolution
• 4 Other simple resolutions
  • 4.1 Nalebuff asymmetric variant
• 5 Bayesian resolutions
  • 5.1 Simple form of Bayesian resolution
  • 5.2 Introduction to further developments in connection with Bayesian probability theory
  • 5.3 Second mathematical variant
  • 5.4 Proposed resolutions through mathematical economics
  • 5.5 Controversy among philosophers
• 6 Smullyan's non-probabilistic variant
  • 6.1 Proposed resolutions
• 7 Conditional switching
• 8 History of the paradox
• 9 See also
• 10 Notes and references

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 "Problems are usually caused by formulating the following types of virtual challenges", there are problems such as:

• Redundant and confusing
  It confuses whether the problem statement written here or the problem statement written in section "1.1 Problem" is the original problem statement.
• Misleading
  The highlighted paragraph that start with "You are given …" and end with "Should you switch?" is the basic setup part of the problem. In such a way of writing, it may be misunderstood that only the basic setup part is the whole problem statement.
• Heterogeneous
  The expression "because you stand to gain twice as much money if you switch while risking only a loss of half of what you currently have" is not using variable symbol. I think such expression may trigger other fallacy such as the discharge fallacy.

As the consequence, I think it is preferable to express only the essence.

Example of the expression of the essence.

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".

• The paragraph "Basic setup" in the section "1.1 Problem" of the English language Wikipedia article "Two envelopes problem" (revision at 21:12, 4 January 2021) refers to the paper.
• The paragraph "Wikipedia's version" in the section "THE PUZZLE" in the paper refers to the English language Wikipedia article "Two envelopes problem" (revision at around 2008). (← Fixed "2018" to "2008" on May 18, 2021)

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.

• The opportunity to change the envelope is given before opening.
• The problem is presented using a reasoning made of numbered steps. This style is common to the paragraph "The switching argument" of the section "1.1 Problem" of the Wikipedia article "Two envelopes problem".
• On April 23, 2021, I searched this paper using Google Scholar and found that it is cited by 27 articles.
• It had been referred by the Wikipedia article "Two envelopes problem" from the early revision at 21:47, 3 October 2005.

☟ 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:

• There are several famous paradoxes that are derived by the expected value.
• Some paradoxes are unique to mathematical discussion.
• Some paradoxes are unique to the case that the opportunity to swap is before opening envelope.

Examples of paradoxes

• Paradox that both player expect positive return
  Early literatures of the two envelopes problem took on two-player game style of the wallet game and they took on paradox too.
  And they mathematically discussed on the case where the swap opportunity is after opening the envelope.
  Literaures :

  • Zabell, S. (1988) (mathematical, opened envelope)
  • Nalebuff, Barry. (1988) (mathematical, opened envelope)
• Paradox that you should swap regardless of the amount
  This paradox is unique to mathematical discussion.
  And I think this paradox is very important, as this lead to breaking of the law of total expectation .
  Literaures :

  • Christensen, R; Utts, J (1992) (Mathemtical, opened envelope)
  • Broome,John.(1995). (Mathemtical, probably opened envelope)
    More mysterious paradox that you should switch without expected value calculation nor opening envelope was presented as a consequence of the above paradox.
• Paradox that having whichever envelope, the other is more preferable
  Literaures :

  • Cargile, J. (1992) (philosophical, unopened envelope)
  • Jackson, F., Menzies, P., & Oppy, G. (1994). (mathematical, unopened envelope)
  • McGrew, T. J., Shier, D., & Silverstein, H. S. (1997) (philosophical, unopened envelope)
• Paradox that switching envelopes due to the switching argument makes you want to switch back due to the same argument
  This paradox is unique to the argument on the case of unopened envelope.
  Literaures :

  • Storkey, Amos. (2000-2005) (philosophical, unopened envelope)
  • Schwitzgebel, E., & Dever, J. (2008) (philosophical, unopened envelope)

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.

• In my eyes, in the literature I read, both mathematicians and philosophers examined each element of the formula, not each reasoning step.
  • From the beginning, mathematicians suspected that the probability 1/2, and fixed it by careful calculation.
  • From the beginning, philosophers suspected the notations "A/2" and "2A".and found that in any cases, using them in the same expected value formula would lead to inconsistencies.
• In my eyes, even McGrew, T. J., Shier, D., & Silverstein, H. S. (1997), which used a similar problem statement and wrote that "An adequate solution must explain exactly what is wrong with this line of reasoning", seemed not to have examined each reasoning step in turn. (← Revised on January 9, 2022)

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 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:

• When will players be given the opportunity to exchange? Before opening the selected envelope? after opening?
• How many players? A niece and a nephew? Only you?
• Are example amounts of money presented? Not presented?

Examples of less important variations:

• Who is the game master? A swami? Uncle of the players? Anonymous?
• How are the money enveloped? Using banknotes? Using sheets of paper on which the amount written?

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 the editors of the article understand that there is only one problem and multiple solutions for it. However, such an understanding is a big misunderstanding.
I think the editors of the article should understand 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 "3 Simple resolution"

Resolutions by philosophers may not be so simple

I think this section 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
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.
goals of the problem	error detection of the expected value calculation formula ※ I think that the formula that derives no paradox is not a goal but a by-product.	confirmation that the probabilty is not always 1/2 fixing of the calculation formula 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.

• Change the title of this section (I'd like "Problem and resolutions discussed by philosophers" or "Paradox of unopened case".)
• Clarify that the chance to switch is before opening the envelope ☜ Important
  (↑ Added on April 10, 2021)
• Clarify that the chance to switch is after opening in the original two envelopes problem which mathematicians put out to the world ☜ Important
  (↑ Added on April 29, 2021)
• Clarify other features above

In the section "3 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.

• They test the expectation formula with the following tests.
  • Test of the rigidity of the symbols assuming the pair of amounts is fixed
  • Test whether any contradiction is exists assuming the symbols is rigid designator
• Depending on the test, they conclude that the symbols "2A" and "A/2" are cannot be used together in the same formula.

And I think it is explained in the section "3 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 "4 Other simple resolutions", as it uses the phrase "any average A".

Section "4 Other simple resolutions"

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.

• Step 7 states that the expected value of the other amount = 1/2 × ( 2 × (the selected amount ) + (the selected amount)/2 ).
• A correct calculation would be:
  Let A and B be the amount in the selected envelope and the amount in the other envelope respectively, then
  Expected value of B = 1/2 ( (Expected value of B, given A > B) + (Expected value of B, given A < B) ).

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

• I think such an opinion cannot be a resolution of the problem described in the section "1.1 Problem".
  Reason :
  I think that the above opinion is applicable to the following cases.
  • The case that the resolver intentionally interprets that the variable symbol in the expectation formula means expected value of the amount of money in the selected envelope.
  • The case that the resolver thinks the problem without looking the expectation formula nor using mathematical notation.
    Example:
    If the selected envelope is the lesser the other envelope is double it, and if it is the greater the other envelope is half of it. It means the expected value of the the other envelope is (1+ 1/2) times the selected envelope.
• The above wrong calculation is derived from very special fallacy called "discharge fallacy".
  And as far as I know, only the following literatures discuss the fallacy.
  • Schwitzgebel, E., & Dever, J. (2008)
  • Jeffrey, R.(1995)
• It is not philosophical.
  The paradox resolved by simple resolutions has been discussed using philosophical terms in philosophical literatures.
  Therefore, even if you are a mathematician, I think you need to discuss the paradox using philosophical terms (eg, "rigid designator", "possible worlds", etc.) rather than using mathematical terms.
• Rather, it's a good fit for other problems such as "the wallet game" that don't use symbols of variable.
  The wallet game was presented by Gardner, M. (1982).
  The characters in the puzzle are math students Joe and Jill at a lunch table with professor Smith. (← Added on April 17, 2021)
  At the end of the puzzle, Gardner suggested that the false assumption of probability is the cause of the paradox. However, I think that the discharge fallacy can occur as often as it. (← Added on April 17, 2021)
  A similar game is presented in the section "8 History of the paradox" of the English Wikipedia article "Two envelopes problem". It is common with the original wallet game in that it does not use symbols of variable.
  (↑ Added on April 14, 2021)

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: discharge fallacy
The concept of "discharge fallacy" is explained in Jeffrey, R.(1995).
And I understood that the fallacy is the careless replacement of unconditional expectations for conditional expectations.
And "the discharge fallacy" is different from the fallacy discussed by philosophers, as most philosophers probably had not think of the expected value of the amount of money contained in the chosen envelope.
I expect that there is some psychological similarity between the discharge fallacy and the confusion between average of rates and rate of averages. ☜ Important
And, based on my own psychological experience, I got the hypothesis that the first half of the psychological process of discharge fallacy would be the confusion between average of rates and rate of averages.
This is another reason to consider the fallacy described in the section "4 Other simple resolutions" to be different from the fallacies described in the section "3 Simple resolution".

In my eyes, it seems that Schwitzgebel, E., & Dever, J. (2008) is not accurately quoted in the section "4 Other simple resolutions"

(Added on April 7, 2021)

In this section, it is claimed that: And the following correct calculation is presented with reference to Schwitzgebel, E., & Dever, J. (2008). However, these contradict to the followings.

• I have not found philosophical literature which considered the expected value in envelope A, other than Schwitzgebel, E., & Dever, J. (2008) and Rawling, P. (1994). (← Revised on February 6, 2022)
• In my eyes, the authors of Schwitzgebel, E., & Dever, J. (2008) apparently did not assume that the symbol X to be the expected value of envelope A, even though they took into account the expected value of envelope A.
  Indeed, we can find the phrase "expected value of X in the «2X» part of the expression" in the literature. This means that the symbol X itself is not the expected value.
  (↑ Added on July 13, 2021)
• In my eyes, the main goal of Schwitzgebel, E., & Dever, J. (2008) is a new diagnostic method to confirm the equality among expected values of X in each cases for error detection of expected value calculation formula. (X means amount of money in the chosen envelope)
  In other words, their goal was not to find a way to fix the formula, but to find a way to test it. (← Added on July 13, 2021)

(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 drawbacks

(On November 26, 2022, this new paragraph 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' ")

The same expression as "E(A|A<B) + (1/4)E(A|A>B)" was written in Jeffrey, R.(1995). However, to my eyes, showing the original fallacious expression "(1/2) 2 E(A) + (1/2) (1/2) E(A)" is more valuable rather than deforming the correct expression to the above expression. And I think that showing that E(A|A<B) is exactly as half as E(A|A>B) is more valuable rather than showing the above deformed expression "E(A|A<B) + (1/4)E(A|A>B)".

☟ 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) 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)

• Most editors seemed to weighted the notability of the opinion, or whether Tsikogiannopoulos first proposed it, or the existence of the journal that published the literature containing the opinion, rather than examining the concept of "success factor" which is the subject of the opinion.
  (← Revised on November 23, 2021)
• In the other discussion titled "Tsikogiannopoulos returns" on the talk page, a mathematician editor commented that the opinion was just a duplication of what the editors already had in the first resolution. However, I wonder he may have overlooked the peculiarity that Tsikogiannopoulos's opinion is the application of the following law.
  If the ratio of two amounts is constant the ratio of their difference to their average is constant.
• Since no fallacy is identified by the opinion, it never be a solution to the two envelopes problem, but just an example of wacky calculations. ☜ Important (← Added on November 27, 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)

• The opinion by Tsikogiannopoulos is that the expected value of the "success factor" of switching envelopes is 0.
  And to calculate so it is necessary that the greater pair and the lesser pair of amounts of money is equally likely.
  Therefore, if the opinion is valid, the expected value of the amount of money in the opposite envelope will always be greater than the amount of money in the chosen envelope, and switching envelopes will be always favorable.
• As far as I understand, in Tsikogiannopoulos, P. (2014), Tsikogiannopoulos said that loss of 50 euros on the case the average is 75 euros offsets gain of 100 euros on the case the average is 150 euros. However, to my eyes, the benefit of 100 euros is greater than the risk of 50 euros whenever it is not a special imminent situation. ☜ Important
  (↑ Added on December 14, 2021)

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".
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.
On November 10, 2004, the author of the blog page himself withdrew the above opinion after reading the mathematical resolution.  ☜ Important

Section "4.1 Nalebuff asymmetric variant"

I think it may be better not to refer to Nalebuff, Barry.(1989) here

(Revised on April 16, 2024)

In the literature, the chance to switch envelope is given after opening the chosen envelope. Therefore it is not appropriate to refer it in this section which locates under the section "4 Other simple resolutions".
Another reason is that Nalebuff is a mathematician and the resolution presented in that literature is mathematical rather than philosophical. (← Added on April 17, 2021)
Yet another reason, in my perception, is that the asymmetric variant is less important in Nalebuff, Barry.(1989). (← Added on April 22, 2021)
If you just would like to refer to Nalebuff, Barry.(1989) as a literature about the asymmetric variant, you may had better to make it an independent section. (← Added on May 11, 2021. Revised on August 14, 2021, November 11, 2021, April 16, 2024)

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

(Added on April 16, 2024)

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

• Cargile, J. (1992)
• Rawling, P. (1994)
• McGrew, T. J., Shier, D., & Silverstein, H. S. (1997)

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 given after 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. As the consequence, the following seems desirable.

• Change the title of this section (I'd like "Problem and resolutions discussed by mathematicians" or "Paradox of opened case)
• Clarify that the chance to switch is after opening the envelope ☜ Important
  (↑ Added on April 10, 2021)
• Clarify other features above

Addition: In the mathematical literatures the chance to switch has been given after opening

The explanation of the fallacy may be wrong

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.
 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)

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.

• "Paradoxical case without mathematical error"
• "Proper but paradoxical probability distribution"

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
The English language Wikipedia article "Two envelopes problem" should have section about the theme of randomized switching

Randomized switching has been very important theme from the early days of the two envelopes problem.
So I think the section "7 Conditional switching" should be replaced with the section "8 Randomized solutions" revision at 12:56, 28 December 2019 .

Addition: (← Added on April 15, 2024)

• I think that "Randomized decision" may be more suitable as the title. ☜ Important
• I think that the important theory by Cover, T. M. (1987) should be referred. ☜ Important

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 miniature version of NOSHI BUKURO which we can't put bills unless folding.
And over time, I've become accustomed to seeing the photo.

(↓ Added on June 18, 2021)
Recently (June, 2021), I found that the miniature version NOSHI BUKURO is sometime used as OTOSHIDAMA BUKURO (Envelopes for New Year's money gifts).
And I found that the photo which was pasted on the lead section on September 18, 2020 is exactly the miniature version. (↑ Revised on August 14, 2021, November 29, 2022)
I had misunderstood that the photo is the simplified version NOSHI BUKURO. (← Added on July 02, 2021)

(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)

• It is a mistake to interpret the photo as a shot of two envelopes. The photo shows the design of miniature NOSHI BUKURO and how to insert banknotes.
• NOSHI BUKURO is not a game toy.
• When using the NOSHI BUKURO as OTOSHIDAMA BUKURO, it is necessary to write the letters "お年玉" in the center of the front surface.
• To my eyes, the floor on which the envelopes are placed looks not so nice and the folded banknote looks stained only just a little. (← Revised on July 02, 2021, August 14, 2021)

(↓ 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)

• Not highlighting a particular culture
• Pasted on some other language Wikipedia articles (eg Russian language Wikipedia article "Задача о двух конвертах", Czech language Wikipedia article "Paradox dvou obálek")

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.

Reference

• Broome,John.(1995).
  The Two-envelope Paradox, Analysis 55(1): 6–11.
   
  
• Cargile, J. (1992)
  On a Problem about Probability and decision
  Analysis 52, 211{216.
   
  
• Christensen, R; Utts, J (1992)
  Bayesian Resolution of the "Excehange Paradox"
  The American Statistician, Vol.46,No.4.(Nov.,1992),pp.274–276.
   
  
• Cover, T. M. (1987)
  Pick the largest number.
  In Open problems in communication and computation (pp. 152-152). Springer New York.
   
  
• English language Wikipedia article "Envelope paradox"
  How to read the English Wikipedia article "Envelope paradox"

  First	Open a page of the English language Wikipedia.
  Second	Enter "Envelope paradox" as the search key word, and click the search button.
  Third	If the article "Two envelopes problem" is shown, click the link on the line "(Redirected from Envelope paradox)".
  Fourth	If the article "Envelope paradox" is shown click the link "View history".

  
  
• Falk, Ruma (2008)
  The Unrelenting Exchange Paradox. Teaching Statistics 30 (3): 86–88.
   
  
• Gardner, M. (1982)
  Aha! Gotcha: Paradoxes to Puzzle and Delight,
  W. H. Freeman and Company, New York.
   
  
• Jackson, F., Menzies, P., & Oppy, G. (1994).
  The two envelope'paradox'.
  Analysis, 54(1), 43-45.
   
  
• Jeffrey, R.(1995)
  Probabilistic Thinking.
  Princeton University.
  
  In this literature, a problem "The problem of the two sealed envelopes" is presented as an example of "the discharge fallacy".
  But the problem is different from the two envelopes problem, as the following formula is used in the problem.
  

  E(Y) will be .5E(.5X) + .5E(2X)

  And it is also different from the wallet game, as it uses symbols of random variable.
  (↑ Added on April 17, 2021)
  
  
• McGrew, T. J., Shier, D., & Silverstein, H. S. (1997)
  The Two‐Envelope Paradox Resolved. Analysis, 57(1), 28-33.
   
  
• Nalebuff, Barry. (1988)
  Puzzles: Cider in Your Ear, Continuing Dilemma, The Last Shall be First, More.
  Journal of Economic Perspectives, 2(2): 149–156.
   
  
• Nalebuff, Barry.(1989)
  The other person'S envelope is always greener. Jounal of Economic Perspectives 3 (1989) 171-181.
   
  
• Rawling, P. (1994)
  A note on the two envelopes problem.
  Theory and Decision, 36(1), 97-102.
   
  
• Schwitzgebel, E., & Dever, J. (2008)
  "The Two Envelope Paradox and Using Variables Within the Expectation Formula" (PDF), Sorites: 135–140
   
  
• Storkey, Amos. (2000-2005)
  Web pages with titles "Amos Storkey - Brain Teasers: Two Envelope Paradox" and "Amos Storkey - Brain Teasers: Two Envelope Paradox - Solution".
   
  
• Tsikogiannopoulos, P. (2014)
  Variations on the Two Envelopes Problem. arXiv preprint arXiv:1411.2823.
   
  
• Zabell, S. (1988)
  One of the additional discussions of Hill, B. M. (1988).
  Bayesian statistics, 3, 233-236.
  
  
  Hill, B. M. (1988).
  De Finetti’s Theorem, Induction, and A (n) or Bayesian nonparametric predictive inference (with discussion).
  Bayesian statistics, 3, 211-241.
  
  
  

Terms

• NAC
  About NAC, please see the page "Wikipedia:Non-admin closure" in the English language Wikipedia.
   
  
• success factor
  The ratio of profit by switching envelope to the average of the two amounts of money in the two envelopes.
  This concept was presented in Tsikogiannopoulos, P. (2014). Variations on the Two Envelopes Problem. arXiv preprint arXiv:1411.2823.
  It is mysterious to me that Tsikogiannopoulos used the average amount as the denominator instead of the total amount. After a long time thinking, I had come to think that it was especially important for him how much the handed amount is above or below the average. (← Added on April 30, 2021)
   
  
• confusion between average of rates and rate of averages
  (Added on December 14, 2021)
  Example of the confusion on the two envelopes problem :
  The amount of money in the other envelope is twice as much as or half of the amount of money in the chosen envelope with same probability. Therefore, the average of these rates is 1.25, so the average of the other amount is greater than the average of the chosen amount, and switching envelopes is always preferable.
  
   
  
• law of total expectation
  In brief it is a proposition which states that E(X) = E(E(X|Y)).
  For details please see the article "Law of total expectation" of the English language Wikipedia.
   
  

Return to the list of my pages written in English about the two envelopes problem

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