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In April, 2021, I wrote a new page
"Maybe flawsome sections of the English Wikipedia article 'Two envelopes problem' (revision at 21:12, 4 January 2021)"
as a sequel to this page.

2018/01/02 14:52:44

First edition 2014/11/25

Inconsistencies in the article 'Two envelopes problem' (revision of November 2014) of the English language Wikipedia

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

On December 1, 2014, I changed the titles of sections.

In my perception, the article titled "Two envelopes problem" (revision 21:39, 23 November 2014) of the English language Wikipedia has some inconsistencies.

Composition of the revision 21:39, 23 November 2014 of the article

Revision 21:39, 23 November 2014 of the article has following composition.

Opening
1 Introduction
  1.1 Discussion
  1.2 Proposed solutions
2 Problem
3 Simple resolutions
  3.1 Nalebuff asymmetric variant
4 Bayesian resolutions
  4.1 Simple form of Bayesian resolution
  4.2 Introduction to further developments
    in connection with Bayesian probability theory
  4.3 Second mathematical variant
  4.4 Proposed resolutions
  4.5 Foundations of mathematical economics
  4.6 Controversy among philosophers
5 Smullyan's non-probabilistic variant
  5.1 Proposed resolutions
6 Extensions to the problem
7 Randomized solutions
8 History of the paradox
9 See also
10 Notes and references

My misunderstanding about the inconsistency among the section "Smullyan's non-probabilistic variant" and the other sections

A few days ago, I wrote as follows. But on December 1, 2014, I noticed that it is not accurate.

Inconsistency 3

In my perception,
from the section "3 Simple resolutions" to the section "4.1 Simple form of Bayesian resolution",
the editors of this article seem to think there are different resolutions for the same mathematical problem.
The section "5 Smullyan's non-probabilistic variant" intimate
a possibility of the existence of two different mathematical problems.

To my eyes, there is an inconsistency.

Smullyan's two arguments had been introduced at the revision 22:05, 3 October 2005.
The editor of the revision seemed to notice the existence of several different paradoxes derived from the two envelopes problem.
So I think that the inconsistency had been brought in after that revision.

In the section "4.2 Introduction to further developments in connection with Bayesian probability theory" of the revision 21:39, 23 November 2014, it has been written that there are two main interpretations and two main resolutions.
So my thought that has been written above is mistake.

Inconsistency among the section "Discussion" and the section "Simple resolutions"

In the section "1.1 Discussion",
two pairs of amounts [$10, $20] and [$20, $40] are presented as a example.
In the main resolution of the section "3 Simple resolutions",
only one pair of amounts [x, 2x] is mentioned.

To my eyes, there is an inconsistency.

Inconsistency about the concept of "expected value"

On January 2, 2018, this paragraph was revised.

The subject matter of this article is
expectation of amounts of money.
A calculation formula by Tsikogiannopoulos was presented in the section "3 Simple resolutions" and its subject matter was
expectation of success factor.

To my eyes, there is an inconsistency.

Inconsistency among the section "3 Simple resolutions" and the section "3.1 Nalebuff asymmetric variant"

In the main resolution of the section "3 Simple resolutions",
only one pair of amounts [x, 2x] is mentioned.
So this resolution corresponds to Smullyan's second argument.
In the sub section "3.1 Nalebuff asymmetric variant",
two pairs of amounts [a, 2a] and [a, a/2] are mentioned.
So this sub section corresponds to Smullyan's first argument..

To my eyes, there is an inconsistency.

Inconsistency in the section "4.2 Introduction to further developments in connection with Bayesian probability theory"

On December 2, 2014, this paragraph was added.

Inconsistency 1

The sentences that begin with "Is the author after … " seem to say that thre are three problems.
  unconditional
  conditional on the smaller amount
  conditional on the amount in envelope A
The sentence that begins with "Thus, there are two main …" says that thre are two problems.

To my eyes, there is an inconsistency.

Inconsistency 2

On December 20, 2014, following was added.

If pushed I'd say, the first sentence of the section seems saying as follows.
The simple resolution assumed that the writer of the switching argument intended to calculate unconditonal expectation.
The first resolution in the section "Simple resolutions" is conditioned on specific one pair of amounts [x, 2x].

To my eyes, there is an inconsistency.

Inconsistency about the interpretations on which the resolutions in the section "Simple resolutions" are based

On December 13, 2014, this paragraph was added.

Resolutions in the section "Simple resolutions" seem based on following interpretation.
Two envelopes problem is pertaining to non-conditional expectation E( B ) and it is not pertaining to conditional expectation E(B | some event).
The first sentence of the section "Bayesian resolutions" describes the interpretation on which the resolution of "Simple resolutions" is based, and the sentence contains following words.
…thinking of the two amounts in the envelopes as fixed (x and 2x).

To my eyes, there is an inconsistency.

Once upon a time

Once upon a time there were not such inconsistencies in the article.
For example, please see the beautiful revision of 18:32, 25 October 2005 of the article.

↓ On January 10, 2015, following was added.

That revision (at 18:32, 25 October 2005) did not refer to minor theories.
Concise
That revision had no inconsistencies.

But I'm afraid that even that revesion (at 18:32, 25 October 2005) had some shortage.

The paragraph which starts with "Now, suppose you reason as follows:" was unique. (I have not found such a description in other articles about the two envelopes problem.)
So I think that this paragraph was not suitable to an encyclopedia.
The section "Proposed solution" said that "The most common way to explain the paradox is to observe that A isn't a constant in the expected value calculation, step 7 above."
But in fact that way is minor rather than common in the 2010s.

So, I had tried to search more beautiful revision among the revisions of the article "Two envelopes problem" in the English language Wikipedia. And unfortunately I have not yet found it.

Addition

Sense of incongruity about the titles "Simple resolustions" and "Bayesian resolutions"

On December 10, 2014, this paragraph was added.
On January 20, 2015, this paragraph was moved to here.

The editors seem to think that the resolutions that they call "Simple resolutions" are simpler than the resolutions they call "Bayesian resolutions".
The main resolution of the "Simple resolutions" seems more complicated than the resolutions of the "Bayesian resolutions".
- Please see the symbol A in the step 6 in the "Problem" section.
  If the value of the symbol A has different values case by case, then A must be "random variable" rather than "variable".
  It is inconsistent to the fact that "two envelope problem" is not an exercise of statistics.
- When I analyze an experiment according to the former resolutions, it is necessary to calculate the total of amounts of money which belong to the same pair of amounts of money.
  It is not necessary when I analyze an experiment according to the later resolutions.
- According to the former resolutions, I must assign different values to the same variable symbol A case by case.
  It requires me to swap the relation among a symbol and number in my mind, but in the later resolution, such a mental action is not required.
- Please see the step 6 in the "Problem" section.
  According to the former resolutions, the expression "the other envelope contains 2A" represents the event "selected envelope contains the lesser amount of the pair of amounts A and 2A" , and at the same time it represents the value of the amount of money in the other envelope.
  According to the later resolutions, the expression "the other envelope contains 2A" does not represents any event, instead it only represents the value of the amount of money in the other envelope at the event "selected envelope contains A and it is the lesser amount".
  ↑ Some phrases were revised on March 28, 2015.
- Please see the expectation formula in step 7 in the "Problem" section.
  According to the former resolutions, the variable symbol A changes value depending on terms.
  But symbol A appears only once in the left side of the formula.
  We must become confused to understand such a equation.
  ↑ It was added on March 28, 2015.

So I feel a sense of incongruity.

Sense of incongruity about some quotations of "Opened version" game

On January 11, 2015, this paragraph was added.
On January 20, 2015, this paragraph was revised.

The section "3 Simple resolutions" quotes the section "1.3 One opened envelope" of Tsikogiannopoulos, P. (2014) .
The section "3.1 Nalebuff asymmetric variant" quotes Nalebuff, Barry. (1988).
But in my perception these quotation might be to be misunderstood.

The game which was introduced in the section "1.3 One opened envelope" of Tsikogiannopoulos, P. (2014) is "Opened version".
(i.e. the opportunity to trade is given after player opened his/her envelope.)
But in the section "3 Simple resolutions", it has not been explained that this game is "Opened version".
The game which was introduced in Nalebuff, Barry. (1988) is "Opened version".
But in the section "3.1 Nalebuff asymmetric variant", it has not been explained that this game is "Opened version".
The rule of the game which was discussed in the section "3 Simple resolutions" is "Closed version".
(i.e. the opportunity to trade is given before players open their envelopes)

Sense of incongruity by an explanation about the ambiguity of the problem

On December 11, 2014, this paragraph was added.

The section "4.2 Introduction to further developments in connection with Bayesian probability theory" contains following sentence.

But the description in step 6 is ambiguous.

On the other hand the description in step 2 is as follows.

step 2.The probability that A is the smaller amount is 1/2, and that it is the larger amount is also 1/2.

In my perception, the ambiguity already starts at step 2 as follows.

To DoublePairian's eyes, the symbol A has a fixed value through the steps after step 2.
To SinglePairian's eyes, the symbol A can have two values after step 2.

In the companion page titled "Inconsistent Variable Theory on The Two Envelope Paradox", the meanings of "DoublePairian" and "SinglePairian" are explained.

Sense of incongruity by the resolution which use an equation "E(B)=E(A|A<B) + (1/4)E(A|A>B)"

On December 13, 2014, this paragraph was added.

One of the resolution in the section "Simple resolutions" says as follows

Step 7 should be
E(B) =
E(B|A<B) P(A<B) + E(B|A>B) P(A>B) =
E(2A|A<B) (1/2) + E((1/2)A|A>B) (1/2) =
E(A|A<B) + (1/4)E(A|A>B).
The writer of the switching argument made two mistakes.
- He forgot he was thinking of expectation values of A.
- He forgot he was thinking of expectation values of A under two different conditions.

Sense of incongruity 1

This resolution premises that there are many person who are as follows.

They are so good at mathematics as they know following mathematical formula.
Let R be a rondom variable, and let E₁ , E₂ be events.
If E₁ and E₂ are complementary event of each other, then
E( R ) = E( R | E₁) P(E₁) + E( R | E₂) P(E₂).
They are not good at mathematics, so they think that following values are equal.
- the mean value of the lesser amounts of money in the two envelopes
- the mean value of the greater amounts of money in the two envelopes

It makes a sense of incongruity in my mind.

Sense of incongruity 2

On January 21, 2015, this paragraph was revised.

From the oldest revision through the revision 20:19, 8 February 2012, the article "Two envelopes problem" in the English language Wikipedia, have not referred to such a resolution ｆor periods more than six years.
The first sentence of this section says that a common way to resolve the paraodox has following properties.
Such a way is common in popular literatures and philosopher's literatures.
Such a way assumes as follows.
- 'A' in step 7 is intended to be the expected value in envelope A.
- The writer of the switching argument intend to calculate the expected value in envelope B using 'A'.
But within my memory, I have read such a way in only one literature Snell, J. L., & Vanderbei, R. (1995).
↑ Revised on March 17, 2015, and April 15, 2017.

So, to my eyes this resolution is minor.

Sense of incongruity 3

On January 10, 2015, this paragraph was added.

In this section, it seems to be described that the equation
E(B|the lower amount is x) = (1/2)2x + (1/2)x
is the special case of the equation
E(B) = E(2A|A<B) (1/2) + E((1/2)A|A>B) (1/2).
But under the circumstances in which the mean value is ∞, the latter equation is not effective but the former equation is still true.

It makes a sense of incongruity in my mind.

Sense of incongruity by the section "4.1 Simple form of Bayesian resolution"

On December 13, 2014, this paragraph was added.
On December 27, 2014, the another paragraph which has same title were merged with this paragraph.

This section contains following word and sentence, and in this section an inappropriate application of the principle of insufficient reason seems to be thought as the cause of paradox.

principle of insufficient reason
No information means that probabilities are equal. ( ← Quotation mistake was revised on January 10, 2016)

But if a person applied the principle to 'Two envelopes problem', he/she seemed to know the importance of the base rate.
It is unacceptable that such a person could not wonder the probability 1/2.

On the other hand, at the revision of 18:32, 25 October 2005, the section titled "Proposed Solution" (solution of "A Harder Problem") had following sentence.

The subjective probability changes when we get new information, so our assessment of the probability that A is the smaller and larger sum changes.

In this section, "base rate fallacy" seems to have been noticed vaguely as the cause of the paradox.

Sense of incongruity by the section "Extensions to the problem"

This paragraph was moved here on January 18, 2015.

The title of the section "6 Extensions to the problem" has the word "extensions".
To my eyes, no extension is described in the section. Only an explanation of the fundamental equations is described.

It makes a sense of incongruity in my mind.

Sense of incongruity by the treatment of "Opened version"

This paragraph was added on January 18, 2015.

Following old articles about the "two envelopes problem" suggest that the "Opened version" (the opportunity to trade is given after players open their envelopes) is the original of the "two envelopes problem".
Nalebuff, Barry. (1988)
Zabell, S.L. (2005).
In the article "Two envelopes problem" (revision 21:39, 23 November 2014) of the English language Wikipedia, "Opened version" of the two envelopes problem seems to be not introduced before the section "6 Extensions to the problem". (← Revised on 6 September, 2015)

It makes a sense of incongruity in my mind.

Sense of incongruity by a reference in the section "Simple resolutions"

This paragraph was added on April 15, 2017.

In the section "Simple resolutions" of the article "The two envelopes problem" (revision of 21:39, 23 November 2014) of the English language Wikipedia, the following expectation formula was described.
Expected value in B = 1/2 ( Expected value in A (given A is larger than B) + Expected value in A (given A is smaller than B) )
And this equotion was acompanied by a reference to "Schwitzgebe, Eric; Dever, Josh (2008)".
But to my eyes, in the article "Schwitzgebe, Eric; Dever, Josh (2008)", expectation formulas which are described as correct formula are almost made of random variables.
An expectation formula which is made of expected values finaly appeared in the later section "the proof".
Therefore I think that for the section "Simple resolutions" the article "Snell, J. L., & Vanderbei, R. (1995)" is more suitable to refer.

It makes a sense of incongruity in my mind.

Reference

Nalebuff, Barry. (1988)
Puzzles: Cider in Your Ear, Continuing Dilemma, The Last Shall be First, More.
Journal of Economic Perspectives, 2(2): 149–156.
Schwitzgebe, Eric; Dever, Josh (2008)
"The Two Envelope Paradox and Using Variables Within the Expectation Formula" (PDF), Sorites: 135–140
Snell, J. L., & Vanderbei, R. (1995)
Three bewitching paradoxes.
Topics in Contemporary Probability and Its Applications, CRC Press, Boca Raton, FL, 355-370.
Zabell, S.L. (2005).
Symmetry and Its Discontents
Cambridge University Press.
Tsikogiannopoulos, P. (2014)
Variations on the Two Envelopes Problem. arXiv preprint arXiv:1411.2823.

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