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Archive of the page "An outline of the Two Envelopes Problem" rev at 2022/09/20 15:41:10

Return to the list of my pages written in English about the two envelopes problem
2022/09/20 15:41:10
First edition 2015/01/26

An outline of the Two Envelopes Problem

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


Skip to Contents

Why do I study the two envelopes problem and the two envelope paradox?

This paragraph was added on April 8, 2017. Title was revised on August 24, 2017, July 8, 2018.

The two envelopes problem is not so difficult, because if we recognise that the probabilities is not necessarily 1/2 then this problem is almost solved.
The two envelope paradox is not so mysterious, because the cause of it is the Base rate fallacy which is psychologically commonplace illusion.
But to my surprise, there are people who think that the correct expectation formula is "E=(1/2)A + (1/2)2A".
This mystery was so big as I could not stop studying this.
↑ Added on May 17, 2018. Revised on August 19, 2018.

Only researchers of psychology can cancel confusion.

This paragraph was added on April 20, 2017. Revised on June 25, 2017.

If an psychological experiment shows the following result, then the following consequence will become certain, and the confusion will be canceled.

The theme of the experiment
Mental model of the amount of money in the opposite envelope.

Method of the experiment
Let the participants do the following mental actions.
Imagine that there are two envelopes one of which contains twice as much money as the other.
Imagine that you have chosen one envelope. (← Revised on January 5, 2020)
Let x denote the amount of money in the envelope you have chosen. (← Revised on January 5, 2020)
Think of amount of money in the other envelope.
Think of the possible lesser value of it and the possible greater value of it.
Calculate the ratio of these values.
Then let them report the ratio.

Expected result of the experiment
I expect that the participants will answer "1 to 4".
(I wrote some ideas of method of experiment in the page "My idea of cognitive psychological experiment about the two envelopes problem".)

Consequence
The expectation formula "E=(1/2)A + (1/2)2A" cannot be the solution of the two envelopes problem, because this equation requires that the above ratio is 1 to 2.

Contents

Introduction

This paragraph was added on April 1, 2015.
Greatly revised on July 3, 2016.

Easily understandable explanation

This section was added on September 9, 2017. Revised on September 13, 2017.

To explain the basic knowledge of the two envelopes problem and the two envelope paradox I have drawn a frequency diagram of an example probability distribution.
I think this diagram is most easy to understand.



I think that we can easily understand the following knowledge from this diagram. On October 22, 2017, I made a page "Experiment for experiencing two envelopes problem".
It provides experiment about the probabilities which are concerned on the two envelopes problem and it may help to realize the above knowledge.



If you have not gotten satisfaction
(Added on June 19, 2018)
Please see the page "Interesting web pages about the Two Envelopes Problem".
Then you will find explanations easy to understand.

Structure of the two envelope paradox

This section was added on April 29, 2016, and revised on October 29, 2016.
Moved here on June 8, 2017.

I think that the two envelope paradox has the following structure.
↓ Revised on June 8, 2017.
Game settings
Added on June 8, 2017.
Common game settings
An unexpected amount of money is placed in one envelope, and twice that amount is placed in another envelope.
One of the two envelopes is handed to player A.
Player A is given an opportunity to change envelope.

Original game settings
The referee C places an unspecified amount of money x in one envelope and amount 2x in another envelope
One of the two envelopes is handed to player A, the other to player B.
Player A opens his envelope and see how much money in it.
↓
Player A first thinks of the amount of money in the own envelope and then thinks of the amount of money in B's envelope.
Added on July 7, 2017.
Thinking before opening A's envelope
If there is $x in my envelope then there is $(x/2) or $2x in B's envelope.

Thinking after opening A's envelope
There is $10 in my envelope so there is $5 or $20 in B's envelope.
↓
Base Rate Fallacy
(Revised on May 3, 2016, September 18, 2016, October 9, 2016, may 27, 2017, July 7, 2017, September 8, 2017.)

During selecting envelope
Steps   The greater envelope chosen The lesser envelope chosen
Envelope selection Amount of money in the chosen envelope amount of money of the greater money amount of money of the lesser money
Likelihood (reminded) 1/2 1/2

During expecting amount of money in the other envelope
Steps   The lesser pair arranged The greater pair arranged
Assigning variable symbol The amount of money in the chosen envelope is denoted by x. the amounts of money are x/2 and x the amounts of money are x and 2x
Pairs of amounts of money x/2, x x, 2x
Odds as the base rate (ignored)
 O1 O2
Base rate fallacy Amount of money in the other envelope x/2 2x
Intuitive Probability
 1/2 1/2
True Probability O1
÷ (O1 + O2) 		O2
÷ (O1 + O2)
↓
The expectation formula which is the seed of the paradox
E=(1/2)(x/2) + (1/2)2x.
↓
Illusion of materialized expectation
Expected value of the amount of money in the opposite envelope is thought as amount of actual money.
↓
Various paradoxes
(Revised on 14, 2017.)
  • Paradox of the broken symmetry (standard version)
    Imagination of all of passible value of amount of money in the own envelope let us think that only choice of envelope makes the other envelope more favorable.

  • Paradox of the broken symmetry (fictitious version)
    The illusion of materialized expectation lets us confuse the problem of expectation and the problem of equivalence of the two envelopes.

  • Paradox of the two envelopes which are greener than each other
    The illusion of materialized expectation lets us feel a paradox when we notice that changing envelopes is favorable for both players.

  • Paradox of the endless switching (closed version only)
    (Added on September 25, 2017. Revised on September 26, 2017.)
    The illusion of resetting expectation makes us think that nothing has changed other than the envelope that is chosen.
    In other words we often forget that we have already made an expectation and switched envelope.
    And as a result to our eyes the original envelope seems more favorable than the new chosen envelope.
    This paradox is peculiar to the closed version problem.

     
  • Paradox of the money pump (closed version only)
    (Revised on September 26, 2017.)
    The illusion of materialized expectation lets us think of the expected value after repeated exchange.


    This paradox is peculiar to the closed version problem.
↓
Two major resolutions
↓
They will not cross forever.
(Added on october 28, 2018)

The standard resolvers and DivideThreeByTwoians resolved inherently different paradoxes.

This paragraph was added on June 11, 2017. The title was changed on July 8, 2017. Revised on July 19, 2017.

In 1988 mathematicians began to solve the paradox of the two envelope problem. The resolved paradox was the standard paradox.

In 1994 philosophers (many of them were DivideThreeByTwoian) began to resolve the two envelopes problem.
They resolved fictitious paradoxes such as
the paradox of the two envelopes which are greener than each other,
the paradox of the endless switching
and the paradox of the money pump.

Why they went separate route?

I imagin the answer as below.
The early articles which presented the standard resolution described the fictitious paradoxes such as above. And the standard resolution resolved the standard paradox and did not resolve these fictitious paradoxes. (← Revised on September 11, 2017.)
In addition, the problem presented by the first article which philosophers encountered was the closed version problem which hindered thinking of expectation conditioned on the amount of money in the chosen envelope.
This feature of these articles made philosophers walk another route.
↑ Revised on June 12, 2017, July 8, 2017.


↑ Added on March 8, 2020.

On March 1, 2018, the paragraph "There are another resolutions in the domain of decision theory" was deleted.

Why are the two envelopes problem and paradox so chaotic ?

This paragraph was added on August 23, 2017. Revised on September 16, 2017. Title was revised on July 8, 2018,March 8, 2020..

I think that the following situation made the two envelopes problem more chaotic than the Monty Hall problem.

Chaos about the paradox corresponding to the opportunity to swap envelopes

This paragraph was added on February 5, 2018.
On May 17, 2018, the contents was revised and the title was changed.
On July 7, 2019, the contents was revised.
opened version problem.
  • Most mathematical literatures about the two envelopes problem discussed the standard paradox of the opened version problem.
  • Some philosophers discussed the paradox of the opened version problem, and some philosophers were not interested in the paradox after opening envelope.

closed version problem.

Examples
  • I have an impression that in Zabell, S. (1988), it was written that there is no paradox before opening envelope.
  • A web page titled "NaClhv: The two envelopes problem and its solution" discussed mathematically on the closed version.
  • A few philosophers mathematically discussed the fallacy of probability in the opened version problem.
    Example: Chalmers, D.J. 1994 .
  • More than a few philosophers mathematically discussed probability or paradoxical distributions on the closed version problem.
    Example:
  • A philosophical article that presented DivideThreeByTwoian's resolution on the closed version problem claimed that on the opened version problem, the two envelopes should be thought equivalent because no information of probability distribution has been given. (← Added on July 14, 2019)
  • A philosophical article that presented DivideThreeByTwoian's resolution for the closed version problem argued that the open version problem is not paradoxical, because the other envelope is more advantageous or disadvantageous depending on the amount revealed. (← Added on July 14, 2019)

Wikipedia articles can be categorized as follows
  • Presenting the standard paradox of the opened version problem.
    The German languge Wikipedia article "Umtauschparadoxon" (Revision at 16:55, 22. Aug. 2016‎)
    The English language Wikipedia article "Envelope paradox" (Revision at 21:38, 17 January 2006) (How to read it)
  • Presenting the DivideThreeByTwoian's paradox of the closed version problem.
    The French languge Wikipedia article "Paradoxe des deux enveloppes" (Revision at 2 avril 2017 à 15:20‎‎)
  • Presenting one of both paradoxes according to the opportunity to swap.
    The Italic language Wikipedia article "Paradosso delle due buste" (Versione del 16 apr 2016 alle 15:12) .
    The English languge Wikipedia. article "Two envelopes problem" (Revision at 22:05, 3 October 2005)
  • Presenting both paradoxes on the same closed version problem.
    The English languge Wikipedia article "Two envelopes problem" (Revision at 00:31, 8 November 2011)
  • Presenting the LesserOrGreaterMeanValuean's paradox on the closed version problem.
    The English languge Wikipedia article "Two envelopes problem" (Revision at 09:49, 17 November 2014) .
    (↑ Added on November 11, 2018)

Roughly speaking,
the opened version problem and the closed version problem are
mathematically same for mathematicians,
but mathematically different for philosophers.

In contrast, in the Monty Hall problem the opportunity to change door is given only after the host opens one of the other doors.
(↑ Added on May 17, 2018)

Chaotic influence of the magical power of the closed version problem

This paragraph was added on June 14, 2018. Title was revised on September 22, 2019.
I think that the closed version problem has a magical power that forces people to turn their mental model into the SinglePairian's mental model. (← Added on May 26, 2019)

In my perception most mathematicians have not been influenced by the magical power of the closed version problem. Therefore on the closed version problem, they think that there is no paradox, or they think the similar paradox as on the opened version problem. (← Revised on May 26, 2019)
However, one or more mathematicians have been influenced by the magical power. (← Added on May 26, 2019)

In my perception many philosophers have been influenced by the magical power of the closed version problem.
However, not a few philosophers have not been influenced. (← Added on May 26, 2019)

I got some hypotheses about the magical power, so, specifically please see "Magical powers of the closed version problem".

In contrast, in the Monty Hall problem, any controversy does not occur even if the opportunity to change is given before opening a door, because nobody will think that the probability changes. (← Added on July 22, 2018)

Another influences

In my perception a few philosophers showed the following unique thought about the symbol "x" in the expected value formula "E=(1/2)(x/2) + (1/2)2x" on the closed version problem.
  • A few philosophers think that the symbol "x" means a random variable, not a variable.
  • A few philosophers think that the symbol "x" itself means an expected value, not a variable.
About this I think as follows.
On the closed version problem, we cannot assign a unique value to the symbol.
This fact gave some philosophers reason to assert that the symbol "x" should not show particular value. (← Revised on August 26, 2018)
(↑ Added on August 12, 2018)

In contrast, in the Monty Hall problem the value of the car as a prize and the value of the goats as a losing lottery are not mentioned. Even if a goat is worth more than a car, the mathematical structure of the Monty Hall problem does not change. (← Revised on July 22, 2018)

Chaos of the problem wording

There are various wordings of the two envelopes problem and none of them are standard.
(Specifically please see the paragraph "Several kinds of wording of the two envelopes problem".)
And there is no uniformity in the problem wordings used by Wikipedia articles in the world (← Added on December 23, 2018)

In contrast, the wording of the Monty Hall problem which has been discussed on the column "Ask Marilyn" of the PARADE magazine is the defacto standard. (← Revised on August 30, 2017)
Needless to say, the English language Wikipedia article "Monty Hall problem" traditionally uses this defacto standard wording. (This article started using this wording at the revision 22:48 on March 12, 2004.)
(↑ Added on December 23, 2018)

Chaos of the existence of fake problems

(Added on March 1, 2018.)
There are fake problems of the two envelopes problem.
For eample.
Problem with wording which tells to think whether to switch or not to switch
Problem with very restrictive wording
Problem with wording which has a menu of reasonings

To my eyes , the problems presented by some revisions of the English language Wikipedia article "Two envelopes problem" seem somewhat close to a fake problem.
Because I think that such problems have a power to invite us to the SinglePairian's problem.
(For details please see the section "Kinds of the wording how the probability 1/2 is combined with amounts of money".)
(↑ Added on March 8, 2018.)

There are fake problems of the Monty Hall problem as well.
For example.
  • Problem presented by demonstration style
    In such a style, the audiences may doubt some of the standard assumptions of the Monty Hall problem.
  • Problem in which a door is accidentally opened
    Under such conditions, the probability cannot be other than 1/2.
(↑ Revised on December 16, 2018)

On July 21, 2019, the paragraph "Chaos about whether to resolve the paradox" was deleted.

Chaos of kind of the paradoxes

(This chaos was added on February 23, 2018.)

Pseudo paradox
The mathematically standard paradox is a pseudo paradox which is derived from a probability illusion.
If you are fond of mathematics you may prefer this paradox.

True paradoxes
The paradox of the paradoxical distribution is true paradox because in my perception no mathematician completely solved this paradox.
And the paradox by the principle of insufficient reason may be true pardox. ← Revised on March 15, 2018.

Fictitious paradoxes
The following paradoxes are fictitious because the paradoxical feelings are illusions.
Paradox with mysterious resolution
DivideThreeByTwoian's paradox and resolution are very mysterious.
(↑ Revised on May 3, 2018.)

Falsely similar paradoxes
The following paradoxes are look like the two envelope paradox but they are different paradoxes. (← Revised on March 8, 2018.)
In contrast, the Monty Hall problem is not anything else except a pseudo paradox.

Chaos of illusions leading to paradoxes

(This chaos was added on September 26, 2017. Revised on February 10, 2019. Title was revised on July 21, 2019)
Case of mathematically standard paradox
illusion derived paradox
Base rate fallacy
the fallacious expectation formula
   ↓
the mathematically standard paradox
(Paradox of
the broken symmetry
(standard version) )

Case of the fictitious paradoxes
original illusion secondary illusion
after looking
"E=(1/2)(x/2)+(1/2)2x" 		derived paradox
Base rate fallacy illusion of materialized expectation
illusion of resetting expectation

Case of DivideThreeByTwoian's paradox
originating paradoxes induced illusion paradox and resolution
DivideThreeByTwoian's paradox and resolution

Case of falsely similar paradoxes
(Added on February 10, 2019)
illusion derived paradox
confusion of mean values under the different conditions LesserOrGreaterMeanValuean's paradox
confusion of ratio of mean values and mean value of ratios MeanRateOfExchangean's paradox


In contrast, the wrong calculation of probability on the Monty Hall problem is the result of only an illusion of probability which is caused by overlooking the necessity to subdivide sample space under new evidence event.

Chaos of the problem domain

(This chaos was added on October 4, 2017. Totally revised on September 25, 2018.)

For some people, mathematics is the problem domain of the two envelopes problem.
Such people think that the expectation formula must be constructed with x/2 and 2x.
So they try to correct the probability.

For some people, decision theory is the problem domain of the two envelopes problem.
Such people try the followings.
  • On the closed version problem, they try to find the way to rationally expect without the information about the probability distribution.
  • On the opened version problem, they try to find the rational expectation formula which guarantee the equivalence of the envelopes.

For some people, philosophy is the problem domain of the two envelopes problem.
Such people try to find the philosophical trap which confuses us. (↑ Revised on November 11, 2018, March 31, 2019)

In contrast, the Monty Hall problem is only a problem of mathematics.

Chaos of the goal of the problem

Some people try to find a fallacy which let us make a wrong calculation formula.
Other people try to find correct calculation formula.
Other people try to judge whether correct calculation formula exists.
Other people try to find diagnostic techniques to screen inappropriate calculation formulas. (← Added on July 8, 2018)
Other people try to reaffirm the effectiveness of expected utility theory. (← Added on July 29, 2018)
Other people try to reveal the trick of the magic which lets the fallacious expectation formula look like plausible. (← Added on October 21, 2018)
Other people try to demonstrate the effectiveness of the trendy philosophical concept such as "possible worlds" and "rigid designator". (← Added on October 21, 2018)
Because the probability 1/2 is reasonable in the domain of decision theory, some people accept the paradox. (← Added on July 21, 2019)

Roughly speaking,
mathematicians try to verify that probability correction resolves the paradox,
but philosophers try to find diagnostic methods to eliminate the expectation formula that leads to the paradox.
(↑ Added on July 7, 2019)

In contrast, the goal of the Monty Hall problem is only to calculate probability.

Is the fallacious expectation formula a result of a fallacy? Or a cause of a fallacy?

(This chaos was added on July 7, 2019.)
We usually think that there is a fallacy which is the cause of the fallacious expectation formula.
However, DivideThreeByTwoian philosophers seem to think that the expectation formula itself is the cause of the fallacy that leads to the paradox.
Indeed they are researching ways to find tricks that lie behind the fallacious expectation formula.

In contrast, the probability 1/2 on the Monty Hall problem is nothing but the result of an illusion of probability.

Chaos about whether it is a paradox or a puzzle or a magic

(This chaos was added on July 8, 2018, revised on October 14, 2018, September 22, 2019)

For the standard resolvers
The problem with the two envelopes certainly be a paradox.
Because the standard resolvers seem to have felt the wonder of the illusion of probability.

For many DivideThreeByTwoian philosophers
(Added on July 21, 2019)
I think that many DivideThreeByTwoian philosophers think that the fallacious expectation formula is the cause of a fallacy leading paradox.
They look like thinking the problem as a kind of magic the trick should be revealed.

For other DivideThreeByTwoians
Many of non-philosopher DivideTreeByTwoians claim that they resolved a paradox.
But I have never found an evidence that they themselves felt a paradox.
They seem to think that some other people felt the paradox they resolved, or they seem to have resolved a philosophical paradox which can not derive paradoxical feeling.
On the other hand, I think that there may be some DivideThreeByTwoians who do not admit the existence of a paradox.
They seem to think the problem is a kind of mistake-searching puzzle.

For the people who cannot doubt the expectation formula
Many of them made various incantations.
Example :
"After repeating the game, difference of result of strategy to always switch and result of strategy to always stay will be shortened."
"It is wrong to doubt advantage of exchange."
"It is an instance of the case that expected value is unreliable."
Et cetera
Therefore I think that they themselves felt a paradox.

In contrast, the Monty Hall problem is certainly a paradox as everyone feels a big paradox.

Chaos of the effect of lack of presentation of the probability distribution

(This chaos was added on May 3, 2018.)
The lack of presentation of the probability distribution results in the following various effects.
  • It can not prevent mathematicians from making the standard resolution.
  • Usual DivideThreeByTwoians think as follows.
    On the closed version problem, they think of one pair of amounts to get the probability be 1/2.
    On the opened version problem, they think that we can not think of expected value.
  • Some DivideThreeByTwoians think as follows.
    The two envelopes remain equivalent even after opening the chosen envelope, because the revealed amount of money gives no information about the probability.
    (Added on April 7, 2019)
  • The researcher of decision theory can think of the expected value using the principle of insufficient reason.
And some mathematicians may remind the "subjective probability" which Bayesian statisticians think of under the lack of information. (← Added on May 31, 2018)

In contrast, the probability distribution of the Monty Hall problem is defined by the standard assumption of it.

Chaos about how to classify the probability distribution

(This paragraph was added on May 31, 2018.)
The probability distribution of the lesser amount of money in the two envelopes is classified in various ways.
  • Some standard resolvers classify the probability distribution as follows.
    • proper distribution with finite mean value
    • proper distribution with infinite mean value
    • improper distribution with unbounded amount of money
    And some of them focus on the law of total expectation .
  • Some standard resolvers classify the probability distribution as follows.
    • proper distribution with bounded amount of money
    • proper distribution with unbounded amount of money
    • improper distribution with unbounded amount of money
    And some of them advocate the theory of assumption of probability distribution.
  • DivideThreeByTwoians have little interest in probability distribution. (← Added on June 19, 2018)
In contrast, in the Monty Hall problem only one probability distribution is determined by the standard assumption of the Monty Hall problem.

Chaos of the research of the cause of the fallacy

(This chaos was added on October 10, 2017.)
People seek the cause of the fallacy of the expectation formula in the following various systems.
  • cognitive mechanism
  • logic system
  • language system
  • decision theory
  • probability theory (← Added on May 31, 2018)

In contrast, researchers of the Monty Hall problem seek the cause of the probability illusion only in the cognitive mechanism. (← Revised on February 23, 2018.)

Lack of common understanding about the standard assumptions

I think that the following assumptions may be the standard assumptions.
  • The expected amount of other envelope should be calculated on the condition that the amount of money of the chosen envelope is a specific value.
  • The paradox which should be mainly resolved is the following contradiction.
    • Regardless of the amount of money of the chosen envelope the other envelope is more favorable.
    • Before opening envelopes the two envelopes are equivalent.
However, I do not know the person other than me who discussed the standard assumptions.
And the following groups will live in the distinct dimensions forever.
  • people who calculate expected value on the condition that the amount of money of the chosen envelope is a specific value
  • people who think that expected value must be primitive
(↑ Added on July 22, 2018)

In contrast, the standard assumptions of the Monty Hall problem has been identified by many researchers of psychology.
In particular, psychologists have shown the robustness of the standard assumptions by verifying that the following opinion is wrong. (← Added on July 22, 2018)
The ambiguity of the Monty Hall problem caused many people to think that the probability is 1/2.

Chaos about the condition on which the expectation should be based

(This paragraph was added on January 18, 2018.)
About the calculation of the expected value, the following points are not determined.
  • Is it rational to calculate expected value even though the probability distribution of the amount of money is not described?
  • Is it rational to calculate expected value even though the chosen envelope was not opened?
  • Is it rational to calculate expected value using probability 1/2 which is given by the principle of insufficient reason or the principle of indifference?
  • Which conditional expectation is rational on the closed version problem?
    • conditional on the amount included in the unopened selected envelope?
    • conditional on the smallest amount of money included in the unopened two envelopes?
    • both not rational?
    (↑ Revised on February 23, 2018.)
The Monty Hall problem also has some options to calculate the conditional probability as follows.
  • Conditional on the fact that the game was started.
  • Conditional on the fact that the contestant chose the door #1 and the host opened one of the other doors.
  • Conditional on the fact that the contestant chose the door #1 and the host opened the door #3.
But on all of these options the probability that switching door gives the contestant a car has same value ⅔ under the standard assumption of the Monty Hall problem.
↑ Revised on March 1, 2018.

There is no experiment which is easy to do and convincing.

(This paragraph was added on November 12, 2017. Revised on December 28, 2017.)
The lack of common understanding of standard assumptions precludes us from doing convincing experiments.
If the standard resolvers do an experiment I think they will use two pairs of amounts of money to prove the fallacy of the probability. (← Revised on February 2, 2020)
If the DivideThreeByTwoians do an experiment I think they will use only one pair of amounts of money to prove the fallacy of the random variable. (← Revised on February 2, 2020)
So their experiments both cannot persuade the opponent.

In contrast, even elementary school students can do persuasive experiments on the Monty Hall problem.

Chaos about which version is original

(This paragraph was added on July 8, 2018.)
The original of the two envelopes problem was the opened version as reported in the paragraph "Original wording".
But some people think that the closed version is the original.
To my surprise, some people present the closed version problem as the original of the two envelopes problem.
I expect that they have been influenced by some Wikipedia articles which present the closed version problem.
(↑ Added on May 3, 2018. Moved here on July 8, 2018)

In contrast, about the Monty Hall problem there is no original other than Selvin, Steve (1975).

Chaos of the worth of the closed version problem

(This paragraph was added on March 22, 2018. Revised on May 3, 2018, July 15, 2018, July 22, 2018.)
There are some people who think that the closed version problem is worth.
But others think that it is not worth so much.
  • The following articles seem to think that there is no paradox before opening envelope. (↑ Revised on July 29, 2018)
  • About half of Wikipedia articles on the two envelopes problem do not explain the closed version problem.
    (Specifically please see "Addition : Which version of the two envelopes problem is presented by the Wikipedias in the world?".)
  • Those who discuss the closed version problem usually are interested in the expected value of the amount of money in spite of no clue given before opening envelope. However, some of the people who discuss the closed version problem are not interested in such expected value.

In contrast, in the Monty Hall problem the opportunity to change door is given only after the host opens one of the other doors.
(↑ Added on May 17, 2018)

Chaos of the worth of the "Ali-Baba" version problem

(This paragraph was added on October 28, 2018)
Usual standard resolver did not discuss the Ali-Baba version problem.
However, as of 2018, several articles clarifying the standard resolution discussed the Ali-Baba version problem. (← Revised on December 23, 2018, February 24, 2019)

DivideThreeByTwoian who advocate the not-three-amounts theory usually discussed the Ali-Baba version problem.
However the other DivideThreeByTwoians did not discuss it.

On the Monty Hall problem there have been not much discussion on game variations.
However, variations of the host behavior of the game in the Monty Hall problem was discussed by people focusing on the ambiguity of the problem wording.

Chaos of the thought of psychologists

(This paragraph was added on February 17, 2019)
Some psychologists thought that the cause of the paradox was a fallacy of probability. (Example : Falk, R., & Konold, C. (1992). )
Some psychologists thought that the cause of the paradox was the DoublePairian's mental model.
A psychologist thought that the cause of the paradox was an inadvertent error of variable symbol.
For some psychologists the two envelopes problem may be difficult to pick up as a subject of psychology, because they think that the domain of it is philosophy or decision theory.

In contrast, on the Monty Hall problem, few people doubt the existence of a very mysterious illusion of probability.

Division of opinion even among mathematicians

(This paragraph was added on March 10, 2019, and was revised on April 7, 2019)
Mathematicians do not necessarily advocate the standard resolution.


I think that these chaoses have let psychological researchers hesitate to study the two envelopes problem, and as a result we have not got scientific image of the two envelopes problem.
(↑ Revised on June 1, 2018)

Have we already gotten a solution which is widely accepted ?

This paragraph was added on June 26, 2016. And the title was changed on April 11, 2017.
Revised on June 11, 2017.
The title and some content were revised on February 23, 2020.

It seems that we cannot get a solution which is widely accepted.

DivideThreeByTwoian's resolution will continue forever.

(On February 23, 2020, this title was changed)

From October 3, 2005 up to the present (Jun 26, 2016), the article "The two envelopes problem" in the English language Wikipedia had showed the theory of "E=(1/2)2a+(1/2)a" without a break.

To my eyes this is a mystery because there are some evidences that the DivideThreeByTwoian's opinions including the theory of "E=(1/2)2a+(1/2)a" may be wrong.
(For details please see "DivideThreeByTwoian's resolutions may be wrong".)
(↑ Revised on September 11, 2017, February 23, 2020)

But there is a possibility that DivideThreeByTwoians had interpreted the two envelope paradox as follows. Nobody can indicate the wrongness of such an interpretation of the paradox.
So nobody can let the DivideThreeByTwoinas change their opinion.

I cannot but foretell that the tide of the DivideThreeByTwoian's resolution continue forever. (← Revised on February 23, 2020)

Standard resolver cannot accept the DivideThreeByTwoian's resolution.

(Revised on October 9, 2016, January 6, 2018, February 24, 2019, February 23, 2020)

The variable symbols "x/2" and "2x" are the key symbols of the formula "E=(1/2)(x/2)+(1/2)2x". So the formula "E=(1/2)2a+(1/2)a" never worth for standard resolvers.Therefore standard resolvers can only doubt the DivideThreeByTwoian's opinion.

DivideThreeByTwoian cannot accept the standard resolution.

(Added on October 9, 2016. Revised on February 24, 2019)

I think that DivideThreeByTwoian cannot get away from the following illusions. So DivideThreeByTwoians (especially philosophers) got an idea that the DoublePairian's mental model is inappropriate for the two envelopes problem.
(↑ Added on July 7, 2017. Revised on July 7, July 8, and August 24, 2017.)

In my perception some of DivideThreeByTwoians understand the two envelopes problem as a magic whose trick should be revealed. (← Revised on October 28, 2018)
So for them switching the interpretation of the problem is not a foul and they does not hesitate to transfer from the DoublePairian's problem to the SinglePairian's problem.

Furthermore no standard resolvers has presented psychological evidence of the following hypothesis.

When people read the expectation formula E = (1/2)(x/2) + (1/2)2x at the first time, they interpreted it along the following sequence.
Their eyes caught this formula.
   ↓
Then their mind interpreted the values x/2 and 2x with conviction and there were two pairs of amounts of money in their mind.
   ↓
Last their mind interpreted the probability 1/2 as the probability of the amount of money in the opposing envelope of each pair. (← revised on October 28, 2018)

Therefore DivideThreebytwoians can only doubt the standard resolver's opinion.

No psychological experiment about mental models had been done.

(Added on March 8, 2017. The title was revised on March 31, 2019)

Without psychological experiment we can not judge whether DivideThreeByTowian's opinion is reasonable.
If an experiment has shown that DivideThreeByTwoians have DoublePairian's mental model when they read the expectation formula "E=(1/2)(x/2) + (1/2)2x", the wrongness of their opinion will become certain.
But I could not find any experiment which was carried out for this purpose.

Consequence

We cannot get a solution which is widely accepted.

Once it seemed that we already had gotten a solution which is semi-widely accepted, but …

This paragraph was added on April 11, 2017. Title was changed on February 6, 2018.

The DivideThreeByTwoians are not a majority.

The following facts suggest that the DivideThreeByTwoians are not so many as we imagin when we read some revisions of the article "The two envelopes problem" of the English language Wikipedia.

The standard resolvers are not a minority

But the standard resolvers are not majority

This paragraph was added on December 3, 2017. Revised on January 9, 2018.

I found a blog which posted the closed version problem in 2017, and in my perception this blog got the following responding comments till December 3, 2017.

opinion frequency
standard resolution 5
pure DivideThreeByTwoian's opinion 5
Not-three-amounts theory 1
Inconsistent-variable theory 4
other 9

Consequence

On April 11, 2017 I wrote as follows.

We cannot get a solution which is widely accepted. But we may already have gotten a solution which is semi-widely accepted.

On December 3, 2017 I have to write as follows.

We have not even a solution which is semi-widely accepted.


Several Classifications of wording of the two envelopes problem

The title of this section was changed on February 1, 2015, July 26, 2015.
The composition of this section was changed on June 19, 2016.

Kinds of the wording about the process how money is placed in envelopes

Version with no description but explaining ratio of the amounts using the word "twice"

(This title was revised on July 1, 2018.)

A typical wording without description about the process of arrangement of money is as follows.
There are two envelopes each of which contains money.
One envelope contains twice as much money as the other.

Version with no description but explaining ratio of the amounts using variable symbol

(Added on December 2, 2017. Revised on April 19, 2018.)

A typical wording is as follows.
An unspecified amount a is placed in one envelope and amount 2a in another envelope.

Version with description of arrangement

A typical wordings with description of the process of arrangement of money are styles using coin toss or coin flipping.

Double coin flipping style wording
Among such wordings, the following style is important, and I call it the Double coin flipping style wording.
(↑ Revised on April 14, 2019)
An amount S of money is placed in an envelope A.
Other amount of money is placed in the other envelope B, this amount is being S/2 or 2S depending on a flip of a coin.
You are handed one of the two envelopes depending on the another flip of a coin.
This wording can be interpreted as a mixing of the Ali-Baba version problem and the two envelopes problem. (← Added on March 24, 2019)
In the article Nalebuff, Barry.(1989), the two envelopes problem was presented after the Ali-Baba version problem with the following phrases.
In the "original" version of the problem, there is no coin toss. We are only told that one envelope contains twice as much money as the other, but not which is which.
The originator of the double coin flipping style may have misunderstood these phrases.
Specifically, they may have skipped the phrase "there is no coin toss", and they may have devised the additional coin toss to realize the phrase "not told which is which".
(↑ Addrd on March 31, 2019. Revised on April 14, 2019)

Kinds of the wording how the opportunity to trade is given

Opened Version

A typical wording of the opened version problem is as follows.
(The former is omitted.)
One of the two envelopes is handed to person A, and the other to person B.
Person A opens his envelope and finds the amount is $10.
Person A reasons as follows.
With a probability 1/2, the amount of money in B's envelope is $5.
With a probability 1/2, the amount of money in B's envelope is $20.
The expected amount of money in B's envelope is (1/2)$5 + (1/2)$20 = $12.5.
(The rest is omitted.)
↑ Revised on March 22, 2015, April 7, 2016, July 11, 2016, March 8, 2017, June 8, 2017.

Closed Version

A typical wording of the closed version problem is as follows.
(The former is omitted.)
Randomly, a person chooses one envelope, say A.
Before opening A the person can change choice.
The person reasons as follows.
Suppose that the amount of money in A is $x.
Then B contains $2x or $0.5x equally likely.
The expected value of the amount in B is 0.5 × $2x + 0.5 × $0.5x = $1.25x.
(The rest is omitted.)
↑ Revised on March 22, 2015, April 3, 2016, April 7, 2016, July 11, 2016, June 8, 2017.

Ambiguous Version

This paragraph was added on February 1, 2015.

A typical wording of the ambiguous version problem is as follows.
(The former is omitted.)
You choose one at random.
You are then given the option of taking the other envelope instead.
↑ After opening? Before opening?
Let the sum of money in your envelope be $n.
The expected sum of money in the other envelope is (1/2)(2n) + (1/2)(n/2) > n.
(The rest is omitted.)
↑ Revised on March 22, 2015, December 26, 2015, April 3, 2016, July 11, 2016, June 8, 2017.

Kinds of the wording of the notation of the amount of money

Version showing concrete amounts of money as example

Example: "If the chosen amount of money is $10, then the opposite amount of money will be $5 or $20.

Version showing only variable symbol of amounts of money

Example: "If the chosen amount of money is $x, then the opposite amount of money will be $x/2 or $2x.

Kinds of the wording of the calculation of the expected value

Version concerning the expected amount of money in the opposite envelope

Most versions were so.

Version concerning Loss and Gain

(This paragraph was created on February 6, 2015, and moved here on June 19, 2016.  Title was changed on July 17, 2016.)

There is another opened version wording of the two envelopes problem.  It concerns the expectation of the gain from the trading of envelopes.
One of the examples of such a wording is the problem in the article "Paradosso delle due buste" (Versione del 17 gen 2015 alle 18:42) of the Italian language Wikipedia.
I can not read Italian language, so I used artificial translator, and I understand that the problem is as follows.
(The former is omitted.)
You open the envelope and find the amount of money is A.
You think the result of exchange as follows.
With a probability 1/2, you will get another A.
With a probability 1/2, you will lose A/2.
The expected gain of money is (1/2)A - (1/2)(A/2) = (1/4)A.
(The rest is omitted.)
↑ Revised on April 7, 2016, July 11, 2016, March 8, 2017, June 8, 2017.
 On July 26, 2015, I found another example which has similar wording in Barron, R. (1989).

Kinds of the wording of the explanation of the existence of a paradox

Paradox of the broken symmetry

(Title was reised on June 9, 2017, September 24, 2017.)

This paradox is derived from both of opened version problem and closed version problem. (← Added on September 24, 2017.)

Sometimes people feel a paradox as follows.
If opposite envelope is favorable than chosen envelope regardless of the amount of money in the chosen envelope, then opposite envelope is always favorable than chosen envelope.
It is unreasonable because that both envelopes are not distinguishable.
Some people will not feel such a paradox.
There are probability distributions of amount of money such that the opposite envelope is always favorable.
(↑ Revised on July 11, 2016.)

Paradox of the envelopes which are greener than each other

(Title was reised on June 10, 2017, September 24, 2017.)

This paradox is derived from both of opened version problem and closed version problem. (← Added on September 24, 2017.)

Sometimes people feel a paradox as follows.
Player A think that the expected amount of money which the player B has is more than the amount of money which A has.
Player B think in similar fashion.
It is unreasonable that the exchange is favorable to both players.
Some people will not feel such a paradox.
It is easy to find the case that the exchange is favorable to both players.

The endless switching paradox which is derived from the closed version problem

(This paradox was added on September 25, 2017. Revised on September 26, 2017.)

Sometimes people feel a paradox as follows.
If you swap the envelopes you can repeat the same calculation and as a result you should swap back.
And you should swap again and again ad-infinitum.
Some people will not feel such a paradox.
After switching envelope the amount of money in your hand is an expectation not an actual money.
So you cannot repeat same calculation after switching envelope.

The money pump paradox which is derived from the closed version problem

(The contents was revised on September 25, 2017.)

Sometimes people feel a paradox as follows.
Because we can apply this calculation after change of envelopes. So if we repeat swapping, the expected amount of money in the chosen envelope will grow up in the rate 25% per every swap.
Some people will not feel such a paradox.
After the first swap the amount of money of the new "the other envelope" is the amount of money of the original "the chosen envelope".
As a result, after the second swap the amount of money can only return to the original amount.

Paradox of the unexpected expected value which is derived from closed version problem

(Added on October 23, 2016. Title was revised on September 24, 2017.)

Sometimes people feel a paradox as follows.
Before seeing the contents of envelope, there is no information about which envelope is more favorable.
Therefore expected value should not suggest switching the envelope, and it should not suggest keeping the envelope.
Some people will not feel such a paradox.
People who are fond of mathematics will not reject conditional expectation which is calculated before seeing the contents of envelope.
And they accept that expected value suggests switching the envelope or suggests keeping the envelope.

Kinds of the wording about the independence from the chosen amount

This paragraph was added on May 25, 2017. Revised on May 26, 2017, June 10, 2017.

I think that independence from the chosen amount is one of very important factor to classify the wording of the problem.

Original type

opportunity of trading independence from the chosen amount example literature wording
after opening not described Zabell, S. (1988) When A offers to exchange envelopes, B readily agrees, since B has already reasoned in similay fashion.
Barron, R. (1989)
 Your expected gain on the exchange is therefore (1/2)(2A) - (1/2)(A/2) = A/4 , which is strictly positive. Given the symmetry inherent in the initial choice of envelopes, this seems absurd.

The history of the two envelopes problem had started with such ambiguous wordings which has no description about the situation that the calculation does not depend on the amount of money in the chosen envelope.

Near original type

Added on June 10, 2017.

opportunity of trading independence from the chosen amount example literature wording
after opening described? Nalebuff, Barry.(1989) But this is paradoxical. The sum of the amount in both envelopes is whatever it is.
<<<Omission>>>
In the "original" version of the problem, there is no coin toss. We are only told that one envelope contains twice as much money as the other, but not which is which.

Unique type

opportunity of trading independence from the chosen amount example literature wording
before opening described Jackson, F., Menzies, P., & Oppy, G. (1994). This means that the first way of doing the calculation involves supposing that for any value of x, if $x is the amount
of money in some particular envelope, it is equally likely that $2x or $0.5x
is the amount in the other envelope.
Article "Two envelopes problem" (Revision at 17:44, 18 August 2013) in the English language Wikipedia Since this is greater than my selected envelope, it would appear to my advantage to always switch envelopes

I think that this type (before opening, described ) is very unique.

Natural type

opportunity of trading independence from the chosen amount example literature wording
after opening described Christensen, R; Utts, J (1992), With a gleam in your eye, you offer to trade envelopes with your opponent
Since she has made the same calculation, she ready agree.
The paradox of this problem is that the rule indicating that one should always trade is [...].
Chalmers, D.J. 1994 Now, this reasoning is independent of the actual amount in envelope 1, and in fact can be carried out in advance of opening the envelope; it follows that whatever envelope 1 contains, it would be better to choose envelope 2.
(Revised on June 24, 2017)
Article "Envelope paradox" (Revision at 10:55, 26 August 2004) in the English language Wikipedia
(How to read it) 		But you could have gone through this same chain of reasoning before you opened the envelope and deduced the same result, that you should always take the other envelope. But that's clearly nonsense.
(Added on June 24, 2017)
Article "Umtauschparadoxon" (am 22. August 2016 um 16:55) in the German language Wikipedia Es kann aber nicht sein, dass der andere Umschlag immer besser ist, da ja beide Umschläge vor dem Öffnen offensichtlich gleichwertig sind.
Article "Paradosso delle due buste" (Versione del 16 apr 2016 alle 15:12) in the Italian language Wikipedia Concludiamo che conviene sempre cambiare busta, a prescindere dal valore che troviamo in quella scelta per prima (e quindi anche senza averci guardato dentro!).
Article "פרדוקס המעטפות" (revision 04:20, 1 במאי 2016‏) in the Hebrew language Wikipedia.
הנימוק הזה נכון לכל Y.

I think that this type is most natural (most primitive? most naive?) .
Many standard resolvers wrote such a wording in their articles.

DivideThreeByTwoian type

opportunity of trading independence from the chosen amount example literature wording
before opening not described An article by DivideThreeByTwoian (1997) The paradox […], whichever envelope I originally selected, it is [...]. And so I should swap.
An article by DivideThreeByTwoian (2001) "x=1,25y", and "y=1,25 x" cannot both be true.
An article by DivideThreeByTwoian (2003) The expected gain of switching is:
 (1) (3/2)n-(3/2)n=0.
 (2) (5/4)x-x=(1/4)x.
 (3) x-(5/4)x=- (1/4)x.
But they cannot all be right.
Article "Two envelopes problem" (Revision at 22:05, 3 October 2005) in the English language Wikipedia. But as the situation is symmetric this can't be correct.
Article "Two envelopes problem" (Revision at 14:56, 28 April 2017) in the English language Wikipedia. To be rational, I will thus end up swapping envelopes indefinitely.

It seems that all DivideThreeByTwoians dislike the description of the situation that the calculation does not depend on the amount of money in the chosen envelope.

Kinds of the wording about the numbered steps

Version with no numbered steps

Most versions had no numbered steps.

Version with numbered steps

This paragraph was added on September 20, 2015, and was moved here on June 19, 2016. Title was changed on June 8, 2017.

A typical wording of the closed version with numbered steps is as follows.
There are two envelopes, one of which contains twice as much money as the other. 
You don't know wich contains the larger sum.
You choose one at random.
Before you open it you are given the option of taking the other envelope instead.
Denote by x the amount of money in your selected envelope. Now, you can reason as follows:
  1. With a probability 1/2, x is the larger amount, and with a probability 1/2, x is the smaller amount.
  2. The other envelope may contain either 2A or A/2.
  3. If x is the larger amount the other envelope contains x/2, and if x is the smaller amount the other envelope contains 2x.
  4. Thus, the other envelope contains x/2 with a probability 1/2 and 2x with a probability 1/2.
  5. Because the expected value of the amount in the other envelope is (1/2)(x/2) + (1/2)2x = 1.25x > x,
    it is rational to swap (Optional phrase : whatever amount of money is in the chosen envelope).
This conclusion contradicts the symmetry of the two envelopes! ← paradox
↑ Revised on April 7, 2016, July 11, 2016, August 19, 2017.
 In my perception, the combination of these steps has some power to lead us to "SinglePairian's problem".
Because it strongly invites us to look at the variable symbol rather than probability 1/2.
↑ Revised at on September 22, 2015.

Kinds of the wording how the probability 1/2 is combined with amounts of money

This paragraph was added on June 3, 2017. Revised on May 17, 2018, July 15, 2018.

opened version problem

I found the following sequences of the frases.

'placed amounts' → 'the chosen amount' → 'probability and the other amounts' → 'expectation'

A typical wording of this version is as follows.
An amount of money x is placed in one envelope and amount 2x is placed in another envelope. ← placed amounts
You open your envelope and see that there is $10 in it. ← the chosen amount
The other envelope contains the lesser amount x , i.e. $5, with probability 1/2. And it contains the greater amount 2x, i.e. $20. with same probability. ← probability and the other amounts
The expected amount of money in the other envelope is (1/2)$5 + (1/2)$20 = $12.50. ← expectation
I think that Zabell, S. (1988) belongs to this type.
And I think that anybody become DoublePairian after reading such a wording.

'placed amounts' → 'the chosen amount' → 'probability and the exchange rates' → 'expectation'

(This wording was added on May 3, 2018.)

A typical wording of this version is as follows.
Two envelopes both contain money. And one envelope contains amount of money twice that of the other. ← placed amounts
You open your envelope and see that there is $A in it. ← the chosen amount
If you exchange envelops, you will , with same probability, get either double your envelope or half of your envelope. ← probability and the exchange rates
The expected gain on the exchange is (1/2)2A - (1/2)(A/2) = A/4. ← expectation
I think that Barron, R. (1989) belongs to this type.
And Sobel, J. H. (1994) may also belong to this type. (← Added on April 14, 2019)
I think that most of us will become DoublePairian after reading such a wording.

'the chosen amount' → 'the other amounts' → 'probability of each situation' → 'expectation'

A typical wording of this version is as follows.
Let's denote the amount of money in the chosen envelope by x. ← the chosen amount
Either the other envelope contains 2x or it contains x/2. ← the other amounts
Either amount must be equally likely. ← probability of each situation
Then the expected amount of money in the other envelope is 1.25x. ← expectation
The first revision (at 10:55, 26 August 2004) of the article "Envelope paradox" of the English language Wikipedia (How to read it) belongs to this type. (← Revised on July 1, 2017, May 17, 2018.)
But I think that it is hard to become SinglePairian after reading such a wording.

'the chosen amount' → 'the other amounts' → 'probability whether the chosen amount is the greater or the lesser' → 'the other amount corresponding to whether the chosen amount is the greater or the lesser' → 'probability and the other amounts' → 'expectation'

Let's denote the amount of money in the chosen envelope by x. ← the chosen amount
Either the other envelope contains 2x or it contains x/2. ← the other amounts
The probability that x is the larger amount is 1/2 and the probability that x is the smaller amount is 1/2. ← probability whether the chosen amount is the greater or the lesser
If x is the larger amount the other envelope contains (1/2)x and if x is the smaller amount the other envelope ontains 2x. ← the other amount corresponding to whether the chosen amount is the greater or the lesser
The other envelope contains x/2 or 2x with same probability 1/2. ← probability and the other amounts
Then the expected amount of money in the other envelope is 1.25x. ← expectation
The problem presented in the revision 21:33, 2 October 2005 of the article "Envelope paradox" of the English language Wikipedia (How to read it) had such a wording.
The main feature of such a wording is that the probability of each situation is presented after the other amounts. (← Added on August 26, 2018)
I think that after reading such a wording many of us will become DoublePairian but some of us will becom SinglePairian.

closed version problem

I found the following sequences of the frases.

'the chosen amount' → 'the other amounts' → 'probability of each situation' → 'the other amounts' → 'expectation'

A typical wording of this version is as follows.
Suppose that the amount of money in the chosen envelope is x. ← the chosen amount
Then the other envelope either contains 2x or x/2. ← the other amounts
Each possibility is equally likely. ← probability of each situation
The expected value of taking the othe envelope is 1.25x. ← expectation
I think that Jackson, F., Menzies, P., & Oppy, G. (1994). belongs to this type. (← Added on May 17, 2018)
And Bruss, F.T. (1996) may also belong to this type. (← Added on April 14, 2019)
I think that most of us become DoublePairian just after reading such a wording.

'the chosen amount' → 'the chosen amount is the greater or the lesser' → 'probability of each situation' → 'the other amounts' → 'expectation'

A typical wording of this version is as follows.
Let's denote the amount of money in the chosen envelope by x. ← the chosen amount
Either the chosen envelope contains the greater amount of money or it contains the lesser. ← the chosen amount is the greater or the lesser
These are equally probable. ← probability of each situation
If x is the lesser amount or the greater amount then the other envelope contains 2x or x/2 respectively. ← the other amounts
Therefore the expected amount of money in the other envelope is 1.25x. ← expectation
I think that McGrew, T. J., Shier, D., & Silverstein, H. S. (1997) belongs to this type. (← Revised on July 1, 2017.)
The main feature of such a wording is that the probability of each situation is presented before the other amounts. (← Added on August 26, 2018)
If I read such a wording several years ago I might think that some people became SinglePairian after reading it.
But now (Jun 3, 20017) I think that anybody becomes DoublePairian just at the moment they have read such a wording.

'the chosen amount' → 'the other amounts' → 'probability whether the chosen amount is the greater or the lesser' → 'the other amount corresponding to whether the chosen amount is the greater or the lesser' → 'probability and the other amounts' → 'expectation'

(Added on July 1, 2017. Revised on July 2, 2017, August 11, 2017, February 9, 2018.)

A typical wording of this version is as follows.
Let's denote the amount of money in the chosen envelope by x. ← the chosen amount
Either the other envelope contains 2x or it contains x/2. ← the other amounts
The probability that x is the larger amount is 1/2 and the probability that x is the smaller amount is 1/2. ← probability whether the chosen amount is the greater or the lesser
If x is the larger amount the other envelope contains (1/2)x and if x is the smaller amount the other envelope ontains 2x. ← the other amount corresponding to whether the chosen amount is the greater or the lesser
The other envelope contains x/2 or 2x with same probability 1/2. ← probability and the other amounts
Then the expected amount of money in the other envelope is 1.25x. ← expectation
The revision 22:05, 3 October 2005 of the article "Two envelopes problem" of the English language Wikipedia had such a wording.
The main feature of such a wording is that the probability of each situation is presented after the other amounts.
I think that after reading such a wording many of us will become DoublePairian but some of us will becom SinglePairian.

'the chosen amount' → 'probability whether the chosen amount is the greater or the lesser' → 'the other amounts' → 'the other amount corresponding to whether the chosen amount is the greater or the lesser' → 'probability and the other amounts' → 'expectation'

(Added on July 2, 2017. Revised on August 11, 2017, May 3, 2018.)
Let's denote the amount of money in the chosen envelope by A. ← the chosen amount
The probability that A is the larger amount is 1/2 and the probability that A is the smaller amount is 1/2. ← probability whether the chosen amount is the greater or the lesser
The other envelope may contain either 2A or A/2. ← the other amounts
If A is the smaller amount the other envelope contains 2A and if A is the larger amount the other envelope ontains A/2. ← the other amount corresponding to whether the chosen amount is the greater or the lesser
The other envelope contains 2A or A/2 with same probability 1/2. ← probability and the other amounts
Then the expected amount of money in the other envelope is 1.25A. ← expectation
In my perception many revisions from the revision 20:51, 9 October 2005 of the article "Two envelopes problem" of the English language Wikipedia belong this type.
The main difference from previous revision is that the probability of each situation is presented before the other amounts. (← Added on August 26, 2018)
I think that after reading such a wording we will become either DoublePairian or SinglePairian.

'probability whether the chosen envelope is the greater or the lesser' → 'the chosen amount' → 'the other amount corresponding each situation' → 'expectation with the other amounts'

(Added on August 26, 2018)

A typical wording of this version is as follows.
The probability that the chosen envelope is the greater is 1/2 and the probability that it is the lesser is 1/2. ← probability whether the chosen envelope is the greater or the lesser
Let's denote the amount of money in the chosen envelope by n. ← the chosen amount
If the chosen envelope is the greater the other envelope contains n/2, and if the chosen envelope is the lesser the other envelope contains 2n. ← the other amount corresponding each situation
Then the expected amount of money in the other envelope is (1/2)(n/2) + (1/2)2n = 1.25n. ← expectation with the other amounts
An article published in 2007 used a wording like the above a wording.
The biggest feature of such a wording is that the notation 'n' of the other amount appear after the description of probability.
I think that after reading such a wording we will become nothing other than SinglePairian.

Kinds of the wording for referring to the lesser amount of money and the greater amount of money

This paragraph was added on August 19, 2017.

Some articles used the following phrases for referring to the amounts of money.
the lesser amount of money
the greater amount of money
I think that such phrases have some power to let us become SinglePairian

Version without these phrases

Most versions did not use such phrases for referring to the amounts of money.

Version with these phrases

The following articles used such phrases for referring to the amounts of money.

Kinds of the wording which summarize the problem

This paragraph was added on November 19, 2017. Revised on November 27, 2017, December 20, 2017.

Some revisions of the English language Wikipedia article "Two envelopes problem" presented summary of the problem.

Some revisions of the English language Wikipedia article "Two envelopes problem" presented summary of the problem before presenting the problem.
I remember that I was confused when I read a revision which has such a summarization in 2013.

History of the article (Minor changes are omitted)
From the first revision 22:36, 25 August 2005 to the revision 21:47, 3 October 2005
These revisions had not index part.
Revision 22:05, 3 October 2005
Index part "Contents" was placed among the problem and solutions.
As a result the article became having lead section (beginning of the article) which describes the problem.
Revision 01:47, 17 March 2008
The index part "Contents" was moved to before the section "The problem".
As a result the lead section became having no expression of the problem.
Revision 15:38, 9 July 2011
A summary of the rule of the game was explained in the lead section.
Following it in the lead section the existence of an argument which recommends swapping envelope was explained without specific expectation formula.
Revision 03:24, 21 June 2016
The following wording which is similar to the wallet game was described in the lead section.
… because you stand to gain twice as much money if you switch while risking only a loss of half of what you currently have, …

In contrast, the lead section of the English language Wikipedia article "Monty Hall problem" (revision at 16:03, 29 November 2018) has full wording of the problem instead of the summary of it.
(↑ Added on december 9, 2018)

Kinds of the wording for identifying each of the two envelopes

This paragraph was added on August 25, 2019.

Contrary to my expectation, it seems that there is no relation between naming the envelopes and being DivideThreeByTwoian.

Version without naming (or numbering) each of the two envelopes

Example:

Version with naming (or numbering) the chosen envelope

Example:

Version with naming (or numbering) the envelope which contains the lesser amount

I have found that the lesser envelope and the greater envelope is called "$b envelope" and "$2b envelope" respectively in the article Brams, S. J., & Kilgour, D. M. (1995). I think that it is remarkable that the authors of the article did not become DivideThreeByTwoian despite using such a wording.

Case of the double coin flipping style wording

In such a case, the envelope containing the sead amount has name "A".
Example:

Wordings which make the problem not the two envelopes problem

This paragraph was added on September 26, 2015.

Wording which concerns mean values rather than particular values

A typical wording of such a kind is as follows.
(The former is omitted.)
Denote the amount of money in your envelope by X, and denote the amount of money in the other envelope by Y.
E(Y) = (1/2)(E(X)/2) + (1/2)2E(X) = 1.25E(X).
(The rest is omitted.)
On the problem which has such a wording, E(X) does not correspond to particular event and the probability can not be conditional.  (← Revised on July 11, 2016.)
Hence the probability in the expectation formula can not have a value except 1/2.
Therefore mathematically it greatly differs from the two envelopes problem.
(Specifically, please look at Is LesserOrGreaterMeanValuean's problem the third problem?)

Wording which concerns mean value of exchange rate

(This paragraph was added on July 17, 2017.)

A typical wording of such a kind is as follows.
Let X and Y be random variables of the amount of money in the chosen envelope and the other envelope respectively.
E(Y/X) =(1/2)(2X/X) + (1/2)((X/2)/X) =1.25.
Therefore E(Y)=1.25E(X) > E(X).
!!! Paradox
On the problem which has such a wording, the equation "E(Y/X) = 1.25" is correct.
Therefore mathematically it greatly differs from the two envelopes problem.
And the paradox on such a wording is caused by a confusion of rate of mean values and mean value of rate.
Therefore psychologically it greatly differs from the two envelopes problem.
(Specifically, please look at Is MeanRateOfExchangean's problem the fourth problem?)

Wording which only demands readers think about whether to switch or not to switch

This paragraph was added on November 21, 2015, and was revised on January 6, 2017.

A typical wording of such a kind is as follows.
There are two envelopes each of which contains money.
One of them contains twice as much amount of money as the other.
You have been given one of them and looked how much it contains.
If you can trade it with the other envelope should you exchange or not? Or does it not matter?
We can make various setting of problem for such a wording. On such problem settings, it can not be the two envelopes problem.  The reason is because the main subject of the two envelopes problem is expected value. Not decision making. (← Revised on April 1, 2017.)
And on such problem settings, it is difficult to find the existence of a paradox.

On May 7, 0218, I found the following funny answer to a similar problem but on the closed version problem.
One expectation of the switching gain is (1/2)(x/2 - x)+(1/2)(2x - x) = 0.25x and the another expectation is (1/2)(-a)+(1/2)a= 0.
Both are not negative, so it is better to exchange.
(↑ Added on May 17, 2018)

On March 1, 2018, the paragraph "If a problem has such a wording then nobody can feel paradox" was deleted.

Wording which is very restrictive

This paragraph was added on June 7, 2017, revised on July 18, 2017 , August 14, 2017, December 9, 2017.
The title was changed on June 7, 2018.

On June 7, 2017, I found the following wording.
There are two envelopes each of which contains money.
You are told that one contains $100 and the other contains $200.
I think that we must become SinglePairian if we read such a wording.

On July 18, 2017, I remembered that I had seen the following wording.
There are two indistinguishable envelopes which contain $x and $2x.
You choose one of the envelopes, knowing that the amount is either $x or $2x. (← Revised on July 21, 2019)
It seems not easy to become DoublePairian after reading such a wording.

On July 21, 2019, I found the following wording.
A certain amount of money is place in an envelope and twice the amount of money is placed in another envelope.
The initial amount is unknown to you.
It seems not easy to become DoublePairian after reading such a wording.

I think that nobody want to create a puzzle with the following mental model given by these wordings.


(↑ Added on March 1, 2018.)

But surprisingly I found the above wordings in some articles which advocated DivideThreebyTwoian's opinion.

Wording which has a menu of reasonings

This paragraph was added on July 1, 2018.

A typical wording of such a kind is as follows.
Reasoning 1
Let x denote the lesser amount of money.
Then the expected value of the amount of money of the each envelope is (1/2)x + (1/2)2x = (3/2)x.
Thus, both envelopes are equally favorable.

Reasoning 2
Let x denote the amount of money of the chosen envelope.
Then the expected value of the amount of money of the other envelope is (1/2)(x/2) + (1/2)2x = (5/4)x.
Thus, the other envelope is advantageous.

Reasoning 3
Let x denote the amount of money of the other envelope.
Then the expected value of the amount of money of the chosen envelope is (1/2)(x/2) + (1/2)2x = (5/4)x.
Thus, the chosen envelope is advantageous.

All these reasonings seem rational. But they are conflicting with each other.
!!! Paradox !!!
I think that the problem with such wording is no longer two envelope problems for the following reasons.

History of the wording of the two envelopes problem

I revised this paragraph on April 3, 2015. And I rewound this on the next day. And I revised this on September 27, 2015.

Original wording

(This paragraph was revised on April 14, 2016 October 23, 2016, and January 11, 2018.)

There are some articles which give some hint to imagine the original problem statement of the two envelopes problem.

A hint by  Nalebuff, Barry. (1988)

In this article, Nalebuff said that he had learned the problem through the following route.

Barry Nalebuff ← H. V. ← S. Zabell ← S. B.

A hint by  Zabell, S. (1988).

Fortunately, on October 22, 2016, I got a copy of the article Zabell, S. (1988).
This historically most important wording of the two envelopes problem was as follows.

One example, closely related to A(n), is a little puzzle which I will call the exchage paradox: 
A, B, and C play the following game. C acts as referee and places an unspecified
amount of money x in one envelope and amount 2x in another envelope. One of
the two envelopes is then handed to A, the other to B.
A opens his envelope and see that ther is $10 in it. He then reasons as follows:
"There is a 50-50 chance that B's envelope contains the lesser amount x (which
would therefore be $5), and a 50-50 chance that B's envelope contains the greater
amount 2x (which would therefore be $20). If I exchange envelopes, my expected
holdings will be (1/2)$5 + (1/2)$20 = $12.50, $2.50 in excess of my present holdings.
Therefore I should try to exchange envelopes."
When A offers to exchange envelopes, B readily agrees, since B has already re-
soned in similar fashion.
It seems unreasonable that the exchange be favorble to both, yet it appears hard to fault the logic of either. Obviously all hinges on A's apparently harmless symmetry assumption that it is equally likely that B holds the envelope with the greater or the lesser amount.

In Zabell, S. (1988) it was written that this form of the two envelopes problem did not originate with S. B. ← Added on November 6, 2016.

A hint by  Barron, R. (1989).

The problem which was written in Barron, R. (1989) is "Opened version with loss and gain".
In Barron, R. (1989), existence of a paradox was explained by the broken symmetry and by the fact that to exchange envelopes are favorable to both players.
And in it, the situation that the calculation does not depend on the amount of money in the chosen envelope is not described.

Conclusion

These findings suggests that the original wording had the following properties. ↑ Revised on may 26, 2017.

History of each kind of wording

Early example of articles which described the independence from the chosen amount

This paragraph was added on May 26, 2017. Title was revised on September 9, 2017.

The following articles seem to be the earliest examples. From this, I expect that such a wording has emerged four years after the original problem.

History of the opportunity to trade

(This header was added on November 11, 2018.)

First "Closed version"

Among the articles which describe "Closed version" problem, the following article seems to be earliest one.

Jackson, F., Menzies, P., & Oppy, G. (1994).
(In the article, existence of a paradox was explained by the broken symmetry.)  ← Added on April 14, 2016.

From this, I expect that the first "Closed version" problem has emerged six years after the first "Opened version" problem.

Early example of "Ambiguous version"

Following articles are early examples of "Ambiguous version" problem.

Evolutionary phylogenetic tree

(Added on November 11, 2018)

I am imagining as follows. 
     Opened  Ambiguous Cloesed
     Verson  Version   Version
      |        |        |
      |        |        |Closed
      |        |        |Version
1994  |        +--------+ 
      |        |      mutation
      |        |
      |        |Ambiguous
      |        |Version
1992  +--------+    
      |      mutation
      |
      |Opened 
      |Version
      |
1988 Original

Early example of "Version with numbered steps"

Title was changed on June 8, 2017, revised on March 15, 2018.

Following articles are early examples.

Early example of "Version with description of arrangement"

Following articles are early example. Remark :
The above articles presented wording of the double coin flipping style. (← Added on March 24, 2019)
I imagine that this style has been originated from Sobel, J. H. (1994). (← Added on April 21, 2019)

Early example of the wording which concerns mean values

On June 19, 2016, unfortunately I had found no example.
However, On June 9, 2019, I have noticed that an expectation formula like "ex(Y) = (1/2)ex(X/2) + (1/2)ex(2X) = 1.25ex(X)" may have been written in a problem presented in Jeffrey, R. (2004).
↑ Revised on June 19, 2016, June 9, 2019.
Remark
Snell, J. L., & Vanderbei, R. (1995) introduced a solution with an expectation formula which was made of expected values. But the problem itself had usual wording. (← Added on April 15, 2017. Revised on August 5, 2018)

For more details

Please see the paragraph "Some fragment of real history of the wording of the two envelopes problem" in my page "A fictional history of the two envelopes problem".  

I have never read a version of the problem which describes no expectation formula.

This paragraph was added on April 1, 2015, revised on June 17, 2017.
This paragraph was moved to here on March 1, 2018.

Taking the wallet game into account, I imagined a version of the two envelopes problem which has the following wording.
There are two envelopes.
One has twice as much money as the other.
Randomly, a person chooses one envelope.
The person can change choice before opening the chosen envelope.
The person reasons as follows.
If I change my choice then with a probability 1/2 I may lose a half of the amount of my money.
If I change my choice then with a probability 1/2 I may obtain the same amount as the amount of money which I have.
Therefore the potential gain is greater than zero and I should change my choice.
But there is no reason of a gap between the envelopes! ← paradox
We can find SinlglePairian's problem and LesserOrGreaterMeanValuean's problem from such a wording.
But we can not find DoublePairian's problem, because the probability 1/2 is the base of such a wording.
This wording may give us fun of other fallacies rather than "Base rate fallacy". So it is favorable to DivideThreeByTwoians.

Strangely I had never read such a wording of the two envelopes problem until I found such a wording in the article "Two envelopes problem" revision 03:24, 21 June 2016 of the English language Wikipedia.

I think that such a wording is not suitable for the title "Two envelopes problem", because such a wording should be called "Two envelopes problem like wallet game".

Wording ranking of the closed version problem

This paragraph was added on July 3, 2017. Revised on June 24, 2018.

I examined wordings of the closed version problem.

About the power to make us SinglePairian

The most powerful one I have ever seen

The English language Wikipedia article "Two envelopes problem" (Revision at 20:51, 9 October 2005).
  1. Denote by A the amount in your selected envelope
  2. The probability that A is the larger amount is ½, and that it's the smaller also ½
  3. The other envelope may contain either 2A or A/2
  4. <<<The rest is omitted.>>>
I think that after reading such wording a few of us will become SinglePairian.
To my eyes this wording resembles the woriding presented in McGrew, T. J., Shier, D., & Silverstein, H. S. (1997).

The most weak one I have ever seen
(On March 1, 2018, This title was changed from "The most top two I have ever seen".)

The English language Wikipedia article "Two envelopes problem" (Revision at 23:56, 18 August 2013).
The problem:
You have two indistinguishable envelopes that each contain money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. You pick at random, but before you open the envelope, you are offered the chance to take the other envelope instead.
It can be argued that it is to your advantage to swap envelopes by showing that your expected return on swapping exceeds the sum in your envelope. This leads to the absurdity that it is beneficial to continue to swap envelopes indefinitely.
Example: Assume the amount in my selected envelope is $20. If I happened to have selected the larger of the two envelopes, that would mean that the amount in my envelope is twice the amount in the other envelope. So in this case the amount in the other envelope would be $10. However if I happened to have selected the smaller of the two envelopes, that would mean that the amount in the other envelope is twice the amount in my envelope. So in this second scenario the amount in the other envelope would be $40. The probability of …
<<<The rest is omitted.>>>
I think that after reading such wording nobody of us will become SinglePairian.

About the power to make us LesserOrGreaterMeanValuean

The most powerful one I have ever seen

The English language Wikipedia article "Two envelopes problem" (Revision at 03:24, 21 June 2016).
However, because you stand to gain twice as much money if you switch while risking only a loss of half of what you currently have, it is possible to argue that it is more beneficial to switch
I think that after reading such wording some of us will become LesserOrGreaterMeanValuean.

History of the paradoxes and resolutions of the two envelopes problem

This paragraph was added on September 20, 2015. Title was revised on June 10, 2017.

Period of mathematical standard resolution

The period of mathematical standard resolution is divided into several parts as follows.

Period of the standard resolution - Part 1

(This paragraph was revised on September 27, 2015, April 3, 2016 and January 14, 2017. Title was revised on August 12, 2018.)

The original wording of the two envelopes problem is "Opened version".  So the variable symbol can have only one value after it has been revealed.
Therefor if a mathematician watched the expectation formula E = (1/2)(x/2) + (1/2)2x, he will soon doubt the probability 1/2, because other parts can not be wrong.
And he will soon find the case that the probability is not 1/2.
(Specifically, please look at The standard resolution and resolved paradoxes.)

Such resolutions seemed to have appeared in 1988 at the latest.
Actually in Zabell, S. (1988), I had found some part of the standard resolution.

Which paradox resolved?
(This paragraph was added on June 10,2017.)
To my surprise, the early articles that gave the standard resolution presented only fictitious paradoxes such as the "paradox of the two envelopes which are greener than each other".
Example On August 5, 2018, the reference to Barron, R. (1989) was deleted, because the main theme of it was Paradoxical distribution.

What cause of the paradox presented?
(This paragraph was added on July 15,2018. Revised on August 5, 2018)
Two kinds of causes are proposed as follows. For details of this theme, please see "The theory of assumption of probability distribution is unusual as a standard resolution")

Period of the standard resolution - Part 2

(This paragraph was added on April 24, 2016. Revised on July 9, 2016. Title was revised on August 12, 2018.)

The fact that there is a case that the probability is not 1/2 is logically enough as a resolution.
But it is a paradox, so even if they were good at mathematics they did not get satisfaction.
Actually, the fact that the probability is not always 1/2 does not guarantee that correct expectation formula does not suggest swapping the envelopes if the opportunity of swapping is given before opening the chosen envelope.
That is why some of them proved that mean values of conditional expectations of the amounts of money in each envelope are same.

Such resolutions seemed to have appeared in 1994 at the latest.
Actually in Chalmers, D.J. 1994 , I had found such a proof.

Period of the standard resolution - Part 3

(This paragraph was added on January 26, 2018. Revised on May 17, 2018. Title was revised on August 12, 2018.)

As the closed version problem became popular, more people resolved the closed version problem with the standard resolution.

One of the early example was the English language Wikipedia article "Two envelopes problem" at the revision 20:17, 26 May 2010.
It had the section "The problem / Solution" which presented the standard resolution on the closed version problem. (← Added on March 15, 2018)
Remarks (Added on March 15, 2018)
  • That section referred a web page "The Two Envelopes Paradox" by Keith Devlin.
    But the problem presented in that page was the opened version problem.
  • In contrast, the section "Proposed Solution 2" of the article "Two envelopes problem" at the revision 22:05, 3 October 2005 presented the standard resolution on the opened version problem.

Period of DivideThreeByTwoian's resolution

The period of DivideThreeByTwoian's resolution is divided into several parts as follows.

Period of resolution with the double coin flipping style wording

(This paragraph was added on March 22, 2018.)
(On March 24, 2019, the title was chaged from "Period of the pure not-three-amounts theory", and contents was revised.)

An article written in 1994 by a philosopher and an article written in 1996 by a mathematician had the following aspects.
I think that such a resolution and DivideThreeByTwoian's resolution are sharing the following concepts. However, I think that the problem resolved by such a resolution is not the pure two envelopes problem.

Period of DivideThreeByTwoian's resolution - Part 1

(This paragraph was revised on January 9, 2016, April 3, 2016, April 24, 2016 and January 14, 2017.)
(This paragraph was revised with new title on July 19, 2017.)

The original wording of the two envelopes problem is "Opened version".  So nobody can have any resolution except the mathematical standard resolution.
But when wording of the "Closed version" had been created, people became able to doubt the variable symbol of the expectation formula rather than probability.
Then some people found a resolution with the not-three-amounts theory and the theory of "E=(1/2)2a+(1/2)a". (← Revised on March 22, 2018.)
(Specifically, please look at the section "DivideThreeByTwoian's paradox and resolution".)

Such resolutions seemed to have appeared in 1997 at the latest. (← Revised on March 22, 2018.)

Which paradox resolved?
In my perception there was no DivideThreeByTwoian's article which represented the independence from the chosen amount.
This strongly indicates that the paradox they resolved is fictitious and their resolution is fake.
(Specifically, please look at My hypotheses about the mind of DivideThreeByTwoians.)

DivideThreeByTwian philosophers seem to have been affected by a famous philosophical article.

(This paragraph was added on May 26, 2017. Title was revised on August 12, 2017, January 13, 2019, February 3, 2019, March 24, 2019.)

I have found that many of the early articles by DivideThreeByTwoians referred to Jackson, F., Menzies, P., & Oppy, G. (1994). .
So I think that the tide of DivideThreeByTwoian's opinion started after it.
And I think that it had such influence because it was published in the journal of philosophy. (← This sentence was moved here on September 7, 2018)
And I think that the following properties behaved important role. (← Added on January 13, 2019)
In my perception it (Jackson, F., Menzies, P., & Oppy, G. (1994). ) described the standard resolution. (← Revised on January 13, 2019)
  • It presented an expected value $1.5x (= (1/2)$x + (1/2)$2x), but I think that the aim of this calculation was to ascertain the equivalence of the two envelopes. (← Added on July 22,2017)
  • It explained that the following assumption is wrong.
    For any value of x (amount of money of the chosen envelope), it is equally likely that $2x or $0.5x is the amount of money of the other envelope.
    (↑ Added on May 3, 2018)

However, I think that the following aspects of this article influenced philosophers. (← Added on January 13, 2019)
  • To be honest, I am afraid that the article (Jackson, F., Menzies, P., & Oppy, G. (1994). ) may have presented an incomplete standard resolution. Because probabilities other than 0, 1, 1/2 were not presented in it. (← Added on October 7, 2018)
  • Without declaration that they had resolved the paradox, this article presented an another paradox with an improper probability distribution (probability is always 1/2).(← Added on January 13, 2019)
    And they did not explicitly state that paradoxes that amount of money is finite and paradoxes that always have a probability of 1/2 are essentially different paradoxes. (← Added on February 14, 2019)

And I think that some philosophers thought from the above influence as follows. (← Added on January 13, 2019)
  • The two envelopes paradox had not been resolved.
  • It is not helpful to correct probability for resolution.

I think that the following objections to the article (Jackson, F., Menzies, P., & Oppy, G. (1994). ) suggest the above my thought.
(Added on February 14, 2019)
  • An objection written in 1997 :
    It is inadequate to challenge the premises of equal probability, for the required resolution must resolve the case that there is neither of upper bound and lower bound.
  • An objection written in 2007 :
    They claimed that for some x it is not equally likely that the amount in the other envelope is 2x or ½x. But you do not nkow what sum of money is in the chosen envelope, therefore it may not be a mistake to suppose that it is equally likely.

Ali-Baba version problem seems to have influenced DivideThreeByTwoians.

(This paragraph was added on July 22, 2017.)

I have found that many of the early articles by DivideThreeByTwoians referred to an article (Nalebuff, Barry.(1989)) which presented the Ali-Baba version problem.
So I think that many of DivideThreeByTwoians advocated their opinion based on difference from the Ali-Baba version.

The trendy philosophical concepts seem to have influenced DivideThreeByTwoians

(This paragraph was added on May 19, 2019)

I have found that many of the early articles by DivideThreeByTwoians used the trendy philosophical concepts such as "possible worlds" and "rigid designator".
So I think that many of DivideThreeByTwoian philosophers have been influenced by the following trendy concepts.

Period of DivideThreeByTwoian's resolution - Part 2

(This paragraph was added on July 19, 2017. Revised on December 4, 2017.)

In my perception, in the period of DivideThreeByTwoian's resolution - Part 1, the theory of non-Ali-Baba version was presented as a logical evidence of wrongness of the fallacious expectation formula. (← Revised on January 8, 2019)
And some DivideThreeByTwoian philosophers explained as follows.
Because thre are only two amounts of money the variable symbol x in "E=(1/2)(x/2)+(1/2)2x" cannot have same value in the two terms.
In this wording the inconsistent variable was a result of the another fallacy. (← Added on February 16, 2018.)

But in my perception, in the period of DivideThreeByTwoian's resolution - Part 2 this explanation was interpreted as follows.
The variable symbol x in "E=(1/2)(x/2)+(1/2)2x" have different values in each of the two terms.
The inconsistent variable were interpreted as the cause of the paradox. (← Added on February 16, 2018.)
This was the birth of the "Inconsistent-variable theory".

In 2005 this theory seemed to have started to be spread through the English language Wikipedia. (← Revised on February 16, 2018.)
(For details please see the paragraph "Inconsistent-variable theory might be a ghost".)

Period of DivideThreeByTwoian's resolution - Part 3

(This paragraph was added on December 4, 2017.)

In this period some DivideThreeByTwoians advocate theory of "E=(1/2)2a+(1/2)a" which is accompanied by no hypothesis about the cause of the fallacy.

This period started before 2017, as one such solution was written on the blog page in 2017. (← Added on June 9, 2019)

Change of the meaning of the equation "E=(1/2)A + (1/2)2A".

(This paragraph was added on April 15, 2017. Revised on March 17, 2019)

The meaning of the paradox of the two envelopes which are greener than each other subtly changes depending on the opportunity to swap envelopes.

(This paragraph was added on August 29, 2017. Title was changed on September 8, 2017, July 22, 2018.)

In the period of the standard paradox the opportunity to trade envelopes is given after opening envelope.
So the paradox of the two envelopes which are greener than each other is fictitious because such a situation is not unusual.

But in the period of DivideThreeByTwoian's resolution the opportunity to trade envelopes is given before opening envelope.
So the paradox of the two envelopes which are greener than each other is not necessarily fictitious because we can not distinguish it from the mathematically standard paradox. (← Revised on August 12, 2018)

Perhaps this change of meaning of paradox might have influenced DivideThreeByTwoian's thinking.

History of the way how to change thinking depending on the opportunity to swap

(This paragraph was added on May 27, 2017. Title was revised on June 7, 2017, September 8, 2017, December 14, 2017.)
(The tables were revised on March 10, 2019)

The following is the early examples which I know.

Case of standard resolvers

early example I know fashion for
"Opened version" 		fashion for
"Closed version"
an article published in 1988. standard
resolution 		No paradox before opening
an article published in 1992. standard
resolution 		No description
an article published in 1994. No description to my eyes
standard resolution
A blog page posted in 2015. No description standard
resolution


Case of DivideThreeByTwoians

early example I know fashion for
"Opened version" 		fashion for
"Closed version"
an article published in 1994. Interpreted the problem as the Ali-Baba version problem. thinking of only one pair of amounts of money
an article published in 1997. No description thinking of only one pair of amounts of money
an article published in 2001. Interpreted the problem as a decision problem, not paradox. thinking of only one pair of amounts of money


Hybrid case
(Added on January 18, 2018.)

early example I know fashion for
"Opened version" 		fashion for
"Closed version"
The English language Wikipedia article "Two envelopes problem" revision at 22:05, 3 October 2005 standard
resolution 		thinking of only one pair of amounts of money
The English language Wikipedia article "Two envelopes problem" revision at 00:31, 8 November 2011 No description thinking of only one pair of amounts of money
standard
resolution


The third and fourth resolutions

(↑ This header was added on March 10, 2019)

The third resolution

This paragraph was added on April 17, 2016, and it was revised on July 11, 2016.

I found the third resolution on April 17, 2016.
I think that combination of the standard resolution and my resolution is complete resolution of the two envelope paradox.

The standard resolution says that probability is not always 1/2.
The non standard resolution says that the expected values of amount of money in the both envelopes are same.
The third resolution which I found yesterday(April 17, 2016) says as below.
The problem of expected value of the amount of money and the problem of equivalence of the two envelopes do not have logical relationship except the law of total expectation .
So in the domain of logic, even if always opposite envelope is favorable the equivalence will not be influenced.


The fourth? resolution

This paragraph was added on May 3, 2017.

If the fallacy about the probability had been arose by the insufficient reason, the following resolution might has some meaning.
Before Opening Envelope
The probability 1/2 is nonsense, because there is no reason to apply the principle of insufficient reason before opening an envelope .
So we should not think of conditional expected value of amount of money in the other envelope.

After Opening Envelope
The probability 1/2 is the result of your propensity about the "insufficient reason".
If yoy did not apply the insufficient reason the paradox did not arise.
So you must accept the result that the other envelope is more favorable which is the result of your own will.

Analogy among the two envelopes problem and quantum dynamics

I do not like such a resolution, but some people as below might like this.

On October 7, 2018, the section "Resolutions may not be only one" was deleted because it overlapped with the section "Why are the two envelopes problem and the two envelope paradox so chaotic ?".

Worries only by paradox

This paragraph was added on April 15, 2016.

The finding that the probability is not necessarily 1/2 is logically enough as a resolution.
(That finding means that even if the opposite envelope is favorable for an amount of money in the chosen envelope, the opposite envelope is not favorable for another amount of money in the chosen envelope.) (← Revised on May 31, 2018)
But the two envelopes problem is a paradox, hence some worries arise as below. So even if they are good at mathematics some people couldn't help proving that these worries are imaginary fears.
(Specifically please see "Fundamental mathematical theories".)

And some mathematicians had found that the following worry is real fear. (Specifically please see "Paradoxical distribution".)

Fundamental mathematical theories

Not necessarily 1/2

(Added on April 6, 2016. Revised on September 4, 2016.)
Let X be the amount of money in the chosen envelope.
Then there is a prior distribution such that
P(X is lower | X=x) ≠ 1/2 and P(X is greater | X=x) ≠ 1/2 for some x.
Popular proof
Let M be the max of possible amount of money.
Let x be the amount of money in the chosen envelope.
Then if x > M/3, the amount of money in the opposite envelope can not be 2x, in other words the probability is zero or undefined.

For reference.

Not always 1/2

Let X be the amount of money in the chosen envelope.
Then there is no prior distribution such that
P(X is lower | X=x) = P(X is greater | X=x) = 1/2 for all x.
Proof by Zabell, S. (1988).
(Added on November 6, 2016)
If P(X is lower | X=x) = P(X is greater | X=x) = 1/2 for all x, then
either the interval [1,2) and R+ would have zero probability mass, or [1, 2) and R+ would hae infinite probability mass.
In either case the probability distribution cannot be proper.
Popular proof for descrete distribution of amounts of money.
If P(X is lower | X=x) = P(X is greater | X=x) = 1/2 for all x, then
there are amount a such that sum of P(X=2n × a) diverges, and the probability distribution cannot be proper.
(↑ Revised on April 7, 2016 and April 24, 2016)
One more proof for descrete distribution of amounts of money.
Let g(x) be the probability of the event "The pair of amounts of money is x and 2x".
Then limx→∞ g(x) = 0.
Therefore for some amount of money x, g(x/2) > g(x).
For reference.
• A blog page "The Universe of Discourse : The envelope paradox".
Is this proof by me for continuous distribution of amounts of money correct?
Let g(x) be the probability density fuction of the event "The pair of amounts of money is x and 2x".
If P(X is lower | X=x) = P(X is greater | X=x) = 1/2 for all x, then g(2x) = 2g(x) for all x.
Let P(n) = P(2n ≤ x ≤ 2n + 1) for natural number n, then P(n) = 2nP(0).
Therefore sum of P(n) diverges, and the probability distribution cannot be proper.

Switching is not always advantageous

Let X be the random amount of money in chosen envelope.
Let Y be the random amount of money in the other envelope.
Then if the prior distribution has a finite mean, then it is false that E(Y|X=x) > x for all x.
(↑ Revised on March 29, 2015)
 For reference.

Two envelopes are not always equivalent

Let X be the random amount of money in chosen envelope.
Let Y be the random amount of money in the other envelope.
Then for some x, E(Y|X=x) ≠ x.
(↑ Added on August 8, 2015)
 For reference.

Switcing is not always non-advantageous

(Added on August 15, 2015. The title was revised on March 3, 2019)
For any probability distribution, for at least one value of x, E(Y|X=x) > x.
For reference.

Mean values of conditional expectations of the amounts of money in each envelope are same

This paragraph was revised on August 22, 2015, February 14, 2016, March 7, 2016 and March 20, 2016.
Let X, Y be the random variables which denote the amount of money in the chosen envelope and the other envelope respectively. And let x be the amount of money in the chosen envelope. Then E[E(X|X)] = E[E(Y|X)].
(↑ Revised on June 21, 2016.)
 For reference. I also wrote some explanations on February 2016.
Please see a companion page "Two methods for the proof of the equivalence of the envelopes of the two envelopes problem".

On May 27, 2016, I found very easy proof for discrete distributions.  (On November 3, 2016, I found a mistake and corrected it.)

Let z be an amount of money in the chosen envelope.
Consider a sub probability space which is as follows.
  • For all integer i, it contains event that the chosen amount is 2ix.
  • In this sub probability space, g(x) is the probability that the lesser amount is x.
Let denote 2iz by zi.
Let denote g(zi) by gi.
Then E(Y|X=zi) = (gi-1 zi-1 + gi zi+1) / (gi-1 + gi).
And P(X=zi) = (1/2)(gi-1 + gi).
Then we can show that E[E(X|X)] = E[E(Y|X)].
 E[E(Y|X)] = ∑ (gi-1 zi-1 + gi zi+1) / 2
= ∑ (gi-1 zi-1 / 2) + ∑ (gi zi+1/ 2) ⋅ ⋅ ⋅ split
= ∑ (gi zi /2) + ∑ (gi-1 zi / 2) ⋅ ⋅ ⋅ change the subscript
= ∑ (gi-1 zi + gi zi) /2 ⋅ ⋅ ⋅ recombine
= E[E(X|X)]

Another calculation
 E[E(Y|X)] = ∑ (gi-1 zi-1 + gi zi+1) / 2
= ∑ (gi-1 zi-1 / 2) + ∑ (gi zi+1/ 2) ⋅ ⋅ ⋅ split
= ∑ (gi zi / 2) + ∑ (gi 2zi/ 2) ⋅ ⋅ ⋅ change the subscript
= ∑ (gi zi (3/ 2)) ⋅ ⋅ ⋅ recombine
= (3/2)E(lesser amount)

On the other hand E[E(X|X)] = ∑ (gi-1 zi + gi zi) / 2
= ∑ (gi-1 zi / 2) + ∑ (gi zi/ 2) ⋅ ⋅ ⋅ split
= ∑ (gi 2zi/ 2) + ∑ (gi zi/ 2) ⋅ ⋅ ⋅ change the subscript
= ∑ (gi zi (3 / 2)) ⋅ ⋅ ⋅ recombine
= (3/2)E(lesser amount)

∴E[E(X|X)] = E[E(Y|X)].

On December 4, 2016, I restudied Chalmers, D.J. 1994 and examined one more another calculation.
 E[E(Y|X)] - E[E(X|X)]] = E[E(Y - X| X)] · · · 1
= ∑ ((gi-1 zi-1 + gi zi+1) - (gi-1 zi + gi zi))/ 2 · · · 2
= ∑ (gi-1 (zi-1 - zi) /2 + ∑ (gi zi+1 - gi zi))/ 2 · · · 3
= - ∑ (gi-1 (zi-1) /2 + ∑ (gi zi)/ 2 · · · 4
= 0 · · · 5


I expect that this drawing help us to understand the following calculation which had been done by Chalmers.
(↓ corresponding row number column was added on 25,2017.)
The final calculation in Chalmers, D.J. 1994 .
corresponding row number
in the above calculation
E(K-A) = integral[0,infinity] h(x) (E(B|A=x) - x) dx · · · 1
= integral[0,infinity] (2g(x) + g(x/2))/4 .
((2x.2g(x) + x/2.g(x/2))/(2g(x)+g(x/2)) - x) dx		 · · · 2
= integral[0,infinity] (2xg(x) - x/2 . g(x/2))/4 dx · · · if anything, 3
= (integral[0,infinity] 2xg(x)dx -
integral[0,infinity] 2yg(y)dy) /4		 · · · 4
= 0. · · · 5

On September 24, 2017, after being inspired by the following articles I examined one more another calculation.
  • A web page titled "Dan’s Geometrical Curiosities - Maximizing your earnings with money envelopes: a mathematical riddle"
  • An answer for a question "If you have two envelopes, and ..." at a question site "Mathematics Stack Exchange"
    (Asked by terrace. Answered by by robjohn♦ on Oct 9, 2014.)
 E[E(Y|X)] · · · 1
= ∑ ((gi-1 zi-1 + gi zi+1))/ 2 · · · 2
= ∑ (gi-1 zi-1) /2 + ∑ (gi-1 zi)/ 2 · · · 3
= ∑ 3 (gi-1 zi-1) /2 · · · 4
= E(X) · · · 5

The original probability space is union of such sub probability spaces.
Therefore in the original probability space, E[E(X|X)] = E[E(Y|X)] too.
(↑ Revised on June , 2016, November 3, 2016)

Expectation formula – Case of discrete distribution –

Let g(m) be the probability that m is the lesser amount.
Let X be the random amount of money in chosen envelope.
Let Y be the random amount of money in the other envelope.
Then expected winning from a trade are
E(Y|X=x) = ( g(x/2) / (g(x) + g(x/2)) )(x/2) + ( g(x) / (g(x) + g(x/2)) )2x.
For reference.

Expectation formula – Case of continuous distribution –

Let g(m) be the probability density function that m is the lesser amount.
Let X be the random amount of money in chosen envelope.
Let Y be the random amount of money in the other envelope.
Then the random variable of the conditional expected value of the other amount on the condition that X is the chosen amount is as follows.
E(Y|X) = (g(X) / (g(X)+(1/2)g(X/2)) )2X + ((1/2)g(X/2) / (g(X)+(1/2)g(X/2)) )(X/2).
↑ Revised on May 3, 2018, November 4, 2018.
For reference. The above literatures presented various methods to calculate the probability density function f(b) of the greater amount b using the probability density function g(a) of the lesser amount a. (↑ Added on September 1, 2019)

The references listed above seem have presented "g(X) / (g(X)+(1/2)g(X/2))" and "g(X/2) / (g(X)+(1/2)g(X/2))" as conditional probabilities.
I think such a calculation is based on a fashion that treats probability density function as a probability. And I myself have used that fashion on the two envelopes problem. However, I have not knowledge about the validity of that fashion,
(↑ Added on August 25, 2019)
So, I made an explanation without conditional probability as follows. Is this correct? (← Revised on August 25, 2019)
Let g(m) be the probability density function that m is the lesser amount.
Let x and X be the amount of money in chosen envelope and the random variable of it respectively.
Let y and Y be the amount of money in the other envelope and the random variable of it respectively.
Let α and β are intervals of non-negative real number that β = α/2.

Then the probability that X∈α is as follows.
P(X∈α)
= (1/2)∫x∈α g(x)dx + (1/2)∫y∈β g(y)dy
= (1/2)∫x∈α g(x)dx + (1/2)∫x∈α (1/2)g(x/2)dx
= (1/2)∫x∈α (g(x) + (1/2)g(x/2))dx.

And if P(X∈α) is not zero, the expected winning from a trade on the condition that X∈α is as follows.
E(Y|X∈α)
= ( (1/2)∫x∈α 2xg(x)dx + (1/2)∫y∈β yg(y)dy ) / P(X∈α)
= ( (1/2)∫x∈α 2xg(x)dx + (1/2)∫x∈α (1/2)(x/2)g(x/2)dx ) / P(X∈α)
= ( (1/2)∫x∈α (2xg(x) + (x/2)(1/2)g(x/2))dx ) / P(X∈α).

Therefore if g is differentiable and P(X∈α) is not zero, the random variable of the conditional expected value of the winning from a trade on the condition that X is the chosen amount is as follows.
E(Y|X) = ( g(X) / (g(X) + (1/2)g(X/2)) )2X + ( (1/2)g(X/2) / (g(X) + (1/2)g(X/2)) )(X/2).
(Added on November 4, 2018)
I think that the section "1.2 Measure-theoretic definition" in the English Wikipedia article "Conditional probability" (Revision at 09:29, 7 December 2019) is related to the above my calculation. (← Added on December 22, 2019)

The factor of the change of probability density function is the inverse number of the factor of change of variables in integral

This paragraph was added on March 18, 2016.
Let r be the ratio of the greater amount of money to the lesser amount of money.
Let g(m) be the probability density function that m is the lesser amount of some pair of amounts of money.
Let f(m) be the probability density function that m is the greater amount of some pair of amounts of money.

Factor of the change of probability density function (f to g)
  f(x) = (1/r)g(x/r).

Factor of the change of variables in integral (x/r to y)
  Let y = x/r,   then ∫ g(x/r) dx = ∫ r g(y) dy.
This rule plays an important role in the proof of the equivalence of the two envelopes.

For reference.

Distribution of the amount of money before switch is same as after switch

This paragraph was added on April 5, 2016.
Let X be a random variable which denotes the amount of money in the chosen envelope.
Let Y be a random variable which denotes the amount of money in new chosen envelope after switching under the condition switching is always done.
Then random variables X and Y have same probability distribution.
I made a drawing to explain this theory.

And I calculated on the case of a discrete probability distribution.
(Added on March 15, 2018.)
Let g(m) be the probability that m is the lesser amount.
Let X be the random amount of money in chosen envelope.
Let Y be the random amount of money in the other envelope after swapping.
Then the relation of the probabilities of X and Y are as follows.
P(Y=x)
= P(X=x/2 and X is the lesser) + P(X=2x and X is the greater)
= P(X=x and X is the greater) + P(X=x and X is the lesser)
= P(X=x).

If there is no limit on the amount, any value is useful as the threshold for switching

This paragraph was added on January 26, 2020.
Let b be an arbitrary positive number, and let y be the amount of money in the envelope you have chosen.
And consider the following switching strategy:
  • If y ≤ b, switch envelope.
  • If else, not switch envelope.
Then the following holds.
If the amount of money has no upper limit and has no lower limit over 0, the average outcome by such a strategy is larger than the average outcome by no switching, whatever the value of b.
For reference.

DoublePairian and SinglePairian, and two 'Two envelopes problems'

DoublePairian and SinglePairian

DoublePairian
Some people make the following mental model about the "Two envelopes problem".



In the following sections , the people with this mental model are called 'DoublePairian'.


SinglePairian
But another some people make the following mental model about the "Two envelopes problem".



In the following sections , the people with this mental model are called 'SinglePairian'.

Two 'Two envelopes problems'

'Two envelopes problem' for the SinglePairians and 'Two envelopes problem' for the DoublePairians differ widely from each other.
Using mathematical notation, we can demonstrate the difference of the two problems.

Let x and y be the amounts in the envelope selected by you and the amount in the another envelope respectively.
Let X and Y be random variables which take x and y as their value respectively.
Let P is a random variable which takes the pair of amounts in the envelopes as it's value.
Let a be the lesser amounts in the two envelopes.

the DoublePairian's problem
(mathematicians prefer this) 		the SinglePairian's problem
(philosophers prefer this)
the condition on which the expectation is calculated the amount of money contained in the first selected envelope
↑ Revised on April 28, 2019 		the pair of amount of money
pairs of the amount two pairs
(x/2, x) and (x, 2x)

x is the amount of the selected envelope 		one pair
(a, 2a)
what are compared
in the Closed version Problem
  		conditional expectation E(Y|X=x)
vs
value x
↓
(considering any x)
↓
random variable E(Y|X)
vs
random variable X
 
conditional expectation E(X|P=(a, 2a))
vs
conditional expectation E(Y|P=(a,2a))

(P is a random variable denoting the pair)

↑ Revised on July 20, 2015, April 28, 2019
what are compared
in the Opened version Problem
  		nothing
invariant the expectation formula must contain term of x/2 , and term of 2x the probabilities of the terms in the expectation formula
must be 1/2

Which mental model is natural?

This paragraph was added on February 23, 2018.

The main sentences of the typical wording of the two envelopes problem is as follows.
Let x denote the amount of money in the chosen envelope, then the amount of money of the other envelope may be x/2 or 2x with equal probability.
Therefore the expected value of the amount of money of the other envelope is (1/2)(x/2) + (1/2)2x.
So if you read the two envelopes problem without prior knowledge, you will get the DoublePairian's mental model.
Therefore the DoublePairian's mental model is natural and the SinglePairian's mental model is not.
It means that the mathematically standard resolution based on the DoublePairian's mental model is natural.

How did DivideThreeByTwoians get the SinglePairian's mental model?

This paragraph was added on February 23, 2018.

I have the following hypotheses. After all I cannot imagine somebody naturally has become SinglePairian. (← Added on March 1, 2018.)

The standard resolution and resolved paradoxes

(Title was changed on August 12, 2018)

Resolved paradoxes

(This paragraph was added on August 12, 2018)

Standard paradox

People who read the two envelopes problem usualy became DoublePairians.

From DoublePairian's point of view we can feel a paradox direct. (← Revised on July 15, 2018)
Let x and X denote the amount of money of the chosen envelope and the random variable of it respectively.
Let y and Y denote the amount of money of the other envelope and the random variable of it respectively.
Then, it becomes as follows.
The odds of pair (x/2. x) and the odds of pair (x, 2x) are equal.
Therefore E(Y|X=x) = (1/2)2x + (1/2)x/2 > x.
For any x E(Y|X=x) > x, in other words E(Y|X) > X. ← independence from the chosen amount
If E(X) and E(Y) are finite, E(Y) > E(X).
Under symmetry E(X) > E(Y).
∴ E(X) > E(X) and E(Y) > E(Y).
!!! Paradox
(↑ Revised on October 28, 2018)
This DoublePairian's paradox is the standard paradox of the two envelopes problem.

The independence from the chosen amount is the essence of the standard paradox

I think that independence from the chosen amount is the essence of the standard paradox.
If this factor is omitted then we can not feel paradox that the law of total expectation is violated.

Paradox of the two envelopes which are greener than each other

(This paragraph was revised on July 21, 2019)

Many of the early articles which present the standard resolution presented this paradox, without presenting resolution.
It means that this paradox was not the main theme of such articles. 

Standard resolution - Part 1

(This paragraph was greatly revised on July 15, 2018. The title was revised on August 12, 2018.)

The period of the standard resolution - Part 1 began from the birth of the two envelopes problem.

Resolution The cause of the paradox
which arise on the
DoublePairian's problem
Many mathematician's thought Many mathematician's opinion is
The cause of the paradox is a wrong assumption like this.
The odds of pair (x/2. x) and the odds of pair (x, 2x) are always equal.
!!! Wrong assumption.
The expectation formula should be corrected as follows.
Let p = P(X is lesser | X=x).
Then with an anknown rate r,
E(Y|X=x) = p (2x) + (1-p) (x/2) = r x.
They have various opinions about the cause of such a wrong assumption as follows.
  • Theory of the intuitive probability
    The cause is one of the following reasons.
    • difficulty of probability
    • ignorance of the prior probability distribution
  • Theory of assumption of probability distribution
    The cause is an assumption of a flat prior probability distribution.
  • No opinion
My thought Nobody can make an assumption about the probability.
Anybody only can be caught by an illusion of probability.
It is the famous illusion named "Base rate fallacy"
 
(↓ Added on March 29, 2018)
If the probability 1/2 has been lead by such an assumption, then the problem domain is decision theory not mathematics.
And we should discuss the paradox by the principle of insufficient reason.
 
(↓ Added on March 29, 2015)
I think that it is also important to claim the following fact.
It is not guaranteed that both of two terms in the expectation formula have meaning.
For example, if the sample space is comprised of multiple of ¥1,000 and the amount x in the chosen envelope is ¥1,000
then the term (1/2)(x/2) has not corresponding event
or the value of (x/2) can not be defined as ¥500.
↑ Revised on April 19, 2015.
A few mathematician's thought
↑ Added on August 8, 2015. 		I found a few mathematicians who think that the cause of the paradox is the wrong use of the prior probabilities to calculate the expectation.

Example
  • A web pages by Amos Storkey
    "Amos Storkey - Brain Teasers: Two Envelope Paradox - Solution".
  • A web page by Keith Devlin
    "The Two Envelopes Paradox".

I understood that their opinions are close to my opinion.
 

The paragraph "Illustration of my thought" was deleted on December 2, 2017.

Thought of the standard resolver
(Added on December 2, 2017)



Mystery of the standard resolution - Part 1
(Added on August 12, 2018)

In the period of the standard resolution - Part 1, it is not clear which paradox was resolved.
Indeed many literatures described the paradox of the two envelopes which are greener than each other.
But many literatures described independence from the chosen amount.
Specifically please see the following paragraphs.

Standard resolution - Part 2

(This paragraph was added on July 9, 2016. Revised on June 19, 2018, July 8, 2018. The title was revised on August 12, 2018.)

 I think that the period of the standard resolution - Part 2 began around 1994.

The part 1 of the standarad resolution explains the reason why we build fallacious calculation formula "E=(1/2)x/2 + (1/2)2x".
But it does not guarantee that correct expectation formula does not suggest swapping the envelopes if the opportunity of swapping is given before opening the chosen envelope.
That is why some people proved that the following two values are same.
(Specifically please see "Mean values of conditional expectations of the amounts of money in each envelope are same".)

Such a proof is the core of the part 2 of the standard resolution. (← Revised on August 12, 2018)

Standard resolution - Part 3

(This paragraph was added on January 26, 2018, revised on June 19, 2018. The title was revised on August 12, 2018.)

 I think that the period of the standard resolution - Part 3 began around 2010.

The part 3 of the standarad resolution resolves the standard paradox on the closed version problem.
However such a solution treats expected value which is conditioned on the imagined amount of money included in an unopened envelope, so it is not comfortable.
Therefore, it is rare to see such a resolution.
Indeed, most mathematicians say that there is no paradox in the closed version problem.

Some articles of the standard resolution did not resolve the written paradox in the problem.

(This paragraph was added on June 9, 2017.)

 The standard resolutions resolve the standard paradox of the two envelopes prbolem.
But many wording of the problem had not described the standard paradox.

Examples of articles which had described the standard resolution but not described the standard paradox (paradox with independence from the chosen amount)
(Revised on September 4, 2017, May 31, 2018.)

articles by standard resolver written pardox independence from the chosen amount
Nalebuff, Barry.(1989) paradox of the two envelopes which are greener than each other The sum of the amount in both envelopes is whatever it is.
(↑ Revised on May 31, 2018
(↑ On May 31, 2018. Zabell, S. (1988) was deleted from the above table.)

To my eyes these articles gave a resolution of the standard paradox but gave no resolution of the written fictitious paradox.
I think that this attribute of these resolutions made them a little hard to understand.

Addition : Examples of articles which had described the standard resolution for the standard paradox (paradox with independence from the chosen amount)
(↑ Revised on September 4, 2017.)

articles by standard resolver written pardox independence from the chosen amount
Christensen, R; Utts, J (1992), paradox of the broken symmetry (standard version) With a gleam in your eye, you offer to trade envelopes with your opponent.
Since she has made the same calculation, she ready agree.
The paradox of this problem is that the rule indicating that one should always trade is [...].
Jackson, F., Menzies, P., & Oppy, G. (1994). paradox of the two envelopes which are greener than each other This means that the first way of doing the calculation involves supposing that for any value of x, if $x is the amount
of money in some particular envelope, it is equally likely that $2x or $0.5x
is the amount in the other envelope.
Chalmers, D.J. 1994 paradox of the broken symmetry (standard version) Now, this reasoning is independent of the actual amount in envelope 1, and in fact can be carried out in advance of opening the envelope; it follows that whatever envelope 1 contains [...]
(This quotation was revised on September 4, 2017.)

Mystery of these standard resolutions

(This paragraph was added on July 17, 2016.)

 I remember the days (2012 or 2013) when I using correct expectation formula tried to prove the equivalence of the two envelopes.
I don't remember whether I succeed or not, but I remember that it was important to me to prove that mean value of the conditional expectation and mean value of the amount of money in the opposite envelope are same.
It is very mysterious for myself because I did not yet know the law of total expectation in those days.
(For the law of total expectation, please see the article "Law of total expectation" of the English language Wikipedia.)
So I have some questions as follows about the standard resolutions as follows.

Is the standard resolution applying Bayesian statistics?

(This paragraph was added on May 31, 2018.)

 I found an article which explains the standard resolution using the word "Bayesian".
But I can not remember famous articles which used the words "Bayesian statistics", "Bayesian probability", "Bayesian theorem" or "Bayes rule".
So I examined the famous articles.

Articles which presented the standard resolution Words
beginning with
"Bayes" 		calculation formula of the conditional expectation
or the conditional probability
presented?
Zabell, S. (1988)
  • Such assumptions go back to Bayes.
  • The Bayesian answer
  • Bayesian analysis
 YES
Nalebuff, Barry.(1989) Nothing YES
Christensen, R; Utts, J (1992),
  • Bayesian resolution
  • Baysian principles
  • Bayesian method
  • …
 YES
Jackson, F., Menzies, P., & Oppy, G. (1994). Nothing NO
Chalmers, D.J. 1994 Nothing YES
Brams, S. J., & Kilgour, D. M. (1995).
  • Bayes' theorem
 YES
Storkey, Amos. (2000-2005)
  • Bayesian interpretation
  • Bayesian perspective
 YES
Devlin, K. (2004).
  • Bayes' theorem
 YES

As a result of this, I think as follows. And I examined some Wikipedia articles which explain the standard resolution.

language of Wikipedia title of the article about the two envelopes problem   revision   Words
beginning with
"Bayes" 		calculation formula of the conditional expectation
or the conditiona probability presented
in the explaianation of the standard resolution?
German Umtauschparadoxon 16:55, 22. Aug. 2016‎ Nothing YES
English Envelope paradox
(How to read it) 		13:49, 14 July 2006 Nothing NO
Two envelops problem 14:56, 28 April 2017‎
  • Bayesian resolutions
  • Baysian interpretation
  • Bayes' rule
  • Bayesian probablity theory
 NO

Calculation formulas appear in the other sections.
Italian Paradosso delle due buste 15:12, 16 apr 2016‎ Nothing NO
Hebrew פרדוקס המעטפות 04:20, 1 במאי 2016‏ Nothing NO
Dutch Enveloppenparadox 13 feb 2014 18:33‎
  • Bayes' rule
 YES
Russian Задача о двух конвертах 05:17, 19 ноября 2016‎ Nothing YES

As a result of this, I got the following impressions.

The theory of assumption of probability distribution is unusual as a standard resolution.

(This paragraph was added on May 31, 2018. Moved here and revised on Julay 15, 2018)

In the period of the standard resolution - Part 1, some literatures presented the following opinion.
The paradox is caused by an assumption about the prior probability distribution that for any chosen amount x, the other amount x/2 and the other amount 2x are equally likely probable.
I call this opinion the theory of assumption of probability distribution.
However the usual standard resolution is based on the theory of the intuitive probability.

These two theories are similar because they state as follows.
But there are big differences.

  The theory of the intuitive probability The theory of assumption of probability distribution
About the cause of fallacy It states that the fallacious expectation formula uses an intuitive probability and the cause of this fallacy is a mistake of treating the prior probability. It states that the fallacious expectation formula assumes an invalid prior probability distribution and the cause of this fallacy is a wrong application of the principle of insufficient reason.
About the correct expectation formula The literatures which presented this theory usually presented the correct expectation formula with the correct conditional probabilities. The literatures which presented this theory often did not present the correct expectation formula.

This theory of assumption of probability distribution is unusual as a standard resolution because of the following reasons.

What is the source of the theory of assumption of probability distribution?

(This paragraph was added on May 31, 2018. Revised on July 15, 2018, July 22, 2018)

My hypothesis about the mind process of the theory of assumption of probability distribution is as follows.
Recognition of sufficient condition
If you assume a probability distribution such that the other amounts are doubled or half of the chosen amount with equal probability, you will get a paradox and the probability distribution must be improper.
  ↓
Opinion of necessary condition
If you got a paradox, you must have assumed an inappropriate probability distribution which derives a paradox.
In other words, I think that the source of the theory of assumption of probability distribution is "Affirming the consequent" (confusion of necessity and sufficiency).

DivideThreeByTwoian's paradox and resolution

The structure of this paragraph was changed on May 14, 2017, August 5, 2017, August 25, 2017, and May 3, 2018.

DivideThreeByTwoian

This paragraph was added on May 3, 2018.

Some people think that thinking of two pairs of amounts of money is the cause of the paradox of the two envelopes.
In other words, they think that it is wrong to simultaneously use the amount x/2 and 2x in the expectation formula. (← Added on July 7, 2019)
And most of them advocate the theory of "E=(1/2)a+(1/2)2a".
So I call them DivideThreeByTwoian because (1/2)a + (1/2)2a = (3/2)a.
However, many of the early DivideThreeByTwoian philosoper did'nt present theory of "E=(1/2)a+(1/2)2a", and they presented various ways to prove notation error of the amounts of money of the fallacious expectation formula instead. (← Addded on February 9, 2020)

I was surprised when I had read DivideThreeByTwoian's opinion

This paragraph was added on August 5, 2017. Moved here on May 5, 2019.

I thought that usual wordings of the two envelopes problem force us to get the DoublePairian's mental model.
In other words the symbols "x/2" and "2x" in the fallacious expectation formula "E=(1/2)(x/2)+(1/2)2x" never be wrong.
Strictly speaking, there may be cases that one of the two terms "(1/2)x/2" and "(1/2)2x" is needless.
Actually there were few mathematicians who doubt the variable symbol x rather than the probability 1/2.
So I was surprised at the opinion that the correct expectation formula is "E=(1/2)a + (1/2)2a".
And I was surprised at all when I found the same opinion in the English language Wikipedia article "Two envelopes problem".

DivideThreeByTwoians had switched the problem to the other problem

This paragraph was added on March 8, 2020.

I think DivideThreeByTwoians had switched the problem as follows.

original problem
<<< The former is omitted. >>>
You choose one envelope at random.
Let x be the amount of money in your envelope, then the amount of money in the other envelope is equally likely x/2 or 2x.
Therefore the expected amount of money in the other envelope is (1/2)(x/2) + (1/2)(2x) = (5/4)x.
<<< The rest is omitted. >>>

switched problem
<<< The former is omitted. >>>
You choose one envelope at random.
Let A be the lesser amount of money, and let x be the amount of money in your envelope, then the amount of money in the other envelope is equally likely x/2 or 2x.
Therefore the expected amount of money in the other envelope is (1/2)(x/2) + (1/2)(2x) = (5/4)x.
<<< The rest is omitted. >>>

DivideNineByEightian's resolution

This paragraph was added on March 24, 2019. The title was changed on May 5, 2019.

Resolution presented for the double coin flipping style problem

This title was added on May 5, 2019.

An article written in 1994 by a philosopher and an article written in 1996 by a mathematician had the following aspects. The problem resolved by this opinion is a mixing of the Ali-Baba version problem and the two envelopes problem.
This resolution looks like the not-three-amounts theory, but not same for the following reasons. (↑ Revised on July 7, 2019)
And in my perception, this resolution did not affect other philosophers that much during the period of "DivideThreeByTwoian's resolution - Part 1".

Remark:
In several revisions of the English Wikipedia article "The two Envelopes problem", the section explaining the inconsistent-variable theory referred to the article by the mathematician above. (← Revised on July 7, 2019)

They might be DivideNineByEightian rather than DivideThreeByTwoian.

(This title was revised on April 21, 2019)

We may have to call their mental model "SingleSeedAmountian's mental model".

The seed amount S is the ruler of their mental model.

deciding
the seed amount S 		S in A
placing
in the envelope B 		S in A
S/2 in B 		S in A
2S in B
handing
a envelope to the player 		S in A
S/2 in B
A is handed 		S in A
S/2 in B
B is handed 		S in A
2S in B
A is handed 		S in A
2S in B
B is handed
the other amount S/2 S 2S S

We may have to call their resolution "DivideNineByEightian's resolution".

The mathematician who wrote a paper about the double coin flipping style problem in 1996 presented an expectation formula "E=(1/2)((1/2)(S/2)+(1/2)S))+(1/2)((1/2)2S+(1/2)S)=1.125S".
Divide nine by eight and you get 1.125.

Why the above mathematician did not think of conditional expectation?

(Added on March 31, 2019)

The complexity of the double coin flipping style wording may have prevent him from escaping the concept of primitive expectation.
As a result of the second coin flip, the same player may be in Ali's position or Baba's position. This finding may have made him think that "the difference from the the Ali-Baba version problem is the key to resolve the paradox". (← Added on April 14, 2019)

I think that if he tried to mathematically get conditional expectation, he would have calculated as follows.
Let t(s) be the probability that the seed amount of money placed in the envelope A is s.
Let g(x) be the probability that the lesser amount of money in the envelope A and B is x.
Let E be the expected value of the amount of money placed in the other envelope.
Then g(x) = (1/4)(t(2x)+t(x/2)).
∴ E = ((2xg(x)+(1/2)xg(x/2)) / (g(x)+g(x/2))
= ((2t(2x)+(1/2)t(x)+2(t(x/2)+(1/2)t(x/4)) x / (t(2x)+t(x)+t(x/2)+t(x/4)).

One more hypothesis about the factor of expecting a kind of magic trick behind the double coin flipping style wording

(This paragraph was added on April 21, 2019. The title was revised on July 7, 2019, August 25, 2019)

I imagine that the double coin flipping style wording has been originated from a mathematical article Sobel, J. H. (1994).
And in that article conditional expected value on uniform probability distributions were discussed on the DoublePairian's mental model.
Therefore, I think whether he wrote the problem wording or read it is a major factor for expecting a kind of magic trick or not. (← Revised on August 25, 2019)

DivideThreeByTwoian's resolution - Part 1

Title was revised on September 2, 2017.

I think that period of DivideThreeByTwoian's resolution - Part 1 began around 1994.

Not-three-amounts Theory
In this period some people thought as follows.
The cause of the paradox is to think of three amounts x/2, x and 2x with fixed value of x, in other words, to calculate expected value on the DoublePairian's mental model.
If you think of only two amounts a and 2a , paradox will vanish.
(↑ Revised on March 31, 2019)
I call this opinion "Not-three-amounts theory".

Not-consistently-interpretable-variable theory
Some of these people presented the following explanation.
Because there are only two amounts of money the variable symbol x in "E=(1/2)(x/2)+(1/2)2x" cannot have same value in the two terms.
I call this explanation the "Not-consistently-interpretable-variable theory".

Theory of non-Ali-Baba version
(This paragrah was added on March 22, 2020)
Some of these people had explained based on difference from the Ali-Baba version.
I call this explanation the "Theory of non-Ali-Baba version".

The Theory of "E=(1/2)a+(1/2)2a"
(Revised on December 1, 2017, December 2, 2017, May 3, 2018)
And some of these people went ahead and said that the correct expectation formula is E=(1/2)a + (1/2)2a.
And they explain their opinion as follows.
The fallacious expectation formula "E = (1/2)2x + (1/2)(x/2)" should be corrected as follows.
Let X be a random variable which denotes the amount of money in the chosen envelope.
Let Y be a random variable which denotes the amount of money in the other envelope.
E(Y| lesser = a)
= P(X is lesser)E(2X | X is lesser ∧ lesser = a) + P(X is greater)E(X/2 | X is greater ∧ lesser = a)
= (1/2)2a + (1/2)a.
I call this opinion the "Theory of "E=(1/2)a+(1/2)2a".

Most of DivideThreeByTwoian advocate this theory.
With that as a hint, I have created the coined word "DivideThreeByTwoian".
However, in the period of DivideThreeByTwoian's resolution - Part 1, many DivideThreeByTwoian philosoper did'nt present theory of "E=(1/2)a+(1/2)2a", and they presented various ways to prove notation error of the amounts of money of the fallacious expectation formula instead. (← Added on February 9, 2020)

Interpretation of the symbol X as a random variable
(Added on September 3, 2017. Revised on September 14, 2017.)
A few years after entering this period some DivideThreeByTwoians who had the following aspects had appeared. I call such people "RandomVariablian" and I think that their logic is a kind of "Affirming the consequent" (confusion of necessity and sufficiency) .
The whole picture of the thinking process by the DivideThreeByTwoian - Part 1 -
(Added on December 2, 2017.)



Examples of the articles written by the DivideThreeByTwoian - Part 1 -
(Added on December 16, 2017. Revised on December 17, 2017.)

Article publised Not-three-amounts Theory presented? Not-consistently-interpretable-variable theory
presented? 		The Theory of "E=(1/2)a+(1/2)2a"
presented? 		Inconsistent-variable theory
presented?
1 1990's yes yes no no
2 1990's yes yes yes no
3 1990's yes yes no no
4 2000's yes no yes no
5 2000's yes no yes no
6 2000's yes no yes no

DivideThreeByTwoian's resolution - Part 2

Title was revised on September 2, 2017.

I think that the period of DivideThreeByTwoian's resolution - Part 2 began around 2005 at the latest.
A philosophical article, published in 2003, presented the inconsistent-variable theory, together with the theory of non-Ali-Baba version. So, the period may have already started in 2003. (← Added on May 12, 2019)

Inconsistent-variable theory
In this period some DivideThreByTwoians interpreted the "Not-consistently-interpretable-variable theory" as follows.
The variable symbol x in "E=(1/2)(x/2)+(1/2)2x" have different values in each of the two terms.
This was the birth of the "Inconsistent-variable theory".
After it many of DivideThreeByTwoians explained as follows.
The cause of the paradox is an inconsistent use of variable symbol in the expectation formula.
E = (1/2)2x + (1/2)(x/2).
!!! Symbol x in the first term and symbol x in the second term denote different values.
This explanation seems to explain the psychological mechanism, but I think it actually does not so. (← Revised on December 23, 2017.)

Difference between the not-three-amounts theory and the inconsistent-variable theory
(Added on March 31, 2019. Revised on April 7, 2019, with new title)
Not-consistently-interpretable-variable theory may be a supplement of not-three-amounts theory.
During the period of "DivideThreeByTwoian's resolution - Part 1", DivideThreeByTwoian philosophers often presented the not-consistently-interpretable-variable theory after presenting the not-three-amounts theory.
This indicates that DivideThreeByTwoian philosophers presented the not-consistently-interpretable-variable theory as a supplement of the not-three-amounts theory.

Hypothesis about the thought of the DivideTreeByTwoian - Part 2 -
(Added on December 2, 2017.)



Examples of the articles written by the DivideThreeByTwoian - Part 2 -
(Added on December 16, 2017.)

The following articles are the early ones I know.

DivideThreeByTwoian's resolution - Part 3

Title was revised on December 3, 2017.

I think that the period of DivideThreeByTwoian's resolution - Part 3 began after 2010 or later.
In this period some DivideThreeByTwoians advocate theory of "E=(1/2)2a+(1/2)a" which is accompanied by no hypothesis about the cause of the fallacy.
In my perception the rate of such DivideThreeByTwoian has grown up year by year.
I think that this trend is spurred by the English Wikipedia article "Two envelopes problem". (← Added on July 7, 2019)
I call such a opinion "Pure DivideThreeByTwoian's opinion".

Hybrid-DivideThreeByTwoian's resolution

This paragraph was added on May 3, 2018.

Some people are DivideThreeByTwoian on closed version problem and is standard resolver on the opened version problem.
I call such a people "Hybrid-DivideThreeByTwoian".

Their mind is very mysterious because of the following reasons.
The following example is the earliest one I know. (↑ Added on March 17, 2019)

DivideThreeByTwoian's resolutions may be wrong

On September 15, 2019, this paragraph was separated from the pargraph "DivideThreeByTwoian's paradox and resolution".

DivideThreeByTwoian's opinion seems logically wrong.

The paragraph was added on December 21, 2017. Revised on December 31, 2017, April 19, 2018, May 17, 2018.

The paragraph "Deviation from the desirable argument" was moved into the section "DivideThreeByTwoian's mind is very mysterious" with new title, on February 16, 2020.

Contradiction to the theory of information

(This paragraph was added on December 9, 2018.)

About the amount of money which the other envelope contains, many DivideThreeByTwoians think as follows. Such a thought contradicts to the theory of information, because of the following reason.
After opening, we get more information than before opening.
Therefore, possible states are reduced after opening, and the above DivideThreeByTwoian's thought contradicts this.

Wrong usage of the trendy philosophy

(This paragraph was added on March 3, 2019.)

DivideThreeByTwoain phirosophers often used the following concepts of the trendy philosophy to rationalise their opinion. They used these concept to prove the wrongness of the simultaneous use of "x/2" and "2x" in the expectation formula.
However it was logically wrong.
Because if they had thought logically, they would prove the above wrongness before using these trendy concepts.

DivideThreeByTwoian's opinion looks like made by illogical reasoning such as "Tautology".

This paragraph was added on July 8, 2017.
Moved here with new titie and revised contents on March 17, 2019.
The title was changed on April, 2019.

The not-three-amounts theory looks like "Tautology".

(This paragraph was added on April 7, 2019)
The not-three-amounts theory may be tautology, because it proves the legitimacy of the SinglePairian's mental model using the same mental model.

The theory of "E=(1/2)2a + (1/2)a" looks like "Affirming the consequent".

No DivideThreeByTwoian who advocated the theory of "E=(1/2)2a + (1/2)a" presented psychological evidence.
This fact suggests that this theory is a fallacy named "Affirming the consequent".
In other words the fact that this expectation formula does not derive paradox seems to have made DivideThreeByTwoians mistake it the answer of the two envelopes problem.

The theory of non-Ali-Baba version looks like "Denying the antecedent".

(This paragraph was added on March 17, 2019)
In the period of DivideThreeByTwoian's resolution - Part 1, many of them claimed the theory of non-Ali-Baba version.
However, the fact the problem is not "Ali-Baba" version does not inplyes that the problem is should be solved on the SinglePairian's mental model.
This means that their logic is "denying the antecedent".

The inconsistent-variable theory may be psychologically doubtful rather than looks like "Tautology".

(This paragraph was added on March 17, 2019, and revised on April 7, 2019)
In the period of DivideThreeByTwoian's resolution - Part 2, they claimed that the inconsistent use of variable symbol the cause of the paradox.
This claim is looks like "Tautology", because expectation formula on the SinglePairian's mental model should be "E=(1/2)A+(1/2)2A".
However, before claiming it they should prove that there is image of just one pair of amounts in our mind when we feel two envelope paradox.

The paragraph "The inconsistent-variable theory looks like Affirming the consequent" was deleted on March 17, 2019.

You can make counter opinions that have same style as their opinions.

This paragraph was added on February 9, 2020.

For the not-consistently-interpretable-variable theory

Not-consistently-interpretable lesser amount theory
If we let A and 2A denote the lesser amount and the greater amount respectively, and think of the case the chosen amount of money is x and the other is x/2, you will find that A means x/2 and the pair of amounts is (x/2, x).
But this interpretation does not match the case the chosen amount of money is x and the other is 2x.

For the theory of non-Ali-Baba version

Theory of non-restrictive wording
The standard two envelopes problem does not have the very restrictive wording.
Therefore, no one is allowed to think of only one money pair.

For the inconsistent-variable theory

Inconsistent-lesser-amount theory
If we let A denote the lesser amount, the symbol "A" will be inconsistent as follows.
If the chosen amount of money is the greater, it is 2A and A means x/2. And if the chosen amount of money is the lesser, it is A and A means x.


DivideThreeByTwoian's opinion is mathematically wrong and psychologicaly doubtful.

The paragraph "Their opinion is composed of some wrong or doubtful hypotheses" was replaced by this paragraph on August 31, 2017.

Not-three-amounts theory

Mathematically wrong

We can (must) think of two pairs of amounts of money in the opened version problem.
Mathematically we can apply same logic to the closed version problem.
Therefore mathematically the "Not-three-amounts theory" has not mathematical inevitability.
So if you want to advocate such a theory, you should show a psychological inevitability.
(↑ Revised on June 24, 2018)

Psychologically doubtful

If we have claimed that thinking three amounts of money is the cause of the paradox, we simultaneously have revealed our following weak point.
Some of us disliked to think of three amounts of money (x/2, x, 2x) after arrangement of money in only two envelopes.
And some of them occasionally thought of these three amounts of money and did not notice it until they found a paradox.
It is very doubtful.

This doubtfulness was strengthened by the fact that they made psychologically incompatible explanations. (← Revised on March 22, 2020)

Some early DivideThreByTwoian philosophers had explained using the theory of non-Ali-Baba version.
(This paragaph was revised on March 22, 2020)
Remark!! This explanation is mathematically wrong, but explains psychological mechanism. (← Revised on January 8, 2019, March 22, 2020)

A box explaining the mystery of this theory has been added here on November 3, 2017. However, it was deleted on July 7, 2019.

On March 22, 2020, the paragraph "Some DivideThreByTwoians had explained based on the number of envelopes." was deleted.

Some DivideThreByTwoians had explained based on number of games.
(This paragraph was revised on February 9, 2020, with new title)
Remark!! This explanation is mathematically wrong, but explains psychological mechanism.

I would like to call this explanation "Theory of multiple games".


The theory of non-Ali-Baba version and the theory of multiple games are psychologically incompatible. (← Revised on January 8, 2019, February 9, 2020)

The theory of "E=(1/2)2a+(1/2)a"

Mathematically wrong

Their opinion is as follows.
The two envelopes are equivalent.
Therefore if an expectation formula is correct it does not suggest swapping the envelopes nor suggest not swapping the envelopes. 
The expectation formula "E=(1/2)2a + (1/2)a" is nothing but the correct formula.
But if it is allowed to think of conditional expected value before opening then it is easy to find a case that switching is favorable to the player.
If it is not allowed then the correct calculation is "E(Y) = (1/2)2E(X|X<Y) + (1/2)(1/2)E(X|X>Y)", not "E=(1/2)2a + (1/2)a".
Therefore in either case the theory of "E=(1/2)2a+(1/2)a" is mathematically wrong.
So if you want to advocate such a theory, you should show a psychological inevitability.
(↑ Revised on June 24, 2018)

Psychologically doubtful

I think that we usually think like below about the two envelopes problem. So I anticipate that many of us refuse the theory of "E=(1/2)2a+(1/2)a".
And as a result of investigation of many articles about the two envelopes problem I am convinced of it.

people who calculate on their own way
(Added on October 7, 2018)

About the theory of "E=(1/2)2a+(1/2)a", I think that there may be some people who did as follows. I would like to call them "people who calculate on their own way".
Some of the DivideThreeByTwoians seemed not have experienced the two envelope paradox in own mind.
I think that some of them belong the type of "people who calculate on their own way".

Inconsistent-variable theory

Mathematically wrong

We can resolve the paradox with three amounts of money x/2, x and 2x.
Therefore mathematically the "Inconsistent-variable theory" has not mathematical inevitability.
So if you want to advocate such a theory, you should show a psychological inevitability.
(↑ Revised on June 24, 2018)

Psychologically doubtful

If we have claimed that the wrong use of variable symbol is the cause of the paradox, we simultaneously have revealed our following weak point.
Some of us occasionally used variable symbol inconsistently and did not notice it until they found a paradox.
It is very doubtful.

Interpretation of the symbol X as a random variable

(Added on September 3, 2017.)

Mathematically wrong

Based on such an interpretation they claim that the two terms "(1/2)(X/2)" and "(1/2)2X" cannot have meaning simultaneously.
But we can make a kind of the standard resolution based on such an interpretation.
Let random variables X and Y denote the amount of money in the chosen envelope and the amount of money in the other envelope respectively.
And let Q(x) = P(X=x and the chosen envelope contains the lesser amount) / P(X=x).
Then E(Y|X)=Q(X)(X/2)+(1-Q(X))2X.
Therefore mathematically such an interpretation has not mathematical inevitability.
So if you want to advocate such an opinion, you should show a psychological inevitability.
(↑ Revised on June 24, 2018)

Psychologically doubtful

Mathematically we don't have to interpret the symbol X as a random variable.
So they might have been misled into such an idea by the "Not-consistently-interpretable-variable theory".

DivideThreeByTwoian's mind is very mysterious.

This paragraph was added on January 4, 2017. And the paragraph "They pretend to be SinglePairians" was deleted.
This paragraph was revised on January 19, 2017, May 14, 2017, May 23, 2017. The title was changed on August 5, 2017.

I can not believe that there is such a paradox.

(This paragraph was added on May 31, 2018. The contents was greatly revised on May 5, 2019)

I cannot imagine the scene that I am experiencing the paradox which DivideThreeByTwoians resolved.
And I think that usual people cannot doubt "x/2" and "2x" in the formula "E=(1/2)(x/2)+(1/2)2x". (← Added on July 7, 2019)
Therefore, I can't stop imagining that DivideThreeByTwoians resolved another paradox which they have not felt.

Should we call their paradox "minor paradox"?

(This paragraph was added on February 16, 2020)

To my eyes, about half those who read the DivideTreeByTwoian's opinion rejected it. Therefore, if DivideThreeByTwoian's problem is a kind of paradox, it should be called "minor paradox".

In contrast, in the Monty Hall problem, most people consider it a paradox of probability.

Their opinions look like logically confused.

(On 5 May 2019, this paragraph was created by collecting a summary of several paragraphs.)
(On February 16, the title and some content were revised)

Their opinion violates cognitive psychology.

(Added on July 16, 2017. Revised on August 12, 2017, August 13, 2017.)

The wording of the two envelopes problem has the following sequence. My knowledge of cognitive psychology tells me like below.
When we read the wording of the problem, the image of amounts of money are more vivid than the probability because the notation of amounts of money have been presented just before the expectation formula.
Therefore we should doubt the probability when we recognize the existence of a paradox because we surely made mistake about the probability rather than about the amounts of money.
On February 16, 2020, a table here was removed.
My knowledge of theories of cognitive development tells me like below.
In child age we learn the concept of probability after learning the concept of quantity.
Therefore in our mind mental object of probability is less vivid than mental object of quantity.
This means that we will make mistake about probability rather than about amount of money.

The following facts suggest that my thought is reliable. Therefore DivideThreebytwoian's opinion is mysterious.

The paragraph "To begin with does the DivideThreeByTwoian's paradox exist?" was moved from here on January 8, 2018.

It seems that they thought it was not important to correct the fallacious expectation formula.

On February 16, 2020, this paragraph was moved from the section "DivideThreeByTwoian's opinion seems logically wrong" and was changed the title.

The fallacious argument which derives paradoxes

The paradox of the two envelopes problem was derived from the following argument.
Let x denote the amount of money of the chosen envelope.
Then the expected amount of money of the other envelope is greater than x.

The desirable arguments

If you are a DoublePairian
your desirable argument may be as follows.
Let x denote the amount of money of the chosen envelope.
Then the expected amount of money of the other envelope is not necessarily greater than x.

If you are a SinglePairian
your desirable argument may be as follows.
(↓ Revised on May 5, 2019, August 4, 2019)
Let X and Y denote the random variable of the amount of money of the chosen envelope and the other envelope respectively.
And let the sample space is composed of the following two kinds of sample.
  kind of sample 1 : X=a and Y = 2a.
  kind of sample 2 : X=2a and Y = a.
Then the expectation of Y depends on X as follows.
  If X = a, then E(Y) = 1 × 2X + 0 × X/2.
  If X = 2a, then E(Y) = 0 × 2X + 1 × X/2.
Integrating this derives as follows.
  E(Y) = (1/2)E(IX=a2X + IX=2a(X/2)).
The following can be gotten from this.
  E(Y)
  = (1/2)E(IX=a2X + IX=2a(X/2))
  = (1/2)E(IX=2aX + IX=aX)
  = E(X).

If you are an anti-DoublePairian
your desirable argument may be as follows.
The cause of the fallacious expectation formula is to use the values x/2 and 2x.
If you don't use these values no paradox happens.
One of such a solution is not to think of any expectation calculation.
Before you open envelope you don't need any expectation calculation.
There is no logical need to think of one pair of amounts of money to avoid the paradox.

If you don't like to think of conditional expected value on the imagined amount of money
your desirable argument may be as follows. (← Added on February 23, 2018.)
It is inappropriate to calculate conditional expected value before getting information by opening the envelope.
Therefore you should not calculate expected value which is conditioned on two pairs of amounts of money.
And you also should not calculate expected value which is conditioned on one pair of amounts of money.
Or your desirable argument also may be as follows. (← Added on June 24, 2018.)
It is inappropriate to calculate conditional expected value before getting information by opening the envelope.
Therefore if you want a calculation formula including both of the case that the chosen amount is the lesser and the case that the chosen amount is the greater, you should calculate as follows.
Let X and Y denote the random variable of the amount of money of the chosen envelope and the other envelope respectively.
Then
E(Y) = (1/2)2E(X| X is the lesser) + (1/2)(1/2)E(X| X is the greater).
Because E(X| X is the lesser) = (2/3)E(X) and E(X| X is the greater) = (4/3)E(X), you get E(Y) = E(X).
Therefore the paradox has been resolved.

If you don't like to to calculate expected value when the probability distribution and the amount of money in the chosen envelope are both unknown
(↑ Added on March 17, 2019.)
your desirable argument may be as follows.
No information, no expectation, no paradox.

DivideThreeByTwoian's argument differs from any of the above

(Revised on February 16, 2020)
(The title was revised on August 18, 2020)
During the period of "DivideThreeByTwoian's resolution - Part 1", DivideThreeByTwoians presented various logics such as "not-three-amounts Theory" to reject the usage of "x/2" and "2x" in the expectation formula.
However, many of them did not present a correct expectation formula.

Any way

First of all, why they reject to think of two pairs of amounts of money only with sensuous reason?
Any way, to my eyes their resolution is very mysterious.

Did the two envelopes problem discussed by DivideThreeByTwoians have special wording?

(Title was revised on August 14, 2017, May 3, 2018, September 1, 2019)

The wordings of the problem they had adopted had special features.

(↑ This header was added August 6, 2017. Revised on August 14, 2017.)

I have checked the following 4 articles which were written by DivideThreeByTwoians.
(They were cited by the article "Two envelopes problem" revision "22:05, 3 October 2005" in the English language Wikipedia.)

An article published in 1997. Cited by 23 articles on June 16, 2017 (Google Scholar).
the process how money is placed in envelopes no description
opportunity to trade before opening
notation of the amount of the lesser money no description
notation of the amount of money in the chosen envelope a certain amount of money x
explanation of the existence of a paradox So I should swap.
independence from the chosen amount no description
numbered steps used
how the probability 1/2 is combined with amounts of money 'the chosen amount'
→ 'probability'
→ 'the other amounts'
→ 'expectation'
special notes none

An article published in 2001. Cited by 1 article on June 16, 2017 (Google Scholar).
the process how money is placed in envelopes Virtually
no discription
opportunity to trade before opening
notation of the amount of the lesser money z
notation of the amount of money in the chosen envelope x
explanation of the existence of a paradox x = 1.25y and y = 1.25;x cannot both be true.
independence from the chosen amount no description
numbered steps not used
how the probability 1/2 is combined with amounts of money 'the chosen amount'
→ 'the other amounts'
→ 'probability'
→ 'expectation'
special notes Reasonable expectation 1.5z was presented before the fallacious expectation 1.25x.

An article published in 2003. Cited by 15 articles on June 16, 2017 (Google Scholar).
the process how money is placed in envelopes no description
opportunity to trade before opening
notation of the amount of the lesser money n
notation of the amount of money in the chosen envelope x
explanation of the existence of a paradox Reasonable expectation (3/2)n
→ not either.
fallacious expectation (5/4)x
→ switching.
independence from the chosen amount no description
numbered steps not used
how the probability 1/2 is combined with amounts of money 'the chosen amount'
→ 'the other amounts with probability'
→ 'expectation'
special notes 1 The problem was defined as the conflict of expectation formula based on SinglePairian's mental model and it based on DoublePairian's mental model.(Added on July 14, 2019)
special notes 2 It was described that the fallacious expectation is reasonable in the "Ali-Baba version" problem.

An article published on web.
the process how money is placed in envelopes no description
opportunity to trade before opening
notation of the amount of the lesser money none
notation of the amount of money in the chosen envelope X
explanation of the existence of a paradox You should switch and you should switch back.
independence from the chosen amount no description
numbered steps not used
how the probability 1/2 is combined with amounts of money 'the chosen amount'
→ 'the other amounts with probability'
→ 'expectation'
special notes none

I found the following special features of their wordings.

Verification of the above findings

I got the following result after examining some other papers and some other web pages which advocate the theory of "E=(1/2)2a+(1/2)a" or the "Not-three-amounts theory".

(↓ Revised on July 14, 2019, July 21, 2019)
year opportunity to trade description of independence from the chosen amount wording suitable for DivideThree-
ByTwoian's opinion 		remark
1994 before opening no description The wording of the problem is double coin flipping style.
Presented resolution is similar to the theory of non-Ali-Baba version.
(↑ Added on July 21, 2019) 		 
1994 before opening no description In the wording of the problem, the MeanRateOfExchangean's paradox is presented.
(However, presented resolution is mean value version of the inconsistent-variable theory.)
(↑ Revised on July 21, 2019) 		not philosopher
2008 after opening "irrespective of the observed value" At the beginning one pair of amounts of money is presented as (θ, 2θ).
(Very special wording) 		not philosopher
2010 before opening no description At the beginning one pair of amounts of money is presented as ($100, $200).
(Very special wording)
↑ Added on August 19, 2017. 		philosopher

I think that this table roughly supports the above findings.

Comparison with non-DivideThreeByTwoian philosophers

(This paragraph was added on July 14, 2019. Revised on July 21, 2019

I got the following result after examining some other articles writen by no-DivideThreeByTwoian philosophers.

year opportunity to trade description of independence from the chosen amount wording suitable for DivideThree-
ByTwoian's opinion 		remark
1992 umbiguous probably no description nothing? Utility is calculated distinguishing the two envelopes given different names like E1 and E2.
1994 before opening "for any value of x" nothing  
1994 before opening
and
after opening 		no description nothing It is claimed that it is not paradoxical before opening.
1994 after opening "independent of the amount of envelope 1" nothing  
1997 before opening "no
matter what
x
is" 		nothing  
1997 before opening "whatever amount r is" nothing  
2000 umbiguous no description nothing  

This table may be supporting the following hypothesis.
DivideThreeByTwoians saw a mirage through wordings suitable for DivideThreeByTwoian's opinion.

Anyway they avoid thinking of the paradox derived from opened version.

(On August 6, 2017, this paragraph was added.)

Almost all standard resolvers do not think of the paradox derived from closed version problem.
There are only a few exceptions. (Jackson, F., Menzies, P., & Oppy, G. (1994). is one of them.)

All DivideThreeByTwoians do not think of the paradox derived from opened version problem.

These facts strongly suggest as follows.

Two envelopes problem for DivideThreeByTwoians and two envelopes problem for the standard resolvers are different things.

True face of DivideThreeByTwoian

This paragraph was added on November 8, 2017.

During the period of "DivideThreeByTwoian's resolution - Part 1", true face of DivideThreeByTwoians might has been DoublePairian.

This paragraph was added on May 14, 2017. Title was revised on November 6, 2017, May 5, 2019.
Contents was greatly revised on July 21, 2019.

Two hypotheses about the true face of DivideThreeByTwoians

(The title was revised on January 26, 2020)

Hypothesis that the true face of DivideThreeByTwoian is DoublePairian
Shortly after the moment when DivideThreeByTwoians read that the amount of money in the chosen envelope is denoted by a variable symbol, they were DoublePairian, in other words they imagined two pairs of amounts of money.
And when reading the paradox, they themselves felt paradoxical feeling.

Hypothesis that DivideThreeByTwoian has failed to change the mental model
When reading the rurle of the game, DivideThreeByTwoians were SinglePairian, in other words they imagined only one pair of amounts of money.
And when reading the fallacious expectation formula, they failed to change mental model to the DoublePairian's mental model. And they felt incongruity about the expression of the other amount using the variable symbol x wich denotes the chosen amount.
And when reading the paradox, they wanted to blow out that incongruity rather than the paradoxical feeling.
I call such a people "hard-headed SinglePairian". (← Added on February 9, 2020)

Which hypothesis is more credible?

(This paragraph was moved here with revised title on January 26, 2020)

I think that the following hints more likely suggest the hypothesis that the true face of DivideThreeByTwoian is DoublePairian.

Hints

After study of the early articles by DivideThreeByTwoians I noticed the following facts.
Recently in 2018, I found the following. (↑ Added on April 19, 2018. Revised on July 1, 2018)

Recently in 2019, I noticed the following. (↑ Added on July 21, 2019)

If I am a hard-headed SinglePairian: (↑ Added on February 9, 2020)

During the period of "DivideThreeByTwoian's resolution - Part 2", true face of some DivideThreeByTwoians might have been SinglePairian.

This paragraph was added on November 8, 2017. The title was revised on July 1, 2018, may 5, 2019, March 1, 2020.

After study of the articles written in the period of DivideThreeByTwoian's resolution - Part 2, I noticed the following facts. These facts suggest the following hypothesis.
They were influenced by existing solutions before examining the problem themselves.
As a result, they became SinglePairian without becoming DoublePairian.
For more detais of this hypothesis, please see the paragraph "Hypothesis that they had read the solution before they read the problem".

Some concrete evidences that resolution by DivideThreeByTwoian is wrong

This paragraph was added on July 18, 2016. Title was changed on May 4, 2017, June 9, 2017. Reised on June 25, 2017. Title was revised on August 5, 2017, September 11, 2017.

On Novenmber 6, 2017, the paragraph "The proof of their correctness is very easy but nobody of them have done." was deleted.

The problem which they solved is highly probably different from the problem they understood.

This paragraph was added on April 21, 2017. Revised on May 4, 2017, May 14, 2017. The title was revised on May 25, 2017, June 10, 2017, July 13, 2017, January 8, 2018.

As discussed in the above paragraph "True face of early DivideThreeByTwoians might be DoublePairian", the early articles by DivideThreeByTwoians suggest the following hypothesis.
Shortly after the moment when DivideThreeByTwoian read that the amount of money in the chosen envelope is denoted by a variable symbol, they imagined two pairs of amounts of money.

From that, we can conclude that DivideThreeByTwoians solved the another problem rather than the problem they had read, and their resolution is fake. (← Revised on July 7, 2019)
(For more detail please see "My hypotheses about the mind of DivideThreeByTwoians".)

On August 5, 2017, paragraph "Explanations by themselves of their opinion is very doubtful" was deleted.

DivideThreeByTowians made contradictory theories.

This paragraph were moved to here on August 5, 2017. Revised on August 31, 2017, October 26, 2017.

DivideThreeByTwoians made the following two theories to explain the mechanism of the cause of the paradox. But these two theories are both doubtful and contradictory. From this contradiction these theories seem both fictitious.

I have never seen a similar explanation for the smullyan's paradox.

This paragraph was added on August 29, 2017. Revised on September 2, 2018.

If DivideThreeByTwoian's opinion is correct for some people somebody would present one of the following explanations for the argument 1 in the smullyan's paradox. But I have never seen such explanations.

Some DivideThreeByTwoians made conflicting explanations for the "Not-three-amounts theory".

This paragraph was added on September 6, 2017.
On January 10, 2018, this paragraph was revised and the title was changed.

Many early DivideThreeByTwoians presented explanation for the "Not-three-amounts theory" as below. But some of them presented the following explanation simultaneously. If we have gotten the former explanation we could not get the latter. Because the symbol x was used consistently.
And if we have gotten the latter explanation we could not get the former. Because we do not need to explain the premise that there was only one pair of amounts of money.
This finding suggests that their explanations did not explain their own real experience but explained a fiction.

Some DivideThreeByTwoians evaluated the inconsistent-variable theory inappropriately.

This paragraph was added on December 20, 2017. Revised on February 13, 2018)

Some DivideThreeByTwoians evaluated inconsistent-variable theory as below. But if I have experienced the inconsistent use of variable symbol in my mind I will say that the inconsistent-variable theory just describes my mind.
Hence the word "acceptable" is inappropriate. (← Added on July 8, 2018)
And I will say that it is enough to just look at my mind to find the inconsistent variable in my mind.
Hence the word "harder" is inappropriate. (← Added on July 8, 2018)

Other evidences which indicate that DivideThreeByTwoian's opinion is wrong

This paragraph was created on August 5, 2017, merging two paragraphs.
Revised on September 6, 2017, January 9, 2018.
The pararaph "Fundamental questions about the mind of DivideThreeByTwoians" was deleted on May 31. 2018.

The pararaph "First of all, why did they want to resolve the closed version problem?" was deleted on January 8, 2018.

The DivideThreeByTwoian's paradox may be a mirage

(Added on December 2, 2017. Moved to here on January 8, 2018.)
(Revised on December 21, 2017, March 3, 2019)
(On July 14, 2019, the title was changed from "To begin with does the DivideThreeByTwoian's paradox exist?".)

Does the closed version problem has no significance ?

The title was changed on September 8, 2019.

On December 2, 2017, I read the following comment about the closed version problem.
Because selecting envelope gives no information so switching does not become advantageous.
And I completely agreed.
But this experience gave me the following doubt about the existence of the paradox which is derived from the closed version problem.
If I want to reject thinking of expected value conditional on the amount contained in the unopened chosen envelope I will say "No information, no expectation, no paradox".   Why don't DivideThreeByTwoian say so?

DivideThreeByTwoian's paradox may be a kind of rumor

This paragraph was added on May 19, 2019.

The closed version problem can be traced back to the article Cargile, J. (1992) which presented the ambiguous version problem.
If the article presented the opened version problem, there may not have been the closed version problem and the DivideThreeByTwoian's paradox.

Is the DivideThreeByTwoian's paradox a mirage ?

On March 3, 2019, I got an idea that the DivideThreeByTwoian's paradox is similar to a mirage, for the following reasons. If this my idea is correct, the DivideThreeByTwoian's paradox exists only as a mirage.

↑ Added on May 19, 2019. Revised on July 14, 2019.

Actually, the followings suggest this my idea.
(↓ Revised on September 1, 2019)
In contrast, the The mathematically standard paradox (Breaking the law of total expectation) is real, despite the weakness of the paradoxical feeling.

DivideThreeByTwoian philosophers have not resolved an existing paradox.
The resolution made by them has created a mirage of a non-existing paradox.
(↑ Added on July 7, 2019)

Even if their opinion is correct it is not worth publishing, or it should be presented as minor opinion.

This paragraph was added on July 24, 2016, and was rvised on December 30, 2016, June 25, 2017.

The paragraph "Their opinion is not a full-fledged solution" was deleted on May 31, 2018.

People who advocate DivideThreeByTwoian's opinion are the minority rather than the majority.

(Title was revised on September 20, 2017.)

To my eyes they are the minority rather than the majority.

Most important thing

(Added on May 14, 2017. Revised on January 8, 2018.)
The opinion of DivideThreeByTwoina is correct only for the minority who have simultaneously the following properties.

On May 5, 2019, the paragraph "If there is standard assumption for the two envelopes problem, DivideThreeByTwoian's opinion is not appropriate for it." was deleted.
On August 13, 2017, the paragraph "If they have solved a problem then they have not resolved any paradox" was deleted.
On May 28, 2017, the following paragraphs were deleted.
"Some people simultaneously advocate both theories"
"Inconsistent-variable theory seems to have been a byproduct of the Not-three-amounts theory"

The "Inconsistent-variable theory" might be a ghost.

This paragraph was added on June 18, 2016, revised and moved here on january 12, 2017 and revised on January 14, 2017, July 2, 2017, July 18.2017, March 24, 2019.

Why I studied the "Inconsistent-variable theory"

The following facts had let me think that there might be some people we could apply the "Inconsistent-variable theory".
So I had studied for several years to ascertain whether some people were caught by inconsistent variable and felt a paradox.

How have I found that the "Inconsistent-variable theory" might be a ghost

Doubt about the independency of the opinion
The following findings let me doubt about the independency of the opinion that inconsistent use of variable symbol is the cause of the paradox.
No explanation of the mental mechanism
The following finding let me recognise that the "Inconsistent-variable theory" does not explain mental mechanism of the inconsistent usage of the variable symbol. (↑ Added on July 2,2017)

Kind of rumor
The following finding let me recognise that the "Inconsistent-variable theory" is only a kind of rumor.
A philosopher presented an opinion very similar to the inconsistent-variable theory on the double coin flipping style problem
(This paragraph was added on July 28, 2019)
An article written by a philosopher in the early 1990s presented an opinion very similar to the inconsistent-variable theory.
If I rewrite it as I have understood, it is as follows.
If the envelope B has been handed, the first x in "(1/2)(x/2) + (1/2)2x" denotes 2s and the second x denotes s/2.
However this opinion had the following features. In the mind of the philosopher, the inconsistency of a designator may has been a fallacy evoked by the expectation formula, rather than the cause of the formula.

Conclusion
The "Inconsistent-variable theory" might be unnecessary addition to the "Not-three-amounts theory". (← Revised on march 31, 2019, and undid on April 7, 2019)
I might have studied a ghost for several years.

How the "Inconsistent-variable theory" appeared

(Title was changed on September 1, 2017.)

I have a hypothesis about the birth of the "Inconsistent-variable theory".

Wording of the problem in the English Wikipedia article "Two envelopes problem"

(Added on August 19, 2017. Title was revised on September 1, 2017.)
(Title and contents were revised on April 7, 2019.)

The wording of the two envelopes problem presented in the English language Wikipedia article "Two envelopes problem" at 20:54, 9 October 2005 was as follows, and it seems to have some power to invite us to the "Inconsistent-variable theory".
Basic setup: You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are given the chance to take the other envelope instead.

The switching argument: Now suppose you reason as follows:

  1. I denote by A the amount in my selected envelope.
  2. The probability that A is the smaller amount is 1/2, and that it is the larger amount is also 1/2.
  3. The other envelope may contain either 2A or A/2.
  4. If A is the smaller amount, then the other envelope contains 2A.
  5. If A is the larger amount, then the other envelope contains A/2.
  6. Thus the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2.
  7. So the expected value of the money in the other envelope is:
    (1/2)(2A) + (1/2)(A/2) = (5/4)A.
  8. <<< The rest is omitted. >>>

To my eyes this wording has unique feature as follows.

It uses phrases "the smaller amount" and "the larger amount" which have definite article "the".
This derives the following ambiguity.
  • If we have a vivid image of one pair of amounts of money before reading these phrases, we will become SinglePairian. And we will easily accept the "Inconsistent-variable theory".
  • If we do not have a vivid image of one pair of amounts of money before reading these phrases we will become either DoublePairian or SinglePairian. (← Revised on April 14, 2019

Similar phrases "the higher amount" and "the lower amount" were used in the article McGrew, T. J., Shier, D., & Silverstein, H. S. (1997), but simultaneously the following phrase was used in it.
The envelope I have selected contains a certain positive amount of money; call it x.
I think that nobody easily become SinglePairian after reading it.
This indicates that the wording used in the English language Wikipedia article "Two envelopes problem" is very unique.

After then, have DivideThreeByTwoian philosophers advocated the inconsistent-variable theory?

(This paragraph was added on September 1, 2019)

I searched philosophical paper published after 2007 and not claiming the mathematical standard resolution, and I found the following papers.
In this way, I could not foind philosopher who was a true-InconsistentVariablian.
From this, I think that the English language Wikipedia article advocating the inconsistent variable theory have not much influenced philosophers.

Recent trend supports my impression that the inconsistent-variable theory might be a ghost

(Title was changed on December 4, 2017, and was revised on April 7, 2019)

Im my perception the rate of the pure DivideThreeByTwoian's opinion is becoming larger than the rate of the inconsistent-variable theory year by year.
I think it suggests that inconsistent-variable theory might be a ghost.

Various opinions DivieThreeByTwoians have about the opened version problem

This section was added on April 14, 2019.

DivideThreeByTwoins usually did not discuss about the opened version problem.
Example:
But there are exceptions.

DivideThreeByTwoian's mind

On September 15, 2019, this paragraph was separated from the pargraph "DivideThreeByTwoian's paradox and resolution".

Usage of the word "rigid designator" by philosopher

This paragraph was added on March 31, 2019.

I examined the usage of the phrase "rigid designator".

Example articles
No. Year of publication Style of problem What it claimed
as the cause
of the paradox 		usage of "possible worlds" usage of "rigid designator" Not-consistently-interpretable-variable theory?
Inconsistent-variable theory?
(Added on March 8, 2020)
1 Early 1990s double coin flipping style Regarding the designation 'the contents of the handed envelope' as rigid in the case the envelope B has been handed. In the case the envelope B has been handed, if the contents of B is fixed, the contents of A must varies across possible worlds against the fact. If the expected contents of A is calculated as '$4= (1/2)((1/2)(contents of B) + (1/2)2(contents of B), '$4' is rigid designator, but 'contents of B' is not.
 In the case the envelope B has been handed, the first 'your amount' denotes 2×$4, and the second 'your amount' denotes $4/2.
(Somewhat alike to the inconsistent variable theory)
2 1990s usual Calculation using fixed amount of money in the selected envelope. If the amounts are regarded as fixed, the two possibilities must be regarded as different possible worlds to which probability principle isn't applied. Not used. x' cannot represent simultaneously the lower amount and the higher amount.
(Not-consistently-interpretable-variable theory)
3 Early 2000s
(This row was added on May 19, 2019) 		It seems that the pair of amounts is restricted to only one. Calculation using the description "the amount envelope A contains".
If it gives correct value for each possibility, then the calculation is illegitimate.
If it has same value throughout the possibilities, then there is not a symmetry between the choices.
 Using "possible world" along with "possibility" or "possible state" Not used. We cannot use a variable which has same value throughout the two possible worlds.
(Not-consistently-interpretable-variable theory)
4 Early 2000s usual Giving the Ali-Baba version-specific reasoning for the two envelopes problem I found an expression like "possible worlds" as follows.
--------
The referenced by a designation of a variable must not vary across different possibilities. 		If a rigid designator 'x' denotes the your amount in the first possibility, then 'x' cannot denote the your amount in the second possibility. If 'x' is an abbreviation of "the your amount", then 'x' denotes different values in each possibility.
(↑ Revised on May 19, 2019)
 If 'x' is a rigid designator and the first 'x' denotes the amount of money in your envelope, the second 'x' does not designate correct value.
(Not-consistently-interpretable-variable theory)

If 'x' is not a rigid designator and the two 'x' denote different values, the expectation formula is nonsense.
(Somewhat alike to the inconsistent variable theory)

From the above table, I think there may be somewhat difference depending the style of problem in the usage of "rigid designator".

Magical powers of the closed version problem

The following paragraphs was moved under this paragraph on December 9, 2017.
The title of this paragraph was chaged on February 16, 2018.

Does "closed version" problem have a magical power to give us the illusion of objective equivalence?

This paragraph was added on January 6, 2018.

The following facts suggest that closed version problem have a psychological power to give us the illusion of objective equivalence.
The image of envelopes is more vivid than the image of money in them because the former is imagined before the later.
I think that this mechanism is the essence of the magical power of the closed version problem.
(↑ Added on March 22, 2018.)

Does "closed version" problem have a magical power to change us into SinglePairian?

This paragraph was added on May 3, 2017. Title was changed on October 3, 2017.

 The following facts suggest that closed version problem have a psychological power to make us not to think of two pairs of amounts of money. I have the following hypothesis which relates to this theme.
On April 12, 2020, the paragraph "Does the principle of insufficient reason on the "closed version" problem have a magical power?" was deleted.

Does "closed version" problem have a magical power to change the subject of the problem?

This paragraph was added on January 6, 2018, revised on January 16, 2018 and April 5, 2018.

 The following facts suggest that the closed version problem has psychological power to make us to think decision as subject matter of the problem and think expectation as the means for it.

Does "closed version" problem have a magical power to change the order of vividity of mental objects?

This paragraph was added on May 17, 2018.

On May 12, 2018, I got the following hypothesis.
The mental images of envelopes, the image of amount of money and et cetera have rank of vividity.
The order of them differs depending on the opened version problem or closed version problem.
For example, like this.
rank opened version closed version
1 the chosen amount
x 		the pair of envelopes
2 the pairs of amounts
(x/2, x), (x, 2x) 		the one pair of amounts
(a, 2a)
3 the pair of envelopes mean of the chosen amount
E(x)
4 the one pair of amounts
(a, 2a) 		the chosen amount
x
5   the pairs of amounts
(x/2, x), (x, 2x)
And I think that we may become DivideThreeByTwoian under the relatively high vividity of the image of the one pair of amounts on the closed version problem.

Does "closed version" problem have a magical power to make us uncomfortable to think with a fixed value of the chosen amount?

This paragraph was added on November 18, 2018.

On November, 2018, I got the following hypothesis.
We often becom SinglePairian when imagining the moment before choosing envelope, under the mechanizm of the rolling back SinglePairian.
After it, the image of unopened envelopes makes us uncomfortable to think with a fixed value of the chosen amount.
Therefore some of us want to drive out the DoublePairian's mental model.
As a result of this motivation, they will have the following ideas and become DivideThreeByTwoina.
  • primitive expectation
  • the idea that it is inappropriate to think of a particular amount of money of the chosen envelope,
    or the idea that the symbol x of the formula "E=(1/2)(x/2)+(1/2)2x" is a random variable
  • et cetera

Is there an illusion that the unseen amount is never fixed?

This paragraph was added on December 9, 2018. The title was revised on February 24, 2019.

On December 9, 2018, I got the following hypothesis.
Before opening envelope the amount of money contained in the chosen envelope is not specified.
From this fact, some of us get an illusion that the amount contained in the chosen envelope is not yet fixed.
And such a people feel uncomfortable to think of specific amount in the chosen envelope.
This illusion is one of the source of the magical power of the closed version problem. (← revised on December 16, 2018.)

Does imagining enveloping money makes us to think pair of amounts is fixed?

This paragraph was added on December 16, 2018. The title was revised on April 12, 2020.

On December 16, 2018, I got the following hypothesis.
After enveloping money in the two envelopes, the pair of amounts of money is determined in the mind of the game master.
From this fact, some of us get an illusion that the pair of amounts is fixed.
And such a people feel uncomfortable to think of multiple pairs of amounts.
This illusion is one of the source of the magical power of the closed version problem.

Does "closed version" problem have a magical power to change us into LesserOrGreaterMeanValuean?

This paragraph was added on May 17, 2018.

A few people get the idea of LesserOrGreaterMeanValuean's problem on the closed version problem.
I have the following hypothesis that explains this magical power of the closed version problem.
Of course, when becoming LesserOrGreaterMeanValuean, we will not be DivideThreeByTwoian.

On March 9, 2018, the paragraph "Does the theory of "E=(1/2)a+(1/2)2a" have a magical power of confusing us?" was deleted.


My hypotheses about the mind of DivideThreeByTwoians are presented in the following sections.

Hypotheses hard to adopt about the mind of DivideThreeByTwoians

This section was added on Decemb11, 2017. Revised on January 11, 2018.
The titles of the paragraphs of this section was revised on May 17, 2018.

I think that the following hypotheses should be rejected.

The hypothesis that DivideThreeByTwoian has never been DoublePairian

On September 1, 2017, this paragraph was added. Revised on May 17, 2018.

In the past the following hypothesis flashed on me.
  • When they have read the rule of the game they become SinglePairian.
  • And when they have read the fallacious expectation formula "E=(1/2)(x/2)+(1/2)2x" they do not become DoublePairian .
  • Therefore they think that the two terms "x/2" and "2x" are cannot coexist in a formula.
But DivideThreeByTwoians are highly probably DoublePairians.
It suggests to reject the above hypothesis.
(For details please see "True face of early DivideThreeByTwoians might be DoublePairian".)

The hypothesis that we can justify DivideThreeByTwoian's opinion because of ambiguity of the problem wording

On December 8, 2017, this paragraph was added.

Almost mathematicians who discussed the two envelopes problem did not accept DivideThreeByTwoian's opinion.
This fact rejects the above hypothesis.

The hypothesis that all DivideThreeByTwoian could not understand the expectation formula well

On December 14, 2017, this paragraph was added.

In the past there was a mathematician DivideThreeByTwoian.
An another DivideThreeByTwoian could understand the expectation formula used to explain the paradoxical distributions. (← Added on July 29, 2018)
These facts reject the above hypothesis.

My Hypotheses about what a concept of expected value DivideThreeByTwoians have

This section was added on March 8, 2020.

Hypothesis that DivideThreeByTwoian objectively think of expected value

illusion of objective equivalence

The following facts suggest that closed version problem have a psychological power to give us the illusion of objective equivalence.

illusion of objective expectation

(The followings came from the deleted paragraph "The old version of my main hypothesis about their mind during the period of DivideThreeByTwoian's resolution - Part 1")

The following facts suggest that closed version problem have a psychological power to give us the illusion of objective expectation.

Hypothesis that their concept of expected value is primitive

(This paragraph was added on March 8, 2018 with the title "Hypothesis that their concept of probability is primitive".)
(This paragraph was revised on March 22, 2018 with new title.)
(This paragraph was moved here on March 8, 2020.)

DivideThreeByTwoians think that expected value is determined by the combination of one situation and one decision.

As a result, they reject the mathematical expectation. (← Added on March 24, 2019)
I think this hypothesis explains the previous hypotheses.
Hints which gave me the above hypothesis
  • One person who discussed the two envelopes problem said as follows.
    About the formula "(1/2)(x/2)+(1/2)2x", the amount x/2 is a selection from x/2 and x, and the amount 2x is a selection from x and 2x.
    It means that these amounts are results of different trials each other.
    So you should not write both in the same expectation formula.
  • Many articles written by DivideThreeByTwoians discussed variations of the rule of the game.
    But few of them discussed variations of the odds of the two pairs of amounts of money.
    This suggests that they did not understand the DoublePairian's problem.
  • Some people who are able to use conditional probability to solve the Monty Hall problem can not accept conditional expectations with two envelope problems. (← Revised on August 19, 2018)

Hypothes of why they like the primitive concept of expected value
(This paragraph was added on February 2, 2020)
The calculation under the primitive concept of expected value is the same as the calculation at the time when the selection of envelope is going to be done.
In other words, the primitive expected value is the same as the expected future value at the selection. (← Revised on February 9, 2020)
Such a similarity let DivideThreeByTwoians prefer the primitive concept of expected value.
This suggests that their probability 1/2 should be called illusional probability, not real probability. (← Added on February 9, 2020)

Hypothesis that they tried to test the expectation formula in comparison with some situations consistent with the concept of primitive probability

(Added on September 20, 2022)

The concept of probabilty that DivideThreeByTwoians had is primitive. In other words, they assigned probablitiies to situations which succeed to the situation which is the origin of a probabilistic phenomenon.
And they tried to test the expectation formula in comparison with some situations consistent with the concept of primitive probability
starting point of the phenomenon probabilistic phenomenon result of the phenomenon
enclosing each amount of a particular pair of amount in a separate envelope probabilistic choice of one of two envelopes whose contents from a particular amount pair the chosen envelope contains the lesser amount of the pair or it contains the greater amount of the pair
one of the two envelope has been chosen, and a specific amount of money has been enveloped based on the amount of the chosen envelope, probabilistically determining the amount of the money going to be put in the other envelope the other envelope contains double or half amount of money in the chosen envelope

Hints which gave me the above hypothesis
Most of the papers by DivideThreeByTwoian philosophers examined the starting points of the probabilistic phenomenon presented in the above table.

My hypotheses about why not interested in the opened version problem while interested in the closed version

This section was added on December 11, 2017.
On Marh 8, 2020, this section was combined with the section "My hypotheses about why they were not interested in the opened version problem", and the title of it was changed from "My hypotheses about why they were interested in the closed version problem.

The paragraph "What hypotheses I need" was deleted on May 5, 2019.

Hypothesis that some of them were influenced by a philosophical article

(Added on February 3, 2019. The title was changed from "An influence of a philosophical article" on April 7, 2019)

There may be the following properties in the article Jackson, F., Menzies, P., & Oppy, G. (1994). I think that these properties may have let DiviideThreeByTwoian philosophers think that the paradox should be resolved in philosophical way instead of mathematical way correcting probability. (← Revised on April 7, 2019)

This section "My hypotheses about why they were not interested in the opened version problem" was added on July 7, 2019, and integrated in the another section on March 8, 2020..

Hypothesis that their concept of "expected value" is not mathematical

(This paragraph was added on July 7, 2019. The title was changed on March 8, 2020)

For DivideThreeByTwoian, the word "probability" means "possibility." In other words, their "probability" means the ratio of frequeny of a phenomenon after only one phenomenon.
In other words, their concept of expected value is primitive. (← Added on March 8, 2020)
Therefore, the opened version problem forcing them to think of two preceding events is inconvenient for them.

Hints which gave me the above hypothesis
  • I found an opinion that we must focus on the probable rather than the possible, in the literature Sutton, P. A. (2010).

Hypothesis that they wanted to reveal the hidden magic trick behind the variable symbols "x/2" and "2x".

(This paragraph was added on Decmeber 10, 2017, revised on November 18, 2018, moved here on July 7, 2019, revised on July 14, 2019)
(On September 8, 2019, the contents was revised and the title was changed form "Hypothesis that for them the two envelopes problem is a problem about logic or language rather than a problem about probability")

Due to the illusion that the standard resolution is wrong some philosophers thought that the two envelopes problem should be resolved by philosophy rather than mathematics.
And they thought that the magic trick should be behind the variable symbols "x/2" and "2x", rather than behind the probability.
Further, they wanted to apply the trendy philosophy like "possible world" and "rigid designator" to resolve the paradox.
As a result they could change interpretation of the variable symbols in the expectation formula without hesitation.
The usage of variable symbols in the opened version problem look appropriate. So, such a problem is not convenient for them.

Hints which gave me the above hypothesis
  • When I read the early articles written by DivideThreeByTwoian philosophers, I could not find indication that the fallacies which they think as the cause of the paradox are their own experience.
    This suggests that the variable symbol in the fallacious expectation formula had not real image in their mind and as a result they could doubt the meaning of the symbol.

Under this hypothesis, the theories presented by DivideThreeByTwoian philosophers may be understood as revealing components of a magic.

(This paragraph was added on March 22, 2020)

Under this hypothesis, I think that the structure of the magic is like below.

Definition of the magic
Not-three-amounts Theory

Proof of the existence of the magic
Ideas of possible trick


My hypotheses about their mind during the period of "DivideThreeByTwoian's resolution - Part 1"

On August 18, 2017, this paragraph was made of the following pargraphs.
  • My main hypothesis that illusion of objective expectation made them be DivideThreeByTwoians
  • Hypothesis that the "Not-three-amounts theory" is the ancestor of the theory of "E=(1/2)a + (1/2)2a"
 The paragraph "What hypotheses I need" was deleted on May 5, 2019.
The paragraph "The old version of my main hypothesis about their mind during the period of DivideThreeByTwoian's resolution - Part 1" was deleted on March 8, 2020.

The new version of my main hypothesis about their mind during the period of "DivideThreeByTwoian's resolution - Part 1"

On February 3, 2019, this paragraph was added.

While reffering the deleted paragraph "An opposed hypothesis to the above my main hypothesis about their mind during the period of DivideThreeByTwoian's resolution- Part 1", I rewrote my main hypothesis.
During the period of "DivideThreeByTwoian Resolution - Part 1", their mind was like below.

Thinking that it is the cause of the paradox to calculate expected value using x/2 and 2x
Being led by the following thoughts, they thought that x/2 and 2x are the key point of the cause of the paradox.
Ignoring opened version problem
On the opened version problem the primitive concept of expected value confuses their mind.
Therefore the opened version problem is inconvenient for them.
As a result they ignored the opened version problem.

Thinking that "E=(1/2)A+(1/2)2A" is the only correct formula
Being led by the following thoughts, they found that "E=(1/2)A+(1/2)2A" is the only correct formula.
Thinking that using x/2 and 2x is magic
(On March 22, 2020, this title was changed)
Based on the SinglePairian's mental model, they thought that the not-consistently-interpretable-variable theory proved that using x/2 and 2x in the expectation formula was magic. (← Revised on March 22, 2020)

(↑ Added on September 8, 2019. Moved here on September 15, 2019)
(For details, please see my hypothesis about the thinking that the problem is a kind of magic.) (← Added on February 10, 2019)

Devising various tricks of the magic
(On March 22, 2020, this title was changed)
They devised various tricks of the magic like below to support their finding. (← Revised on March 22, 2020) (↑ Revised on March 22, 2020)
Their aim is to rationarize their opinion, so the psychological confflict of these explanations is no problem.

Unconsciously rewriting the problem
They unconsciously rewrote the problem from the DoublePairian's problem to the SinglePairian's problem and don't notices that they did so.

Difference from my old main hypothesis
Unlike my old main hypothesis, the following illusions do not play major role in this hypothesis. Instead, primitive concept of expected value play a major role.

My sub hypotheses about their mind during the period of "DivideThreeByTwoian's resolution - Part 1"

On June 10, 2017, this paragraph was created gathering some paragraphs. On August 18, 2017, title was revised.

Hypothesis that magical power of the closed version problem let DivideThreeByTwoians seek an expectation formula that indicates equivalence of the two envelopes.

On September 10, 2017, this paragraph was created combining the following two paragraphs.
  • Hypothesis that DivideThreeByTwoians took fictitious paradoxes seriously.
  • Hypothesis that a phantom of single pair of amounts of money strengthened their fake resolution
"Magical power of the closed version problem" let DivideThreeByTwoians think that any correct expectation formula must indicate the eqivalence of the two envelopes regardless of amount of money of the chosen envelope.
In other words, the magical power of the closed version problem gave DivideThreeByTwoians the illusion of objective equivalence.
And this illusion let DivideThreeByTwoians take the following fictitious paradoxes seriously. (← Revised on January 6, 2018.) As a result, these fictitious paradoxes became real paradoxes in the mind of them. (← Added on March 22, 2018.)

Therefore they misunderstood that the standard resolution is not adequate because it only resolved the standard paradox. (← Added on September 11, 2017.)
In addition to it rolling back the problem gave them a vivid image of one pair of amounts. (← Revised on February 13, 2018.)
And they didn't like to think of a conditional expected value conditioned on the imaginary amount of money of the un-opened envelope. (← Added on February 13, 2018.)
So they did as follows.

Hints which gave me the above hypothesis
(This paragraph was revised on February 10, 2019)
  • Almost articles written by DivideThreeByTwoians presented the two envelopes problem without the description of independence from the chosen amount.
  • To my surprise, a mathematician said that the expected value after opening envelopes is
     p(1/2)(the chosen amount)  + (1 - p)2(the chosen amount),
    but the expected value before opening envelopes is
     (1/2)(the lesser amount)  + (1/2)2(the lesser amount)
    It indicates that even a capable mathematician might get infections of the phantom of single pair of amounts of money.
  • In the early 1990s, on the opened version problem a psychologist showed an opinion similar to the standard resolution.
    But after about ten years, the same psychologist showed the DivideThreeByTwoian's resolution for the closed version problem.
    ↑ Added on January 26, 2017. Revised on May 31, 2018.


The paragraph "Hypothesis that a misreading strengthened their fake resolution" was deleted on September 1, 2017.

The paragraph "Hypothesis that they advocated the "Not-three-amounts theory" to escape from uncanny posterior probability" was deleted on July 29, 2018, because it overlaps with the paragraph "Hypothesis that their concept of probability is primitive".

Hypothesis that rolling back the problem gave them a vivid image of one pair of amounts

(This paragraph was added on January 6, 2018, revised on March 8, 2018.)

They rolled back the problem and found a vivid image of one pair of amounts of money at the stage of setting money.

As a result they became not able to think of two pairs of amounts of money.
I call such DivideThreeByTwoians "Rolling back SinglePairian".
(↑ Added on Februry 23, 2018.)

Hints which gave me the above hypothesis
  • One DivideThreeByTwoian said that it is hard to recognise the step on which the fallacy occur.

The paragraph "Hypothesis that their concept of probability is primitive" was deleted on March 22, 2018.
The paragraph "'Hypothesis that their concepts of 'expected value' is not mathematical concept" was deleted on March 22, 2018.

The paragraph "Hypothesis that they thought that the coin flip illusion was the cause of the fallacious expectation formula" was deleted on December 25, 2017.

Hypothesis that it is hard to calculate conditional expectation with a vivid image of only one pair of amounts

(This paragraph was added on May 12, 2019)
We can get the standard resolution under a vivid image of only one pair of amounts of money as follows.
Let X and Y denote the random variable of the amount of money of the chosen envelope and the other envelope respectively.
And let the sample space is composed of the following two samples.
  sample 1 : X=a and Y = 2a.
  sample 2 : X=2a and Y = a.
Then the expectation of Y depends on X as follows.
  If X = a, then E(Y) = 1 × 2X + 0 × X/2.
  If X = 2a, then E(Y) = 0 × 2X + 1 × X/2.
Thus, in this case the probability is 1 or 0, rather than 1/2.
The cause of the paradox is a wrong belief that the probability is always 1/2.
But this resolution is too complex compared to the simplicity of the image of a pair of amounts.
Therefore, under such an image, it is psychologically biased to calculate expected values ​​with a and 2a instead of x/2 and 2x.

Hypothesis that the primitive concept of expectation let them invent the idea of illusion of flipping coin imitating the Ali-Baba version problem

(This paragraph was added on Decmeber 25, 2017.)
(Revised on March 8, 2018 with new title. Revised again on May 17, 2018 with new title.)
(Revised on January 8, 2019 with new title)

The two envelopes problem does not fit the primitive expectation though the Ali-Baba version problem fits the primitive expectation.
So they investigated differences among the two envelopes problem and the Ali-Baba version problem and as a result they thought as follows.
On the Ali-baba version problem the DoublePairian's mental model is correct.
But on the two envelopes problem the DoublePairian's mental model is not correct.

Hints which gave me the above hypothesis
  • The DivideThreeByTwoians who advocated the theory of non-Ali-Baba version appear to have not experienced an illusion of flipping coin themselves. (← Revised on January 8, 2019)


The paragraph "Hypothesis that DivideThreeByTwoian' mind is dominated by the probability illusion" was deleted on July 29, 2018.

Hypothesis that DivideThreeByTwoian philosophers thought that the problem is a kind of magic with hidden tricks

On October 14, 2018, this paragraph was added.
On October 28, 2018, the contents was revised, and the title was changed from "Hypothesis that DivideThreeByTwoian thought that the problem is a kind of magic whose trick should be revealed".
On February 3, 2019, this paragraph was moved here.
On September 8, 2019, the contents and tne title was revised.

On October, 2018, I got a hypothesis that DivideThreeByTwoian thought that the problem is a kind of magic whose trick should be revealed.
Some DivideThreeByTwoians did not think the fallacious formula "E=(1/2)(x/2)+(1/2)2x" as a result of a fallacy.
Conversely, they thought that the formula was the cause of the confusion which caused a paradox.
They sought the hidden tricks, and found the following tricks.
  • mental model change
  • confusion of designation
    • There is a confusion of designation using the variable symbol "x/2" and "2x".
    • Trendy philosophy like "possible world" and "rigid designator" should resolve the confusion.
Therefore they imagined that the formula was created to confuse us rather than to solve a mathematical problem.
As a result, they could not imagine that they replaced the problem.

Hints which gave me the above hypothesis
  • Many of the early DivideThreeByTwoian philosophers proposed Not-three-amounts Theory.
    This fact suggests that almost DivideThreeByTwoians understood DoublePairian's mental model.
  • Some DivideThreeByTwoian philosophers discussed game variation of the two envelopes problem.
    This fact reminds me that variations of the host behavior in the Monty Hall problem was discussed by people who focused on the ambiguity of the problem wording.
  • Some DivideThreeByTwoian philosophers simultaneously proposed the following resolutions that were contradictory to each other.
  • Some DivideThreeByTwoian philosophers preferred to diagnose the expectation formula rather than to calculate carefully like mathematicians.
    For example, many of them tried to check whether the variable symbol is rigid designator. (← Added on February 10, 2019)
  • In my perception DivideThreeByTwoians were not interested in the psychological mechanism of the fallacy which they thought. (← Added on September 8, 2019)
  • In the past there was a DivideNineByEightian who said that the two envelope paradox and the missing dollar riddle have a common fallacy. (← Added on September 8, 2019)
  • I imagine that the double coin flipping style wording has been originated from a mathematical article Sobel, J. H. (1994). However, in that article conditional expected value on uniform probability distributions were discussed with the DoublePairian's mental model not with the SingleSeedAmountian's mental model. Therefore, I think that it is a big factor whether he has written the problem wording or has read of it. Namely, we may tend not to doubt our own wording and tend to doubt given wording.
    (↑ Added on April 21, 2019)


On September 8, 2019, the paragraph "Hypothesis that DivideThreeByTwoians thought that the problem is a problem about logic or language" was deleted.

Hypothesis that DivideThreeByTwoians didn't like to think of a conditional expected value conditioned on the imaginary amount of money of the un-opened envelope

On February 13, 2018, this paragraph was added.

DivideThreeByTwoians didn't like to think of a conditional expected value conditioned on the imaginary amount of money of the un-opened envelope.
However they could get a vivid image of the one pair of amounts of money which is going to be enveloped in the two envelopes.
So they switched the problem to the SinglePairian's problem.

Hints which gave me the above hypothesis
  • Some DivideThreeByTwoians could think of two pairs of amounts of money on the opened version problem.


On November 11, 2018, the paragraph "A very big question : DivideThreeByTwoians have not explained what the cause of the fallacious expectation formula is?" was deleted.

On February 3, 2019, the paragraph "A basic question : If the probability is expressed using symbols of probability is it easy to become DivideThreeByTwoian?" was deleted.

Remaining questions about their mind during the period of "DivideThreeByTwoian's resolution - Part 1"

On May 12, 2019, this paragraph was added.



My hypotheses about their mind during the period of "DivideThreeByTwoian's resolution - Part 2"

On August 18, 2017, this paragraph was added.

The paragraph "What hypotheses I need" was deleted on May 5, 2019.

First of all, we should reject the hypothesis that some DivideThreeByTwoians spontaneously themselves interpreted the variable symbol inconsistently and thought that the "Inconsistent-variable theory" is true.

(This paragraph was added on September 15, 2017. Title was revised on September 18, 2017.)
(Moved to here on December 11, 2017.)
Revised on July 29, 2018.)

On September 15, 2017, the following hypothesis flashed on me.
However, after examining articles written by DivideThreeTwoians I could not find an author who spontaneously had used variable symbol inconsistently, so it should be rejected.
DiviideThreeByTwoians who advocate the "Inconsistent-variable theory" have the following properties.
  • When they read the fallacious expectation formula "E=(12)(x/2)+(1/2)2x" they interpreted the variable symbol x inconsistently.
  • But they did not feel any paradox because they soon noticed the mistake of their interpretation.
  • However when they encountered the "Inconsistent-variable theory" they accepted it.
    In other word they thought that there were some people who used variable symbol inconsistently and felt some paradox.
    So they thought the theory true.

My main hypothesis about their mind during the period of "DivideThreeByTwoian's resolution - Part 2"

This paragraph was added on July 6, 2017. The title was revised on June 24, 2018.
On February 10, 2019, this paragraph was greatly revised, and the title as chnaged from "Hypotheses that explain the birth of the Inconsistent-variable theory".
Hypotheses that explain the birth of the Inconsistent-variable theory
I examined some articles written by early DivideThreeByTwoians who were philosophers.
And I compared these and the early revisions of the English language Wikipedia article "The two envelopes problem".
And as a result I got the following hypothesis.
Philosophers advocated the "Not-three-amounts theory" based on difference from the Ali-Baba version.

And they had strengthened their opinion by the following wording.
Because thre are only two amounts of money the variable symbol x in "E=(1/2)(x/2)+(1/2)2x" cannot have same value in the two terms.
In this wording the inconsistent variable was a result of the another fallacy. (← Added on February 13, 2018.)

Some editors of the English language Wikipedia article "The two envelopes problem" misread this wording and they interpreted it as below.
The variable symbol x in "E=(1/2)(x/2)+(1/2)2x" has different values in each of the two terms.
The inconsistent variable were interpreted as the cause of the paradox. (← Added on February 13, 2018.)
This is the birth of the "Inconsistent-variable theory" which have bothered me for several years.

My second main hypothesis about their mind during the period of "DivideThreeByTwoian's resolution - Part 2"

This paragraph was added on June 24, 2018.
The title was changed from "Hypothesis that the inconsistent-mean-value theory is an ancestor of the incosistent-variable theory", on February 10, 2019..

Some of the editors of the above English language Wikipedia articles may had read the inconsistent-mean-value theory as the inconsistent-variable theory.
This is the another birth of the "Inconsistent-variable theory".

Hints which gave me the above hypothesis

In 2005, the English wikipedia articles "Envelope paradox" (How to read it) and "Two envelopes problem" referred to an article which presented the following thinking.

That article presented :
  • The first term of the expectation formula "(1/2)(X/2)+(1/2)2X" corresponds to the envelope with the greater amount and the second term corresponds to the envelope with the lesser amount.
    (This thinking seems to be derived from the SinglePairian's mental model.)
  • The fact that the expected values of the symbol X of the each terms are different (briefly, the fact that X of the first term is different from X of the second term) indicates that the expectation formula is wrong.
    (This thinking is similar to the inconsistent-variable theory.)
  • Letting "X" denote the lesser amounts, a calculating formula "(1/2)X + (1/2)2X") has not such property. Therefore this formula is the correct one.
Such a resolution is partially similier to the resolution of LesserOrGreaterMeanValuean's paradox, and partially similier to the inconsistent-variable theory.
Therefore I call such an opinion "inconsistent-mean-value theory".


My sub hypotheses about their mind during the period of "DivideThreeByTwoian's resolution - Part 2"

On June 10, 2017, this paragraph was created gathering some paragraphs.
On September 10, 2017, the title was changed from "Other Hypotheses" to the new title.

Hypothesis that they had read the solution before they read the problem

(The paragraph "Hypothesis that DivideThreeByTwoian's opinion is a fiction made of "Affirming the consequent" (confusion of necessity and sufficiency)" was replaced by this hypothesis on September 19, 2017.)

They might have read one of the following solutions before they examined the expectation formula "E=(1/2)(x/2)+ (1/2)2x" well. Because of the influence of these solutions they lost the chance to be DoublePairian.
And they had an illusion that they did not consistently use the variable symbols.
And they created a fiction based on one of the following concepts.
  • There are some people who felt a paradox from an expectation formula which is a mistake of "E=(1/2)2a + (1/2)a".
  • There are some people who inconsistently used the variable symbol and made the fallacious expectation formula "E=(1/2)(x/2)+(1/2)2x".
I call such DivideThreeByTwoians "Influenced SinglePairian".
(↑ Added on December 30, 2017.)

Hints which gave me the above hypothesis
  • A few DivideThreeByTwoians presented their opinion for the opened version problem.
    This fact indicates that DivideThreeByTwoinas had not searched own mental phenomenon that derived the fallacious formula "E=(1/2)(x/2)+(1/2)2x".
  • There was an article which was as follows.
    • It presented the two envelopes problem which contained an expesion "A maths student did the fallacious expected value calculation".
    • The author presented the "Inconsistent-variable theory".
    But maths student uses variable symbol consistently, so I think that the author of the article did not think the problem carefully.
    It is highly probable that the article was made of secondhand knowledge.

This hypothesis explains some of the mysteries of DivideThreeByTwoian's mind.
The paragraph "Hypothesis that before DivideThreeByTwoians were DoublePairian they were ProbabilityFirstian" was deleted on August 14, 2017.

Hypothesis that to DivideThreeByTwoian's eyes the two envelopes problem is a puzzle rather than paradox

This paragraph was revised on July 5, 2017.

On July 5, 2017, the following hypothesis flashed upon me.
DivideThreeByTwoians think that there are only two amounts of money A and 2A.
Why do they think so?
The answer is that to their eyes, the two envelopes problem is a puzzle like the following puzzle.
There are a few men.
Two of them are fathers of some of the others.
Two of them are sons of some of the others.
How many men?
Answer is three (one grandfather, one father and son).
(↓ Added on July 6, 2017.)
And to their eyes the two envelopes problem is like below.
There is an expectation formula E=(1/2)(x/2)+(1/2)2x.
How many amounts of money?
Answer is two (the lesser amount and the greater amount).

TwoEnvelopesPuzzlian
If DivideThreeByTwoians have such a property I call them "TwoEnvelopesPuzzlian".

Hints which gave me the above hypothesis
  • DivideThreeByTwoians reject the thinking of three amounts of money x, (x/2) and 2x. (← revised on July 6, 2017.)
  • Many DivideThreeByTwoians stubbornly reject the thought that a fallacy of probability is the cause of the paradox.

This hypothesis explains some of the mysteries of DivideThreeByTwoian's mind.

Hypothesis that there is a mechanism like the image of "My Wife and My Mother-in-Law"

On February 24, 2019, this paragraph was added.　On February 25, 2019, this paragraph was moved here,

The mechanism how DivideThreeByTwoians interpret the expectation formula is similar to the mechanism of ambiguous images such as the image of "My Wife and My Mother-in-Law".
  • Usually we will interpret the expectation formula based on the DoublePairian's mental model when initially reading the formula without prior knowledge.
  • However, if we imagin one pair of amount, some of us try to switch the interpretation of the expectation formula from the one based on the DoublePairian's mental model to the one based on the SinglePairian's mental model.
  • And for some people, it is hard to switch back to the original interpretation after switching the interpretation.

(↑ Added on May 19, 2019)

Hints which gave me the above hypothesis
  • When I read the blog page that posted the DivideThreeByTwoian's opinion on February, 2019, I carelessly became SinglePairian and accepted their opinion for a few seconds.
  • When I looked the image of "My Wife and My Mother-in-Law" first time, I understdood it as an image of young woman. But after reading explanation of the ambiguity of it, I understood it as an image of old woman. And it required a conscious effort to switch back to the original interpretation.
  • There are various versions of the image "My Wife and My Mother-in-Law", and the initial interpretation varies from version to version.
  • A blog page presented a wording which was similar to the following wording.
    Let x denote the chosen amount.
    Equally likely the other envelope may has a greater or lesser amount.
    If it contains 2x, switching gain is x. If it contains x/2, switching loss is x/2.
    The expected switching gain is : E = ½ ( x ) + ½ ( –½ x ) = ¼ x.
    The sentences "If it contains 2x, switching gain is x. If it contains x/2, switching loss is x/2" let us make a vivid image of the chosen amount denoted by "x", and the symbol "x" can not denote different values among these two sentences.
    However, to my surprise the opinion of the blog page is the inconsistent-variable theory.
    After examining the book referenced by the blog page, I imagined as follows.
    • When first read the problems presented in the referenced book, he became DoublePairian.
    • When read the the inconsistent-variable theory written in the referenced book, he became SinglePairian.
    • When he wrote the above wording in the blog page, he did not notice that the wording he wrote was not fit for the Singlepairian's mental model.

Hypothesis that they have experienced Eureka effect

On September 20, 2022, this paragraph was added.

The DivideThreeByTwoian was originally DoublePairian. However, when he realized that the meaning of variable symbols became unclear when he stood on the SinglePairian' mental model, the Eureka effect kicked in and he became unable to leave the SinglePairian's mental model.
  • The first time they saw the expectation formula they became DoublePairian.
  • However, when they read the opinion that the first variable symbol X and the second symbol X means different value, the Eureka effect worked on them.
  • They became so addicted to the pleasant Eureka effect that they couldn't go back to the DoublePairian's mental model.

Hints which gave me the above hypothesis
  • I saw an endless debates between a DivideThreeByTwoian mathematician and a non-DividThreeBytwoian mathematician. This suggests that even mathematicians may not be acceptable of the idea of the probability fallacy when they become DivideThreeByTwoain.
  • In my own experience, even though I grammatically understand that the amount represented by the variable X is not necessarily fixed, it is not intuitively acceptable. It may be that the Eureka effect when noticing the fallacy of probability makes the idea of the fallacy of variables intuitively unacceptable for me.

My hypotheses about their mind during the period of "DivideThreeByTwoian's resolution - Part 3"

On December 11, 2017, this paragraph was added.

Hypothesis that during the period of "DivideThreeByTwoian's resolution - Part 3" their goal is to find an expectation formula which derives no paradox

They did read that the expectation formula "E=(1/2)2a + (1/2)a" does not derive a paradox.
Then they misunderstood that they got answer of the problem.
And they began spreading the pure DivideThreeByTwoian's opinion.

My hypotheses about the mind of Hybrid-DivideThreeByTwoians

On May 3, 2018, this paragraph was added.

Hypothesis that the magical power of the closed version problem made them Hybrid-DivideThreeByTwoian

On July 29, 2018, this paragraph was added. On February 10, 2019, the content and the title was revised.
The magical power of the closed version problem made them not be able to think on the DoublePairian's mental model before opening envelope.
  • On the closed version problem which is under the magical power of the closed version problem they can not think on the DoublePairian's mental model.
  • However on the opened version problem which is free from the magical power they can can think on the DoublePairian's mental model.

Hints which gave me the above hypothesis
  • A DivideThreeByTwoian who resolved paradox on the closed version problem said as follows.
    • Before opening envelope the two envelopes are can not be distinguished.
    • After opening envelope the probabilities depend on the amount of money of the chosen envelope.
    He is not Hybrid-DivideThreeByTwoian, but his opinion gives a hint.


On July 29, 2018, the paragraph "Hypothesis that vividity of the objects in the mind of the Hybrid-DivideThreeByTwoian depend on whether the chosen envelope is opened" was deleted.

Hypothesis that Hybrid-DivideThreeByTwoian have been influenced by the English language Wikipedia article "Two envelopes problem"

On May 17, 2018, this paragraph was added.
Many revisions of the English language Wikipedia article "Two envelopes problem" presented the DivideThreeByTwoian's opinion on the closed version problem and simultaneously presented the mathematically standard resolution on the opened version problem.
So some people who have read such revisions may become Hybrid-DivideThreeByTwoian.

Hints which gave me the above hypothesis
(Added on February 24, 2019)
  • The English language Wikipedia article "Two envelopes problem" had the stance of the Hybrid-DivideThreeByTwoian from 2005.
  • The Italic language Wikipedia article "Paradosso delle due buste" clarified the stance of the Hybrid-DivideThreeByTwoian since March 27, 2006.
  • A book which took the stance of the Hybrid-DivideThreeByTwoian was published in In the early 2010's.

Note：
In 2005, the English language Wikipedia article "Two envelopes problem" was referring to a philosophical paper which took the stance of Hybrid-DivideThreeByTwoian.
(↑ dded on March 17, 2019)

Hypothesis that the mind as teacher rather than researcher lets them be Hybrid-DivideThreeByTwoians

On April 14, 2019, this paragraph was added.
The Hybrid-DivideThreeByTwoians I know are as below.
  • Some of the editors of the Italian language Wikipedia.
  • A writer of a book about the history of probability theory.
  • An author of papers published in a journal on teaching.
This indicates that the Hybrid-DivideThreeByTwoians prioritizes to inform the main theory about the two envelope problem.


On April 14, 2019, the paragraph "Hypothesis that primitive expectation lets them be Hybrid-DivideThreeByTwoians" was deleted.

On December 11, 2017, the paragraph "My hypotheses about the mind of DivideThreeByTwoians who presented the very restrictive wording" was deleted.

My hypotheses about the DivideThreeByTwoian treating the two envelopes problem as a paradox

This section was added on February 2, 2020. The title was revised on March 8, 2020.

Hypothesis that the naturalness of the symbols x/2 and 2x in the DivideThreeByTwoian's mind has let them think the problem as a paradox

The title was revised on March 8, 2020.
The symbols x/2 and 2x are natural in the DivideThreeByTwoian's mind as their true face is DoublePairian.
On the other hand, the magical power of the closed version problem makes the vividity of the SinglePairian's mental model stronger.
Therefore, they think that the DoublePairian's mental model is an illusion rather than a misalculation.
This fictitious illusion makes them think the two envelopes problem is a kind of paradox rather than a miscalculation.
And they wrote their papers using various philosophical concept like "rigid designator" and "possible worlds" which explain the relation of the two mental models,
However, the naturalness of "x / 2" and "2x" remains in their mind, and this contradiction gives them a strong paradoxical feeling. (← Added on March 22, 2020)
This hypothesis means that the DivideThreeByTwoian philosophers have created a new paradox rather than a resolution to the two envelope paradox. (← Added on March 22, 2020)

On March 22, 2020, the contents of the paragraph "Hypothesis that DivideThreeByTwoian's paradox gives them a strong wonder" was integrated into the another paragraph.

My hypotheses about the reason why DivideThreeByTwoians didn't notice that they had switched the problem to the another problem

This section was added on September 13, 2017. Title was changed on December 5, 2017.

The paragraph "What hypotheses I need" was deleted on May 5, 2019.

My main hypothesis

Hypothesis that the illusion of objective equivalence and the illusion of objective expectation let DivideThreeByTwoians think that they have solved the original two envelopes problem.

On October 10, 2017, this paragraph was added.
The illusion of objective equivalence and the illusion of objective expectation have gave DivideThreeByTwoians an illusion that they have found the one and only correct expectation formula.
The illusion of objective expectation let them think that "E=(1/2)(x/2) + (1/2)2x" and "E=(1/2)2a + (1/2)a" have been made under same purpose.
The illusion of objective equivalence let them think that "E=(1/2)2a + (1/2)a" is the exact correct expectation formula.
(↑ Added on March 29, 2018.)
This illusion prevents them from noticing the possibility that the problem which they have solved is the another problem.

Other my hypotheses

Hypothesis that DivideThreeByTwoians thought that it is permitted not to use x/2 and 2x in the expectation formula, because they thought the usage of them is the root cause of the paradox

On October 3, 2017, this paragraph was added.
On July 29, 2018, the cntents was revised and the title was changed.
On July 7, 2019, the cntents was revised and the title was changed.

The following reasons have let them refuse the thought of two pairs of amounts of money. So DivideThreeByTwoians thought that the DoublePairian's mental model is inappropriate to calculate expected values.
As a result of it, they thought that the variable symbol "x" in the expectation formula was not the condition for the calculation, and it is permitted to replace the variable symbol in the expected formula.

On August 12, 2018, the paragraph "Hypothesis that conviction of the wrongness of the standard resolution let DivideThreeByTwoians think that the problem itself is wrong" was deleted.

Hypothesis that DivideThreeByTwoians think that the oened version problem and the closed version problem are mathematically distinct problems

On August 12, 2018, this paragraph was added.

I got the following hypothesis on August 6, 2018.
DivideThreeByTwoians thought that before opening the chosen envelope any expectation formula should not suggest exchange of envelopes.
Therefore to their eyes the standard resolution does not fit the closed version problem.
So they thought that the closed version problem is the another problem which is differ from the opened version problem. And they thought that the fallacy of the closed version problem must differ from the fallacy of the opened version problem.
I got one more another hypothesis on December 23, 2018.
Before opening envelope, primitive expectation forces them to think epectation based on the pair of amounts setted in the each envelope.
On the other hand, after opening envelope, primitive expectation forces them to think epectation based on the revealed amount in the opened envelope.
As a result, they don't hesitate to switch problem depending on the opportunity to swap envelope.

Hints which gave me the above hypotheses
An article described the opened version problem as "a harder problem".

Hypothesis that they could not imagine another mental model other than SinglePairian's mental model

On July 7, 2019, this paragraph was added as a revised version of the paragraph "Hypothesis that they could not imagine another problem because they were the influenced SinglePairian".

I got the following hypothesis on August 6, 2018.
DivideThreeByTwoians cannot imagine other than SinglePairian's mental model for various reasons. As a result they could not notice the possibility that they have solved another problem.

Hints which gave me the above hypotheses
  • DivideThreeByTwoian philosopher who have not proved the wrongness of the mathematically standard resolution may have been affected by the mechanism like the image of "My Wife and My Mother-in-Law".
  • Many DivideThreeByTwoians who did not refer to the DoublePairian's mental model may be the influenced SinglePairian.
  • Many DivideThreeByTwoians who did not try to understand the mathematical resolution may have been affected by cognitive dissonance.

Hypothesis that DivideThreeByTwoian interpreted the two envelopes problem as a problem in the domain of decision theory

On November 16, 2017, this paragraph was added.

I got the following hypothesis on November 16, 2017.
Some people became DivideThreeByTwoian through the following process.
  1. They read the two envelopes problem in a journal of philosophy (not mathematics).
  2. In an article of philosophy they read, the problem to be solved is that an expected formula recommends switching the envelopes.
  3. They interpreted the two envelopes problem as a problem in the domain of decision theory.
  4. They thought that finding an expectation formula which does not recommend swapping the envelopes is the solution which the problem demands.
  5. They did not hesitate to switch to the mathematically another problem because doing so does not switch to the another problem in the domain of decision theory.

Hints which gave me the above hypothesis
  • There is an article of philosophy which has the above property.
  • Almost of DivideThreeByTwoians does not recognise the need to understand the standard resolution.


On July 7, 2019, the paragraph "Hypothesis that DivideThreeByTwoian have unconsciously rewritten the problem to the SinglePairian's problem" was deleted.

My hypotheses about the reason why it is so hard to change DivideThreeByTwoians mind

This section was added on December 5, 2017. Title was changed on December 11, 2017.

The paragraph "What hypotheses I need" was deleted on May 5, 2019.

My new hypotheses

Hypothesis that they have mental barrier to understand the DoublePairian's mental model

On August 18, 2019, this paragraph was added.

If they are the rolling back SinglePairian, mechanism like the image of "My Wife and My Mother-in-Law" prevent them from accepting the DoubulePairian's mental model.
As a result, they think DoublePairian's opinion is correct only in the case of Ali in the Ali-Baba version problem.

If they are the influenced SinglePairian, they cannot imagine the DoublePairian's mental model.
As a result, they interpret DoublePairian's opinion from the SinglePairian's view point, and they gues that DoublePairian's opinion is wrong.

Hints which gave me the above hypothesis
  • Example of a paper by DivideThreeByTwoin philosophers who appeared to be the influenced SinglePairian.
    The paper argued that the following propositions were incompatible.
    • The envelope A contains the leeser amount, then the amount is n.
    • The envelope A contains the greater amount, then the amount is n.
    When their this claim was objected in an paper, they wrote new paper to reply that the objection is wrong. But their response indicated that they had not understand the DoublePairian's mental model.

Hypothesis that the standard resolution destroys the premise of their thinking

On March 8, 2018, this paragraph was added.

The premise of their thinking may be one of the following beliefs.
  • The two envelopes problem is not a problem of probability but a problem of logic or language.
  • It is not permissible to doubt the probability 1/2.
  • The expected value must be calculated on the SinglePairian's mental model.
Therefore they can not accept the standard resolution which destroys such a premise.

Hints which gave me the above hypothesis
  • In my perception, DivideThreeByTwoians have not read the standard resolution or was not trying to understand it.

My other hypotheses

On July 7, 2019, the paragraph "Hypothesis that in the period of DivideThreeByTwoian's resolution - Part 1, they thought DoublePairian's mental model inappropriate on the closed version problem." was deleted.

The paragraph "Hypothesis that the hallucination of the inconsistent-variable is infectious" was deleted on July 29, 2018.

The paragraph "Hypothesis that mathematical explanation and experiment can not let DivideThreeByTwoians change their opinion" was deleted on July 29, 2018.

Hypothesis that the primitive concept of expected value prevents them from understanding the mathematically standard resolution

On March 22, 2018, this paragraph was added.

The primitive concept of expected value prevents DivideThreeByTwoians from understandig the DoublePairian's problem.
Therefore they cannot understand the mathematically standard resolution which is for the DoublePairian's problem.

Hypothesis that DivideThreeByTwoians simply do not try to read the mathematically standard resolution

On March 3, 2019, this paragraph was added.

DivideThreeByTwoian think that it is not necessary to understand the mathematical standard resolution when they are discussing about the two envelopes problem.

Hints which gave me the above hypothesis
  • I know two cases that DivideThreeByTwoian never tried to read the explanation written by a standard resolver when discussing.


Can somebody let DivideThreeByTwoians change their opinion?

This paragraph was added on April 7, 2016.
The title was changed from "What is the true resolution of the paradox DivideThreeByTwoians feel?" on May 7, 2016.
The title was changed on June 12, 2016.
The composition of this paragraph was revised on March 1, 2018.
Many redundant paragraphs were deleted on July 29, 2018.
The paragraph "Trying to explain that the probabilities are not necessarily 1/2" was deleted on March 31, 2019.
The header "Trying to explain that they have not resolved the original paradox" was deleted on March 31, 2019.

I have found only two cases that DivideThreeByTwoians changed their opinion.

Case 1
A DivideThreeByTwoian changed own opinion by the following explanation.
After open your envelope, the amount of money in your envelope is fixed.
But he/she came to think that the expected value of the opposite amount of money must be same as the chosen amount of money, even after opening the chosen envelope. (← Revised on December 12, 2017.)

Case 2
A DivideThreeByTwoian changed own opinion by the following question.
Did you read his (a standard resolver) explanation?
This case suggest that many DivideThreeByTwoians object to the standard resolution without reading it.

The paragraph "I have tried to make good explanations and failed" was deleted on March 31, 2019.

On August 5, 2017, the paragraph "Still I have to say to DivideThreeByTwoians." was deleted.

On May 12, 2019, the paragraph "Stepwise explanation to let DivideThreeByTwoians change their opinion" was deleted.

My ideas of explanations to make DivideThreeByTwoians think their paradox as a mirage

This paragraph was added on May 12, 2019. Title was revised on September 1, 2019, February 16, 2020.

I think that the following explanations may let DivideThreeByTwoians think that their paradox may be a mirage.
In other word, these explanations may let them think that instead of resolving the existing paradox they may have created a fictional paradox.

Psychological experiment

Statistics on philosopher opinions

(The title was changed on February 16, 2020)

Paradox about the DivideThreeByTwoian's paradox

This paragraph was added on March 31, 2019.

I noticed that the following paradox continues to bother me.
I think that the paradox resolved by DivideThreeByTwoians has been proven to be a kind of mirage, as I have written so far.
On the other hand, I can not refuse the possibility that the DoublePairian's mental model is not appropriate for the closed version problem.
I do not know if this confusion in my mind is the result of the magical power of the closed version problem.

If some psychology researcher finds a way to change the mind of ​​DivideThreeByTwoians, this my confusion will disappear.

After all

This paragraph was added on July 29, 2018. Revised on August 19, 2018, March 17, 2019

I think that the mind of DivideThreeByTwoians can be summarized as follows.
Their mind during the period of "DivideThreeByTwoian's resolution - Part 1"
Some of the following illusions led them to think that the paradox should be resolved on the SinglePairian's mental.model. (↑ Revised on Februry 2, 2020)

And the following ideas made them think that the usage of "x/2" and "2x" in the fallacious expectation formula is the source of confusion which lead to the paradox. (← Added on July 7, 2019. Revised on February 2, 2020) (↑ Revised on Februry 2, 2020)

And based on the SinglePairian's mental model, they thought that the not-consistently-interpretable-variable theory proved that using x/2 and 2x in the expectation formula was magic. (← Revised on March 22, 2020)
However, the following illusions prevented them from becoming aware that the problem they solved was not the two envelopes problem. And they devised various tricks of the magic like below to support their finding. (← Revised on February 2, 2020, March 22, 2020) And they could to keep the idea that the problem was not a miscalculation but a kind of paradox, because their true face is DoublePairian. (← Added on February 2, 2020)
It must be said that they have not resolve the two envelope paradox but they have created a different paradox. Because they felt paradox by different mechanism. (← Added on March 22, 2020)
Their mind during the period of "DivideThreeByTwoian's resolution - Part 2"
The following mental phenomena happened. As a result, in their mind, "E=(1/2)(x/2)+(1/2)2x" is a mistake of "E=(1/2)A+(1/2)2A".
And the following illusions kept them away from noticing that the problem they solved was not the two envelopes problem.
Conclusion
The binding of the following ideas is the source of such a chaos.
  • The opportunity to swap envelopes is given before opening envelope.
  • The problem raises a paradox worthy of solution. (← Revised on February 9, 2020)
The problem which DivideThreeByTwoians were looking at was a kind of mirage.
And a mechanism like the image of "My Wife and My Mother-in-Law" prevented them from noticing that they solved another problem.

However, several psychological questions remain as described in the paragraph Psychological problems about DivideThreeByTwoian's mind.

In any case

(This paragraph was moved here on October 28, 2018)

They who think of only two amounts of money look like sorcerers.

(A drawing was deleted on Aapril 21, 2019)

(A drawing was deleted on July 29, 2018)

(A drawing was deleted on April 7, 2019)

Or they look like seeing a mirage.

↑ Added on April 7, 2019.

Or they look like thinking that they are looking at magic tricks. ( ← Revised on May 5, 2019)

↑ Added on Novenber 4, 2018.


Psychological problems

Following problems have not been studied by researchers of cognitive psychology.

Psychological problems about DoublePairian's paradoxes

(This paragraph was revised on February 21, 2017.)

Which is the cause of DoublePairian's paradoxes?

I think that the theory of assumption of probability distribution is psychologically unlikely.
And I think that this opinion is supported by the result of the experiment 3 reported by Burns, Bruce D.
(↑ Added on August 19, 2018)

Psychological problems about the standard paradox and resolution

(This paragraph was added on July 31, 2016, and was revised on February 21, 2017.)

Why standard resolvers could feel standard paradox?

Are the paradox of the part 1 of the period of the standard resolution and the paradox of the part 2 same?

(↑ This title was revised on June 16, 2017, March 22, 2020)

For the part 1 and the part 2 of the period of the standard resolution, please see History of the resolutions of the two envelopes problem.

Psychological problems about DivideThreeByTwoian's mind

(Revised on February 20, 2017.)

How powerful is the magical power of the closed version problem?

(Added on September 25, 2018)
On March 22, 2020, the paragraph "Have DivideThreeByTwoians felt paradox by themselves?" was deleted.

Does the DivideThreeByTwoian's paradox really exist?

(Added on July 29, 2018. Revised on August 19, 2018.)

Which did let them be DivideThreeByTwoian?

Is DivideThreeByTwoian DoublePairian?

(This paragraph was added on March 10, 2017. Revised on June 16, 2017.)

How many DivideThreeByTwoians did find the not-three-amounts theory by themselves?

(This paragraph was added on March 29, 2018.)

Why were they convinced of their opinion, when their opinion is logically wrong?

(This paragraph was added on August 26, 2018. The title was changed on March 22, 2020)

Reversal of role between psyochologist and mathematician

(This paragraph was greatly revised on March 24, 2019, March 31, 2019)

To my eyes there is a reversal of role between psyochologist and mathematician. I wish psychologists verify hypotheses which the mathematician made.
Because, I think that usual people don't interpret like the above. (← Added on March 24, 2019)

Mutation of the problem through long thinking

Considering the two envelopes problem for days, some mutations of the problem may occur in our head.
(↑ Added on March 10, 2019)

Is LesserOrGreaterMeanValuean's problem the third problem?

LesserOrGreaterMeanValuean's problem

On December 2014, I found a opinion that there are some people who have the following thought.
The subject matter of the "two envelopes problem" is the magnitude relation of the mean values of the chosen envelope and the other envelope under the following situations.
  • the amount of money in the chosen envelope is lesser than it in the other envelope
  • the amount of money in the chosen envelope is greater than it in the other envelope
(↑ Revised on March 18, 2015, September 22, 2019)

If such people exist and are not minority, they should be called "LesserOrGreaterMeanValuean".

LesserOrGreaterMeanValuean's paradox

Let X and Y be random variables of the amount of money in the chosen envelope and the other envelope respectively.
E(Y)=(1/2)2E(X) + (1/2)(1/2)E(X)
=1.25 E(X) > E(X).
!!! Paradox
↑ Revised on February 11, 2018, September 2, 2018.

Resolution of LesserOrGreaterMeanValuean's paradox

On June 3, 2019, the format was greatly revised.

  The cause of the paradox
which arise on the
LesserOrGreaterMeanValuean's problem
A few mathematician's thought (Revised on July 21, 2017, February 8, 2018, June 3, 2019)
LesserOrGreaterMeanValueans forgot they were thinking of expectation values under two different conditions.
E(Y|Y=2X)=2E(X|Y=2X) and E(Y|Y=(X/2))=(1/2)E(X|Y=(X/2)).
Therefore, E(Y)=(1/2)2E(X) + (1/2)(1/2)E(X).

!!! Each mean values in the two terms must be conditioned on different conditions.
The expectation formula should be corrected as follows.
E(Y)=(1/2)2E(X|X<Y) + (1/2)(1/2)E(X|X>Y).

On June, 2019, I found that this kind of fallacy was called "the discharge fallacy" in Jeffrey, R. (2004). (← Added on June 3, 2019)

Some mathematics

Verification of the above resolution
(Added on December 23, 2018)
E(Y)
= (1/2)E(Y|X<Y) + (1/2)E(Y|X>Y)
= (1/2)2E(X|X<Y) + (1/2)(1/2)E(X|X>Y).
A verification of E(Y)=E(X) under the resolution
(Added on March 29, 2016. Title changed on December 23, 2018, June 3, 2019)
2E(X|X<Y) = E(X|X>Y)   and   E(X|X<Y) + E(X|X>Y) = 2E(X).
∴ E(X|X<Y) = (2/3)E(X)   and   E(X|X>Y) = (4/3)E(X).
Therefore,
E(Y)
= (1/2)2E(X|X<Y) + (1/2)(1/2)E(X|X>Y)
= (1/2)2(2/3)E(X) + (1/2)(1/2)(4/3)E(X) = E(X).
↑ Revised on April 26, 2017, July 20, 2017, February 8, 2018.
One more verification of E(Y)=E(X) under the resolution
(Added on July 20, 2017. Title changed on December 23, 2018, June 3, 2019)
E(Y)
= (1/2)2E(X|X<Y) + (1/2)(1/2)E(X|X>Y)
= (1/2)E(X|X<Y) + (1/2)E(X|X<Y) + (1/2)(1/2)E(X|X>Y)
= (1/2)E(X|X<Y) + (1/2)(1/2)E(X|X>Y) + (1/2)(1/2)E(X|X>Y)
= (1/2)E(X|X<Y) + (1/2)E(X|X>Y) = E(X).
Relation to the another random variable
(Added on December 23, 2018. Revised on June 3, 2019)
Let A be the random variable of the lesser amount. Then it becomes as follows.
E(A)
= (1/2)E(A|X<Y) + (1/2)E(A|X>Y)
= (1/2)E(X|X<Y) + (1/2)E(X/2|X>Y)
= (1/2)E(X|X<Y) + (1/2)E(Y|Y<X).
Therefore, E(A)= E(X|X<Y) = E(Y|Y<X).
Remark:   Because E(X+Y) = 3E(A) and E(X) = E(Y), E(X) = E(Y) = (1/2)E(A) + (1/2)E(2A).

Is the paradox which has the above resolution common?

  opinion
A few mathematician's thought
(Added on June 3, 2019) 		If having the SinglePairian's mental model, the equation of the above resolution coinsides the equation of the theory of "E=(1/2)a+(1/2)2a".
So, some mathematicians thought that the paradox which has the above resolution is common among the advocates of the theory of "E=(1/2)a+(1/2)2a".
 
My thought about it I can not imagine that there can be one who have such a complicated mental model and make such a simple confusion.
So to my eyes their opinion is only distortion.
(↓ Added on August 12, 2018)
I remember that I have thought the LesserOrGreaterMeanValuean's problem several years ago.
However it was after the experience of the usual two envelopes paradox.
 
(↓ Added on September 17, 2017.)
The equation
"E(Y) = (1/2)E(Y|X<Y) + (1/2)E(Y|X>Y)"
is an application of the theorem
"E(Y)
= P(event e) E(Y|e) + (1 - P(e)) E(Y|the complementary event of e)".
And they did it as a matter of course.
But this theorem is not so simple, so to my eyes their opinion is doubtful.
 

On September 22, 2019, the paragraph "If the expectation formula is not described in the problem" was delteted.

My hypotheses about the mechanism how people become LesserOrGreaterMeanValuean

(This paragraph was added on April 19, 2018.)

Hypothesis 1
The people who have read the wallet game before reading the two envelopes problem may become LesserOrGreaterMeanValuean.

Hypothesis 2 (cognitive mutation of the problem by languageization)
Some people including me thought the two envelopes problem too many times.
Think, think, think, ⋅ ⋅ ⋅
At last in their mind the expectation formula mutates as follows.
E=(1/2)(x/2)+(1/2)2x.
  ↓
Swapping will give half of the amount or double the amount.
The image of the probability fades away and the amounts of money become obscure.
As a result they become LesserOrGreaterMeanValuean unconsciously.
I call this mechanism "cognitive mutation of the problem by languageization".

Hypothesis 3 (If the expectation formula is not described in the problem)
(This hypothesis was created on September 22, 2019, with the idea of the deleted paragraph "If the expectation formula is not described in the problem")
If the expectation formula is not described in the problem, we may get a mental model in which the amounts of money are not mathematical quantity but a vague concept which resembles both certain value and average value.
If so, we will feel the LesserOrGreaterMeanValuean's paradox.
However, until recently I have never read a version of the problem which describe no expectation formula.

Essential difference between the LesserOrGreaterMeanValuean's resolution and the theory of "E=(1/2)2a+(1/2)a"

(This paragraph was added on May 3, 2018. The title was revised on February 10, 2019)

If we denote E(X|X<Y) by "a" then we get the following.
(↑ Revised on May 17, 2018.)
E(Y) = (1/2)2E(X|X<Y) + (1/2)(1/2)E(X|X>Y) = (1/2)2a + (1/2)a.
It looks like the theory of "E=(1/2)2a+(1/2)a".
And some people have the opinion that the theory of "E=(1/2)2a+(1/2)a" is a special case of the LesserOrGreaterMeanValuean's resolution.
But I think that their opinion is wrong because of the following reasons. (↑ Revised on May 17, 2018, October 14, 2018)

Is MeanRateOfExchangean's problem the fourth problem?

On May 3, 2018. I changed my coined word "MeanRateOfTheExchangean" to "MeanRateOfExchangean".

MeanRateOfExchangean's problem

Some people may think that the subject matter of the "two envelopes problem" is expectation of rate of the exchange.
If such people are not minority, they should be called "MeanRateOfExchangean".

MeanRateOfExchangean's paradox

Let X and Y be random variables of the amount of money in the chosen envelope and the other envelope respectively.
E(Y/X) =(1/2)(2X/X) + (1/2)((X/2)/X) =1.25.
Therefore E(Y)=1.25E(X) > E(X).
!!! Paradox

Resolution of MeanRateOfExchangean's paradox

  The cause of the paradox
which arise on the
MeanRateOfExchangean's problem
Anybody's thought Mean value of rate of the exchange
is not always equal to
the ratio of mean value before exchange and mean value after exchange.
↑ revised on March 18, 2015
 
My thought about it MeanRateOfExchangean's paradox seems to occur more easily than LesserOrGreaterMeanValuean's paradox.
 
Why do I know it?
Because I have experienced both paradoxes.
↑Added on August 28, 2017. Revised on April 5, 2018.
 

For reference.

The paragraph "This is not the fifth problem" was deleted on October 5, 2017.

Relevant true paradoxes

(↑ This header was added on March 10, 2019)

The two envelope paradox is a pseud paradox caused by fallacy of probability calculation.
However, some relevant paradoxes are true paradox, not pseud.

Paradoxical distributions which have infinite mean value

On March 10, 2019, this title was revised.

Paradixical distribution

If infinite mean value is allowed, there can be "Paradoxical distributions" that switching the envelopes is always advantageous for all amount of money in the chosen envelope.
Following example is most famous among such distributions.

pair of amounts probability
20 and 21 (2/3)0 / 3
21 and 22 (2/3)1 / 3
·
·
· 		·
·
·
2n and 2n+1 (2/3)n / 3
·
·
· 		·
·
·

For reference. In addition to the paradoxical discrete probability distribution, continuous probability distribution was also discussed in some of the above literatures.
For example, a probability density function "f(s)=1/(s+1)2 for s > 0" was presented in Broome,John.(1995).
(↑ Added on July 22, 2018)

Addition: (Added on September 15, 2019)
The most famous paradoxical distribution above is a special case of the following distributions.
Let r denote a number where 0 < r < 1.
Let n denote a natural number where n ≥ 0 and let (2n, 2n+1) be a pair of amounts of money placed in the two envelopes.
Let X and Y be random variables representing the amounts of money in the chosen envelope and the other envelope respectively.
And consider a probability distribution that the probabiltiy of (2n, 2n+1) is rn(1-r).
Then :
  • If r < 1/2, E(X) converges and E(X) = E(Y). (Not paradoxical)
  • If r = 1/2, E(X) diverges and E(Y|X) = E(X|X) for X ≠ 20. (The other envelope is almost always as favorable as the chosen envelope)
  • If r > 1/2, E(X) diverges and E(Y|X) > E(X|X). (Paradoxical distribution)

Paradox about the equivalence of the two envelopes on the paradoxical distribution

This paragraph was added on December 19, 2017.

There is no wonder even if the other envelope is more favorable for an amount of money of the chosen envelope.
But it is paradoxical that the other envelope is always more favorable for any amount of money of the chosen envelope.

Nothing would blow paradoxical feeling away

This paragraph was added on March 31, 2016.

Following analysis could not soften my paradoxical feeling of the paradoxical distribution.

Paradoxical distribution is not paradoxical in logic
We feel a paradox from the two envelopes problem as follows.
Let X be a random variable which denote the amount of money in the chosen envelope.
Let Y be a random variable which denote the amount of money in the opposite envelope.
If for any x E(Y|X=x) > x, then E(Y|X) > E(X) and E(Y) > E(X). · · · (1)
(1) was not held under the condition that mean value of the amount of money is infinite. Therefore paradoxical distribution is not paradoxical in logic.
But this finding does not blow paradoxical feeling away.

Distribution of the amount of money before switch is same as after switch
Let X be a random variable which denotes the amount of money in the chosen envelope.
Let Y be a random variable which denotes the amount of money in new chosen envelope after switching under the condition switching is always done.  ( ← Revised on April 5, 2016)
Then random variables X and Y have same probability distribution. · · · (1)
From (1) we get a conviction of the equivalence of the two envelopes.
But this finding does not blow paradoxical feeling away.

If a experiment has been done
This paragraph was added on April 3, 2016, and was Revised on May 5, 2016.
Let n be the number of games.
Let Xn be a random variable which denote the cumulative amount of money in the chosen envelope.
Let Yn be a random variable which denote the cumulative amount of money in the opposite envelope.
Then  (Xn/n) - (Yn/n)  and  Xn/Yn will not converge.
But this finding does not blow paradoxical feeling away.

If we think of the difference of finite world and infinite world
This paragraph was added on July 17, 2016.

Even if the probability distribution of the amounts of money is a paradoxical distribution, the non-equivalence of envelopes in the finite world is not conflict to the equivalence in the infinite world.

But this finding does not blow paradoxical feeling away.

Paradoxical distributions are similar to the Zeno's paradox.

This paragraph was added on May 22, 2016, revised on May 29, 2016.

Let's think of the paradox of Achilles and the tortoise. This is one of the Zeno's paradoxes

In the paradox of Achilles and the tortoise, there is a infinite sequence of pair of positions of Achilles and the tortoise.
Each position of Achilles is same as the previous position of the tortoise.
But Achilles can reach the tortoise.

In the paradoxical distributions on the two envelopes problem, there is a infinite sequence of pair of the chosen amount of money and the expected opposite amount of money.
In each pair, the latter is greater than the former.
But if we think of all of possible games we should think that the two envelopes are equivalent.

This similarity suggests me that the paradox of the paradoxical distributions will be never resolved like Zeno's paradox.

Distinction between considering infinite sets and considering all elements of infinite set

This paragraph was added on January 13, 2018.

The following thinkings may be essentially different. And we seem unconsciously assume the law of large numbers when we try to find a knowledge from thinking of the all elements. ← Revised on March 15, 2018.
Because on the paradoxical distribution the law of large numbers is not satisfied it is nonsense to think of all amounts of money of the chosen envelope .
This means that there is not a paradox.

(↓ Added on March 15, 2018. Revised on March 22, 2018.)
From the above my thought I found the following correspondence between these paradoxes as follows.

paradox The nature of the infnite set The nature of each element of the infinite set
the two envlope paradox Probability distribution does not change by always swapping. Swapping is advantageous regardless of the amount of money of the chosen envelope.
the Zeno's paradox The sum of the time required for each interval cannot exceed a certain value. Any interval has next interval which requires non-zero time.

But these paradoxes have a big difference.
Everyday we experience the Zeno's paradox. As an example, today at a convenience store, I was overtaken by other customer.
But nobody have experience a phenomenon which has infinite mean value.

My another thought about paradoxical distributions

This paragraph was added on October 4, 2015. The title was changed on May 22, 2016.

Amounts over the mean value
If we can imagine amounts of money over the mean value ( = ∞ ) these amounts will recover the equivalence of the two envelopes.

Sequence of non-paradoxical distributions
If we think of a sequence of probability distribution which has the following aspects , then the paradoxical feeling softens.

Related works

This paragraph was added on March 28, 2015, and was moved here with new title on August 14, 2016.

Paradox that exchange on a paradoxical distribution is advantageous even before expecting

This paragraph was added on December 19, 2017. The title was changed on May 31,2018.

On the paradoxical distribution if we imagine an amount of money of the chosen envelope we should swap envelopes.
Therefore we can think that we should swap envelopes even if we do not imagine an amount of money of the chosen envelope.
This means that we should swap envelopes just after the first choice. !!! Paradox !!!

The essence of this paradox

If we do not imagine an amount of money of the chosen envelope, we must have imagined that we imagined an amount of money.

I would like to resolve this paradox as follows.

Forget your argument, and you don't need to swap.


NotSpecificDistributionian's paradox

On May 17, 2018, the title of this paragraph was changed to "NotSpecificDistributionian's problem". On March 10, 2019, the title was re-changed.

Some people think that we must solve the Two envelopes problem on the uncertainty of distribution of the amount of money.
I call such people "NotSpecificDistributionian", and call such problem "NotSpecificDistributionian's problem".

NotSpecificDistributionian's problem is not the matter of mathematics

NotSpecificDistributionian's paradox

But some paradoxes will occur even on such a problem setting.

Paradox by the lack of information

Because the distribution of the amount of money is uncertain, you get no hint of whether you should trade or not even when your envelope has been opened.
However if you find $1 in your envelope, and know that it is possible that the another envelope contains $1, 000, 000, you will best fast trade.
!!! Paradox !!!
I have written this in reference to vos Savant, Marilyn (1996). ← Added on March 15, 2018.

Paradox by the principle of insufficient reason

This paragraph was added on July 23, 2017.
This paragraph was moved to here on October 5, 2017.

We are allowed to think that the probability is 1/2 if we take the principle of insufficient reason and throw away the proper probability distribution.
And we will not feel any paradox, because our own will is the cause of the result.
But strangely we can feel the following paradox.
If I imagine the amount of money of my envelope, I should try to change envelope because the expected amount of money in the other envelope is 1.25 times the amount of money of my envelope.
This conclusion has no wonder because it is the result of my decision to take principle of insufficient reason.
In other words the expectation of amount of money is subjective, not objective.
But I will get same conclusion for any amounts of money of my envelope, so I should try to change envelope without imagination of amount of money.
When I am handed an envelope, automatically it is less favorable than the other.
!!! Paradox !!!
My resolution is as below.
(Revised on October 5, 2017, January 26, 2018.)
The decision on the principle of insufficient reason is the matter of decision theory and the equivalence of the two envelopes is the matter of mathematics.
Especially the probability which is made by the principle of insufficient reason has not relation to the true probability distribution.
And it has not relation to any mean values. (← Added on March 29, 2018.)
So we can not discuss the equivalence of the two envelopes in relation to that principle before opening the chosen envelope.
Conclusion : Forget your argument, and they shall be equivalent.

On October, 2017, I found a similar solution as below in a puzzle book.
Even if always you should switch, the equivalence of the two envelopes is kept as a whole.

In my perception, the following articles discussed the similar theme.
(Revised on July 8, 2018)

Common fallacy behind

(This paragraph was added on March 15, 2018.)

I think that a coommonplace fallacy named "the fallacy of composition" is behind the Zeno's paradox, the paradox of the paradoxical distribution and the paradox by the principle of insufficient reason.

Paradox like "Unexpected hanging paradox"

This section was added on June 17, 2017. The title was revised on March 8, 2018.

About "Unexpected hanging paradox" please see the English language Wikipedia.

This type of analysis was presented in the following articles.

Thinking like the prisoner in the "Unexpected hanging paradox"

Assumptions

The amont of money has upper boundary.
Both player A and B know the value of this upper boundary.
Both players are allowed to look privately at the amount of money of their own envelopes.

Terms

Suppose the max value of the amount of money in the two envelope is M.
In other words the greatest pair of amounts is M and M / 2.
And let Mn denote an amount of money x that M / 2n+1 < x <= M / 2n.

Case 1
sub case
player A
player B 		Should A offer to trade? Should B accept A's offer to trade?
1-1
M0
M -1 		this case not happens this case not happens
1-2
M1 		No
(A has greater amount) 		uncertain


Case 2
sub case
player A
player B 		Should A offer to trade? Should B accept A's offer to trade?
2-1
M1
M0 		No
(If B accepts B has M2.) 		No
(B is like A in case 1-2.)
2-2
M2 		uncertain


Case 3
sub case
player A
player B 		Should A offer to trade? Should B accept A's offer to trade?
3-1
M2
M1 		No
(If B accepts B has M3.) 		No
(B is like A in case 2-2.)
3-2
M3 		uncertain
(↑ Revised on October 14, 2018)

And so on.

Consequence of this thinking.

Player A should not offer to trade.
Symmetrically player B should not offer to trade.

Paradox like "Unexpected hanging paradox"

The above thinking suggests keeping the chosen envelope regardless the amount of money.
But if the player A offers to exchange according to the suggestion of the expected amount, player B will be confused.
(↑ Revised on January 6, 2018.)


The section 'Classification of "two envelope paradoxes"' was deleted on March 29, 2018.

Smullyan's paradox

Outline

In Smullyan, Raymond (1992)  two contradictory propositions are proven about two envelopes problem. Each proposition is proven by each argument respectively.
In an abbreviated form, these arguments are as follows. In the page "Smullyan's paradox on the two envelopes problem", I wrote my thought about this paradox.

I think that same mental mechanism is hidden behind DivideThreeTwoian's oppinion and Smullyan's paradox.

To my eyes DivideThreeByTwoians seem to think as bellow.

The following expectation formulas have same purpose.
E=(1/2)(x/2)+(1/2)2x
E=(1/2)a + (1/2)2a
The expectation formula "E=(1/2)(x/2)+(1/2)2x" is wrong.
Therefore the expectation formula "E=(1/2)a + (1/2)2a" is the only one correct expectation formula.

I think that we can not easily make the following distinctions on the Smullyan's paradox.

Distinction among the possible gain x in the argument 1 and the possible gain d in the argument 2
Distinction among the possible loss x/2 in the argument 1 and the possible loss d in the argument 2
(↑ Revised on July 29, 2018)

I think that these two confusions have same pattern.
So I think that same mental mechanism is hidden behind DivideThreeTwoian's oppinion and Smullyan's paradox.

Ancestors of the two envelopes problem

Similar problems which were created before the two envelopes problem

This section was added on February 5, 2015.

Necktie paradox
Kraitchik,M.(1943) showed the "Necktie paradox".

Problem by Schrödinger
Nalebuff, Barry.(1989) described that Littlewood, J. E. (1953) showed a problem which Schrödinger created.

Wlallet game
Merryfield, K. G., Viet, N., & Watson, S. (1997) described that Gardner, M. (1982) showed the "Wallet game".

Comparison of the "Two envelopes problem" and the "Wallet game"

This section was added on February 5, 2015, and greatly revised on December 2, 2018.

Outline of the "Wallet game"

(This paragraph was revised on October 2, 2017.)

Wallet game is said to be an ancestor of the two envelopes problem.
According to Gardner, M. (1982), the paradox of the wallet game has the following scenario.

There is a big difference among the wallet game and the two envelopes problem

In the two envelopes problem, the equivalence of the two envelope is guaranteed.   But in the wallet game, the equivalence of the two wallets is not guaranteed.
Example.
  • Both of the mean values of the amount of money in the A's wallet and B's wallet are ¥1, 000.
  • A's wallet usually contains ¥1,000.
  • B's wallet equally likely contains ¥500 or ¥1,500.
In this case A will gain ¥250 on the average.
Condition that the game is fair for each players was discussed in Merryfield, K. G., Viet, N., & Watson, S. (1997).

Mental models on the wallet game

ManyPairian's mental model
If a paradox of the wallet game arise from the image of the amount of money in hand,  people may have made ManyPairian's mental model.



DoublePairian's mental model for the two envelopes problem is very alike to ManyPairan's mental model.

WinLosePairian's mental model
If a paradox of the wallet game arise from the image of mean value of the amount of money in hand,  people may have made WinLosePairian's mental model.



There is a theory that people may make LesserOrGreaterMeanValuean's mental model on the two envelopes problem. (However, I can't believe such a theory.)
I think that LesserOrGreaterMeanValuan's mental model for the two envelopes problem is somewhat alike to WinLosePairian's mental model.

Mathematical and psychological comparison

(This paragraph was added on December 2, 2018.)

Symbols used below
Two envelopes problem Wallet game
x, X : the chosen amount and the random variable of it.
y, Y : the other amount and the random variable of it. 		x, X : the price of your necktie and the random variable of it.
y, Y : the price of the opponent's necktie and the random variable of it.
w, W : your wining money and the random variable of it.

Comparison from the standard resolver's view point
  Two envelopes problem Wallet game
expectation formula arising paradox E(Y|X=x)
= (1/2)2x + (1/2))(x/2)
> x. 		E(W|X=x)
> (1/2)x - (1/2)x
= 0.
mental model DoublePairian's mental model ManyPairian's mental model
cause of the fallacious expectation illusion that probability is 1/2 wrong assumption that probability is 1/2
solution of the problem E(Y|X=x)
= P(X<Y|X=x)2x
+ P(X>Y|X=x)(x/2). 		E(W|X=x)
= P(X<Y|X=x)E(Y|X<Y and X=x)
- P(X>Y|X=x)x.
resolution of the paradox
  • It depend on X whether E(Y|X) is greater than X or equal to X or lesser than X.
  • Because E[E(Y|X)] = E(X), the equivalence of the two envelopes is guaranteed.
 Whether E(W|X=x) is plus or minus depends on the probability distribution of the prices of the two neckties.

Comparison from the LesserOrGreaterMeanValuean's view point
  Two envelopes problem Wallet game
expectation formula arising paradox E(Y)
= (1/2)2E(X) + (1/2))(E(X)/2)
> E(X). 		E(W)
> (1/2)E(X) - (1/2)E(X)
= 0.
mental model LesserOrGreaterMeanValuean's mental model WinLosePairian's mental model
cause of the fallacious expectation Oblivion of different condition
  • Oblivion of different condition
  • wrong assumption that probability is 1/2
solution of the problem E(Y)
= P(X<Y)2E(X|X<Y)
+ P(X>Y)(1/2)E(X|X>Y). 		E(W)
= P(X<Y)E(Y|X<Y)
- P(X>Y)E(X|X>Y).
resolution of the paradox
  • Mean values of the lesser amount and the greater amount are not same.
  • Because E(X) = E(Y), the equivalence of the two envelopes is guaranteed.
 Whether E(W) is plus or minus depends on the probability distribution of the prices of the two neckties.

the WinLosePairians often make a fallacious resolution for the paradox of the wallet game

(This paragraph was added on December 2, 2018.)

In my perception, for the paradox of the wallet game, the WinLosepairians often make a fallacious resolution as follows.
E(W)
= P(X<Y)E(Y|X<Y) -P(X>Y)E(X|X>Y)
=(1/2)E(X|X>Y) -(1/2)E(X|X>Y)
= 0.
Therefore, the game is fair.
The following wording is typical in such a resolution.
My wallet contains the less money if I win. And it has the more money one if I lose.

I think that such a resolution is the result of a wrong assumption that both amount have same probability distribution.

The necktie paradox often referred to as an ancestor of the two envelopes problem

This section was added on December 2, 2018.

Variety of the situation in the necktie paradox

This paragraph was added on December 2, 2018.

What motivates the players
(Added on December 9, 2018)

According to Kraitchik,M.(1943) and Albers, C. J., Kooi, B. P., & Schaafsma, W. (2005), in the original version, each player claims to have the finer necktie. (← Revised on February 10, 2019)
However in some versions, the players want to see who has the cheaper necktie.

How the judgement done
According to Kraitchik,M.(1943) and Albers, C. J., Kooi, B. P., & Schaafsma, W. (2005), in the original version, the judgement is done by a third person. (← Revised on February 10, 2019)
However in some versions, the judgement is done according to the information from the wives who had given the neckties to the husbands as Christmas gift.

What the winner gets
According to Kraitchik,M.(1943) and Albers, C. J., Kooi, B. P., & Schaafsma, W. (2005), in the original version, the winner gets the opponent's necktie. (← Revised on February 10, 2019)
However in some versions, the winner gets money as much as the price of the opponent's necktie.

The situation of the necktie paradox seems to be less realistic

This paragraph was added as a section on November 25, 2018, and revised on December 2, 2018.

It is not strange that two men compete with the amount they have spent to get their neckties.
However, because of the following reasons, the situation of the necktie paradox seems to be less realistic. In contrast, the situation of the two envelopes problem is more realistic even though the game in the problem is more artificial.

How resolved originally

This paragraph was added on December 2, 2018.

Referring Kraitchik,M.(1943), I guessed that an explanation like below was written. (← Revised on February 10, 2019) I think that such an explanation is based on the ManyPairian's mental model. (← Revised on February 10, 2019)
However, the following assertions were written in Kraitchik,M.(1943). (← Added on February 10, 2019) Under such conditions, the contestants look like identical twins.

Problem by Schrödinger

This section was added on April 28, 2019.

Problem

I found a problem owed to Schrödinger in Littlewood, John Edensor / Edited by Bollobás, Béla (1986).
If I rewrite it as I have understood, it is as follows. This problem is somewhat similar to the two envelopes problem where two players always want to trade.

Is it closely related to the two envelopes problem?

In Nalebuff, Barry. (1988), the above problem were referred as "an early statement of a closely related problem".
But I do not think so, as it is more closely related to the another problem presented in Littlewood, John Edensor / Edited by Bollobás, Béla (1986).
If I rewrite the another problem as I have understood, it is as follows. The above "the point" can be proven using induction as follows. I think this problem is more interesting than the problem by Schrödinger.

The "Ali-Baba" version (The most famous variation of the two envelopes problem)

This section was added on February 5, 2015.
The title was changed on March 10, 2019)

'"Ali-Baba" version' is my coined word. (← Added on January 25, 2018.)

This type of problem was introduced in Nalebuff, Barry. (1988) and Nalebuff, Barry.(1989).
If I summarize it, it is as follows. From Ali's point of view
If the mean value of the original amount of money is finite, then trading envelopes gives Ali an expected gain of 25% on the average.
(↑ Added on February 6, 2015)
Whether or not the mean value finite, trading envelopes gives Ali a conditional expected gain of 25% of the amount of money given her.
(↑ Revised on May 19, 2019)

From Baba's point of view
If the mean value of the original amount of money is finite, then trading envelopes gives Baba an expected loss of 25% on the average.
But without information of the probability distribution of the original amount of money, he can not compute conditional expectation of the loss from trading.

The possibility that this problem had influenced philosophers

(Added on April 15, 2017. Revised on July 16, 2017, July 1, 2018, November 25, 2018.)

 The "Ali-Baba" version problem has the following aspects. I think that these aspects may have confused people who wanted to resolve the two envelope paradox as follows Many philosphers who were DivideThreeByTwoians refered Nalebuff, Barry.(1989). So I think that the above aspects of the "Ali-Baba" version may have affected on philosophers' thinking and may have made them DivideThreeByTwoians. (← Revised on July 1, 2018)

The possibility that this problem influenced philosophers indirectly but more heavily

(Added on March 24, 2019)

 In the article Nalebuff, Barry.(1989), the "Ali-Baba" version problem was presented before presenting the two envelopes problem.
And a philosopher who read that article wrote a paper with the double coin flipping style wording, and he presented a unique resolution.
(Specifically, please see "DivideNineByEightian's resolution".)
Therefore the "Ali-Baba" version problem may have influenced philosophers through such a indirect pass.
Remark:
To my eyes, the article Nalebuff, Barry.(1989) did not present the double coin flipping style wording. it presented the two envelopes problem with the phrase "In the original version of the problem, there is no coin toss".

Paradoxes with some similarity to the two envelope paradox

This title was added on August 11, 2019.

The Siegel paradox

This section was added on June 7, 2018. The contents was revised on October 1, 2019.

In my perception, there are two kinds of Siegel paradoxes as follows. To my eyes, the original paradox is not a paradox but a hypothesis about the relation among the anticipated exchange rate and the forward exchange rate.
And, the later paradoxes look like parodies of the two envelope paradox. (← Added on October 8, 2019)

For details please see my page "The Siegel paradox and the two envelope paradox".
(The contents of that page was significantly revised on October 1, 2019)

The shooting room paradox

This section was added on August 11, 2019.

As far as I have understood, this paradox is as follows.
At first round, some people enter a room and two dice are rolled.
If the result is double sixes, they are shot, and if not they leave and the next round starts with ten times many new people.
At any round, the probability that an individual person is shot is 1/36.
However, about 90 percent of participants in this game are shot.
The above room and people seem to symbolize the earth and humanity.
Because it is said that this paradox has been developed by John Leslie in connection with the Doomsday argument. (← Added on August 18, 2019)
This paradox has a similarity to paradoxical distributions.
However, I think this paradox is not as paradoxical as the two envelope paradox for the following reasons. It is interesting that the above percentage 90 relates to the following equation. (← Added on August 18, 2019)
0.9 = 1 / 1.1111…

Incantations that were spelled to illogically resolve the paradox

This section was added on February 1, 2015, and was revised on April 14, 2016.

Incantations which were spelled by DivideThreeByTwoians

This paragraph was added On February 13, 2016. Revised on August 20, 2016.

It is probable that DivideThreeByTwoians felt the paradox by the illusion of objective expectation.
This paradox is fictitious because it is based on an illusion.
Therefore their resolutions are a kind of incantation.

On March 31, 2019, the paragraph "Their opinion itself is incantation" was deleted.

Incantations to explain the Not-three-amounts Theory

This paragraph was moved from paragraph "Claims that it is wrong to be DoublePairian" and was revised on March 29, 2016.

Incantations to explain the Not-consistently-interpretable-variable theory

This paragraph was added on March 29, 2016. The title was revised on March 31, 2019.

Incantations to explain the Inconsistent-variable theory

This paragraph was revived on March 31, 2019 with some of the old contents of it.

Incantation to pretend not to notice paradox derived from the opened version problem.

This paragraph was added on June 1, 2017. Revised on February 4, 2018.

Incantations which claim that we must apply different logic to the closed version problem and the opened version problem

This paragraph was added on March 28, 2015. Revised on January 4, 2017.
And was moved to here on February 23, 2018.  

Incantations which were spelled by other people

Incantations which claim that envelopes are equivalent even after opening

Some people claim that the equivalence of envelopes are not lost even if the amount of money in one envelope is revealed.

(↑ Added on March 3, 2019)

In the page Incantations used by equivalent-expectationian on the two envelopes problem, some of such various claims are illustrated.

Incantations which claim that it is all right even if exchange is always advantageous

Some people claim that the fact that exchange is always advantageous does not contradict with the equivalence of the two envelopes.

Examples

Incantations which claim that there is no paradox

Some people say that there is no paradox but a feeling of paradox. ← revised on January 9, 2016.

Example

Incantations which claim that it is wrong to think about calculation formula of expectation

This paragraph was added on March 20, 2015.

Some people say that it is wrong to think about calculation formula of expectation.

Examples

Randomized switching

This paragraph was added on September 19, 2015.

If we can play opened version game repeatedly, which is the best strategy?

Some mathematicians study the strategies to earn more on average than the strategy not to exchange any time.
They take the condition that the distribution of the amount of money is unknown.
And they study how to decide depending on the amount of the revealed money.
Strategies which use random number are called "Randomised switching".

For reference.

An experiment

On July 12, 2015, referring to the article by Emin Martinian, I tried to see the effect of "randomized switching", and got the following result.
Condition of the experiment
  • Amounts of money have double-precision floating-point values.
  • The lesser amount uniformly distributes between 0.0 and 1.0.
  • For the chosen amount Y, the decision to switch will be made with a probability Exp(-Y / 2).
Method
 I used Excel.

Result

An explanation

On july 15, 2016, referring to Ross, S. M., Christensen R. and Utts, J.(1994)., I created a simple explanation.
Let g(x) be a function which has the following characteristic.
If b > a,   0 < g(b) < g(a) < 1.
And let y be the amount of money in the chosen envelope.
And let S1 be a strategy to exchange in probability g(y), and let S2 be a strategy not to exchange
Then S1 will make more earnings than S2.

Why?

Let's think of a pair of amounts of money (a, 2a).
Then g(2a) < g(a).
It means that under the strategy S1, the probability to get the greater amount is (1/2) (g(a) + (1 - g(2a)). It is larger than 1/2 which is the probability under the strategy S2. (← Revised on January 5, 2020)
( For detail, please read "An alternative randomized solution" written in the section "Randomized solutions" of the English language Wikipedia article "Two envelopes problem" (Revision at 22:46, 28 December 2019). ) (← Added on January 5, 2020)
Since this argument holds for any pairs of amounts, you can expect that the strategy S1 will give you more gain. (← Added on January 5, 2020)
I had applied this explanation to the above experiment.
g(a) = Exp(-a / 2)   and   g(2a) = Exp(-2a / 2).
∴ g(2a) = g(a)2.
g(2a) < g(a)   because   g(a) < 1. (← Revised on July 1, 2017.)

If people play game of the two envelopes problem

This section was added on February 6, 2015. Revised on June 7, 2018.

An imaginary case of a class of students where the instructor offers money enveloped in two envelopes

This case was described in Christensen, R; Utts, J (1992),

A real case of a class of probability theory where the professor offers money enveloped in two envelopes

This case was reported in a web page titled "Numberplay: Your Money or Your Logic - NYTimes.com".

An imaginary case of a professional soccer player who were invited from two teams

This paragraph was added on May 29, 2016.

This case was described in a web page titled "Matifutbol: Probability and sunrise".
The story of the soccer player is like a short novel.

An imaginary case of a television game show

This paragraph was added on August 12, 2016.

This case was described in a web page titled "NaClhv: The two envelopes problem and its solution".

Other issues

Research of psychlogy of decision making

Some researchers of psychology of decision making use the two envelopes problem as a material.

For reference.  

More than one English language Wikipedia article about the two envelopes problem

(This paragraph was added on July, 2017. The title was changed on July 23, 2017, October 3, 2017, June 16, 2019, June 23, 2019.)

More than one article

(Added on June 16, 2019)

As of June 16, 2019, the English language Wikipedia has the following articles about the two envelopes problem.
How to read an article which is redirected to the article "Two envelopes problem"
(Revived with new title on June 17, 2019)

Example: Case of "Envelope paradox"
FirstOpen a page of the English language Wikipedia.
Second  Enter "Envelope paradox" as the search key word, and click the search button.
ThirdIf the article "Two envelopes problem" is shown, click the link on the line "(Redirected from Envelope paradox)".
FourthIf the article "Envelope paradox" is shown click the link "View history".

How to read all articles redirected to the article "Two envelopes problem"
(Revised with new title on June 17, 2019)
FirstOpen the page of the article "Two envelopes problem" of the English language Wikipedia.
Second  Click the link "What links here" on the left side bar.
ThirdIf you see a page titled "Pages that link to 'Two envelopes problem'", search the links labeled "redirect page" on the page.
FourthClick the searched link.
FifthIf a redirected page is shown, click the link "View history".

Comparison of the article "Envelope paradox" and the article "Two envelopes problem"

(This paragraph was added on July 23, 2017. The title was changed on June 16, 2019)

To my eyes differences of the two articles are as follows. For details please see another my page "Two English language Wikipedia articles on the two envelope paradox".

Features of the article "Two-envelope paradox"
(This paragraph was added on June 23, 2019)
To my eyes this article has the following features.
Features of the article "Exchange paradox"
(This paragraph was added on June 16, 2019)
To my eyes this article has the following features.

My hope

(This paragraph was added on July 25, 2017.)
I think that the following two articles are creatures of distinct dimensions.
"Envelope paradox" is a creature in the mathematical dimension.
"Two envelopes problem" is a creature in the philosophical dimension.
↑ Added on October 4, 2017.
And I know that at least four editors of the article "Envelope paradox" wanted the survival of it in 2006.
So I hope that some editor cancels the redirection from "Envelope paradox" to "Two envelopes problem" and makes the two articles coexist.

Digression : Some of the remarkable events on the English language Wikipedia article "Envelope paradox"

This paragraph was added on June 23, 2019.

Vote for Deletion

On August 18, 2005, this article survived a vote for deletion. Three days ago, this vote was raised by an editor who had an opinion that it is ridiculous to expect based on unknown amount contained in the chosen envelope.
(This opinion was wrong from the beginning, because the problem presented in the article is the opened version problem.)

Final rediret

(Added on June 30, 2019. Revised on July 7, 2019)

A redirect to the article "Two envelopes problem" was edited on August, 2006.
I have found no indication that the editors of this article accepted this redirect, but strangely the redirect has not been removed.

Transfer of the talk page of this article to the archive of the talk page of the article "Two envelopes problem"

(Added on July 7, 2019)

Around June 12, 2009 (after three years of the final redirect), the talk page of the article "Envelope paradox" was transferred to the archive of the talk page of the article "Two envelopes problem" and given the page name "Talk:Two envelopes problem/Archive 1".
I think that I could not notice the existence of the article "Envelope paradox" if this transfer had not been done.


Digression : Some of the remarkable events on the English language Wikipedia article "Two envelopes problem"

This paragraph was added on January 20, 2019.

Suggenstion to merge

(Added on June 30, 2019. Revised on 30, 2019, July 7, 2019)

The next day of the creation of this article, a merge tag suggesting to merge it into the article "Envelope paradox" was placed.
This suggestion seems correct, but the merge has not took place.

Replacement of the wording

(Added on June 30, 2019)

The wording of the problem presented in the original revision (August 25, 2005) was the opened version problem.
About one month later, at the revision 22:05, 3 October 2005, the wording was replaced with the wording presented in the article "Envelope paradox" changing to the closed version problem.
And at the same revision the opened version problem became called "A Second Paradox". (← Revised on January 7, 2020)
Remark:
The original revision presented a book (Williams, David. (2001)) as a reference. This literature seems to explain two envelopes problem mathematically in detail.

Changing meaning of the opened version problem

(Added on January 7, 2020. Revised on January 12, 2020)

At the revision 18:42, 8 October 2008, the opened version problem became not described.
At the revision 20:37, 8 April 2011, in the section "Extensions to the problem", the opened version problem became presented as the subject of the switching strategy rather than the two envelope paradox. (← Revised on January 12, 2020)
At the revision 18:36, 1 May 2011, the new section "Randomized solutions" that describes Cover's principle was added. (Reference : Cover, T. M. (1987). )
At the revision 17:23, 13 February 2012, the section "Extensions to the problem" was rewritten to present only calculating formula of conditional expected value.

Appearance of the standard resolution on the closed version problem

(Added on February 2, 2020)

At the revision 18:42, 8 October 2008, the opened version problem and the standard resolution on it became not described.
At the revision 23:46, 3 May 2011, the standard resolution was written in the section "informal solution" and the section "formal solution" both discussing the closed version problem.
In my eyes, these editions are very strange. That's because almost articles describing the standard resolution discussed the opened version problem.

A new expectation formula which was regarded as equivalent to the known expectation formula

In a section created at the end of 2014 on the talk page of the article "Two envelopes problem", many editors discussed about quotation of a new expectation fomula. (Even now in January 2019, we can read that section on the talk page, thanks to an editor who NACed the section.)

The new formula is a calculation on the two pairs of amounts, so, it is essentially different from the DivideThreeByTwoian's formula. And the calculated expectated value does not have a dimension of a usual quantity.
To my eyes, the new formula is comparing the ratio of the losing (case of the lesser pair) and the ratio of the gaining (case of the greater pair) to the each mean value, despite the mean values are different. (Remark : With the new formula, these losing and gaining always offset, because if the ratio of two amounts is constant the ratio of their difference to their average is constant.)
(↑ Added on January 27, 2019. Revised on February 10, 2019)
This new formula seems very unique, despite the following facts.
  • Exactly the same formula except the currency unit and the number of zeros was presented on a blog written in 2004.
  • One of the participants of the discussion found the same idea on a blog written in 2007.
(↑ Added on February 10, 2019)
However, none of the participants of the discussion (except the proposer) did not discuss the actual meaning of the formula, while discussing notability or benefit of the formula and the reliability of information source.
In addition, some of the participants of the discussion (except the proposer) including the major editors said that it has essentially same meaning as the known DivideThreeByToian's formula, with no explanation how to verify it. (← Revised on January 27, 2019)

Such a tone of discussion may have influenced the later editings of the article, and even now in February 2019, we can see traces in the article. (← Revised on February 10, 2019)

Sudden decrease of the frequency of edition

From 2005 to 2014, the article "Two Envelope Problems" was edited quite frequently.
However, since December 2014, the frequency drastically decreased after a major revision after 17 days article locking after an edit warring.

Removed picture of two envelopes

A picture of two envelopes was pasted at the revision 11:42, 27 August 2009.
To my disappointment, it was removed at the revision 22:34, 18 January 2016 after no less than 6 years have passed.

Complete removal of the "Randomized solutions" section

(Added on January 5, 2020)

On December 28, 2019, with a title "An alternative randomized solution", a new strategy was written in the section "Randomized solutions". The strategy is to use a monotonically increasing function as the probability not to switch. I think such strategy is one of the most common randomized switching strategies.

However, to my surprise, just after the edition, on December 29, 2019, the section "Extensions to the problem" and this section "Randomized solutions" were replaced by completely different section "Conditional switching". (← Revised on January 7, 2020)

The new section "Conditional switching" does not mention the effect of randomizing. And the term "Conditional switching" seems to only mean that the player is allowed to open the chosen envelope before deciding whether to switch.

On May 1, 2011, the section "Randomized solutions" was created by one of the major editors of the article "Two envelopes problem". And main parts of the content were written by him in May 2011.
Anyway, the article "Two envelopes problem" has lost the section describing randomized switching strategies.

Addition : Which version of the two envelopes problem is presented by the Wikipedias in the world?

This paragraph was added on June 2, 2017. Revised on September 7, 2017. Title was changed on November 5, 2017.
This paragraph was moved to here with new title on March 1, 2018.

Definition of terms :
 "standard resolution"
It means the opinion that the probability is not always 1/2.
"resolution of 3/2"
It means the DivideThreeByTwoian's opinion that "E=(1/2)A+(1/2)2A" is the correct expectation formula.
"resolution using mean value"
(Revised on May 17, 2018)
It means the opinion that the correct expectation formula is as follows.
"E(Y) = (1/2)E((X/2) | X is the greater) + (1/2)E(2X | X is the lesser)"

language of Wikipedia title of the article about the two envelopes problem revision opened version problem closed version problem
Czech Paradox dvou obálek 5. 4. 2013, 12:06‎ nothing problem
with no resolution
German Umtauschparadoxon 16:55, 22. Aug. 2016‎ standard resolution nothing
English Envelope paradox
(How to read it) 		13:49, 14 July 2006 standard resolution nothing
Two envelops problem 14:56, 28 April 2017‎ problem
with no paradox
but with randomized switching 		resolution of 3/2
resolution using mean value
standard resolution?
(↑ Revised on
May 17, 2018)
Spanish Paradoja de los dos sobres 22:00 9 feb 2017‎ unique resolution
(↑ Revised on
May 17, 2018) 		problem
with no resolution
Farsi رادوکس دو پاکت ‏۱۲ فوریهٔ ۲۰۱۷، ساعت ۰۶:۵۰‏ nothing resolution of 3/2
French Paradoxe des deux enveloppes 2 avril 2017 à 15:20‎ problem with no paradox
(in the section
"Modifications de l'énoncé") 		resolution of 3/2
of various type
Italian Paradosso delle due buste 15:12, 16 apr 2016‎ standard resolution resolution of 3/2
Hebrew פרדוקס המעטפות 04:20, 1 במאי 2016‏ standard resolution? nothing
Hungarian Kétborítékos paradoxon 2016. szeptember 5., 19:35‎ resolution of 3/2
for opened version problem 		nothing
Dutch Enveloppenparadox 13 feb 2014 18:33‎ standard resolution
(↑ Revised on
May 17, 2018) 		resolution of 3/2
Russian Задача о двух конвертах 05:17, 19 ноября 2016‎ standard resolution nothing
Serbian Проблем две коверте 01:46, 4. јануар 2017. nothing resolution of 3/2
standard resolution?
(↑ Revised on
May 17, 2018)
Ukrainian Задача про два конверти 05:17, 19 ноября 2016‎ standard resolution nothing


Major revision history of this page

(This paragraph was added on July 29, 2018)

On March 10, 2019, the style of the index of this page was changed to the style using javascript.

On February 10, 2019, the paragraph "The new version of my main hypothesis about their mind during the period of DivideThreeByTwoian's resolution - Part 1" was added., and the paragraph "An opposed hypothesis to the above my main hypothesis about their mind during the period of DivideThreeByTwoian's resolution- Part 1" was deleted.

On October 7, 2018, the section "Resolutions may not be only one" was deleted because it overlapped with the section "Why are the two envelopes problem and the two envelope paradox so chaotic ?".

In 2017, the following sections were added.
On February 13, 2016 ,this page was greatly revised and titles of some sections was changed.

Reference

Terms



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