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Return to the list of my pages written in English about the two envelopes problem
2024/04/15 5:05:32
First edition 2015/01/26

An outline of the Two Envelopes Problem


On March 17, 2024, a new section "How did DivideThreeByTwoians come up with their resolutions? New." was added.
On September 22, 2022, I made an archive of this page as 'Archive of the page "An outline of the Two Envelopes Problem" rev at 2022/09/20 15:41:10', and I started great revising.

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.

On March 17, 2024, the section "Easily understandable explanation" was moved from here.

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.
Totally revised on June 3, 2023.

Composition of the problem description

Part that describes the game settings

This part describes the situation just before exchanging envelopes.
common description
  • there is two envelopes each containes money
  • amount of money in one envelope is double as the other
  • the player is handed one envelope randomly selected from these aenvelopes
  • the player does not know his envelope containing which amount
  • the player is given the right to swap envelopes
variations
  • Who decides the first choice (← Added on April 01, 2024)
    • the game master
    • the player oneself
    • e.t.c
  • Whether player opens own envelope before exchange
    • the player opens (opened virsion)
    • the player does not open (closed virsion)
  • nunber of players
    • one
    • two
  • who is the player
    • you
    • game master nephews
    • e.t.c

Part that explains how the other amount expected

This part explains how the expected amount in the other envelope and the expected profit from the swap are calculated.
variations
  • Are specific amounts used as example? Or are only variable symbols used?
    • Specific example amounts are used
      An example from Zabell, S. (1988): 
      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.
    • Only variable symbols are used
      An example from Jackson, F., Menzies, P., & Oppy, G. (1994). 
                                                          Suppose that the amount of money
      in A is $x. Then B either contains $2x or $0.5x. Each possibility is equally likely,
      hence the expected value of taking B is 0.5.$2x + 0.5.$0.5x =$1.25x, a gain of
      $0.25x.

Part that describes paradox

This part describes various paradoxes induced by the expectation.
Examples of paradoxes

The psychological mechanism for noticing paradoxes

The following are inherited from older revisions.

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

  
Standard resolution and resolved paradoxes

(Added on June 14, 2017. Totally revised on August 5, 2018.)
Interpretation of the problem (← Added on April 01, 2024)
The standard resolvers understood that the problem was about conditional expectations. And they started to examin the expectation formula in the context of double pairs of amounts.
↓
Detection of incorrect part of the expectation formula
There is no possibility of mistaking except probability 1/2.
↓
The standard resolution
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.
↓
Resolved paradoxes (← Hyper link was deleted on November 11, 2018)
Paradox of the broken symmetry (standard version)
The law of total expectation can be confirmed by the correct expectation formula.
Paradox of the two envelopes which are greener than each other (on the opened version problem)
The correct expectation formula says that there is no problem even if they are greener than each other.
Paradox of the two envelopes which are greener than each other (on the closed version problem)
The paradox converges to the Paradox of the broken symmetry (standard version).
 
DivideThreeByTwoian's paradox and resolution

(Revised on June 14, 2017.)
Mental impact of the image of unopened envelopes

For details please see "The closed version problem has magical powers".
↓
DivideThreeByTwoian's paradox
Illusion of objective equivalence
This illusion let some people think that correct expectation formula should suggest that exchange of envelopes is not advantageous nor disadvantageous.
Illusion of objective expectation
This illusion let some people think that there is only one correct expectation formula.

The paragrph "The primitive concept of expected value" was deleted on April 01, 2024.
↓
DivideThreeByTwoian's resolution
The above Illusions made them seek a hidden a magic trick behind the symbol x/2 and 2x. And they got an image of one pair of amounts of money enveloped in the two envelopes, and they examined the expectation formula in the context of one pair of amounts.
(↑ Revised on March 8, 2020, April 01, 2024)

They claimed that they had resolved the paradox with various arguments as follows which created to rationalize their opinion. (↑ Added on December 23, 2018. Revised on January 8, 2019, April 7, 2019)

For details please see "My hypotheses about the mind of DivideThreeByTwoians".

On April 01, 2024, the pargraph "They will not cross forever." was deleted.

Ambiguity of the problem description and a mystery about it

Multiple major ambiguities in the part that explains how the other amount expected

  • Ambiguity of the situation for the expected value calculation
    • situation that a specific amount is in the handed envelope?
    • situation that a specific pair of amounts are in the envelopes?
    • situation that the ratio of the lesser and the greater amounts is 1 to 2?
  • Ambiguity of the specification of the amount of the money in the handed envelope
    • the amount is fixed regrdless it is revealed or not?
    • the unrevealed amount is not fixed?
  • Ambiguity of the meaning of the probability 1/2 (← Revised on April 3, 2024)
    • It means the probability that the handed amount is less/great than the other amount?
      In other words, it means the probability that the handed envelope is the lesser/greator envelope?
    • It means the probability that the handed amount is the lesser/greater amount of the mentioned pair of amounts?
    • It means the probability that the mentioned handed amount is the lesser/greater amount of some pair of amounts?

Variaty of the fallacy depending on the interpretation of the problem

The two major interpretations
  • DivideThreeByTwoians interpletaion and fallacies
    Ambiguas point Interpretation
    the situation for the expected value calculation specific pair of amounts are in the envelopes
    the amount in the handed envelope not fixed value
    meaning of the probability 1/2　 probability that the handed amount is the lesser/greater amount of the mentioned pair of amounts
      (In this interpretation the probability is fixed at 1/2)
    fallacy non rigid variable symbol is used in the calculation formula
    • in "x/2", x denotes the greator amount of the pair of amounts
    • in "2x", x denotes the lesser amount of the pair of amounts
    expected amount they think correctly calculated (1/2)2A + (1/2)A
      (A is the lesser amount of the pair of amounts)
  • Standard resolver's interpletaion and fallacies (← Revised on June 6, 2023)
    Ambiguas point Interpretation
    the situation for the expected value calculation specific amount is in the handed envelope
    the amount in the handed envelope fixed value
    meaning of the probability 1/2　 probability that the handed amount is less/greator than the other amount
     (In this interpretation, the probability is not necessarily 1/2 if the handed amount is fixed)
    fallacy probability 1/2 is an illusion of probability
    (in my opinion it is a kind of Base rate fallacy)
    expected amount they think correctly calculated (probablity that X is the lesser)2X +
    (probablity that X is the geator)(1/2)X

Interpretations not so popular
(The paragraph "LesserOrGreaterMeanValuean's interpretaition and fallacies" was replaced with this paragraph, on June 4, 2023)
  • Possible but unseen interpretation
    Ambiguas point Interpretation
    the situation for the expected value calculation situation that the ratio of the lesser and the greater amounts is 1 to 2
    the amount in the handed envelope not fixed value
    meaning of the probability 1/2　 probability that the handed amount is the lesser/greater amount of some pair of amounts
     (In this interpretation the probability is fixed at 1/2)
    fallacy non rigid variable symbol is used in the calculation formula
    • in "x/2", x denotes the greator amount of some unknown pair of amounts
    • in "2x", x denotes the lesser amount of some unknown pair of amounts
    expected amount they think correctly calculated Nothing
      (In the first place, it is also questionable whether the two amounts represented by "x/2" and "2x" are in the same currency unit)

The biggest mystery about the two envelopes problem

I think following is the biggest mystery about the two enveloes problem.
Incredibly few people accept the idea that there are many interpretations that lead to different fallacies and different resolutions.
(↑ Revised on June 6, 2023)

Reasons for such my opinion
  • I have never read an opinion which accepts both of the above the two major interpretations on the web.
    However, strangely enough, I couldn't find any essential difference in the problem sentences that were the basis of each interpretation, except for whether or not the handed envelope was opened.
    ↑ added on June 6, 2023
     
  • Reading the talk page of the English language Wikipedia article "Two Envelope Problem", it seems that there have been many edit warings between the two major interpretations. In my eyes, it looks like many editors of the sections by philosophical interpretation (DivideThreeByTwoian's interpretation) and many editors of the sections by mathematical interpretation (Standard resolver's interpletaion) were unable to accept each other's interpretation.
    ↑ revised on June 6, 2023
     
  • At the most beautiful revision at 18:32, 25 October 2005 of the article "Two envelopes problem" presented philospical interpretaion on the case of not opened envelope, and presented mathematical interpretation on the case of opened envelope. Without such segregation, it might be difficult for the two interpretations to coexist in the same person's mind.
     
  • Actually, I myself understand DivideThreeByTwoian's interpretation, but I cannot accept it.

My hypothesis about this mystery
They have experienced "Eureka effect" to resolve this mystery and the effect prevents them to accept the other interpretation of the problem".


On June 3, 2023, the paragraph "The standard resolvers and DivideThreeByTwoians resolved inherently different paradoxes." was deleted.

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 of the problem wording

There are various wordings of the two envelopes problem and none of them are standard.
(Specifically please see the section "Composition of the problem description".)
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)

A small difference in wording can make a big difference in meaning.

This paragraph was added on April 15, 2024.

I think there is no big difference in expression between the following wordings but a big difference in meaning.

Wording in Jackson, F., Menzies, P., & Oppy, G. (1994).
………
Suppose that the amount of money in A is $x. Then B either contains $2x or $0.5x. Each possibiity is equly likely, hence the expected value of taking B is 0.5・$2x + 0.5・$0.5x= $1.25x, a gain of $0.25x.
………

Wording that I made from the wording in the English language Wikipedia article "Envelope paradox" (revision at 10:55, 26 August 2004)

I got the following wording from the article by removing the phrase "you open one envelope and find in it."
………

Suppose you open one envelope and find in it the amount of money A.  You reason as follows:

  1. The other envelope may contain either ½A or 2A
  2. Because the envelopes were indistinguishable, either amount must be equally likely.
  3. So the expected value of the money in the other envelope is ½(½A + 2A) = 1¼A.
  4. This is greater than A, so you gain by swapping.
………

Wording in the English language Wikipedia article "Two envelopes problem" (revision at 22:05, 3 October 2005)
………

Denote by A the amount in your selected envelope. Now, suppose you reason as follows:

  1. The other envelope may contain either 2A or A/2
  2. The probability that A is the larger amount is ½, and that it's the smaller also ½
  3. If A is the smaller amount the other envelope contains 2A
  4. If A is the larger amount the other envelope contains A/2
  5. Thus, the other envelope contains 2A with probability ½ and A/2 with probability ½
  6. So the expected value of the money in the other envelope is ½(2A) + ½(A/2) = 1¼A
  7. This is greater than A, so you gain
………

I think that people will be prone to DoublePairian after reading the above former wordings. And I think that people will be prone to SinglePairian after reading the above latter wordings.

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 have developed several hypotheses regarding the psychology created by this magical power. (Please refer to The closed version problem has magical powers") (← Revised on March 17,2024)

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 calculation 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 existence of fake problems

(Added on March 1, 2018.)
(Greatly revised on April 15, 2024)
There are fake problems of the two envelopes problem.
For eample:
I think such problems have a power to invite us to the SinglePairian's problem. 

Addition:
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 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)

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 over the full range of the amount in the envelopes. (← Revised on April 01, 2024)

For some people, strategic judgment is the problem domain of the two envelopes problem.
(This paragraph was added on April 01, 2024)
Such people try the followings.
  • On the closed version problem, some people try to find the rational expectation formula which guarantee the equivalence of the envelopes over the full range of the amount in the envelopes.
  • On the opened version problem, some people try to find a computational method to decide whether to exchange or notto maximize gain.

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.


(The paragraph "Chaos of the research of the cause of the fallacy" was deleted on April 15, 2024.

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 and opened version problem

(This paragraph was added on March 22, 2018. Revised on May 3, 2018, July 15, 2018, July 22, 2018.)
(This paragraph was Tottally revised on April 15, 2024 with new title)
  • The following standard resolvers seemed to think that there is no paradox before opening envelope.
  • A revision of the English language Wikipedia article "Two envelopes problem" revised on 18:32, 25 October 2005 seemed equally weighted on the both of closed version and opened version.
  • Apparently some DivideThreeByTwoians thought there was no paradox after opening envelope.
    • Case of a paper published in a statistical journal in 2008.
    • From the revision at 20:17, 26 May 2010, there has been not described the standard resolution on the opened version problem.

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.


The paragraph "Chaos of the thought of psychologists" was detelete on April 15, 2024.

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)

There never seems to be a widely accepted solution

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.
The title was changed from "Have we already gotten a solution which is widely accepted ?" on March 17, 2024.
The title was changed a little on April 15, 2024.

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

No one can prove the wrongnes of the DivideThreeByTwoian's opinion

(On February 23, 2020, this title was changed)
(On April 15, 2024, this title was changed)
From October 3, 2005 up to the present (March 2024), the article "The two envelopes problem" in the English language Wikipedia had showed the DivideThreeByTwoian's opinions without a break.

To my eyes this is a mystery because the DivideThreeByTwoian's opinions seem strange.   (For details please see "How did DivideThreeByTwoians come up with their resolutions?")
(↑ Revised on September 11, 2017, February 23, 2020, March 17, 2024)

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 wrote as follows.

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

On March 17, 2024, I have to write as follows.

I've never seen anyone accept both standard resolution and DivideThreeByTwoian's resolution.


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.
………
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.
………
↑ 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.
………
Randomly, a person chooses one envelope, say A.
Before opening envelope A the person can change choice.
The person reasons as follows.
Let $x denote the amount of money in envelope A.
If $x is the lesser or the greater amount the other envelope B contains $2x or $0.5x respectively.  These possibilities are equally likely.
The expected value of the amount in envelope B is 0.5 × $2x + 0.5 × $0.5x = $1.25x.
………
↑ Revised on March 22, 2015, April 3, 2016, April 7, 2016, July 11, 2016, June 8, 2017, April 15, 2024.

Ambiguous Version

This paragraph was added on February 1, 2015.

A typical wording of the ambiguous version problem is as follows.
………
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.
………
↑ Revised on March 22, 2015, December 26, 2015, April 3, 2016, July 11, 2016, June 8, 2017.

Kinds of the wording about the ownership of the selected envelope.

(Added on November 06, 2022)

Clearly determined before the opportunity to trade is given

An example is as follows.
You may choose one or the other envelope, and keep the cheque it contains.
(From Broome,John.(1995).)

Probably determined when the player select one of the envelopes

An example is as follows.
… , and you select one at random. You are then offered the chance to swap and take the other instead.
(from Clark, Michael. & Shackel, Nicholas. (2000).)


On April 15, 2024, the paragraph "Kinds of the wording of the notation of the amount of money" was deleted.

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.

Paradox that several reasonings differently conclude about whether you should switch or not

(Added on April 15, 2024)

A tipical wording is the case of Priest, G., & Restall, G. (2003).

The fragment which referes to a pardox in it
………
Here are three forms of reasoning about this situation, which we shall call Forms 1, 2 and 3, respectively.

Form 1 Let n be the minimum of the quantities in the two envelopes. Then there are two possibilities, which we may depict as follows:
  Possibility 1 Possibility 2
Your Envelope n 2n
Other Envelope 2n n
………
Conclusion: SWITCHING IS A MATTER OF INDIFFERENCE.

Form 2 Let x be the amount of money in your envelope. Then there are two possibilities, which we may depict as follows:
  Possibility 1 Possibility 2
Your Envelope x x
Other Envelope 2x x/2
………
Conclusion: SWITCH.

Form 3 Let y be the amount of money in the other envelope. Then there are two possibilities, which we may depict as follows:
  Possibility 1 Possibility 2
Your Envelope 2y y/2
Other Envelope y y
………
Conclusion: KEEP.

Prima facie, Forms 1, 2 and 3 seem equally good as pieces of reasoning.
Yet it seems clear that they cannnot all be right. What should we say?


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.
This paragraph was totally revised on April 15, 2024.

The earliest example I know of is the case of McGrew, T. J., Shier, D., & Silverstein, H. S. (1997) is as follows.

The numbered steps in it
………
(1)  The envelope I have selected contains a certain positive amount
of money; call it x.
(2)  Either my envelope contains the higher amount or it contains the
lower amount - and these are eqully probable.
(3)  If my envelope contains the lower amount, then the other enve-
lope contains 2x; thus, by swapping I would gain x.
(4)  If my envelopecontains the higher amount, then the other enve-
lope contains x/2; thus by swapping I would lose only x/2.
(5)  Hence, the expected utility of swapping - namely, 1.25x (.5 times
2x plus .5time x/2) - is greater than the expected utility of stick-
ing with the original pick -namely, x.
………

Another early example:

On November 06, 2022, the section "Kinds of the wording how the probability 1/2 is combined with amounts of money" was removed as it appeared to be of low value.

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:

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

This paragraph was added on August 19, 2017.
This paragraph was totally revised with new title on April 15, 2024.

Wording which representing amounts of money with specific example amounts

A typical wording of such a kind is the case of Zabell, S. (1988).

A fragment in it which refers to amounts of money
………
A opens his envelope and see that there is $10 in it. He then reasons as follows:
"Thre is a 50-50 chance tht 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).
………

Meaning of symbols :
A, B  players
Creferee
xthe lesser amount of the pair of amounts
Remark: If $10 is the lesser amount the pair (x, 2x) is ($10, $20). And if $10 is the greator amount the pair (x, 2x) is ($5, $10).

Wording which denotes the amount of money in the selected envelope using a variable symbol

Allmost literatures are this type.
An Example is the case of Barbeau, E. (1990).

The fragment in it which referes to amounts of money
………
Suppose that the number revealed to you is A. Then the other cards has the number 2A or 0.5A, each with equal probaility. If you stick with the card shown, your expected winnings are A. If you switch, your expected winnings are 0.5(2A) + 0.5(0.5A) = 1.25A. Thus you should always select the card other than the one revealed to you.
………

Wording which denote the amounts of money in the selected envelope and the other envelope using two variable symbols

A typical wording of such a kind is the case of Broome,John.(1995).

The fragment in it which referes to amounts of money
………
Let the amount of the cheque in your chosen envelope be x. Whatever x may be, there is a probability of 1/2 that the cheque in the other envelope - call it y - is 2x and a probability of 1/2 that it is x/2. So, whatever x, the expectaion of y is (1/2)2x + (1/2)(x/2). This comes to 5x/4, which is more than x.
………


Kinds of the wording for expresion of the case the selected amount is the lesser and the ase it is the greator

This paragraph was added on April 15, 2024.

Wording which focused on envelopes (Example: the lesser envelope)

One example of this kind is the case of Cargile, J. (1992)

A fragment in it which referes to amounts of money
………
Let m be the amount in e1 and n be the amount in e2. We know that either n=2m or m=2n, and that U(E1/L∨￢L) is m and U(￢E1/L∨￢L) is n. Also, U(E1/L) = m = 2n, and U(E1/￢L) = m = n/2, and U(￢E1/L) = n = m/2, and U(￢E1/L) = n =2m.
………
Since m＝2n when L and n/2 when ￢L, we have:  U(E1/L∨￢L) = 2nPr(L) + (n/2)Pr(￢L). And for Pr(L) = 1/2, this is (1.25)n. That is, it is better, 25% better, than U(￢E1/L∨￢L). At the same time, we know that U(￢E1/L∨￢L) = (m/2)Pr(L) + 2mPr(L), which, for Pr(L) = 1/2 is (1.25)m. Which is to say that U(￢E1/L∨￢L) is 25% better than U(E1/L∨￢L).
………

Meaning of symbols :
e1, e2 two envelopes
E1the propoition that you choose e1
Lthe proposition that e1 contains the larger amount
U(E1/L), U(E1/￢L)the utility of E1 in case L, the utility of E1 in case ￢L
Pr(L), Pr(￢L)the probability of L, the probability of ￢L

Another example of this kind is the case of the English language Wikipedia article "Two envelopes problem" (revision at 22:58, 25 August 2005)

A fragment in it which referes to amounts of money
………
Each brother goes to his own room to opens his envelope. The first one sees that the amount of money in the envelope is y.
………
If he doesn't swap envelopes, he will have a profit of y. Now there is 50% chance that he has the smaller envelope, and 50% chance he has the larger one. Therefore, the expected value of swapping is 50% (y/2) + 50% (2y) = 1.25y.
………

Remark:
With this kind of wording we cannot think about possibilities with probabilities other than 1/2.

Wording which focused on amount of money

Allmost literatures are this type.

For example, the case of Jackson, F., Menzies, P., & Oppy, G. (1994). is as follows.

A fragment in it which referes to amounts of money
………
Suppose that the amount of money in A is $x. Then B either contains $2x or $0.5x. Each possibiity is equly likely, hence the expected value of taking B is 0.5・$2x + 0.5・$0.5x= $1.25x, a gain of $0.25x.
………


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)

On April 15, 2024, the paragraph "Kinds of the wording for identifying each of the two envelopes" was integratied to another paragraph.

Wordings which make the problem not the two envelopes problem

This paragraph was added on September 26, 2015.
This paragraph was moved here on March 17, 2024.

Wording which concerns mean values rather than particular values

A typical wording of such a kind is as follows.
………
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).
………
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.

Wordings which are 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.

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)

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 rather than SinglePairian after reading such a wordings.
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.

On April 15, 2024, the paragraph "Wording which has a menu of reasonings" was deleted.  

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 would give us some fun with other fallacies, such as "Discharge fallacy" in Jeffrey, R. (2004)." rather than "Base rate fallacy". (← Reivised onMarch 17, 2024)

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.
The found wording was as follows.
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
(↑ Added on March 17, 2024)
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".

On April 15, 2024, the paragraph "Ranking of closed version problem about the power that makes us SinglePairians" was deleted.

On March 17, 2024, the pragraph "About the power to make us LesserOrGreaterMeanValuean" was deleted.

Comparison of wordings preferred by DivideThreeByTwoians and Non-DivideThreeByTwoians

This paragraph was added on November 06, 2022.

I compaired wordings used by DivideThreeByTwoians and them used by Non-DivideThreeByTwoians.

Cases of Unopened envelope

Cases of Non-DivideThreeByTwoian Philosophers
case Steps to put each amount in each envelope Who does select envelope? Does the money in the chosen envelope belong to the player? Does the player have an opponent? Who thinks about expected value? Is the expectation formula made of an example amount of money? What are the probabilities in the expectation formula?
A paper published in 1994 No description The player Yes No The author of the problem statement No (It is made imagining that the chosen envelope contains $x) probabilities that the opposite amount is double or half of the chosen amount
A paper published in 1995 No description The player Yes No The readers of the problem imagining that they are the player Yes (It is made imagining that the chosen envelope contains $100) probabilities that the opposite amount is the lesser or the greater
A paper published in 1995 No description The player Yes No The readers of the problem imagining that they are the player No (It is made imagining that the chosen envelope contains x) probablities that the oposite amount is double or half of the chosen amount
A paper published in 2000 No description The player Yes No The readers of the problem imagining that they are the player No (It is made imagining that the chosen envelope contains x) probabilities that the opposite amount is double or half of the chosen amount

Cases of DivideThreeByTwoian Philosophers
case Steps to put each amount in each envelope Who does select envelope? Does the money in the chosen envelope belong to the player? Does the player have an opponent? Who thinks about expected value? Is the expectation formula made of an example amount of money? What are the probabilities in the expectation formula?
A paper published in 1997 No description The player Yes No "I" (the writer of the problem) who is the player No (It is made imagining that the chosen envelope contains x) probablities that the chosen envelope is the lesser or the greater
A paper published in 2003 No description unspecified Yes No The readers of the problem imagining that they are the player No (It is made imagining that the chosen envelope contains x) probablities that the chosen envelope is the lesser or the greater
A paper published in 2007 No description The player Yes No The readers of the problem imagining that they are the player No (It is made imagining that the chosen envelope contains n) probablities that the chosen envelope is the lesser or the greater

Surprising findings

For a long time I thought there was no noticeable difference between the problem statements used by DivideThreeByTwoian philosophers and Non-DivideThreeTwoian philosophers.
However, from the above tables, I found that the meaning of the probabilities in the expectation formula were significantly different between the two.
DivideThreeByTwoian philosophers used the probabilities of selected envelope, while the Non-DivideThreeByTwoian philosophers used the probabilities of combinations of amounts.

An example by DivideThreeByTwoian philosopher
Either my envelope contains the higher amount or it contains the lower amount - and these are equally probable.
An example by Non-DivideThreeByTwoian philosopher
Then B either contais $2x or $0.5x. Each possibility is eually likely, hence the expected value of …

Addition: Cases of Opened envelope

Cases of Mathematicians
case Steps to put each amount in each envelope Who does select envelope? Does the money in the chosen envelope belong to the player? Does the player have an opponent? Who thinks about expected value? Is the expectation formula made of an example amount of money? What are the probabilities in the expectation formula?
Zabell, S. (1988) The refferee C places an unspecified amount of money x in one envelope and amount 2x in another envelope. The refferee C handed one of the two envelope to A, the other to B. Yes Yes (Players A and B) The player A Yes (The envelope handed to A contains $10) probabilities that the opposite amount is double or half of the chosen amount

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.)

On March 17,2024, the paragraph "DivideThreeByTwian philosophers seem to have been affected by a famous philosophical article." was deleted.

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.)
(One row was deleted on March 17, 2024.)

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

On March 17, 2024, settion "The third and fourth resolutions" haｓ been moved around the end of this page.

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 ?".

On March 17, 2024, the section "Worries only by paradox" was deleted.

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. (↑ One row was deleted on March 17, 2024)
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).

Easily understandable explanation

This section was added on September 9, 2017. Revised on September 13, 2017.
On March 17, 2024, this section was moved here.

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.

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.

This section was greatly revised on March 17, 2024.
Some paragraphs were deleted, and another paragraphs were moved to different locations.

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)

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)

DivideNineByEightian's resolution

This paragraph was added on March 24, 2019. The title was changed on May 5, 2019.
This paragraph was moved here on March 17, 2024.
 

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)

How did DivideThreeByTwoians come up with their resolutions? New (Mrch, 2024)

On March 17, 2024, this section had been created integlating and totaly revising two sections "DivideThreeByTwoian's resolutions may be wrong" and "DivideThreeByTwoian's mind".

Resolution by DivideThreeByTwoian is strange.

DivideThreeByTowians made contradictory theories.

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.

They did not state the basis for the assumption of the Single-pair-amount model.

There was no explicit instruction in the problem text to assume the Single-pair-amount model as a premise for calculating the expected value, so it was necessary to show the basis for this.

There was no indication in their arguments that they themselves have experienced fallacies.

They didn't express the mental state of mind, such as "There was not anything strange when I first looked at the expected value calculation formula, but after close analysis, I found something strange."
It seems like they were trying to uncover a magic trick hidden in the expected value formula from the beginning.

DivideThreeByTwoian philosophers may have thought the problem was a kind of magic with hidden tricks.

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 candidates.
  • Trick 1 : mental model change
  • Trick 2 : 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.
As a result they thought the challenge was not to fix the expected value calculation formula, but to reveal these tricks.

Mind of the early DivideThreeByTwoian philosophers

Many early DivideThreeByTwoian philosopherss advocated the DivideThreeByTwoian's resolution - Part 1".

Charcterlistic of the early DivideThreeByTwoian philosophers

Which model is used? What fallacies are pointed out? Which paradox is discussed?
This paragraph was added on April 01, 2024) Did they think the expected value calculation formula requires fixed selected amount?
They viewed that thinking about the chosen amount as fixed was the cause of the problem.
I think it suggests that they have a consciousness that the expected value calculation formula requires to fix the amount.
Did they consider the opened version problem meaningful?
They haven't touched on the topic.
Can't they leave from the probability of selection of the envelopes? In other words, can they accept a expectation formula which leads that exchanging is favorable more staying.?
They thought it mistake to discuss cases where the probability is not 1/2.

Hypotheses about the early DivideThreeByTwoian philosophers

Hypothesis that their true face is it of the DoublePairian
This hypothesis holds as follows.
They experienced an illusion of probability with the Double-pair-amount model at the beginning of the discussion, but the influence of the Alibaba-type problem led them to stray into the DivideThreeByTwoian's resolution on the Single-pair-amount model.
Hypothesis that their true face was it of the SinglePairian and they tried to reveal the trick the expected value calculation formula had
This hypothesis holds as follows.
They consistently took the Single-pair-amount model, and they examine the expected value calculation formula and revealed the trick of it.
They have discovered a trick, not a fallacy.

Mind of the early DivideThreeByTwoian editors of Wikipedia articles

Many early DivideThreeByTwoian editors of Wikipedia articles edited the DivideThreeByTwoian's resolution - Part 2".

Charcterlistic of the early DivideThreeByTwoian editors of Wikipedia articles

Which model is used?
Consistently Single-pair-amount model (SinglePairian's mental model)
What fallacies are pointed out? Which paradox is discussed?
This paragraph was added on April 01, 2024) Did they think the expected value calculation formula requires fixed selected amount?
There is no indication that they saw any validity or paradox in the expected value calculation formula with a fixed selected amount.
Did they consider the opened version problem meaningful?
Some editors thought that the opened version problem was a problem in the another dimension.
Another editors thought that the opened version problem was meaningless.
Can't they leave from the probability of selection of the envelopes? In other words, can they accept a expectation formula which leads that exchanging is favorable more staying.?
They haven't touched on the probability of combination of amounts.
And they persistently fought editing wars with mathematical resolution editors.

Hypotheses about the early DivideThreeByTwoian editors of Wikipedia articles

Hypothesis that they should be called the influenced SinglePairian
This hypothesis holds as follows.
They were influenced by some DivideThreeByTwoian's literature to become SinglePairian.

The early DivideThreeByTwoians' findings are something of a mirage.

The two envelope problem was just a riddle to them not a paradox.

If their true faces were DoublePairan
They were trapped in a wrong hypothesis that the Double-pair-amount model is the cause of the confusion while their real problem was the illusion of probability.

If their true faces were SinglePairan
They did not experience the polysemy of the variable symbol. And they could find no paradox as the expected value calculation formula is meaningless.

My hypotheses about the mind of DivideThreeByTwoian

How they (DivideThreeByTwoian) developed their theory.?

This paragraph was added on April 01, 2024.

I came up with the following explanation about the process that the early DivideThreeByTwoian philosophers and Wikipedia editors developed their theory.
The problem to be solved
They thought the problem to be solved is that the exchange is advantageous no matter which envelope is chosen.  Therefore, to avoid this, they thought it necessary to show that the correct expected value formula does not suggest a favorable exchange.
↓
Diagnosis of fallacies in the fallacious expected value formula
↓
Discovering the formula "(1/2)A + (1/2)2A"as a correct formula
They obtained a formula "(1/2)A + (1/2)2A" by repairing the fallacious expected value formula "(1/2)(x/2) + (1/2)2x" using Inconsistent-variable theory, and they found that the obtained formula guarantees that the two envelopes are equally favorable.
↓
Declaration that the paradox has been resolved
Since repaired expected value formula obtained by their theory guarantees the equivalence, they thought the paradox was resolved.

Why they (DivideThreeByTwoian) avoid the idea of probability miscalculation?

This header was added on April 01, 2024.

I have developed several hypotheses to explain why they (philosophers and Wikipedia editors) stubbornly avoid the idea of probability miscalculation.

Vivid image of the envelopes gotten by changing the problem from opened version to closed version

This hypothesis holds as follows.
If the problem changed to closed version from opened version the envelopes become more vivid than amounts inside. As a result, they become unaware of the probability of combinations of amounts.

Rollback which gives you a vivid image of one pair of amounts

This hypothesis holds as follows.
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".

Reading sentence by sentence of the problem text

This hypothesis holds as follows.
If you read the following text sentence by sentence, different values will be assigned to the symbol x.
In contrast If, you read the following text with one glance, same value will be assigned to the symbol x. 
Ler x denote the amount in the handed envelope.
If the envelope is the lesser the amount in the another is 2x.
If the envelope is the greator the amount in the another is x/2. 

Aversion to the probability of past events

This hypothesis holds as follows.
They felt strong cleepiness about Bayesian probability as it calculate certainty of past events. As a result they avoided the idea of probability illusion as it requires to calculate Bayesian probability..

Expected value as a kind of conserved quantity

This hypothesis holds as follows.
They assumed that the expected value calculated before choosing the envelope would hold after the choice. Threfore the probability illusion theory was not accepted as the varying probabilities could destroy equality.
Addition: Probability is never a type of conserved quantity as it is a type of ratio.

Primitive concept of expected value

This hypothesis holds as follows.
DivideThreeByTwoians think that expected value is determined by the combination of one situation and one decision.

As a result, they reject the mathematical expectation.

Eureka effect

This hypothesis holds as follows.
Human who have experienced eureka effect become not able to accept the another solution. So if you experienced eureka effect by DivedThreeByTwoians solution, you will be not able to accept the theory of probability illusion.
Addition 1:
There is a person who wrote a paper enlightening of the theory of polysemy of the variable symbol after being impressed by it.
Addition 2:
Those who advocate the illusion of probability cannot imagine themselves experiencing the fallacy expressed in the DivideThreeByTwoian opinion.

Motivation to get assurance that you won't have to switch

This hypothesis holds as follows.
They thought the purpose of solving the problem was to make sure that switching envelope is not better than holding. So they could not accept any opinion that the expected value calculation formula may suggests to switch.

Motivation to write a paper on linguistics

For them, the two-envelope problem is a good topic to write a paper on the linguistics they have learned. In order to write such a paper, it was necessary to avoid ideas that focused on probability theory.

A mechanism like the image of "My Wife and My Mother-in-Law"

This hypothesis holds as follows.
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.

The closed version problem has magical powers

To my eyes, the resolution of DivideThreeByTwoians looks very mysterious as follows.
  • To my eye, the problem representation they are using looks more familiar to the Double-pair-amount model than the Single-pair-amount model.
  • The problem text written by DivideThreeByTwoians has no notable features except that the trading opportunity is given before opening envelope.
  • Despite the above, the population of DivideThreByTwoians is unexpectedly large.
    (Please see paragraph "But the standard resolvers are not majority".)
Why? Does the closed version problem have magical powers?

I think this might be explained by some of the above hypotheses about DivideThreeByTwoians' minds.

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.

The third and fourth resolutions

(↑ This header was added on March 10, 2019) (&nbp; On March 17, 2024, this section was moved here.

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.
 

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