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The Way to the Inconsistent Variable Theory about the Two Envelope Paradox

I thought about the Inconsistent Variable Theory on The Two Envelope Paradox. 
And I have read some articles written by SinglePairians and some Wikipedia articles that describe a resolution that was led in SinglePairian's viewpoint , and found that there may be two stages on the way to the Inconsistent Variable Theory.

…  But on March 29, 2015, I have gotten yet another opinin.

…	But on April 4, 2016, I finally comprehended that the Inconsistent Variable Theory was only one of many fictions they made. Please see below for the details. Theory of "E = (1/2)2a + (1/2)a" on The Two Envelope Paradox An outline of the Two Envelopes Problem

DoublePairian & SinglePairian

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



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

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



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

What decide whether we become a DoublePairian or SinglePairian

I predict that wording of the expression of 'the two envelopes problem' affects whether we become DoublePairian or SinglePairian.

I analyze the characteristic of the problem expression as follows.

structure of the problem expression

A problem expression can be divided into 'Rule part' and 'Paradox part'.

In the Rule part the rule of the game is described.
In the Paradox part the way to get a paradox is described.

An example of Rule part

An example of Paradox part

Properties of Rule part

Facets of classification	Classification	Meaning of the classification
Closed or Opened	Closed	You do not know the amount in your envelope at trading of envelopes.
	Opened	You find the amount in your envelope before trading of envelopes.
Setting process	No setting process	The process to enclose money with envelopes is not written.
	Second is double	The amount of money to secondly enclose is as the twice the first envelope.
	Second is double or half	The amount of money to secondly enclose can be as the twice the first envelope or as the half.
	Only One Pair	Example: $100 and $200

Properties of Paradox part

Facets of classification	Classification	Meaning of the classification
Values of amount in the Paradox part	no specific, no variable	Example: The amount in the another envelope may be half of yours or double of yours
	Specific 1:4	Example: The amount in the another envelope may be $10 or $40
	Specific 1:2	Example: The amount of money may be $20 or $40. If the amount in your envelop is $20 then the amount in the another envelope shall be $40.
	Variable 1:4	Example: The amount in the another envelope may be A/2 or 2A
	Variable 1:2	Example: If the amount in yours is A then it in the another envelope shall be 2A
	Variable without probability	Example: You will gain X , or loss X/2. In any case you will gain more than X/2
	Rate of gain without probability	Example 1: You will gain 100% , or loss 50 %. In any case you will gain more than 50%
		Example 2: You will gain same amount as yours , or loss half amount as yours. In any case you will gain more amount than half of yours.
	Rate of gain with probability Added on March 14, 2015.	Example: After exchange, your money will be doubled with a probability 1/2 , and it will be halved with a probability 1/2.
	Using variable symbole to easily produce Inconsistent Variable	Example 1: In a probability 1/2, x = 2M and y = M,  and in the same probability, y = x/2. In a probability 1/2, x = M and y = 2M,  and in the same probability, y = 2x.
		Example 2: With a probability 1/2, the onother envelope contains the lesser amount x, and with a probabilty 1/2 it contains the greater amount 2x. Added on March 14, 2015.

Properties of the assertion by SinglePairian

The conculusion of the articles written by SinglePairian has following propertie.

Facets of classification	Classification	Meaning of the classification
Assertion	Only rejecting the expectation formula	Example; The another envelope is either the larger or the smaller, but not both. There are not three possibilities of amount.
	Advocating IVT	Advocating the Inconsistent Variable Theory.

Example of the Articles Written by SinglePairians

I have read some articles written by SinglePairians. Following is the result.

No.	Closed or Opened	Setting process	Values of amount in the Paradox part	Assertion
2	Closed	No setting process	Variable 1:4	Advocating IVT
3	Closed	No setting process	Specific 1:4	Only rejecting the expectation formula
5	Opened	No setting process	Variable 1:4	Advocating IVT
6	Opened	No setting process	Specific 1:4	Only rejecting the expectation formula
7	Closed	Only One Pair	Variable 1:4	Advocating IVT
8	Closed	Second is double or half	Variable 1:4	Advocating IVT

↑ I removed case 4 on March 27, 2015, and added case 8 on April 3, 2015, and removed case 1 on July 11, 2015.

Example of the Wikipedia Articles which describe resolution within SinglePairian's View

The case of English language Wikipedia

The Inconsistent Variable Theory is described as "Common Resolution" in the English language Wikipedia article titled "Two envelopes problem" (revision of 21:21, 14 July 2014).
And the article refers to some paper which is advocating the theory.
After reading this, I became unable to deny the possibility that the "Inconsistent Use of Variable" in the expectation formula of the two envelopes problem can happen in real human brain.

Remark !
The oldest revision of the article which describe the Inconsistent Variable Theory is the revision 22:05, 3 October 2005. This revision is very short and simple. But I think that it includes almost all contents of the revision of 21:21, 14 July 2014. 
↑ Added on October 11, 2014.

The cases of another language Wikipedia

On July, 2014, I examined another language Wikipedia articles.
Following table is the result.


The term "Open" in this table was changed to "Opened" on May 3, 2015.

language	Title of Article (revision) (original revision)	Closed or Opened	Values of amount in the Paradox part	Is the equivalence of the two envelopes explained on SinglePairian's mental model ?	Is Inconsistent Variable Theory described ?	Is resolution on DoublePairian's mental model described ?
Deutsch	"Die Umtauschsituation' in the article "Umtauschparadoxon" (am 14. Juni 2014 um 08:43) (27. März 2005 um 09:48)	Opened	Specific 1:4 50, 100, 200 Euro	NO	–	YES
English	"Two envelopes problem" (21:21, 14 July 2014) (22:36, 25 August 2005)	Closed (Opened virsion at original revision) (switch to closed virsion at the revision 22:05, 3 October 2005)	Specific 1:4 $10, $20, $40	YES	YES	YES
Français	"Paradoxe des deux enveloppes" (du 23 avril 2014 à 18:53) (du 22 mars 2006 à 20:11)	Closed	Variable 1:4 M/2, M, 2M	YES	YES ?	NO
Italiano	"Paradosso delle due buste" (5 lug 2013 alle 13:01) (20 feb 2006 alle 11:47)	Opened	Variable 1:4 A/2, A, 2A	YES	YES ?	YES
Magyar	Kétborítékos paradoxon (2013, március 8, 11:10-kor) (2006. április 28., 11:04-kor)	Opened	Variable 1:4 A/2, A, 2A	YES	YES	YES
Nederlands	"Enveloppenparadox" (op 13 feb 2014 18:33.) (op 8 mrt 2005 om 22:41)	Closed (Opened virsion at original revision) (switch to closed virsion at the revision op 9 mrt 2005 om 13:33)	Rate of gain without probability gain 100%, loss 50%	YES	NO ?	YES
Русский	"Задача о двух конвертах" (14:19, 10 июля 2014) (13:39, 20 августа 2009)	Opened	Variable 1:4 X/2, X, 2X	NO	–	YES

Seeing the above tables, I have found following fact.

• I have expected that wording of the two envelopes problem affects whether we become a DoublePairian or SinglePairian.  But this expectation may be wrong.
• Even if the "Closed or Opened" property is "Opened", and the "Values of amount in the Paradox part" property is "Specific 1:4", some SinglePairians continue holding SinglePairian's mental model.
• Not all of the people who continue holding SinglePairian's mental model advocate the Inconsistent Variable Theory.
• Some SinglePairians understood the paradox with three specific amounts and advocate the Inconsistent Variable Theory.
  This fact suggest that the people who advocated the theory did not use Inconsistent Variable in their brains.

There may be two stages on the way to the Inconsistent Variable Theory

These findings let me think that there may be two stages on the way to the Inconsistent Variable Theory.

step	DoublePairians	SinglePairians
Read Rule part	They imagine SinglePairian's mental model	They imagine SinglePairian's mental model
Read Paradox part	They switch mental model to DoublePairian's mental model	They can not switch mental model to DoubleParin's mental model.
Thinking	Someone doubt the probability. Someone think infinite number of tries of the game. Et cetera	They doubt the expectation formula
		Stage 1 They reject the expectation formula.
		They wonder why other people can find a paradox.
		Stage 2 They conceive of the Inconsistent Variable Theory.

But on March 29, 2015, I have gotten yet another opinin.

On March 14, 2015, .I had gotten another opinion about the way to the Inconsistent Variable Theory from the findings in "Example of the Articles Written by SinglePairians"  and "Example of the Wikipedia Articles which describe resolution within SinglePairian's View".
And on March 29, 2015. I have gotten yet another opinion.

The following table describes these new opinions.

↓The header of this table was revised on May 3, 2015 and August 11, 2016.

step	DoublePairians Self-styled SinglePairian Make no story	DoublePairians Self-styled SinglePairian Make a story of IVT	DoublePairians Self-styled SinglePairian Influenced by the "Not three amounts theory"	DoublePairians Self-styled SinglePairian Influenced by IVT
Read Rule part	They imagine SinglePairian's mental model	They imagine SinglePairian's mental model	They imagine SinglePairian's mental model	They imagine SinglePairian's mental model
Read Paradox part	They switch mental model to DoublePairian's mental model	They switch mental model to DoublePairian's mental model	They switch mental model to DoublePairian's mental model	They switch mental model to DoublePairian's mental model
Thinking	They can't doubt the probability.	They can't doubt the probability.	They can't doubt the probability.	They can't doubt the probability.
Reject two pairs of amounts of money	They find that if the expectation formula is comprised of only two amount of money A and 2A then the paradox will vanish.	They find that if the expectation formula is comprised of only two amount of money A and 2A then the paradox will vanish.
	They think that the cause of the paradox is to think of amounts X/2 and 2X.	They think that the cause of the paradox is to think of amounts X/2 and 2X.
Make a story	They don't go ahead.	They go ahead and make a story of IVT to rationalize their thought.
Influenced by suggestion			They read the "Not three amounts theory" which says that the cause of the paradox is to think of amounts X/2 and 2X. And they think themselves SingleParian, but they don't advocate IVT.	They read the IVT. And they think themselves SingleParian, and advocate IVT.

But on April 4, 2016, I finally comprehended that the Inconsistent Variable Theory was only one of many fictions they made.

(This paragraph was added on August 11, 2016.)

On April 4, 2016, I conceived folowing idea.

And I finally comprehended that the Inconsistent Variable Theory was only one of many fictions they made.

Please see below for the details.

• Theory of "E = (1/2)2a + (1/2)a" on The Two Envelope Paradox
• An outline of the Two Envelopes Problem

Terms

• DivideThreeByTwoian
  When I think of the people who think that the expectation formula E=(1/2)A + (1/2)2A" is the correct version of the paradoxical formula E=(1/2)(x/2) + (1/2)2x, I call them "DivideThreeByTwoian", because (1/2)A + (1/2)2A = (3/2)A.
   
  

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