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2018/10/23 9:15:24
First edition 2014/09/27

A fictional history of the two envelopes problem

The word "two envelope" in this title was changed to "two envelopes" on January 23, 2015.

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

I substantially revised this page on April 3, 2015. And I rewound it on the next day.

I wonder how the original of the two envelopes problem was

I think that possibly we may be caught by inconsistent variable when we read the original of the two envelopes problem.  Using search engine, I had searched the original problem. But I could not find it. So, I imagined the original of the problem and a fictional history of the two envelopes problem.

Explanation of terms used in this page

I use following terms to describe the history of the Two Envelope Paradox.

Term Meaning Example
Rule Part In the Rule part the rule of the game is described. One of two envelope contains some amount of money and another contains twice as much as it.
At random one of them become your envelope (you pick it or you are given it).
You are able to trade your envelope with another.
Paradox Part In the Paradox part the way to get a paradox is described Let A be the amount of money in your envelope.
The Expectation of the amount of another envelope is (1/2)(A/2) + (1/2)2A = 1.25A.
Therefore the expectation of the amounts in the another envelope is always larger than yours.
It contradicts the equality of the two envelopes.
Setting process The process of placing money in the wallets or envelopes The amount of money to secondly enclose is as the twice the first envelope.

A fictional history of the two envelopes problem

Attention
The contents of this section is just a figment of my imagination.

Wallet Game Period

The Wallet Game was the ancestor of the two envelopes problem, and it had following properties.
  • Two players do not know the amounts of money in his own wallet and opponent's wallet.
  • No variable symbol was used in setting process.
  • No specific example amounts of the money in the two wallets are described.
  • Expected value was calculated without variable symbol.
The problem was like this.
Equally rich two persons play following game.

RULE PART
They place their wallets on the table.
Each of the two person do not know the amount of money in his wallet.
Each of the two person do not know the amount of money in the opponent's wallet.
Whoever has wallet which contains the smaller amount of money wins all the money in the other wallet. (← Revised on October 10, 2016.)

PARADOX PART
Each of the players reason as follows.
"I may lose what I have but I may also win more than I have. So the game is to may advantage."
Taking following article into account, I wrote this.
The readers of the problem could be caught by the illusion of probability,  
and they also could be caught by the inconsistent mental variable.
 
I imagine that the original two envelopes problem is the Opened version problem

This section was greately revised on April 2, 2017.

I imagine that an original of the two envelopes problem may be as follows.
It has following properties.
  • The opportunity to swap the envelopes is given before the chosen envelope is opened.
  • No variable symbol was used in rule part.
  • One specific example amount in the chosen envelope and two specific example amounts in other envelope are described.
  • Expected value was calculated without variable symbol.
The problem might be like this.
There are two envelopes.

RULE PART
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.

PARADOX PART
He reasons as follows:
"There is a 50-50 chance that B's envelope contains the lesser amount which would be $5,
and a 50-50 chance that B's envelope contains the greater amount which would be $20.
If I exchange envelopes, my expected holdings will be (1/2)$5 + (1/2)$20 = $12.50.
Therefore I should try to exchange envelopes."
When A offers to exchange envelopes, B readily agrees, since B has already reasoned in similar fashion.
It seemes unreasonable that the exchange be faorable to both.
Taking following article into account, I wrote this.
  • Zabell, S. (1988)
    Note: In this article, the expected value was calculated with mathematical expectation formula using specific amounts.

The readers of the problem could be caught by the illusion of probability,  
but they could not be caught by the inconsistent mental variable nor by the inconsistent explicit variable.
 
Introduction of variable symbol into the expectation calculating formula

This section was greately revised on April 2, 2017.

The wording of the two envelopes problem was changed to contain explicit expectation formula.
  • The opportunity to swap the envelopes is given after the chosen envelope is opened.
  • A variable symbol is introduced in rule part.
  • No specific example amount is described.
  • A variable symbol is used in the expectation formula.
The problem might be like this.
You are to select one of two envelopes that each contains a check.

RULE PART
One of the checks has a face value twice that of the other.
After making a selection, you open the envelope and learn the face value of the check is $A.
You are now offered the option of exchanging the check you hold for the one in the remaining envelope.

PARADOX PART
If you exchange envelopes, with even odds, you will either double or halve your initial wininngs.
Your expected gain on the exchange is (1/2)(2A) + (1/2)(A/2) - A = A/4, which is strictry positive.
Because the symmetry in the initial choice of envelopes is given, this seemes absurd.
Taking following article into account, I wrote this.
The readers of the problem could be caught by the illusion of probability,  
but they could not be caught by the inconsistent mental variable nor by the inconsistent explicit variable.
 
I think that
in this period any people thought that
the Closed version problem does not induce a paradox.

(↑ Revised on April 8, 2017.)
Philosophers started to submit papers of philosophy about the Closed version problem

Beginning of thing
(Revised on October 23, 2018)


The first ambiguous version problem
In 1992, a philosopher presented a "Ambiguous version" problem on which no description of the opportunity to trade envelopes is described.
I think that the paper triggered discussion of the Closed version problem.

The early closed version problem
In 1994, some philosophers who were good at mathematics submitted papers to journals of philosophy. They were not DivideThreeByTwoians, but their papers have magical power to let other philosophers be DivideThreeByTwoians
  • The wording they used was "Closed version" in which the opportunity of trade is given before opening envelope.
    (This item was added on August 14,2014.)
  • Their paper discussed cases that always opposite envelope is favorable and there are no mathematical resolution.
    • improper distribution of amounts of money in the two envelopes
    • proper but paradoxical distribution
  • Some of them referred to the expectation formula "E=(1/2)A + (1/2)2A" which DivideThreeByTwoians idolise.

Thereafter some other philosophers became DivideThreeByTwoians and wrote papers
(Revised on Jun 26, 2017.)


Some of the philosophers who had read the above-mentioned papers had thought as below and became DivideThreeByTwoians.
  • The two envelopes paradox had not been yet resolved by mathematicians.
  • The paradox would be resolved with the expectation formula "E=(1/2)A + (1/2)2A".
In the late 1990s and the early 2000s, they submitted papers.
I expect that their papers influenced English language Wikipedia. And I expect that they had indirectly influenced French language Wikipedia and others.

The possibility that an article "Nalebuff, Barry.(1989)" had influenced philosophers.
(Added on April 16, 2017. Revised on July 6, 2017.)

The game described in the first problem presented in "Nalebuff, Barry.(1989)" is like below.
  • After the amount of money in the envelope which is handed to Ali is determined the amount of money in the envelope which will be handed to Baba is determined to be twice or half the Ali's amount depending on the result of a coin flip.
I call this problem the"Ali-Baba" version problem.

The "Ali-Baba" version problem has the following aspects.
  • From Ali's point of view the expected amount of money in Baba's envelope is calculated by the expectation formula E=(1/2)(x/2)+(1/2)2x which has been presented in the two envelopes problem.
  • This calculation gives Ali a correct expectation while this gives wrong expectation in the two envelopes problem.
Many philosphers who were DivideThreeByTwoians refered this article, so I think that the aspect of this problem might have unintendedly effect on philosophers thinking and might have made them DivideThreeByTwoians.
 
↑ Added on August 12, 2016.
Change after that
(Revised on April 16, 2017.)


In the English language regions
(Revised on Jun 26,2017.)

At 10:55, 26 August 2004, the article "Envelope paradox" in the English language Wikipedia was created as the first article abut the two envelopes problem. (How to read it)
On it Mathematical standard resolution was presented by the main editor, and sevral edit warring broke out among the main editor and some DivideThreeByTwoians.

At 22:36, 25 August 2005, the article "Two envelopes problem" in the English language Wikipedia was created as the second article abut the two envelopes problem.
The first revision of it presented Opened version problem, but at 22:05, 3 October 2005 the contents was replaced with DivideThreeByTwoian's opinion.

At 07:47, 24 August 2006, the article "Envelope pardox" was redirected to the article "Two envelopes problem".
Since then, Mathematical standard resolution has been treated as a supplementary opinion of the DivideThreeByTwoian's opinion in the article "Two envelopes problem" of theEnglish language Wikipedia.

As the time goes on, in the English language regions, Closed version problem has become more popular than Opened version problem.
And the following wording of the two envelopes problem were spread by the people who prefered theory of "E=(1/2)2a + (1/2)a".
  • The opportunity to swap the envelopes is given before the chosen envelope is opened. · · · (1)
  • A variable symbol is used in rule part.
  • No specific example amount of money in the chosen envelope is presented and no specific example amounts of money in other envelope are presented. · · · (2)
  • Expected value is calculated with mathematical expectation formula.
  • A variable symbol is used in the expectation formula.
This kind of typical problem sentence has been written in the revision of 22:05, 3 October 2005 of the article "The two envelopes problem" in the English language Wikipedia.
In this revision of that article, there is an expression which derives inconsistent explicit variable.
The probability that A is the larger amount is ½, and that it's the smaller also ½
If A is the smaller amount the other envelope contains 2A
If A is the larger amount the other envelope contains A/2
(↑ Added on April 4, 2015)
And the same revision of 22:05, 3 October 2005 started presenting the inconsistent variable theory.
I think that this theory made the two envelopes problem a puzzle rather than paradox.
→ How the paradox turned into a puzzle
(↑ Added on July 6, 2017.)

In other language regions

In some of other language regions, Closed version problem has not become major. And Opened version problem has been more popular.

Caution
I can not read German, Russian, and Italian languages, so I used artificial translator to examine following Wikipedia articles.

Example:
country Which is more popular?
Germany The two envelopes problem in the article "Umtauschparadoxon" (revision of am 14. Juni 2014 um 08:43 Uhr) in the German language Wikipedia is the Opened version problem.
Russia The two envelopes problem in the article "Задача о двух конвертах" (revision of 14:19, 10 июля 2014) in the Russian language Wikipedia is the Opened version problem.
Italy The two envelopes problem in the article "Paradosso delle due buste"(revision of 5 lug 2013 alle 13:01) in the Italian language Wikipedia is the Opened version problem.
Japan In Japan,
the Opened version problem is more popular
than the Closed version problem.


Addition

Some fragment of real history of the wording of the two envelopes problem

This section was added on January 28, 2015, and revised on August 22, 2015, October 9, 2016, October 23, 2016.

Following table shows some fragment of real history which I know.
Literature The opportunity to swap the envelopes
After opening?
Before opening?
Description of how the amount of money is assigned Description of the amount of money in the chosen envelope Specific sample of two pairs of amount
(↓ Revised on April 2, 2017)
Expectation formula Phrase about all possible amounts of money in the chosen envelope
(This column was added on April 3, 2016)
Paradox resolved
Zabell, S. (1988)
After x and 2x $10 ($10, $20)
and
($20, $40)
(1/2)$5 + (1/2)$20 None Greener than each other (Double‐tongued expectation)
Barron, R. (1989). After one of the
checks has a face value twice that of the other
$A   expected gain
(1/2)(2A) - (1/2)(A/2)
None Greener than each other (Double‐tongued expectation)
Paradoxical distribution
(The words "Money Pump" here means paradoxical distributions.)
Nalebuff, Barry.(1989)
(Added on Jun 25, 2017)
After one envelope contains twice as much money as the other X ($5, $10)
and
($10, $20)
0.5[0.5X + 2X] = 1.25X None Greener than each other (Double‐tongued expectation)
Christensen, R; Utts, J (1992), After m and 2m x dollars   (1/2)(x/2 + 2x) "one should always trade" The mathematically standard paradox
Falk, R., & Konold, C. (1992). After Ther are two cards on the table. One of them has written on it a positive number; the other, half that number. A   0.5×(2A) + 0.5×(0.5A) Thus, you should always select the card other than the one revealed to you. Money pump or endless switching
Jackson, F., P. Menzies and G. Oppy (1994), Before one envelope contais twice as much money as the other $x   0.5×$2x + 0.5×$0.5x probably
None
Expectation which's coming before opening
Paradox by an improper probability distribution
Castell, P., & Batens, D. (1994). Before and After
(↑ Revised on April 3, 2016)
the figure on one cheque is the double of the figure on the other $x   ½ · $2x + ½ · $0.5x "regardless of what she finds in her envelope" Paradox by an improper probability distribution
Rawling, P. (1994). whether or not you have opened your envelope the sum in envelope B is either double or half the sum in envelope A emv(yours)

("emv" means "expected monetary value")
  (ⅰ) emv(other)
=0.5(0.5emv(yours))+0.5(2emv(yours))
=1.25emv(yours).
(ⅱ) emv(B)
=0.5(0.5emv(A))+0.5(2emv(A))
=1.25emv(A).
(ⅲ) emv(yours)
=0.5emv(envelope A)
+0.5emb(envelope B)
=emb(other)
None You can reason that you might as well swap and also can reason that you might as well stand pat.
Chalmers, D.J. 1994. probably
After
one contains twice as much as the other $100 ($50, $100)
and
($100, $200)
a 50% chance $200,
and a 50% chance $50.
expected value is $125.
"this reasoning is independent of the actual amount in envelope 1" The mathematically standard paradox
Linzer.E.(1994).
It quoted to
vos Savant, Marilyn (1992).
After one contains twice as much money as the other $100   no description "switching seems to increase the average take by 25%" Paradoxical distribution
Broome,John.(1995). probably
Before
one cheque is twice the other x   (1/2)2x + (1/2)(x/2) "you shold switch, whatever x may be" The mathematically standard paradox
Brams, S. J., & Kilgour, D. M. (1995). After $b and $2b $100 ($50, $100)
and
($100, $200)
(1/2)$200 + (1/2)$50 "it would always be better to switch" The mathematically standard paradox
Bruss, F. T. (1996). Before
(same as game-1 in Sobel, J. H. (1994))
Envelope A contains a randomly determined sum.
Envelope B contains either half or twice that sum as a consequence of the flip of a coin.
(same as game-1 in Sobel, J. H. (1994))
S
(same as game-1 in Sobel, J. H. (1994))
 
(same as game-1 inSobel, J. H. (1994))
0.5 (2S + (1/2)S)
(same as game-1 in Sobel, J. H. (1994))
probably
None
Money pump or endless switching
McGrew, T. J., Shier, D., & Silverstein, H. S. (1997). Before one twice as much as the other x   .5 times 2x
plus
.5 times x/2
None Greener than each other (Double‐tongued expectation)
Norton, J.D. 1998. Before a randomly chosen amount and twice that amount x   (1/2)(2x - x)
+
(1/2)(x/2 - x)
"no matter what amount x I find in my envelope" Greener than each other (Double‐tongued expectation)
Opening problem in
Chase, J. (2002).
no description one envelope contais twice as much money as the other $n   0.5(2n) + 0.5(n/2) None Expectation which's coming before opening
The article "Envelope paradox" (Revision at 10:55, 26 August 2004) in the English language Wikipedia (How to read it)
(Added on Jun 24, 2017.)
After One envelope contains twice as much as the other. A   ½(½A + 2A) = 1½A. "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." The mathematically standard paradox
The article "Umtauschparadoxon"
in the German language Wikipedia
(Revision of am 27. März 2005 um 10:48 Uhr)
After In der einen Truhe befände sich das Zehnfache der anderen. 100 Gulden (10, 100)
and
(100, 1000)
(Gulden)
(900 - 90) : 2 Gulden
(The symbol ":" has the meaning of division.)
None Greener than each other (Double‐tongued expectation)
Section
"1. INRODUCTION"
of
Albers, C. J., Kooi, B. P., & Schaafsma, W. (2005).
After y in envelope 1
and
2y in envelope 2
x = zy
(z is envelope number)
  ½·½x + ½·2x
"you should not use the marginal or prior probability P(Z = 1),but the conditional or posterior probability P(Z = 1|X = x)"  
The ariticle "Two envelopes problem" in English language Wikipedia.
(Revision 22:36, 25 August 2005)
After Into one he places a sum of money, and into the other, he places twice that amount. y   the expected profit of swapping
50% (y/2) + 50% (2y)
None Greener than each other (Double‐tongued expectation)
The ariticle "Two envelopes problem" in English language Wikipedia.
(Revision 20:51, 9 October 2005)
Before one envelope contains twice as much as the other A   ½(2A) + ½(A/2) "you should swap whatever you see in the first envelope" The mathematically standard paradox
The ariticle "Two envelopes problem" in English language Wikipedia.
(Revision 06:43, 30 November 2005)
Before one envelope contains twice as much as the other A   ½(2A) + ½(A/2) None Money pump or endless switching
Falk, Ruma (2008). Before one envelope contains twice as much as the other A   ½(2A) + ½(A/2) None Money pump or endless switching
Bliss, Eric. 2012. no description one has twice as much as the other X   .5(.5) + .5(2X) None Greener than each other (Double‐tongued expectation)
The section
"2 The Classical Two-Envelope Paradox" in
Powers, M. R. (2014).
After X and 2X y   (1/2)(y/2) + (1/2)(2y) "the decision maker's behavior is the same for all values of y" The mathematically standard paradox

How to read the article "Envelope paradox" of the English language Wikipedia

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


How the paradox turned into a puzzle

This section was added on July 6, 2017.

I have the following hypothesis.
Philosophers advocated the "Not three amounts theory" based on difference from the Ali-Baba version.

And they had strengthened their opinion by the following wording.
Because thre are only two amounts of money the variable symbol x in "E=(1/2)(x/2)+(1/2)2x" cannot have same value in the two terms.
Some editors of the English language Wikipedia article "The two envelopes problem" misread this wording and they interpreted it as below.
The variable symbol x in "E=(1/2)(x/2)+(1/2)2x" have different values in each of the two terms.
This was the birth of the "Inconsistent variable theory".
And in the mind of these editors of the English language Wikipedia article "Two envelopes problem" the paradox turned into a puzzle.


Kind of paradox of the two envelopes problem

Reference

Terms



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