Return to the list of my pages written in English about the two envelopes problem
I substantially revised this page on April 3, 2015. And I rewound it on the next day.
Following table shows some fragment of real history which I know.
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".
I have the following hypothesis.
Return to the list of my pages written in English about the two envelopes problem
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 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 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.
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.
Equally rich two persons play following game.
Taking following article into account, I wrote this.
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."
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.
There are two envelopes.
Taking following article into account, I wrote this.
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.
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.
You are to select one of two envelopes that each contains a check.
Taking following article into account, I wrote this.
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. 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
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.
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.
The "Ali-Baba" version problem has the following aspects.
| ||||||||||
↑ 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
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:
|
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 |
Description of how |
Description of the amount of money in the chosen envelope |
Specific sample of (↓ Revised on April 2, 2017) |
Expectation formula |
Phrase about (This column was added on April 3, 2016) |
|
---|---|---|---|---|---|---|---|
Zabell, S. (1988) |
After | x and 2x | $10 |
($10, $20) and ($20, $40) |
|
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 |
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) |
|
None | Greener than each other (Double‐tongued expectation) |
Christensen, R; Utts, J (1992), | After | m and 2m | x dollars |
|
"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 |
|
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 |
|
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 |
|
"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 |
|
"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) |
|
"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)) |
(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 |
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 |
+ |
"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 |
|
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 |
|
"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) |
(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) |
|
"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 |
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 |
|
"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 |
|
None | Money pump or endless switching | |
Falk, Ruma (2008). | Before | one envelope contains twice as much as the other | A |
|
None | Money pump or endless switching | |
Bliss, Eric. 2012. | no description | one has twice as much as the other | X |
|
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 |
|
"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.
And in the mind of these editors of the English language Wikipedia article "Two envelopes problem" the paradox turned into a puzzle.
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
-
The mathematically standard paradox
Thinking of expectations for all values of the amount of money in the chosen envelope breaks symmetry between envelopes.
Whatever the amount of money in the chosen envelope is, it is smaller than the expected amount of money of the opposite envelope
So you should swap whatever you see in the first envelope. But as the situation is symmetric this can't be correct.
-
Paradox by an improper probability distribution
If we imagine improper probability distribution the expected gain on exchange of envelopes can be positive for all values of chosen amount of money.
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Paradoxical distribution
There exist proper , normalized probability distibutions for the total purse such that the expected gain on exchange of envelopes is positive for all values of chosen amount of money.
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Greener than each other (Double‐tongued expectation)
Because this calculation does not depend on the envelope, so you should change your choise even if you have chosen the oposite envelope.
If you choose envelope A the calculation will suggest swapping.
And even if you choose the opposite envelope B it will suggest swapping.
-
Money pump or endless switching
Because we can apply this calculation after swap of envelopes, we should switch back, and so on.
And if we repeat swapping, the expected amount of money in the chosen envelope will grow up in the rate 25% per every swap.
(↑ Revised on April 2, 2017.)
-
Expectation which's coming before opening
Without opening, the mere choosing of envelope cannot give any information.
But whichever envelope you originally selected, the reasoning suggests swapping.
(This paradox can be derived only on the closed version problem.)
Reference
-
Albers, C. J., Kooi, B. P., & Schaafsma, W. (2005).
Trying to resolve the two-envelope problem. Synthese, 145(1), 89-109.
-
Barron, R. (1989)
The Paradox of the Money Pump: A Resolution.
Maximum entropy and Bayesian methods, Cambridge, England, 1988
edited by J. Skilling
Dordrecht : Kluwer Academic Publishers, c1989
pp. 423-428
In this article paradoxical distributions are denoted by the words "Money Pump".
-
Bliss, Eric. 2012.
A Concise Resolution to the Two Envelope Paradox
-
Brams, S. J., & Kilgour, D. M. (1995).
The box problem: to switch or not to switch.
Mathematics Magazine, 27-34.
-
Broome,John.(1995).
The Two-envelope Paradox, Analysis 55(1): 6–11.
-
Bruss, F. T. (1996).
The fallacy of the two envelopes problem.
Mathematical Scientist, 21(2), 112-119.
-
Castell, P., & Batens, D. (1994).
The two envelope paradox: the infinite case.
Analysis, 46-49.
-
Chalmers, D.J. 1994.
The two-envelope paradox: A complete analysis?
-
Chase, J. (2002).
The non-probabilistic two envelope paradox. Analysis, 157-160.
-
Christensen, R; Utts, J (1992),
Bayesian Resolution of the "Excehange Paradox"
The American Statistician, Vol.46,No.4.(Nov.,1992),pp.274–276.
-
Falk, R., & Konold, C. (1992).
The psychology of learning probability.
Statistics for the twenty-first century, 151-164.
-
Falk, Ruma (2008).
The Unrelenting Exchange Paradox. Teaching Statistics 30 (3): 86–88.
-
Jackson, F., P. Menzies and G. Oppy (1994),
The Two Envelope 'Paradox'
Analysis 54. 43–45
-
McGrew, T. J., Shier, D., & Silverstein, H. S. (1997)
The two-envelope paradox resolved. Analysis, 28-33.
-
Kent G. Merryfield, Ngo Viet, and Saleem Watson (1997).
The Wallet Paradox
American Mathematical Monthly,104,1997,647–649.
-
Linzer.E.(1994).
The Two Envelope Paradox, American Mathematical Monthly 101. pp.417–19.
-
Nalebuff, Barry.(1989)
The other person's envelope is always greener. Jounal of Economic Perspectives 3 (1989) 171-181.
-
Norton, J.D. 1998.
When the sum of our expectations fails us: The exchange paradox.
Pacific Philosophical Quarterly 79:34–58.
-
Powers, M. R. (2014).
Preserving Dominance Reasoning in Two Two-Envelope Paradoxes.
Working paper,Tsinghua University School of Economics and Management.
-
Rawling, P. (1994).
A note on the two envelopes problem.
Theory and Decision, 36(1), 97-102.
-
Sobel, J. H. (1994)
Two envelopes. Theory and Decision, 36(1), 69-96.
-
vos Savant, Marilyn (1992).
Ask Marilyn, Parade, (September 1992) 20.
-
Zabell, S. (1988)
Discussion related to Hill, B. M. (1988).
Bayesian statistics, 3, 233-236.
Hill, B. M. (1988).
De Finetti’s Theorem, Induction, and A (n) or Bayesian nonparametric predictive inference (with discussion).
Bayesian statistics, 3, 211-241.
Terms
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Opened version problem
In the opened version problem the opportunity to swap the envelopes is given after the chosen envelope is opened.
-
Closed version problem
In the closed version problem the opportunity to swap the envelopes is given before the chosen envelope is opened.
-
illusion of probability
The illusion of probability which cause the paradox of two envelopes problem is "Base Rate Fallacy."
Because of it, We forget the odds of the possible two pairs of amount of money in the two envelopes.
That why we thoughtlessly assign probability1/2 to the terms in the expectation formula.
-
theory of
"E=(1/2)2a + (1/2)a"
The theory of"E=(1/2)2a + (1/2)a" is a theory that the expectation formula"E=(1/2)2a + (1/2)a" is the correct version of the formula"E=(1/2)(x/2) + (1/2)2x" .
For details pleas see An outline of the Two Envelopes Problem.
-
Mathematical standard resolution
"Mathematical standard resolution" means the following opinion.
The probability that the other envelope contains x/2 and the probability that it contains 2x are not necessarily 1/2.
Actual probabilities correspond to the odds of the pair (x/2, x) and the odds of the pair (x, 2x).
So there is no wonder. even if the opposite envelope is more favorable for a value of the amount of money in the chosen envelope.
And if the opposite envelope is favorable for a value of the amount of money in the chosen envelope, the opposite envelope must be unfavorable for some other value of the amount of money in the chosen envelope.
The equivalence of the two envelopes is surely kept.
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DivideThreeByTwoian
I call the people who advocate the theory of"E=(1/2)2a + (1/2)a" "DivideThreeByTwoian" because"(1/2)2a + (1/2)a = (3/2)a" .
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inconsistent variable theory
It is as follows.The variable symbol of the expectation formula denotes the smaller amount in the one term , while it denotes larger amount in the another term.
To mix different instances of a variable in the same formula is the cause of the paradox.
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inconsistent mental variable
It is my coined word.
If a same element in a mental model simultaneously denote different values in human brain, I call it "inconsistent mental variable"
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inconsistent explicit variable
It is my coined word.
If a same variable symbol simultaneously denote different values in an explicit formula, I call it "inconsistent explicit variable"
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