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In my perception this article had the following features at almost revisions.
In my perception this article had the following features at almost revisions.
(Added on January 27, 2019)
On July 3, 2009, the last redirection from "Envelope paradox" to "Two envelopes problem" has been done after several redirection and canceling redirection.
(↑ Revised on October 4,,2017.)
And after it no edit warring had broken out about the redirection.
Since then usual reader may not notice the existence of the article "Envelope paradox".
I think that these two articles are creatures of distinct dimensions.
So I hope that some editor cancels the redirection from "Envelope paradox" to "Two envelopes problem" and makes the two articles coexist.
The mathematically standard resolution is exprressed with the following phrase.
The mathematically standard resolution is exprressed with the following phrase.
Surprisingly the mathematically standard resolution was not presented about twenty months.
Standard resolution revived on the closed version problem
The mathematically standard resolution is exprressed with the following phrase.
Standard resolution revived on the closed version problem
The mathematically standard resolution is exprressed with the following phrase.
As a result it became harder to find the links to the following important articles.
The mathematically standard resolution is exprressed with the following phrase.
In my perception many big editing has been done in 2011.
The mathematically standard resolution is exprressed with the following phrase.
In my perception many big editing about the mathematically standard resolution has been done after 2012.
As a result it became harder for readers to find important literatures.
The explanation of the mathematically standard resolution has many sentences so it is not easy-to-understand for me.
(↑ Revised on May 15, 2018)
Especially, there is a sentence which is confusing with the theory of assumption of probability distribution. When I write it in my word, it will be the following sentence.
(↑ Revised on January 5, 2020)
Return to the list of my pages written in English about the two envelopes problem
2021/04/07 12:11:30
First edition 2017/07/25
Two English language Wikipedia articles on the two envelope paradox
This title was revised on November 21, 2017, May 5, 2018.On April 6, 2021, there are two English language Wikipedia articles on the two envelope paradox.
The older article is hidden behind the redirection. (← Confirmed on April 7, 2021)
The older article is hidden behind the redirection. (← Confirmed on April 7, 2021)
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.
How to read the older 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. |
Third | If the article "Two envelopes problem" is shown, click the link on the line "(Redirected from Envelope paradox)". |
Fourth | If the article "Envelope paradox" is shown click the link "View history". |
Comparison of the two articles
(Revised on July 27, 2017.)Major difference
To my eyes the difference of the article "Envelope paradox" and the article "Two envelopes problem" is as follows.- Wording of the problem in the former article is near to the original wording of the two envelopes problem but wording in the later article is not.
Especially the problem presented in "Envelope paradox" is the opened version problem and the problem presented in "Two envelopes problem" is the closed version problem.
(↑ Revised on October 4, 2017.) - Explanation of the mathematically standard resolution is more intelligible in the former article than in the later article.
Features of the article "Envelope paradox"
(Revised on May 12, 2018.)In my perception this article had the following features at almost revisions.
- The problem wording was near to the original wording of the two envelopes problem.
- The mathematically standard resolution was presented but the DivideThreeByTwoian's resolution was never presented.
- The essence of the mathematically standard resolution is the finding that a fallacy of probability is the cause of the paradox.
This article seems have presented it explicitly.
- The article "Umtauschparadoxon" (revision am 7. März 2016 um 02:20) in the German language Wikipedia.
- The article "Задача о двух конвертах" (revision в 12:55, 19 июля 2016) in the Russian language Wikipedia.
- The article "פרדוקס המעטפות" (revision 04:20, 1 במאי 2016) in the Hebrew language Wikipedia.
Features of the article "Two envelopes problem"
(Revised on May 12, 2018.)In my perception this article had the following features at almost revisions.
- The problem wording seemed have come from articles written by early DivideThreeByTwoian philosophers.
- Contrary to the order of history of the two envelopes problem, the DivideThreeByTwoian's resolution was presented before the mathematically standard resolution.
- Historically important opened version problem was being lightly treated.
- The essence of the mathematically standard resolution is the finding that a fallacy of probability is the cause of the paradox.
However, in some revisions of this article, this essence seems explained not so clearly.
(↑ Revised on May 22, 2018)
(Added on January 27, 2019)
- The mathematically standard resolution was explained with a sentence which confuses me. When I write it in my word, it will be the following sentence.
In order that the probability is always 1/2 whatever amount in the chosen envelope, we apparently believe in advance that all the amounts in the geometric progression equally likely to be the smaller amount.I feel some sense of incongruity, because such a phrase is confusing with the theory of assumption of probability distribution.
- In my perception, the explanation of the DivideThreeByTwoian's resolution may contain heterogeneous elements.
Specifically, the DivideThreeByTwoian's resolution was explained by presenting the following expectation formula as the correct formula.Let random variables A and B denote the amount in the chosen envelope and the other envelope, respectively.I feal some sense of incongruity because I think such an explanation is the resolution of the LesserOrGreaterMeanValuean's paradox, not the resolution of the DivideThreeByTwoian's paradox.
ThenE(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) .
The explanation of "E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2)" was accompanied by reference to an article which presented the following diagnostic methods.
"E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2)" .
The former essence is "giving rationale to other formula", and the latter essence is "correction of the formula". (← Added on March 24, 2019)
Let random variables X and L denote the chosen amount and the lesser amount, respectively.
Then"E=(1/2)(X/2) + (1/2)2X)" is wrong and the expectation formula "E=(1/2)L + (1/2)2L" is correct.
However, I think that it is inappropriate to refer this method, because the essence of this method differ from the essence of the formula Then
E(X| (X/2) is the lesser amount) ≠ E(X| 2X is the greater amount) and
E(L| L is the lesser amount) = E(L| 2L is the greater amount).
As a result of this diagnosis, the expectation formula E(L| L is the lesser amount) = E(L| 2L is the greater amount).
The former essence is "giving rationale to other formula", and the latter essence is "correction of the formula". (← Added on March 24, 2019)
How the article "Envelope paradox" had been hidden
(Added on July 26, 2017. Revised on July 27,2017.)On July 3, 2009, the last redirection from "Envelope paradox" to "Two envelopes problem" has been done after several redirection and canceling redirection.
(↑ Revised on October 4,,2017.)
And after it no edit warring had broken out about the redirection.
Since then usual reader may not notice the existence of the article "Envelope paradox".
My hope
(Revised on July 27, 2017.)I think that these 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.
And I know that at least four editors of the article "Envelope paradox" wanted the survival of it in 2006.
(← Added on October 4, 2017.)
"Two envelopes problem" is a creature in the philosophical dimension.
So I hope that some editor cancels the redirection from "Envelope paradox" to "Two envelopes problem" and makes the two articles coexist.
Appendix : Transition of the presentation of the standard resolution in the article "Two envelopes problem" (Minor changes are omitted)
This appendix was added on November 24, 2017. Revised on January 14, 2018, January 16-18, 2018, February 12, 2018, May 17, 2018, May 22, 2018.From the first revision 22:36, 25 August 2005 to the revision 21:47, 3 October 2005
No resolutionsection | contents |
---|---|
– | opened version problem with no solution |
Revision 22:05, 3 October 2005
Both resolutionssection | contents |
---|---|
Lead section | closed version problem |
Proposed Solution 1 |
DivideThreeByTwoian's resolution with inconsistent variable theory |
A Second Paradox | opened version problem |
Proposed Solution 2 | mathematically standard resolution |
The subjective probability changes when we get new information, so our assessment of the probability that A is the smaller and larger sum changes
Revision 20:51, 9 October 2005
(Added on January 15, 2018.)
The order of the problem wording changed.
section | contents |
---|---|
Lead section |
closed version problem (More suitable for the DivideThreeByTwoian's resolution) |
Proposed Solution |
DivideThreeByTwoian's resolution with inconsistent variable theory |
A Harder Problem | opened version problem |
Proposed Solution | mathematically standard resolution |
Revision 01:47, 17 March 2008
Both resolutionssection | contents |
---|---|
The problem | closed version problem |
The problem / Proposed solution |
DivideThreeByTwoian's resolution with inconsistent variable theory |
A harder problem | opened version problem |
A harder problem / Proposed solution | mathematically standard resolution |
The subjective probability changes when we get new information, so our assessment of the probability that A is the smaller and larger sum changes.
Revision 18:42, 8 October 2008
Standard resolution deletedsection | contents |
---|---|
The problem | closed version problem |
The problem / Proposed solution |
DivideThreeByTwoian's resolution with inconsistent variable theory |
mathematically standard resolution deleted!!! |
Surprisingly the mathematically standard resolution was not presented about twenty months.
Revision 20:17, 26 May 2010
DivideTreeByTwoian's resolution deletedStandard resolution revived on the closed version problem
section | contents |
---|---|
The problem | closed version problem |
DivideThreeByTwoian's resolution deleted!!! | |
The problem / Solution |
mathematically standard resolution? Or theory of assumption of probability distribution? on the closed version problem |
Applying the same logic with different ratios shows that we are implicitly assuming that every positive amount is equally likely - however since this implies a uniform probability over an infinite set of values, it is not a valid prior distribution.
Revision 22:01, 20 December 2010
No resolutionsection | contents |
---|---|
The problem | closed version problem |
The problem / Solution | No solution presented |
Revision 03:12, 1 January 2011
DivideThreeByTwoian's resolution revivedsection | contents |
---|---|
The problem | closed version problem |
The problem / Solution |
DivideThreeByTwoian's resolution with inconsistent variable theory |
Revision 23:46, 3 May 2011
DivideThreeBytwoian's resolution deletedStandard resolution revived on the closed version problem
section | contents |
---|---|
The problem | closed version problem |
DivideThreeByTwoian's resolution deleted!!! | |
Informal solution Formal solution |
mathematically standard resolution on the closed version problem |
To be specific, it could never be the case that the other envelope is equally likely to be double or half the first envelope, whatever the amount in the first envelope.
Revision 23:30, 15 August 2011
(Added on February 17, 2018.)
The section "External links" was replaced to a link to the page "Two envelopes problem/sources" which contains these external links.
As a result it became harder to find the links to the following important articles.
- David J. Chalmers, The Two-Envelope Paradox: A Complete Analysis?, Department of Philosophy, University of Arizona
- Keith Devlin, The Two Envelopes Paradox, Mathematical Association of America, July–August 2004
- Amos Storkey, Money Trouble and Money Trouble – Solution, 2005
- etc
Revision 01:31, 8 November 2011
DivideThreeBytwoian's resolution revivedsection | contents |
---|---|
Problem | closed version problem |
First resolution |
DivideThreeByTwoian's resolution with inconsistent variable theory |
Second resolution | mathematically standard resolution |
… then it is simply impossible that whatever the amount A=a in the first envelope, it is equally likely that the second contains a/2 or 2a.
In my perception many big editing has been done in 2011.
Revision 18:44, 8 February 2012
The mathematical standard resolution as an uncommon resolutionsection | contents |
---|---|
The problem | closed version problem |
The problem/A common resolution |
DivideThreeByTwoian's resolution with inconsistent variable theory |
Alternative interpretation/Resolution of alternative interpretation |
mathematically standard resolution (Probably not theory of assumption of probability distribution) (← Revised on January 27, 2019) |
… then it is impossible that whatever the amount A=a in the first envelope might be, it would be equally likely, according to these prior beliefs, that the second contains a/2 or 2a.
Revision 10:47, 18 March 2012
Mathematical standard resolution categorised "Bayesian"section | contents |
---|---|
The problem | closed version problem |
A common resolution |
DivideThreeByTwoian's resolution with inconsistent variable theory |
Introduction to resolutions based on Bayesian probability theory/Proposed resolutions |
mathematically standard resolution (Probably not theory of assumption of probability distribution) (← Revised on January 27, 2019) |
In my perception many big editing about the mathematically standard resolution has been done after 2012.
Revision 03:15, 10 October 2012
(Added on February 17, 2018.)
The link to the page "Talk:Two envelopes problem/Literature" was deleted from the section "See also".
As a result it became harder for readers to find important literatures.
Revision 09:49, 17 November 2014
(Added on November 25, 2017. Revised on November 27, 2017.)
The mathematically standard resolution explained using geometric progression
section | contents |
---|---|
Lead section |
closed version problem (as a summary of the problem) |
Problem | closed version problem |
Logical resolutions |
DivideThreeByTwoian's resolution with inconsistent variable theory |
E(B)= (↑ Revised on June 26, 2018) (LesserOrGreaterMeanValuean's resolution) |
|
|
|
Mathematical resolutions/Simple mathematical resolution |
mathematically standard resolution? Or theory of assumption of probability distribution? (explained using geometric progression) |
(↑ Revised on May 15, 2018)
Especially, there is a sentence which is confusing with the theory of assumption of probability distribution. When I write it in my word, it will be the following sentence.
In order that the probability is always 1/2 whatever amount in the chosen envelope, we apparently believe in advance that all the amounts in the geometric progression equally likely to be the smaller amount.
(↑ Added on January 27, 2019)
Revision 16:15, 22 November 2014
(On January 5, 2020, the revision date was revised)
The statements about the inconsistent variable theory was replaced by other statements regarding the relationship between the smaller amount average and the larger amount average.
(↑ Revised on January 5, 2020)
section | contents |
---|---|
Lead section |
closed version problem (as a summary of the problem) |
Problem | closed version problem |
Logical resolutions |
E(B)= (↑ Revised on June 26, 2018) (LesserOrGreaterMeanValuean's resolution) |
|
|
Mathematical resolutions/Simple mathematical resolution |
mathematically standard resolution? Or theory of assumption of probability distribution? (explained using geometric progression) |
Revision 16:46, 12 April 2016
(Revised on November 25, 2017, November 27, 2017.)
Pure DivideThreeByTwoian's opinion presented
section | contents |
---|---|
Lead section |
closed version problem (as a summary of the problem) |
Introduction/Problem | closed version problem |
Simple resolution |
(pure DivideThreeByTwoian's opinion) |
Simple resolutions |
E(B)= (↑ Revised on June 26, 2018) (LesserOrGreaterMeanValuean's resolution) |
Bayesian resolutions/Simple form of Bayesian resolution |
mathematically standard resolution? Or theory of assumption of probability distribution? (explained using geometric progression) |
Revision 03:24, 21 June 2016
(Added on January 16, 2018)
The following wording was added 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, …
section | contents |
---|---|
Lead section |
closed version problem (as a summary of the problem) |
Introduction/Problem | closed version problem |
Simple resolution |
(pure DivideThreeByTwoian's opinion) |
Other simple resolutions |
E(B)= (↑ Revised on June 26, 2018) (LesserOrGreaterMeanValuean's resolution) |
Bayesian resolutions/Simple form of Bayesian resolution |
mathematically standard resolution? Or theory of assumption of probability distribution? (explained using geometric progression) |
Revision 20:02, 15 October 2016
(Added on February 16, 2018)
The not-three-amounts theory was added to the beginning pf the section "Other simple resolutions".
section | contents |
---|---|
Lead section |
closed version problem (as a summary of the problem) |
Introduction/Problem | closed version problem |
Simple resolution |
(pure DivideThreeByTwoian's opinion) |
Other simple resolutions |
not-three-amounts theory (in just one sentence "The step 7 assumes that the second choice is independent of the first choice. ") |
E(B)= (↑ Revised on June 26, 2018) (LesserOrGreaterMeanValuean's resolution) |
|
Bayesian resolutions/Simple form of Bayesian resolution |
mathematically standard resolution? Or theory of assumption of probability distribution? (explained using geometric progression) |
Revision 14:55, 28 April 2017
(Added on June 26, 2018)
The following wording was added in the section "Solutions".
… any apparent paradox is generally due to treating what is actually a conditional probability as an unconditional probability …
I think this sentence conforms to the mathematically standard resolution.
section | contents |
---|---|
Lead section |
closed version problem (as a summary of the problem) |
Introduction/Problem | closed version problem |
Simple resolution |
(pure DivideThreeByTwoian's opinion) |
Other simple resolutions |
not-three-amounts theory (in just one sentence "The step 7 assumes that the second choice is independent of the first choice. ") |
E(B)= (↑ Revised on June 26, 2018) (LesserOrGreaterMeanValuean's resolution) |
|
Bayesian resolutions/Simple form of Bayesian resolution |
mathematically standard resolution? Or theory of assumption of probability distribution? (explained using geometric progression) |
Revision 02:15, 3 November 2017
(Revised on November 25, 2017, November 27, 2017.)
section | contents |
---|---|
Lead section |
closed version problem (as a summary of the problem) |
Introduction/Problem | closed version problem |
Simple resolution |
(pure DivideThreeByTwoian's opinion) |
Other simple resolutions |
not-three-amounts theory (in just one sentence "The step 7 assumes that the second choice is independent of the first choice. ") |
E(B)= (↑ Revised on June 26, 2018) (LesserOrGreaterMeanValuean's resolution) |
|
Bayesian resolutions/Simple form of Bayesian resolution |
mathematically standard resolution? Or theory of assumption of probability distribution? (explained using geometric progression) |
State of the revision 16:51, 12 December 2018
(Added on January 13, 2018. Revised on March 15, 2020)
section | contents |
---|---|
Lead section |
closed version problem (as a summary of the problem) |
Introduction/Problem | closed version problem |
Simple resolution |
Equality of the gain and loss by exchange (Added on March 15, 2020) |
|
|
The not-three-amounts theory is explained in the paragraph "In simple words", using the following findings.
|
|
Other simple resolutions | The not-three-amounts theory is explained with the sentence "The step 7 assumes that the second choice is independent of the first choice. " |
E(B)= (LesserOrGreaterMeanValuean's resolution) |
|
Bayesian resolutions/Simple form of Bayesian resolution |
mathematically standard resolution? Or theory of assumption of probability distribution? (explained using geometric progression) |
State of the latest revision 19:48, 6 March 2020
(Added on March 15, 2020)
The "simple solution" section was given a new format, but the content remained the same.
State of the revision as of 17:49, 17 September 2020
(Added on April 7, 2021)
The paragraph beginning with "The step 7 assumes that the second choice is independent of the first choice" was removed from the section "Other simple resolutions".
section | contents |
---|---|
Lead section |
closed version problem (as a summary of the problem) |
Introduction/Problem | closed version problem |
Simple resolution | Equality of the gain and loss by exchange |
|
|
The not-three-amounts theory is explained using the following findings.
|
|
Other simple resolutions |
E(B)= (LesserOrGreaterMeanValuean's resolution) |
Bayesian resolutions/Simple form of Bayesian resolution |
mathematically standard resolution? Or theory of assumption of probability distribution? (explained using geometric progression) |
Terms
-
mathematically standard resolution
Resolution which explains that the cause of the paradox of the two envelopes problem is a fallacy of probability.
For details please see my page "An outline of the Two Envelopes Problem".
-
DivideThreeByTwoian
There are people who have the following opinion.Let (a, 2a) is the pair of amount of money in the two envelopes.I call them "DivideThreeByTwoian" because
Then"E=(1/2)a + (1/2)2a" is the correct expectation formula."E=(1/2)a + (1/2)2a" is (3/2)a.
For details please see my page "An outline of the Two Envelopes Problem".
-
opened version problem
The type of problem in which the opportunity to switch envelope is given after opening the first chosen envelope.
-
closed version problem
The type of problem in which the opportunity to switch envelope is given before opening the first chosen envelope.
-
inconsistent variable theory
I call the following theory the "inconsstent variable theory".In the fallacious expectation fromulaFor details please see my page "An outline of the Two Envelopes Problem"."E=(1/2)(x/2)+(1/2)2x" the variable symbol x has different values in the term"(1/2)(x/2)" and the term"(1/2)2x" . This fallacy is the cause of the paradox.
-
LesserOrGreaterMeanValuean
If a person think as follows I call him/her LesserOrGreaterMeanValuean.The subject matter of the "two envelopes problem" is the magnitude relation of the following values.For details please see my page "An outline of the Two Envelopes Problem".- mean value of the amounts of money in the envelopes which have lesser amount
- mean value of the amounts of money in the envelopes which have greater amount
- mean value of the amounts of money in the other envelopes
-
not-three-amounts theory
Some DivideThreeByTwoians thought as follows.The cause of the paradox is to think of three amounts x/2, x and 2x.I call this opinion the "not-three-amounts theory".
If you think of only two amounts a and 2a , paradox will vanish.
For details please see my page "An outline of the Two Envelopes Problem".
-
theory of assumption of probability distribution
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 "theory of assumption of probability distribution".
This opinion says that the fallacious expectation formula assumes an invalid prior probability distribution. Thus, this opinion essentially differ from the mathematically standard resolution.
-
Ali-Baba version
"Ali-Baba version" is my coined word.
On this type of problem, after the amount contained in Alli's envelope is determined, either 2A or A/2 is assigned as the amount contained in Baba's envelope.
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