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In the following sections , the people with this mental model are called 'DoublePairian'.
But another some people make following mental model about the Two Envelope Problem.
In the following sections , the people with this mental model are called 'SinglePairian'.
I call the former problem "DoublePairian's problem" and call the later problem "SinglePairian's problem".
On October and November, 2014, in the talk page of the article "Two envelopes problem" in the English language Wikipedia, professor Richard Gill explained that there are two kinds of "Two envelopes problem".
But within editors of this article, some who have SinglePairian's mental model could not understand his explanation.
In this example, the two kinds of problem differ as follows.
DoublePairian's mathematical model
SinglePairian's mathematical model
DoublePairian's mathematical model changed to another form
Having been inspired by Clark, Michael. & Shackel, Nicholas. (2000)., we can change the DoubleParian's mathematical model to another form as follows.
This paragraph was revised on August 14, 2016.
On January 10, 2015, this paragraph was revised.
On December 2014, when I read the section "Simple resolutions" in the article "Two envelopes problem" (at the revision 21:39, 23 November 2014) in the English language Wikipedia, I have learned that there can be following theory.
But I am suspicious of such a theory.
I can not imagine that there can be one who can have such complicated mental model and make such a confusion.
To my eyes this theory is only distortion.
If such people are not minority, they should be called "LesserOrGreaterMeanValuean", and their problem should be called "The third problem of the two envelopes problem".
LesserOrGreaterMeanValuean's mathematical model
Return to the list of my pages written in English about the two envelopes problem
2016/08/14 12:52:41
First edition 2014/11/12
Two kinds of the Two Envelopes Problem
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.
DoublePairian & SinglePairian
Some people make following mental model about the Two Envelope Problem.
In the following sections , the people with this mental model are called 'DoublePairian'.
But another some people make following mental model about the Two Envelope Problem.
In the following sections , the people with this mental model are called 'SinglePairian'.
DoublePairian's problem & SinglePairian's problem
The problem which DoublePairians consider and the problem which SinglePairians consider differ from each other.I call the former problem "DoublePairian's problem" and call the later problem "SinglePairian's problem".
On October and November, 2014, in the talk page of the article "Two envelopes problem" in the English language Wikipedia, professor Richard Gill explained that there are two kinds of "Two envelopes problem".
But within editors of this article, some who have SinglePairian's mental model could not understand his explanation.
Places where we can find differences of the two kinds
Has switch advantage?
Let's consider an example of the probability distribution.|
Pairs of amounts of money |
probability |
|---|---|
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20% |
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20% |
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20% |
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20% |
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20% |
In this example, the two kinds of problem differ as follows.
| Question |
Answer for |
Answer for |
|---|---|---|
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Imagine that the envelope which you pick encloses ¥4,000. |
¥5,000 | SinglePairians seem to dislike such a question. |
| Are always the two envelopes equivalent each other? |
No. If the envelope I pick has ¥4,000, then the expected gain from exchange is Therefore I should exchange. |
Yes. If the pair includes ¥2,000 and ¥4,000, then the expected gain from exchange is In this way, the expected gain is always 0. |
A question inspired from Smullyan, Raymond (1992).
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Answer for |
Answer for |
|---|---|---|
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Consider the amount of the money in the envelope that you have not picked. How much is the ratio of the smallest amount to the largest amount? |
1 to 4 | 1 to 2 |
Mathematical model
These two problems use different mathematical model to prove the equivalence of the two envelopes which both are not opened.DoublePairian's mathematical model
SinglePairian's mathematical model
DoublePairian's mathematical model changed to another form
Having been inspired by Clark, Michael. & Shackel, Nicholas. (2000)., we can change the DoubleParian's mathematical model to another form as follows.
Why a resolution of either of the two paradoxes can not resolve the other paradox?
On March 18, 2015, title was changed.| Resolutions |
Can it resolve the paradox on the |
Can it resolve the paradox on the |
|---|---|---|
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The expetaion formula is The probability ½ in this formula is a result of a fallacious thought that |
Yes. |
No. This can not be a reasoning, because it is mentioning three amounts |
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If we think only two amounts A and 2A, then the equivalence of the two envelopes will be verified. |
No. This can not be a reasoning of the problem that mentions three amounts |
Yes. |
The cause of the fallacy which derived each paradox of these two problems
This paragraph was added on November 22, 2014, and its title was changed on March 18, 2015.This paragraph was revised on August 14, 2016.
|
The cause of the fallacy which derived paradox on the |
The cause of the fallacy which derived paradox on the |
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|---|---|---|
| Mathematician's thought |
The wrong assumption that always the odds of each pair of amounts of money are equal. My thougt Nobody can make such a assumption. Anybody only can be caught by a illusion of probability. It is the famous illusion named "Base Rate Fallacy". |
(Mathematicians can not feel any paradox from such a elementary problem.) |
| Philosopher's thought | (Many philosophers can not understand such a problem and refuse it.) |
Many philosophers were not interested in the cause of the fallacy. Some philosophers described two contradictory two causes as follows 1）Careless thought of two pairs of amounts of money 2)Inconsistent use of a variable symbol My thougt Have they really experienced themselves the two envelope paradox? |
Addition
Is there the third problem?
On December 20, 2014, this paragraph was added.On January 10, 2015, this paragraph was revised.
On December 2014, when I read the section "Simple resolutions" in the article "Two envelopes problem" (at the revision 21:39, 23 November 2014) in the English language Wikipedia, I have learned that there can be following theory.
- Some people have following mental model.
- They think that the subject matter of the "two envelopes problem" is the magnitude relation of the following values.
- 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
- And they forgot they were thinking of expectation values under two different conditions.
I can not imagine that there can be one who can have such complicated mental model and make such a confusion.
To my eyes this theory is only distortion.
If such people are not minority, they should be called "LesserOrGreaterMeanValuean", and their problem should be called "The third problem of the two envelopes problem".
LesserOrGreaterMeanValuean's mathematical model
Reference
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Clark, Michael. & Shackel, Nicholas. (2000).
The Two Envelope Paradox, Mind Magazine (Vol 109.435.July 2000)
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Smullyan, Raymond (1992).
Satan, Cantor, and infinity and other mind-boggling puzzles.
Alfred A. Knopf. pp. 189–192. ISBN 0-679-40688-3.
日本語訳「スマリヤンの無限の論理パズル : ゲーデルとカントールをめぐる難問奇問」
レイモンド・スマリヤン著 ; 長尾確訳. 白揚社
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