Return to the list of my pages written in English about the two envelopes problem
After studying the articles which advocate IVT ( Inconsistent Variable Theory on The Two Envelope Paradox) , I have found following facts.
The author of it claimed following opinion.
But after reading this paper, I found that the main opinion of this paper was that thinking two pairs of amount of money is the cause of the paradox
The base of his thought
He might have understood the process of the placing money in the two envelopes as follows.
Not three amounts theory
To my eyes, the main opinion of his paper is as follows.
Explanation like IVT
To my eyes, he have made following explanation to prove his main opinion.
Some people think that we unconsciously interpret the two envelopes problem as follows.
The answer of this problem is as follows.
Examples of these articles
Examples of these articles
In my perception, in the articles presented above, psychological mechanism of fallacy which causes the paradox was not discussed.
The authors of them might be as follows.
In the mathematical standard resolution of the two envelope paradox, two pairs of amounts of money are considered.
In my perception, in the articles presented above, they sought the reason to avoid thinking of two pairs of amounts of money,
So they sought flaws of the mathematical standard resolution.
In other words, in my perception, they seem to be products of the imagination.
I think that each of them resolve each fictitious paradoxes.
Return to the list of my pages written in English about the two envelopes problem
This page is too old.
So please see the page "An outline of the Two Envelopes Problem" on this site instead.
So please see the page "An outline of the Two Envelopes Problem" on this site instead.
Last edition 2015/09/23 6:59:13
First edition 2015/06/14
The relatives of IVT on the two envelopes problem
Some parts of this page was greatly revised on September 22, 2015.After studying the articles which advocate IVT ( Inconsistent Variable Theory on The Two Envelope Paradox) , I have found following facts.
- IVT (Inonsistent variable theory) is as follows.
- Same symbol is used in two terms of the expectation formula, but these terms correspond to different situations.
- Hence the variable symbol denotes values of different random variables.
- This is the cause of the paradox.
- Articles which advocate IVT are less than my expectation.
- But there are many articles which claim that thinking two pairs of amount of money is the cause of the paradox.
I call it "Not three amounts theory". - There are some articles which claim as follows.
- We unconsciously interpret that the variable symbol denotes a conditional mean value of the amount of the first envelope.
- We often confuse the following conditional mean values.
The mean value of the amount of money on the greater or smaller side.
The mean value of the amount of money on the greater side.
The mean value of the amount of money on the smaller side. - This confusion of the conditional mean values is the cause of the paradox.
Caution
I who am Japanese wrote this page in English, but I am not so good at English.
I who am Japanese wrote this page in English, but I am not so good at English.
Some articles which advocate "Not three amounts theory"
Some people think as follows.- In the arranged two envelope, there are only two amounts of money.
- For example, if these amounts are $10 and $20, then we are allowed to think one of two pairs of amounts of money [$10, $20].
- But the expectation formula which cause the paradox concerns two pairs of amounts of money.
For example, if your envelope contains $10, then the expected value of the amount of money in the other envelope is(1/2)$5 + (1/2)$40.
This expectation formula concerns two pairs of amounts of money [$5, $10] and [$10, $40] simultaneously. - Thinking such a two pairs of amounts of money simultaneously is the cause of the paradox.
A paper written in the early 2010s
This paper was written in the early 2010sThe author of it claimed following opinion.
- We should think as if we just have done the arrangement of money.
- If we have arranged $10 and $20, and if chosen envelope contains $10, it is wrong to think that the other envelope probably contains $5. (← This quotation was revised on September 23,2015.)
The paper to which many articles referred as the pioneer of IVT
This paper by a mathematician was published in the 90's. And many articles referred to it as the pioneer of IVT. (← Revised on September 22, 2015.)But after reading this paper, I found that the main opinion of this paper was that thinking two pairs of amount of money is the cause of the paradox
The base of his thought
He might have understood the process of the placing money in the two envelopes as follows.
- Place an amount S of money in an envelope.
- Then with a probability 1/2 place an amount S/2 of money or with same probability place an amount 2S of money in the other envelope.
Not three amounts theory
To my eyes, the main opinion of his paper is as follows.
The cause of the paradox is to think the expected value at the phase of placing money.
During this phase pairs of amounts of money (S/2, S) and (S, 2S) are possible. But after the phase only one pair is possible.
During this phase pairs of amounts of money (S/2, S) and (S, 2S) are possible. But after the phase only one pair is possible.
Explanation like IVT
To my eyes, he have made following explanation to prove his main opinion.
During the phase of selecting a envelope, the symbol S in the first term denotes the greater amount of money, and the symbol S in the second term denotes the lesser amount of money.
Therefore the unique symbol S mistakenly simultaneously denotes the values of different random variables.
Some articles which advocate IVT
A paper by a researcher of psychology
This paper written in the 2000s claimed as follows.- In the expectation formula "(1/2)(A/2) + (1/2)2A" has not algebraic consistency.
The first term corresponds to the situation that your envelope contains greater amount of money.
The second term corresponds to the situation that your envelope contains smaller amount of money.
- One can denote by A a constant but unseen amount.
But the probabilities that the other envelope contains either A/2 or 2A can not be same to each other for all possible values of A.
Some articles which claim that a confusion of the conditioned mean values is the cause of the paradox.
This paragraph was revised on June 21, 2015.Some people think that we unconsciously interpret the two envelopes problem as follows.
Express the expected value of the amount of money in the other envelope with some conditional expected values of the amount of money in the first envelope.
The answer of this problem is as follows.
Case that the pair of amounts is fixed like IVT
- Let C and O be the chosen envelope and the other envelope respectively.
- Let X and Y be the random variable of the amount of money in the envelope C and the envelope O respectively.
- Let M be the random variable of the smallest amount of money in the pair of amounts of money.
In other words, if X = M then Y = 2M, and if Y = M then X = 2M. - Then
E(X|X = M ∧ M = m ) = m andE(X|X = 2M ∧ M = m) = 2m. - Using these equations, E(Y) is calculated as follows.
E(Y) = (1/2)2E(X|X = M ∧ M = m) + (1/2)(1/2)E(X|X = 2M ∧ M = m) = (3/2)m. - On the other hand
E(X) = (1/2)m + (1/2)2m = (3/2)m. - These results meet the fact E(X) = E(Y).
We often confuse conditional mean values and construct a expectation formula as follows.
E(Y) = (1/2)2E(X|X = M ∧ M = m) + (1/2)(1/2)E(X|X = 2M ∧ M = m) = (1/2)(1/2)E(X) + (1/2)2E(X) = (5/4)E(X).
This confusion of the conditional mean values is the cause of the paradox.
Examples of these articles
- An article which was written by researchers of mathematical sciences and was published in middle 1995.
- Some revision at early 2012 of the article "Two envelopes problem" in the English language Wikipedia.
Case that the pair of amounts is not fixed
- Let C and O be the chosen envelope and the other envelope respectively.
- Let X and Y be the random variable of the amount of money in the envelope C and the envelope O respectively.
- Let A = E(X|C is the smaller side).
- Then
E(X|C is the greater side) = 2A. - Using these equations, E(Y) is calculated as follows.
E(Y) = (1/2)2E(X|C is the smaller side) + (1/2)(1/2)E(X|C is the greater side) = (3/2)A. - On the other hand
E(X) = (1/2)A + (1/2)2A = (3/2)A. - These results meet the fact E(X) = E(Y).
We often confuse conditional mean values and construct a expectation formula as follows.
E(Y) = (1/2)(1/2)E(X) + (1/2)2E(X) = (5/4)E(X).
This confusion of the conditional mean values is the cause of the paradox.
Examples of these articles
- Some revision at late 2014 of the article "Two envelopes problem" in the English language Wikipedia.
doubtfulness of these relatives
The row about the claim that the symbol in the expectation formula denotes a random variable is deleted on September 22, 2015.relative | logical doubt | psychological doubt |
---|---|---|
IVT The claim that the symbol in the expectation formula denotes different values in each terms |
no doubt |
please see Inconsistent Variable Theory on The Two Envelope Paradox |
The claim that thinking two pairs of amount of money is the cause of the paradox |
The advocators of such a opinion did not prove that if we think of two pairs of amounts of money we cannot avoid paradox. |
If we have thought of two pairs of amount of money, after feeling of a paradox, will we wonder why we have thought of two pairs of amounts? |
The claim that confusion of the conditional mean values is the cause of the paradox (Pair of amounts is fixed) |
no doubt |
On such a setting of problem, we more easily make a confusion of mean value of rates and rate of mean values. Is there anyone who really made a confusion of conditional mean values? |
The claim that confusion of the conditional mean values is the cause of the paradox (Pair of amounts is not fixed) |
no doubt |
This confusion will more easily happen when we try to solve a problem with no explicit expectation formula. But I have never read such a problem. Is there anyone who really made such a confusion? |
They had never sought the cause of their fallacy
This was added on June 20, 2015.In my perception, in the articles presented above, psychological mechanism of fallacy which causes the paradox was not discussed.
The authors of them might be as follows.
- They felt no paradox as their own experience. So they had no need to find psychological mechanism of the fallacy which causes the paradox.
- They thought that if they find a fashion of thinking which does not cause paradox, then the paradox is resolved.
They sought the flaw of the mathematical standard resolution
This was added on September 22, 2015.In the mathematical standard resolution of the two envelope paradox, two pairs of amounts of money are considered.
In my perception, in the articles presented above, they sought the reason to avoid thinking of two pairs of amounts of money,
So they sought flaws of the mathematical standard resolution.
Conclusion
The fact that IVT has many relatives suggests that none of the relatives is based on real experience.In other words, in my perception, they seem to be products of the imagination.
I think that each of them resolve each fictitious paradoxes.
They had realy resolved a paradox.
But the resolved paradox was a fiction.
But the resolved paradox was a fiction.
Return to the list of my pages written in English about the two envelopes problem