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This page is too old.
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. In this way IVT is only one of these relatives.

Caution
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.

A paper written in the early 2010s

This paper was written in the early 2010s
The author of it claimed following opinion.

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.
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.

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. But in the same paper, another opinion which was in conflict with it were simultaneously described as follows.

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

And some articles claim as follows.

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

Case that the pair of amounts is not fixed

And some articles claim as follows.

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

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
 
 
Not three amounts theory
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 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.


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