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Composition of this page was changed on March 21, 2015, and February 26, 2017.
I call them the "equivalent-expectationian."
They spell various incantations to prove their hypothesis.
There is a variety of equivalent-expectationian.
(x/2, x) and (x, 2x) from beginning of the game, and the amount of the opened envelope is x, following propositions are mathematically equivalent.
Among these incantations, the naive incantation is most pure, because it contains no trick.
The title was re-changed on March 29, 2018.
Some people spell an incantation to apply the principle of "no information no change" to expectation.
I found incantation 2 in a research paper of psychology.
Addition
This section was revised on February 26, 2017 with new title.
When I express this in a mathematical form.
Indeed the above equation means nothing but the following theorem.
(Added on February 2017)
This expectation formula is incomprehensible, because of the following reasons.
(This paragraph was added on October 18, 2014, and was revised on February 28, 2016, September 4, 2016, and September 10, 2016)
I have notice the following tricks."(1/2)(100/150) + (1/2)(-50/75)" is an expectation.
There was similar calculation formula in an article which was posted to a web page in 2009.
I will summarize it as follows.
An article represented exactly the same opinion in 2004.
I will summarize that opinion as follows.
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2019/01/08 8:55:19
First edition 2014/10/11
Incantations used by equivalent-expectationian on 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.
Composition of this page was changed on March 21, 2015, and February 26, 2017.
Variety of equivalent-expectationian on the two envelopes problem
Some people think the expectation of the amounts of money in the two envelopes must be same, even one of these envelopes has been opened.I call them the "equivalent-expectationian."
They spell various incantations to prove their hypothesis.
There is a variety of equivalent-expectationian.
- ExpectedAmountEqualsChosenAmountian
Some people think that the expected amount in the other envelope must be equal to the amount of money which is in the chosen envelope.
(On April 19, 2018, I replaced the name ExpectedAmountIsChosenAmountian by the name ExpectedAmountEqualsChosenAmountian.)
- OpeningAnEnvelopeGivesNoInformationian
Some people think that we can not get any information from revealed amount of money in chosen envelope
(On March 29, 2018 and April 19, 2018, I replaced the name.)
Incantations spelled by ExpectedAmountEqualsChosenAmountian
Mathematical incantation spelled by ExpectedAmountEqualsChosenAmountian
Let x be the amount of money in the opened envelope.
The possible pairs of amounts are(x/2, x) and (x, 2x).
If we restrict the pairs to these two pairs from beginning, then the prior expectation of the amount of the opened envelope is
p(3x/4) + (1-p)(3x/2).
If that expectation is equal to x, then
p = (2/3).
If that expectation is equal to x, then the expectation of the amount in the closed envelope is
E = p(x/2) + (1-p)2x = x.
Therefore, the two envelopes are equivalent !
This incantation has some mathematical flavor, so I think this has some dubiousness.
The possible pairs of amounts are
If we restrict the pairs to these two pairs from beginning, then the prior expectation of the amount of the opened envelope is
If that expectation is equal to x, then
If that expectation is equal to x, then the expectation of the amount in the closed envelope is
Therefore, the two envelopes are equivalent !
Naive incantation spelled by ExpectedAmountEqualsChosenAmountian
Let x be the amount in the opened envelope, and let Y be the random variable of the amount in the another envelope.
The expectation of the amount in the another envelope must be the same as the amount in the opened envelope.
ThereforeProvability(Y= x/2) = 2/3, and Provability(Y=2x) = 1/3.
This incantation is very naive, but it contains the statement of why they think so. I think this incantation has no dubiousness.
The expectation of the amount in the another envelope must be the same as the amount in the opened envelope.
Therefore
Equivalence of these incantations
If the pairs of amounts are restricted to only two pairs- The prior expectation of the amount of money in the envelope which is to be opened is x.
- When the amount of money in the opened envelope is x, the conditional expectation of the amount of money in the another envelope is x. (← Revised on October 14, 2014)
If one of the incantations mentioned above is an absolute incantation, then the rest also are absolute incantations.
Among these incantations, the naive incantation is most pure, because it contains no trick.
Incantations spelled by OpeningAnEnvelopeGivesNoInformationian
This paragraph was added on February 1, 2015. And its title was changed on March 21, 2015.The title was re-changed on March 29, 2018.
Some people spell an incantation to apply the principle of "no information no change" to expectation.
Incantation 1
Incantation 2
In a famous book, I found an incantation which is similar to incantation 1.
Before opening, it has already been known that the other envelope contains twice or half as much money as your envelope.
So opening an envelope does not give new information about whether you should trade or not.
So opening an envelope does not give new information about whether you should trade or not.
Incantation 2
Opening an envelope cannot change the amounts in the envelopes,
So it should not matter whether you keep or trade envelopes.
So it should not matter whether you keep or trade envelopes.
I found incantation 2 in a research paper of psychology.
Addition
A calculation which is not an incantation but may be a magic
This section was revised on February 26, 2017 with new title.
A calculation which appeared in English language Wikipedia
Recently, following expectation formula was added to the revision of 12:20, 28 May 2014 of the article "Two envelopes problem" in English language Wikipedia.
The amount in the opened envelope is 100.
If the pair of amounts is 50 and 100, the gain of trading is minus 50, and the average of the two amounts is 75. · · · (1)
If the pair of amounts is 100 and 200, the gain of trading is plus 100, and the average of the two amounts is 150. · · · (2)
The expected return from trading is
E = (1/2)(100/150) + (1/2)(-50/75) = 0. · · · (3)
Therefore the two envelopes are equivalent !
The subject of this expectation formula
If the pair of amounts is 50 and 100, the gain of trading is minus 50, and the average of the two amounts is 75.
If the pair of amounts is 100 and 200, the gain of trading is plus 100, and the average of the two amounts is 150.
The expected return from trading is
Therefore the two envelopes are equivalent !
According to the paper which presented the above formula, the subject of this expectation is called "Success Factor", and as far as I understood it, the success factor is the ratio of the following two values.
(1) The difference (positive or negative) of the amount which will be gotten after trading and the amount which has been gotten before trading
(2) The mean value of the amount which will be gotten after trading and the amount which has been gotten before trading
(1) The difference (positive or negative) of the amount which will be gotten after trading and the amount which has been gotten before trading
(2) The mean value of the amount which will be gotten after trading and the amount which has been gotten before trading
When I express this in a mathematical form.
Let x be the chosen amount of money.
Let r be the ratio of the greater amount of money to the lesser amount of money in a pair of amount of money.
Then the success factor F is as follows.
For the case that the chosen amount is the lesser
F = (rx - x)/((rx + x)/2) = 2(r - 1)/(r + 1). – – – constant value
For the case that the chosen amount is the greater
F = (x/r - x)/((x + x/r)/2) = - 2(r - 1)/(r + 1). – – – constant value
If r = 2 then each of them is 2/3 or -2/3.
The case of the English language Wikipedia article "Two envelopes problem" at the revision of 12:20, 28 May 2014 is as follows. (← Revised on April 12, 2018.)
Let r be the ratio of the greater amount of money to the lesser amount of money in a pair of amount of money.
Then the success factor F is as follows.
For the case that the chosen amount is the lesser
For the case that the chosen amount is the greater
If r = 2 then each of them is 2/3 or -2/3.
The case of the English language Wikipedia article "Two envelopes problem" at the revision of 12:20, 28 May 2014 is as follows. (← Revised on April 12, 2018.)
If the lesser has been chosen, F = (200 - 100)/((200 + 100)/2) = 100/150 = 2/3 .
If the greater has been chosen,F = (50 - 100)/((100 + 50)/2) = -50/75 = -2/3
If the greater has been chosen,
Indeed the above equation means nothing but the following theorem.
(Added on February 2017)
Let x, y denote two numbers.
Then if the ratio of x and y is constant, the ratio of (x - y) and (x + y) is constant.
Then if the ratio of x and y is constant, the ratio of (x - y) and (x + y) is constant.
This expectation formula is incomprehensible, because of the following reasons.
- Success factor is a kind of magnification. However, the amounts corresponding to the denominators of the two success factors are different. Therefore, there is no point in adding or subtracting them. (← Added on January 8, 2019)
- Can a constant value be expected value? (← Added on September 10, 2016, revised on February 26, 2017)
- If this equation compare feeling of loss and feeling of earning, why no utility function has been used? (← Added on January 10, 2015)
- In this formula, both of the terms which denote probability are constant 1/2, therefore this formula does not calculate conditional expectation.
So this formula is not suitable for the section "Alternative interpretation" of the article "Two envelopes problem" (revision of 12:20, 28 May 2014) in English language Wikipedia.
(This paragraph was added on October 18, 2014, and was revised on February 28, 2016, September 4, 2016, and September 10, 2016)
I have notice the following tricks.
- Using specific values of amount of money to conceal the fact that the success factors (the subjects of the expected value) have constant values
(2/3 and -2/3). - Using mean values as denominator.
Coefficient (1/2) as probability.
Similar calculation
This paragraph was added on March 19, 2015.There was similar calculation formula in an article which was posted to a web page in 2009.
I will summarize it as follows.
You choose one envelope and find $20.
There are two situations { $10, $20 } and { $20 , $40 }.
Let Y be the lesser amount in each pair of amounts.
If you have 2Y, then you will lose Y after switching.
If you have Y, then you will earn more Y after switching.
You will either gain or lose Y.
Expected gain is(1/2)(-Y) + (1/2)(Y) = 0 .
It is essentially the same as the above calculation formula except the following differences.
There are two situations { $10, $20 } and { $20 , $40 }.
Let Y be the lesser amount in each pair of amounts.
If you have 2Y, then you will lose Y after switching.
If you have Y, then you will earn more Y after switching.
You will either gain or lose Y.
Expected gain is
- It concerns lesser amount of each pair of amounts rather than mean value of amounts of each pair of amounts.
- It has forgotten to divide the chosen amount by the lesser amount in each pair of amounts.
And it has a wrong usage of the variable symbol.
So it is not correct mathematically, but it is simple and easy to understand.
An article that presented exactory the same opinion 10 years ago
This paragraph was added on March 29, 2018. The title was revised on April 12, 2018.An article represented exactly the same opinion in 2004.
I will summarize that opinion as follows.
Suppose that the amount of money in the chosen envelope is 10,000.
Then the expected gain by exchange is(1/2)(-2,500/7,500) + (1/2)(5,000/15,000) = 0.
And the expected loss by no-exchange is(1/2)(2,500/7,500) + (1/2)(-5,000/15,000) = 0.
From this, I realized that the people who advocate such an opinion are not as rare as I thought.
Then the expected gain by exchange is
And the expected loss by no-exchange is
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