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Different mental models and different illusions make the same expectation formula for the Wallet Game. But there are two different 'Wallet Game Paradoxes'.
The Rule Part of this Paradox
In the following sections , the people with this mental model are called 'DoublePairian'.
If a DoublePairian think about the Wallet Game, he or she probably have a mental model such as follows.
In the following sections , the people with this mental model are called 'ManyPairian'.
Some people make following mental model about the Two Envelope Problem.
In the following sections , the people with this mental model are called 'SinglePairian'.
If a SinglePairian think about the Wallet Game, he or she probably have a mental model such as follows.
(This figure was revised on December 26, 2014)
In the following sections , the people with this mental model are called 'WinLosePairian'.
But if the ManyPairians have been caught by a probability illusion and the WinLosePairians have been caught by the Inconsistent Variable, they may make same expectation formula as above.
Even if such a thing happened., their paradoxes are different each other.
(Please see the page "Inconsistent Variable Theory on The Two Envelope Paradox")
But I think that a WinLosePairian of the Wallet Game will easily be caught by the Inconsistent Variable.
Reason
Return to the list of my pages written in English about the two envelopes problem
2015/01/17 23:21:27
First edition 2014/07/26
The Mental Models for the Wallet Game
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.
Different mental models and different illusions make the same expectation formula for the Wallet Game. But there are two different 'Wallet Game Paradoxes'.
The Wallet Game
According to some documents, the paradox of the Wallet Game seems to have following scenario.The Rule Part of this Paradox
- Each of two persons places his or her wallet on the table.
- They have forgotten the amount of the money which is in the wallets.
- The person who is the owner of the smallest amount will win all amount in these wallets.
- Each of them think as follows:
I may lose the amount of my money and I may win more amount than mine.
The game is to my advantage because the potential gain is greater than zero. - But there is no reason of a gap between their wallets. It is a paradox !
Mental Models for the Two Envelope Paradox and the Wallet Game
Some people make following mental model about the Two Envelope Problem.In the following sections , the people with this mental model are called 'DoublePairian'.
If a DoublePairian think about the Wallet Game, he or she probably have a mental model such as follows.
In the following sections , the people with this mental model are called 'ManyPairian'.
Some people make following mental model about the Two Envelope Problem.
In the following sections , the people with this mental model are called 'SinglePairian'.
If a SinglePairian think about the Wallet Game, he or she probably have a mental model such as follows.
(This figure was revised on December 26, 2014)
In the following sections , the people with this mental model are called 'WinLosePairian'.
Each pairians make same expectation formula
If a ManyPairian has been caught by probability illusion, he or she may think as follows.- Let x be the amount in your wallet.
- Let X be the random variable of the amount in your wallet
and let Y is the random variable of the mount in another wallet. - Let G be the random variable of the gain.
- Let e be the expectation of your gain.
e = E(Y | Y > X and X = x) + E(-X | Y < X and X = x) = E(x + G | Y > X and X = x) + E(-X | Y < X and X = x) = (1/2)(x + α) + (1/2)(-x) ← probability illusion = (1/2) α > 0 - The expected gain is always greater than zero.
- But there is no reason of a gap between their wallets. It is a paradox !
- Let X be the random variable of the amount in your wallet
and let Y is the random variable of the mount in another wallet. - Let G be the random variable of the gain.
- Let e be the expectation of your gain.
e = E(Y | you win) + E(-X | you lose) = E(X + G | you win) + E(-X | you lose) = (1/2) E(X) + (1/2) α - (1/2) E(X) ← the Inconsistent Variable = (1/2) α > 0 - The expected gain is always greater than zero.
- But there is no reason of a gap between their wallets. It is a paradox !
But if the ManyPairians have been caught by a probability illusion and the WinLosePairians have been caught by the Inconsistent Variable, they may make same expectation formula as above.
Even if such a thing happened., their paradoxes are different each other.
Is there a chance that a WinLosePairian be caught by the Inconsistent Variable?
I think that a SinglePairian of the Two Envelope Paradox will hardly be caught by the Inconsistent variable.(Please see the page "Inconsistent Variable Theory on The Two Envelope Paradox")
But I think that a WinLosePairian of the Wallet Game will easily be caught by the Inconsistent Variable.
Reason
- The wording of the description of the paradox of the Wallet Game has some properties that induce the Inconsistent Variable.
- When we think about the Two Envelope Paradox, we must think of trading rate and the amount of money. Therefore it is hard to be caught by the Inconsistent Variable. But when we think about the Wallet Game, it is enough if we think of only the amount of money. Therefore it is easy to be caught by the Inconsistent Variable.
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