My idea of cognitive psychological experiment about the two envelopes problem

Caution
I who am a Japanese wrote this page in English, but I am not so good at English.

An experiment about the cause of the paradox

Hypothesis which I want to prove

The cause of the two envelope paradox is base rate fallacy and is not careless application of the principle of insufficient reason.

Method

	Type 1 problem NO BASE RATE	Type 2 problem WITH LIKELIHOOD	Type 3 problem WITH BASE RATE	Type 4 problem WITH BASE RATE AND LIKELIHOOD
Base rate which participants read (Probability distribution of pair of amount)	not expressed	not expressed	pair of amounts : probability (10, 20) : 4/10 (20, 40) : 1/10	pair of amounts : probability (10, 20) : 4/10 (20, 40) : 1/10
Likelihood which participants read (Probability that the chosen envelope contains lesser amount, and the probability that it contains greater amount)	amount : probability lesser : 1/2 greater : 1/2	amount : probability lesser : 1/3 greater : 2/3	amount : probability lesser : 1/2 greater : 1/2	amount : probability lesser : 1/3 greater : 2/3
Revealed amount in the chosen envelope	20	20	20	20
question which participants answer	Assign values to P₁ and P₂ of the following expectation formula. E = P₁ × 10 + P₂ × 40	Assign values to P₁ and P₂ of the following expectation formula. E = P₁ × 10 + P₂ × 40	Assign values to P₁ and P₂ of the following expectation formula. E = P₁ × 10 + P₂ × 40	Assign values to P₁ and P₂ of the following expectation formula. E = P₁ × 10 + P₂ × 40
P₁, P₂	uncertain	uncertain	P₁ 4/5 P₂ 1/5	P₁ 8/9 P₂ 1/9
P₁, P₂ by Base rate fallacy (My expectation)	P₁ 1/2 P₂ 1/2	P₁ 2/3 P₂ 1/3	P₁ 1/2 P₂ 1/2	P₁ 2/3 P₂ 1/3
P₁, P₂ by a person who make careless application of the principle of insufficient reason (My expectation)	P₁ 1/2 P₂ 1/2	P₁ 2/3 P₂ 1/3	P₁ 4/5 P₂ 1/5	P₁ 8/9 P₂ 1/9

The result that I expect

I expect as follows.

Participants will fail about the type 3 problem and type 4 problem.

I hope that these result prove that the cause of the paradox is base rate fallacy.

An experiment about the DivideThreeByTwoian's opinion

On April 28, 2017, this paragraph was revised and the title was changed.

About the DivideThreeByTwoian's opinion, please see a paragraph "DivideThreeByTwoian's Resolution" in my page "An outline of the Two Envelopes Problem".

Hypotheses which I want to prove

Almost people get a mental model which is made of two pairs of amounts of money when they have read the paradoxical formula "E=(1/2)(x/2) + (1/2)2x".
Some of them accept the opinion that the correct expectation formula is "E=(1/2)A + (1/2)2A", in other words some of them become DividethreeByTwoians.
Some of DivideThreeByTwoians will accept one of the not three amounts theory and inconsistent variable theory or will not accept both.

Method

step 1
Participants read following expression.
There are two envelopes.
One envelope contains twice as much money as the other.
Randomly a person chooses one envelope.
Before opening the chosen envelope the person can change choice.
The person thinks as below.
Suppose that the amount of money in the chosen envelope is $x.
Then the other envelope contains $2x or $(x/2).
step 2 (Question 1)
Participants answer following question.
Consider the amount of money in the non-chosen envelope.
Which is the ratio of the least amount and the greatest amount?
1 : 2 ? or 1 : 4 ? or else ?
step 3
Participants read following expression.
And the person thinks as below.
$2x and $(x/2) are plased in the non-chosen envelope with same probability 0.5.
So the expected value of the amount money in the non-chosen enbelope is 0.5 × $2x + 0.5 × $(x/2) = $1.25x.
step 4 (Question 2)
Participants answer following question.
Do you understand this calculating formula of expected value?
step 5 (Question 3)
Participants answer following question.
Consider the amount of money in the non-chosen envelope.
Which is the ratio of the least amount and the greatest amount?
1 : 2 ? or 1 : 4 ? or else ?
step 6 (Explaining of paradox)
Participants read following expression.
The expectation indicates that the person should change choice regardless of the amount of money in the chosen envelope.
This conclusion contradicts the Symmetricalness of the two envelopes!
step 7 (Question 4)
Participants answer following question.
Do you think that this is a paradox?
step 8 (Question 5)
Participants answer following question.
Consider the amount of money in the non-chosen envelope.
Which is the ratio of the least amount and the greatest amount?
1 : 2 ? or 1 : 4 ? or else ?
step 9 (correct calculation formula)
Participants read the following resolution.
This paradox will be resolved as follows.
Let A be the lesser amount of money in the two envelopes.
Then the expected value of the amount of money in the non-chosen envelope is (1/2)A+(1/2)2A=(3/2)A.
And the expected value of the amount of money in the chosen envelope is also (1/2)A+(1/2)2A=(3/2)A.
In this way the two envelopes are equally favorable.
Therefore the paradox has been resolved.
step 10 (Question 6)
Participants answer following question.
Do you agree that this resolves the paradox?
step 11 (Question 7)
Participants answer following question.
Consider the amount of money in the non-chosen envelope.
Which is the ratio of the least amount and the greatest amount?
1 : 2 ? or 1 : 4 ? or else ?
step 12 (hypotheses about the cause of the paradox)
Participants read the following hypotheses.
Hypothesis 1 (not three amounts theory)
The pairs of amount of money (x, x/2) and (x, 2x) belongs different games.
In other words, only two envelopes can not envelope three amounts of money x, x/2 and 2x.
The paradoxical calculating formula "E=(1/2)(x/2) + (1/2)2x" has been made by forgetting this.
Hypothesis 2 (inconsistent variable theory)
Look carefly at the variable symbol in the paradoxical calculating formula "E=(1/2)(x/2) + (1/2)2x".
Symbol x in the first term and symbol x in the second term are denoting different values.
This fallacy is the cause of the paradox.
step 13 (Question 8)
Participants answer following question.
Which hypothesys is most reliable?
- Hypothesis 1
- Hypothesis 2
- None

The result that I expect

I expect as follows.

	question	the most answer
question 1	the ratio (the ratio of the least amount and the greatest amount of money in the non-chosen envelope)	"1 : 4"
question 2	Do you understand this calculating formula of expected value?	yes
question 3	the ratio	"1 : 4"
question 4	Do you think that this is a paradox?	yes
question 5	the ratio	"1 : 4"
question 6	Do you agree that this resolves the paradox?	mostly "no" a few "yes"
question 7	the ratio	mostly "1:4" a few "1:2"
question 8	Which hypothesys is most reliable?	uncertain

I hope that these result prove that only the standard resolution is the true resolution of the two envelope paradox.

If the result did not comply with the above expectation

(This paragraph was added on April 29, 2017.)

I think that if the result did not comply with the above expectation the another experiment like below should be done.

New method

step 1
Participants read following expression.
There are two envelopes.
One envelope contains twice as much money as the other.
Randomly a person chooses one envelope.
The person opens it and notices the amount of money.
Then the person can change choice.
The person thinks as below.
Suppose that the amount of money in the chosen envelope is $x.
Then the other envelope contains $2x or $(x/2).
step 2 and the following steps
<<< same as original experiment >>>

The new result that I expect

I expect as follows.

	question	the most answer
question 1	the ratio (the ratio of the least amount and the greatest amount of money in the non-chosen envelope)	"1 : 4"
question 2	Do you understand this calculating formula of expected value?	yes
question 3	the ratio	"1 : 4"
question 4	Do you think that this is a paradox?	yes
question 5	the ratio	"1 : 4"
question 6	Do you agree that this resolves the paradox?	no
question 7	the ratio	"1 : 4"
question 8	Which hypothesys is most reliable?	"None"

If the new result complied expectation

I must more study about the case that the opportunity to trade is givven before opening the chosen envelope.

If the new result did not comply expectation

I must change my opinion.

Terms

standard resolution
It is as follows.
The cause of this paradox is a fallacy of probability.
The probability that the other envelope contains x/2 and the probability that it contains 2x are not necessarily 1/2.
Actual probabilities correspond to the odds of the pair (x/2, x) and the odds of the pair (x, 2x).
So there is no wonder. even if the non-chosen envelope is more favorable for a value of the amount of money in the chosen envelope.
And if the non-chosen envelope is favorable for a value of the amount of money in the chosen envelope, the non-chosen envelope must be unfavorable for some value of the amount of money in the chosen envelope.
The equivalence of the two envelopes is surely kept.
not three amounts theory
It is as follows.
The cause of the paradox is to think of three amounts x/2, x and 2x.
If you think of only two amounts A and 2A , paradox will vanish.
inconsistent variable theory
It is as follows.
The variable symbol of the paradoxical expectation formula denotes two different values. In one term it denotes the lesser amount of money, and in the another term it denotes the greater amount of money.
To mix different instances of a variable in the same formula is the cause of the paradox.

Return to the list of my pages written in English about the two envelopes problem