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

Is the principle of insufficient reason the cause of the paradox of the two envelopes problem?

The process through which the Two Envelope Paradox arise.

The process through which the Two Envelope Paradox (Exchange Paradox) arise is as follows.

• Let A be the envelope which you first select and let B is the another envelope.
• Let e be the expectation of the amount in the envelope B while the amount of A is x.
e = (probability 1/2) (x/2) + (probability 1/2) 2x = 1.25x · · · (1)
• It means that e is greater than x, and you should trade envelope A with B.
• Therefore always the envelope that you chose first has less expected amount than another envelope.
• But there is no reason of a gap between two envelopes.
• It is a paradox !

The principle of insufficient reasen

The mechanism of this paradox is disappointingly simple.

We forget the odds of the pair (x/2 , x) , and the odds of the pair (x , 2x).
That why we thoughtlessly assign probability 1/2 to the terms in equation (1).
This is a kind of probability illusion called 'Base Rate Fallacy', and it is the cause of the two envelope paradox.

But some people don't think so.

They think that an improper application of  "the principle of insufficient reason"  is the cause why we assign a probability 1/2 to the event that on the condition the chosen envelope contains a specific amount of money, the chosen envelope contains lesser amount of money and why we assign a probability 1/2 to the event that on the same condition, it contains greater amount of money.

Comparison between 'base rate fallacy' and 'insufficient reason'

Terminology

I will use following terminology, in the following paragraphs and sections.

• The likelihood
  The probability that the chosen envelope contains lesser amount of money on the condition of a certain pair of amount,  and the probability that the chosen envelope contains greater amount of money on the condition of a certain pair of amount.
• The conditional probability
  The probability that the chosen envelope contains lesser amount of money on the condition that the chosen envelope contains a certain amount of money,  and the probability that the chosen envelope contains greater amount of money on the same condition.

Majority

I think that the base rate fallacy is the cause of the paradox, but I have found no articles that say base rate fallacy is the cause of the paradox of the two envelopes problem. So my opinion is minor.

complexity

By a base rate fallacy, we simply confuse the likelihood and the conditional probability.
But to apply the principle of insufficient reason, we must think there is no information about the conditional probability except the likelihood.

consistency

I think that we can not unconsciously apply the principle of insufficient reason.
When we find a contradiction after we apply such a principle,  can we continue believing the probability?

If the base rate is described in the problem statement

I imagine a psychological experiment which use problem statement which describe following probability distribution.

pairs of amount of money	probability
¥1,000, ¥2,000	1%
¥2,000, ¥4,000	49%
¥4,000, ¥8,000	49%
¥8,000, ¥16,000	1%

I expect that such a experiment will end with following result.

• If the cause of the paradox is the principle of insufficient reason, any participants of the experiment will hesitate to assign a probability 1/2 to the conditional probability.
• If the cause of the paradox is the base rate fallacy,  some participants will assign probability 1/2 to the conditional probability.

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

---