Apopuli.136
net.math
utzoo!decvax!ucbvax!G:cliff
Sat Apr 10 12:36:29 1982
Newcombe's paradox debunked (maybe)
There is no answer to the question of how to maximize your wealth,
because the situation is set up to make that question meaningless.
Newcombe's paradox is interesting, I bet it was dreamed up by a
Bayesian statistician to confuse everyone... Once having understood
it, I would be more inclined to call it a "mental illusion"
(by analogy to an "optical illusion").
The illusion is caused by two contradictory assumptions made in the
statement of the problem:
1) The computer can predict with that incredible
degree of accuracy.
2) When you enter the room you have total freedom of choice
as to which box to pick up.
"Why are those two assumptions contradictory?" you ask. I will first
attempt to show that 2) cannot exist in the presence of 1):
Suppose that when you enter the room you are so confused you decide
to flip a (fair) coin to decide which box to pick up. That means
that you have a 50-50 chance of taking out box B. That means that
the computer had to predict the outcome of a fair coin toss with
99.9% accuracy. This is impossible unless the computer is actually
clairvoyant (with 99.9% accuracy). This means that at the time
that the computer made the prediction your course of action
was already decided and you actually have NO CHOICE when you enter
the room!
If that is not clear to you, the other way is easier to understand.
1) cannot exist in the presence of 2):
Suppose that all the people in line decide to flip fair coins
to decide which box to pick up. If their coin lands heads they will
take both boxes, if tails, take box B. Each one has a 1/2 chance
of picking box B. Nothing can predict the outcome of a fair
coin toss with better than 50% accuracy (let alone a sequence of
fair coin tosses!), so the computer's prowess has been grossly
exaggerated!
QED
=====================
Discussion:
I personally prefer the first argument above, I see the whole situation
in terms of looking down on a long line of robots, all going in and
out of the room. As each one exits I see that all the robots (except
for maybe 1 in 1000) who leave with only one box get a million
dollars, while those who take both only get $1000.
This makes me sad for the poor robots and I try to tell them as they
go in that they should take just one box, but they are all
pre-programmed and can only do what was decided for them a week
earlier.
Thinking about this problem is something like counting
sheep.......
-Cliff Frost (ucbvax!populi!cliff)
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