[extropy-chat] Re: monty hall paradox again: reds and green gorfs

[Eliezer Yudkowsky]

Spike wrote:
 >>Eliezer Yudkowsky
 >>
 >>> ...The probability that
 >>>the other envelope is larger must go to 1/3, it doesn't
 >>>matter how it gets there."
 >>
 >>Your mathematician friend is flat wrong, and needs to study Bayesian
 >>probability theory. --  Eliezer S. Yudkowsky
 >
 > OK cool, I was hoping someone would say that.  So where
 > is the error?  Are you saying that Bayesian reasoning
 > predicts that there *is* a profit in swapping?  Even
 > if you have no idea how much a zorg is?  How do you
 > set up the Bayesian priors?  How do you set up a
 > sim to prove it?

Here's a slightly edited version of a post I sent to the evol-psych mailing 
list:

Herbert Gintis wrote:
  > If you thought the Monty Hall problem was tough, try the following.
  >
  > A man has two sons, whom he want to give money to. He says to the sons,
  > "I am giving you each a sealed envelope. One envelope has twice as much
  > money as the other in it. I will allow you look inside the envelope. If
  > neither of you is satisfied with your gift, you can exchange envelopes."
  > Each son looks in his own envelope. The first says to himself, "Dad gave
  > me $10,000. So my brother has either $5000 or $20,000, the average of
  > which is $12,500, which is more than what I have. So I'll switch if he
  > is willing." The other brother has either $5000 or $20000. Call this
  > amount X. The brothers makes the same argument to himself, saying "My
  > brother has either X/2 or 2X, the average of which 1.25 X, which is
  > greater than what I have, which is X. So I'll trade if he's willing." So
  > both trade.
  >
  > But how could this be? It makes no sense that they would both want to
  > trade no matter how much money is in their envelope, because it is
  > random who got which envelope. Moreover, the brothers could use the same
  > reasoning to trade, even without looking in the envelope!!!!

This *is* a really tough puzzle...

I think it's fair to assume that the dollar amounts of $X and $2X directly
translate into player's utilities, rather than answering that the dollar
amounts should be adjusted on a logarithmic scale or whatever - I don't
think that's the intended problem being posed, although psychologically,
as opposed to mathematically, dollars as such are (a) logarithmic
incremental utilities (b) inconsistently valued depending on whether they
are gains or losses.  Let us suppose that the problem is being posed to a
Bayesian decision system rather than a human - it still seems troublesome.

It seems to me that this problem derives from having an improper prior
probability distribution on the dollar amounts offered, such that the
total expected amount offered is infinite.  In other words, suppose that
when I look inside the envelope, then no matter what $X I see, it still
seems to me exactly 50% likely that the other brother has $2X as that he
has $X.  Intuitively (I am not doing the math, just thinking as I go
along) this seems to require a prior probability distribution which is
completely uniform across all real values stretching from zero to infinity.

It is like the old paradox of randomly picking a number between zero and
infinity - no matter which number you pick, almost all other numbers will
be larger.  So given one of two finite dollar amounts randomly
picked from between 0 and infinity, you should prefer the other one.

If we assume a saner prior probability distribution for the total amount
of money $3X - one where the total possible range of 3X is finite - then a
Bayesian decision system should never have any trouble.  The next question
is what this looks like from the viewpoint of the person holding the envelope.

It seems to me that the basic error is saying, "I am holding $X, and my
brother is equally likely to be holding $X/2 or $2X, therefore I should
switch."  For each possibility in the prior probability distribution for
the total money $T, it is equally likely that you are holding the envelope
containing $T/3 or $2T/3.  As soon as you look in the envelope and find
out that you are holding a specific amount $X, you will have to update
your prior probability distribution based on that new information, and it
may no longer be equally likely that the other person is holding $2X or $X/2.

You cannot, at the beginning of the problem, say that you are holding a
constant amount (when you do not know what it is) and ask whether your
brother is holding $2X or $X/2, assuming equal probabilities, because this
requires a different prior probability distribution.  Why?  Let's suppose
there's a uniformly distributed set of possible total amounts $T from $3
to $300.  If you open the envelope and see $1, you *know* the other person
has $2.  If you open the envelope and see $200, you *know* the other
person has $100.  So given any finite prior distribution, it cannot be
possible, for all specific dollar amounts $X, that the other person is
equally likely to have $2X or $X/2.

If you think to yourself, "The total amount of money in these total
envelopes is $T, and I am holding the envelope with either $2T/3 or $T/3,
with equal probability," your math is straightforward.

If you think to yourself, "I am holding some amount of money $X, and the
other person has a probability P of holding $X/2 and a probability 1-P of
holding $2X," you must then use an unsimple probability distribution for X
and an unsimple X-dependent distribution for P in order to calculate your
expected utility from switching.

A specific example:  Suppose the prior value of $T is distributed evenly
from $3 to $300, and you open up the envelope and find $100?  It is
equally likely that the other person's envelope contains $50 or $200, and
you should switch (assuming the dollar amounts represent your utilities).

But if you see a value between $101 and $200, you should definitely refuse
to switch.  If you see $101, you know the other envelope cannot contain
$202.  So most of the times you open the envelope you will somewhat want
to switch, but a substantial portion of the times you will *definitely*
not want to switch.  And it balances exactly, so before you open the
envelope, you have no particular motive to switch one way or the other.

-- 
Eliezer S. Yudkowsky                          http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence