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

[Hal Finney]

A logic paradox is a situation where there are two arguments that reach
opposing conclusions, yet both arguments seem equally valid.

In this case, one argument goes as follows.  We have the two envelopes,
and when they are closed it is obvious that there is no point in
switching.  We know in advance that when we open one we will see some
number, so after doing so we gain no new information, hence there should
still be no point in switching.

The other argument says that once you open the envelope and see X, the
other envelope has X/2 or 2X with equal likelihood, hence you have an
expected gain of X/4 if you switch, so you should switch.

Both arguments seem valid on their own.  But they can't both be right.
(Or can they?  See below.)

One of the problems in discussing paradoxes is that people tend to talk
past each other.  Each side recites his argument, as if that refutes
the other.  But that doesn't resolve it, no matter how loudly you
shout one argument or the other.  The nature of a paradox is two sided.
Both arguments appear valid.  Emphasizing the validity of one of them
merely deepens the paradox.

The only way to resolve a paradox is to point out a flaw in one of the
arguments.  You have to look for weaknesses, not strengths.  You have
to say why one of the arguments doesn't work.  And again, this can't be
done merely by going back and emphasizing the validity of the opposing
argument.  You have to find a flaw in the internal logic of one of the
arguments.  That's how you make progress in a paradox.  If people don't
understand this rule, then arguing about paradoxes is unproductive.

Now, in this particular case, things are a little different, because both
arguments really are valid, in my opinion.  How can this be, when they
reach opposite conclusions?  Isn't that a contradiction, which should
never happen in logic?

The problem is that the paradox as stated is simply logically impossible.
It contains a contradiction.  And once you start with a contradiction,
it should be no surprise that you can derive one.

The contradiction is very simple.  There is supposedly a uniform and equal
probability distribution over the values which could be in the envelope.
But in fact there can be no uniform, nonzero probability distribution
over an infinite number of values.  Hence the circumstances in the
paradox cannot arise.

If you change the rules to say that there could be any value in the
envelope less than one billion dollars, then the correct strategy is
to switch if the value you see is less than half a billion, else don't
switch.  This is simple and logical and no one will disagree with it.

Things are more complicated if you don't know the probability
distribution, but again, there is really no such thing as a totally
unknown probability distribution.  A totally unknown probability
distribution would have to give nonzero probability to an infinite number
of values, which is again impossible.  The truth is that you can always
make some estimate based on your experience with life circumstances and
with logic puzzles for what is a reasonable probability distribution
over the values in the envelopes.  Given any such distribution, you can
logically derive an optimum rule for when to switch, similar to the rule
for the billion dollar cap example.  And no one will disagree with what
the optimal rule is for a given distribution, because it requires only
unambiguous math to derive it.  No paradox arises in such cases.

Hal