[extropy-chat] monty hall paradox again

[Spike]

Eliezer has proposed a puzzle, a variation of
which we have discussed here before, but with
a maddening twist.

Suppose an unknown but whimsical benefactor has chosen 
to give you a monetary gift.  A messenger is sent with 
two identical envelopes and offers to give you one of
them.  The messenger knows not the amounts of money
in either envelope, but tells you that one of the
envelopes contains twice as much as the other.  You
are to choose an envelope.  You choose, and inside you
find ten dollars.  Now the messenger offers to
trade your ten dollars for the contents of
the other envelope.  Would you trade?  Why?

I reason that there is a 50% chance the other envelope
contains 5 dollars and 50% chance it contains twenty,
so mathematical expectation value of the other envelope
is .50*5 + .50*20 = 12.50 so I would trade.  Same reasoning
applies if the first envelope contained 500 or 5000
dollars or a billion, all under the assumption that seems
so natural to me, that money is good, so more is
better and too damn much is just right.  You trade
5 dollars for a 50% shot at 20.  Such a deal!

Nowthen, since we have concluded that for each dollar
in the envelope you choose, the other envelope contains
an expected buck twenty five, you would *always* trade,
regardless.  For that reason, there really is no reason
to bother opening and looking in the envelope you chose first.  
Regardless of the amount therein, you will immediately trade 
it away for the other one since you expect it to contain more.
So why not skip the step of choosing the first envelope
and subsequently trading it (opened or unopened) for the second?  
Why not just decide which one you would choose, then take 
the other one instead?  Or if you choose one, then trade
it, you might go thru the same line of reasoning that
you did before and conclude that regardless of the amount
in the envelope you now hold, the other one contains 25%
more, and since you still haven't opened either envelope,
you can still trade back.  Then of course the same line
of reasoning *still* applies, so you trade again.  And
again.  And so on to infinity and beyond.

So, is this not a strange situation?  Ideally of course
you could smite the messenger and run off with both
envelopes.  Or you could keep trading unopened envelopes
until you lose track of which one you chose first.  Or
you could trade envelopes until the messenger perishes
of age-related infirmities, then run off with both.  But
you notice that the messenger is both stronger and younger
than yourself, and so would be unlikely to precede you
in death, and if you were to smite her she would likely
knock you silly and take both envelopes herself.  So a
choice must be made.

Could it be that it somehow doesn't matter if one chooses
then trades, or chooses then sticks?  For that to be the
case, then one must somehow explain how it is that if one
chooses and opens to find a ten spot, then the probability
that the other envelope contains a fiver has somehow
mysteriously increased to 2/3, and the probability it
contains a 20 has dropped to 1/3, so that 5*2/3 + 20*1/3
= 3.33 + 6.67 = 10.  But how?

Eliezer has suggested that if the first envelope contains
a sufficiently large sum, say a million bucks, one could
make an educated guess that no one is likely to give you
two million.  I say this argument is irrelevant.  Regardless
of the amounts in the envelopes, I see no reason for it 
to be more likely that you chose the larger amount the
first time.  

Emil Gilliam suggested a clever variation.  Suppose the
envelopes contain sums expressed in some unfamiliar
foreign currency, zorgs, again with one envelope containing
twice as many zorgs as the other.  There is a currency 
exchange down the street where you can trade your zorgs 
for dollars, euros, gold, sex, whatever you want, but you
have no idea how much you have.  You open your first
envelope and find you have 10 zorgs, but this can
represent any amount between pocket change and 10 tall
piles of dough, you know nothing.  So now would
you trade?  Does it matter now if you open the
first envelope?  Why?  

If you decide it does matter if you open or not, 
what have you actually learned from seeing you 
have 10 zorgs?  It might be large enough to 
use Eliezer's argument, but you don't know that.  
Does it matter if the messenger knows
how much is 10 zorgs?  Why?  Would you trade for
the other envelope after opening yours?  Would you 
trade without opening yours first?  How does the
law of averages somehow know if you looked in
your envelope, so as to readjust the probabilities
to make it of no value to open your envelope before
trading?

spike