[extropy-chat] monty hall paradox again

[Mike Lorrey]

THe solution comes in terms of you figuring your personal marginal
risk.

If the envelope just has ten bucks in it, you figure, "eh, that'll buy
me lunch, not a big risk for me," so you trade it for the second
envelope. Twenty bucks is twice as nice, would allow you to schmooze
your boss by buying him or her lunch, while five bucks would allow you
to upgrade from the cheap $5 lunch you were already planning on having.

If, instead, there is a million bucks, you figure, "this is more money
than I'll be able to save on my own over the next 15-30 years, and
while the odds of the other envelope being $2 million vs $500k are
50-50, the $500k difference between this and the lesser choice means
the difference between having a really nice house AND being able to
retire, versus having to choose one or the other. I'll stick with the
million."

--- Spike <spike66 at comcast.net> wrote:
> 
> 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
> 
> 
> 
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=====
Mike Lorrey
Chairman, Free Town Land Development
"Necessity is the plea for every infringement of human freedom.
It is the argument of tyrants; it is the creed of slaves."
                                         -William Pitt (1759-1806) 
Blog: http://www.xanga.com/home.aspx?user=Sadomikeyism


	
		
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