[extropy-chat] monty hall paradox again

[Alan Eliasen]

Spike wrote:
> 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!

   The fact that this leads to a paradox is probably the tip-off that the
reasoning isn't quite right.

   Say that the envelopes contain n and 2n dollars.  Then, you'll either pick
the envelope that contains n or 2n on your first guess.

   Draw the payoff matrix for staying or switching:

   First pick  ---->  n    2n
--------------------------------
      Stay            n    2n
     Switch          2n     n

   Projected earnings for either case is 3/2n, and no option (always staying,
switching, whatever) improves that.

-- 
  Alan Eliasen                 | "You cannot reason a person out of a
  eliasen at mindspring.com       |  position he did not reason himself
  http://futureboy.homeip.net/ |  into in the first place."
                               |     --Jonathan Swift