[Phys-l] Monty Hall problem

[John Denker <jsd@av8n.com>]

On 06/29/2010 05:33 AM, Philip Keller wrote:

> BTW, the Monty Hall problem is best understood by realizing that you
> have a one in three chance of having picked the right curtain
> initially and EVERY other time, the car ends up behind the one you
> can switch to!

That's true for the "standard" version of the problem, in 
which the host is required to always show what's behind a 
door different from your initial pick and also different
from the winning door.

Slight changes in the rules lead to dramatically different
conclusions.

Game 1:  You never get the option to switch.
Your strategy (1):  No strategy required.
Result:  You win 1 time out of 3.


Game 2:  Monty is required to make the offer every time.
Your strategy (2):  Always switch.
Result:  You win 2 times out of 3.


Game 3:  Monty is your friend.  If your initial pick is
  not a winner, he offers you the option to switch.  If
  your initial pick is a winner, he "forgets" to make the
  offer.
Your strategy (3):  Always switch if given the option.
Result:  You win every time.


Game 4:  Monty is your adversary, and pursues a "fool's
  mate" strategy.  That is, if and only if your initial
  choice is a winner, he offers you the option to switch.
Your "strategy" (4a):  Always switch if given the option.
Result:  You lose every time.  This goes on as long as
  you adhere to the "always switch" strategy.  I don't
  recommend it under these conditions.


REMARK:  According to
  http://en.wikipedia.org/wiki/Monty_Hall_problem

> > In her book The Power of Logical Thinking, vos Savant (1996:15)
> > quotes cognitive psychologist Massimo Piattelli-Palmarini as saying
> > "... no other statistical puzzle comes so close to fooling all the
> > people all the time"

I conjecture that this behavior is not because people are
stupid, but rather because they have a deep-seated fear of
scenario (4a) ... especially if the rules have not been
clearly stated.  Even if the rules /have/ been clearly stated,
there remains the question of whether you can really trust
the other party to adhere to such rules.  People may not be 
able to clearly articulate what's bugging them, but still
I am inclined to give them credit for a well-founded innate
suspicion of the "always switch" strategy.


Game 4, continued:  Same as game (4) above.
Your strategy (4b): _________________________________
Result: ____________________________

So here is a puzzle for you:  Fill in the blanks above.  We
have seen that "always switch" is a very bad strategy if
Monty is your adversary and has the option of making the
offer or not.  So ... can you do better, under these
conditions?


Game 5:  Monty is your adversary, and is free to pursue any
  strategy he likes.  He gets to choose whether or not to
  make the offer.  His best strategy is: _________________
Your strategy (5):  _____________________________
Result: ________________________

Again, fill in the blanks.  What is his best strategy?
What is your best strategy?  What is the result if each
of you pursue your best strategy?  

Most importantly, suppose Monty pursues some other strategy.  
Does your strategy take advantage of this?  You need to
check this in order to verify that his strategy is in 
fact optimal.

Remark:  Note that the game-theory analysis is dramatically
more involved than a simple probability calculation.

Spoiler, or not:  The wikipedia article addresses some of
these points, but doesn't do a very good job, so don't
think you can solve these puzzles simply by quoting the
results given there.