[extropy-chat] monty hall paradox again

[Alan Eliasen]

Alejandro Dubrovsky wrote:
> No, i wasn't implying any kind of psychological analysis was wanted.
> This paradox seems especially interesting to me because unlike most
> paradoxes i hear about, this one has a nice sharp practical effect which
> cannot be "thought away" like others.  That is, if someone comes with
> two envelopes, one containing y and the other containing 2y, you choose
> one and get amount x (where x is either y or 2y but you don't know
> which), do you swap?  

   That was the original problem.  It doesn't matter what you do.

> Now what about the following case:  someone comes and gives you an
> envelope which contains x, then says you can return it and get another
> envelope with amounts 2x or x/2 (equal probability of either), do you
> swap in this case?

   In this case, if you stay with x, your projected earnings are x.  Easy.

   If you swap, you have a 1/2 chance of getting x/2, and a 1/2 chance of
getting 2x.  Your projected earnings are:

     PE = 1/2 * x/2  +  1/2 * 2x

        =      x/4   +     x

        =      5/4 x

    So always swap in this case, since 5/4 x is larger than x.

> If the answers are no to the first and yes to the second question, what
> is the significant difference in scenarios? 

   The first problem was equivalent to predicting a coin-flip; you're either
going to pick the bigger envelope, or you aren't.  No matter how much you
think about it, you're not going to improve your odds of guessing it right.
The payoff matrix was perfectly symmetrical, giving you no reason to favor one
strategy over the other.

   The second option gives you a non-symmetrical payoff matrix.  All you need
to do is look at the projected earnings to see that 5/4 is more than 1.

   If the payoffs were double or *nothing*, it's easy to see that the strategy
becomes even--projected earnings for swapping would become x also, and it
wouldn't matter if you stayed or swapped.  But, hell, I'll play "double or
half" on a coin-flip any time that I start out ahead!

-- 
  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