Re: [Phys-L] Inference Lab Design

[John Denker <jsd@av8n.com>]

On 08/18/2012 11:29 PM, D.V.N. Sarma wrote:
> The candidates were blindfolded and three white caps were placed on 
> their heads. The blindfolds were removed. All the three raised hands. 
> Presently one of them said that his cap is white. How did he guess that?

Wow.  That is a sophisticated, tricky question.

I vaguely recall seeing that question (or something similar) many years
ago, along with an alleged solution.  For some reason I did not trust
the solution, although back then I could not have told you why not.

Having thought about it off-and-on for the last day, I think I now have
a better understanding of what the trick is here.  It purports to be a 
pure logic problem, but really it's much more than that.

To briefly outline how I see it, the analysis proceeds in /stages/ :
 1) There are some inferences that can be made in a single stage.  For 
  example, if I see a black hat on player A and yet player B raises his 
  hand, I can in one stage infer the color of my hat.
 2) Now, if it turns out that nobody can make any first-stage inferences,
  I can use that fact to make a second-stage inference.

It is a classic childish mistake to play the game assuming the other guy
will make the wrong move.  On the other hand, it is also a mistake to assume 
the other guy will always instantly make the best move.  (For example:  you 
should drive defensively.)  Sophisticated game theory says you should play 
in such a way that you do well /no matter what/ the other guy does.

In the game in question, there is a reward for answering quickly, but you
must not answer too quickly!  You need to give the other players time to
make their first-stage inferences ... but then you need to make your second-
stage inference quicker than they do.  So this requires psychology as well
as logic:  you need to appraise the capabilities of the other players.

This is not a worst-case scenario, because there is no incentive for the
other players to bluff.  If they could make a first-stage inference, they
would have every incentive to blurt it out promptly.  In contrast, there 
are plenty of real-world scenarios where bluffing is possible, such that
the other guy will go to great lengths to disguise his capabilities, and
thereby trick you into making a wrong move.

The OP asked for inference exercises that could be used during the first
week of class.  I would not put this example in that category.  Instead
I would file it away for later.  I would certainly not forget about it,
because it remains quite valuable.  It is neither super-simple nor worst-
case-difficult.  I most earnestly hope and recommend that by the /end/ of 
the course, all students should be able to handle problems as hard as this,
or even harder.

========

See also next message.