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Re: [Phys-L] Inference Lab Design

• From: John Denker <jsd@av8n.com>
• Date: Sun, 19 Aug 2012 11:23:22 -0700

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.

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