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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.
Consider: These are supposedly "logicians" so it's almost impossible
to imagine any of them not instantaneously making the first stage
inference--i.e., recognizing the implication of seeing one black hat,
one white hat, and two raised hands. Thus, the moment one of them
makes the second stage inference (assuming only perhaps that it
wasn't worked it out in advance), there is no need for further
delay.
...... I'd say that it involves psychology in an almost trivial
if still interesting way, which is why I think this *would* be a good
example to use even in the first week of class.