I've read a book where the author used the following puzzle to illustrate the concept of common knowledge in logic:
"Three girls (call them A, B, and C) ride a train that passes through a very dirty tunnel, so each gets her face stained by dirt. Each sees that the two other girls have dirty faces (but doesn't see her own face) and starts laughing at them. However, then a conductor comes in and says "at least one of you has a dirty face". Then the girls understand that their own faces are dirty and stop laughing."
I feel that this version of the puzzle is wrong – indeed, let's consider the reasoning of girl A immediately after she sees the other two girls laughing: She thinks "OK, let's assume that my face is clean. In this case, girl B will have the following reasoning: "I see that girl C has a dirty face and is laughing at something, but everything is fine with girl A, so something must be wrong with me. I better stop laughing and wash my face". But because girl A observes that girl B is laughing, she must then conclude that the conjecture "A's face is clean" must be wrong – so she would stop laughing without any new info from the conductor.
So this problem is different from the famous example of blue-eyed and green-eyed islanders because in this case, all the girls are laughing in the first place – so they put the information about someone having a dirty face into common knowledge before the conductor does that himself (while the islanders in the original puzzle were not communicating with each other in any way). Am I right?
Best Answer
Yes, your reasoning seems correct.
In general, scenarios with perfect logical reasoners that involve continuous time like this are sort of tricky, because all inferences are conditioned on whether other people have done something by time t, and so it seems that everything should happen instantaneously - but then, the perfect logical reasoners don't actually have any time to make their observations! (Perhaps you could design some kind of puzzle with instantaneous perfect logicians, but where one must take into account speed-of-light delays...)
This is why, in famous problems like the blue-eyed islander puzzle, each inferential step is made on a discrete timescale (every night at midnight, and each morning when the islanders see each other again).