Zylon
1841
So many frickin’ hats.
As far as I’m concerned, logic riddles like the islander eyes are on a par with proofs that the sum of infinity is -1/12 and other such hand-wavy legerdemain that accomplish little beyond demonstrating how otherwise logical systems can have imperfections that yield nonsensical results when twisted in just the right way.
There’s no collusion involved in the (different) hats puzzle in your link. The point is the prisoners are able to figure out what to do without communicating.
It’s not logically equivalent to the islanders puzzle, though it does share similarities. The crux of the islanders puzzle is that mutual knowledge and common knowledge are different: everyone knowing a fact is not the same as that fact being common knowledge. That seeming paradox isn’t present in the four hats puzzle. They are similar puzzles in that the solutions involves players inferring other players’ observations from other players’ actions, and in that the surprising solutions hinge on players having common knowledge of both some fact – in the four hats puzzle, it is common knowledge that there are two hats of each color and that everyone is rational.
So in the base case, n=1, clearly it is not common knowledge that everyone can see a blue-eyed islander, because the blue-eyed islander cannot.
Now assume it doesn’t work out for (n-1). Then for the case of n, any brown-eyed islanders (or green-eyed, etc) can see all n, but each blue-eyed islander can only see (n-1) blue-eyed islanders, from which they conclude this is insufficient for each of those islanders to draw the conclusion of common knowledge IF this islander (whose perspective we’re considering) doesn’t have blue eyes (even though they do). So it doesn’t work without the explorer by induction.
It occurred to me when considering n=3 that each of the 3 could only assume that everyone else saw at most 2, and even though everyone saw at least 2 (enough to draw the conclusion that everyone else saw at least 1), anyone who saw only 2 couldn’t assume that the others saw enough to assume common knowledge. In effect, the explorer synchronizes the blue-eyed islanders with the rest.
I was mistaken earlier in thinking I understood the necessary condition in the epistemic logic for common knowledge. That was only a formal condition for mutual knowledge.
Would the islanders puzzle’s answer still be the same had the explorer said, “It’s nice to be around brown-eyed people again”?
Of course, in the general sense that if there are n X-eyed people and the visitor announces there is at least one X-eyed person, all the X-eyed people will learn they have X-eyes on day n.
In fact, much, much more complex and general situations yield the same result. Suppose there are 1,000 named colors and there are in turn 1,000 islanders with each of these eye colors, so there are a million islanders with a thousand possible eye colors. A visitor unwittingly announces, “I see that at least one of you has hazel eyes.” On this island, people don’t kill themselves when they learn their eye color, instead, on each day each family, and families come in various sizes, is asked, “Does anyone know with certainty what their eye color is?” and announces to everyone else what the answer is. Eventually, someone will “yes.” This is also true if the visitor says something like, “Ah, I see Fred over there doesn’t have hazel eyes,” or any other statement which is (1) public and (2) rules out at least one configuration of islanders / eye colors.
Proof: follows from the Proposition proved by Aumann (1976).
