## Blue Eyes with Superrationality

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Xias
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### Re: Blue Eyes with Superrationality

Vytron wrote:Or I don't get the argument at all? Is it that superrational beings don't leave at all because the guru speaking doesn't change anything? But how does that compare to my case posted? Because it's clear a single islander still leaves at D1 and that forces 2 islanders to leave D2 and such. The guru clearly changes the base case and superrationality doesn't change anything.

That's what I think it is. It's obvious to us that the proof by induction still applies, but for some reason Flewk is convinced that

a) Given a number of Blues, X, those Blues have some shared minimum knowledge that there is X-N blue islanders. Flewk believes that N=2, I think.
b) If the number that the Guru says is X-N or less, then no new information is given, and the islanders don't leave.

We all need to get on the same page though, because it seems that a lot of the replies in this thread are almost irrelevant to what Flewk is arguing. Which is why we have a half-dozen "Let me see if I can explain it better..." posts and nobody has been able to actually get anywhere.

emlightened
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### Re: Blue Eyes with Superrationality

If the base assumption is that it's possible (i.e. B>0), then that's equivalent to the guru speaking. If we have B>n, for some n, agreed by everyone (I guess this is where 'superrational' comes in), then it works for all B>n by assumption of such, and the people can leave n days earlier. The problem is that that doesn't work for all n, and n has to be independent of B.

flewk assumes that n can be arbitrary, or a function of B. The former breaks for any B<n. The latter breaks for any case such that f(B-1)!=f(B).

I think, though, that the 'superrational' part comes in in the scenario 'If there is a blue eyed person, then the guru could speak, so let's just say she has'. I think it works whenever there's a blue-eyed person, but not if there aren't any.

Still, I'd like to know what page flewk is on.

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Vytron
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### Re: Blue Eyes with Superrationality

It'd be bizarre if superrationality was a disadvantage (perfect logicians leave day n - but supperrational perfect logicians conclude they already had the bit given by the guru, so they never figure their own eye color...)

Xias
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### Re: Blue Eyes with Superrationality

But Flewk isn't arguing that the addition of superrationality changes the solution. Read this quote:

flewk wrote:2) Think about what you mean by "cooperating the solution". How do you think the original islanders were waiting together and leaving together?

3) Every islander knows
I) the parameters of the game
II) everyone else knows the parameters
III) everyone is a perfect logician

This is a superrational problem.

The problem with the original solution is that it did not consider other conclusions a superrational population might make. A cascade caused by the nested logic only works if the trigger is new information.

Regardless of the definition of superrationality, it seems we agree that they are perfect logicians, which means they will make every possible logical conclusion. Can you find a flaw in the normalized perspective argument? If not, then "at least 1" will not be new information.

Emphasis mine.

Flewk's arguments are that:

a) The original problem (Randall's Blue Eyes Problem) actually involves superrationality: the islanders are defined as superrational by the fact that they are all perfectly rational and aware of this fact.
b) The original solution (Randall's solution) is incorrect because "there is someone with blue eyes" is already common knowledge.

Flewk uses his own understanding of superrationality while trying to demonstrate both (a) and (b), but also seems to think that (b) is true strictly given the initial conditions of the problem and holds even if you disagree with (a).

--

I'm not sure how useful it is to have this meta discussion, but it doesn't look like anyone was getting anywhere when nobody seems to actually understand the position that Flewk is taking. Everyone is approaching the problem from weird angles because of misunderstandings and disagreements, but the real problem Flewk has is of the same form as Objection #1.

Vytron
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### Re: Blue Eyes with Superrationality

Xias wrote:I'm not sure how useful it is to have this meta discussion

It is very useful, thanks. It should save a lot of work because of the misunderstandings that were going on.

I still restate my case in which I ask flewk at which n does it break (i.e. if we agree that the guru makes a difference when there's only one blue eyed person, the point at which it doesn't is critical.)

Cauchy
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### Re: Blue Eyes with Superrationality

flewk wrote:The superrational part is that the islanders know everyone else must arrive at the same conclusions with the same given information. Even the original solution required superrationality as the islanders moved en masse and even waited together based on the same piece of information.

But the islanders don't have the same information, so how will they know they're coming to the same conclusion? For each other person, I can see their eyes, but they can't, so why should I expect that they'll come to the same conclusions that I do? I don't even know that our situations are analogous. If my eyes are, say, red, then I wouldn't be in the same state as any of the other islanders, who all have either blue eyes or brown eyes. Even if I somehow knew that my eyes were definitely either blue or brown, there's no single islander I could point to and say "I'm in a situation that's analogous to that islander's situation, so I know we'll arrive at the same conclusions.". If I could do that, I'd know my own eye color, and just leave.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
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rmsgrey
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### Re: Blue Eyes with Superrationality

Hofstadter's application of super-rationality to the Prisoner's Dilemma is to say "Since we are both in the same situation, with the same information and the same reasoning power, we will necessarily both come to the same conclusions. Therefore, rather than considering the situations where one of us co-operates and the other defects, we only have two scenarios to consider - we both co-operate and gain; or we both defect and lose. Since gaining is better than losing, it's clearly better to co-operate".

There does not seem to be a parallel to this reasoning that applies to the blue eyes problem - every person on the island has a unique set of data, and knows that they do, so they cannot be confident of drawing the same conclusions as anyone else, so cannot base further reasoning on the assumption that they would behave the same way as some other person.

Applying super-rationality here would mean starting from some statement like "I will leave on the same night as everyone else with the same eye colour as me will" and drawing some conclusion from that. Saying "everyone will either reach the same conclusions as me, or differ by 1 on a couple of numbers" isn't super-rationality because you're breaking the symmetry. In fact, that "differs by one" is what makes the solution work.

Vytron
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### Re: Blue Eyes with Superrationality

Yes, if collusion helped in the original puzzle, then super-rationality would help as well (say, if there's multiple optimal cases of collusion to leave earlier, and as early as possible, they could'nt be used because other people don't have a way to know what other people are doing. Super-rationality would fix this, as whatever optimal method of reasoning you choose to use would be used by everyone else, as if you were colluding), but since it doesn't, it doesn't.

(I know the OP is talking about something else...that super-rationality allows everyone to have common knowledge about the existence of at least one blue eyed person so the guru is ignored. The point is super-rationality isn't changing anything)

PTGFlyer
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### Re: Blue Eyes with Superrationality

I've got it, but it doesn't really use logic, just [spoiler]extrapolation of a pattern at low numbers. I don't see how it works if there are 100 people on the island, since it seems that the only way to solve this is to imagine that there is only 1 person with blue eyes and then step up, when everybody knows that there are at least 99 blue-eyed people on the island.