1. Everyone on the island is superrational, if they played the Prisoner Dilemma, all of them would cooperate.
2. The guru says "I can see someone who has blue eyes. I can see everyone is superrational." So this is common knowledge.
3. Before they're put on the island, they're able to speak as much as they like with each other, make a plan, set rules, and anything that would let them escape earlier. However, once on the island, they're not able to communicate, which means they're not able to help other people realize what's their eye color (for instance, patting on the shoulder the people with blue eyes wouldn't be allowed.)
With these changes, are these people able to realize what is their eye color and leave the island sooner?
Cauchy wrote:You can't shortcut a day, because each day the islanders all learn actual new information during the public reveal that no one left.
Superrational beings know that all other people will not leave the first 90 days, they will gain no information because they already had that information, so they should be able to skip them. There should be N amount of days that is safe to skip, mainly, because it's safe to skip 1 day.
douglasm wrote:Suppose that you have some superrational rule that, given a number N of blue-eyed people that you see, you skip D = f(N) days. For this rule to be worth anything, there has to be at least one value for N where f(N) > 0. For this rule to work for the single blue-eyed person case, f(0) and f(1) must be 0. Therefore, there are at least two distinct output values for f(N). Therefore, there exists an N where f(N) != f(N+1).
Consider the case of N where f(N) != f(N+1). This can be true in two different ways, either f(N) < f(N+1) or f(N) > f(N+1).
Consider f(N) < f(N+1). Suppose that there are, in fact, N+1 blue-eyed people on the island. The people with blue eyes will see N blue-eyed people and will skip f(N) days. Everyone else (with brown eyes) will see N+1 blue-eyed people and will skip f(N+1) days. The brown-eyed people are waiting for day N+1 - f(N+1). The blue-eyed people are waiting for day N - f(N). Because f(N) < f(N+1), the former will be at or before the latter, and all the brown-eyed people will conclude that they have blue eyes and attempt to leave. Puzzle failed.
So, there cannot be any N where f(N) < f(N+1). This means f(N) cannot increase. Combine this with the known value f(1) = 0, which is required for the single blue-eyes case, and f(N) cannot exceed 0 for any positive N.
It doesn't matter how clever your rule is, even if everyone's superrational or discussed it ahead of time, if there is any case where your rule has you skipping a positive number of days then there is a case where your rule causes the solution to fail.
Well, you said my previous attempt failed only with 750 people. If it worked with less than 750 or more than 750, then superrational beings should be able to make a plan to escape early, they just need to abort the plan when they're close to this number.
Okay, if superrational beings can't leave earlier than perfect logicians (i.e. superrationality changes nothing, unlike in the Prisoner's Dilemma) I'd like to understand why, to me, the main difference is that superrational beings knows what everyone else know without having to wait.
Suppose that after the guru talked, some random superrational being talked, and said:
"I see at least 2 people with blue eyes!"
Would superrational beings then be able to leave one day sooner? Because, if they see 99, they see at least 2, and know that everyone sees at least 2, so they pretend that someone yelled that and leave one day earlier.
Again, superrational beings would not do that, because doing so was arbitrary, but if one is able to find a strategy that leaves earlier, that would work if the prisoners were able to communicate and leave the island, then it could be used by superrational beings without needing to communicate, because they know such a strategy exists and know all other people are using it.
Since it's possible to form a plan that makes it likely for people to leave, but it's not for certain, this variation where the prisoners wouldn't risk their life is presented:
they're all prisoners, and the guru is a warden that gives them a chance to be set free. Every day, the people of the prison have these options:
1- Stay on the prison for one more day.
2- Try to guess their eye color.
If they go with 2:
a) If they guess wrong they're executed.
b) If they guess right they're set free.
The conditions of the prison aren't good, so they really want to be set free, and so, whenever they're certain of their eye color, they'll guess it. However, nobody wants to die, so they won't do it unless they know it for certain. This paragraph is common knowledge.