Prisoner dilemma

[borisp]

Induct on the number of people with blue eyes. If one person has blue eyes, that person takes the pill on the first night, since now they know they have blue eyes, since they only see people with brown eyes. On the second night, everyone else takes the pill, since the only person they have known on the island to have blue eyes has taken the pill. If any of those still living had had blue eyes, the lone person with blue eyes would not have done this. But he did do it, and now they know they all have brown eyes.

Now assume the statement "If there are N people on the island with blue eyes, then they wake up every day until the (N+1)st day, at which point they do not wake up, and everyone on the island knows this" is true. This is what we actually argue by induction.

(I admit here it is tricky because you have to slip in the auxiliary "everyone knows this" clause to complete the argument. But remember that we are just trying to prove that everyone dies, the actual statement we use for induction can be any true statement that will allow us to deduce this. The truth of this auxiliary statement in the base case follows from the rationality of the inhabitants.)

Suppose there are N+1 people on the island with blue eyes. On the (N+1)st day, they will all wake up. (Here we are tacitly assuming that the above statement is the only way they can deduce to kill themselves, which is really the content of the follow up question. I.e. in the absence of new information, everyone lives happily.) Each person with blue eyes only sees N people with blue eyes, so they deduce they must have blue eyes, since they know that it cannot be true that there are N people with blue eyes. (At this point, all of the blue eyed people know the above statement in the (N+1)st case, taking care of the "clause.") Now these N+1 people take the pill, don't wake up the (N+2)nd day, at which point the brown eyed people learn of the truth of the theorem in the (N+1)st case, they die, etc.

Sorry, I guess it's a bit arrogant to call this "easy" [img]/images/graemlins/wink.gif[/img]

[borisp]

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So the follow up question is to prove that "If the traveler never comes, then no one dies (from the custom)."

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It's already been worked here:

http://forumserver.twoplustwo.com/showfl...rue#Post7711296

I don't know why you find this solution simple, to me it's very complicated. But I also don't understand why you are asking this once you have the solution. If you understand the induction then it becomes obvious that if the traveller doesn't come, the chain thinking event never gets started.

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Here's a puzzle. Can you figure out the Phrase soon2bepro Searched on to bring up that Thread?

PairTheBoard

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"some of you have horns"? That's just from reading the URL on the link...

[PairTheBoard]

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So the follow up question is to prove that "If the traveler never comes, then no one dies (from the custom)."

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It's already been worked here:

http://forumserver.twoplustwo.com/showfl...rue#Post7711296

I don't know why you find this solution simple, to me it's very complicated. But I also don't understand why you are asking this once you have the solution. If you understand the induction then it becomes obvious that if the traveller doesn't come, the chain thinking event never gets started.

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Here's a puzzle. Can you figure out the Phrase soon2bepro Searched on to bring up that Thread?

PairTheBoard

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"some of you have horns"? That's just from reading the URL on the link...

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I should have specified, "by looking at the Bolded words in the Thread".

PairTheBoard

[soon2bepro]

hehe, got me. That was what I remembered that would bring up only that thread [img]/images/graemlins/smile.gif[/img]

It's been 8 months, what did you expect? =)

[soon2bepro]

I still don't understand why the people with brown eyes would die.

Once the last day has passed, and everyone with blue eyes dies, the brown eyed people know that none of them has blue eyes.

Btw, it's highly likely I could be making a mistake somewhere, because I find it extremely hard to think about this situation, even knowing the solution. I guess it's just because I want to go about it step by step.

[borisp]

This should have been more clearly stated in the assumptions: They all know that there are only two possibilities for eye color. So, when they have ruled out blue as a possibility, then they know they are all brown - eyed, which means they know their own eye color, which means it's suicide time.

Was this what was unclear? This might also shed light: when the brown eyed people see N+1 folks not wake up, they know that there were not N+2 people with blue eyes (i.e. that none of them have blue eyes) because if there were, then the N+1 blue eyed people would have had no way to draw the suicidal conclusion. I.e. they are all perfectly rational, and they all know that everyone else is perfectly rational, which is another necessary assumption that I should have made clearer.

[soon2bepro]

uh, I got confused because in the other thread DarrylP said something (probably a joke) about all the guys without horns becoming paranoid and leaving town also. Thought in this problem only the blue eyed people should die. My bad.

[MaxWeiss]

Actually I read the problem wrong. I thought there was only one switch. I stand by my statement if that were true. But I think misreading the original problem is far worse than sucking at doing the right problem, so I put on the dunce hat either way...

Edit: I just realized that what I responded to was in fact a one switch question. Ok, now I am puzzled. You say I am wrong and have a poor imagination, so now I'm curious <u>and</u> I feel bad! Please tell me the answer!!! I want to know how the one switch one would work!

[PairTheBoard]

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Actually I read the problem wrong. I thought there was only one switch. I stand by my statement if that were true. But I think misreading the original problem is far worse than sucking at doing the right problem, so I put on the dunce hat either way...

Edit: I just realized that what I responded to was in fact a one switch question. Ok, now I am puzzled. You say I am wrong and have a poor imagination, so now I'm curious <u>and</u> I feel bad! Please tell me the answer!!! I want to know how the one switch one would work!

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In the One-Switch problem you have the option of doing nothing with the switch. In the Two-Switch problem you must Flick a Switch. So you Flick the B switch instead of doing nothing. The B-Switch just becomes a "Noise" switch then which carries no information. When the two switches are worked in that way it is equivalent to the One-Switch problem with the option of doing nothing.

PairTheBoard

[LeadbellyDan]

I not quite sure if this is right but I think if one of the islanders, Maud, one day announces:

"Some of you have blue eyes"

then everyone apart from Maud dies.

If Dave and Maud were they only islanders with blue eyes then Dave would immediately know his eye colour and kill himself on day 1.

If Maud has blue eyes and two other guys, Mark and Dave, also have blue eyes. Mark and Dave would wait to see if the other killed themself on day 1 and then if they didnt they would both realise they had blue eyes and would die on day 2. Meanwhile Maud is none the wiser of her own eye colour. The same thing would have happened if she had brown eyes.

On day 3 everyone else would realise they had brown eyes and would die. Maud would still be clueless.

Im pretty sure the induction works in the same way as the previous example for three or more blue eyes non-Maud islanders.