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[Leafsnail]

Let me rephrase the question, and slightly change the answer, in a way that makes it make at least a *little* more sense.  Still not where I want to go with it, but still.

Instead of "I see someone with blue eyes", the guru says "I like pie", and everyone decides that since the guru said something, it's time to star thinking.  Or, better yet...

100 blue-eyed, 100 brown-eyed, no guru.  Everyone meets for the first time on day 1, so there IS new information on that day.  After a hundred days, everyone leaves at once.

No they wouldn't.  There's no logical line of reasoning that would allow you to magically guess your own eye colour from everyone elses alone.  Again, if you were on this island and saw 93 people with blue eyes and 34 with brown eyes, how would you arrive at any conclusion without the guru's message?

Okay.  Here's a puzzle.  It's stupid and simple but surprisingly it's (barely) related, plus it at least sounds kind of like Monty Hall's bastard child:

There's a huge pile of envelopes that contain various amounts of money, from one dollar up to 65 thousand (give or take).  Someone gives you two envelopes and tells you to pick one.  Before you open it, they tell you that the other one has either half as much or twice as much, and offers to let you change your selection.  What do you do?  Would your answer change if you could open the first envelope and look inside before you switched?

You should switch, I think.  Let's say the amount of money in the first envelope is $X.  Switching to one with half loses you $X/2  Switching to one with double gains you $X.  So assuming there's a 50-50 chance of either option, it'll be better on average to switch.

For looking inside... shouldn't make a difference if we're thinking like economists, but it does make sense to be loss averse sometimes.

[Earthquake Damage]

There's a huge pile of envelopes that contain various amounts of money, from one dollar up to 65 thousand (give or take).  Someone gives you two envelopes and tells you to pick one.  Before you open it, they tell you that the other one has either half as much or twice as much, and offers to let you change your selection.  What do you do?  Would your answer change if you could open the first envelope and look inside before you switched?

The only time the one you chose is relevant is if it's worth at least half the maximum value or less than twice the minimum.  If the most you can win is 65k and your envelope holds 40k, you can't possibly win 80k so that changes things.  The same applies if your first choice is worth less than twice the minimum value.

Otherwise, normalize the amount of your first pick to 1.  The expected value for switching is [(2 + 0.5)/2] = 1.25.  Assuming a 50-50 chance of either, it's in your best interest to switch.

[Sowelu]

Disregard this I need to L2Logic apparently

[G-Flex]

There's a huge pile of envelopes that contain various amounts of money, from one dollar up to 65 thousand (give or take).  Someone gives you two envelopes and tells you to pick one.  Before you open it, they tell you that the other one has either half as much or twice as much, and offers to let you change your selection.  What do you do?  Would your answer change if you could open the first envelope and look inside before you switched?

The only time the one you chose is relevant is if it's worth at least half the maximum value or less than twice the minimum.  If the most you can win is 65k and your envelope holds 40k, you can't possibly win 80k so that changes things.  The same applies if your first choice is worth less than twice the minimum value.

Otherwise, normalize the amount of your first pick to 1.  The expected value for switching is [(2 + 0.5)/2] = 1.25.  Assuming a 50-50 chance of either, it's in your best interest to switch.

Of course, the problem with simple logic/probability puzzles like this is that they tend to be oversimplistic. In the real world, there might be extenuating circumstances. For instance, let's say you really need a thousand dollars for something specific, and switching will either get you two thousand or five hundred. In a case like that, you might be better off not switching, if the possible results of switching are either "an extra thousand dollars" or "you don't have the money you need for whatever thing you need it for, with whatever consequences that entails".

Of course, from a strict expected-value perspective, you should switch; things just aren't always that simple in reality.

[Darvi]

@Sowelu: Wrong. Totally wrong.

The only way that someone can leave on the first day is if they know two things:  A) I see 0 people with blue eyes, B) I know there is one person with blue eyes, C) therefore it's me.  If they see > 0   people with blue eyes, they can't make any assumptions about   themselves.  The guru has only said "I see greater than zero people".    That is the only constraint you have to work with.

This is right, but now imagine there were two people. You would know that if you hadn't blue eyes, the other would leave at day 1. We agree on that, right? But he doesn't leave on day 1, meaning you do have blue eyes, and you leave on day 2.

And so on. You see X people with blue eyes. Either they leave on day X, meaning you don't have blue eyes, or you do have blue eyes, and you leave a day later. Even if you see that there are 99 other blue eyed people, you are all logical enough that you can collectively come up with the strategy to leave on day X+1.

Now imagine the Guru wouldn't speak. a single person couldn't leave, since he has no information to work with. So in a 2 blue eyed people scenario, if the other doesn't leave, it's not because you don't have blue eyes. It's because he knows jack about himself.

Again with three people. Staying during day 2 doesn't mean you don't have blue eyes. The others just don't know better. And so on.

[Sowelu]

This is right, but now imagine there were two people.

This is your error.  There are not two people, there are more.  You are ignoring this fact.  Induction is not valid here, because no new information is added with each step.  For N=2, the events of the first day add information: Someone COULD have left, IF you had brown eyes, so you gain information by the fact that they did not.  For N>3, nobody could have left on the first day, so no information is gained.

Tell me what information is gained between day 1 and 2 when there are 100 people with blue eyes.  If information is gained, then you can build by induction.  If people choose to do nothing purely due to convention, you cannot.

What information is gained?

[G-Flex]

This is right, but now imagine there were two people.

This is your error.  There are not two people.  You are ignoring this fact.  Induction is not valid here

Ergh. Yes it is. Read the solutions page again. You can determine what would happen if there were one person, and what would happen if there were two people, and can extrapolate what would happen if there were three, four, or any other number of people from there.

[Darvi]

This is right, but now imagine there were two people.

This is your error.  There are not two people.  You are ignoring this fact.  Induction is not valid here, because no new information is added with each step.  For N=2, the events of the first day add information: Someone COULD have left, IF you had brown eyes, so you gain information by the fact that they did not.  For N>3, nobody could have left on the first day, so no information is gained.

Wrong. Assume you had brown eyes. Then the other two would make the N=2 assumptions, meaning they would leave at day 2. But if they don't leave at day 2, you have blue eyes, and you leave at day 3. Repeat as necessary. QED.

[Sowelu]

Bleh, okay, you win.  I'll go edit my embarassing post out of existance now.

[Darvi]

Don't be ashamed. It's very complicated and contains tons of recursing. Even I was skeptical, and I was the one who solved the riddle :/

[Sowelu]

See, I still don't buy the whole guru thing.  That's my problem.  I can see how information is added by each step of recursion, but...there's no information added by the guru.  Everyone knows that everyone sees blue eyes.  It seems like when people use recursion, they pretend that nobody knows that until the last step...but that's silly, because that's the critical piece of information and it's always available.

[Darvi]

But the recursion relies on the fact that the N=1 case gets resolved by the guru. Read my pre-previous post for what would happen if the Guru wouldn't talk.
Unless they'd all do the "leave on day x" thing. Which they wouldn't because it wouldn't be based on logical deductions anymore.

[G-Flex]

See, I still don't buy the whole guru thing.  That's my problem.  I can see how information is added by each step of recursion, but...there's no information added by the guru.

The guru is necessary for the initial theorem that the recursion gets down to (the n=1 and n=2 scenarios).


I admit I have trouble understanding just what information the guru gives them, but I can understand that it works.

[Darvi]

The guru doesn't really add any information beyond the N=1 case. But the islanders, being Vulcans, refuse to use strategies that aren't completely based on logic.

[Leafsnail]

I think I've got a piece of inducted reasoning for what the guru adds, actually.  This will... be a bit weird, so bear with me.

For 1 blue eyed person, it's easy.  You don't know anything about how many blue eyed people there are on the island until the guru tells you.

For 2 blue eyed people, it's still quite simple.  You already know there's a blue eyed person on the island, but you don't know that person 2 knows there is until the guru says so.  That's the crucial piece of information the guru provides.

For 3 blue eyed people, it's harder.  You already know there's a blue eyed person, and you know that person 2 knows there's a blue eyed person.  However, you don't know that person 2 knows that person 3 knows that there is until the guru says so (think about it - if you theoretically had brown eyes, person 2 is put in the same situation as the example above, and doesn't have that information).  This is the crucial piece of information the guru provides.

For 4 blue eyed people... the same again.  You know there's a blue eyed person, you know that person 2 knows there's a blue eyed person, you know that person 2 knows that person 3 knows there's a blue eyed person, but you don't know that person 2 knows that person 3 knows that person 4 knows there's a blue eyed person.  This is the crucial piece of information the guru provides.

This can be extended all the way up to 100 blue eyed people.  Sure, you know there's a blue eyed person, but you don't know that person 2 knows that person 3 knows that...

The guru doesn't really add any information beyond the N=1 case. But the islanders, being Vulcans, refuse to use strategies that aren't completely based on logic.

It's nothing to do with that.  Assuming no communcation is possible, there really is no way for them to work out their eye colours without the announcement.