SERVING THE QUANTITATIVE FINANCE COMMUNITY

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CBuster
Topic Author
Posts: 4
Joined: August 11th, 2009, 12:33 pm

### The isolated village

twofish / mikei think a key assumption is that the villagers behave in a predictable way when killing themselves (lol!).clearly if a villager takes a random amount of time (with no upper bound) to kill themselves after discovering their dot colour, then no other villager gains any knowledge from another villager not dying (i.e. can't determine whether the villager is about to die or not dying at all).

alexrem
Posts: 12
Joined: April 3rd, 2009, 4:57 pm

### The isolated village

I guess another way to explain that the white-dotted villagers die at time n if there are n white dots in total is the following:Consider one white-dotted person at time n. He sees n-1 dots and also that all those guys with the n-1 dots are alive. He assumes first he has a black dot. He considers what one of those white-dotted guys thinks at time n-1. That guy then sees n-2 guys with white dots and all of them are alive. He assumes that he has a black dot and considers what one of those n-2 white-dotted guys thinks at time n-3. etc. So the second-last white dotted guy in the chain, considered by the third-last guy, sees the only 1 guy with 1 dot; and that guy has not yet killed himself.But then this second last guy must have realized he has the white dot by time n and so killed himself. But he didn;t kill himself, therefore the third last guy's assumption that he has the black dot fails and so he then would have realized before time n that he has the white dot and would have killed himself.In a similar fashion, all of the previous villagers' assumptions about having the black dot actually fail, so we continue in this fashion to get that the first guy's assumption that he has the black dot must be false. Therefore on time n he realizes he has the white dot and kills himself.By the way, if one villager reveals any information about the number of dots he sees, all the other villagers will also kill themselves (i.e. we don't even need anybody to come, just an evil villager )
Last edited by alexrem on August 12th, 2009, 10:00 pm, edited 1 time in total.

alexrem
Posts: 12
Joined: April 3rd, 2009, 4:57 pm

### The isolated village

QuoteOriginally posted by: twofishIs there a requirement that people die in an "orderly" way? If people die in a random way, does this throw off the induction?I think as long as the villagers know each others' dying times, they are ok, because then depeding on the amount of time passed, you can deduce how mnay decision-making steps have occured for the villagers. But then you also need to have each villager know that all villagers know each other's dying times, etc. lol
Last edited by alexrem on August 12th, 2009, 10:00 pm, edited 1 time in total.

MCarreira
Posts: 1724
Joined: July 14th, 2002, 3:00 am

### The isolated village

QuoteOriginally posted by: alexremBy the way, if one villager reveals any information about the number of dots he sees, all the other villagers will also kill themselves (i.e. we don't even need anybody to come, just an evil villager )That's why one has never found such a tribe ... they only exist now in lore.It's up to modern day archeologicians (pun intended) to connect the dots about their existence.

twofish
Posts: 4944
Joined: February 18th, 2005, 6:51 pm

### The isolated village

I think another key assumption is that the villagers know why they are dying.One variation on the problem is how many "stupid villagers" do you need to save the village?

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### The isolated village

QuoteOriginally posted by: MCarreiraQuoteOriginally posted by: alexremBy the way, if one villager reveals any information about the number of dots he sees, all the other villagers will also kill themselves (i.e. we don't even need anybody to come, just an evil villager )That's why one has never found such a tribe ... they only exist now in lore.It's up to modern day archeologicians (pun intended) to connect the dots about their existence.That pun was spot on

kevin08
Posts: 109
Joined: October 24th, 2008, 11:42 pm

### The isolated village

The system is stable if there are no more than 3 of any kind. The village couldn't have more then 4 people when the explorer visited them. Therefore his is liable for their death. Right?
Last edited by kevin08 on August 17th, 2009, 10:00 pm, edited 1 time in total.

wileysw
Posts: 593
Joined: December 9th, 2006, 6:13 pm

### The isolated village

common knowledge

kevin08
Posts: 109
Joined: October 24th, 2008, 11:42 pm

### The isolated village

QuoteOriginally posted by: wileyswcommon knowledgeFor k>2, why the villagers have to wait for the outsider's public announcement?

wileysw
Posts: 593
Joined: December 9th, 2006, 6:13 pm

### The isolated village

it's a little cumbersome to write down the exact statement:e.g., when k=3, say A, B and C have white dots on their forehead. denote statement X as "there is at least one person with white dot on his head".if no outsider's public announcement, A, B and C all know X, and they all know that the other two both know X, but A is not sure whether B knows that C also knows X. from A's point of view, B could be seeing 1 or 2 white dot(s), and he would not or would know that C knows X for these two cases respectively.and not sure if this might help.
Last edited by wileysw on August 25th, 2009, 10:00 pm, edited 1 time in total.

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