A fun exercise from classic game theory

Originally posted by AThousandYoung
If everyone has slept with everyone, then every woman knew adultery had been committed before the prophet came, because they committed it!

The idea is that (once you remove common knowledge) when the prophet proclaims to everyone that adultery has been committed then common knowledge results from perfect rationality. That is every woman knows that every woman knows that adultery has been committed.

With this and a years time every wife is simultaneously able to eliminate the possibility that only her husband is faithful, leaving only that every husband is cheating. Tragedy and mayhem insue . . .

Originally posted by telerion
The idea is that (once you remove common knowledge) when the prophet proclaims to everyone that adultery has been committed then common knowledge results from perfect rationality. That is every woman knows that every woman knows that adultery has been committed.

With this and a years time every wife is simultaneously able to eliminate the possibility t ...[text shortened]... sband is faithful, leaving only that every husband is cheating. Tragedy and mayhem insue . . .

But it is still confusing because before the prophet spoke, all the women could see 363 men with A's on their heads, and they knew that all other women could see at least 362 A's, so all women knew adultery had been comitted, and all women knew that all the other women knew adultery had been comitted. Yet the prophet saying that adultery has been comitted comes as some startling thing to them and starts the countdown!

Before the prophet arrives, everyone knows that adultery has been committed. All the men (except maybe one) have giant A's on their heads!

The prophet adds nothing to the situation. Nothing changes. He's stating the obvious.

Try it with one wife and one husband on the island - she kills her husband that night.

Well she's been cheating, too, so this fails. If she'd been faithful then she'd kill him.

Try it with two wives. Each knows after the first night that the other one didn't kill her own husband, and therefore must be able to see an A. Therefore on the second night both wives kill their husbands.

If the women are faithful, then sure.

Try it with three wives - each reckons if their husband is innocent then the other two wives will kill their husbands on the second night - when that doesn't happen they know their husband must have a A and so they all kill their husbands on the third night.

Flawed. If Wife 1 sees an A on Husband 2's head and Husband 3's head, then no matter how long she waited, she's not going to kill her husband. Why would she? She knows the answer, and knows each of the other women see an A also. Every woman is aware that someone else's husband has an A, and every woman knows this. Nobody dies.

Excellent, give tiger a pretzel.

What?

That is every woman knows that every woman knows that adultery has been committed.

Every woman already knew this, right? If Joe has an A on his head, then Mary, who is Sam's wife, can see the A, just like Sally, Mike's wife, can. Sally knows Mary won't kill anyone, because she knows Mary can see Joe's A, even if she can't see Sam's.

Originally posted by AThousandYoung
Try it with three wives - each reckons if their husband is innocent then the other two wives will kill their husbands on the second night - when that doesn't happen they know their husband must have a A and so they all kill their husbands on the third night.

Flawed. If Wife 1 sees an A on Husband 2's head and Husband 3's head, then no ma ...[text shortened]... woman is aware that someone else's husband has an A, and every woman knows this. Nobody dies.

Ok, we have Anne and Andrew, Bertha and Bob, Cathy and Cedric.

Each wife can see an A on the other two husbands heads, but not on their own head. They figure there might well not be an A on their husbands head, so they don't kill him.

Now the prophet tells them adultery has been comitted.

Anne thinks: "Hmm - Let me assume Andrew doesn't have an A then Bertha and Cathy can each see one A - Bertha will be wondering if Bob has an A and she will now know that if Cathy can't see an A she will kill Cedric on the first night. If Cathy doesn't kill Cedric then Bertha will know that Cathy can see an A, and so she will kill Bob on the second night.

When, on the second night, no husbands are killed then Anne knows her initial premise must be wrong and Andrew does have an A - so she kills Andrew on the third night.

Each wife can see an A on the other two husbands heads, but not on their own head. They figure there might well not be an A on their husbands head, so they don't kill him.

OK.

Now the prophet tells them adultery has been comitted...Anne thinks: "Hmm - Let me assume Andrew doesn't have an A then Bertha and Cathy can each see one A

Agreed.

Bertha will be wondering if Bob has an A

OK.

and she will now know that if Cathy can't see an A she will kill Cedric on the first night.

Well, there's the problem that maybe Cathy or Bertha committed the adultery with a single man. Assuming the prophet said "a married man has committed adultery" then you are correct.

If Cathy doesn't kill Cedric then Bertha will know that Cathy can see an A

OK.

and so she will kill Bob on the second night.

Because we're assuming Andrew lacks an A. OK.

[/b]When, on the second night, no husbands are killed then Anne knows her initial premise must be wrong and Andrew does have an A - so she kills Andrew on the third night.[b]

OK, I think I buy it now...as long as the original problem is changed to say "a currently married man has committed adultery on this island while he was married" and each man has been only married once.

Remember I screwed it all up by declaring common knowledge in the OP. Common knowledge is what is achieved after the prophet makes his announcement. Before that time, a wife cannot rule out for certain that her husband is the only faithful partner. And yeah, the prophet should have said, "by a married man."

To get a feel for it, think of this simple problem.

3 men are blindfolded and then told to draw hats from a barrel. They draw hats and place their hat upon their head.

Then they are guided to seats. They sit in a room facing one another like this:

*

* *



Now the blindfolds are removed so each can see the other two. They CANNOT see their own hat, only those of the two across from them.

Now the moderator announces, "The hat that you drew is either blue or red. There were fewer than three red hats in the barrel."

He then asks them all simultaneously, "Do you know the color of your hat?"

They all simultaneously answer, "No."

The mod says, "Stupid fools. Let me ask you again. Do you know the color of your hat?"

They all simultaneously answer, "No."

The mod says, "Morons. Do you know the color of you hat?"

They all answer simultaneously, "Yes."

Now assuming that all three men are perfectly rational, what are the colors of the men's hats?

The answer to this is analagous to that of the island question (when posed correctly).


This is not necessarily for AThousand or iamtiger, but anyone who is still wondering.

Originally posted by telerion
This is adapted from a old economics article. Been searching for the exact reference to no avail. I'll try to find it and post it after the problem has been here a while.

Problem: The Island of Unfaithful Husbands

An island is inhabited by 365 couples. Now a special property of this island is that if a husband cheats on his wife, then an 'A' ap ...[text shortened]... ted upon this island!

The question is what happens next and why? There is a detailed answer.

Each woman thinks that there man is the adulterer bcause he dosent have a A. All men killed. The prophet must repopulate the island.

Hat's problem:

Let's call the men A, B, C.

First question: If A sees two red hats he would know his hat was blue. He doesn't know, therefore B and C now know they don't both have red hats. B and C also make this judgement.

Second question: B knows that he and C don't both have red hats. Then, if C has a red hat then he could say he had a blue hat. After B's answer C knows he has a blue hat. Same goes for A and C.

At the third question: They all know they all have blue hats.

Originally posted by Palynka
Hat's problem:

Let's call the men A, B, C.

First question: If A sees two red hats he would know his hat was blue. He doesn't know, therefore B and C now know they don't both have red hats. B and C also make this judgement.

Second question: B knows that he and C don't both have red hats. Then, if C has a red hat then he could say he had a blue h ...[text shortened]... e hat. Same goes for A and C.

At the third question: They all know they all have blue hats.

Yep. That's essentially how the island problem (when correctly posted) works.

Originally posted by telerion
This is adapted from a old economics article. Been searching for the exact reference to no avail. I'll try to find it and post it after the problem has been here a while.

Problem: The Island of Unfaithful Husbands

An island is inhabited by 365 couples. Now a special property of this island is that if a husband cheats on his wife, then an 'A' ap ...[text shortened]... ted upon this island!

The question is what happens next and why? There is a detailed answer.

I think we have to change it so that the women will only countenance murder once the prophet has arrived. Perhaps beforehand they didn't know the 'A' signified adultery, so the prophet could say something to that effect.

I think we need to claim all the women were faithful, or the prophet needs to say "a man has committed adultery." Otherwise a woman might have done it.

I think we also need to claim all the men were only married once. Otherwise they might have committed adultery with a previous wife.

I think also the prophet needs to put the A's on the men's heads, or the significance of them needs to become obvious only when the prophet speaks (as Acolyte pointed out). Otherwise the prophet adds nothing new to the situation, and there might be an additional complication of the A's appearing one at a time.

Also, it needs to be clear that a currently married man who is currently on the island committed the adultery. Otherwise the man who committed adultery might have died or left the island.