# A fun exercise from classic game theory

telerion
Posers and Puzzles 24 Sep '04 23:25
1. telerion
True X X Xian
24 Sep '04 23:25
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' appears on the husband's forehead. This 'A' is plainly visible to everyone on the island except for the cheater's wife, which is lucky for him because if his wife finds out she will slit his throat while he sleeps.

Now everything is going hunky-dorey. All the husbands are cheating and each wife is clueless to her husbands infidelity, until one day . . .

A prophet from the mainland arrives on the island and pronouces,
Adultery has been committed upon this island!

The question is what happens next and why? There is a detailed answer.
2. 25 Sep '04 01:33
Well, assuming that the women on the island are of average intelligence, I wonder why it took a prophet to get the ball rolling.

I'll assume that every woman on the island knows what the A stands for.

Each woman sees an A on the forehead of every man except for her own husband. Each woman must therefore feel special that their husband is faithful. Among the women, there would be one that would feel like bragging about how faithful her husband is. The moment that she utters this declaration, the other women will each interject that this woman's husband has an A and that it is their husband that is loyal. From this point on, it is a matter of minutes until the women figure out what is going on and run home to commit gendercide later that night.

Besides, if every husband on the island is cheating, then who are they cheating with? Any husband may be cheating with one or more wives or one or more husbands. I doubt that there are very many women that are unaware of cheating since most are engaging in cheating themselves.

-Ray.
3. telerion
True X X Xian
25 Sep '04 01:582 edits
Yes assume that all the women know what 'A' means. And while your scenario is far more likely, let's go ahead and assume that neither the women nor the men rat out any cheater to his wife.

Also assume that each man has slept with every other person on the island (including his wife). This would explain why nobody talks about it.

AH YES, I ALMOST FORGOT! One final and all important assumption (This is economics ya know. We need unrealistic assumptions.) Assume all the women are perfectly rational and that each woman knows all the other women are perfectly rational and that each woman knows that each woman knows that all other women are perfectly rational and that each woman knows . . . ad infinitum.

In other words assume common knowledge. Sorry for that little slip up. It's an important component.

4. Acolyte
25 Sep '04 10:12
Originally posted by telerion
Also assume that each man has slept with every other person on the island (including his wife). This would explain why nobody talks about it.
In that case I don't think the women would be too fussed. You've got a regular orgy going on that island! ðŸ˜²
5. telerion
True X X Xian
25 Sep '04 13:51
Originally posted by Acolyte
In that case I don't think the women would be too fussed. You've got a regular orgy going on that island! ðŸ˜²
Yeah, seems like a situation where you shouldn't rock the boat. I think the prophet was jealous. I'll have to consider that a bit.
6. Aynat
Dog Companion
26 Sep '04 20:27
Originally posted by telerion

AH YES, I ALMOST FORGOT! One final and all important assumption (This is economics ya know. We need unrealistic assumptions.) Assume all the women are perfectly rational and that each woman knows all the other women are perfectly rational and that each woman knows that each woman knows that all other women are perfectly rational and that each woman know ...[text shortened]... ds assume common knowledge. Sorry for that little slip up. It's an important component.

[/b]
While we are making assumptions...let us assume that all the men on the island are also perfectly rational and that each man knows that all the other men are perfectly ration and that each man also knows that each men knows that all other men are perfectly...etc. etc.

Perhaps the men are adulterous with each other?
7. telerion
True X X Xian
26 Sep '04 20:45
Originally posted by Aynat
While we are making assumptions...let us assume that all the men on the island are also perfectly rational and that each man knows that all the other men are perfectly ration and that each man also knows that each men knows that all other men are perfectly...etc. etc.

Perhaps the men are adulterous with each other?
Yeah, sure that works. Anybody gonna take a stab at solving this? If nobody's working on it, I'll post another question along this line that is much easier to solve and will make this one easier.
8. skeeter
515 + 30 days
26 Sep '04 21:19
Originally posted by telerion
Yeah, sure that works. Anybody gonna take a stab at solving this? If nobody's working on it, I'll post another question along this line that is much easier to solve and will make this one easier.
Hmmm...I think that the pivitol assumption has to be that the men can see each others 'A's but are also aware that their wives cannot see their spouses. All the prpophet has to do is apply the same ctiterium to the women. ie 'A's for the women adulterers unable to be seen by their spouses but only if both are faithful.

skeeter.
9. telerion
True X X Xian
26 Sep '04 21:34
Originally posted by skeeter
Hmmm...I think that the pivitol assumption has to be that the men can see each others 'A's but are also aware that their wives cannot see their spouses. All the prpophet has to do is apply the same ctiterium to the women. ie 'A's for the women adulterers unable to be seen by their spouses but only if both are faithful.

skeeter.
Well let's just say women don't get an 'A' on their forehead. In the original problem, the women got 'A's, and the men did not. The men were also the ones who would kill their wife. I think the article was written in the 50's or very early 60's in the US so you can see where this double standard comes from.

10. 27 Sep '04 15:29
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.
Try it with one wife and one husband on the island - she kills her husband that night.

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.

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.

With N wives, each knows that N-1 guilty husbands will be killed in n-1 nights, and when that doesn't happen they know there must be N guilty husbands.

So In the scenario given, all the husbands sail off the island on the 365th day to avoid being killed on the 365th night.
11. 27 Sep '04 18:22
Originally posted by iamatiger
Try it with one wife and one husband on the island - she kills her husband that night.

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.

Try it with three wives - each reckons if their husband is ...[text shortened]... all the husbands sail off the island on the 365th day to avoid being killed on the 365th night.
OR:

they kill the prophit, and one of them dresses up like him, and says he was mistaken.
12. telerion
True X X Xian
27 Sep '04 20:07
Originally posted by iamatiger
Try it with one wife and one husband on the island - she kills her husband that night.

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.

Try it with three wives - each reckons if their husband is ...[text shortened]... all the husbands sail off the island on the 365th day to avoid being killed on the 365th night.
Excellent, give tiger a pretzel.

Very good job. I believe the article ended much more tragically.
13. 28 Sep '04 14:44
Originally posted by telerion
Excellent, give tiger a pretzel.

Very good job. I believe the article ended much more tragically.
Thanks, the confusing thing to me about this is that they all knew there was at least 364 adulterers before the prophet arrived, and they all new that all the other wives knew there was at least 363, but him telling them that there is at least on adulterer - even though it doesn't change any of their knowledge one jot - starts this countdown.
14. telerion
True X X Xian
29 Sep '04 19:45
Originally posted by iamatiger
Thanks, the confusing thing to me about this is that they all knew there was at least 364 adulterers before the prophet arrived, and they all new that all the other wives knew there was at least 363, but him telling them that there is at least on adulterer - even though it doesn't change any of their knowledge one jot - starts this countdown.
Yeah, I took another look over my opening post and discovered that I had nested the answer inside the question!

Bummer. I should not have assumed common knowledge. Only perfect rationality. So then once the prophet comes along, common knowledge takes effect. Good job anyway. And thanks for catching that!
15. AThousandYoung