15
Sep
2014
Game Theory, Rational Women, and Dead Husbands
I also manage to contradict Tyler Cowen in this one. There’s a lot going on, it’s a jumbled mess. Just click through if you’re interested.
I also manage to contradict Tyler Cowen in this one. There’s a lot going on, it’s a jumbled mess. Just click through if you’re interested.
I don’t get where the justification comes from in thinking of this problem according to “Day 1″, then ” Day 2″, etc.
That’s the standard assumption made in puzzles like this. If you want it to be more precise, we can add the following requirement: you’re only allowed to shoot your husband exactly at sunset. That way people can only react to the choice of others to shoot or not shoot their husbands the next day.
It is all nonsense. We are missing the point which is to say that 100 cheating females are allowed to murder their equally cheating husbands and this is somehow acceptable. The thought experiment is that both women and men seem to have little regard for men
Err, it could be one female doing all the cheating . . .
How does her husband “cheat” then? And what exactly is he “cheating” anyhow?
I don’t get where the justification comes from in thinking of this problem according to “Day 1″, then ” Day 2″, etc.
It’s because if a woman finds out her husband has cheated, she is required to kill him on the day she finds out.
So after a day ends, you know that on that day there was no woman who found out her husband cheated.
If no husband was killed that day.
Huh? let us say 40 people observed 39 wives cheating (sorry I am old and a misogynist) and 60 people saw 40 wives cheating, how would the 40th know to kill his wife without knowing the total cheats? I know my wife is absolutely faithful and I look out on day 98 and see all those deluded fools who believe their wife and my wife are the only faithful ones, knowing, of course, my wife is actually the only faithful wife. No wonder I used WordPerfect and Lotus right up until there was no other choice.
See my comment. I think it should clarify things.
Bob, in case of the MSFT question, they said all women ‘instantly know” when another man cheated. They don’t need to be present or see it happen.
Bob. You ARE special. It’s beginning to dawn. Thank you.
Raja I was trying to come up with a more realistic scenario that would make sense of it. If you think I changed the assumptions in a way that messes up the analysis, let me know; you might be right.
Bob, your modification definitely doesn’t change anything. After all, “the world outside the bathroom” is exactly the same as “the realm of things that women know about instantly”.
The only complication is that we have to assume that two women never go to the bathroom at the same time, so that a woman never misses an adultery that’s not her own.
Keshav not only is that the best comment left on this blog, it may be the best comment left on any blog.
Keshav, it seems to me that in the original the women got the information because they were all cheating too. But in Bob’s modification It could have just as well been a goat tied to a well in the middle of town square. Was I supposed to assume esp?
The original says “Every wife in the village instantly knows when a man other than her husband has cheated.” How can the explanation for that be anything other than ESP? How does the fact that a woman is cheating allow her to instantly know when a man she’s not even interacting with at the moment is cheating?
The original says “Every wife in the village instantly knows when a man other than her husband has cheated.” How can the explanation for that be anything other than ESP? How does the fact that a woman is cheating allow her to instantly know when a man she’s not even interacting with at the moment is cheating?
I’m not reading that the same as you and that’s probably a mistake on my part but if every woman cheated then instantly they would know when A man that was not their husband cheated. Not that they would instantly know when all men that were not their husband had cheated.
Oh, I see. You’re reading the sentence as “for all woman X there exists a man Y who is not the husband of X such that X knows that Y is cheating” whereas I’m reading it as “for all woman X, for any man Y who is not the husband of X, X knows that Y is cheating.” I think my reading is more plausible, because if your reading was correct they would have simply said “every woman is cheating with someone” rather than talking circuitously about what the woman knows.
It’s more plausible that they all have 99% accurate esp but are mysteriously unaware of their own husbands debauchery than a thought puzzle having deliberately vague phrasing? I won’t say I’m right or I know but it seems that my understanding makes a whole lot more sense, original writers intent be damned. I suspect you are probably reading it the way someone intended I’m just not convinced it is more plausible.
Grane, your interpretation makes more sense, in terms of the laws of physics and biology, but I think the puzzle clearly relies on the other interpretation. FWIW, that’s why I introduced the women going to the restroom while everybody else can see everything, etc.
Maybe when a man cheats, he cheats with every woman in the village?
There is only one bathroom in the village.
That also explains their unusual social setup, but this is about logic not common sense.
The queen slept with someone’s husband. The queen is a hussy and a home wrecker. No woman has proof it is all she said, she said, that’s good enough for a conviction in the US.
Hussy, home wrecker AND gossip. No wonder she went into politics.
Bob, let’s consider the case where there are only three couples instead of one hundred. (The basic logic doesn’t change.) Before the Queen arrives, Woman 1 thinks that 2 women have been cheated on. Woman 1 thinks that Woman 2 thinks that 1 woman has been cheated on. And Woman 1 thinks that Woman 2 thinks that Woman 3 thinks that no one has been cheated on. (Note: in all of this I’m assuming that everyone assumes they’ve not been cheated on until they find proof otherwise.)
Now consider after the Queen’s announcement. Woman 1 reasons that if Woman 2 sees Woman 3 not kill her husband, then Woman 2 will think to herself, “If Woman 3 had seen no adultery, then after the Queen’s announcement she would have killed her husband. But she didn’t kill her husband, so she must have seen some adultery. And since I know Woman 1 was never cheated on, I must have been cheated on, so I have to kill my husband.” So Woman 1 concludes that if Woman 2 sees Woman 3 not kill her husband, then Woman 2 must kill her husband. And then if Woman 1 sees Woman 2 not killing her husband after seeing Woman 3 not killing her husband, then Woman 1 must conclude that her assumption that she was not cheated on is wrong. So then she’ll kill her husband.
So the Queen’s announcement very clearly does have an effect on things.
The key point is, even if no one thinks an absurdly false thing (like “there are almost no adulteries in the village”), there may still be someone who thinks that someone thinks that someone thinks that …. that someone thinks an absurdly false thing.
Just to finish my solution, on the third day Woman 1 will kill her husband, and by symmetry that’s when Women 2 and 3 will kill their husbands as well. So I think Microsoft is right that in the case of a hundred couples, all the women will kill their husband on day 100.
Keshav I hope you’re right, and I admit I didn’t write it out on a piece of paper. But are you saying it is NOT common knowledge (in the original scenario) that at least one husband has cheated on his wife?
Yes, I am saying that. Before the queen’s announcement, everyone knows that there has been adultery in the village, but person 1 thinks that person 2 thinks that person 3 thinks … that person 100 thinks that there is no adultery in the village.
The puzzle states “whenever a wife uses the restroom, her husband has the option of cheating on her, in full view of everyone else in the village”
Which I took to mean everyone (apart form the wife in the restroom ) sees all cheating incidents.
If this is not the assumption then its possible that no-one happens to be around for any of the cheatings – in which case there is no grounds for any husband killings – so there are at least some situations where the queens advice has no effect. If you assume random viewings then (I think) a wife still can’t draw any conclusions from the Queens statements since she (and any other wife) may have avoided seeing any cheatings purely by luck – so can’t kill her husband on that basis.
“Which I took to mean everyone (apart form the wife in the restroom ) sees all cheating incidents.” Yes, I’m assuming the same thing you are.
So each woman knows that all the other woman have been cheated on. But Woman 1 thinks that Woman 2 has only seen 98 women being cheated on. And Woman 1 thinks that Woman 2 thinks that Woman 3 has only seen 97 women being cheated on. Etc, all the way to the fact that Woman 1 thinks that Woman 2 thinks that Woman 3 thinks … that Woman 100 has seen no one being cheated on.
Look at my comment above, where I discuss the case where there’s only 3 couples in the village rather than 100.
I think I may have stated this slightly wrong.
I think it more accurate to say: If cheating-sitings are random then no wife can kill her husband with certainty of his guilt. Even if she has seen no incidences and the Queen assures her there has been at least one , she will never know for sure if she has avoided seeing any purely by luck.
By the way, I just wanted to clarify that when I said yes to Bob, I was using the term “common knowledge” in the technical sense, not in the sense we use in every day English. I hope that’s not causing any confusion.
Well, it is now !
What is the technical sense of “common knowledge”.
Well, now it’s a moot point, since you understand the solution. But if you want to know, here is the technical definition of common knowledge. If you have group of people G, and a sentence P, then P is said to be “common knowledge” for G if everyone in G knows that P is true, everyone in G knows that everyone in G knows that P is true, everyone in G knows that everyone in G knows that everyone in G knows that P, etc.
See this Wikipedia article:
http://en.wikipedia.org/wiki/Common_knowledge_(logic)
Great job, Keshav, this is definitely what the puzzle is going after. I will amend my Liberty Chat post and point people to your long comment.
Do you think that the fact that Microsoft thinks this a good question to ask job applicants might explains something about the usability of the software that Microsoft turns out ?
I see Bitter Clinger made the same joke, only better.
Google found out long ago that brainteasers are useless:
http://www.nytimes.com/2013/06/20/business/in-head-hunting-big-data-may-not-be-such-a-big-deal.html?smid=pl-share
“On the hiring side, we found that brainteasers are a complete waste of time. How many golf balls can you fit into an airplane? How many gas stations in Manhattan? A complete waste of time. They don’t predict anything. They serve primarily to make the interviewer feel smart.”
Assume, say, that there were 3 adulterers then every wife has seen at least 2 cheating incidents, so everyone knows (if they are logical) that there will be no killings by wife’s who have seen only 1 cheating incident, and will draw no conclusions from the lack of such killings.
As the puzzle states “it just so happens that in this particular village, every husband has in fact cheated on his wife.” its pretty obvious that there will be no more husband killing incidents on day 99 than on day 1.
So the temptation “to conclude that the Queen’s announcement surely has no effect” seems correct.
Transformer, what’s wrong with my reasoning in the comment above?
http://consultingbyrpm.com/blog/2014/09/game-theory-rational-women-and-dead-husbands.html#comment-935000
Well, you posit a set of beliefs that would lead to husband killings – but those beliefs seem to be exogenous to the model and are not based upon pure logic.
What are you , a post-Keynesian or something ? (that’s a joke ,BTW)
“Well, you posit a set of beliefs that would lead to husband killings – but those beliefs seem to be exogenous to the model and are not based upon pure logic.” The only thing that I’m “positing” is that each woman believes that she has not been cheated on until she has proof that she has been cheated on. Do you see any other unstated assumptions I’m making?
“Woman 1 thinks that 2 women have been cheated on. Woman 1 thinks that Woman 2 thinks that 1 woman has been cheated on. And Woman 1 thinks that Woman 2 thinks that Woman 3 thinks that no one has been cheated on. ”
Do you mean that Woman 1 KNOWS that 2 women have been cheated on (because she has seen it happen ?), if not why does she have this belief ?
But why would “Woman 1 thinks that Woman 2 thinks that 1 woman has been cheated on.”?
What possible logical basis for this belief could she have if sitings are random , and data sharing is dis-allowed ?
I’m not seeing where these beliefs come from if not from logical deductions from personal observations ?
Let me start off by saying I’m not assuming sitings are random, I’m assuming that every woman sees all adulteries other than her husband’s adulteries. Now that that’s out of the way, let me answer your questions:
“Do you mean that Woman 1 KNOWS that 2 women have been cheated on (because she has seen it happen ?)”. No, I meant that she thinks that exactly two women have been cheated on, because she has seen two women being cheated on, and she assumes that she herself has not been cheated on.
“What possible logical basis for this belief could she have if sitings are random , and data sharing is dis-allowed ?” Again, I’m assuming sitings of cheating is universal (everyone sees everything that happens as long as they’re not in the bathroom), not random.
Here is the logic: Woman 1 thinks she has cheated on, and she knows that Woman 2 has not seen herself being cheated on, because otherwise she would have killed her husband. Therefore, Woman 1 concludes that the only act of cheating that Woman 2 has witnessed is Woman 3 being cheated on. And since by assumption all women assume that they’ve not been cheated on until they find proof to the contrary, Woman 1 concludes that Woman 2 thinks that only Woman 3 has been cheated on.
That was the basis of my assertion that Woman 1 thinks that Woman 2 thinks that one woman has been cheated on. Does that make sense?
yes, that makes sense – thanks for clarifying.
Are you keeping the ” it just so happens that in this particular village, every husband has in fact cheated on his wife.” assumption from the puzzle ?
Because if you maintain that assumption then (even in your 3 couples version) I see no rational basis for husband-killing.
“Are you keeping the ” it just so happens that in this particular village, every husband has in fact cheated on his wife.” assumption from the puzzle ?” Yes, I am.
“Because if you maintain that assumption then (even in your 3 couples version) I see no rational basis for husband-killing.” Well, let me explain it again.
Before the announcement, Woman 1 thinks two women have been cheated on, Woman 1 thinks Woman 2 thinks one woman has been cheated on, and Woman 1 thinks Woman 2 thinks Woman 3 thinks no woman have been cheated on. Do you understand how to prove all that?
Assuming that you do, here is what happens after the Queen’s announcement: On day 1, Woman 1 thinks Woman 2 thinks “Woman 3 is going to kil her husband, because she’s seen no adultery and yet she knows that there is adultery.”
So if Woman 3 fails to kill her husband on Day 1, then Woman 1 will think Woman 2 thinks “Woman 3 heard the announcement and yet didn’t kill her husband, which means she witnessed adultery other than her hsuband’s. But Woman 1 was never cheated on, so that means I was cheated on.” So Woman 1 concludes that if Woman 3 fails to kill her husband on Day 1, then Woman 2 will kill her husband on Day 2.
So then if Woman 3 fails to ki her husband on Day 1 and Woman 2 fails to kill her husband on Day 2 in response, then on Day 3 Woman 1 will conclude that her assumption that she’s never been cheated on is wrong, so she’ll kill he’d husband. And by symmetry that’s when Woman 2 and Woman 3 will also kill their husbands.
Tell me if any step doesn’t make sense.
It all make sense now, and I agree with your initial comment now that I understand it.
transformer, FWIW, Keshav is definitely doing what the puzzle officially needs for its solution. That’s how game theorists solve these things. By all means, click the links in my original article to read my criticism of this type of reasoning, but Keshav is doing this the “right” way.
yes, I finally got it after reading Garrett’s comment below.
Based on that, I think Keshav 3-couple model (above) is an easy way to understand this – I just missed the point when I first read it – I’ll blame my lack of grounding in game-theory 🙂
Bob, I assume after your edit, you believe my solution is not only “right” but also right? 🙂
Keshav is definitely right on this if we make standard game-theoretic assumptions: All women know everything, except whether their husbands have cheated.
Each woman believes that 99 adulteries have occurred.
Therefore, each woman believes that each other woman wrongly believes that 98 adulteries have occurred. Therefore, each woman believes that each other woman wrongly believes that each other woman wrongly believes that 97 adulteries have occurred.
Therefore, each woman believes that each other woman wrongly believes that each other woman wrongly believes that each other woman wrongly believes that 96 adulteries have occurred.
…
Therefore, each woman believes that each other woman wrongly believes that each other woman wrongly believes that each other woman wrongly believes [repeat “that each other woman wrongly believes” 96 more times] that zero adulteries have occurred.
So each woman, following the logic of assuming that her own husband has not cheated and applying the same logic to each other woman, needs to revise her beliefs (about other people’s beliefs about other people’s beliefs, and so on) in light of the common knowledge that nobody believes that nobody has committed adultery. Further, she has to revise them every day until she revises her own belief on day 100.
This assumes, of course, that there’s common knowledge of all women’s flawless rationality.
Yes, exactly.
Ah, now I get your point Keshav – sorry for being slow.
I don’t understand the regression. Why wouldn’t it stop at each woman knowing that 99 husbands cheated, but believing all other women think it is 98. I mean if every woman knows that every other woman knows about all cheaters except their own, and you know there were 99, then that would mean you would also know that each individual woman would be aware of at least 98 cheaters. What am I missing?
Never mind, I should’ve read the post first. I get it now.
Each woman believes that 99 adulteries have occurred.
Therefore, each woman believes that each other woman wrongly believes that 98 adulteries have occurred.
No, each woman *knows* that at least 99 and possibly 100 adulteries have occurred. Each woman also knows that each other woman knows that.
Each woman also knows that each other woman knows that.
Oops. That was wrong. Each woman knows that:
– when there are 99 adulteries, the other women know there are at least 98 adulteries and possibly 99 adulteries
– when there are 100 adulteries, the other women know there are at least 99 adulteries and possibly 100 adulteries
Nice way of putting it Garrett.
In your scenario, why wouldn’t all the women kill their husband on the second day? If every woman saw each man cheat while with every other woman, then after nobody killed their husband the first day, wouldn’t they all realise at that moment that their husband must’ve cheated, too?
Or, I guess it would be the third day, since they all know for sure that every other woman is for sure aware of 98 cheaters.
Dan, good question, that’s what was tripping me up originally. But if you really care, write it out on a piece of paper, assuming just 3 couples:
Woman A knows that Man B and Man C have cheated.
Woman B knows that Man A and Man C have cheated.
Woman C knows that Man A and Man B have cheated.
Woman A knows that Woman B knows that Man C has cheated.
Woman B knows that Woman C knows that Man A has cheated.
BUT, Woman A does NOT know if Woman B knows that Woman C knows at least one man has cheated. Woman A would say, “From my observations and deductions, I admit it is possible that Woman B herself cannot be sure that Woman C has seen any adultery. It’s true, *I* know that Woman C saw Man B cheat. But Woman B doesn’t know Man B cheated, let alone know that Woman C knows this.”
So, I’ve just shown Dan that Woman A can’t be sure that Woman B thinks Woman C has seen any adultery. Therefore, if Woman C doesn’t kill anybody, ever, then Woman A won’t conclude anything by watching Woman B’s lack of a response.
I get it with three people. But if you add in another 97 then woman A would know that woman C and B and everyone else both saw 98 cheaters for sure. She would know this because she was with each woman for 98 times it happened. So I think at day three they would all realize their husband cheated.
Never mind, I get it. This is the same thing that was tripping me up yesterday.
Wait, are you saying that woman 1 believes woman 100 thinks nobody cheated? Because that clearly wouldn’t be the case. She would have saw 98 separate instances of cheating with each woman.
No, it’s not that Woman 100 thinks no body has cheated. It’s that Woman 1 thinks Woman 2 thinks Woman 3 thinks … Woman 100 thinks nobody has cheated.
Dan, Woman 1 knows that all the other woman have seen at least 98 women get cheated on. But Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Women 3 has only seen 97 adulteries. And Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that … Woman 100 has seen 0 adulteries. Do you understand that?
That’s what is tripping me up. I see how it works with three people, but not 100. I mean woman 1 knows that 99 cheating a occurred. She also knows woman 2-100 have for sure saw 98.
Why would she think any woman thinks it is possible that another woman could think nobody cheated? I mean, they all know for sure that each woman saw 98 people cheat because they were with each other when it happened.
Dan wrote:
Why would she think any woman thinks it is possible that another woman could think nobody cheated?
It’s not; that’s one way of explaining why I made my initial mistake, because I thought this fact ended the discussion. But in order to get everybody to kill their husbands on Day 100, it’s not enough just for every woman to know that every woman has witnessed at least one adultery.
If you really want to see it, extend it to four couples and then you’ll find (I think?) that A knows that B knows that C knows that at least one man has cheated, but A can’t be sure that B knows that C knows at least 2 men have cheated. This is the case, even though all women know that 3 men have cheated, and all women know that all women know that at least 2 men have cheated.
(I’m doing this on the fly, I hope you get the big picture, I might be botching the specifics.)
I’ll have to write it out for ten people and see what I’m missing.
Whoa! Just go from 3 to 4 couples and that should be enough.
I.e. with the Queen, the village with 3 couples kill all 3 husbands on Day 3, but with 4 couples, it happens on Day 4. And that without the Queen, nobody dies.
Dan I suggest you try to understand the case of three couples, the case of four couple, five couples, etc., until you understand that increasing the number of couples doesn’t change the fundamental logic.
I get the case for three couples. That makes perfect sense to me. The only thing tripping me up is this “It’s that Woman 1 thinks Woman 2 thinks Woman 3 thinks … Woman 100 thinks nobody has cheated.”
I don’t see how that works out.
For example,
Woman 1 knows that Man 2 and Man 3 and Man 4 and Man 5 cheated.
Woman 2 knows that Man 1 and Man 3 and Man 4 and Man 5 cheated.
Woman 3 knows that Man 1 and Man 2 and Man 4 and Man 5 cheated.
Woman 4 knows that Man 1 and Man 2 and Man 3 and Man 5 cheated.
Woman 5 knows that Man 1 and Man 2 and Man 3 and Man 4 cheated.
Also,
Woman 1 knows Woman 2-5 knows that 3 men cheated.
Also,
Woman 1 knows that Woman 2, 4, 5 knows that Woman 3 knows that 2 men cheated.
Woman 1 knows that Woman 3-5 knows that Woman 2 knows that 2 men cheated.
Woman 1 knows that Woman 2, 3, 5 knows that Woman 4 knows that 2 men cheated.
Woman 1 knows that Woman 2-4 knows that Woman 5 knows that 2 men cheated.
I just don’t see how you get to Woman 1 thinks it is possible that woman 2 thinks it is possible that woman 3 thinks it is possible that woman 4 thinks it is possible that woman 5 thinks nobody cheated.
OK, I assume you’re talking about the case of a village with five couples.
Let me write things out explicitly, rather than in terms of numbers of women.
1. Woman 1 thinks it’s possible that she was not cheated on.
2.. Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that neither Woman 1 nor Woman 2 as cheated on.
3. Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that neither Woman 1 nor Woman 2 nor Women 3 were cheated on.
4. Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that Woman 4 thinks it’s possible that neither Woman 1 nor Woman 2 nor Women 3 nor Woman 4 were cheated on.
5. Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that Woman 4 thinks it’s possible that Woman 5 thinks it’s possible that neither Woman 1 nor Woman 2 not Woman 3 nor Woman 4 nor Woman 5 were cheated on.
Whew! Now tell me, out of my five statements, what is the earliest statement you have trouble understanding?
(It may be easier to do the four couple case, by the way.)
“4. Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that Woman 4 thinks it’s possible that neither Woman 1 nor Woman 2 nor Women 3 nor Woman 4 were cheated on.”
This is where I start getting tripped up. Woman 1 knows for sure that Woman 2 knows for sure that woman 3 knows for sure that woman 4 knows for sure of 1 cheater.
“Woman 1 knows for sure that Woman 2 knows for sure that woman 3 knows for sure that woman 4 knows for sure of 1 cheater.” Yes, that’s true. Woman 1 knows that Woman 2 knows that Woman 3 knows that Woman 4 knows that Woman 5 has been cheated on.
But how does that contradict statement 4?
Do you want me to explain the logic of how to go from Statement 3 to Statement 4?
Alright, here is what is tripping me up in the five person example. Woman 1 knows for sure that each other woman is aware that the other women know of 2 cheaters. She knows this because she was in the room with them when it happened. If she knows this for sure, it would be a contradiction for her to believe it possible that there is any progression where any of them could possibly think nobody cheated.
Side note: This is both annoying and stimulating for me at the same time.
Yes, it’s true that Woman 1 knows that Woman 2 knows that the Woman 3 knows that there are at least two cheaters, namely Man 4 and Man 5. And we know that Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 think it’s possible that Man 3 and Man 4 are the only two cheaters. But if Man 4 and Man 5 were the only two cheaters, then Women 4 would only know of one cheater, namely Man 5. Therefore, we can conclude that Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that Woman 4 only knows of one cheater.
Does that make sense?
Sorry, I made a typo. Here is what the comment should say (I put the corrected part in bold):
Yes, it’s true that Woman 1 knows that Woman 2 knows that the Woman 3 knows that there are at least two cheaters, namely Man 4 and Man 5. And we know that Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 think it’s possible that Man 4 and Man 5 are the only two cheaters. But if Man 4 and Man 5 were the only two cheaters, then Women 4 would only know of one cheater, namely Man 5. Therefore, we can conclude that Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that Woman 4 only knows of one cheater.
Does that make sense?
Yeah, I get how that progression works, which is why this has tripped me up.
“But if Man 4 and Man 5 were the only two cheaters”
But they weren’t the only cheaters. That is why this doesn’t make any sense to me. Woman 1 knows with 100% certainty that every woman is aware of 3 cheaters. She also knows with 100% certainty that every woman knows that every other woman is aware of two cheaters. So, knowing this, why would woman 1 introduce a false narrative like “if man 4 and 5 were the only cheaters…”? While doing that would make that progression work out, it wouldn’t make sense to do that considering she already knows with 100% certainty the things I said above.
Dan read my latest post.
OK Dan, let me write things out more systematically.
A. Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that Man 4 and Man 5 are the only two cheaters.
B. If Woman 3 thinks it’s possible that Man 4 and Man 5 are the only two cheaters, then that means that Woman 3 thinks it’s possible that Woman 4 knows of only one cheater (namely Man 5).
C. Therefore, Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that Woman 4 knows of only one cheater..
Out of these three statements, which is the earliest one you’re having trouble understanding? Note that Statement A is the same as Statement 3 in my list of five statements from before, and Statement C is the same as Statement 4.
All of those make sense to me.
Whoops, sorry.
“B. If Woman 3 thinks it’s possible that Man 4 and Man 5 are the only two cheaters, then that means that Woman 3 thinks it’s possible that Woman 4 knows of only one cheater (namely Man 5).”
Woman 3 doesn’t think it is possible that there are only two cheater. She saw three cheater with her own eyes.
Check that, woman 3 saw 4 cheaters with her own eyes.
Yeah, I know that the if part of the sentence is false, but that doesn’t affect the truth of the conditional statement. They key point is, the statement “Man 4 and Man 5 are the only cheaters.” And the statement “Woman 4 knows of only 1 cheater, namely Man 5” are equivalent statements. Whether those statements are true or false, do you at least agree that they’re equivalent?
If you agree that they’re equivalent, then we can start with e statement “Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that Man 4 and Man 5 are the only cheaters.” And then we can substitute “Man 4 and Man 5 are the only cheaters.” with the equivalent statement “Woman 4 knows of only 1 cheater, namely Man 5” . Thus we get “Woman 1 thinks it’s possible that Woman 2 thinks it’s possible that Woman 3 thinks it’s possible that Woman 4 knows of only 1 cheater, namely Man 5.”
Does that make sense?
Dan, I think this is all easier to think about in the four-couple case. See my comment in Bob’s new post: consultingbyrpm.com/blog/2014/09/couples-therapy.html#comment-937689
To be clear, the comment I’m referencing is the second reply I made to the original comment you made. Specifically, the comment where I give Woman 3 an extended speech.
I need to think on this more, and try to figure out where my mind is getting jumbled up.
I just wanted to say that the technical notion of “common knowledge” (which is very different from what the term means in common English) which underlies this puzzle is an absolutely beautiful concept on mathematics. See the Wikipedia article:
http://en.wikipedia.org/wiki/Common_knowledge_(logic)
And it has major consequences; see Aumann’s agreement theorem.
To Tylers’s example. I think a lot of people would recognise your assessment of men as petulant, self destructive and ignorant. However, it does not change the picture much. Whether you describe it as “soft”, the women *are* prepared to settle for less. Also your assessment of women as more rational does not entirely stack up.
In the dictator game, the rational response is to give away nothing. Women gave away more, so they are less rational. (Or that they care about more than their monetary returns from the game.)
In the ultimatum game, it is rational to accept any amount offered, otherwise you get nothing. We find that : “Players 2 of both genders choose a higher minimum acceptable offer when facing a female player 1” i.e. player two is not rationally accepting any offer, but player two of either gender requires a higher payment if player 1 is female. The extent to which player two is irrational is greater if player 1 is a woman.
People are not both rational and informed, of course. In the absence of the ability to calculate the outcomes of every situation, it is advantageous to adopt general strategies which one applies to many situations, even when that would not be the optimum in any particular case. From the information in the abstract, we can glean that the same strategy would not work for men and women. If an individual woman player 1 adopts the same strategy as the average man, then they will leave with nothing more often than the man. In order for their offer to be accepted, it musty on average be higher than the man’s.
From your other articles:
The game where everyone picks a number form 0-100, and the winner is the closest to half the average of the numbers. You say the logic remains the same if you include the subjects choice, but I am not sure that is the case. Particularly for the two player game.
“One-half the average of their picks is 1, and therefore Jane wins the prize; she has played perfectly.” Surely there is a difference between playing perfectly and chancing to win. If she had picked zero, she would also have won. However, if she picked 1 and John had picked zero, she would have lost, so picking 1 is not perfect play.
“We have seen that a player may choose a number different from zero in this simple game and still win” Yes, but it is rational to maximise your chances of winning if you have a choice of numbers to pick. In the two player game, the only number where you cannot lose is zero – you must win or draw. If she picked 20 and he picked 80, she would win (half the average is 25), but that would not be a rational number to pick if she is capable of working out the probabilities. It is really just a working out odds exercise, and not interesting form a game theory perspective. Rather like saying whoever picks the lowest number wins – every rational person would pick 0.
However, if you do not include the selector’s choice, you have a better representation. You need a three player game. Jane must try to pick closest to half the average of Bob and Alice. (Perhaps I see why you avoid the “traditional” choice of subject names!). Between Bob and Alice, the one who picks the lowest number will win. So they must pick the lowest number, and still beat Jane.
I would think a different scenario unfolding on day 100. All men on that day would know they are heading for execution. Each man would then kill his wife before the woman got a chance to kill him. Therefore, on day 100 we have 100 dead women.
But that would be sexist.
Men are having affairs with other village women aren’t they? So village women are also having affairs with other men, yet they kill only the man for the transgression. Sounds like a socialist society to me, and sexist, but biased against men.
I am beginning to have this feeling that I am getting smarter the more time I spend around Bob!