If you enjoy this joke (which is discussed here, and comes from the folks at *Spiked Math Comics**) *as much as I do, you might be a gearhead.

It illustrates one of the many surprising and subtle impacts of common knowledge. Yale’s **John Geanakoplos** provides an even more perverse version of the bar cartoon, in this incredibly helpful chapter :

Imagine three girls sitting in a circle, each wearing either a red hat or a white hat. Suppose that all the hats are red. When the teacher asks if any student can identify the color of her own hat, the answer is always negative, since nobody can see her own hat. But if the teacher happens to remark that there is at least one red hat in the room, a fact which is well-known to every child (who can see two red hats in the room) then the answers change. The first student who is asked cannot tell, nor can the second. But the third will be able to answer with confidence that she is indeed wearing a red hat.

You can find out the reasoning behind the inference here.

Back in 1997, **Barry Nalebuff** ** **and I published an article showing how the common knowledge about insulting or threatening bits of information can destroy relationships and thereby be a “barrier to negotiation.” The beginning of the article lays out a simple description of the double infinity of knowledge that constitutes common knowledge:

When we communicate one thing, we often unavoidably send other messages. To start with a simple example, imagine that Ian says to Barry, “My mother’s name is Karen.” From Ian’s communication, Barry learns more than just the underlying bit of information (mom’s name). The communication also lets Barry know that: (a) Ian knows his mom’s name, and (b) Ian knows that Barry knows Ian’s mom’s name.

What is less well understood is that when we *teach, we learn. *When Ian tells Barry about his mom, Ian learns several things. For example, by telling Barry, Ian now knows that Barry knows Karen is the name of Ian’s mom. Direct communication of a fact can potentially create two infinite series of knowledge. If** **we symbolize the underlying bit of information (Karen is Ian’s mom) as** ***K*,** **then Ian’s communication might create the following hierarchy of beliefs:

Table 1: Potential Hierarchy of Beliefs | |

Ian’s Knowledge |
Barry’s Knowledge |

1a. Ian knows K |
1b. Barry knows K |

2a. Ian knows Barry knows K |
2b. Barry knows Ian knows K |

3a. Ian knows that Barry knows that Ian knows K |
3b. Barry knows that Ian knows that Barry knows K |

and so on . . . | and so on . . . |

If Ian’s communication succeeds in creating both of these infinite series, economists would say that the underlying fact is “common knowledge.” Table 1 makes clear how directly communicating a simple fact can produce other types of knowledge. Ian begins with what we will call “first-order” information (la) and wants to convey this to Barry (1b). But in doing so, Ian may be teaching Barry “higher-order” information as well (2b, 3b, 4b, etc.). Moreover, by teaching Barry, Ian may unavoidably acquire higher-order information himself (2a, 3a, 4a, etc.).

[HT: **Peter Siegelman**]

This is taken to its extreme in the Blue-Eyes puzzle:

http://www.xkcd.com/blue_eyes.html

The bit that messes me up, is that the additional information (“I can see someone with blue eyes”, or above “I can see someone with a red hat”) doesn’t add anything new to the situation, but changes the outcome. How is that possible?

It adds information because it begins a string of iterative inferences.

The “blues eye puzzle” isn’t quite well formed as stated on his website, because he really needs to add the premise that in addition to “all BEING perfect logicians”, they also need to “KNOW that the others are perfect logicians”. But he omits that piece of information because it would seem odd, would thus tip some people off to the answer making the overall puzzle easier.

In normal human interactions it is not advisable to assume others around you are perfect logicians, it will probably get you maimed or killed eventually. So that really needs to be added to the premises.

A related problem is a “surprise exam problem”. It seems that a teacher cannot give a surprise exam to logic students. If he says ” I will give a surprise exam in period Y” (e.g. I will give a surprise exam in October) the exam cannot actually be a surprise.

If the exam is on the 31st if wouldn’t be a surprise, and since it cannot be on the 31st it clearly cannot be on the 30th either, and so on and so on. Until they decide there can be no surprise test. Then they are surprised when there is indeed a test…

I’m not getting the red hat example. Red hats are worn by A, B, and C. All are told that at least ONE is wearing a red hat. A is asked her hat color, but she can’t tell since she can see at least one red hat. B can’t tell since she also can see at least one red hat. In the example, C can deduce that she must be wearing a red hat. But I think this is wrong.

A can’t tell if she’s wearing a red hat, because she can see at least one red hat. Her answer tells both B and C that at least one of them is red. But this doesn’t inform either B or C; B can see that C is red, and C can see that B is red. Then when B says she doesn’t know, this tells A and C that at least one of them is red. But this doesn’t further inform C, since B could be talking about A, who C can see.

The logic is challenging, so modify the situation and see what answers change. Change C’s hat to white. Now the teacher has said that at least one hat is red – that’s the same. A sees at least one red hat – on B – so her answer doesn’t change. B sees at least one red hat – on A – so her answer doesn’t change. C can see at least one red hat – she still sees two – and so her answer does not change.

Her answer must therefore be “I don’t know. ”

Am I missing something?

David,

Your alternative scenario is wrong. In that one, B would know that she is wearing a red hat, because A let her know that there is at least 1 red hat on B and C, and since C doesn’t have one, B knows she has one herself.

In scenario 1, C knows she has a red hat herself because if she didn’t, B would have known that B was wearing a red hat.

Got it. Thanks.

You were on the right track by looking at what happens if C’s hat is white. If so, then …

A sees one red and one white so she says “IDK”.

B now has more info. B knows that if her own hat were white then A would have seen two white hats and known her own hat was red, therefore B would know her own hat was red.

However, since B answers IDK after A’s IDK, C knows that the hypothetical situation above (C has a white hat) is not possible. Therefore she must have a red hat.

Yeah, I guess I’m missing the same thing.

Here’s what C knows:

From A’s comment: Either B or C is Red OR both B & C are Red

From B’s comment: Either A or C is Red OR both A & C are Red

From C’s sight: Both A & B are Red.

So C can still be Red or White, right?

Girl A looks out and thinks, well, if I see two white hats, this thing is over bc i’ll know that i am the only one wearing a red hat. But shoot, they are both wearing red hats. Well crud, that doesn’t tell me anything. “I don’t know”

Now, B knows that A saw at LEAST ONE red hat. She sees C’s hat is red, so now her only question is, is my hat red or white. Well I don’t know bc my hat could also be red. Once again, B thinks to herself, I know that A saw at least one red hat, but she could have seen two red hats. “I don’t konw”

So now C says, well this is easy! I know that A saw at least one red hat, and I know that B knows that that A saw one red hat. B can see my hat. If MY HAT had been white, B would have known for sure that she was wearing a red hat. But she didn’t know. And because she didn’t know that can only mean that my hat is red.

Does this help?

To be completely fair to the people who don’t get it.

Relying on your fellow humans to make correct inferences, even inferences as simple as the one “C assumes B made” is not generally a good day to day life strategy.

People are really really not very good with logic generally.

Dear Joshua; I once taught math for a few months. This was relatively new to me as I had only tutored a boy whose native tongue was Spanish before that. And I taught to each individual. Allow questions to be raised and answering THEIR questions about the lesson. So the only lesson plan I had was to teach the lesson and in a way that every individual understood it in their way. Grades on standardized tests went up around 25%. What did I get out of this experience. People can be good with logic. One has to reach each individual in their own terms. Schools are set up to do the opposite- teach to an arbitrary standard.

Try teaching an introduction symbolic logic at a university. You will see some of the most bizarre grade distributions you ever saw. Highly bimodal. And these are college students, granted not all geniuses, but most of them are reasonably intelligent.

40% of the class intuitively grasps logic, they are distributed around a B+/A-. Interestingly you will see several people who normally don’t get good grades in this group.

60% of the class just really really doesn’t get it, distribution around C-/D+. Likewise you will see people who normally get good grades, but just cannot wrap their mind around the abstract.

You can alleviate it somewhat with teaching/tutoring et cetera, but a large portion of even the college going population just isn’t cut out for logic (at least by age 18-20) . Maybe if you got them at a younger age it would be easier to reach more of them.

Dear Joshua; I have in a way as well for approx. 10 years (the same course). I Required class of 15 students to write papers (every 2-3 weeks) analyzing a text. And if they got lower than a C, they had an allotted amount of time to rewrite. Alot of work. Average grades B to B+ and papers were thoughtful, if not the first time, then the second time around.. when they rationally understood the requirements. And due to subject matter, there was no way in which one could cheat.

Got it. Thanks.

My husband did this add for a client once. I wrote the copy. I think I had the `idea’, came from that Russian children’s book about a man wearing many,many hats at once who walks into town, ends up falling asleep and waking up. Pro-found.

A’s answer of I don’t know means that B or C has a red hat.

If B sees a white hat on C then B knows that she has a red hat.

However, B does not see a white hat on C, therefore she does not know which color hat she has.

Since B does not know then C knows she must have a red hat.

The order of the answers matter.

–

JimFive

What’s odd about this kind of recursive thinking is that people appear to be pretty bad at generally, but really good at it in specific circumstances. Pedagogy, for instance, requires what Michael Tomasello calls a “shared intentionality.” For me to teach you, you have to understand my intent to teach, and I must know that you know that. Then the information I give you after that will be more greatly emphasized. That kind of recursive thinking is actually a key difference between apes that, well, ape and we humans who can engage in shared intentionality.

Dear Joshua;

Not preying at all. At least, that was not my intention. A lesson— yes. Teaching math and having your life and integrity threatened daily and simultaneously was a good experience (I learned a lot and did acquire an ace of a wonderful student in the process, but overall it still was a hugely stressful experience of which I would mostly like to forget about). Teaching Theories of Social Order over a ten year period was enlightening. Loved every minute of it and do hope to have an opportunity to do so again. We (my students and I) learned so much from each other. I had one student (a black woman) of about 25 and a “gentle reader” who may not ever find herself. She writes this amazing paper with quotes all over the place. I questioned the quotes (since she never cited a source) Turns out, she was the source and did not know that one need not cite one’s self. I asked her about going on to graduate school. Seems like she had too much to do to survive i.e., as in make sure there is food on the table.

As for teaching someone a lesson- I never read Comte from that standpoint until I left myself open to learning the real lesson that he had in mind. And If I thought that he was lying to me i.e., presenting me with a false statement from the start, I might not have read Comte or Spencer or Martineau with an open mind to sorting the matter of their enterprise out for myself. The truth hurts. I was oblvious to Martineau’s achievments for a very, very long time. We all learn from our mistakes. That is how we learn by acknowledging them.

Great comic!

The red hat one is a tricky one, basically preying on the same trap as the Monty Hall problem (people assuming superficially uninformative answers contain no information even though with a little inference they actually do).

I have seen fairly smart people fight Monty Hall for quite a while before conceding. This red hat one seems a little more straightforward.

Yes, I’ve seen the same. Very smart people who just cannot understand the Monty Hall problem. One guy was so angry at me that we bet using a deck of cards. At $1 a guess, he quit when I got up by $20. Then he understood.

Err … so what?

Sorry, but I don’t get the point of the point. I am willing to admit that I am quite possibly being obtuse and oblivious and would be happy to learn something useful (or even interesting) from this article, but I’m not.

Is the point that there are a lot of folks who are not good at (read, have been badly taught) logic?

There is also a very large population that cannot change a tire …

(Really not trying to be a bore, here. But all I’m seeing is a discussion of the sky is up and other bits of ‘knowledge’.)