Are you joking?

The only way ANY version of the problem works is if we suspend reality and treat it as a thought experiment. When the announcement is described as *spoken* we must make many assumptions, including that everyone heard and understood it individually. Changing the communication format to email, handwritten notes, or telepathy, changes absolutely nothing in that thought experiment. It still requires that perfect communication be assumed.

If you want to let the practicalities of reality intrude on the problem then we can find many ways to make it fail. But this problem uses green eyed dragons who are also all perfect logicians. So get ****ing real man. This tangent is silly.

Well, perfect communication is not the same thing as common knowledge. Common knowledge, what is required in these cascading logic problems, requires an infinite "I know that you know that I know that you know ...". An email or any non-verbal message cannot provide that.

Here is the simplest example. Suppose two green eyed dragons. Suppose a person emails them both that there is at least one green eyed dragon on the island. There is nothing pertaining to the email that can 100% guarantee to each recipient that the other recipient received (and read) the email. (The receipt message is typically sent to the sender, not the other recipients.) So neither changes to a sparrow (wins the million dollars).

The communication loop can be closed if both recipients tell one another that they received (and read) the email from the human, but that is an extra step. Then the email acts the same as the communal voice message, but they are not the same thing. Anybody who has two kids knows the difference.

And this type of communication will not provide the common knowledge if there are many dragons. How can each communally convey to all the others that they have received/read the email message? The only way is via each one getting up in front of the entire group and proclaiming that they have received/read the message. But that gets us right back to a communal voice messaging vehicle. And the one outsider's message is "equivalent" to the 100 dragon messages. That's the power of common knowledge.

Anyway, this was a hot topic during WWII when an air force squadron leader would need to communicate key information to each man in his squadron (while flying in separate airplanes) with less than 100% reliable communication systems. Many of the leaders in the modern communication theory revolution worked for the government during the war and wrestled with thorny problems like this.

You are still talking about practical realities in a perfect thought experiment where they don't matter and are totally irrelevant to the point of the exercise. This is a logic puzzle, not a communication or logistics problem.

I still think the common starting point is important. For example, they can all be told separately on different days, and all told that each of the others were told, and if we assume perfect communication then there is common knowledge shared between them that at least one dragon has green eyes and they all know they all know, but if they don't all know when the last dragon was told and that each of the others know as well, how can any of them know when to expect the other 99 to change?

I don't like this definition, because there is nothing about the problem that says every dragon must have green eyes. It should be "there are a bunch of dragons on an island, n of which have green eyes", not "there are n dragons on and island and they all have green eyes". This thread is the only time I've seen the problem written the second way.

Clearly if there are 200 dragons on the island then saying there aren't 110 with green eyes isn't trivial by your definition.

QFT

If I'm right on this, it's another reason the email scenario doesn't work. Even if we assume everyone read it (perfect communication), none of them know when each of the others read it. Maybe some of them only check email every other day, or once a week. So what happens? Do the dragons who read the message on the day it was sent realize after day 106, and the other 99 haven't changed, that they all have green eyes? If not, how about after day 110? What about the dragons who read it three days after it was sent? Surely if nobody has changed after day 200 the island explodes.

The problem with the problem is that dragon 1 already knows that dragons 2-100 know that there is at least one green-eyed dragon. Dragon 1 also knows that dragons 2-100 also know that he (and all of the rest of the dragons) knows (know) that there is at least one green-eyed dragon. This is the same for dragon-x.

If we make the assumptions inherent to the common knowledge part of the thought problem, they ought to have all turned into sparrows long ago unless we add a condition that they are infinitely unobservant until the visitor shows up.

This is not a problem with the problem; this is what makes it a great problem.

Edit: Keep trying, you'll get it.

Yeah sorry....I meant saying less than/not exactly n dragons have green eyes where n is greater than the number of dragons is trivial and doesn't change anything.

What happens if everyone knows 1 particular dragon is illogical and unpredictable?

It depends on exactly what the stranger says and what color the illogical dragon's eyes are. Sometimes all the other green eyed dragons will change, sometimes none of them will.

I understand the common knowledge problem. I won't claim that I independently derived the proper answer, but it is child's play once you have heard it.

As stated in this specific problem, the common knowledge was already there. BrianTheDragon already knows that lastcardcharlieDragon (and every other dragon) knows that lastcardcharlieDragon both knows that there is at least one green-eyed dragon. BrianTheDragon (and every other dragon) already knows that lastcardcharlieDragon (and every other dragon) knows that BrianTheDragon (and every other dragon) knows that there is at least one green-eyed dragon.

Just because someone wants something to make a common knowledge problem, doesn't mean that they did make a common knowledge problem.

Maybe I am not following your line of thinking. Are you saying that the dragons should have turned into sparrows long before the human said anything?

Consider the very simplest case of two green eyed dragons. They would live forever on the island without turning into sparrows, right? There is no way that either (both) could logically conclude that he has green eyes. Of course, each thinks that he could have blue eyes -- seeing that the other dragon has green eyes tells each nothing about their own eye color.

The minute a human says to them (in each other's presence) that at least one green eyed dragon lives on the island, each (both) can use the common knowledge cascade logic to infer that he himself (call it me) has green eyes -- after waiting till past midnight and seeing that the other dragon did not change into a sparrow, which would have been the case if I had had blue eyes.

Again, sorry if I am misinterpreting what you are saying.

No. Before the announcement, the dragons have 99th order knowledge.

http://en.wikipedia.org/wiki/Common_knowledge_(logic)

How is the OP not a special case (k = the number of people on the island) of the more general problem in the wiki?

I don't know if it was linked, but Terry Tao had a blog on this topic a while ago:http://terrytao.wordpress.com/2008/0...anders-puzzle/

Yes. Before the announcement:

Dragon1 knows that dragons 2-100 have common knowledge that there are at least 98 green-eyed dragons.

Dragon1 knows that dragonX has common knowledge that all other dragons have common knowledge there are at least 97 green-eyed dragons.

So, dragon1 knows that it is common knowledge that there are at least 97 green-eyed dragons. He and the other dragons ought to have turned into sparrows 100 days after their first get together.

It is the same. It has a presumption that the people (and dragons) have no capability of figuring out what common knowledge exists in the others without being explicitly told. That is unreasonable unless we are to presume that the people/dragons are perfectly logical yet incredibly stupid in figuring out what other people/dragons know.

No. Just work it out. After any number of days all any dragon can say is "I might have green eyes or I might not have green eyes".


Again, no. Before the announcement there is no way for a dragon to determine his own eye color by observing other dragons. If we don't get that its not worth even talking about common knowledge because the mistakes are far more basic.

What definition of "common knowledge" are you using, then? It obviously isn't this one:

Since you are using a different definition of "common knowledge", your following remark does not apply.

Maybe the number 100 it too big for you (not enough fingers and toes?). Or maybe you don't yet realize that 98 is not 97 or that "ad infinitum" doesn't mean you can just stop when you want to. I don't know which it is.

Just do this logic with two dragons. What is the statement P that both dragons know and that both dragons know that the other dragon knows?

No, you are wrong. You are not understanding the concept of common knowledge. It is not simply that two people believe the same thing, it is far deeper than that. It is the infinite "I know that you know that I know that ...".

One more example may help convince you. Go to the three dragon example (since, clearly, you did not learn from the simple two dragon example). Call them A, B, and C (you are A). A, B, and C are sitting around a table having lunch. Each can clearly see the eye color of the others.

You see that B and C both have green eyes (you don't know your own eye color). So, logically, you infer that both B and C either each see two other dragons with green eyes (in the case you have green eyes) or each see one other dragon with green eyes and one other dragon with blue eyes.

So, to your point above, each of B and C knows that there is at least one green eyed dragon on the island. They share this knowledge. But it is not COMMON knowledge in the technical sense of the term (maybe you think "common" knowledge means the same as shared knowledge; it does not).

Okay, continuing the story. Based upon initial sightings, no green-eyed dragon is 100% of their own eye color, so none of them turn into a sparrow the first midnight.

What about on day two. Again, the three dragons sit around the table and have lunch. They see that no dragon has turned into a sparrow. At midnight of the second night does any dragon turn into a sparrow?

Consider B's thinking (from A's point of view). He knows that C is green, but does not know his own (B) color. Suppose he (B) is blue. If I (A) am blue, then C is the only green. But so what -- C would not magically turn into a sparrow. There is nothing that precludes the eye colors from being blue, blue, green (from C's perspective). So B cannot infer anything about C not turning into a sparrow on the second midnight, and A cannot infer anything about B not turning into a sparrow on the third midnight.

Anyway, I don't know why the two dragon example is not persuasive. Both dragons KNOW that that there is at least one green eyed dragon. Yet they never turn into sparrows, clearly, in this case. Shared knowledge is not the same as common knowledge.

If you can convince yourself that both green-eyed dragons on an isolated island would turn into sparrows on the second day, please elucidate.

You aren't getting my criticism of the problem.

There is a limiting factor that they ought know to turn into sparrows on the Kth day after being told that there is at least one green-eyed dragon.

They only get to the Kth day post-some-random-nonautistic-person-mentioning-that-common-knowledge-exists if they are infinitely unobservant. They ought to have turned into sparrows K days after their first get-together.

I think the last possibility can safely be eliminated.

Okay, I apologize for not understanding your criticism. Now I think I do.

The standard solution to the problem when there are N green-eyed dragons on the island is that they will all turn into sparrows on the Nth night after the human's announcement.

Instead, you are saying that all of the dragons (or some subset?) will turn into sparrows on the Kth night after the human's announcement, where K<N.

Could you show us an example where this is true? How about the simplest cases where N=2 or N=3? These cases can be pretty simply solved by straightforward logic. If these cases are "too small" for some reason, I'm open to any other case you'd like to present.

Thanks much.