Ugh. After all that, I think I haven't sufficiently broken the logic of the situation to make things fail to move forward.

I guess I still don't see the issue...."Every dragon can see green eyes" means "At least 2 dragons have green eyes". If the statement is made fully public there can't be a difference in the reactions of the dragons to either statement.

Yeah. It didn't work. The entire logic is eliminative, so extra information does not cause any problems for the logic.

I don't think I can break the problem without breaking the symmetry of knowledge. Someone has to know something that someone doesn't in order for this to fail.

Here comes another try: "Every green-eyed dragon can see a green-eyed dragon."

This statement does not appear to allow a "the dragons to know that all the other dragons know that all the other dragons know..." cascade because the statement doesn't logically preclude the possibility that all the dragons have blue eyes.

Maybe.

If there are two GED, they will both see green eyes and know the other must see GE too, so both instantly know. If there are three GED, and D1 imagines D2 sees the GE of D3... and they each imagine this, I guess that stops the chain?

Although the statement implies there is at least one GED, so maybe it does cause a logic chain reaction in the same way as the original statement unless preceded by, "If there are any GED."

No, I think I still fail. Suppose there are 3 dragons.

* If one sees 2 GEDs, then nothing can be concluded immediately.
* If one sees a GED and a BED, he would immediately know that he has GE and would be gone after the first night.
* If one sees 2 BEDs, then the dragon would know he is a BED because he can't see a GED.

Since all dragons see 2 GEDs, they're all around on the second day, and they know that the other dragons did not see a GED and a BED. Since each dragon knows that the other dragons see at least one GED, he knows that they could not have seen 2 BEDs.

So even though there are more logical possibilities when everyone is still around on the second day, the fact that the dragon can see a GED still causes the logic to move forward.

"Every green-eyed dragon can see a dragon that can see a green-eyed dragon."

At two dragons, nobody knows anything.

This is of no help to anyone because it's always true regardless of the distribution of dragons.

"Every green-eyed dragon can see a green-eyed dragon that can see a green-eyed dragon."

At two dragons, both dragons know that they have green eyes.

But at three dragons, I think this stops.

* If you see 2 GEDs, you can't conclude anything.
* If you see 1 GED and 1 BED, you can't conclude anything.
* If you see 0 GEDs, you can't conclude anything.

This means that when everyone shows up on the next day, we cannot exclude any combinations of that the other dragons saw.

Well, if I've learned anything it's that it's best to watch what you say around dragons.

Why can't you conclude you have green eyes?

Well that phrase is just silly anyway. It's like saying any GED can see someone who can see him too. Yeah, that doesn't tell them anything new at all I hope.

I'm all about silly though, don't get me wrong. My eyes are seeing green right now, if you know what I mean, eh.

What am I missing? It seems that that statement rules out the exactly 1 green eyed dragon, right? Every green eyed dragon can see another green eyed dragon. So any dragon that sees one green and one blue knows that they must have green. So the logic cascades and they all correctly deduce that they have green eyes.

Maybe I am totally misinterpreting what you are doing here.

P.S. Coke to dessin!

For more fun, start with either the original situation or the three dragon situation. What if the informer, instead of addressing the entire crowd, texted each dragon privately with the same message "at least one dragon has green eyes" , without each dragon knowing the others got the same text? What happens? Nothing changes I think.

If you add to the text message “all dragons received and read this same message”, what happens? Eventually all dragons will change to sparrows.

I know you are just having fun, but the issue of trying to generate common knowledge among several people via messaging is actually a well-known issue in communication theory. Actually nothing will happen in your second case since you have not created common knowledge.

Awwwwwww crap.

If the dragons don't know that the other dragons know something, it doesn't really move forward.

Let's just look at two dragons.

Day 0: Every dragon is informed that there is a green-eyed dragon, but they don't know that they other dragons know. Each dragon can see a green-eyed dragon.

Day 1: Dragon A cannot conclude anything about the fact that Dragon B is still on the island. As far as Dragon A knows, nothing is different for Dragon B. And there's nothing inconsistent about the statement and the fact that he sees a GED.

I don't see why.

Just to be super clear, for the second case I proposed a text message that says "at least one dragon has green eyes. All dragons received and read this same message” Wouldn't that be the equivalent to the original post case where it was announced to a live full crowd "at least one dragon has green eyes?"

Oh... LOL... my brain must be fried this weekend. I should just stop posting in this thread for a few days.

No. Think about it.

Yeah, that doesn't look to be any different than saying it in front of everybody.

Common knowledge is harder to achieve than you think. There is no way that a SENDER of a message can know (and convey with 100% certainty) that multiple RECEIVERS have (will -- in the future) received a message.

Believe me, this issue has been the topic of a great deal of study in communication theory.

That's a good point, but don't they have email receipts for just this, or the iphone message that shows "read" after the recipient opens it? Anyway, it's kind of beside the point, as it was brought up earlier there's really no way to know a group of 100 dragons all heard the message either, because everyone knows at least one dragon will have blasted a huge stanky one and distracted the others. So the problem just assumes perfect communication for the sake of understanding the logic.

This isn't any different than the original problem.

I don't know if you guys are joking or just not thinking about this.

If you cannot see the difference between communication among people all in the same room and separate emails sent to these same people, then you don't have any idea what the concept of common knowledge is all about.

For the third time, this is a well-known topic of study in communication theory. I studied this at Stanford when the concept of common knowledge was fairly new (Bob Aumann, John Geanakoplos). It is really not hard to understand if you try.

I can see a difference, but not much if there are email receipts showing they were all read. Then you just have to trust they were all actually read, which is no different than trusting everyone in the room was paying attention. I guess you can argue not every recipient can know every other recipient read the email, even though they know it was sent to all.