For $k = 2$, it is merely "first-order" knowledge. Each
blue-eyed person knows that there is someone with blue eyes, but each
blue eyed person does ''not'' know that the other blue-eyed person has
this same knowledge. (from http://en.wikipedia.org/wiki/Common_knowledge_(logic))
I am not getting this. If there are more than one people that have blue eyes, each person can see that there is a person with blue eyes and people that have green eyes can clearly see that there are people with blue eyes. So even before the common knowledge annoucement, isn't it natural to say that everyone knows that everyone knows there is at least one person with blue eyes – common knowledge?
How am I mistaken? I am not getting how announcement sets common knowledge – as it seems for me that there already is common knowledge.
Edit: OK, I get it for $k = 2$. But what about $k>2$? Then, everyone would be sure to know that everyone knows that there at least exists one person with blue eyes, right? Doesn't this already constitute as common knowledge?
Best Answer
Consider the two persons $A, B$ with blue eyes.
$A$ sees $B$, so she knows there is somebody with blue eyes. (And so does $B$.)
But $A$ has to consider the possibility that $B$ is the only person with blue eyes. (After all, $A$ does not know the colour of her own eyes.)
In this case (that is, if $B$ is the only person with blue eyes, a case $A$ cannot rule out), before the announcement, $B$ wouldn't know that there are people with blue eyes. So $A$ cannot be sure whether $B$ knows.
The same holds for $B$ concerning $A$.
So before the announcement $A$ and $B$ do not know whether everyone knows that there are blue-eyed people - they are precisely in doubt about what the other knows.