# Common knowledge of “there exist at least”

Suppose there are 100 blue-eyed agents. Let it be common knowledge that all agents are rational. No agent knows the color of his own eyes. They all see each other's eyes and they all see each other seeing each other's eyes.

Is it correct to state that it is common knowledge that at least 98 agents have blue eyes?

Or is there some subtlety of epistemic modal logic that should have prevented me claiming i) this and/or ii) that this is equivalent to the public announcement that at least 98 agents have blue eyes?

Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum.1

1: Osborne, Martin J., and Ariel Rubinstein. A Course in Game Theory. Cambridge, MA: MIT, 1994. Print.

(source)

Everyone sees 99 agents with blue eyes. Therefore, everyone knows that there are at least 99 agents with blue eyes. However, does everyone know that everyone knows this?

Suppose one of the agents (A) wants to know what any other agent B knows. He knows that every other agent knows that there are at least 98 agents (all 99 that are seen by A except B himself) with blue eyes.

Now suppose that A wants to know what B knows about what a third agent C knows. A knows that B sees at least 98 agents that have blue eyes, of which one is C. Therefore, A knows that B is sure that C knows that at least 97 agents (the 100 except A, B and C) have blue eyes.
We cannot count A because in a worst-case for A there might actually be only 99 blue-eyed agents (everyone except A), which means that for B the number of blue-eyed agents is 99 or 98 (100, without A, and with or without himself). Then if B wants to know what C knows in this worst-case scenario he has to subtract C as well, which makes the number 98 or 97. Therefore A is not sure if B is sure that C knows that there are 98 agents with blue eyes.

We can go a level deeper, in which we will see that A can only be sure that B is sure that C is sure that D is sure about there being at least 96 blue-eyed agents.

Because common knowledge is "ad infinitum", and we need to subtract one from 100 for every level we go deeper, because we can never count the agents in this chain A, B, C, ..., there is no common knowledge except that there are at least 0 blue-eyed agents.