Common Knowledge and Recursion

There is a huge discontinuity in epistemology and capabilities when you go from n-1 to n perfect logicians in certain environments. With one mind missing, the recursive ladder of “I know that you know…” collapses and knowledge fizzles. But with all n, common knowledge ignites: the recursion sustains itself, and collective action becomes inevitable. The green eyes puzzle captures this perfectly:

“On an island of 100 perfect logicians, some people have green eyes but no one knows their own eye color. They can see others’ eyes but can’t communicate. A rule says anyone who knows they have green eyes must leave at midnight. A visitor mentions “At least one of you has green eyes.” From that one statement, everyone deduces on exactly the 100th midnight that they themselves have green eyes. How?”

If everyone knows that everyone is a perfect logician, the single statement “At least one of you has green eyes” creates common knowledge (everyone knows, everyone knows that everyone knows, everyone knows that everyone knows that everyone knows, …). It is as though the nth node of a neural network allows it to leap pass some critical threshold. If there is any gap in everyone knowing everyone is a perfect logician, everyone is stuck on the island forever.