I know that he knows that I know that he knows

Question

Not sure if this is the right place for this question but here goes...

Some time in the past I've stumbled upon some site/article/post/etc that discussed the issue of 2 parties having (or not) some information and being aware of the other party knowing it. Something like:

A: knows X

B: knows that A knows X

A: knows that B knows that A knows X

and so on...

Now in the source I've seen there was a statement that with enough of those "cycles" (like 4 or 5 maybe?) both parties have like a perfect knowledge of the situation - they know and they're aware that the other party knows and so on... So this doesn't (need to) go infinitely, at some point everyone is aware of everything. Apparently, it's a known problem and there's already been some work/reasoning devoted to it.

Unfortunately, I wasn't able to find anything on the internet that concerned this topic and I'm curious. So my question is: has anyone ever heard about something like this? Does this problem have an established name? Can you point me to any articles/sources of information? I'd appreciate it.

Answer 1

I don't know of any articles on the internet, your question was the first one to show up when I googled "he knows that I know that he knows that I know". I just thought about this and found it interesting, but my brain doesn't seem to be able to handle it when I get beyond 4 cycles. It is interesting to know that it does not go on forever and that you eventually get perfect information within a few cycles! I suspected this but could not focus enough to confirm it in my mind. I want to find out at what point perfect information is achieved!

Sidenote: This reminds me of that riddle where 30 green eyed prisoner are trapped in a prison with no reflective surfaces and where they can see each other but not communicate. And each day they get a chance to leave but if they don't have green eyes they will get shot. But if you tell them something that they all already know - that at least one of them has green eyes - they will eventually figure it out and realize they all have green eyes. The crux is that though they all knew it, they did not know that everybody else knew that everybody else knew that everybody else knew, and so on until the last person, knew that there was at least one green eyed person.

Answer 2

If B knows X, and tells A “I know X”, then the complete stack of information is created instantaneous.

If the knowledge gets revealed bit by but, that’s different

Say B knows X. B tells C, and C tells B. Now B knows X, A knows that B knows, but B doesn’t know that A knows. If A told B directly, then everyone would know everything again.

But A talks in his sleep and that’s how B finds out - now B knows X, A knows that B knows X, B knows that A knows that B knows. So every time only partial information is handed over, the knowledge grows a little bit.