Knowing that they know that you know that they know

Question

Suppose there are two spies, A and B. There is also a secret, s. The following situation unfolds:

(1) A learns s.

(2) B learns that A knows s.

(3) A learns that B knows that A knows s.

(4) B learns that A knows that B knows that A knows s.

(5) A learns that B knows that A knows that B knows that A knows s.

(6) This situation goes on forever, with both spies finding more and more information about each other but accomplishing absolutely nothing practical.

I know similar questions have been asked before (and I guess now you know that I know), but these are the specific things I'm wondering about this situation.

• At some point, the cycle of knowing that someone else knows that you know something, etc., becomes unintuitive. What in the world does it mean, in common sense terms, to "know that they know that you know that they know, etc."?
• Is there some point at which adding another layer of "knowledge" ceases to affect that actual knowledge of either party?
• Is it possible to traverse infinite steps of this knowledge cycle in a finite amount of time? Since gaining a piece of knowledge requires conscious thought, and conscious thought requires time, it would seem not.
• There is the case where A and B are the same person. It seems that in this case, this person would have to take a sequence of steps: first knowing s, then knowing they know s, then knowing they know they know s. Again, in this case, each piece of knowledge requires a conscious thought that takes time. This case is interesting because it takes higher order thinking to know that you know something. For example, a cat may know what a window looks like, but has it ever pondered the fact that it knows what a window looks like? Is there some point in the self-knowledge of knowledge that humans are rationally incapable of reaching?

Answer 1

The SEP entry on common knowledge is about exactly these issues. It doesn't seem to discuss self-knowledge, but one of its references does seem to: "Backward Induction and Beliefs About Oneself" by Michael Bacharach.

"Common knowledge" is the name of the infinite (or cyclic) form of mutual knowledge, and it seems most popular to believe that it is an attainable state of knowledge—e.g., if Alice announces a fact to Bob in a face-to-face conversation.

On the other hand, it is provably impossible to achieve this state over an unreliable communication channel—this is the two generals problem. One could argue that all channels are unreliable to an extent, even face-to-face conversation.

You can contrive examples where arbitrarily many levels of "I know you know" nesting lead to observable differences in behavior if everyone is perfectly rational. A nice example is the blue-eyed islanders paradox, whose resolution hinges on the fact that the islanders know (other islanders know)ⁿ something, but don't know (other islanders know)ⁿ⁺¹ that thing, and n can be chosen arbitrarily.

Answer 2

Time and Human Limits on Working Memory are the limits

What you are talking about is knowing about knowing. There is no logical limit (as you can go on forever with that train of thought), but there is a practical limit, as eventually you will run out of time to think about that (and/or forget what number you are on). You can repeat that sequence over and over without end, but the only limit is your memory and time. Time because we are not immortal, and thus we die, and thus we do not have infinite time, and thus our time is finite, and thus we cannot think an infinite set. If you are doing this in your head, then the limit is how many words you can keep in your head. For instance, I am confident that no human on Earth can fit 100000000^10000000000 words in their head. Thus, you cannot go on with this train of thought forever.