Why is infinite recursion on the common knowledge assumption necessary? If something is common knowledge in a game, that means that every player knows it, and every player knows that every player knows it, and so on. Are there cases where only one such level of knowing that others know is necessary? 

What happens if such 'knowledge of knowing' stops at a certain level? What breaks down if at some level, the 'knowledge of knowing' doesn't hold? E.g., I know that you know, but you don't know that I know that you know. 

On the same note, I'd like to understand how this infinite recursion (for lack of a better term...) of knowledge figures mathematically into say, equilibrium concepts. 

Thanks in advance!
Ariel Rubinstein has a fabulous paper in which he illustrates that Common Knowledge and Almost Common Knowledge are very different!

http://www.cs.cornell.edu/courses/cs6764/2018fa/The_Electronic_Mail_Game.pdf