I am trying to understand the formalism of common knowledge. The sense behind the concept is already well explained here, however I struggle with linking the formalism to a practical use case.
When reading the definitions it first appears straight forward: We have a set of states $S$, an event $E\subseteq S$ as a subset of these states and a partition $P_i=\{\{\ldots\},\{\ldots\},\ldots,\{\ldots\}\}$ of $S$ representing the knowledge of a decision maker $i$ in a state. Here I am not quite clear how I should practically understand the partition's elements (i.e. these subsets of $S$). The puzzle pieces seem to fall into place when reading on: In state $s\in S$, decision maker $i$ knows that one of the states in $P_i(s)$ occurs, but he doesn't know which one. Here $P_i(s)$ is the unique element in $P_i$ that contains $s$.
The knowledge function $K_i(e)=\{s\in S|P_i(s)\subset e\}$ is the set of states (a subset of the event $e$), where the decision maker knows that event $e$ occurs. In other literature, instead of the proper subset, the improper subset is also used: $K_i(e)=\{s\in S|P_i(s)\subseteq e\}$. The operator for the idea "everyone knows $e$" is defined by intersecting the knowledge of all decision makers $i$ as follows: $E(e)=\bigcap_{i}K_i(e)$. Iterating the function $E$ is understood as the well known function composition $E^1(e)=E(e)$ and $E^{n+1}=E(E^n(e))$. Finally the common knowledge function is given by:
$$C(e)=\bigcap_{n=1}^{\infty}E^n(e)=E^1(e)\cap E^2(e)\cap E^3(e)\ldots$$
So good so far, but when trying to applicate this formalism to a practical use case, I'm missing the right impetus. I found a very simple and illustrative example in this YT video "Is It Common Knowledge?" (by James Miller):
I would really appreciate it if someone can help me how to use the (admittedly simple) formalism to describe this use case.
My rough ideas are:
- I have to define the three "decision makers", the boy ($i=1$), the robot ($i=2$) and the girl ($i=3$). This is our starting point.
- I have to define $S$ and the partitions $P_1,P_2,P_3$. Here I already have difficulties.
- For our boy, robot and girl we have to define the knowledge functions $K_1,K_2,K_3$.
- To describe that "$Y=3$" is not common knowledge, we have to intersect $K_1(e)\cap K_2(e)\cap K_3(e)$, use the iteration $K_1(K_2(K_3(e)))$ and come up with an result that is an empty set $\emptyset$.
I would be grateful if one could help me to put the puzzle pieces together.
