# How to interpret the intersected information partition element of different actors?

In this paper, it said that given $$\omega$$ is the current state of the world and $$P_1(\omega)$$ is the unique element from the information partition $$\textit{P_1}$$ of actor 1 in which the player 1 is informed that $$P_1(\omega)$$ contains $$\omega$$. According to the aforementioned paper (p.1237), it says that "To say that 1 knows that 2 knows (Event) E means that E includes all $$P_2$$ in the information partition $$\textit{P_2}$$ that intersect $$P_1(\omega)$$." I wonder why is the aforementioned intersection in this passage so important, since intersection here only mean that two information partition elements share some states.

I think there are two thing I need to understand: The interpretation of the shared states of two information partition elements from different people; the interpretation of the case when two informational partition elements intersect.

Also why is it seem to be inapplicable for the case when the partition elements mentioned are partially in $$E$$

I'm not an expert in epistemic game theory but let me see if I can help with an example.

Suppose the state space is $$\Omega = \{1,2,3,4,5\}$$, and there are two players with information partitions $$\mathcal{P}_1 = \left\{\{1,2,3\},\{4,5\} \right\} \\ \mathcal{P}_2 = \left\{\{1,2\},\{3,4\},\{5\}\right\}$$ Suppose the true state is $$\omega = 5$$. Let us consider the event $$E = \{4,5\}$$.

Player 1 "knows" $$E$$, since $$P_1 = \{4,5\} \subset E$$.

Player 2 "knows" $$E$$, since $$P_2 = \{5\} \subset E$$.

However, does Player 1 know that Player 2 knows $$E?$$ No.

What are the elements of $$\mathcal{P}_2$$ that intersect $$P_1$$? $$\{3,4\}$$ and $$\{5\}$$.

However, $$\{3,4\} \not \subset E$$, so Player 1 does not know that Player 2 knows $$E$$.

On the other hand, does Player 2 know that Player 1 knows $$E$$? Yes.

What are the elements of $$\mathcal{P}_1$$ that intersect $$P_2$$? $$\{4,5\}$$, and $$\{4,5\} \subset E$$, so Player 2 knows that Player 1 knows $$E$$.

• I understand the concept algebraically. However, I still do not understand intuitively why (in your example) {3,4} also has to belong to E for Player 1 to know that Player 2 knows E. Also what about the case when a partition element is partially in $E$? – Aqqqq Nov 17 '19 at 14:50