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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 $E$

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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 $E$.

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  • $\begingroup$ 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 $E$? $\endgroup$ – Aqqqq Nov 17 '19 at 14:50

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