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

Question

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 $E$

Walrasian Auctioneer · Answer

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 $E$.

Michael Greinecker · Answer

The intersection itself is not terribly interesting, what matters here is only that it is not empty- that it contains some state.
To know event $E$ means that one does not consider it possible that the event $E$ does not obtain and that $E$ does indeed obtain. Since the true state is always considered possible, that $i$ knows $E$ at $omega$ means that $mathcal{P}_i(omega)subseteq E$- all states considered possible by $i$ at $omega$ are in $E$. Now, that states at which $i$ knows $E$ are an event itself. Write $K_i(E)$ for this event. Then $$K_i(E)={omegainOmegamid mathcal{P}_i(omega)subseteq E}$$ is the set of states at which $i$ knows $E$. We are now looking at the event $$K_1big(K_2(E)big),$$ the event that $1$ knows that $2$ knows that $E$ holds. Now,
$$K_1big(K_2(E)big)={omegainOmegamid mathcal{P}_1(omega)subseteq K_2(E)}$$
$$={omegainOmegamid mathcal{P}_1(omega)subseteq K_2(E)}=big{omegainOmegamid mathcal{P}_1(omega)subseteq {omega'inOmegamid mathcal{P}_2(omega')subseteq E}big}$$
$$=big{omegainOmegamid text{ if }omega'inmathcal{P}_1(omega)text{ then } mathcal{P}_2(omega')subseteq Ebig}$$
$$=big{omegainOmegamid text{ if }omega'inmathcal{P}_1(omega), Pinmathcal{P_2}, omega'in Ptext{ then } Psubseteq Ebig}$$
$$=big{omegainOmegamid text{ if }omega'inmathcal{P}_1(omega)cap P, Pinmathcal{P_2}text{ then } Psubseteq Ebig}$$
$$=big{omegainOmegamid text{ if }mathcal{P}_1(omega)cap Pneqemptyset, Pinmathcal{P_2}text{ then } Psubseteq Ebig}.$$

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

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