I have a problem with the usage of a Belief operator $B_i$ in the derivation of a result on a common belief operator $CB$.
First of all, some basic definitions (where $i$ is an individual), that will be useful:
$E = \prod_i E_i$
$E_{-i} = \prod_{j \neq i} E_i$
$B(E) = \prod_i B_i (E_{-i})$
$B^k (E) = B ( B^{k-1} (E)) \text{ with }k>1$
$CB (E) = \bigcap_{k \geq 1} B^k (E)$
Here there is my problem.
$$\begin{align} CB (E) = \bigcap_{k \geq 1} B^k (E) & = B(E) \cap \bigcap _{k\geq2} B^k (E)=\\ & = \prod_i B_i (E_{-i}) \cap \bigcap_{k \geq 2} B^k (E) = \\ & = \prod_i B_i (E_{-i}) \cap \bigcap_{k \geq 2} B (B^{k-1} (E)) = \\ & = \prod_i B_i (E_{-i}) \cap \bigcap_{k \geq 2} \prod_i B_i (B_{-i}^{k-1} (E)) = \\ & = \prod_i \left( B_i (E_{-i}) \cap \bigcap _{k \geq 2} B_i (B_{-i}^{k-1} (E)) \right) \equiv \prod_i CB_i (E) \end{align}$$
My problem is with what the last step does not make explicit. My guess is that this is the following
$$\begin{align} \prod_i \left( B_i (E_{-i}) \cap \bigcap _{k \geq 2} B_i (B_{-i}^{k-1} (E)) \right) & = \prod_i \left( B_i (E_{-i}) \cap \bigcap _{k \geq 2} B_{i}^k (E) \right) = \\ & = \prod_i \bigcap_{k \geq 1} B^{k}_{i} (E_{-i}) \end{align}$$
However, assuming my guess is correct, there are two things I don't see completely:
1) the relation between $B$ and $B^{k-1}$ in presence of an index $i$;
2) the logic behind the presence of both $E$ and $E_{-i}$, and how they end up in $\prod_i CB_i (E)$.
Thanks a lot for any feedback.
It looks to me like the last step is definitional (hence the $\equiv$ symbol); in other words, the formula inside the parentheses is the definition of $CB_{i}(E)$.
Of course, this still leaves open the question of what the heck $B_{-i}^{k-1}(E)$ is. Going by the equality between the 3rd and 4th line, it seems like it is just an abbreviation for $$(B^{k-1}(E))_{-i} = \prod_{j \neq i} B_{j}(B_{-i}^{k-2}(E)).$$ The real source of confusion here then might actually be the definition of $B(E)$. Usually this is read "everyone believes $E$", but the formulation you provided says something apparently weaker, namely that each agent $i$ believes $E_{-i}$. Why is this? I'm not sure—I'd need to look at the paper you took this from—but my guess is that $E = \prod_{i} E_{i}$ is assumed to be such that each $E_{i}$ is something that $i$ is sure of (i.e. something that satisfies $B_{i}(E_{i}) \leftrightarrow E_{i}$); for example, $E_{i}$ could be an action that $i$ is choosing to take, or (assuming introspection) a belief that $i$ has about some other event.