Bayes correlated equilibrium of Bergemann and Morris

Question

The paper of Bergemann and Morris proves a theorem based on some foundations about the information sets and their expansions. I am trying to understand theorem one intuition, more precisely I cite the appropriate parts that we need to know, that is:

$\textbf{Definition 4:}$ (Combination). The information structure $S^∗ = (T^∗, \pi^∗:\Theta\to\Delta(T_i^*))$ is a combination of information structures $S^1 = (T^1, \pi^1)$ and $S^2 = (T^2, \pi^2)$ if

$$T_i^*=T_i^1\times T_i^2,\quad\text{for each $i$}$$ and $$\sum_{t_i^2\in T_i^2}\pi^*(t_i^1,t_i^2|\theta)=\pi(t_i^1|\theta),\quad\text{for each $t_i^1\in T_i^1$ and $\theta\in\Theta$} $$

$$\sum_{t_i^1\in T_i^1}\pi^*(t_i^1,t_i^2|\theta)=\pi(t_i^2|\theta),\quad\text{for each $t_i^2\in T_i^2$ and $\theta\in\Theta$} $$

Note that the above definition places no restrictions on whether signals $t_i^1\in T_i^1$ and $t_i^2\in T_i^2$ are independent or correlated, conditional on $\theta$, under $\pi^∗$. Thus any pair of information structures $S^1$ and $S^2$ will have many combined information structures.

$\textbf{Definition 5:}$ (Expansion). An information structure $S^∗$ is an expansion of $S^1$ if $S^∗$ is a combination of $S^1$ and some other information structure $S^2$.

Suppose strategy profile $\beta$ was played in $(G, S^∗)$, where $S^∗$ is a combination of two information structures $S^1$ and $S^2$. Now, if the analyst did not observe the signals of the combined information structure $S^∗$, but only the signals of $S^1$, then the behavior under the strategy profile $\beta$ would induce a decision rule for $(G, S^1)$. Formally, the strategy profile $\beta$ for $(G, S^∗)$ induces the decision rule σ for $(G, S^1)$:

\begin{equation}\sigma(a|t_i^1,\theta):=\frac{\sum_{t_i^2\in T_i^2}\pi^*(t_i^1,t_i^2|\theta)\Pi_{j=1}^i\beta_{j}(a_j|t_i^1,t_i^2)}{\pi(t_i^1|\theta)}\end{equation} for each $a\in A$ whenever $\pi^1(t_i^1|\theta) > 0$.

Based on the above the authors give the following therem that makes a connection between the Bayesian Nash equilibrium and the Bayesian Correlated equilibrium, that is

$\textbf{Theorem 1:}$ A decision rule $\sigma$ is a Bayes correlated equilibrium of $(G, S)$ if and only if, for some expansion $S^∗$ of $S$, there is a Bayes Nash equilibrium of $(G, S^∗)$ that induces $\sigma$.

I have the following questions

$\textbf{Question 1:}$ They define the the rule $\sigma(a|t_i^1,\theta)$ in a way to make the connection between the sets of BNE and BCE and I can not understand why they did so?

$\textbf{Question 2:}$ What is the interpretation of $\sigma(a|t_i^1,\theta)$, in terms of porbability someone can say it is the Bayesian rule, however all these parameters as $\pi^*$, $\beta$ and $\pi(t_i^1|\theta)$ have some intuitive interpretation what is it?

$\textbf{Question 3:}$ Why it is so special this obedient rule $\sigma(a|t_i^1,\theta)$ for both the BNE and BCE and does it suffice to make the transition from the one set of solutuions to the other? I am a little confused because it seems like the two sets are somehow the one the subset of the other.

$\textbf{Question 4:}$ Are BNE and BCE directly or indirectly connected? If not, what is the purpose to do this here?

Answer 1

Suppose you are an analyst studying a Bayesian game. You know the players, the possible states of nature, the common prior, the action spaces, the payoff functions, and you know about some information channels available to the players, the latter given via some information structure. However, you can't rule out that the players have additional information channels available to them. This additional information can be modelled via expansions.

Now, you are sure that they play a Bayes-Nash equilibrium with respect to all information channels they have, only some of which you are aware of. Then, by Theorem 1 of the paper, the players will play a Bayes correlated equilibrium with respect to those information structures you as an analyst are aware of.

Now, you, the analyst, might be able to observe $\theta$, the signals $t_i$, and the chosen actions. From the joint distribution of these actions, you can derive the conditional distribution over action profiles. This conditional distribution is exactly the decision rule $\sigma$.