# Bayes correlated equilibrium of Bergemann and Morris

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?

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 $$\sigma$$.
• two questions 1. I think there is a theorem that says, any Nash equilibrium is a correlated equilibrium but it does not hold the opposite (if you have any details to check it somewhere feel free to give me a link), deos this mean that the same hold for a BNE and the BCE, namely by theorem $1$ Bergemann and Morris may want to do this parallelism right? 2. so the decision rule $\sigma: \Theta\times T\to \Delta(A)$ is behavioral strategy profile? Namely it gives a pdf on the set of the mixed action of the players and this is a correlated strategy as well? Oct 2 at 10:03