I would like your help to understand the definition of Bayes Correlated Equilibrium (BCE) in an incomplete information game at p.7 of this paper.
Let me summarise the definition provided in the paper.
There are $N\in \mathbb{N}$ players, with $i$ denoting a generic player.
There is a finite set of states $\Theta$, with $\theta$ denoting a generic state.
A basic game $G$ consists of
for each player $i$, a finite set of actions $A_i$, where we write $A\equiv A_1\times A_2\times ... \times A_N$, and a utility function $u_i: A\times \Theta \rightarrow \mathbb{R}$.
a full support prior $\psi\in \Delta(\Theta)$.
An information structure $S$ consists of
for each player $i$, a finite set of signals $T_i$, where we write $T\equiv T_1\times T_2\times ... \times T_N$.
a signal distribution $\pi: \Theta \rightarrow \Delta(T)$.
A decision rule of the incomplete information game $(G,S)$ is a mapping $$ \sigma: T\times \Theta\rightarrow \Delta(A) $$
Definition of BCE: The decision rule $\sigma$ is a BCE for the game $(G,S)$ if, for each $i=1,...,N$, $t_i\in T_i$, and $a_i\in A_i$, we have $$ \sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i}|t_i, \theta) \sigma(a_{-i}| a_i,t_i, t_{-i}, \theta) u_i(a_i, a_{-i},\theta) $$ $$ \geq \sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i}|t_i, \theta) \sigma(a_{-i}| a_i,t_i, t_{-i}, \theta) u_i(\tilde{a}_i, a_{-i},\theta) $$ $\forall \tilde{a}_i\in A_i$.
Question:
1) I don't follow how the conditional expectation is computed in the definition. Consider for example the left hand side (LHS) and let me go through each step by using the notation $Pr(\cdot)$ to generically denote any probability distribution.
The LHS is the expected payoff of player $i$ where the expectation is computed wrto anything that is unknown to him conditional on what he knows. Hence, $$ \sum_{a_{-i}, \theta, t_{-i}} u_i(a_{-i},a_i, \theta) \times Pr(a_{-i}, \theta, t_{-i}| t_i, a_i)= $$ $$ \sum_{a_{-i}, \theta, t_{-i}} u_i(a_{-i},a_i, \theta) \times Pr(a_{-i}| \theta, t_{-i}, t_i, a_i)\times Pr(\theta, t_{-i}| t_i, a_i)= $$ $$ \sum_{a_{-i}, \theta, t_{-i}} u_i(a_{-i},a_i, \theta) \times \underbrace{Pr(a_{-i}| \theta, t_{-i}, t_i, a_i)}_{\equiv \sigma(a_{-i}| a_i,t_i, t_{-i}, \theta) \text{ [OK!]}}\times \underbrace{Pr(t_{-i}|\theta, t_i, a_i)}_{\equiv \pi(t_{-i}|t_i, \theta)? \text{ Where is $a_i$?} }\times \underbrace{Pr(\theta| t_i, a_i)}_{\equiv \psi(\theta)?\text{ Where are $t_i, a_i$?}}= $$
Are we assuming
1) $t_{-i}\perp a_i $, conditional on $\theta, t_i$
2) $\theta \perp t_i, a_i$
?
2) How does the definition of BCE simplify to when $N=1$?
From reading at p.25 of the linked paper, it seems that a BCE is still intended as mapping from state and signal to a probability distribution over actions. From reading at p.26 of the linked paper, the authors then say "[...] In that case, the set of BCE correspond to joint distribution of actions and states [...]". I'm confused.
Also, when $N=1$, how does the definition of BCE differ from the definition of Bayesian Nash Equilibrium?
3) Just as a curiosity, what it is the reason of including the adjective "correlated"?