Clarification on the definition of Bayes Correlated Equilibrium 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"?
I think the reason you're having difficulty is that your definition is actually not equivalent to Bergemann and Morris' definition of BCE, except under specific assumptions which include the ones you state regarding independence. However, in general, we do not wish to make these assumptions. (For example, assuming that $t_{-i}$ is independent of $a_i$ conditional on $\theta$ and $t_i$ means that the mediator cannot reveal any additional information about other players' types once they have revealed the state to a player. This seems unnecessarily restrictive.)

In the paper, a BCE induces a probability distribution $\Pr$ that satisfies

$$ \sum_{a_{-i},t_{-i},\theta}\Pr(a_i,a_{-i},t_i,t_{-i},\theta) u_i(a_i,a_{-i},\theta) \ge \sum_{a_{-i},t_{-i},\theta} \Pr(a_i,a_{-i},t_i,t_{-i},\theta) u_i(a^\prime_i,a_{-i},\theta) $$

for all $a_i^\prime \in A_i$. If we wish to express this inequality in terms of player $i$'s expected utility given their information, then we divide both sides by $\Pr(a_i,t_i)$. Similarly, from the inequality 

$$ \sum_{a_{-i},t_{-i},\theta}\Pr(a_{-i},t_{-i},\theta \vert a_i,t_i) u_i(a_i,a_{-i},\theta) \ge \sum_{a_{-i},t_{-i},\theta} \Pr(a_{-i},t_{-i},\theta \vert a_i, t_i) u_i(a^\prime_i,a_{-i},\theta), $$

we can multiply both sides by $\Pr(a_i,t_i)$ to recover the first expression above. To see how this squares with your calculation, notice that

$$ \begin{align*} \Pr(a_{-i}\vert a_i,t_i,t_{-i},\theta) \Pr(t_{-i}\vert a_i,t_i,\theta)\Pr(\theta\vert a_i,t_i) \Pr(a_i,t_i) &amp;= \Pr(a_{-i},t_{-i},\theta \vert a_i, t_i) \Pr(a_i,t_i) \\
&amp;= \Pr(a_i,a_{-i},t_i,t_{-i},\theta) \\
&amp;= \Pr(a_i,a_{-i}\vert t_i,t_{-i},\theta) \Pr(t_i,t_{-i} \vert \theta) \Pr(\theta)\\
&amp;= \sigma(a_i,a_{-i}\vert t_i,t_{-i},\theta) \pi(t_i,t_{-i} \vert \theta) \psi(\theta).
\end{align*} $$

In particular, we have that $$ \Pr(a_{-i}\vert a_i,t_i,t_{-i},\theta) \Pr(t_{-i}\vert a_i,t_i,\theta)\Pr(\theta\vert a_i,t_i) = \frac{\sigma(a_i,a_{-i}\vert t_i,t_{-i},\theta) \pi(t_i,t_{-i} \vert \theta) \psi(\theta)}{\Pr(a_i,t_i)}. $$ This makes it difficult to get the decomposition you want without making needlessly strong independence assumptions.

When $N=1$, there is only one player, so the equilibrium is (in some sense) degenerate. However, the difference between one-player BCE and one-player BNE is that the BCE set is larger, and includes any BNE. (Indeed, this is true even when $N&gt;1$.)

For your second question:


  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. 


Formally, a BCE is a probability distribution of actions for each realisation of the state and signals. This probability distribution induces a joint distribution over actions and states. Just integrate out $(t_i,t_{-i})$ in $\Pr(a_i,a_{-i},t_i,t_{-i},\theta)$ to get that joint distribution.

Finally, this equilibrium concept is called Bayes Correlated Equilibirum because it extends our definition of correlated equilibrium (defined by Aumann for complete information games) to games of incomplete information, in the same way that Bayes-Nash equilibrium extends Nash equilibrium.