Features of the set of Bayes Correlated Equilibrium in a single-agent decision problem with uncertainty I would like your help to better understand the notion of Bayes Correlated Equilibrium (BCE) in a single-agent decision problem with uncertainty The notion of BCE is provided in this paper for a generic $N$-player game. 



Consider the following single-agent choice problem under uncertainty. 

Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{V})$. First, let nature draw a realisation $v$ of $V$ from $P_V$. 
Then, let the decision maker choose an action $y\in \mathcal{Y}$, with $\mathcal{Y}$ finite, without observing $v$. Upon the decision has been made, the decision maker gets a payoff $u(y,v)$.

Before choosing an action, the decision maker can receive some signal to refine her prior about the state of the world.



Now, I want to use Theorem 1 in Bergemann and Morris (2016) to characterise the set of optimal strategies under minimal assumptions on the amount of information that is processed by the decision maker. To do that, I introduce the notion of one-player BCE.

A one-player BCE of the game described is a probability distribution $P_{Y,V}\in \Delta(\mathcal{Y}\times \mathcal{V})$ such that:

1) $\forall v \in \mathcal{V}$
$$\sum_{y\in \mathcal{Y}}P_{Y,V}(y,v)=P_V(v)$$

2) $\forall y\in \mathcal{Y}$ and $\forall \tilde{y}\neq y$
$$
\sum_{v\in \mathcal{V}} [u(y,v)-u(\tilde{y},v)] \times P_{Y,V}(y,v)\geq 0
$$



Let $\mathcal{P}$ be the set of BCE. 
Let 
$$
\mathcal{Q}\equiv \{P_Y\in  \Delta(\mathcal{Y}): P_Y(y)\equiv \sum_{v\in \mathcal{V}}P_{Y,V}(y,v) \text{ }\forall y \in \mathcal{Y}, P_{Y,V}\in \mathcal{P}\}
$$



Questions:

1) Is $\mathcal{Q}$ convex? I guess it is because $\mathcal{P}$ is convex. Could you confirm?

2) Let $ \mathcal{B}\equiv \{b\in \mathbb{R}^{|\mathcal{Y}|}: b^Tb\leq 1\}$ be the unit ball in $\mathbb{R}^{|\mathcal{Y}|}$. Let us stack in a $|\mathcal{Y}|\times 1$ vector the image set of every probability distribution $P_{Y}\in \mathcal{Q}$. With some abuse of notation, let us still denote any such a vector by $P_{Y}\in \mathcal{Q}$. Does 
$$
\max_{P_Y\in \mathcal{Q}} b^T P_Y
$$
exists $\forall b\in \mathcal{B}$? I guess the answer to this question depends on the features of $\mathcal{Q}$. For example, if $\mathcal{Q}$ is compact, then the maximum above exists. Could you help?
Yes,  $Q$ is convex and closed precisely because $P$ is and Q is its image under a continuous function. Therefore, $Q$ is compact.

The function you are maximizing is continuous as well, so the maximum must exist.