Consider a game where a decision maker (DM) has to choose action $l\in \mathcal{Y}$ possibly without being fully aware of the state of the world $V$.
The choice set $\mathcal{Y}$ has cardinality $L$. The state of the world is an $L\times 1 $ vector and we denote its $l$-th element by $V_l$.
The state of the world has support $\mathcal{V}$.
When DM chooses action $l\in \mathcal{Y}$, she receives the payoff $V_l$. That is, she receives a payoff equal to the $l$-th element of the vector $V$.
Let $P_V\in \Delta(\mathcal{V})$ be the DM's prior.
The DM may also process some signal (formalised by the concept of information structure) to refine his prior and get a posterior.
Let us define the concept of 1 player Bayesian Correlated Equilibrium provided in Bergemann and Morris (2013,2016,etc.).
$P_{Y,V}\in \Delta(\mathcal{Y}\times \mathcal{V})$ is a 1 player Bayesian Correlated Equilibrium if
1) $\sum_{l\in \mathcal{Y}}P_{Y,V}(l,v)=P_V(v)$ for each $v\in \mathcal{V}$
2) $\sum_{v\in \mathcal{V}}V_l P_{Y,V}(l,v)\geq \sum_{v\in \mathcal{V}}V_k P_{Y,V}(l,v)$ for each $l$ and $k\neq y$.
Bergemann and Morris show that the set of Bayesian Correlated Equilibrium is equal to set of optimal optimal behaviours under a range of information structures.
QUESTION: Consider a search model, in which we design a protocol according to which the DM discovers information about the state of the world.
For example, suppose that the DM engages in a sequential search where he discovers the $l$-th element of $V$ if and only if the maximum utility secured up to that moment is lower than a reservation value.
Can this model always be written as prior/information structure/posterior? In other words, does the framework of Bergemann and Morris nest search models?