Set of Bayes Correlated Equilibria when complete information is not available

Question

Model

Consider a game where a decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world.

The state of the world has support $\mathcal{V}$.

When DM chooses action $y\in \mathcal{Y}$ and the state of the world is $v\in \mathcal{V}$, she receives the payoff $u(y,v)$.

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.

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Question

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

2) $\sum_{v\in \mathcal{V}}u(y,v) P_{Y,V}(y,v)\geq \sum_{v\in \mathcal{V}}u(k,v) P_{Y,V}(y,v)$ for each $y$ and $k\neq y$.

Theorem 1 in Bergemann and Morris (2016) claims that the set of 1-player Bayesian Correlated Equilibrium is equal to set of optimal behaviours under a range of possible information structures (this result holds in fact for any n-player game, thus also for $n=1$ as in this case).

Such information structures can go from the degenerate information structure (i.e., no information whatsoever on that state of the world and, hence, prior is equal to posterior) to the complete information structure (i.e., full revelation of the state of the world).

My question is: can we characterise the collection of 1-player Bayesian Correlated Equilibria for the model above under the assumption that the complete information structure is not available to agents (i.e., agents are not able discover the exact value of the state)? If yes, how? I believe it should amount to inserting a third constraint in the definition above but I can't see which one.

Does Theorem 1 in Bergemann and Morris (2016) still hold in that case?

Answer 1

You can certainly do it. However, keep in mind that the BCE will not be a lot "smaller". This is because there are many information structures that are almost perfectly informative, or that fully reveal some states but not all of them. Therefore, by disallowing the complete information structure you are simply removing one element in the boundary of the set of BCE's. Keep also in mind that this set has, in general, infinitely many elements. Since it is a convex set, any convex combination of two elements in it is also in the set, for example.

Going back to your question, there are many ways to rule out complete information structures. The one that comes to mind stems from the fact that a fully informative signal will correspond to a BCE where for each $y\in Y$, the support of $P_{V|Y}$ is a singleton. Notice that if this is the case, learning what action you should take implies learning the state of the world.

Therefore, the extra constraint would be that $P_{Y,V}$ must satisfy that there exists a $y^*\in Y$ such that the cardinality of the support of the conditional probability of the states given that action is bigger than 1, i.e. $|P_{V|y^*}|>1$. This is consistent with the idea that at least in some cases the DM remains uncertain about the state of the world after receiving the recommendation to choose action $y^*$.

Notice that a signal that fully reveals all but one state of the world, and instead induces a posterior belief for such a state of $99.999\%$ will not be ruled out by the extra constraint, so including the constraint will have very little effect.