Robust predictions in single-agent decision problem with uncertainty

Question

I would like your help to better understand the possibility of using the notion of Bayes Correlated Equilibrium (BCE) in a single-agent decision problem with uncertainty to make predictions on optimal strategies that are robust to the belief environment. The notion of BCE is provided in this paper for a generic $N$-player game.

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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 $u(y,v)$.

For example, suppose that $\mathcal{Y}\equiv \{1,2,3\}$. $V$ is a $3\times 1$ random vector, $V\equiv (V_1,V_2,V_3)$. $P_V$ is the 3-variate standard normal distribution. $u(y,v)\equiv v_y$.

Before choosing an action, the decision maker can receive at least one signal to refine her prior (minimal amount of information). I assume that this minimal signal is completely uninformative (degenerate information structure).

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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 (degenerate in this case). To do that, I introduce the notion of one-player Bayesian Correlated Equilibrium (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 $\forall \tilde{y}\neq y$ $$ \sum_{v\in \mathcal{V}} (v_y-v_{\tilde{y}}) \times P_{Y,V}(y,v)\geq 0 $$

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Question:

1) Is it obvious that a BCE exists and is unique in my example?

2) Suppose now that I know for sure that the decision maker does not have any information in addition to the degenerate information structure. How would I characterise an optimal strategy in such a case (without the necessity of being robust to possibly richer belief environments)? How would that definition differ from the BCE definition given above?

Answer 1

1) Given that $\mathcal{Y}$ is finite, it is obvious that the BCE is not an empty set (or it exists). Remember that the set of BCE's is the set of decisions that are optimal for some belief about $\mathcal{V}$ that could be formed (as a posterior belief) after observing some signal and having a prior $P_V$.

To show there is at least one distribution satisfying the definition if BCE, you can consider an uninformative signal. Given that there is only a finite set of actions, one of them must maximize the expected utility given the prior. If there is more than one, randomize among them as you please. Let the optimal action be $P_Y^*$ (in this notation I capture both situations where the maximizer is unique and when there is more than one maximizer) and define $P_{Y,V}=P_Y^*P_V$, then such a distribution constitutes a BCE.

Now, you should not expect uniqueness because if there was only one distribution that constitutes a, or if BCE was a singleton (since we usually think of BCE as a set), then information is essentially irrelevant. It would mean that there is an action in $\mathcal{Y}$ that is the best action regardless of any extra information you might receive.

2) To characterize this, simply find the optimal strategy when the belief is equal to the prior, $P_V$. If you want to put it in terms of the two expressions you have, then $P_{Y,V}$ must be a product distribution, i.e $P_{Y,V}(y,v)=P_Y(y)\cdot P_V(v)$ for some $P_Y(y)\in\Delta(\mathcal{Y})$ (This will capture the fact that the player has no further information beyond the prior and so it cannot correlate his actions with the state of the world). The second inequality will pin down $P_Y$. In fact, using the notation from the previous bullet point, it should be equal to $P_Y^*$.