Exact definition of one-player Bayesian Correlated Equilibrium

Question

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 $V$. The state of the world has support $\mathcal{V}$. The DM receives the payoff $u(y,v)$ depending on the chosen action $y$ the realisation $v$ of $V$. Let $P_V\in \Delta(\mathcal{V})$ be the DM's prior.

Is the following the correct definition 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) $\sum_{v\in \mathcal{V}}u(y,v)P_{Y,V}(y,v)\geq \sum_{v\in \mathcal{V}}u(\tilde{y},v)P_{Y,V}(y,v)$ for each $y$ and $\tilde{y}\neq y$.

In particular, I have doubts about $2)$: what if there is a $y$ such that $P_{Y,V}(y,v)=0$ for each $v\in \mathcal{V}$? Am I missing something?

Answer 1

The concept of the BCE from their 2016 paper is similar to what you have. I think Bergemann and Morris' intuitive explanation is valuable so I'll paraphrase it here.

Each player in the game has a decision rule that chooses an action, $y$, dependent on the state of the world $V$, and the player's information set, which we'll call $S$. This information set includes both a finite set of signals for each player, $T_i$, and a signal distribution, $\pi: \mathcal{V} \rightarrow \Delta T$. As you wrote your example, you assume the set of signals is a singleton, leaving us with only a player's prior. This is a possible information structure, but is not necessary.

We can thus write the the decision rule as a mapping, $\sigma$,

\begin{align*} \sigma : S \times V \rightarrow \Delta Y \end{align*}

The lone criteria for a CBE in this setting is that each players' decision rule is ``obedient''. By obedient we simply mean that the action, $y$, chosen by the decision rule must be the optimal action for the player. Thus, a player will always follow the action chosen by their decision rule.

I believe you are confusing the information structure and the decision rule. My information set is not a function of the action I choose in this setting, so $P_{V,Y}(y,v)$ does not have any meaning. Thus, you don't need to be concerned about the existence of a $y$ such that $P_{V,Y}(y,v)=0$ for all $v$.

It is possible to be in this setting that there exists an action $y$ such that $\sigma(y_i|t_i)=0$ for all signals, $t$. But this would simply mean the player never chooses that action in equilibrium.

Is it possible that there exists a signal $t$ such that $\sigma(y_i|t_i)=0$ for all actions, $y$? No, and it would follow for the basic Nash existence proof, given certain constraints on $u(\cdot),$ $\mathcal{Y}$ and $\mathcal{V}$.

Answer 2

You have specialized the definition of BCE in two dimensions: there is only one player, and the player has no private information. If you want to allow for private information you can let the player have some signal $\pi:\mathcal{V}\rightarrow\Delta(T_i)$

And let the decision rule $P_{\mathcal{Y},\mathcal{T},\mathcal{V}}\in\Delta(\mathcal{Y}\times \mathcal{T}\times \mathcal{V})$ be a single-player BCE if

1. $\sum_{y\in Y}P_{\mathcal{Y},\mathcal{T},\mathcal{V}}(y,t,v)=\pi(t|v)P_{\mathcal{V}}(v)$

2. For each $t\in \mathcal{T}$, and $y\in \mathcal{Y}$: $$\sum_{v\in \mathcal{V}}u(y,v)P_{\mathcal{Y},\mathcal{T},\mathcal{V}}(y,t,v)\geq\sum_{v\in \mathcal{V}}u(\tilde y,v)P_{\mathcal{Y},\mathcal{T},\mathcal{V}}(y,t,v)$$ for all $\tilde y\neq y$