Exact definition of one-player Bayesian Correlated Equilibrium

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?

• Shouldn't 1 be a summation over $y$? As for 2, if your notation is correct, then if $P_{Y,V}(y,v) = 0$ for each $v$, both left and right hand sides are $0$? I do not see the problem? Nov 21, 2019 at 0:31
• Yes, changed it sorry
– Star
Nov 21, 2019 at 10:19

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 $$\mathcal{V}$$.

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 $$\tilde y\neq y$$