<p>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 <span class="math-container">$\pi:\mathcal{V}\rightarrow\Delta(T_i)$</span></p>

<p>And let the decision rule <span class="math-container">$P_{\mathcal{Y},\mathcal{T},\mathcal{V}}\in\Delta(\mathcal{Y}\times \mathcal{T}\times \mathcal{V})$</span> be a single-player BCE if</p>

<ol>
<li><p><span class="math-container">$\sum_{y\in Y}P_{\mathcal{Y},\mathcal{T},\mathcal{V}}(y,t,v)=\pi(t|v)P_{\mathcal{V}}(v)$</span></p></li>
<li><p>For each <span class="math-container">$t\in \mathcal{T}$</span>, and <span class="math-container">$y\in \mathcal{Y}$</span>: <span class="math-container">$$\sum_{v\in \mathcal{V}}u(y,v)P_{\mathcal{Y},\mathcal{T},\mathcal{V}}(y,t,v)\geq\sum_{v\in \mathcal{V}}u(\tilde  y,v)P_{\mathcal{Y},\mathcal{T},\mathcal{V}}(y,t,v)$$</span> for all <span class="math-container">$\tilde y\neq y$</span></p></li>
</ol>

CC BY-SA 4.0
2019-11-21T18:55:44.293Z
2019-11-21T19:01:06.087Z
2019-11-21T19:01:06.087Z
Regio
<p>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.</p>

<p>Each player in the game has a decision rule that chooses an action, <span class="math-container">$y$</span>, dependent on the state of the world <span class="math-container">$V$</span>, and the player's information set, which we'll call <span class="math-container">$S$</span>. This information set includes both a finite set of signals for each player, <span class="math-container">$T_i$</span>, and a signal distribution, <span class="math-container">$\pi: \mathcal{V} \rightarrow \Delta T$</span>. 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.</p>

<p>We can thus write the the decision rule as a mapping, <span class="math-container">$\sigma$</span>,</p>

<p><span class="math-container">\begin{align*}
\sigma : S \times V \rightarrow \Delta Y
\end{align*}</span></p>

<p>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, <span class="math-container">$y$</span>, 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.</p>

<p>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 <span class="math-container">$P_{V,Y}(y,v)$</span> does not have any meaning. Thus, you don't need to be concerned about the existence of a <span class="math-container">$y$</span> such that <span class="math-container">$P_{V,Y}(y,v)=0$</span> for all <span class="math-container">$v$</span>.</p>

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

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

CC BY-SA 4.0
2019-11-21T06:13:30.427Z
2019-11-21T06:13:30.427Z
corran_horn
<p>Consider a game where a decision maker (DM) has to choose action <span class="math-container">$y\in \mathcal{Y}$</span> possibly without being fully aware of the state of the world <span class="math-container">$V$</span>. The state of the world has support <span class="math-container">$\mathcal{V}$</span>. The DM receives the payoff <span class="math-container">$u(y,v)$</span> depending on the chosen action <span class="math-container">$y$</span> the realisation <span class="math-container">$v$</span> of <span class="math-container">$V$</span>. Let <span class="math-container">$P_V\in \Delta(\mathcal{V})$</span> be the DM's prior.</p>

<p>Is the following the correct definition of 1 player Bayesian Correlated Equilibrium provided in Bergemann and Morris (2013,2016,etc.)?</p>

<p><span class="math-container">$P_{Y,V}\in  \Delta(\mathcal{Y}\times \mathcal{V})$</span> is a 1 player Bayesian Correlated Equilibrium if</p>

<p>1) <span class="math-container">$\sum_{y\in \mathcal{Y}}P_{Y,V}(y,v)=P_V(v)$</span> for each <span class="math-container">$v\in \mathcal{V}$</span></p>

<p>2) <span class="math-container">$\sum_{v\in \mathcal{V}}u(y,v)P_{Y,V}(y,v)\geq \sum_{v\in \mathcal{V}}u(\tilde{y},v)P_{Y,V}(y,v)$</span> for each <span class="math-container">$y$</span> and <span class="math-container">$\tilde{y}\neq y$</span>.</p>

<p>In particular, I have doubts about <span class="math-container">$2)$</span>: what if there is a <span class="math-container">$y$</span> such that <span class="math-container">$P_{Y,V}(y,v)=0$</span> for each <span class="math-container">$v\in \mathcal{V}$</span>? Am I missing something?</p>

CC BY-SA 4.0
2019-11-21T00:31:06.910Z
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?
Walrasian Auctioneer
CC BY-SA 4.0
2019-11-21T10:19:22.383Z
Yes, changed it sorry
Star
<game-theory><decision-theory><bayesian-game><choice-theory>
Exact definition of one-player Bayesian Correlated Equilibrium
Star
# Exact definition of one-player Bayesian Correlated Equilibrium

(Asked by: username_0 on 2019-11-20)

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?

**Tags:** game-theory, decision-theory, bayesian-game, choice-theory

> **Comments:**
> **username_1**: 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?  
> **username_0**: Yes, changed it sorry  

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## 2 answers

### Answer by username_3 on 2019-11-21. Score: 3 | Accepted answer

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 $\mathcal{V}$.


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### Answer by username_2 on 2019-11-21. Score: 1

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 $\tilde y\neq y$


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