<p>Suppose you are an analyst studying a Bayesian game. You know the players, the possible states of nature, the common prior, the action spaces, the payoff functions, and you know about some information channels available to the players, the latter given via some information structure. However, you can't rule out that the players have additional information channels available to them. This additional information can be modelled via expansions.</p>
<p>Now, you are sure that they play a Bayes-Nash equilibrium with respect to all information channels they have, only some of which you are aware of. Then, by Theorem 1 of the paper, the players will play a Bayes correlated equilibrium with respect to those information structures you as an analyst are aware of.</p>
<p>Now, you, the analyst, might be able to observe <span class="math-container">$\theta$</span>, the signals <span class="math-container">$t_i$</span>, and the chosen actions. From the joint distribution of these actions, you can derive the conditional distribution over action profiles. This conditional distribution is exactly the decision rule <span class="math-container">$\sigma$</span>.</p>

CC BY-SA 4.0
2021-10-02T10:03:20.647Z
two questions


1. I think there is a theorem that says, any Nash equilibrium is a correlated equilibrium but it does not hold the opposite (if you have any details to check it somewhere feel free to give me a link), deos this mean that the same hold for a BNE and the BCE, namely by theorem $1$ Bergemann and Morris may want to do this parallelism right? 


2. so the decision rule $\sigma: \Theta\times T\to \Delta(A)$ is behavioral strategy profile? Namely it gives a pdf on the set of the mixed action of the players and this is a correlated strategy as well?
Hunger Learn
CC BY-SA 4.0
2021-10-02T12:35:04.443Z
1. The parallel is right: With complete information,  a BNE is a NE and a BCE is a CE. Then Theorem 1 says that CE are exactly the NE distributions for some additional information. 2. A behavior strategy maps a single players information states to distributions over their actions. So these are different things.
Michael Greinecker
CC BY-SA 4.0
2021-10-01T09:39:01.227Z
2021-10-01T09:39:01.227Z
Michael Greinecker
<p>The <a href="https://econtheory.org/ojs/index.php/te/article/viewFile/20160487/15457/452" rel="noreferrer">paper of Bergemann and Morris</a> proves a theorem based on some foundations about the information sets and their expansions. I am trying to understand theorem one intuition, more precisely I cite the appropriate parts that we need to know, that is:</p>
<p><span class="math-container">$\textbf{Definition 4:}$</span> (Combination). The information structure <span class="math-container">$S^∗ = (T^∗, \pi^∗:\Theta\to\Delta(T_i^*))$</span> is a combination of information structures <span class="math-container">$S^1 = (T^1, \pi^1)$</span> and <span class="math-container">$S^2 = (T^2, \pi^2)$</span> if</p>
<p><span class="math-container">$$T_i^*=T_i^1\times T_i^2,\quad\text{for each $i$}$$</span>
and
<span class="math-container">$$\sum_{t_i^2\in T_i^2}\pi^*(t_i^1,t_i^2|\theta)=\pi(t_i^1|\theta),\quad\text{for each $t_i^1\in T_i^1$ and $\theta\in\Theta$} $$</span></p>
<p><span class="math-container">$$\sum_{t_i^1\in T_i^1}\pi^*(t_i^1,t_i^2|\theta)=\pi(t_i^2|\theta),\quad\text{for each $t_i^2\in T_i^2$ and $\theta\in\Theta$} $$</span></p>
<p>Note that the above definition places no restrictions on whether signals <span class="math-container">$t_i^1\in T_i^1$</span> and <span class="math-container">$t_i^2\in T_i^2$</span> are independent or correlated, conditional on <span class="math-container">$\theta$</span>, under <span class="math-container">$\pi^∗$</span>. Thus any pair of information structures <span class="math-container">$S^1$</span> and <span class="math-container">$S^2$</span> will have many combined information structures.</p>
<p><span class="math-container">$\textbf{Definition 5:}$</span> (Expansion). An information structure <span class="math-container">$S^∗$</span> is an expansion of <span class="math-container">$S^1$</span> if <span class="math-container">$S^∗$</span> is a combination of <span class="math-container">$S^1$</span> and some other information structure <span class="math-container">$S^2$</span>.</p>
<p>Suppose strategy profile <span class="math-container">$\beta$</span> was played in <span class="math-container">$(G, S^∗)$</span>, where <span class="math-container">$S^∗$</span> is a combination of two information structures <span class="math-container">$S^1$</span> and <span class="math-container">$S^2$</span>. Now, if the analyst did not observe the signals of the combined information structure <span class="math-container">$S^∗$</span>, but only the signals of <span class="math-container">$S^1$</span>, then the behavior under the strategy profile <span class="math-container">$\beta$</span> would induce a decision rule for <span class="math-container">$(G, S^1)$</span>. Formally, the strategy profile <span class="math-container">$\beta$</span> for <span class="math-container">$(G, S^∗)$</span> induces the decision rule σ for <span class="math-container">$(G, S^1)$</span>:</p>
<p><span class="math-container">\begin{equation}\sigma(a|t_i^1,\theta):=\frac{\sum_{t_i^2\in T_i^2}\pi^*(t_i^1,t_i^2|\theta)\Pi_{j=1}^i\beta_{j}(a_j|t_i^1,t_i^2)}{\pi(t_i^1|\theta)}\end{equation}</span>
for each <span class="math-container">$a\in A$</span> whenever <span class="math-container">$\pi^1(t_i^1|\theta) &gt; 0$</span>.</p>
<p>Based on the above the authors give the following therem that makes a connection between the Bayesian Nash equilibrium and the Bayesian Correlated equilibrium, that is</p>
<p><span class="math-container">$\textbf{Theorem 1:}$</span> A decision rule <span class="math-container">$\sigma$</span> is a Bayes correlated equilibrium of <span class="math-container">$(G, S)$</span> if and only if, for some expansion <span class="math-container">$S^∗$</span> of <span class="math-container">$S$</span>, there is a Bayes Nash equilibrium of <span class="math-container">$(G, S^∗)$</span> that induces <span class="math-container">$\sigma$</span>.</p>
<p>I have the following questions</p>
<p><span class="math-container">$\textbf{Question 1:}$</span> They define the the rule <span class="math-container">$\sigma(a|t_i^1,\theta)$</span> in a way to make the connection between the sets of BNE and BCE and I can not understand why they did so?</p>
<p><span class="math-container">$\textbf{Question 2:}$</span> What is the interpretation of <span class="math-container">$\sigma(a|t_i^1,\theta)$</span>, in terms of porbability someone can say it is the Bayesian rule, however all these parameters as <span class="math-container">$\pi^*$</span>, <span class="math-container">$\beta$</span> and <span class="math-container">$\pi(t_i^1|\theta)$</span> have some intuitive interpretation what is it?</p>
<p><span class="math-container">$\textbf{Question 3:}$</span> Why it is so special this obedient rule <span class="math-container">$\sigma(a|t_i^1,\theta)$</span> for both the BNE and BCE and does it suffice to make the transition from the one set of solutuions to the other? I am a little confused because it seems like the two sets are somehow the one the subset of the other.</p>
<p><span class="math-container">$\textbf{Question 4:}$</span> Are BNE and BCE directly or indirectly connected? If not, what is the purpose to do this here?</p>

CC BY-SA 4.0
2021-09-30T17:16:48.977Z
To be proactive and provide a clarification for everyone. This question 
https://economics.stackexchange.com/questions/29903/the-notions-of-bayes-correlated-equilibrium-and-bayesian-nash-equilibrium-in-a-g
 
does not answer my question, because it refers to the same setup.
Hunger Learn
<microeconomics><mathematical-economics><game-theory><nash-equilibrium><bayesian-game>
Bayes correlated equilibrium of Bergemann and Morris
Hunger Learn
# Bayes correlated equilibrium of Bergemann and Morris

(Asked by: username_0 on 2021-09-30)

The [paper of Bergemann and Morris](https://econtheory.org/ojs/index.php/te/article/viewFile/20160487/15457/452) proves a theorem based on some foundations about the information sets and their expansions. I am trying to understand theorem one intuition, more precisely I cite the appropriate parts that we need to know, that is:

$\textbf{Definition 4:}$ (Combination). The information structure $S^∗ = (T^∗, \pi^∗:\Theta\to\Delta(T\_i^\*))$ is a combination of information structures $S^1 = (T^1, \pi^1)$ and $S^2 = (T^2, \pi^2)$ if

$$T\_i^\*=T\_i^1\times T\_i^2,\quad\text{for each $i$}$$
and
$$\sum\_{t\_i^2\in T\_i^2}\pi^\*(t\_i^1,t\_i^2|\theta)=\pi(t\_i^1|\theta),\quad\text{for each $t\_i^1\in T\_i^1$ and $\theta\in\Theta$} $$

$$\sum\_{t\_i^1\in T\_i^1}\pi^\*(t\_i^1,t\_i^2|\theta)=\pi(t\_i^2|\theta),\quad\text{for each $t\_i^2\in T\_i^2$ and $\theta\in\Theta$} $$

Note that the above definition places no restrictions on whether signals $t\_i^1\in T\_i^1$ and $t\_i^2\in T\_i^2$ are independent or correlated, conditional on $\theta$, under $\pi^∗$. Thus any pair of information structures $S^1$ and $S^2$ will have many combined information structures.

$\textbf{Definition 5:}$ (Expansion). An information structure $S^∗$ is an expansion of $S^1$ if $S^∗$ is a combination of $S^1$ and some other information structure $S^2$.

Suppose strategy profile $\beta$ was played in $(G, S^∗)$, where $S^∗$ is a combination of two information structures $S^1$ and $S^2$. Now, if the analyst did not observe the signals of the combined information structure $S^∗$, but only the signals of $S^1$, then the behavior under the strategy profile $\beta$ would induce a decision rule for $(G, S^1)$. Formally, the strategy profile $\beta$ for $(G, S^∗)$ induces the decision rule σ for $(G, S^1)$:

\begin{equation}\sigma(a|t\_i^1,\theta):=\frac{\sum\_{t\_i^2\in T\_i^2}\pi^\*(t\_i^1,t\_i^2|\theta)\Pi\_{j=1}^i\beta\_{j}(a\_j|t\_i^1,t\_i^2)}{\pi(t\_i^1|\theta)}\end{equation}
for each $a\in A$ whenever $\pi^1(t\_i^1|\theta) > 0$.

Based on the above the authors give the following therem that makes a connection between the Bayesian Nash equilibrium and the Bayesian Correlated equilibrium, that is

$\textbf{Theorem 1:}$ A decision rule $\sigma$ is a Bayes correlated equilibrium of $(G, S)$ if and only if, for some expansion $S^∗$ of $S$, there is a Bayes Nash equilibrium of $(G, S^∗)$ that induces $\sigma$.

I have the following questions

$\textbf{Question 1:}$ They define the the rule $\sigma(a|t\_i^1,\theta)$ in a way to make the connection between the sets of BNE and BCE and I can not understand why they did so?

$\textbf{Question 2:}$ What is the interpretation of $\sigma(a|t\_i^1,\theta)$, in terms of porbability someone can say it is the Bayesian rule, however all these parameters as $\pi^\*$, $\beta$ and $\pi(t\_i^1|\theta)$ have some intuitive interpretation what is it?

$\textbf{Question 3:}$ Why it is so special this obedient rule $\sigma(a|t\_i^1,\theta)$ for both the BNE and BCE and does it suffice to make the transition from the one set of solutuions to the other? I am a little confused because it seems like the two sets are somehow the one the subset of the other.

$\textbf{Question 4:}$ Are BNE and BCE directly or indirectly connected? If not, what is the purpose to do this here?

**Tags:** microeconomics, mathematical-economics, game-theory, nash-equilibrium, bayesian-game

> **Comments:**
> **username_0**: To be proactive and provide a clarification for everyone. This question  https://economics.stackexchange.com/questions/29903/the-notions-of-bayes-correlated-equilibrium-and-bayesian-nash-equilibrium-in-a-g   does not answer my question, because it refers to the same setup.  

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## 1 answers

### Answer by username_1 on 2021-10-01. Score: 4 | Accepted answer

Suppose you are an analyst studying a Bayesian game. You know the players, the possible states of nature, the common prior, the action spaces, the payoff functions, and you know about some information channels available to the players, the latter given via some information structure. However, you can't rule out that the players have additional information channels available to them. This additional information can be modelled via expansions.

Now, you are sure that they play a Bayes-Nash equilibrium with respect to all information channels they have, only some of which you are aware of. Then, by Theorem 1 of the paper, the players will play a Bayes correlated equilibrium with respect to those information structures you as an analyst are aware of.

Now, you, the analyst, might be able to observe $\theta$, the signals $t\_i$, and the chosen actions. From the joint distribution of these actions, you can derive the conditional distribution over action profiles. This conditional distribution is exactly the decision rule $\sigma$.

> **Comments:**
> **username_0**: two questions   1. I think there is a theorem that says, any Nash equilibrium is a correlated equilibrium but it does not hold the opposite (if you have any details to check it somewhere feel free to give me a link), deos this mean that the same hold for a BNE and the BCE, namely by theorem $1$ Bergemann and Morris may want to do this parallelism right?    2. so the decision rule $\sigma: \Theta\times T\to \Delta(A)$ is behavioral strategy profile? Namely it gives a pdf on the set of the mixed action of the players and this is a correlated strategy as well?  
> **username_1**: 1. The parallel is right: With complete information,  a BNE is a NE and a BCE is a CE. Then Theorem 1 says that CE are exactly the NE distributions for some additional information. 2. A behavior strategy maps a single players information states to distributions over their actions. So these are different things.  

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