<p>The common prior <span class="math-container">$p\in\Delta L$</span> and the transition probability <span class="math-container">$q:L\to\Delta A$</span> induce a joint distribution on <span class="math-container">$L\times A$</span> in which the pair <span class="math-container">$(l,a)$</span> is selected with probability <span class="math-container">$p_l\cdot q_l(a)$</span>. You can also recover <span class="math-container">$q$</span> outside (strategically irrelevant) type profiles <span class="math-container">$l$</span> such that <span class="math-container">$p_l$</span> by the usual formula for conditional probabilities.</p>
<p>Now, an equilibrium of the extended game leads to eventually (possibly &quot;at infinity&quot;, this is long cheap talk) both players choosing an action. You can then calculate the joint distribution of types and chosen actions. If it is the same as the distribution above, we can say that the equilibrium of the extended game induces <span class="math-container">$q$</span>.</p>
<p>If the randomization device would not be independent of the type distribution, you would need to actually define a joint distribution on <span class="math-container">$\Omega\times L$</span>. If this distribution is not independent, a player may learn about the other player's type than what they learn from their own type alone. In that case, <span class="math-container">$\Omega$</span> would be a signal of types and not just a correlation device used to implement correlated equilibria of the extended game.</p>

CC BY-SA 4.0
2021-12-11T08:48:51.110Z
So, you mean that by adding the state space $(\Omega,\mathcal{B},\mu)$ it is like we are adding one more variable to our model, that randomizes somehow the game. Thus, instead of having $p(l)q(a|l)$, we have a new factor added and this becomes, $\mu(\omega)p(l)q(a|l)$? What if $l$ was based on $\omega$ in the sense that given $\omega\in\Omega$ then $l$ is drawn according to $p:\omega\to\Delta(l)$ and hence $(\omega,l,a)$ were selected according to $\mu(\omega) p(l_i,l_{-i}|\omega)q(a|l_i,l_{-i},\omega)$? This type or randomiation still hilds right?
Hunger Learn
CC BY-SA 4.0
2021-12-11T09:28:07.863Z
In the model, the correlation between $\omega$ and $l$ comes from the strategies of the players. Why should a player's type be drawn according to a correlation device? This makes little sense for the interpretation.
Michael Greinecker
CC BY-SA 4.0
2021-12-11T10:32:58.003Z
The correlation comes between $\omega$ and $l$ comes from the strategies, but the authors have not seen the connection somehow in the maths. It's just an assumption, but where does this help? In the correlated equilibrium of the cheap talk game, the mediator (or device) does not receive any information from the players. He makes recommendations on how to exchange messages, but remains fully ignorant of the players’ types. (see in page $98$). But from the $(ext_M\Gamma)^{\mu}$ game $q$ is induced. How is this so? How can one probability measure $\mu$ induce another $q$ without any dependence?
Hunger Learn
CC BY-SA 4.0
2021-12-11T10:36:11.450Z
With the caononical game, they take recommendations according to $q$ which in essence means that this $q$ is the correlated strategy of the extended game by the device when the players are sincere and obednient in equilibrium. But what is the connection between $\mu$ and $q$ in cheap talk game? Intuitively and in terms of maths... I don't know if I am clear....in my question...in the previous comment....
Hunger Learn
CC BY-SA 4.0
2021-12-11T11:17:57.947Z
"How is this so? How can one probability measure μ induce another q without any dependence?" Again, correlation comes from strategies. To answer the second question, the very point of the paper is that you can achieve what you can get ins a communication equilibrium with a mediator who collects information from the players first and then makes recommendations, by a correlated equilibrium of a cheap talk extension in which agents get signals that do not depend on any private information. Of course, the players have their private information and their ultimate action choices will depend on it.
Michael Greinecker
CC BY-SA 4.0
2021-12-10T19:18:43.780Z
2021-12-10T19:35:53.813Z
2021-12-10T19:35:53.813Z
Michael Greinecker
<p>Taking a look at this paper of <a href="https://onlinelibrary.wiley.com/doi/pdf/10.3982/TE944" rel="nofollow noreferrer">Forges and Vida</a> the authors define a correlation device in page <span class="math-container">$102$</span>, that is a standard probability space <span class="math-container">$\left(\Omega,\mathcal{B},\mu\right)$</span>, They assume that the probability space represents the extraneous events, select in <span class="math-container">$\left(\Omega,\mathcal{B}\right)$</span> according to <span class="math-container">$\mu$</span> and are idepedent to the game. Well one fisrt question here is, what changes if the extraneous events are not indepedent to the game?</p>
<p>The correlation device is added to the extended game by cheap talk phase, but I can not understand the reason. Finally in page <span class="math-container">$103$</span> in the sketch or proof parapgraph, the following senteces is quoted. ``Let <span class="math-container">$q$</span> be a communication equilibrium of <span class="math-container">$\Gamma$</span> (it is the basic game). We gradually construct a set of messages <span class="math-container">$M$</span>,  a correlation device <span class="math-container">$\mu$</span> for  <span class="math-container">$ext_M\Gamma$</span>, and equilibrium strategies in <span class="math-container">$(ext_M\Gamma)^{\mu}$</span> that induce the transition probability <span class="math-container">$q$</span>&quot;</p>
<p>Note that the transition probability is a mapping that assigns the profile of types send by the players to the communication mechanism <span class="math-container">$q$</span> to the set of actions.</p>
<p><span class="math-container">$\textbf{Question:}$</span> What does it mean that they construct a correlation device <span class="math-container">$\mu$</span> for  the extended by cheap talk phase game <span class="math-container">$ext_M\Gamma$</span>, and equilibrium strategies in <span class="math-container">$(ext_M\Gamma)^{\mu}$</span> that induce the transition probability <span class="math-container">$q$</span>? What does the term induce denotes? Are <span class="math-container">$\mu$</span> and <span class="math-container">$q$</span> connected mathematically somehow? How are tese two types of  equilibria, the communication equilibria with a mediator and the equilibrium with the correlation device are connected?</p>

<mathematical-economics><game-theory><nash-equilibrium><bayesian-game><dynamic-games>
Correlation device that induces a specific transition probability
Hunger Learn
# Correlation device that induces a specific transition probability

(Asked by: username_0 on 2021-12-10)

Taking a look at this paper of [Forges and Vida](https://onlinelibrary.wiley.com/doi/pdf/10.3982/TE944) the authors define a correlation device in page $102$, that is a standard probability space $\left(\Omega,\mathcal{B},\mu\right)$, They assume that the probability space represents the extraneous events, select in $\left(\Omega,\mathcal{B}\right)$ according to $\mu$ and are idepedent to the game. Well one fisrt question here is, what changes if the extraneous events are not indepedent to the game?

The correlation device is added to the extended game by cheap talk phase, but I can not understand the reason. Finally in page $103$ in the sketch or proof parapgraph, the following senteces is quoted. ``Let $q$ be a communication equilibrium of $\Gamma$ (it is the basic game). We gradually construct a set of messages $M$, a correlation device $\mu$ for $ext\_M\Gamma$, and equilibrium strategies in $(ext\_M\Gamma)^{\mu}$ that induce the transition probability $q$"

Note that the transition probability is a mapping that assigns the profile of types send by the players to the communication mechanism $q$ to the set of actions.

$\textbf{Question:}$ What does it mean that they construct a correlation device $\mu$ for the extended by cheap talk phase game $ext\_M\Gamma$, and equilibrium strategies in $(ext\_M\Gamma)^{\mu}$ that induce the transition probability $q$? What does the term induce denotes? Are $\mu$ and $q$ connected mathematically somehow? How are tese two types of equilibria, the communication equilibria with a mediator and the equilibrium with the correlation device are connected?

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


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## 1 answers

### Answer by username_1 on 2021-12-10. Score: 5 | Accepted answer

The common prior $p\in\Delta L$ and the transition probability $q:L\to\Delta A$ induce a joint distribution on $L\times A$ in which the pair $(l,a)$ is selected with probability $p\_l\cdot q\_l(a)$. You can also recover $q$ outside (strategically irrelevant) type profiles $l$ such that $p\_l$ by the usual formula for conditional probabilities.

Now, an equilibrium of the extended game leads to eventually (possibly "at infinity", this is long cheap talk) both players choosing an action. You can then calculate the joint distribution of types and chosen actions. If it is the same as the distribution above, we can say that the equilibrium of the extended game induces $q$.

If the randomization device would not be independent of the type distribution, you would need to actually define a joint distribution on $\Omega\times L$. If this distribution is not independent, a player may learn about the other player's type than what they learn from their own type alone. In that case, $\Omega$ would be a signal of types and not just a correlation device used to implement correlated equilibria of the extended game.

> **Comments:**
> **username_0**: So, you mean that by adding the state space $(\Omega,\mathcal{B},\mu)$ it is like we are adding one more variable to our model, that randomizes somehow the game. Thus, instead of having $p(l)q(a|l)$, we have a new factor added and this becomes, $\mu(\omega)p(l)q(a|l)$? What if $l$ was based on $\omega$ in the sense that given $\omega\in\Omega$ then $l$ is drawn according to $p:\omega\to\Delta(l)$ and hence $(\omega,l,a)$ were selected according to $\mu(\omega) p(l_i,l_{-i}|\omega)q(a|l_i,l_{-i},\omega)$? This type or randomiation still hilds right?  
> **username_1**: In the model, the correlation between $\omega$ and $l$ comes from the strategies of the players. Why should a player's type be drawn according to a correlation device? This makes little sense for the interpretation.  
> **username_0**: The correlation comes between $\omega$ and $l$ comes from the strategies, but the authors have not seen the connection somehow in the maths. It's just an assumption, but where does this help? In the correlated equilibrium of the cheap talk game, the mediator (or device) does not receive any information from the players. He makes recommendations on how to exchange messages, but remains fully ignorant of the players’ types. (see in page $98$). But from the $(ext_M\Gamma)^{\mu}$ game $q$ is induced. How is this so? How can one probability measure $\mu$ induce another $q$ without any dependence?  
> **username_0**: With the caononical game, they take recommendations according to $q$ which in essence means that this $q$ is the correlated strategy of the extended game by the device when the players are sincere and obednient in equilibrium. But what is the connection between $\mu$ and $q$ in cheap talk game? Intuitively and in terms of maths... I don't know if I am clear....in my question...in the previous comment....  
> **username_1**: "How is this so? How can one probability measure μ induce another q without any dependence?" Again, correlation comes from strategies. To answer the second question, the very point of the paper is that you can achieve what you can get ins a communication equilibrium with a mediator who collects information from the players first and then makes recommendations, by a correlated equilibrium of a cheap talk extension in which agents get signals that do not depend on any private information. Of course, the players have their private information and their ultimate action choices will depend on it.  

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