<p>First, <span class="math-container">$l(\tau)(y)$</span> is a function of both <span class="math-container">$\tau$</span> and <span class="math-container">$y$</span>, and the dependence on <span class="math-container">$\tau$</span> is essential; this is how communication happens. Since <span class="math-container">$T$</span> and <span class="math-container">$Y$</span> are finite, there is no point in introducing integrals.</p>
<p><span class="math-container">$\mathbb{E}_{l(\tau)}g(F(y))$</span> is the expectation over the function when the random value <span class="math-container">$y$</span> is distributed according to <span class="math-container">$l(\tau)$</span>. That is,
<span class="math-container">$$\mathbb{E}_{l(\tau)}g(F(y))=\sum_{y\in Y}\Big(g(F(y))\cdot l(\tau)(y)\Big).$$</span>
Here, <span class="math-container">$l(\tau)$</span> is a distribution over <span class="math-container">$Y$</span> that assigns probability <span class="math-container">$l(\tau)(y)$</span> to <span class="math-container">$y$</span>.</p>

CC BY-SA 4.0
2021-09-28T07:13:54.513Z
The point is that the index $l(\tau)$ as it is written is not convenient to work. it says $\mathbb{E}_{l(\tau)}$ but instead the probability measure is $l(\tau)(y)$. I'd prefer the notation to be $\mathbb{E}_{l(y|\tau)}$. Even further, $l(\tau)(y)=l(y|\tau)$, so by conditioning on $\tau$ it means that $\tau$ is "known" and the only "unkown" parameter here is $y$, isn't it?
Nav89
CC BY-SA 4.0
2021-09-28T07:18:11.327Z
As for the integrals, I am trying to think how could someone write all these in terms of integrals...this does not have to do if they are finite or not. How could the notation become? Say taht $T$ and $Y$ are some intervals that are subsets of real numbers...
Nav89
CC BY-SA 4.0
2021-09-28T10:52:35.963Z
$l(τ)(y)$ is a number, not a measure.
Michael Greinecker
CC BY-SA 4.0
2021-09-28T12:46:06.523Z
Wait....what? if $P$ is the probability measure then $\mathbb{E}_{P}$ represents the expectation operator over $P$. Furthermore $l$ denotes a pdf if you read the definition carefully and since this is the case it is a probability measure. If you understand something else, tell me what am I missing here....
Nav89
CC BY-SA 4.0
2021-09-28T13:01:13.223Z
Well, $l(\tau)$ is a probability measure on $Y$ and it assigns the value $l(\tau)(y)$ to the point $y$.
Michael Greinecker
CC BY-SA 4.0
2021-09-27T21:31:40.163Z
2021-09-27T21:31:40.163Z
Michael Greinecker
<p>In the spirit of the previous question that I have done, <a href="https://economics.stackexchange.com/questions/47649/difficulty-in-understanding-the-notation-related-from-probability-theory-with-ga?noredirect=1#comment76748_47649">here</a> considering the paper <a href="https://econ-papers.upf.edu/papers/169.pdf" rel="nofollow noreferrer">here</a> I am trying to make the matching <a href="https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.685.4204&amp;rep=rep1&amp;type=pdf" rel="nofollow noreferrer">definition <span class="math-container">$2.2$</span> here.</a></p>
<p>I will give two definitions and I would like to clarify how are they connected</p>
<p><span class="math-container">$\textit{Definition of a communication mechanism:}$</span> A communication mechanism is a triple <span class="math-container">$\mathcal{C}=((T^i)_i, (Y^i)_i , l )$</span>, where <span class="math-container">$T^i$</span> is <span class="math-container">$i's$</span> finite set of messages, <span class="math-container">$Y^i$</span> is <span class="math-container">$i's$</span> finite set of signals, and <span class="math-container">$l: T=(\Pi_{i\in I} T^i)\to \Delta(Y)$</span> is the signal function. When <span class="math-container">$t$</span> is the profile of messages sent by the players to the mechanism, <span class="math-container">$y\in Y$</span> is drawn according to <span class="math-container">$l(t)$</span> and player <span class="math-container">$i$</span> is informed of <span class="math-container">$y_i$</span>. Furthermore, <span class="math-container">$\mathcal{T}_i=\Delta(T_i)$</span> represents the set of mixed messages for player <span class="math-container">$i$</span> and <span class="math-container">$l$</span> is extended to <span class="math-container">$\mathcal{T}$</span> by</p>
<p><span class="math-container">$$\underbrace{l(\tau)( y)}_{l(y|\tau)}=\mathbb{E}_{\tau} l(t)( y)=\mathbb{E}_{\tau}l(y|t)=\sum_{\tau\in \mathcal{T}}l(y|t)\times \tau(t)$$</span>.</p>
<p>In this point I clarify that,
<span class="math-container">$$l(\tau)(y)=l(y|\tau),\quad\text{some different way of defining the conditional expectation}$$</span>
and hence since <span class="math-container">$l(\tau)(y)$</span> is a function of <span class="math-container">$y$</span> we could write for sipmlicity</p>
<p><span class="math-container">$$\nu(y):=\sum_{\tau\in \mathcal{T}}l(y|t)\times \tau(t)=l(\tau)(y)$$</span></p>
<p><span class="math-container">$\textit{Q1:}$</span> Is it right to redifine <span class="math-container">$l(\tau)(y)$</span> to <span class="math-container">$\nu(y)$</span> and how could I change the <span class="math-container">$\sum$</span> to <span class="math-container">$\int$</span>? I mean could it be right to ew-write <span class="math-container">$\nu(y)=\int_{\mathcal{T}}l(t|y)d(\tau(t))?$</span></p>
<p>the next definition says</p>
<p><span class="math-container">$\textit{Definition of a game with a communication mechanism:}$</span> Given a compact game <span class="math-container">$G$</span> and a communication mechanism <span class="math-container">$\mathcal{C}$</span>, then if <span class="math-container">$\Gamma(\mathcal{C}, G)$</span> is the game <span class="math-container">$G$</span> extended by <span class="math-container">$\mathcal{C}$</span>, it unfolds as follows:</p>
<ul>
<li>each player <span class="math-container">$i$</span> sends a message <span class="math-container">$t^i$</span> to the mechanism</li>
<li><span class="math-container">$y\in Y$</span> is drawn according to <span class="math-container">$l(t)$</span> and each player <span class="math-container">$i$</span> is informed of <span class="math-container">$y^i$</span></li>
<li>each player <span class="math-container">$i$</span> chooses <span class="math-container">$\sigma^i\in \Sigma^i$</span> according to <span class="math-container">$y^i$</span></li>
<li>the vector payoff is <span class="math-container">$g(\sigma)$</span>
A strategy for player <span class="math-container">$i$</span> is given by a mixed message <span class="math-container">$\tau^i\in \mathcal{T}^i$</span> and by a
mapping <span class="math-container">$F^i: Y^i\to \Sigma^i$</span>. The payoff function is given by
<span class="math-container">$$g_{\mathcal{C}}(\tau, F) =\mathbb{E}_{l(\tau)}g(F(y))$$</span></li>
</ul>
<p><span class="math-container">$\textit{Q1:}$</span> I assume that since y is drawn according to l(\tau) then the index in the last payoff function should be <span class="math-container">$\mathbb{E}_{l(\tau)(y)}$</span> instead of <span class="math-container">$\mathbb{E}_{l(\tau)}$</span> why is this not the case?</p>
<p><span class="math-container">$\textit{Q3:}$</span> By writing donw the the payoff function in terms of sums or integrals (instead of terms in expextations) if the probability disrtibution is continuous, what is going to be the formuation?</p>

CC BY-SA 4.0
2021-09-25T11:53:13.833Z
in this point $\mathbb{E}_{l(\tau)}l(t)( y)$ you should write $\mathbb{E}_{\tau}l(y| t)$....i think
Hunger Learn
CC BY-SA 4.0
2021-09-25T11:58:48.673Z
I corrected this...
Nav89
<microeconomics><mathematical-economics><bayesian-game><probability><information>
Mechanisms of communication in game theory
Nav89
# Mechanisms of communication in game theory

(Asked by: username_0 on 2021-09-25)

In the spirit of the previous question that I have done, [here](https://economics.stackexchange.com/questions/47649/difficulty-in-understanding-the-notation-related-from-probability-theory-with-ga?noredirect=1#comment76748_47649) considering the paper [here](https://econ-papers.upf.edu/papers/169.pdf) I am trying to make the matching [definition $2.2$ here.](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.685.4204&rep=rep1&type=pdf)

I will give two definitions and I would like to clarify how are they connected

$\textit{Definition of a communication mechanism:}$ A communication mechanism is a triple $\mathcal{C}=((T^i)\_i, (Y^i)\_i , l )$, where $T^i$ is $i's$ finite set of messages, $Y^i$ is $i's$ finite set of signals, and $l: T=(\Pi\_{i\in I} T^i)\to \Delta(Y)$ is the signal function. When $t$ is the profile of messages sent by the players to the mechanism, $y\in Y$ is drawn according to $l(t)$ and player $i$ is informed of $y\_i$. Furthermore, $\mathcal{T}\_i=\Delta(T\_i)$ represents the set of mixed messages for player $i$ and $l$ is extended to $\mathcal{T}$ by

$$\underbrace{l(\tau)( y)}\_{l(y|\tau)}=\mathbb{E}\_{\tau} l(t)( y)=\mathbb{E}\_{\tau}l(y|t)=\sum\_{\tau\in \mathcal{T}}l(y|t)\times \tau(t)$$.

In this point I clarify that,
$$l(\tau)(y)=l(y|\tau),\quad\text{some different way of defining the conditional expectation}$$
and hence since $l(\tau)(y)$ is a function of $y$ we could write for sipmlicity

$$\nu(y):=\sum\_{\tau\in \mathcal{T}}l(y|t)\times \tau(t)=l(\tau)(y)$$

$\textit{Q1:}$ Is it right to redifine $l(\tau)(y)$ to $\nu(y)$ and how could I change the $\sum$ to $\int$? I mean could it be right to ew-write $\nu(y)=\int\_{\mathcal{T}}l(t|y)d(\tau(t))?$

the next definition says

$\textit{Definition of a game with a communication mechanism:}$ Given a compact game $G$ and a communication mechanism $\mathcal{C}$, then if $\Gamma(\mathcal{C}, G)$ is the game $G$ extended by $\mathcal{C}$, it unfolds as follows:

* each player $i$ sends a message $t^i$ to the mechanism
* $y\in Y$ is drawn according to $l(t)$ and each player $i$ is informed of $y^i$
* each player $i$ chooses $\sigma^i\in \Sigma^i$ according to $y^i$
* the vector payoff is $g(\sigma)$
  A strategy for player $i$ is given by a mixed message $\tau^i\in \mathcal{T}^i$ and by a
  mapping $F^i: Y^i\to \Sigma^i$. The payoff function is given by
  $$g\_{\mathcal{C}}(\tau, F) =\mathbb{E}\_{l(\tau)}g(F(y))$$

$\textit{Q1:}$ I assume that since y is drawn according to l(\tau) then the index in the last payoff function should be $\mathbb{E}\_{l(\tau)(y)}$ instead of $\mathbb{E}\_{l(\tau)}$ why is this not the case?

$\textit{Q3:}$ By writing donw the the payoff function in terms of sums or integrals (instead of terms in expextations) if the probability disrtibution is continuous, what is going to be the formuation?

**Tags:** microeconomics, mathematical-economics, bayesian-game, probability, information

> **Comments:**
> **username_1**: in this point $\mathbb{E}_{l(\tau)}l(t)( y)$ you should write $\mathbb{E}_{\tau}l(y| t)$....i think  
> **username_0**: I corrected this...  

---

## 1 answers

### Answer by username_2 on 2021-09-27. Score: 1

First, $l(\tau)(y)$ is a function of both $\tau$ and $y$, and the dependence on $\tau$ is essential; this is how communication happens. Since $T$ and $Y$ are finite, there is no point in introducing integrals.

$\mathbb{E}\_{l(\tau)}g(F(y))$ is the expectation over the function when the random value $y$ is distributed according to $l(\tau)$. That is,
$$\mathbb{E}\_{l(\tau)}g(F(y))=\sum\_{y\in Y}\Big(g(F(y))\cdot l(\tau)(y)\Big).$$
Here, $l(\tau)$ is a distribution over $Y$ that assigns probability $l(\tau)(y)$ to $y$.

> **Comments:**
> **username_0**: The point is that the index $l(\tau)$ as it is written is not convenient to work. it says $\mathbb{E}_{l(\tau)}$ but instead the probability measure is $l(\tau)(y)$. I'd prefer the notation to be $\mathbb{E}_{l(y|\tau)}$. Even further, $l(\tau)(y)=l(y|\tau)$, so by conditioning on $\tau$ it means that $\tau$ is "known" and the only "unkown" parameter here is $y$, isn't it?  
> **username_0**: As for the integrals, I am trying to think how could someone write all these in terms of integrals...this does not have to do if they are finite or not. How could the notation become? Say taht $T$ and $Y$ are some intervals that are subsets of real numbers...  
> **username_2**: $l(τ)(y)$ is a number, not a measure.  
> **username_0**: Wait....what? if $P$ is the probability measure then $\mathbb{E}_{P}$ represents the expectation operator over $P$. Furthermore $l$ denotes a pdf if you read the definition carefully and since this is the case it is a probability measure. If you understand something else, tell me what am I missing here....  
> **username_2**: Well, $l(\tau)$ is a probability measure on $Y$ and it assigns the value $l(\tau)(y)$ to the point $y$.  

---

