<p>Given that</p>
<blockquote>
<p><span class="math-container">$T^i$</span> is <span class="math-container">$i's$</span> finite set of messages</p>
</blockquote>
<p>and the mechanism uses the signal function to map <span class="math-container">$T$</span> to a signal, the architects of the mechanism can use any arbitrary label for the individual messages. In simple game theoretical examples, the strategies of player 1 are often denoted <span class="math-container">$a_1,a_2...$</span> This does not tell us anything about what player 1 is doing, these are just labels.</p>
<p>E.g., in the Prisoner's dilemma player 1 can either Defect <span class="math-container">$(D)$</span> or Cooperate <span class="math-container">$(C)$</span>. Yet you could also denote these strategies by <span class="math-container">$a_1$</span> and <span class="math-container">$a_2$</span>, these are just labels. The game in some sense is a mechanism that will map the strategy profile to a payoff vector. This mechanism can be such that <span class="math-container">$(D,D)$</span> maps to <span class="math-container">$(-1,-1)$</span>, and in a differently phrased but essentially identical mechanism the mapping can be <span class="math-container">$(a_1,b_1) \to (-1,-1)$</span>.</p>
<p>Another example: <br>
If we shared a language other than English (and if English were not the language of the SE), I could write this answer to you in the other language, using different symbols and words, but conveying the same &quot;message&quot;.</p>
<p>Yet another example: <br>
Suppose you want to send me a secret signal (we are spies or counting cards). We can agree that the signal will be a wink. Or we could agree it would be a nod. Or a sneeze. It does not really matter - we just have to agree on the mapping beforehand.</p>
<hr />
<p><em>Okay, but what is a &quot;message&quot; in economics?</em> <br>
This is kept broad on purpose. As you can see above what passes as a &quot;message&quot; depends on the context. It could be your expectations about inflation which you send to someone making a survey, it could be the sales numbers of your business units you send to the corporate sales officer, etc. The surveymaker/sales officer may then send you the results of their survey, resulting in a &quot;signal&quot;, which may inform your strategy going forward.</p>

CC BY-SA 4.0
2021-10-03T07:26:59.017Z
Thanks for your answer I think it is helpful....i will change the name of the question to what can be considered as a "message"
Nav89
CC BY-SA 4.0
2021-10-03T05:53:59.907Z
2021-10-03T09:02:53.723Z
2021-10-03T09:02:53.723Z
Giskard
<p>As a sequel of some other questions that I have done for communication and information mechanisms with respect to the paper of <a href="https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.685.4204&amp;rep=rep1&amp;type=pdf" rel="nofollow noreferrer">Gossner</a> I have a question for the signal function.</p>
<p>Fix a finite set of states <span class="math-container">$\Omega$</span> and set the full support common prior as <span class="math-container">$\psi\in\Delta_{++}(\Omega)$</span>.</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">$l: T\times\Omega\to \Delta(Y)$</span> (where <span class="math-container">$T=(\Pi_{i\in I} T^i)$</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
<span class="math-container">$$l:\mathcal{T}\times\Omega\to \Delta(Y)$$</span></p>
<p>According to the communication mechanism, the definition reminds Lehrer's definition and more precisely it says that</p>
<p>Definition 2.2. Given a compact game G and a communication mechanism <span class="math-container">$(C, G,\psi)$</span> is the game <span class="math-container">$G$</span> extended by <span class="math-container">$C$</span> that unfolds as follows:</p>
<ul>
<li>the state of the world <span class="math-container">$\omega\in\Omega$</span> is realized</li>
<li>each player <span class="math-container">$i$</span> sends a message <span class="math-container">$\tau_i$</span> to the mechanism</li>
<li><span class="math-container">$y\in Y$</span> is drawn according to <span class="math-container">$l(\tau)$</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> s.t. <span class="math-container">$\sigma_i:\Omega\times\Delta(Y)\to\Delta(S^i)$</span> according to <span class="math-container">$y_i$</span>;</li>
<li>the vector payoff is realized.</li>
</ul>
<p>My question is the following:</p>
<p><span class="math-container">$\textbf{Question:}$</span> The signal function takes a profile of messages <span class="math-container">$\tau$</span> and in return we take some kind of signal <span class="math-container">$y$</span>. I can not think of something that can be considered as &quot;message&quot;. What can this be? I mean the term message is too general. For example Yuval Heller in a papers says that it is an alphabet, however it does not seem to me so easy to think of an economic alphabet that gives back some kind of signal. Also, I think it can be a random variable that denotes some characteristic of the economic environment or the individual characteristics of the player. What does this <span class="math-container">$\tau$</span> represents? Is this something like a random variable or some kind of singal or a word like &quot;short&quot;, &quot;long&quot;, &quot;indifferent&quot;?</p>

<mathematical-economics><game-theory><signaling>
What can be considered as a "message"
Nav89
# What can be considered as a "message"

(Asked by: username_0 on 2021-10-02)

As a sequel of some other questions that I have done for communication and information mechanisms with respect to the paper of [Gossner](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.685.4204&rep=rep1&type=pdf) I have a question for the signal function.

Fix a finite set of states $\Omega$ and set the full support common prior as $\psi\in\Delta\_{++}(\Omega)$.

$\textit{Definition of a communication mechanism:}$ 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 $l: T\times\Omega\to \Delta(Y)$ (where $T=(\Pi\_{i\in I} T^i)$) 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
$$l:\mathcal{T}\times\Omega\to \Delta(Y)$$

According to the communication mechanism, the definition reminds Lehrer's definition and more precisely it says that

Definition 2.2. Given a compact game G and a communication mechanism $(C, G,\psi)$ is the game $G$ extended by $C$ that unfolds as follows:

* the state of the world $\omega\in\Omega$ is realized
* each player $i$ sends a message $\tau\_i$ to the mechanism
* $y\in Y$ is drawn according to $l(\tau)$ and each player $i$ is informed of $y\_i$;
* each player $i$ chooses $\sigma\_i\in\Sigma^I$ s.t. $\sigma\_i:\Omega\times\Delta(Y)\to\Delta(S^i)$ according to $y\_i$;
* the vector payoff is realized.

My question is the following:

$\textbf{Question:}$ The signal function takes a profile of messages $\tau$ and in return we take some kind of signal $y$. I can not think of something that can be considered as "message". What can this be? I mean the term message is too general. For example Yuval Heller in a papers says that it is an alphabet, however it does not seem to me so easy to think of an economic alphabet that gives back some kind of signal. Also, I think it can be a random variable that denotes some characteristic of the economic environment or the individual characteristics of the player. What does this $\tau$ represents? Is this something like a random variable or some kind of singal or a word like "short", "long", "indifferent"?

**Tags:** mathematical-economics, game-theory, signaling


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## 1 answers

### Answer by username_1 on 2021-10-03. Score: 2 | Accepted answer

Given that

> $T^i$ is $i's$ finite set of messages

and the mechanism uses the signal function to map $T$ to a signal, the architects of the mechanism can use any arbitrary label for the individual messages. In simple game theoretical examples, the strategies of player 1 are often denoted $a\_1,a\_2...$ This does not tell us anything about what player 1 is doing, these are just labels.

E.g., in the Prisoner's dilemma player 1 can either Defect $(D)$ or Cooperate $(C)$. Yet you could also denote these strategies by $a\_1$ and $a\_2$, these are just labels. The game in some sense is a mechanism that will map the strategy profile to a payoff vector. This mechanism can be such that $(D,D)$ maps to $(-1,-1)$, and in a differently phrased but essentially identical mechanism the mapping can be $(a\_1,b\_1) \to (-1,-1)$.

Another example:   
If we shared a language other than English (and if English were not the language of the SE), I could write this answer to you in the other language, using different symbols and words, but conveying the same "message".

Yet another example:   
Suppose you want to send me a secret signal (we are spies or counting cards). We can agree that the signal will be a wink. Or we could agree it would be a nod. Or a sneeze. It does not really matter - we just have to agree on the mapping beforehand.

---

*Okay, but what is a "message" in economics?*   
This is kept broad on purpose. As you can see above what passes as a "message" depends on the context. It could be your expectations about inflation which you send to someone making a survey, it could be the sales numbers of your business units you send to the corporate sales officer, etc. The surveymaker/sales officer may then send you the results of their survey, resulting in a "signal", which may inform your strategy going forward.

> **Comments:**
> **username_0**: Thanks for your answer I think it is helpful....i will change the name of the question to what can be considered as a "message"  

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