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As a sequel of some other questions that I have done for communication and information mechanisms with respect to the paper of Gossner 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"?

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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.

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  • $\begingroup$ 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" $\endgroup$
    – Nav89
    Oct 3, 2021 at 7:26

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