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"?


1 Answer 1


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.

  • $\begingroup$ 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
    Commented Oct 3, 2021 at 7:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.