# What can be considered as a "message"

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"?

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.

• 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" Oct 3 '21 at 7:26