# Example of exchanging information

I am searching for a simle model that can simulate the following procedure.

Suppose that $$i$$ and $$j$$ are two agents that each one obtains her state dependets signal $$s_i(\omega)$$ and $$s_j(\omega)$$. After observing their own signals with probability $$1$$, they do not know anything about the signal that the other agent has, but they do know the common prior $$\pi$$ about the signals, s.t. $$\pi:\Omega\to \Delta(S)$$.

To make it simply, keep the state $$\omega\in\Omega$$ wise and let us assume that $$S_i=\{s_1(\omega),s_2(\omega)\}=S_j$$ namely there are two signals and the signal space is the same for both plyers. I refer to each players signal with the index $$i$$ or $$j$$ because they may both observe either the same signal, say $$s_1(\omega)$$ or $$s_2(\omega)$$ or each of them may observe different signals.

The playes can communicate each other directly, sending a message and let suppose that $$M$$ is an arbitrary message space wher $$\emptyset\in M$$ denotes that the players do not send any message.

If player $$i$$ send a message and given that he will be truthful, say $$m_{i\to j}(s_i(\omega))$$ (the other agent sends $$m_{j\to i}(s_j(\omega))$$ resp.)then he needs to follow some procedure of encodind and decoding, which is not known to me where some keys are drawn according to a know probability distribution know as $$U[0,1)$$ which I assume refers to the uniform $$[0,1)$$?(correct me if I am mistake). So the players know somehow the keys to decrypt the messages when they take it.

My questions are the following:

$$\textbf{Q1:}$$ Can anybody provide the process of encoding and decoding between the two plaeyrs exchange of messages?

$$\textbf{Q2:}$$ Can this communication be perfectly secure? For example none esle could intervene to steal their" information or create some noise changing these encryption-decryption keys that will make the agnets misintepreat their messages?

$$\textbf{Q3:}$$ After the players receive their messages, then could they use some veryfication procees to be sure about the encrypted messages that they received and find out if someone stole their" information or create the noise that I am assuming in $$\textbf{Q2:}$$?

$$\textbf{Q4:}$$ If agent $$i$$ (or $$j$$ or both) decide to mislead the other agent, by sending a message with a wrong key or lying about his type, is there any chance that the other agent could detect the lie based on some procedure and the fact that he knows the set of signals and the common prior of them?

$$\textbf{Q5:}$$ The probaility distribution U[0,1) is also common knolwde, but could it be replaced with another, like a normal distribution or whatever?

$$\textbf{P.S:}$$ I would like to thank in advance whoever is going to answer all these questions, but I do not know any of the cryptograpgic theory mathematical tools, that may be simple or common sense for most people here, but it is far away from my field. I am interested in the design of information exchange that is decentrized where the agetns can communicate directly to each other, no mediators or trusted parties. I find it very interesting, but I do not understand the cryptographic tools that are used and in most cases evey author uses different theory. I am writing my questions above in a way to understand the most simple procedures of them that are used.

• @moderators I could not tag information theory as well, but I think all these are fine. Please any help could be valuable at the moment, although I know that I ask for too much in only one question. Nov 11, 2021 at 11:16