A problem in game theory is that of communication. Taking into account the classic approach of Myerson and Forges, the agents communicate each other, however indirectly, through a communication mechanism that receives messages from them and replies to them a recommendation according to a rule. Say that $m$ denotes the profile of messages and $q(\cdot|m)$ is the rule of the mechanism such that the recommendation $q:M\to\Delta(A)$ is a profile of mixed actions, that denotes the plan of the mediator provided by the mechanism. Every player $i$ will learn only $a_i$ and not the whole profile of actions. The challenge in such problems is to find an efficient mechanism that replicates the this one, replacing the device or mediator that takes messages and gives back to the players recommendations, with a scheme of plain conversation. For example, the players could talk directly exchanging information and they could generate this rule in such a way where each of them at the end of the communication phase, they would only know their own recommendation $a_i$, while not cheating with their message $m$ at the beginning of the conversation.
A very specific technique in cryptography that helps us to design such mechanisms is the one of secure multiparty computation. Suppose that every player $i$ has a prior information $s_i$ that is private to her. In case the mediator exists there is nothing to be solved in the problem, and it is easy for the agent to trust her. However, when the mediator will be replaced only with cheap talk, the players are not sure that they will not be cheated so there must be a specific encryption-decryption scheme which will help them encrypt the inputs and send encrypted messages, proceed in the stage where they will make the appropriate calculations with the shared parts of the individual information to obtain the whole information $s=(s_1,s_2,...,s_I)$ encrypted, and finally in the last stage where they take the output of the process they reconstruct the rule function $q$. At the end of the process every player $i$ will only know her prior information $s_i$ and her recommendation to play $a_i$ or some mixed action.
The exaple is the following. Each player has some initial information $s_i$, she sends a message $m_i(s_i)=z_i$ that is a function of $s_i$, but whoever learns it, she will learn anything about $s_i$. So, player $i$ sends to the other players $m_i$ and the other players say $j=-i$ send her $m_j(s_j)=\left(m_1(s_1),\dots,m_{i-1}(s_{i-1}),m_{i+1}(s_{i+1}),\dots,m_{I}(s_{I})\right)=\left(l_1,\dots,l_{i-1},l_{i+1},\dots,l_{I}\right)=l_j$. Therefore every player $i$ obtains the tuple $(m_i(s_i),m_j(s_j))$. At the next stage some calculations are made. At the output stage the players want the probability distribution
$$P(f)=P\left(f(s)_s{\in S}\right)=\Pi_{s\in S}q(f(s)|s)$$
More precisely, the inpout will be some function $g(l_1,l_2,\dots,l_I)=f\left(m_1^{-1}(l_1),m_2^{-1}(l_2),\dots,m_I^{-1}(l_I)\right)$, where $q$ should be reconstructed in such a way that will be a composition of functions of the previous stages and every player will learn $g_i(l_i)=a_i$.
Could such a protocol designed with the help of multiparty computation? What are the assumptions that must be made for the encryption functions $m_i$ and what extra assumptions do I need? Is there in the literature of computer since any similar scheme?
$\textbf{Hint:}$ The protocols of Rabin and Ben-or has some of the above properties and Françoise Forges, but how could someone combine them? Can we? Any help or idea how to search in the literature for similar protocols is appreciated beyond this two papers.
