# I present a communication game - Could you please make comments on my assumptions, notation and properties that I may have not considered yet?

I consider the following communication game. Suppose that we have $$I$$ players and each one of them learns a private signal $$s_i=(s_{i,1},s_{i,2},...,s_{i,k})$$, where $$k$$ is finite and also, every player $$i$$ learns a private code name $$l_i$$, which serves as her "name" to refer to, in the game. Imagine that $$l_i$$ is like a unique id (positive integer) for every player $$i$$ such that $$l=(l_1,l_2,\cdots,l_i)\in L$$ is the profile of ids and $$L$$ is of the same dimension with $$I$$. The game is played in three rounds that consist of two talk phases and one play phase.

1. In the first round the players talk, sending the following message $$(l_i,s_{i,1},s_{i,2},...,s_{i,k})=(l_i,m_i)\in \mathbb{Z}_p^{k+1}$$, which is interpreted in the sense that "I am player $$l_i$$ and I report that my private information is $$s_{i,1}, s_{i,2}, ..., s_{i,k}$$". The message space $$m_i\in \mathbb{Z}_p^k$$ is the space of truthful reporting signals. Note that $$\mathbb{Z}_p$$ is the set of integers modulo $$p$$, where $$p$$ is a large prime positive integer.
2. In the second round they also talk and after gathering the whole information they respond back to each other a mixed action to play, that is modeled in the following sense. Each player $$i$$ will learn all the distribution $$m$$ of messages and will give back the message $$r_i=\pi_i\circ(1_{L}\times1_{M}\times g_i)$$ such that $$\pi_i$$ is a permutation, $$1_L$$ is the identity on $$L$$, $$1_M$$ the identity on $$M$$ and $$1_{L}\times1_{M}\times g_i:L\times M\to L\times M\times \Delta(A^i): (l,m)\to(l,m,g_i(l,m))$$. We denote $$\Delta(A^i)$$ with the profile of mixed actions send by the palyer $$i$$ to the rest of the players. Permutation $$\pi_i$$ serves as an encryption so as every $$j$$ will learn her own coordinate and we define this in the sequel. To be more precise

$$\pi_i=\begin{pmatrix}(l_1,m_1) & (l_2,m_2) & \cdots & (l_j,m_j) & \cdots & (l_I,m_I)\\ g_1(l_1,m_1) & g_2(l_2,m_2) & \cdots & g_j(l_j,m_j) & \cdots & g_I(l_I,m_I)\end{pmatrix}$$

the above representation shows that every $$(l_j,m_j)$$ is associated with exactly one $$g_i(l_j,m_j)$$ which is a mixed action that is instructed by player $$i$$ as a recommendation to player $$j$$. Also $$g_i(l,m)$$ is a vector of $$|I|$$ dimension assuming that the recommendation $$g_i(l_i,m_i)$$ is the message that she sends to herself.

1. In the third phase the players play their recommended strategies based on a honest majority since they are truthful in the beginning of the game, namely every player $$j$$ will play the recommendation according to the decision mapping $$\tau_i:L\times M\times \Delta(A^j)\to \Delta(A_i)$$ where by $$\Delta(A_i)$$ we denote the space of mixed actions of player $$i$$

$$\tau_i(l,m,g_j)=pr_i\circ\pi^{-1}_j$$

In order to play the recommended strategy player $$i$$ must receive the same recommendation from the majority (excluding herself although her opinion by the message she sends to herself helps as a vitrification scheme).

This is a proposed game that extends the game of Universal mechanisms $$(1990)$$. More precisely I present a small extension of the proof. My worries are the following.

1. I define $$g_i$$ as a function that takes as input a vector of $$(l,m)\in\mathbb{Z}_p^{Ι(k+1)}$$ where $$p$$ is a large prime positive integer and $$g_i\in\Delta(A^i)$$ is a profile of mixed actions (vector) of dimension $$I$$ where every player $$j$$ will learn her own coordinate by $$i$$ and also player $$i$$ can send a message to herself (namely $$g_i$$ is of dimension $$|I|$$). What assumptions do I need to do so as $$g_i$$ is injective? I think that I need two things to examine here, the first one linearity of the profile $$g_i(l,m)\in\Delta(A^i)$$ of mixed strategies that are proposed by the player $$i$$ and that it is 1-1.

2. I just want an opinion if anyone of you see something missing in my assumptions or in the formalization in general.

P.S. I know that it is not as much detailed as it should be, but it is the best I can do write know.

P.S. For reasons of formalization, I assume that the id name of every player $$i$$ is $$l_i$$.

• Having checked only 1. at the top: What is $p$? What is $\mathbb{Z}_p$? The last entry of $l$ should be $l_I$. If $(l_i,m_i)\in\mathbb{Z}_p^{k+1}$, then $l_i\in\mathbb{Z}_p$ and $L=\mathbb{Z}_p^I$, so why introduce $L$ at all? Do you assume $m_i=s_i$ for all players? If yes, then why give them different names? If no, then you shouldn't write down the same profile for both. Mar 31 at 7:24
• $\underbrace{\mathbb{Z}_{p}^{k+1}\times\mathbb{Z}_p^{k+1}\times\cdots\mathbb{Z}_p^{k+1}}_{\text{$I$times}} = (\mathbb{Z}_{p}^{k+1})^I = \mathbb{Z}_{p}^{I(k+1)}$, right? Mar 31 at 7:27
• @VARulle For any prime $p$ , the set $\mathbb{Z_p}$ with the addition mod p and multiplication mod p, and congruence mod p, is a field. Mar 31 at 7:47
• @VARulle for your second comment yes you are right. Mar 31 at 7:48
• I re-edited the $1$. question and I think that in order for my solution to be well defined I need to check two things, the linearity of $g_i \in \Delta(A^i)$ which is a profile (namely vector) of mixed strategies, and this has to be a $1-1$ mapping. In essence $g_i:\mathbb{Z}_p^{I(k+1)}\to \Delta(A^i)$ where $\Delta(A^i)$ is a profile of mixed strategies. Note again that the exponent i refers to the player who sends the recommendation, namely $A^i=A_1^i\times A_2^i\times\cdots\times A_I^i=\Pi_{j=1}^IA_j^i$, which differs from $A_i$ Mar 31 at 11:52