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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 $l_i$.

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    $\begingroup$ 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. $\endgroup$
    – VARulle
    Mar 31 at 7:24
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    $\begingroup$ $\underbrace{\mathbb{Z}_{p}^{k+1}\times\mathbb{Z}_p^{k+1}\times\cdots\mathbb{Z}_p^{k+1}}_{\text{$I$ times}} = (\mathbb{Z}_{p}^{k+1})^I = \mathbb{Z}_{p}^{I(k+1)}$, right? $\endgroup$
    – VARulle
    Mar 31 at 7:27
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    $\begingroup$ @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. $\endgroup$
    – studen21
    Mar 31 at 7:47
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    $\begingroup$ @VARulle for your second comment yes you are right. $\endgroup$
    – studen21
    Mar 31 at 7:48
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    $\begingroup$ 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 $A_i$ $\endgroup$
    – studen21
    Mar 31 at 11:52

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