<p>I consider the following communication game. Suppose that we have <span class="math-container">$I$</span> players and each one of them learns a private signal <span class="math-container">$s_i=(s_{i,1},s_{i,2},...,s_{i,k})$</span>, where <span class="math-container">$k$</span> is finite and also, every player <span class="math-container">$i$</span> learns a private code name <span class="math-container">$l_i$</span>, which serves as her &quot;name&quot; to refer to, in the game. Imagine that <span class="math-container">$l_i$</span> is like a  unique id (positive integer) for every player <span class="math-container">$i$</span> such that <span class="math-container">$l=(l_1,l_2,\cdots,l_i)\in L$</span> is the profile of ids and <span class="math-container">$L$</span> is of the same dimension with <span class="math-container">$I$</span>. The game is played in three rounds that consist of two talk phases and one play phase.</p>
<ol>
<li>In the first round the players talk, sending the following message
<span class="math-container">$(l_i,s_{i,1},s_{i,2},...,s_{i,k})=(l_i,m_i)\in \mathbb{Z}_p^{k+1}$</span>, which is interpreted in the sense that &quot;I am player <span class="math-container">$l_i$</span> and I report that my private information is  <span class="math-container">$s_{i,1}, s_{i,2}, ..., s_{i,k}$</span>&quot;. The message space <span class="math-container">$m_i\in \mathbb{Z}_p^k$</span> is the space of truthful reporting signals. Note that <span class="math-container">$\mathbb{Z}_p$</span> is the set of integers modulo <span class="math-container">$p$</span>, where <span class="math-container">$p$</span> is a large prime positive integer.</li>
<li>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 <span class="math-container">$i$</span> will learn all the distribution <span class="math-container">$m$</span> of messages and will give back the message <span class="math-container">$r_i=\pi_i\circ(1_{L}\times1_{M}\times g_i)$</span> such that <span class="math-container">$\pi_i$</span> is a permutation, <span class="math-container">$1_L$</span> is the identity on <span class="math-container">$L$</span>, <span class="math-container">$1_M$</span> the identity on <span class="math-container">$M$</span> and <span class="math-container">$1_{L}\times1_{M}\times g_i:L\times M\to L\times M\times \Delta(A^i): (l,m)\to(l,m,g_i(l,m))$</span>. We denote <span class="math-container">$\Delta(A^i)$</span> with the profile of mixed actions send by the palyer <span class="math-container">$i$</span> to the rest of the players. Permutation <span class="math-container">$\pi_i$</span> serves as an encryption so as every <span class="math-container">$j$</span> will learn her own coordinate and we define this in the sequel. To be more precise</li>
</ol>
<p><span class="math-container">$$\pi_i=\begin{pmatrix}(l_1,m_1) &amp; (l_2,m_2) &amp; \cdots &amp; (l_j,m_j) &amp; \cdots &amp; (l_I,m_I)\\
g_1(l_1,m_1) &amp; g_2(l_2,m_2) &amp; \cdots &amp; g_j(l_j,m_j) &amp; \cdots &amp; g_I(l_I,m_I)\end{pmatrix}$$</span></p>
<p>the above representation shows that every <span class="math-container">$(l_j,m_j)$</span> is associated with exactly one <span class="math-container">$g_i(l_j,m_j)$</span> which is a mixed action that is instructed by player <span class="math-container">$i$</span> as a recommendation to player <span class="math-container">$j$</span>. Also <span class="math-container">$g_i(l,m)$</span> is a vector of <span class="math-container">$|I|$</span> dimension assuming that the recommendation <span class="math-container">$g_i(l_i,m_i)$</span> is the message that she sends to herself.</p>
<ol start="3">
<li>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 <span class="math-container">$j$</span> will play the recommendation according to the decision mapping <span class="math-container">$\tau_i:L\times M\times \Delta(A^j)\to \Delta(A_i)$</span> where by <span class="math-container">$\Delta(A_i)$</span> we denote the space of mixed actions of player <span class="math-container">$i$</span></li>
</ol>
<p><span class="math-container">$$\tau_i(l,m,g_j)=pr_i\circ\pi^{-1}_j$$</span></p>
<p>In order to play the recommended strategy player <span class="math-container">$i$</span> 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).</p>
<p>This is a proposed game that extends the game of <a href="https://www.jstor.org/stable/2938319" rel="nofollow noreferrer">Universal mechanisms <span class="math-container">$(1990)$</span></a>. More precisely I present a small extension of the proof. My worries are the following.</p>
<ol>
<li><p>I define <span class="math-container">$g_i$</span> as a function that takes as input a vector of <span class="math-container">$(l,m)\in\mathbb{Z}_p^{Ι(k+1)}$</span> where <span class="math-container">$p$</span> is a large prime positive integer and <span class="math-container">$g_i\in\Delta(A^i)$</span> is a profile of mixed actions (vector) of dimension <span class="math-container">$I$</span> where every player <span class="math-container">$j$</span> will learn her own coordinate by <span class="math-container">$i$</span> and also player <span class="math-container">$i$</span> can send a message to herself (namely <span class="math-container">$g_i$</span> is of dimension <span class="math-container">$|I|$</span>). What assumptions do I need to do so as <span class="math-container">$g_i$</span> is injective? I think that I need two things to examine here, the first one linearity of the profile <span class="math-container">$g_i(l,m)\in\Delta(A^i)$</span> of mixed strategies that are proposed by the player <span class="math-container">$i$</span> and that it is 1-1.</p>
</li>
<li><p>I just want an opinion if anyone of you see something missing in my assumptions or in the formalization in general.</p>
</li>
</ol>
<p>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>
<p>P.S. For reasons of formalization, I assume that the id name of every player <span class="math-container">$i$</span> is <span class="math-container">$l_i$</span>.</p>

CC BY-SA 4.0
2022-03-31T07:24:24.847Z
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.
VARulle
CC BY-SA 4.0
2022-03-31T07:27:20.353Z
$\underbrace{\mathbb{Z}_{p}^{k+1}\times\mathbb{Z}_p^{k+1}\times\cdots\mathbb{Z}_p^{k+1}}_{\text{$I$ times}} = (\mathbb{Z}_{p}^{k+1})^I = \mathbb{Z}_{p}^{I(k+1)}$, right?
VARulle
CC BY-SA 4.0
2022-03-31T07:47:17.717Z
@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**.
studen21
CC BY-SA 4.0
2022-03-31T07:48:03.380Z
@VARulle for your second comment yes you are right.
studen21
CC BY-SA 4.0
2022-03-31T07:56:22.077Z
I know what $\mathbb{Z}_p$ for a prime number $p$ is, but you used $\mathbb{Z}_p$ before you said anything about $p$, which is confusing. That's what my first comment refered to.
VARulle
CC BY-SA 4.0
2022-03-31T07:58:52.460Z
@VARulle yes you are right, but when i saw this i added the extra detail in the main body of my text. I know that what I write is too technical but trust me, this is just a part of what I have in my notes. And it is the exact part that I want someone third to check if it is ok...
studen21
CC BY-SA 4.0
2022-03-31T08:00:49.220Z
I don't understand the thing about $l_i$ being a "code only know to player $i$". $l_i$ is a number, and if $i$ sends a message containing this number in the first element of the vector, then it can be read by the receivers and they also know $l_i$.
VARulle
CC BY-SA 4.0
2022-03-31T08:08:45.987Z
@VARulle Yes exactly. When he will send the message everyone else knows $l_i$ for every $i$, but before they communicate this, it is also private information. Imagine that this code is like the codes that the traders have in trading desks. Each one of them is know with a number like an id. Instead of giving a name, they use this id. everything will be revealed form $i$ to $j=-i$, even her code name, but what I mean is that in the beginning this was also private information to her.
studen21
CC BY-SA 4.0
2022-03-31T08:21:46.563Z
O.k., but a "*code (positive integer) that is unique for every player*" is already there, it is $i$ itself. So why introduce this $l_i$ then instead of just using $i$ as the id?
VARulle
CC BY-SA 4.0
2022-03-31T09:06:06.960Z
@VARulle for formalization reasons, let us assume that the id name is $l_i$ instead of $i$. Nothing more. I am not gonna change the main body of the text again.
studen21
CC BY-SA 4.0
2022-03-31T11:52:42.030Z
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 $A_i$
studen21
<microeconomics><mathematical-economics><game-theory><bayesian-game><linear-algebra>
I present a communication game - Could you please make comments on my assumptions, notation and properties that I may have not considered yet?
studen21
# I present a communication game - Could you please make comments on my assumptions, notation and properties that I may have not considered yet?

(Asked by: username_0 on 2022-03-30)

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.

3. 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 $(1990)$](https://www.jstor.org/stable/2938319). 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 $l\_i$.

**Tags:** microeconomics, mathematical-economics, game-theory, bayesian-game, linear-algebra

> **Comments:**
> **username_1**: 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.  
> **username_1**: $\underbrace{\mathbb{Z}_{p}^{k+1}\times\mathbb{Z}_p^{k+1}\times\cdots\mathbb{Z}_p^{k+1}}_{\text{$I$ times}} = (\mathbb{Z}_{p}^{k+1})^I = \mathbb{Z}_{p}^{I(k+1)}$, right?  
> **username_0**: @username_1 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**.  
> **username_0**: @username_1 for your second comment yes you are right.  
> **username_1**: I know what $\mathbb{Z}_p$ for a prime number $p$ is, but you used $\mathbb{Z}_p$ before you said anything about $p$, which is confusing. That's what my first comment refered to.  
> **username_0**: @username_1 yes you are right, but when i saw this i added the extra detail in the main body of my text. I know that what I write is too technical but trust me, this is just a part of what I have in my notes. And it is the exact part that I want someone third to check if it is ok...  
> **username_1**: I don't understand the thing about $l_i$ being a "code only know to player $i$". $l_i$ is a number, and if $i$ sends a message containing this number in the first element of the vector, then it can be read by the receivers and they also know $l_i$.  
> **username_0**: @username_1 Yes exactly. When he will send the message everyone else knows $l_i$ for every $i$, but before they communicate this, it is also private information. Imagine that this code is like the codes that the traders have in trading desks. Each one of them is know with a number like an id. Instead of giving a name, they use this id. everything will be revealed form $i$ to $j=-i$, even her code name, but what I mean is that in the beginning this was also private information to her.  
> **username_1**: O.k., but a "*code (positive integer) that is unique for every player*" is already there, it is $i$ itself. So why introduce this $l_i$ then instead of just using $i$ as the id?  
> **username_0**: @username_1 for formalization reasons, let us assume that the id name is $l_i$ instead of $i$. Nothing more. I am not gonna change the main body of the text again.  
> **username_0**: 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 $A_i$  

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## 0 answers

