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Suppose that we have a Bayesian game, where $t_i\in T_i$ denotes the type of player $i$. Say that we have a communication game (communication equilibrium). The players do send each other an encrypted message about their type. If $L_i$ is an isomorphic space of $T_i$ and $\phi_i:T_i\to L_i$ is an permutation (injection + surjection= bijection), then every player $i$ instead of sending their type to each other player they can send the message $\phi_i(t_i)=l_i$. Also, in order to protect herself from cheating, let $\rho_i:L_i\times Y_i\to X_i$ be a cipher, that encodes the private information of player $i$, that is, $y_i\in Y_i$ is the key and $x_i\in X_i$ is the code, where again $\rho_i(\cdot,y_i)$ is a bijective so that the pair $(x_i,y_i)$ is associated with exactly one $l_i$. For technical reasons we made the asumption $|Y_i|\geq|T_i|$ (but why? Is this Shannon's property?).

In our game we have $I$ players, with the above representation we can use a lemma from probability theory, that is:

$\textbf{Lemma:}$ If $\phi_i$ is a random variable with support on $\{1,2,\dots,n_i\}$, and $y_i$ is uniformly distributed over $\{1,2,\dots,n_i\}$ indepedent of $\phi_i$, then the random variable $x_i$ defined as $x_i=\phi_i\ominus_{n_i}y_i$ (where $\phi_i\ominus_{n_i}y_i=\phi_i-y_i(mod{n}_i)$) is also uniformly distributed over $\{1,2,\dots,n_i\}$.

In other words $l_i=\phi_i(t_i)=x_i\oplus_{n_i}y_i$. Then every player $i$ instead of sending $l_i$ to the other agents as a message, she sends in the half of them $x_i$ and the rest of them (we do not know if $I=2k$ or $I=2k+1$, with $k\neq 0$ a potitive integer) $y_i$. Then in a subsequent phase they communicate each one to unite the pieces and verify that they only learn $l_i=\phi_i(t_i)=x_i\oplus_{n_i}y_i$ (however they still haven't learned $t_i$ but $\phi_i(t_i)=l_i$.

My questions are the following

  1. Is this mechanism of information transmission secure? If not, how can I make it?
  2. Could I use a scheme as in secret sharing, where every player $i$ could distribute shares of the key $y_i$ to all of the other players $j\in I-\{i\}$? For example could I further assume that $y_i$ is written as a linear combination of some $w_i$ that all these $w_i$ are non-zero and independent such that $y_i=\sum_{j=1}^{I-1}w_jy_j$? Is this right or wrong? Could anybody provide some help-references-guidance or either show some maths that could make it possible to make such a construction?

In general, how can I enrich this mechanism of communication to become more efficient and secure?

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