Although there is no use of cryptographic protocols in Gossner (1998), the author refers to protocols of communication and he has a main result that I struggle to prove, because he does not use a specific way to do it. If anyone has seen the paper and could give some intuitive explanation from where to start or he\she could prove the theorem or give some steps to do so I would appreciate it.
The following definitions are taken from the paper of Gossner (1998). I want to understand how he proves the theorem that is the main result in his work. I give four definitions and the theorem which is based on them.
$\textit{Definition 1:}$ $I$ is a finite set of players and and $G=((S_i)_{i=1}^I,g)$ is a compact game, that is given by a compact set of strategies $S_i$ for each player $i$ and by a continuous payoff function $g:S(=\times S_i) \to \mathbb{R}^{I}$. Also the mixed set of strategies is defined as $\Sigma_i=\Delta(S_i)$ which is a standard way in game theory.
$\textit{Definition 2:}$ The information structure $\mathbb{I}=((X_i)_{i=1}^{I},\mu)$ is given by a finite set of signals $X_i$ for each $i$ and by a probability measure $\mu$ over $X$. When $x$ is drawn according to $\mu$, player $i$ is informed about the coordinate $x_i$.
$\textit{Definition 3:}$ A communication mechanism is a triple $\mathbb{C}=((T_i)_{i=1}^I, (Y_i)_{i=1}^I , l )$, where $T_i$ is $i's$ finite set of messages, $Y_i$ is $i's$ finite set of signals, and $l: T\to \Delta(Y)$ is the signal function. When $t$ is the profile of messages sent by the players to the mechanism, $y\in Y$ is drawn according to $l(\cdot|t)$ and player $i$ is informed of $y_i$. $\mathbb{T}_i=\Delta(T_i)$ represents the set of mixed messages for player $i$ and $l$ is extended to $\mathbb{T}$ by $l(y|\tau)=\mathbb{E}_{\tau} l(y|t)$.
In this place I can define the protocol
$\textit{Definition 4:}$ For a given communication mechanism $\mathbb{C}$, the protocol is a pair $(\tau,\phi)$, such that:
- a translation $\phi=(\phi_i)_{i=1}^I$ is a family of mappings $\phi_i: Y_i\to \Delta(X_i)$
- and $\tau$ is a profile of mixed messages
In his paper Gossner proves a theorem which is his main result and it is the following
$\textit{Theorem:}$ $(\tau,\phi)$ is secure if and only if
For every player $i$ and $\tau_i^{'}\in\mathbb{T}_i$, $m(\tau_i^{'})=m(\tau_i)$
For every player $i$ and $\tau_i^{'}\in\mathbb{T}_i$, $\alpha_i\supset \gamma_{\tau_{i}^{'}}$
the function $m(\tau_i^{'})$ denotes the marginal of $P_{(\tau_{i}^{'},\tau_{-i})}$ on $X_{-i}$ where $$P_{(\tau_{i}^{'},\tau_{-i})}(y,x)=l(y|(\tau_{i}^{'},\tau_{-i}))\phi(x|y)$$ and $m(\tau_i^{'})$ is the probability over players other than $i$'s translated signals when they follow the protocol $(\tau,\phi)$ and when $i$'s messages are distributed according to $\tau_{i}^{'}$.
The second bullet of the theorem refers to the comparison of statistical experiments due to Blackwell theorem in statistical experiments, in order to compare different information of a player on other's signals. It's meaning is that if $\alpha_i\supset \gamma_{\tau_{i}^{'}}$ then the statistical experiment $\alpha_i$ is more informative than $\gamma_{\tau_{i}^{'}}$.
$\textit{Question:}$ Can anybody prove the theorem above?
$\textit{Hint:}$ In order to define incentives for the players not to change their messages or compute the new messages from the communication procedure differently the author makes a comparison between the N.E. of the basic game extended by the communication mechanism with respect to the extended game by the information generated by the protocol.
References
Gossner (1998). Secure protocols or how communication generates correlation. Journal Of economic Theory, 83(1), 69-89.
