The following definitions are taken from the paper of Gossner (2000). 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 messsages
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 expiriments, in order to compare different informations of a player on other's signals. It's meaning is that if $\alpha_i\supset \gamma_{\tau_{i}^{'}}$ then the statistical expiriment $\alpha_i$ is more informative than $\gamma_{\tau_{i}^{'}}$.
$\textit{Question:}$ Can anybody proove the theorem? In the paper he uses different methods to make the proof, but I do not care which one could someone use to proove it or even he\she could find an intuitive way to explain one of them. I strugle to understand his notation.
$\textit{Hint:}$ In order to defince incentives for the players not to change their messages or compute the new messages from the communication procedure differently the authos makes a comparision between the N.E. of the basic game extended by the communicaition mechanism with respect to the extended game by the information generated by the protocol.