Equivalence from correlated/communication equilibrium to Nash Equilibrium?

Question

Taking into account the seminal papers of Forges and Imre Bárány, they proove a very strong result that gives an exact connection among the communication and the correlation equilibrium solution concept with the correlation and (Baysian) Nash equilibrium solution concept. The latter result gives the theoretic solution of unmediated communication. The basic idea comes from the following theorem of Barany's paper below.

$\textbf{Theorem:}$ Let $G_o$ be a $n$-person game and $x$ be a correlated equilibrium payoff of $G_o$ with rational valued underlying probability distribution. Then there exists a direct communication game $G$ extending $G_o$ (i.e. one where plain conversations allowed before moving) such that $x$ is a Nash equilibrium payoff of $G$.

In this point I give the notions of distribution protocols and sure protocols according to Barany.

A distribution protocol (DP) that can used to replace fortune should satisfy the following properties. For each $k=1,2,...,n$, $I_k$ determines the letter $a_k\in A_k$ uniquely, i.e. there is a map $f_k$ knwon to $P_k$ with $f_k(I_k)=a_k$ such that

$1.$ $Prob(f_1(I_1)=a_1,f_2(I_2)=a_2,\cdots,f_k(I_n)=a_n)=p(a_1,a_2,\cdots,a_n)$

$2.$ $Prob(f_1(I_1)=a_1,f_2(I_2)=a_2,\cdots,f_k(I_n)=a_n|I_k)=p(a_1,a_2,\cdots,a_n|a_k)$ (which means that no more information is available to player $k$ than $a_k$ because he knows $I_k$

$3.$ Any unilateral deviation in probability does not influence conditions $1.$ and $2.$

$4.$ Any unilateral deviation from the rules in detected with probability $1$

A sure protocols (SP) is satisfying all $1.$ $2.$ $3.$ and $4.$

Could anyone provide help or details how this is proved after all? In the case of Barany, no explicit proof is given, while Forges tries to make a set-theoretic proof making a transition from the pre-play communication equilibrium with respect th the pre-play plain scehme of conversation (the latter is simple cheap talk with no mediation while the former needs a device that coordinates the players and helps the trnamission of the meesages).

Answer 1

You take $p$ to be the corresponding correlated equilibrium with $A_k$ being the strategy space of player $k$ Conditions 1. and 2. mean that each player can compute the prescribed action they should play and that they do not know more than this prescription would give them. This is exactly what a correlated equilibrium requires (together with optimality, of course). Now, it might still be possible to manipulate these probabilities. Condition 3. says that misreporting probabilities will not change the resulting distribution, so there is no advantageous deviation in which probabilities are misreported. Condition 4. covers the remaining possible deviations. Since they are detectable, a player who deviates this way can be punished (jointly minmaxed) by the other players. Of course, the protocol does not guarantee that carrying out the punishment is in the interest of the punishers; no condition of sequential rationality is imposed. You only get a Nash equilibrium.