A Detail-free Mediator and the 3 Player Case

Two players can make use of a trusted third party who mediates and partially resolves their conflict. Usually, the mediator should be aware of the situation and give suggestions to the players accordingly. However, a corrupt mediator can have a big influence on the outcome of the game. We single out a transparent mediator which can be safely applied in any two player game without loss of efficiency. That is, the mediator is independent of the game and the desired outcome. Technically, we show that any correlated equilibrium of any two player game can be obtained as Nash equilibria of the game, extended with cheap, pre-play communication, where players can communicate through the proposed mediator. The key idea is that after the mediated communication the players can have a plain conversation. In particular, the mediating communication device is transparent, controllable and is the same for all games and for all equilibrium distributions. We extend the result to three player games and show that one of the players can play the role of the mediator. We implement the set of correlated equilibrium in Nash equilibria of an extended game where the players have a plain conversation. The central assumption is that players can be invited to eavesdrop a private conversation. We extend the analysis to games with incomplete information and to the set of communication equilibria.