A detail-free mediator

We present an extension to any finite complete information game with two players. In the extension, players are allowed to communicate directly and, additionally, send private messages to a simple, detail-free mediator, which, in turn, makes public announcements as a deterministic function of the private messages. The extension captures situations in which people engage in face-to-face communication and can observe the opponent's face during the conversation before choosing actions in some underlying game. We prove that the set of Nash equilibrium payoffs of the extended game approximately coincides with the set of correlated equilibrium payoffs of any underlying game.