Int J Game Theory 2001) 30:41±60 9999 2001 Repeated communication through the mechanism and Olivier Gossner*, Nicolas Vieille y * THEMA, Universite  Paris X-Nanterre, 200 avenue de la Re Âpublique, 92001 Nanterre, France and CORE, 34 Voie du Roman Pays, Universite  Catholique de Louvain, Belgique. y Laboratoire d'e Âconome Âtrie, Ecole Polytechnique, and De Âpartement Finance et Economie, HEC, 1 rue de la Libe Âration, 78 351 Jouy en Josas, France e-mail: vieille@hec.fr) Received December 1998/Final version November 2000 Abstract. We consider the ``and'' communication device that receives inputs from two players and outputs the public signal yes if both messages are yes, and outputs no otherwise. We prove that no correlation can securely be implemented using this device, even when in®nitely many stages of commu- nication are allowed. 1. Introduction Our goal in this paper is to analyze the intrinsic correlation opportunities o¨ered by a given communication device Forges [3]). It thus relates closely to the literature on preplay communication, and more precisely to the literature on mediated talk, initiated by Lehrer [8]. In this strand of literature, a game is given, and one wishes to implement correlated equilibrium distributions c.e.d. thereinafter) of the game as the outcome of communication equilibria, using communication devices of a simple form. In Lehrer and Sorin [9], it is shown that any c.e.d. with rational entries) coincides with the distribution induced by some communication equilibrium, where the communication device sends public outputs that depend deterministically on the inputs. It is natural to allow for repeated preplay communication; namely, to consider situations in which preplay communication proceeds in stages. At each stage, the players send inputs to the device, that sends back outputs. In that case, a stronger result is obtained. Given a game, there exists a commu- nication device with public and deterministic output, that has the following property: every c.e.d. can be approximated by the outcome of an equilibrium of the game extended by ®nitely many stages of preplay communication. Thus, the same communication device is used for every c.e.d.; only the length of the preplay communication depends on the particular c.e.d.