Communication, correlated equilibria and incentive compatibility

Publisher Summary In principle, anything that a player can do to communicate and coordinate with other players could be described by moves in an extensive-form game, so that planning these communication moves would become part of the player's strategy choice itself. Adding a communication system does not eliminate any of the equilibria of the original game because there are always equilibria of the communication game in which reports and messages are treated as having no meaning and hence are ignored by all players. Such equilibria of the communication game are called “babbling equilibria.” The set of correlated equilibria of a strategic-form game Г has a simple and tractable mathematical structure because it is closed by convex and is characterized by a finite system of linear inequalities. The revelation principle for strategic-form games asserted that any equilibrium of any communication system can be simulated by a communication system in which the only communication is from a central mediator to the players, without any communication from the players to the mediator. The one-way nature of this communication is not surprising, because the players have no private information to tell the mediator about, within the structure of the strategic-form game. However, players in a Bayesian game may have private information about their types, and two-way communication would then allow the players' actions to depend on each other's types as well as on extraneous random variables, such as coin tosses. Thus, in Bayesian games with communication, there may be a need for players to talk as well as to listen in mediated communication systems.