Implementability of Correlated and Communication Equilibrium Outcomes in Incomplete Information Games

In a correlated equilibrium, the players’ choice of actions is directed by random, correlated
messages that they receive from an outside source, or mechanism. This allows for more
equilibrium outcomes than without such messages (pure-strategy equilibrium) or with
statistically independent ones (mixed-strategy equilibrium). In an incomplete information
game, the messages may also convey information about the types of the other players,
either because they reflect extraneous events that affect the types (correlated equilibrium)
or because the players themselves report their types to the mechanism (communication
equilibrium). Thus, mechanisms can be classified by the connections between the messages
that the players receive and their own and the other players’ types, the dependence or
independence of the messages, and whether or not randomness is involved. These
properties may affect the achievable equilibrium outcomes, i.e., the payoffs and joint
distributions of type and action profiles. Whereas for complete information games there are
only three classes of equilibrium outcomes, with incomplete information the number is 14–
15 for correlated equilibria and 15–17 for communication equilibria. Each class is
characterized by the properties of the mechanisms that implement its members. The
majority of these classes have not been described before.

JEL Classification: C72.

Keywords: Correlated equilibrium; Communication equilibrium; Incomplete information;
Bayesian games; Mechanism; Correlation device; Implementation