Several meaningful properties of new definitions can be easily verified.
21The set of BCE and CCE outcomes are numerically calculated and plotted using
First, correlated cheap talk and correlated communication equilibrium outcomes are invariant to adding redundant types to the given type space in an incomplete information game G. That is, the set of correlated cheap talk and correlated communication equilibrium outcomes depend only on play- ers’ belief hierarchies regarding Θ, which are determined by players’ types (Ti)i∈I in given G.22 This property is shared with Liu’s (2015) belief invari-
ant Bayes correlated equilibrium and Dekel, Fudenberg and Morris’ (2007) interim correlated rationalizability.
Observation 5. Given an incomplete information game G, the set of cor- related cheap talk equilibrium outcomes and the correlated communication equilibrium outcomes are invariant to the addition of redundant types to the type structure T in G.
Proof. Established by definitions.
Next, we know from Forges (1986) that the set of communication equilib- rium outcomes is a convex polyhedron because it can be described by linear inequalities. For a given G, if we fix the correlating device or the correlated communication device, then the set of correlated cheap talk equilibrium out- comes and the set of correlated communication equilibrium outcomes are convex polyhedron because they are described by sets of linear inequalities. The sets of equilibrium outcomes given G with varying correlating devices or correlated communication devices are unions of convex polyhedra. I strongly suspect that the set of correlated cheap talk equilibrium outcomes and the set of correlated communication equilibrium outcomes are also convex poly- hedra, as we can observe from Figure 8-(b) for the example. However, I do not know the general proof of the claim.
22A type space T in an incomplete information game G models players’ belief hierarchies
Forges (1990) justifies her definition of communication equilibrium by showing that any equilibrium of an extension of G with some form of pre- play communication (which does not explain the correlated communication examined in this paper) can be replicated by a communication equilibrium, in which a communication device receives reports from players about their orig- inal types (ti ∈ Ti)i∈I, and it sends out action recommendations to players.
The correlated communication device introduced in this paper combines two types of mediators: a mediator which represent players’ beliefs in communi- cation games and a mediator which represents a communication protocol as in Forges (1990). Therefore, a correlated communication device can represent any pre-play correlated communication (a communication with uncertainties about players’ strategies) given an incomplete information game as follows. Observation 6. Let us extend an incomplete information game G with the following arbitrary communication ¯C, thereby define GC¯. Before and after
observing ti ∈ Ti, players are engaged in finite communication stages. At
each stage, players receive extra signals (these do not change their beliefs about Θ), send messages to a communication device, and receive communi- cation outcomes from the communication device according to a predetermined rule. The communication protocol described here includes both ex-ante and interim communication, and it is the most general form of finite pre-play communication for a given G. Then any perfect Bayesian equilibrium out- come of some communication extension GC¯ of G as described above can be
obtained by a correlated communication equilibrium outcome of G.
Proof. Given an incomplete information game G, I show that any equilibrium outcome of an arbitrary finite communication extension GC¯ as described
above can be replicated in an epistemic type space of GC, which is a 2-
stage game of G extended by some communication protocol C with properties described in the Proposition 2. Define Ω as the set of final histories of GC¯.
For all ω ∈ Ω, define ˆt(ω) and ˆθ(ω) given by G, and define ˆmi(ω) to be player
i’s reporting strategy in the history ω. Let player i’s information partition PMi
i consist of sets of final histories which share same reporting strategies of
player i. Let information partitionPAi
i to be player i’s information partition
at the final histories. Also, define ˆβi(ω) so that it coincides withPAii. Players’
action strategiesAˆOi
i are defined to coincide with action strategies with given
information ˆβi(ω). Check that rationality conditions of players are satisfied
in the constructed epistemic type space.
A large literature23 discusses how to emulate equilibrium outcomes with
mediated communication through some combination of direct communica- tions. Direct communication, or cheap talk, is the most familiar form of communication, but the existence of a communication device as a mediator can seem to be artificial and unnatural. As noted in the literature, various extended protocols of cheap talk are powerful enough to mimic mediated talk in most cases: Given that the correlated communication equilibrium is identical to the communication equilibrium after belief invariant signals are added, observations from the literature regarding relationships between mediated and direct communication are applicable to relationships between correlated communication equilibrium and some extended protocols of cor- related cheap talk.
Observation 7. In most cases, a correlated communication equilibrium out- come of game G can be achieved by players who observe belief invariant sig-
23See Forges (1990), B´ar´any (1992), Ben-Porath (2003), Gerardi (2004), Krishna (2007),
and Vida and Forges (2013). Note that Vida and Forges (2013) defines a “correlated cheap talk” that is different from what is defined in this paper. Vida and Forges (2013) allow extra signals that are independent to original signals before repeated cheap talk game, in order to emulate a mediated communication by a repeated cheap talk preceded by independent extra signals. In contrast, the correlated cheap talk suggested in this paper allows players to observe extra signals that can be correlated to the original signals with belief invariance, and extra signals represent players’ beliefs.
nals C and then engage in some extended direct communication protocol.