Costly Communication in Repeated Interactions

Olivier Gossner 

We introduce a model of dynamic interactions with asymmetric
information and costly communication. One player, the
forecaster, has superior information to another player, the
agent, concerning the realizations of a stream of states of
nature.

A repeated game takes place between the sequence, the
forecaster, and the agent. The agent chooses at each stage
an action from a finite set that depends on the past
history. The forecaster's stage decisions may depend not
only on past history, but also on future realizations of
nature. Hence, there are two aspects in the forecaster's
stage decisions. First, this player can take strategic
decisions that may affect both player's stage payoff.
Second, the forecaster may choose messages from a set that
will be subsequently observed by the agent. Our model
encompasses these two aspects in a unified action set for
the forecaster.

Our model allows player's stage payoffs to depend
arbitrarily on their actions and on the state of nature.
This includes the particular case in which the forecaster's
action set is separable in a strategic decision space and a
message space, and in which payoffs do not depend on the
second.

In order to achieve cooperative outcomes in this repeated
game, it may be necessary for the forecaster to use actions
to inform the agent of future realized states of nature.
Using information theory, we measure the amount of
information sent by the forecaster to the agent at each
stage, and the amount of information used by the agent in
the course of the game.

Intuitively, during repetitions of the game, the total
information used by the agent cannot exceed the total
information received.  This can be easily formalized and
expressed using our measures for information. This in turn
has implications on the set of empirical distributions over
triples (state of nature, forecaster's actions, agent's
actions) that are attainable by some strategies of the
players in the repeated game. We express this constraint via
a single formula, that we call the information constraint.

On the other hand, we prove that, given any distribution $Q$
over this triple that fulfills the information constraint,
there exists a communication scheme between the forecaster
and the agent such that the induced distribution is $Q$.

Therefore, our results establish that the set of achievable
empirical distributions given the communication constraints
of the game is fully characterized by the information
constraint.

Given any payoff function, the set of feasible payoffs in
the repeated game can be computed by taking the image of the
set of feasible distributions. Therefore, the information
constraint also characterizes the set of feasible payoffs
for all possible payoff functions.

We provide applications of the above approach to team
problems and to situation with non-common objectives. When
both the forecaster and the agent share common preferences,
we characterize the best expected payoff that their team can
guarantee. When preferences are not aligned, we characterize
the set of equilibrium payoffs of the repeated game.