The economics of information transmission
2002
It is a well known phenomenon that communication between a group of
players affects the set of equilibria of a repeated game. This communication
can be costly. Its cost may be measured in terms of complexity, or payoffs.
In particular, a payoff cost of communication occurs when players communicate
through their actions, and one is interested in characterizing the optimal
communication.
We study a class of games that models the communication cost of online
coordination. For example: A sequence X = (X(t)) of zeroes and ones is chosen
at random, and announced to a player, the seer, but not to his teammate, the
follower. A repeated game then proceeds in which the follower and the seer
choose actions Y (t) and Z(t) in {0, 1} at stage t, and their (common) stage
payoff is 1 whenever X(t) = Y (t) = Z(t).
We address the question of the optimal communication level from the seer
to the follower in order to maximize payoffs. We characterize this level using
the notion of entropy, and provide construction of optimal strategies for the
team: the seer and the follower. In particular, we prove that against a family
of i.i.d. random variables (1/2, 1/2), the maximum expected payoff v to the
team is the solution of the equation
H(x) + log(3) (1 - x) = 1;
where H is the entropy function and log is taken in basis 2. Moreover, we prove
the existence of pure strategies of the prophet and the seer that guarantee v
against all sequences X = X(t). Hence, the zero-sum game in which player I
is the Team (no correlation is allowed between its members), player II is the
opponent, and the payoff to I is as above, has a value, and this equals v. Our
result has direct implications for repeated games played by finite automata, or
by players with bounded recall.
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