  We consider a Bayesian persuasion problem where the persuader and the
decision maker communicate through an imperfect channel that has a fixed and
limited number of messages and is subject to exogenous noise. We provide an
upper bound on the payoffs the persuader can secure by communicating through
the channel. We also show that the bound is tight, i.e., if the persuasion
problem consists of a large number of independent copies of the same base
problem, then the persuader can achieve this bound arbitrarily closely by using
strategies that tie all the problems together. We characterize this optimal
payoff as a function of the information-theoretic capacity of the communication
channel.
