  The literature on strategic communication originated with the influential
cheap talk model, which precedes the Bayesian persuasion model by three
decades. This model describes an interaction between two agents: sender and
receiver. The sender knows some state of the world which the receiver does not
know, and tries to influence the receiver's action by communicating a cheap
talk message to the receiver.
  This paper initiates the algorithmic study of cheap talk in a finite
environment (i.e., a finite number of states and receiver's possible actions).
We first prove that approximating the sender-optimal or the welfare-maximizing
cheap talk equilibrium up to a certain additive constant or multiplicative
factor is NP-hard. Fortunately, we identify three naturally-restricted cases
that admit efficient algorithms for finding a sender-optimal equilibrium. These
include a state-independent sender's utility structure, a constant number of
states or a receiver having only two actions.
