Converging Better Response Dynamics in Sender-Receiver Games

We consider information transmission between a sender, who has finitely many types, and a receiver, who must choose a decision in a real interval. The payoffs depend on the sender’s type and the receiver’s decision. We assume that the payoff functions are well-behaved. We characterize the pure strategy perfect Bayesian equilibrium outcomes as incentive-compatible partitions of the sender’s types. We propose an algorithm, which starts from the finest partition. Then, at every step, if the current partition is not incentive compatible, a random type of the sender improves its payoff, and the receiver best responds. We show that every possible run of the algorithm converges to a unique incentive-compatible partition ${\mathrm{\Pi }}_{*}$. This partition ${\mathrm{\Pi }}_{*}$ is such that any partition with more cells than ${\mathrm{\Pi }}_{*}$ is not incentive compatible, so the algorithm determines to which extent information transmission can be effective. The partition ${\mathrm{\Pi }}_{*}$ also satisfies some refinement criteria for perfect Bayesian equilibria in sender-receiver games. Furthermore, in a discrete version of a popular class of examples (namely, if the sender’s type is uniformly distributed and payoff functions are quadratic, with a constant upward bias for the sender), ${\mathrm{\Pi }}_{*}$ ex ante Pareto dominates every other incentive-compatible partition.