Signaling with Private Monitoring

A sender signals her private information to a receiver who privately monitors the sender’s behavior, while the receiver transmits his private inferences back through an imperfect public signal of his actions. In a linear-quadratic-Gaussian setup in continuous time, we construct linear Markov equilibria, where the state variables are the players’ beliefs up to the sender’s second order belief. This state is an explicit function of the sender’s past play—hence, her private information—which leads to separation through the second-order belief channel. We examine the implications of this effect in models of organizations, reputation, and trading. We also provide a fixed-point technique for finding solutions to systems of ordinary differential equations with a mix of initial and terminal conditions, and that can be applied to other dynamic settings.