Signaling with Private Monitoring

TL;DR

This paper analyzes dynamic signaling with private monitoring, where an informed long-run player signals her type over time to a myopic observer who privately monitors her actions, leading to a novel separation effect in equilibrium.

Contribution

It introduces a linear Markov equilibrium framework using second-order beliefs to account for private monitoring, revealing new signaling dynamics.

Findings

Private monitoring creates a separation effect in signaling.

Equilibrium characterized by belief states based on second-order beliefs.

Applications to leadership, reputation, and trading models.

Abstract

We study dynamic signaling when the informed party does not observe the signals generated by her actions. A long-run player signals her type continuously over time to a myopic second player who privately monitors her behavior; in turn, the myopic player transmits his private inferences back through an imperfect public signal of his actions. Preferences are linear-quadratic and the information structure is Gaussian. We construct linear Markov equilibria using belief states up to the long-run player's $\textit{second-order belief}$. Because of the private monitoring, this state is an explicit function of the long-run player's past play. A novel separation effect then emerges through this second-order belief channel, altering the traditional signaling that arises when beliefs are public. Applications to models of leadership, reputation, and trading are examined.