Signaling Equilibria for Dynamic LQG Games With Asymmetric Information

Abstract

In this article, we consider a finite-horizon dynamic game with two players who observe their types privately and take actions that are publicly observed. Players' types evolve as conditionally independent, controlled linear Gaussian processes, and players incur quadratic instantaneous costs. This forms a dynamic linear quadratic Gaussian game with asymmetric information. We show that under certain conditions, players' strategies that are linear in their private types, together with Gaussian beliefs, form a perfect Bayesian equilibrium (PBE) of the game. This is a signaling equilibrium due to the fact that future beliefs on players' types are affected by the equilibrium strategies. We provide a backward-forward algorithm to find the PBE. Each step of the backward algorithm reduces to solving an algebraic matrix equation for every possible realization of the state estimate covariance matrix. The forward algorithm consists of Kalman filter recursions, where state estimate covariance matrices depend on equilibrium strategies. We provide necessary and sufficient conditions for the existence of the solution of the algebraic matrix equation for scalar actions. Finally, we present numerical examples.