Linear Equilibria for Dynamic LQG Games with Asymmetric Information and Dependent Types

TL;DR

This paper characterizes perfect Bayesian equilibria in a complex dynamic LQG game with asymmetric, dependent private information, showing that players can summarize hierarchical estimates into their own state estimate, enabling a backward/forward solution approach.

Contribution

It demonstrates that hierarchical estimates in asymmetric information LQG games can be reduced to individual state estimates, allowing for a tractable equilibrium characterization.

Findings

Players' estimates on others' estimates can be summarized into their own estimate on the state.

A backward/forward algorithm similar to dynamic programming characterizes the equilibrium.

Kalman filter covariances depend on observations, complicating offline evaluation.

Abstract

We consider a non-zero-sum linear quadratic Gaussian (LQG) dynamic game with asymmetric information. Each player observes privately a noisy version of a (hidden) state of the world $V$, resulting in dependent private observations. We study perfect Bayesian equilibria (PBE) for this game with equilibrium strategies that are linear in players' private estimates of $V$. The main difficulty arises from the fact that players need to construct estimates on other players' estimate on $V$, which in turn would imply that an infinite hierarchy of estimates on estimates needs to be constructed, rendering the problem unsolvable. We show that this is not the case: each player's estimate on other players' estimates on $V$ can be summarized into her own estimate on $V$ and some appropriately defined public information. Based on this finding we characterize the PBE through a backward/forward algorithm akin to dynamic programming for the standard LQG control problem. Unlike the standard LQG problem, however, Kalman filter covariance matrices, as well as some other required quantities, are observation-dependent and thus cannot be evaluated off-line through a forward recursion.