Linear-quadratic Gaussian Games with Asymmetric Information: Belief Corrections Using the Opponents Actions

TL;DR

This paper analyzes two-player linear-quadratic Gaussian games with asymmetric information, demonstrating how players can improve state estimation and outcomes by using opponent actions, and providing a closed-form belief update solution.

Contribution

It introduces a belief correction method leveraging opponent actions in asymmetric information LQG games and derives a closed-form Bayesian update for the belief process.

Findings

Players can enhance state estimates using opponent actions.

A linear Nash equilibrium is characterized with estimation errors included.

Application to bargaining illustrates the benefits of belief corrections.

Abstract

We consider two-player non-zero-sum linear-quadratic Gaussian games in which both players aim to minimize a quadratic cost function while controlling a linear and stochastic state process {using linear policies}. The system is partially observable with asymmetric information available to the players. In particular, each player has a private and noisy measurement of the state process but can see the history of their opponent's actions. The challenge of this asymmetry is that it introduces correlations into the players' belief processes for the state and leads to circularity in their beliefs about their opponents beliefs. We show that by leveraging the information available through their opponent's actions, both players can enhance their state estimates and improve their overall outcomes. In addition, we provide a closed-form solution for the Bayesian updating rule of their belief process. We show that there is a Nash equilibrium which is linear in the estimation of the state and with a value function incorporating terms that arise due to errors in the state estimation. We illustrate the results through an application to bargaining which demonstrates the value of these information corrections.