To What Extent do Open-loop and Feedback Nash Equilibria Diverge in General-Sum Linear Quadratic Dynamic Games?

TL;DR

This paper compares open-loop and feedback Nash equilibria in linear quadratic dynamic games, providing conditions for their equivalence and bounds on their divergence, aiding understanding of strategic interactions over time.

Contribution

It introduces a method to synthesize OLNE strategies via auxiliary Riccati equations and establishes conditions and bounds for their divergence from FBNE in LQ games.

Findings

OLNE strategies can be derived from auxiliary Riccati equations.

Conditions are identified under which OLNE and FBNE coincide.

An upper bound on the deviation between OLNE and FBNE is established.

Abstract

Dynamic games offer a versatile framework for modeling the evolving interactions of strategic agents, whose steady-state behavior can be captured by the Nash equilibria of the games. Nash equilibria are often computed in feedback, with policies depending on the state at each time, or in open-loop, with policies depending only on the initial state. Empirically, open-loop Nash equilibria (OLNE) could be more efficient to compute, while feedback Nash equilibria (FBNE) often encode more complex interactions. However, it remains unclear exactly which dynamic games yield FBNE and OLNE that differ significantly and which do not. To address this problem, we present a principled comparison study of OLNE and FBNE in linear quadratic (LQ) dynamic games. Specifically, we prove that the OLNE strategies of an LQ dynamic game can be synthesized by solving the coupled Riccati equations of an auxiliary LQ game with perturbed costs. The construction of the auxiliary game allows us to establish conditions under which OLNE and FBNE coincide and derive an upper bound on the deviation between FBNE and OLNE of an LQ game.