Nash equilibria in scalar discrete-time linear quadratic games

TL;DR

This paper characterizes all Nash equilibria in scalar discrete-time infinite-horizon LQ games with two agents using algebraic geometry tools, providing bounds on their number and conditions for uniqueness.

Contribution

It formulates a polynomial system to find all Nash equilibria in simple LQ games and applies algebraic geometry to analyze their number and properties.

Findings

Maximum of three Nash equilibria in the studied setting

Sufficient conditions for the existence of at most two equilibria

Conditions for the uniqueness of Nash equilibrium

Abstract

An open problem in linear quadratic (LQ) games has been characterizing the Nash equilibria. This problem has renewed relevance given the surge of work on understanding the convergence of learning algorithms in dynamic games. This paper investigates scalar discrete-time infinite-horizon LQ games with two agents. Even in this arguably simple setting, there are no results for finding $\textit{all}$ Nash equilibria. By analyzing the best response map, we formulate a polynomial system of equations characterizing the linear feedback Nash equilibria. This enables us to bring in tools from algebraic geometry, particularly the Gr\"obner basis, to study the roots of this polynomial system. Consequently, we can not only compute all Nash equilibria numerically, but we can also characterize their number with explicit conditions. For instance, we prove that the LQ games under consideration admit at most three Nash equilibria. We further provide sufficient conditions for the existence of at most two Nash equilibria and sufficient conditions for the uniqueness of the Nash equilibrium. Our numerical experiments demonstrate the tightness of our bounds and showcase the increased complexity in settings with more than two agents.