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Nash equilibria in scalar discrete-time linear quadratic games

Giulio Salizzoni, Reda Ouhamma, Maryam Kamgarpour

TL;DR

The paper addresses the problem of characterizing all Nash equilibria in scalar discrete-time infinite-horizon two-agent LQ games. By formulating a polynomial system from best-response dynamics and applying Gröbner-basis techniques, the authors reduce the search to a univariate quintic $g(k_2)$ whose real roots in the stabilizing interval correspond to all Nash equilibria. They prove existence and derive tight bounds: at most three equilibria, with discriminant-based conditions yielding at most two or a unique equilibrium. The approach enables exact computation of all equilibria and offers insights into how the equilibrium count changes with system parameters, validated through numerical experiments. This work advances understanding of equilibrium structure in discrete-time LQ games and connects algebraic geometry to game-theoretic analysis, providing practical tools for equilibrium enumeration and design.

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 Gröbner 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.

Nash equilibria in scalar discrete-time linear quadratic games

TL;DR

The paper addresses the problem of characterizing all Nash equilibria in scalar discrete-time infinite-horizon two-agent LQ games. By formulating a polynomial system from best-response dynamics and applying Gröbner-basis techniques, the authors reduce the search to a univariate quintic whose real roots in the stabilizing interval correspond to all Nash equilibria. They prove existence and derive tight bounds: at most three equilibria, with discriminant-based conditions yielding at most two or a unique equilibrium. The approach enables exact computation of all equilibria and offers insights into how the equilibrium count changes with system parameters, validated through numerical experiments. This work advances understanding of equilibrium structure in discrete-time LQ games and connects algebraic geometry to game-theoretic analysis, providing practical tools for equilibrium enumeration and design.

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 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 Gröbner 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.

Paper Structure

This paper contains 12 sections, 3 theorems, 20 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM STATEMENT
  3. COMPUTING THE NASH EQUILIBRIA
  4. A system characterizing Nash equilibria
  5. A simpler system using Gröbner Basis
  6. CHARACTERIZING THE NUMBER OF NASH EQUILIBRIA
  7. Main result
  8. Proof of Theorem \ref{['thm:main']}
  9. Best response
  10. Proof of Theorem \ref{['thm:main']}
  11. NUMERICAL EXAMPLES
  12. CONCLUSION

Key Result

Lemma 1

A pair of policies $(k_1,k_2)$ is a Nash equilibrium of the game with dynamics eq:dynamic and cost functions eq:costFunction if and only if it is a solution of the following system

Figures (1)

  • Figure 1: Top: The discriminant as a function of $a$ for different values of $r_2$, the $y-$axis is in symmetric log scale. Bottom: The Number of Nash equilibria under the same conditions.

Theorems & Definitions (12)

  • Definition 1
  • Lemma 1
  • proof
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Lemma 2
  • proof
  • Remark 7
  • ...and 2 more