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Feedback Nash equilibria for scalar N-player linear quadratic dynamic games

Benita Nortmann, Mario Sassano, Thulasi Mylvaganam

TL;DR

The paper addresses infinite-horizon, discrete-time, scalar-state $N$-player linear-quadratic dynamic games and proposes a graphical interpretation of Feedback Nash Equilibria (FNE) to reveal how many equilibria may exist and what their stability properties are. It rewrites the coupled FNE conditions in terms of auxiliary functions $f_ au( au)$ and a mapping $a_{cl}=0hat f(0 exti)$, turning the problem into a tractable intersection problem. By imposing assumptions on the parameters $oldsymbol{ ext{oldsymbol{ ext}}}= rac{b_i^2 q_i}{r_i}$ and exploiting the geometry of these functions, the authors derive precise statements on existence, multiplicity (up to $2^N-1$), and uniqueness of FNE under different open-loop stability regimes and cost structures, including several special cases. A numerical example with $N=3$ demonstrates the predicted multiplicity and confirms the effectiveness of the graphical method, while also contrasting discrete-time results with known continuous-time analogues.

Abstract

Considering infinite-horizon, discrete-time, linear quadratic, N-player dynamic games with scalar dynamics, a graphical representation of feedback Nash equilibrium solutions is provided. This representation is utilised to derive conditions for the number and properties of different feedback Nash equilibria a game may admit. The results are illustrated via a numerical example.

Feedback Nash equilibria for scalar N-player linear quadratic dynamic games

TL;DR

The paper addresses infinite-horizon, discrete-time, scalar-state -player linear-quadratic dynamic games and proposes a graphical interpretation of Feedback Nash Equilibria (FNE) to reveal how many equilibria may exist and what their stability properties are. It rewrites the coupled FNE conditions in terms of auxiliary functions and a mapping , turning the problem into a tractable intersection problem. By imposing assumptions on the parameters and exploiting the geometry of these functions, the authors derive precise statements on existence, multiplicity (up to ), and uniqueness of FNE under different open-loop stability regimes and cost structures, including several special cases. A numerical example with demonstrates the predicted multiplicity and confirms the effectiveness of the graphical method, while also contrasting discrete-time results with known continuous-time analogues.

Abstract

Considering infinite-horizon, discrete-time, linear quadratic, N-player dynamic games with scalar dynamics, a graphical representation of feedback Nash equilibrium solutions is provided. This representation is utilised to derive conditions for the number and properties of different feedback Nash equilibria a game may admit. The results are illustrated via a numerical example.

Paper Structure

This paper contains 6 sections, 5 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphical interpretation of FNE
  4. Conditions for existence, number and properties of FNE
  5. Example
  6. Conclusion

Key Result

Theorem 1

Consider the game eq:system, eq:cost, $i = 1, \ldots, N$. The set of strategies $\{u_1^\star, \ldots, u_N^\star \}$, where for $i = 1,\ldots, N$, constitutes a FNE of the game if and only if where $a_{cl} := a + \sum_{j=1}^N b_j k_j$, and there exist $p_i \in \mathbb{R}$, for $i = 1,\ldots, N$, satisfying the set of equations for $i =1,\ldots, N$. The cost incurred by player $i$ starting from i

Figures (2)

  • Figure 1: Plot of the auxiliary functions for $\tilde{q}_2 = 0.05$ and $\tilde{q}_3= 0$, $f_1(\xi)$ (red), $f_2(\xi)$ (green), $f_3(\xi)$ (blue), $f_\ell(\xi)$, $\ell = 4, \ 5$ (grey), $f_{6}(\xi)$ (cyan), $f_{7}(\xi)$ (dark green) and $f_{8}(\xi)$ (magenta), and their intersection points (yellow crosses) with the horizontal lines at $a = \tilde{a}$ (yellow) for $\tilde{a} = 0.3$ and $\tilde{a} = -5$. The black dashed lines indicate the linear asymptotes of $f_\ell(\xi)$, $\ell = 1, \ldots,8$, and the grey dotted lines indicate $\xi = \pm \sqrt{\sigma_1} = \pm 0.3162$.
  • Figure 2: Plot of the auxiliary functions for $\tilde{q}_2 = -0.8$ and $\tilde{q}_3= -0.9$, $f_1(\xi)$ (red), $f_2(\xi)$ (green), $f_3(\xi)$ (blue), $f_\ell(\xi)$, $\ell = 4, \ 5$ (grey), $f_{6}(\xi)$ (cyan), $f_{7}(\xi)$ (dark green) and $f_{8}(\xi)$ (magenta), and their intersection points (yellow crosses) with the horizontal lines at $a = \tilde{a}$ (yellow) for $\tilde{a} = 0.3$ and $\tilde{a} = -5$. The black dashed lines indicate the linear asymptotes of $f_\ell(\xi)$, $\ell = 1, \ldots,8$, and the grey dotted lines indicate $\xi = \pm \sqrt{\sigma_1} = \pm 0.3162$.

Theorems & Definitions (16)

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