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On Two-Player Scalar Discrete-Time Linear Quadratic Games

Chiara Cavalagli, Alberto Bemporad, Mario Zanon

Abstract

For the characterization of Feedback Nash Equilibria (FNE) in linear quadratic games, this paper provides a detailed analysis of the discrete-time discounted coupled best-response equations for the scalar two-player setting, together with a set of analytical tools for the classification of local saddle property for the iterative best-response method. Through analytical and numerical results we show the importance of classification, revealing an anti-coordination scheme in the case of multiple solutions. Particular attention is given to the symmetric case, where identical cost function parameters allow closed-form expressions and explicit necessary and sufficient conditions for the existence and multiplicity of FNE. We also present numerical results that illustrate the theoretical findings and offer foundational insights for the design and validation of iterative NE-seeking methods.

On Two-Player Scalar Discrete-Time Linear Quadratic Games

Abstract

For the characterization of Feedback Nash Equilibria (FNE) in linear quadratic games, this paper provides a detailed analysis of the discrete-time discounted coupled best-response equations for the scalar two-player setting, together with a set of analytical tools for the classification of local saddle property for the iterative best-response method. Through analytical and numerical results we show the importance of classification, revealing an anti-coordination scheme in the case of multiple solutions. Particular attention is given to the symmetric case, where identical cost function parameters allow closed-form expressions and explicit necessary and sufficient conditions for the existence and multiplicity of FNE. We also present numerical results that illustrate the theoretical findings and offer foundational insights for the design and validation of iterative NE-seeking methods.
Paper Structure (22 sections, 13 theorems, 96 equations, 4 figures)

This paper contains 22 sections, 13 theorems, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Statement
  4. Mathematical Analysis of the Game
  5. Symmetric Setting
  6. Saddle FNE
  7. Symmetric Setting
  8. Value Function
  9. Conclusions
  10. Proofs
  11. Proof of Lemma \ref{['lemma:exclude_null']}
  12. Proof of Proposition \ref{['prop:ONLY_ONE_ROOT']}
  13. Proof of Proposition \ref{['prop:SUFF_ONLY_ONE_ROOT']}
  14. Proof of Proposition \ref{['prop:existence']}
  15. Proof of Lemma \ref{['lemma:facto_FNE_law']}
  16. ...and 7 more sections

Key Result

Lemma 4.0.1

Let $(x_i^\star, x_{-i}^\star)$ be a stable FNE. Then

Figures (4)

  • Figure 1: Two cases in which we have respectively unique and multiple stable FNE given $\sigma_i$ fixed. Each stable FNE is the intersection between the curve $x_{-i}=h^{(i)}_-(x_i)$ and the FNE law $x_{-i}=r^{(-i)}_\pm(x_i)$.
  • Figure 2: Factorization of the FNE law in standard and symmetric setting. In the second case the two curves $r^{(-i)}(x_i)_\pm$ contribute equally to the hyperbolic and symmetric union.
  • Figure 3: Display of value functions of the three stable FNE per different game configurations.
  • Figure 4: Community Cost evaluation of outer equilibria $X_1$ (pink dots), $X_3$ (green dots) with respect to the central equilibrium $X_2$.

Theorems & Definitions (14)

  • Definition 3.1: Stable Feedback-Nash-Equilibrium
  • Lemma 4.0.1
  • Lemma 4.0.2
  • Proposition 4.1: Necessary Condition
  • Proposition 4.2: Sufficient Condition
  • Proposition 4.3
  • Lemma 4.0.3: Geometric structure of the FNE law
  • Theorem 4.1: Uniqueness and Multiplicity
  • Theorem 4.2
  • Theorem 4.3
  • ...and 4 more