2508.20675
Bridging Finite and Infinite-Horizon Nash Equilibria in Linear Quadratic Games
Giulio Salizzoni, Sophie Hall, Maryam Kamgarpour
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:57 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves that fixed points of the finite-horizon Riccati map f are exactly the stationary infinite-horizon Nash equilibria (Proposition 1), that any such equilibrium can be recovered by choosing QT = P (Corollary 1), and that cycles of f correspond to stabilizing periodic Nash equilibria (Lemma 1 and Theorem 1). These statements and their key identities match the candidate’s arguments: both use the completion-of-squares relation to obtain (Acl)^T P^i Acl − P^i = −(Qi + Ki^T Ri Ki) ≺ 0 (yielding stability) and the telescoping identity over a period for cycles. The candidate gives fuller completion-of-squares proofs (including the periodic Nash property via average-cost completion-of-squares), whereas the paper invokes detectability/stabilizability and a textbook periodic LQR theorem. Hence, the results agree and are correct, with different proof styles.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper’s core contributions are correct, relevant, and clearly motivated: fixed points of the finite-horizon map equal stationary equilibria, any such equilibrium can be recovered by terminal-cost selection, and cycles correspond to periodic equilibria. The empirical section is informative. A few proofs are compressed; inserting the completion-of-squares steps and a short deviation-based argument for periodic equilibria would improve readability and self-containment. With minor clarifications, the manuscript is suitable for a specialist audience.