{"id":"83e7243b-a9c7-4373-9c6e-6a773ace0749","arxiv_id":"2508.20675","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The finite-horizon Riccati recursion's fixed points equal infinite-horizon Nash equilibria, and its cycles, if any, are periodic Nash equilibria.","lead":"Finite-horizon and infinite-horizon multi-player linear-quadratic games can have different Nash equilibria. This paper proves the finite-horizon equilibrium recursion is a dynamical system whose fixed points are exactly the infinite-horizon equilibria, and that cycles of the recursion, when they exist, are periodic equilibria.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 1's cycle-stability proof omits non-real eigenvalues; as written, Theorem 1's periodic-Nash claim is not fully rigorous, though the gap is likely patchable.","rationale":"The paper's main bridge, Proposition 1, is sound under the stated assumption Q_i ≻ 0: the forward inclusion is by definition, and the reverse inclusion uses the Lyapunov identity (Acl)^T P_i Acl - P_i = -(Q_i + K_i^T R_i K_i) ≺ 0 to obtain stability, which then implies stabilizability and detectability. Corollary 1 follows by setting QT = P. The honestly flagged limitation about Q_i semidefinite is a scope restriction, not an internal inconsistency. The weakest part of the distinctively new cycle contribution is Lemma 1, whose proof only treats real eigenvalues. Since Theorem 1 depends on Lemma 1 for the stabilizability and detectability conditions needed to invoke the periodic Riccati theorem, the proof of the periodic-Nash claim is incomplete for generic (complex-eigenvalue) cases. The gap is very likely repairable by a Hermitian-form argument, so this does not invalidate the theorem; it does justify the reader's conditional verdict rather than full acceptance. The numerical evidence for cycles is also not verified against the exact algebraic conditions (13), but Remark 2 explicitly concedes this, so it is a limitation rather than a hidden defect. Overall, no reason to move the reader's CONDITIONAL verdict, but the complex-eigenvalue omission in Lemma 1 is the most load-bearing technical concern and should be fixed before publication.","tokens_in":11270,"tokens_out":17049,"duration_ms":187488,"concrete_test":"Independently re-derive Lemma 1 allowing complex eigenpairs: let Θ_L v = λ v with v ∈ C^n, ||v||=1, and take the Hermitian form of equation (14), i.e. multiply on the left by v^H and on the right by v. If the resulting identity is (1 - |λ|^2) v^H P_i1 v = Σ_{l=1}^L (Θ_{l-1}v)^H (Q_i + K_l^T R_i K_l)(Θ_{l-1}v), then the right-hand side is strictly positive whenever |λ| ≥ 1 because Q_i ≻ 0 and v ≠ 0, so |λ| < 1 follows. This check settles whether Lemma 1's omission is only an exposition gap or a genuine logical break in the proof of Theorem 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step for the cycle result is Lemma 1, which proves that a cycle {P_l} satisfying (13) is stabilizing, i.e. ρ(Θ_L) < 1 with Θ_L = Acl_L ... Acl_1. Its proof by contradiction only considers an eigenvector v ∈ R^n with real eigenvalue λ, |λ| > 1. A real matrix can have complex eigenvalues outside the unit circle, and the argument as written does not rule these out. Consequently the conclusion in the final sentence of Lemma 1, '|λ| < 1 for every eigenvalue', does not follow from the real-eigenvalue case alone. This is not a trivial stylistic point: Theorem 1 explicitly relies on Lemma 1 to claim stabilizability of the periodic pairs (Acl,-i_l, B_i) and to invoke the periodic Riccati uniqueness theorem [24, Thm 6.11]. Without a valid Lemma 1, the theorem that a cycle is a periodic Nash equilibrium of the infinite-horizon game is formally incomplete. The gap appears fixable: replacing v^T by the conjugate transpose v^H in the Hermitian form of (14) yields (1 - |λ|^2) v^H P_i1 v = Σ_l (Θ_{l-1}v)^H (Q_i + K_l^T R_i K_l)(Θ_{l-1}v), whose right-hand side is strictly positive if |λ| ≥ 1 and Q_i ≻ 0, giving a contradiction. But this repair is not present in the paper, so the proof as written is incomplete. The separate issue of Q_i ≻ 0 is an explicit assumption (footnote 1, Remark 1), not an internal gap; the central result is conditional on it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the backward Riccati recursion of finite-horizon LQ games as a discrete-time dynamical system f. The main theoretical claims are: (i) the fixed points of f coincide with the stationary linear state-feedback Nash equilibria of the infinite-horizon game, under Qi,Ri ≻ 0 (Proposition 1); (ii) every such equilibrium can be reproduced as a stationary finite-horizon equilibrium by setting the terminal cost to the equilibrium matrix P (Corollary 1); and (iii) any cycle of f satisfying (13) is stabilizing (Lemma 1) and defines a periodic Nash equilibrium of the time-invariant infinite-horizon game (Theorem 1). Simulations then classify the recursion's behavior into convergence to stationary equilibria, convergence to cycles, and bounded non-convergent trajectories.","tokens_in":11662,"tokens_out":5781,"duration_ms":66301,"significance":"If the central claims hold, the paper gives a clean algebraic explanation of the finite/infinite-horizon gap in LQ games and identifies a new phenomenon—periodic Nash equilibria in time-invariant LQ games—that is absent in the single-agent LQR case. The fixed-point equivalence in Proposition 1 is simple but useful, and Corollary 1 gives a constructive identity, even if its practical value is limited by the need to know the target equilibrium in advance. The paper is transparent about its main assumptions and limitations: Remark 1 explicitly flags that Qi ≻ 0 is load-bearing, and Remark 2 correctly cautions that the numerical evidence does not by itself prove existence of cycles. The main defect is a proof gap in Lemma 1 that currently leaves Theorem 1 formally incomplete, although the gap appears readily fixable.","major_comments":[{"comment":"","section":"§IV-B, Lemma 1 (proof following Eq. (14))"},{"comment":"","section":"§IV-B, Theorem 1"},{"comment":"","section":"§IV-A, Proposition 1 and Remark 1"}],"minor_comments":[{"comment":"Typo: 'i∈[i,N]' should be 'i∈[1,N]'.","section":"§IV-B, Lemma 1"},{"comment":"Duplicate word: 'by setting the the terminal costs'. Also, the interval [0, 1, ..., T−1] should be written [0, T−1].","section":"§IV-A, Corollary 1"},{"comment":"The abstract says finite-horizon LQ games 'admit a unique Nash equilibrium', but §III-A correctly states that uniqueness holds iff equation (7) has a unique solution. Rephrase the abstract to avoid an unconditional claim.","section":"Abstract and §II"},{"comment":"The numerical criteria for declaring convergence to a cycle, convergence to a fixed point, or non-convergence are not specified. Please state the tolerances, horizon lengths, and random sampling distributions (including seeds) so the results are reproducible. Figure 2 would also benefit from clearer axis labels and a unified color scale.","section":"§V, Simulations"},{"comment":"The observation that individual policies in a cycle need not be stabilizing is valuable for receding-horizon games. Consider stating this as a formal remark after Theorem 1 rather than only in the simulation discussion.","section":"§V-B and Fig. 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope and the main idea is attractive. The central fixed-point equivalence is sound under the stated assumptions, but the periodic-equilibrium result currently rests on an incomplete proof (complex eigenvalues in Lemma 1) and a somewhat compressed invocation of the periodic Riccati theorem. Both issues are patchable, so I recommend major revision rather than rejection. The authors should also calibrate the novelty claim: the recovery result in Corollary 1 is constructive but essentially tautological, and the numerical cycle evidence is not a proof of existence; the paper is appropriately cautious in Remark 2, but this caution should be reflected in the abstract and introduction as well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper gives a clean conceptual statement—fixed points of the finite-horizon Riccati recursion are exactly the stationary infinite-horizon Nash equilibria, and cycles of that recursion correspond to periodic Nash equilibria. The first claim is correct under Q_i,R_i ≻ 0 and the paper is honest about why that assumption is load-bearing. The second claim is genuinely new but rests on a lemma whose proof only treats real eigenvalues; the complex case is missing. I think it's a quick fix, but as written the proof is incomplete.\n\nWhat I liked: Proposition 1 is sound. The reverse inclusion relies on positive definite Q_i to get a strict Lyapunov decrement; Remark 1 explicitly concedes that semi-definite Q_i can break it. That's the right way to handle a delicate assumption. Corollary 1 is almost definitional—choose terminal cost equal to the equilibrium value—but it makes the selector role of terminal costs explicit and the paper does not oversell it. The numerical catalog showing three regimes (convergence to stationary, convergence to cycles, bounded non-convergence) is useful, and the observation that individual policies in a cycle need not be stabilizing is a nice practical warning for receding-horizon applications.\n\nSoft spots. The main one is Lemma 1. The proof by contradiction takes a real eigenvector v and derives a contradiction from (lambda^2-1) v^T P v > 0 against a negative right-hand side. Nothing rules out complex eigenvalues with |lambda|>1. The stress-test note has the repair: use v^H and the Hermitian form to get a contradiction. That is standard. But as written, the final 'therefore |lambda|<1 for every eigenvalue' does not follow. Since Theorem 1 leans entirely on Lemma 1 to invoke the periodic Riccati result, this needs to be patched.\n\nSecond, the paper does not verify the numerical cycles against the exact algebraic condition (13), and Remark 2 openly says so. So the periodic-Nash existence claim rests on simulation only. That is a limitation, not a deception, but readers should not mistake the numerics for a proof.\n\nMinor: the numerical experiments are described but no code or data are released, so the 10^5 random-games table is hard to reproduce.\n\nOverall: the central fixed-point bridge holds up; the cycle result is promising but formally incomplete in one step. This deserves a serious referee—the fix is easy, the paper is honest, and the conceptual contribution is real. I would suggest accepting conditional on a corrected Lemma 1 and ideally an exact cycle example.","headline":"Clean fixed-point bridge between finite and infinite-horizon LQ Nash equilibria; the cycle-stability lemma has a patchable missing complex-eigenvalue case.","tokens_in":12122,"tokens_out":2514,"would_cite":true,"duration_ms":26073,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A25","93C55","49N10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Riccati fixed points equal all stationary Nash equilibria of infinite-horizon LQ games.","keywords":["linear quadratic games","Nash equilibria","Riccati recursion","fixed points","periodic equilibria","terminal cost selection","receding-horizon games","dynamical systems"],"falsifier":"Find a game with Q_i positive semidefinite (e.g., two-agent scalar, Q_1=0) with a fixed point P=f(P) but the closed-loop matrix has spectral radius >= 1; Proposition 1's reverse inclusion would fail. Alternatively, verify a reported cycle: if a sequence satisfies (13) but the product of its closed-loop matrices has spectral radius >= 1, Lemma 1 fails.","tokens_in":1629,"feed_emoji":"🎮","tokens_out":7983,"duration_ms":92781,"temperature":0.7,"pith_summary":"This paper bridges finite-horizon linear quadratic games, which have a unique Nash equilibrium, and infinite-horizon games, which can have many. Treating the finite-horizon backward recursion as a nonlinear dynamical system, the authors prove that its fixed points are exactly the stationary Nash equilibria of the infinite-horizon game, and any such equilibrium can be reproduced in a finite-horizon game by appropriate terminal costs. Cycles of the recursion correspond to periodic non-stationary Nash equilibria. Simulations document three asymptotic regimes: convergence to a stationary equilibrium, convergence to a cycle, and bounded non-convergent trajectories. This matters for tuning finite-horizon approximations, because terminal costs act as equilibrium selectors and stability is not guaranteed.","feed_headline":"Riccati fixed points equal all stationary Nash equilibria","feed_subtitle":"Terminal costs select any infinite-horizon equilibrium, and cycles yield periodic ones in LQ games.","key_machinery":"The key object is the map f : (S^n_++)^N -> (S^n_++)^N that sends the terminal cost tuple P_{t+1} to the previous-step Riccati solution P_t via (4)-(5). Its fixed points are equated with stationary Nash equilibria; its periodic orbits, satisfying P_l = f(P_{l+1}), are equated with periodic Nash equilibria. The reverse inclusion Fix(f) subset of P_NE_stat relies on positive definiteness of every Q_i, giving detectability of (A_cl^{-i}, Q_i^{1/2}) and a strict Lyapunov decrement; this is the load-bearing assumption.","core_discovery":"The central claim is Proposition 1: for the discrete-time infinite-horizon LQ game with strictly positive definite state-cost matrices Q_i, the set of stationary Nash equilibria P_NE_stat is exactly the fixed point set Fix(f) of the finite-horizon Riccati recursion (10). Corollary 1 states that for any equilibrium P, setting the terminal cost Q_T = P yields a finite-horizon equilibrium K_t = g(P) for all t, recovering the infinite-horizon policy. Theorem 1 extends to cycles: any sequence {P_l} satisfying (13) yields a periodic Nash equilibrium that stabilizes the closed-loop system over one period, though individual policies in the cycle need not be stabilizing. The paper thus shows that the","pith_inferences":["A testable extension is that for Q_i positive semidefinite, the inclusion Fix(f) subset of P_NE_stat can fail; one could search for a fixed point whose induced closed-loop matrix is unstable, which would directly falsify the unregularized version of Proposition 1.","The cycle-length distribution in simulations suggests that low-period cycles dominate, and cycle length appears largely independent of state dimension or number of agents; a theoretical explanation of this insensitivity could be pursued.","The bounded non-convergent regime may correspond to strange attractors or quasi-periodic behavior of the Riccati map; connecting this to known results in nonlinear dynamics could predict when receding-horizon implementations fail.","Incentive design could exploit the selector property: by choosing terminal costs, a planner may steer a population of agents toward a desired equilibrium without altering the game's infinite-horizon structure."],"forward_implications":["Terminal costs in finite-horizon LQ games act as equilibrium selectors: choosing Q_T = P selects the infinite-horizon equilibrium P for any P in P_NE_stat.","A receding-horizon game that applies policies from a periodic cycle may be stable only over one full period; individual policies can have spectral radius above 1, which can cause transient instability.","If the recursion converges, its limit is automatically an infinite-horizon Nash equilibrium, so finite-horizon approximations are consistent for attractive equilibria.","The existence of multiple attractors means convergence cannot be inferred from the existence of an equilibrium; stability of equilibria determines which, if any, are reached.","Periodic Nash equilibria, previously not identified in time-invariant LQ games, are shown to be possible and are observed numerically."],"supporting_citations":[{"why":"Supplies the existence and uniqueness result for finite-horizon feedback Nash equilibria and the characterization via coupled Riccati equations (Corollary 6.1, Remark 6.5), which defines the recursion studied.","marker":"[7]"},{"why":"Provides the algorithm used in simulations to compute the three Nash equilibria of the scalar two-agent example and the conditions for their existence.","marker":"[8]"},{"why":"Gives the continuous-time scalar two-agent result that the recursion always converges and that the limit depends on terminal costs, the direct precedent this paper extends.","marker":"[13]"},{"why":"Supplies the periodic discrete Riccati equation theory used in Theorem 1 to prove that a cycle satisfying (13) yields a stabilizing periodic Nash equilibrium.","marker":"[24]"},{"why":"Supports Remark 1's claim that without detectability, the infinite-horizon Riccati equation may admit multiple positive semidefinite solutions and the optimal one may not be stabilizing.","marker":"[23]"}],"fun_headline_variants":["Finite-horizon Riccati fixed points exactly infinite-horizon equilibria","Terminal costs tune LQ games to any infinite-horizon equilibrium","Periodic orbits in Riccati map yield periodic Nash equilibria","Three regimes: stationary, periodic, or non-convergent LQ trajectories","Riccati recursion fixed points characterize all stationary Nash equilibria"],"cache_read_input_tokens":13824,"weakest_assumption_plain":"Every player's state-cost matrix Q_i must be strictly positive definite; if Q_i is only positive semidefinite, a fixed point need not be a Nash equilibrium, so the finite/infinite bridge can break.","fun_headline_variants_meta":{"raw":{"variants":["Finite-horizon Riccati fixed points exactly infinite-horizon equilibria","Terminal costs tune LQ games to any infinite-horizon equilibrium","Periodic orbits in Riccati map yield periodic Nash equilibria","Three regimes: stationary, periodic, or non-convergent LQ trajectories","Riccati recursion fixed points characterize all stationary Nash equilibria"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000257,"raw_usage":{"total_tokens":1385,"prompt_tokens":684,"completion_tokens":701,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":604}},"tokens_in":428,"tokens_out":701,"duration_ms":7297,"temperature":1.0,"reasoning_tokens":604,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:54:32.944944+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a game with Q_i positive semidefinite (e.g., two-agent scalar, Q_1=0) with a fixed point P=f(P) but the closed-loop matrix has spectral radius >= 1; Proposition 1's reverse inclusion would fail. Alternatively, verify a reported cycle: if a sequence satisfies (13) but the product of its closed-loop matrices has spectral radius >= 1, Lemma 1 fails.","supporting_citations":[{"cited_title":"Basar and G","cited_arxiv_id":null,"evidence_quote":"Supplies the existence and uniqueness result for finite-horizon feedback Nash equilibria and the characterization via coupled Riccati equations (Corollary 6.1, Remark 6.5), which defines the recursion studied."},{"cited_title":"Nash equil ibria in scalar discrete-time linear quadratic games,","cited_arxiv_id":null,"evidence_quote":"Provides the algorithm used in simulations to compute the three Nash equilibria of the scalar two-agent example and the conditions for their existence."},{"cited_title":"Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Q uadratic Differential Games,","cited_arxiv_id":null,"evidence_quote":"Gives the continuous-time scalar two-agent result that the recursion always converges and that the limit depends on terminal costs, the direct precedent this paper extends."},{"cited_title":"The period ic riccati equation,","cited_arxiv_id":null,"evidence_quote":"Supplies the periodic discrete Riccati equation theory used in Theorem 1 to prove that a cycle satisfying (13) yields a stabilizing periodic Nash equilibrium."},{"cited_title":"The discrete riccati equation of optimal c ontrol,","cited_arxiv_id":null,"evidence_quote":"Supports Remark 1's claim that without detectability, the infinite-horizon Riccati equation may admit multiple positive semidefinite solutions and the optimal one may not be stabilizing."}],"review_version":1}