{"id":"a13aaf49-7e38-4125-a758-57fdb6e83f9d","arxiv_id":"2506.19565","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite-horizon 'plan T steps, use one step' strategies converge in total cost to an infinite-horizon feedback Nash equilibrium, conditional on the coupled Riccati iteration converging, with an explicit cubic error bound.","lead":"This paper shows that in multi-player linear-quadratic games, each player can approximate an infinite-horizon equilibrium by solving a short finite-horizon game and using only the first move, with a computable error bound. This offers a tractable shortcut for control problems where solving the coupled infinite-horizon equilibrium equations is difficult.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem depends on an unproved convergence assumption (Assumption 1(ii)); until that assumption is proven or replaced by a checkable condition, the approximation guarantee is conditional.","rationale":"After reading the manuscript in good faith, the central theorem is internally consistent given Assumption 1. The weakest point is not the algebra in Appendix B but the unproved convergence of the coupled backward Riccati recursion. The paper's own Conclusions identify this as open, so flagging it is not manufacturing a flaw. I also examined the claim that theta_i(epsilon) is a cubic polynomial: the displayed formula for M depends on epsilon, so the coefficients are not literally parameter-free constants; this is a secondary presentational issue that can be repaired by restricting epsilon to a compact range and does not affect the convergence conclusion. Compared with that, Assumption 1(ii) is more load-bearing because without it neither Lemma 2 nor Theorem 3 applies at all. The reader's CONDITIONAL verdict matches this assessment, so no change is needed.","tokens_in":18318,"tokens_out":9491,"duration_ms":104815,"concrete_test":"Search over scalar two-player LQ instances (the setting of reference [7]) that satisfy Assumptions 1(i) and 1(iii), and iterate (3)-(5) backward from P_i,1 = 0 for a large number of steps. If any instance yields a non-convergent (periodic or divergent) trajectory, Assumption 1(ii) is not implied by (i),(iii), confirming the gap; if all tested instances converge, attempt an analytic proof of convergence, since a single counterexample would settle the question either way.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Assumption 1(ii) (Section 3) is the load-bearing step: Lemma 2 defines the infinite-horizon FNE as the limit of the backward Riccati recursion (3),(5), and Theorem 3's convergence and cubic bound both require epsilon -> 0, which follows only from this convergence. The paper does not prove Assumption 1(ii) and explicitly states in the Conclusions that parameter-based conditions guaranteeing convergence remain open. The other assumptions (i) and (iii) do not obviously imply (ii), and the numerical example only checks a single parameter instance. Therefore, as it stands, the central claim is a conditional theorem whose main hypothesis cannot be verified for a given model; if the backward Riccati iteration cycles or diverges, the cost gap need not vanish and the bound does not apply.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a receding-horizon approximation of infinite-horizon feedback Nash equilibria (FNEs) in discrete-time linear-quadratic games. For finite-horizon games with input/output/state dynamics, the authors derive a sufficient invertibility condition under which the unique FNE can be computed by solving a sequence of linear equations (Algorithm 1). For the infinite-horizon game, they consider a 'look-T^i-steps-ahead, move-one-step' strategy and prove (under Assumption 1) that the resulting total costs converge to the limiting FNE cost as all prediction horizons go to infinity, with an explicit upper bound on the cost gap in terms of the deviation of the strategy matrices. A numerical example with two players illustrates the convergence.","tokens_in":18440,"tokens_out":13363,"duration_ms":125290,"significance":"The paper addresses a computationally challenging problem: solving coupled algebraic Riccati equations for infinite-horizon LQ games. The proposed finite-horizon strategy is tractable, and the error bound is derived rather than fitted. The linear-equation algorithm for finite-horizon games is a useful practical contribution. However, the main theorem relies on a strong convergence assumption (Assumption 1(ii)) that is acknowledged as an open problem, and the numerical example does not directly match the theoretical setup; these issues substantially temper the significance unless addressed.","major_comments":[{"comment":"Assumption 1(ii) is the load-bearing hypothesis: Lemma 2 identifies the infinite-horizon FNE with the limit of the backward Riccati recursion, and Theorem 3's limit and bound require epsilon -> 0, which follows only from this convergence. The paper does not prove Assumption 1(ii) and states in the Conclusions that parameter-based conditions guaranteeing it remain open. Since the hypothesis is exactly the difficult part of the problem, the central guarantee is conditional on an unverifiable condition. Please either prove the convergence for a nontrivial class of games, provide a checkable sufficient condition, or restructure the paper's claims to make the conditional nature explicit in the title and abstract.","section":"Section 3, Assumption 1(ii)"},{"comment":"The infinite-horizon theory in Section 3 assumes zero reference trajectories (l_i^t = 0), as in the cost (12). The numerical example, however, uses constant nonzero references l_1 = [1,1]^T and l_2 = [-1,-1]^T. Thus the simulation does not validate the theorem's hypotheses; it illustrates convergence for a different (affine) problem. Please either extend the theoretical analysis to cover constant affine terms or modify the example to the zero-reference case so that the simulation directly tests Theorem 3.","section":"Section 4, numerical example"},{"comment":"Theorem 3 states that the bound is a cubic polynomial theta_i(epsilon) = theta_i1 epsilon + theta_i2 epsilon^2 + theta_i3 epsilon^3 with coefficients determined by fixed parameters. However, the definitions of G_i1 and G_i2 given after the theorem involve M = max_{t in N+} (t-1)(lambda + b epsilon)^{t-2}, which depends on epsilon. Consequently, the coefficients are not fixed; they vary with epsilon, and M diverges as lambda + b epsilon -> 1. This makes the advertised 'parameter-free' cubic bound inaccurate. Please either provide a uniform upper bound on M over the admissible range of epsilon (which would need to be stated as an additional condition), or revise the statement to make the epsilon-dependence of the coefficients explicit.","section":"Theorem 3 and its bound"}],"minor_comments":[{"comment":"In the proof, the limits are written as t -> +infinity but should be T -> +infinity (e.g., 'lim_{t->+infinity} K_i^*(T)'). This is a typographical error that may confuse readers.","section":"Appendix A, proof of Lemma 2"},{"comment":"The identity 'K_i^*(T) = K_i^*(T0-T+t(T0))' is not properly defined; the subscripts are missing. Please clarify the notation so that the dependence on the stage index is explicit.","section":"Section 3, Fact 2"},{"comment":"Assumption 1 begins with sequences indexed by t = 0, -1, -2, ... and an initial condition P_i^1 = 0. It would be helpful to explain that this index denotes time-to-go in the backward iteration, to avoid confusion with the physical time index used elsewhere.","section":"Section 3, Assumption 1"},{"comment":"The proof of Lemma 1 is omitted with a reference to standard results. A concise sketch or a more precise citation than 'modifying Corollary 6.1 in [4]' would improve accessibility.","section":"Section 2, Lemma 1"},{"comment":"The caption states that the horizontal dashed lines represent the limiting FNE strategy matrices; since the paper does not prove convergence for this example, it would be helpful to state explicitly that these are the limits of the iterations as T -> infinity, not independently verified equilibria.","section":"Section 4, Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its main assumption, but the gap between the theorem's hypothesis and a verifiable condition is substantial. The numerical example should be aligned with the theory, and the cubic-polynomial claim needs correction. With these revisions, the paper could be a useful contribution. The novelty relative to the existing literature on iterative methods for LQ games is moderate; the main appeal is the simple linear-equation algorithm and the explicit error bound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the paper in one take: it proves a clean conditional result—if the backward coupled Riccati recursion converges, then the receding-horizon FNE strategy approximates an infinite-horizon FNE with an explicit cubic cost-gap bound. The proof of the bound is a direct matrix-norm estimate in Appendix B and the algebra checks out. The paper is also honest about what it does not prove: Assumption 1(ii), the convergence condition, is stated as an assumption and the conclusions explicitly call parameter-based conditions for it an open question.\n\nWhat is actually new is the explicit cubic bound and the convergence statement for heterogeneous discount factors with i/o/s dynamics. The finite-horizon strategy itself is an MPC-style idea, and the uniqueness condition based on invertibility of H(P) plus the linear-equation algorithm for computing the finite-horizon FNE is a useful practical contribution.\n\nThe soft spots are in proportion. The main one is that Assumption 1(ii) is load-bearing and unproven. Without it, Lemma 2 and Theorem 3 do not apply, and you cannot check the assumption for a given model. That caps the practical force of the result, though it does not invalidate the conditional theorem. The numerical example uses nonzero reference trajectories, while the theorem assumes zero references. The simulation does illustrate convergence of the K matrices and costs, but it is not a direct test of the stated theorem. No code or data are included, so reproduction requires reimplementation. Lemma 2 also leans on a cited theorem from Monti et al. without reproducing the argument; that's standard practice, but it means the lemma inherits whatever conditions that theorem carries.\n\nOverall, this is a serious theory paper aimed at researchers working on LQ dynamic games and multi-agent MPC. It deserves a proper referee. The result is correctly stated, the proof is checkable, and the limitations are declared. I would send it out, with the request that the authors either adjust the simulation to match the theorem's hypotheses or extend the theorem to affine reference signals, and that they add a discussion of what is known about when Assumption 1(ii) holds.","headline":"A clean conditional result: finite-horizon LQ game strategies approximate an infinite-horizon FNE with an explicit cubic cost-gap bound, but the main assumption (backward Riccati convergence) is unproven and the simulation does not match the theorem's hypotheses.","tokens_in":18981,"tokens_out":2521,"would_cite":false,"duration_ms":27566,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A25","91A50","93C55","49N10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Finite-horizon 'look-ahead' strategies approximate infinite-horizon feedback Nash equilibria, with an explicit cubic cost-gap bound.","keywords":["feedback Nash equilibrium","linear-quadratic dynamic game","Riccati difference equation","finite-horizon approximation","infinite-horizon game","heterogeneous discount factors","cost bound","receding-horizon strategy"],"falsifier":"Choose a small LQ game satisfying the invertibility and stability parts of Assumption 1 and run the backward Riccati iteration (3)--(5) from $P^i_1=0$; if the iterates never settle at finite limits, Assumption 1(ii) fails and the theorem's conclusion is not guaranteed. For any instance where Assumption 1 holds, simulate the finite-horizon strategy for growing common horizon and compare the empirical cost gap with the right-hand side of the Theorem 3 inequality; a violation would falsify the stated bound.","tokens_in":18081,"feed_emoji":"🎮","tokens_out":10362,"duration_ms":101975,"temperature":0.7,"pith_summary":"In infinite-horizon discrete-time linear-quadratic games, finding a feedback Nash equilibrium (a state-feedback strategy profile from which no player wants to deviate) means solving coupled algebraic Riccati equations, which is computationally hard. This paper proposes that each player, at every stage, solve a short $T^i$-stage game and apply only its first control, and asks whether this cheap strategy approximates an infinite-horizon equilibrium. Under assumptions that the backward Riccati iteration is invertible, convergent, and stabilizing, the paper proves that each player's total cost under these finite-horizon strategies converges to the cost at the limiting feedback Nash equilibrium as every prediction horizon grows. It also gives an explicit cubic-polynomial upper bound on the cost gap in terms of the distance between strategy matrices. A nonscalar numerical example illustrates the predicted convergence.","feed_headline":"Finite look-ahead play converges to infinite-horizon Nash equilibrium","feed_subtitle":"Each player solves a short game and uses one step; total costs approach the full-game equilibrium with an explicit error bound.","key_machinery":"The engine is the coupled generalized discrete Riccati difference equations (3)--(5), plus their affine partners (4), (6), (7) in the finite-horizon case. At each backward step, given $P^j_{t+1}$, the feedback gains come from one linear system $H(P_{t+1})K_t = g(P_{t+1})$, and the same matrix $H(P_{t+1})$ also enters the affine terms; the cost matrices then update by simple quadratic formulas. Because the horizon $T$ only shifts the iteration count, the first-stage matrices $K^{i*}_1(T^i)$ are constant over time, which lets each player reuse the same gain at every stage. Letting the backward iteration run to $-\\infty$ turns these recursions into coupled algebraic Riccati equations whose solution supplies the limiting FNE, and the proof of Theorem 3 bounds each term of the infinite cost sum by comparing $(\\tilde F^*,\\tilde G^*)$ with $(F^*,G^*)$ through telescoping powers and the stability margin $\\lambda + b\\epsilon < 1$.","core_discovery":"The central claim is that the 'look $T^i$ steps ahead, move one step' strategy is a certified approximation of one infinite-horizon FNE. Formally, with $\\epsilon = \\max_i \\|K^{i*}_1(T^i) - K^{i*}\\|_2$ and $\\theta_i(\\epsilon)$ a cubic polynomial in $\\epsilon$ built from fixed model parameters, Theorem 3 states that when $\\|F^*\\|_2 + (\\sum_j \\|B^j\\|_2)\\epsilon < 1$, the per-player cost gap satisfies $$|\\tilde{J}^i(x_1) - J^i(x_1)| \\le \\tfrac12 \\|x_1\\|$_2^{2}$\\, \\theta_i(\\epsilon)/(1-\\delta^i),$$ and $\\epsilon \\to 0$ as $T_h = \\min_i T^i \\to \\infty$. The limit matrices $(K^{i*},P^{i*})$ are the fixed point reached by the backward Riccati recursion and solve the coupled algebraic Riccati equations, so the limiting strategies form an FNE. A sequence of easily computed finite-horizon equilibria therefore approximates a generally hard infinite-horizon equilibrium, with a quantitative performance certificate.","pith_inferences":["Editorial extension: if the Riccati iteration converges geometrically, then $\\epsilon$ decays geometrically and the cubic bound implies geometric decay of the cost gap; the paper does not quantify the rate, but rate analysis would yield a horizon-selection rule.","Editorial extension: the first-stage-control scheme could be tried on time-varying or nonlinear dynamics, but the telescoping-power argument is specific to LQ structure, so a different proof would be needed.","Editorial extension: since the bound carries $1/(1-\\delta^i)$, patient players need longer horizons (or tighter gain convergence) to reach the same relative accuracy; heterogeneous horizons could be calibrated to discount factors."],"forward_implications":["Replacing the coupled algebraic Riccati equations by a sequence of linear solves makes the finite-horizon FNE computable in practice; Algorithm 1 verifies the invertibility condition as it runs.","As the shortest prediction horizon grows, the first-stage gains approach the limiting gains and each player's total cost approaches the infinite-horizon FNE cost.","The cost gap is bounded by a cubic polynomial in the strategy-matrix distance, so small gain errors translate into small cost errors, and the bound vanishes as $T_h\\to\\infty$.","The result covers heterogeneous discount factors $\\delta^i\\in(0,1]$, so players with different patience levels may use different look-ahead lengths."],"supporting_citations":[{"why":"Supplies the definition of feedback Nash equilibrium and the Riccati-based characterization that Lemma 1 adapts to the input/output/state finite-horizon game.","marker":"[4]"},{"why":"Documents the computational difficulty of infinite-horizon FNEs and supplies iterative algorithms that the finite-horizon strategy is meant to bypass.","marker":"[5]"},{"why":"Provides the infinite-horizon FNE characterization and the proof template, specifically its Theorem 3.2, used in Lemma 2 to show the limiting matrices form an FNE.","marker":"[6]"},{"why":"Defines the approximate equilibrium against which the paper positions its alternative to solving the nonconvex semidefinite program formulation.","marker":"[24]"},{"why":"Justifies restricting attention to linear time-invariant feedback strategies in the infinite-horizon game.","marker":"[26]"}],"fun_headline_variants":["Short-sighted players reach long-run Nash equilibrium","Finite horizon play approximates infinite-horizon equilibrium","Cubic error bound for finite-horizon Nash approximation","No Riccati needed: finite lookahead suffices","Look ahead, act now: close to Nash"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole guarantee rests on an assumed convergence: as the backward Riccati recursion is run further, the strategy and cost matrices must settle at finite limits, and the paper does not prove when that happens. The paper itself leaves parameter conditions for this convergence as an open question.","fun_headline_variants_meta":{"raw":{"variants":["Short-sighted players reach long-run Nash equilibrium","Finite horizon play approximates infinite-horizon equilibrium","Cubic error bound for finite-horizon Nash approximation","No Riccati needed: finite lookahead suffices","Look ahead, act now: close to Nash"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000534,"raw_usage":{"total_tokens":2616,"prompt_tokens":1041,"completion_tokens":1575,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":657,"completion_tokens_details":{"reasoning_tokens":1497}},"tokens_in":657,"tokens_out":1575,"duration_ms":13409,"temperature":1.0,"reasoning_tokens":1497,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T23:06:55.921957+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a small LQ game satisfying the invertibility and stability parts of Assumption 1 and run the backward Riccati iteration (3)--(5) from $P^i_1=0$; if the iterates never settle at finite limits, Assumption 1(ii) fails and the theorem's conclusion is not guaranteed. For any instance where Assumption 1 holds, simulate the finite-horizon strategy for growing common horizon and compare the empirical cost gap with the right-hand side of the Theorem 3 inequality; a violation would falsify the stated bound.","supporting_citations":[{"cited_title":"Basar, G","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of feedback Nash equilibrium and the Riccati-based characterization that Lemma 1 adapts to the input/output/state finite-horizon game."},{"cited_title":"Nortmann, A","cited_arxiv_id":null,"evidence_quote":"Documents the computational difficulty of infinite-horizon FNEs and supplies iterative algorithms that the finite-horizon strategy is meant to bypass."},{"cited_title":"Monti, B","cited_arxiv_id":null,"evidence_quote":"Provides the infinite-horizon FNE characterization and the proof template, specifically its Theorem 3.2, used in Lemma 2 to show the limiting matrices form an FNE."},{"cited_title":"Nortmann, T","cited_arxiv_id":null,"evidence_quote":"Defines the approximate equilibrium against which the paper positions its alternative to solving the nonconvex semidefinite program formulation."},{"cited_title":"Engwerda, Algorithms for computing nash equilibria in determin- istic lq games, Comput","cited_arxiv_id":null,"evidence_quote":"Justifies restricting attention to linear time-invariant feedback strategies in the infinite-horizon game."}],"review_version":1}