{"id":"2545fe40-d65a-4d80-9d1e-c9a75f2cb934","arxiv_id":"2411.10297","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The authors derive offline and online inverse differential game algorithms that recover all cost-function parameter sets consistent with observed Nash equilibrium trajectories, with convergence guarantees for the online version.","lead":"This paper presents two methods for figuring out what each player in a multi-agent control game is trying to minimize, based only on observed trajectories. The methods work for nonlinear systems and explicitly map out the family of cost functions that could explain the same behavior.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed equivalence set is not established: Theorem 1's sets (25)/(26) are only shown to satisfy HJB residuals at finitely many sampled states, and the 'unique OC solution' restriction on the free vector w̄ is not characterized.","rationale":"The reader's weakest assumption correctly identifies the uncharacterized w̄ condition: Theorem 1 restricts the free vector only by requiring the decoupled HJB equation to be necessary and sufficient for a unique OC solution, with no constructive test. My stress-test adds a sharper, more concrete gap: because Assumption 6 is only a rank condition on finitely sampled states, the affine sets (25)/(26) are not proven invariant under a change of evaluation grid. The claim that every parameter in the computed set solves Problem 1 requires exactly this invariance, or an explicit guarantee that HJB residuals vanish everywhere on X (or at least along the relevant trajectories). The paper makes a credible contribution: the separate analysis of value-function versus cost-function approximation errors is valuable, and the numerical example illustrates the main behaviors, including the failure mode for cost approximation. However, the central solution-set theorem is conditional on a verification condition that is neither stated constructively nor tested. The grid-dependence test proposed above would settle whether the gap is real. Since the reader already issued a CONDITIONAL verdict that matches this assessment, no verdict change is recommended.","tokens_in":25009,"tokens_out":4867,"duration_ms":54027,"concrete_test":"Recompute the offline solution set (25) for the Section V example using two different grids of K̄ states, both satisfying Assumption 6, e.g. the reported 21×21 grid and a shifted 25×25 grid; enumerate the affine set (25) for the same w̄ and compare. If the (αᵢ, βᵢ) sets are not identical, or if a parameter from one grid has nonzero HJB residual at a state in the other grid, then Theorem 1 overclaims. Then simulate the FNE for such a parameter and compare the resulting trajectories to the GT trajectories.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that every parameter in sets (25)/(26) reproduces the observed GT trajectories. The proof of Theorem 1 only shows that the affine families solve the decoupled HJB equations at the K̄ chosen states (Assumption 6) and that μ̂ᵢ = μᵢ*. Two gaps make the implication unsupported. First, Assumption 6 controls only the rank of the sampled regressor matrix, not the equality of the affine solution sets across different finite grids. When the regressor is rank-deficient, the null space of M_HJB(r) over a finite grid need not coincide with the null space of the same operator over all of X; a parameter drawn from (25) can therefore have nonzero HJB residual at unsampled states and need not define the same optimal-control solution. Second, even if the residual vanished everywhere, Theorem 1 restricts w̄ only by the non-constructive condition that the decoupled HJB be necessary and sufficient for a unique OC solution; no characterization (for example, positive definiteness of R̂ᵢᵢ and Q̂ᵢ, or closed-loop stability) is given. Lemma 4's Assumption 7 presupposes the same kind of exact alignment between identified strategies and an exactly representable DG. Thus the solution-set claim is not proven; the numerical example tests only one element of (25), not the whole set.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes offline and online inverse differential game (IDG) methods for nonlinear differential games, aiming to recover all cost-function parameters that reproduce given ground-truth trajectories of a feedback Nash equilibrium. The offline method (Section III) uses the observed trajectories to first identify the Nash equilibrium strategies and parts of the value functions (Lemmas 1 and 2), then reformulates the coupled Hamilton-Jacobi-Bellman equations as decoupled linear equations, and solves quadratic programs to obtain affine solution sets (Theorem 1, Eqs. (25) and (26)). The online method (Section IV) uses gradient-descent updates for the same parameters and proves convergence to a fixed element of the offline solution set (Theorem 2). The paper also analyzes the effect of approximating the value functions (bounded strategy error) and the cost functions (in general, unbounded or biased parameter estimates). A two-player numerical example illustrates the offline and online methods and the approximation-error cases.","tokens_in":25293,"tokens_out":5960,"duration_ms":61022,"significance":"If the central equivalence claim of Theorem 1 is established, the paper would make a substantial contribution: it would provide a formal treatment of non-uniqueness in nonlinear multi-player inverse differential games beyond the known scaling ambiguity, give the first online nonlinear IDG method with a convergence guarantee to the offline solution set, and provide a separate approximation-error analysis for value and cost functions that is genuinely informative for practitioners. The paper's strengths include the clean derivation of the decoupled HJB reformulation under the stated parametric assumptions, the explicit handling of per-player scaling non-uniqueness in Lemma 2, the machine-checkable linear-algebra formulation of the solution sets, and an honest negative result for cost-function approximation (Lemma 5). The numerical example is reproducible and demonstrates the proposed algorithms on a nontrivial two-player nonlinear example.","major_comments":[{"comment":"The central claim that every parameter in the sets (25) and (26) solves Problem 1 is not fully proven. Equations (21) and (22) are required to hold for every x in X, but the QP (28) only enforces them at the K-bar sampled states x-bar_k. Assumption 6 ensures that further data points do not increase the column rank of the sampled regressor, but it does not imply that the null space of the finite-sample matrix equals the null space of the operator on all of X. A parameter satisfying the finite-grid equations can therefore have nonzero HJB residual at unsampled states and does not necessarily define the same optimal-control solution. To support the claim, the authors need either a genericity or analyticity condition that forces equality of the solution sets, or a revised statement that the computed sets are only the finite-data versions and the global equivalence holds under an additional verifiable condition.","section":"Theorem 1, Eqs. (25)-(26)"},{"comment":"The restriction on the free vectors w-bar_i (or w-bar_i^(r)) in (25) and (26) is non-constructive: the condition that the decoupled HJB equation be 'necessary and sufficient for a unique OC solution' is not characterized by any checkable condition, such as positive definiteness of R-hat_ii and Q-hat_i or closed-loop stability of the resulting feedback law. Without such a characterization, the algorithm cannot actually compute the claimed 'set of all equivalent cost function parameters', because one cannot decide which affine parameters belong to the set. The numerical example implicitly uses positive definiteness of Q_i to define the set, but this criterion is not connected to the theorem; the claim should be either proved or made conditional on an explicit, verifiable condition.","section":"Theorem 1, w-bar conditions"},{"comment":"Assumption 7 presupposes exactly the alignment that the paper argues is necessary: it assumes that there exist cost parameters R-tilde_ij and beta-tilde_i such that the approximated value functions theta*_i^T phi_i(x) are the exact value functions of a differential game with those cost functions. Lemma 4 then establishes only that the parameters from (25)/(26) yield a FNE equal to the identified control laws mu-tilde*_i, under this assumption. This is a conditional result, and Section V-B shows that Assumption 10 (the online analogue) can fail. The paper should state in the main text, not only in the numerical example, that the approximation-error bound for the offline method rests on an unverifiable alignment assumption and that when the assumption fails the HJB-based identification may not produce a FNE matching the identified strategies.","section":"Lemma 4 and Assumption 7"},{"comment":"The proof of Theorem 2 introduces an additional condition that is not listed among the theorem's assumptions: it is stated that 'we can assume that the GT trajectory x*(t0 to infinity) is excited such that Assumption 6 is fulfilled as well'. Assumption 6 is defined for the offline grid (23)-(24) and is not a standing assumption on the online trajectory or the probing signal. Furthermore, the proof uses m_HJB(t) evaluated along the probing signal x_HJB(t) of Remark 3, but the theorem does not state that this signal stays inside X or that the continuous-time rank condition holds. These points need to be clarified and made part of the hypotheses, or the proof needs to be revised to derive them from Assumptions 8 and 9.","section":"Theorem 2, proof after Eq. (41)"}],"minor_comments":[{"comment":"The bound in (29) uses the norm of the gradient of the approximation error function epsilon-bar_i, but the Stone-Weierstrass theorem only provides boundedness of the function itself. If X is not compact, boundedness of the gradient does not follow; the assumption should state that X is compact or that the derivative of epsilon-bar_i is bounded.","section":"Lemma 3, Eq. (29)"},{"comment":"The stopping criterion (35) depends on a time window T and a threshold that are not specified; it would be helpful to state how these are chosen in practice and whether the stopping time affects any of the theoretical guarantees, since Lemmas 6-8 discuss exponential convergence but the algorithm terminates at a finite time.","section":"Section IV, stopping criterion (35)"},{"comment":"The NSAE values in the approximation-error examples (e.g., delta_x ≈ 404.8 and delta_u ≈ 920.3) exceed 1, meaning the normalized error is larger than the maximum absolute value of the signal; while this is consistent with the theory's guarantee of only boundedness, a remark explaining that the theoretical bound is not tight and that these values indicate a large practical error would improve the presentation.","section":"Section V-B, NSAE values"},{"comment":"The 'highest possible column rank' conditions in Assumptions 5 and 6 are phrased in terms of saturation with respect to data points, but no finite procedure to verify them from data is described; for nonlinear basis functions, this condition is not directly checkable and deserves a comment on how a practitioner would confirm it.","section":"Assumptions 5 and 6"},{"comment":"The caption of Figure 1 refers to 'the last two initial state resets' but the time axis begins at 12 s; the initial reset times should be identified to make the comparison between ground-truth and estimated trajectories easy to follow.","section":"Figure 1 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of IEEE Transactions on Cybernetics and addresses a timely problem. The main concern is that the abstract's strong claim—that the offline method 'computes the sets of all equivalent cost function parameters'—is not currently fully supported by the proof of Theorem 1, which relies on a finite-grid residual condition and an uncharacterized unique-OC condition. The issues are technical and appear fixable: the authors could add a genericity/analyticity condition to bridge the finite grid to the whole domain, give a constructive characterization of the admissible free parameters, or explicitly narrow the claim to finite-data solution sets. The approximation-error analysis is a valuable contribution and is presented honestly, including the negative result for cost-function approximation. I recommend major revision and would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a substantive step forward for nonlinear inverse differential games, but the central solution-set theorem is not fully proven. The gap is real but addressable, so it deserves refereeing rather than rejection.\n\nWhat's new and good: the paper lifts HJB-based inverse differential games from linear-quadratic settings to nonlinear systems with known basis functions, and it treats non-uniqueness explicitly by computing affine solution sets rather than a single parameter vector. The separate analysis of value-function approximation versus cost-function approximation errors is genuinely useful and honest: it shows that value-function approximation errors stay bounded while cost-function errors generally do not, and the numerical example demonstrates that failure mode clearly. Lemma 1's reformulation and Lemma 2's identification are clean under the stated assumptions. The online method with convergence to the offline set is a real novelty, as are the error bounds in Lemmas 7 and 8.\n\nSoft spots: the main claim of Theorem 1—that every parameter in (25)/(26) reproduces the observed trajectories—only holds if the decoupled HJB equation is necessary and sufficient for a unique optimal-control solution, and that condition is stated but never characterized. No constructive condition, like positive definiteness of the recovered matrices, is given. Relatedly, the proof of Theorem 1 works with HJB residuals at finitely many sampled points (Assumption 6), and the step from 'residual zero on the grid' to 'residual zero on all of X' is not justified. A parameter drawn from the affine set can have nonzero HJB residual off the grid. Assumption 7, used in Lemma 4, presumes the exact alignment between identified strategies and an exactly representable differential game—which is precisely the kind of thing the paper tries to establish. The online convergence proof (Theorem 2) hinges on Assumption 9, and it also uses an unstated 'we can assume the GT trajectory is excited' step to extend Assumption 6 to the online setting; that is an extra assumption, not a consequence. The numerical example tests only one element of the solution set, so it doesn't validate the set claim.\n\nNone of this is a takedown. The core machinery is sound, and the approximation-error analysis is valuable on its own. The authors have been candid about several limitations. The gaps are about characterization and proof completeness, not about a fundamentally wrong approach.\n\nWho it's for: control theorists working on inverse optimal control, differential games, and human-robot interaction; also anyone building data-driven cost identification for multi-agent systems. It deserves a serious referee, with the expectation of a major revision to pin down Theorem 1 and the online excitation conditions.","headline":"Real progress on nonlinear inverse differential games, but the solution-set claim rests on an uncharacterized condition and a finite-sample assumption; referee it with a demanded revision.","tokens_in":25831,"tokens_out":2737,"would_cite":true,"duration_ms":26395,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49N70","49N45","91A23","93B30","93C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Inverse differential games can be solved by computing the full set of equivalent cost parameters, with online convergence to one element.","keywords":["inverse differential games","inverse optimal control","Hamilton-Jacobi-Bellman equations","feedback Nash equilibrium","non-uniqueness solution set","online learning","approximation error","nonlinear systems"],"falsifier":"Compute the solution set for a nonlinear game with two distinct free-vector choices that both satisfy the paper's conditions, then solve the forward differential game for each parameter pair and compare trajectories to the observed ones; if any parameter inside the set produces a trajectory that deviates from the observations, the claim that the set contains all equivalent parameters fails. Alternatively, construct a game with value-function approximation error where no parameterized cost function can realize the identified strategy, which would show that Assumption 7 and Lemma 4 do not apply.","tokens_in":24739,"feed_emoji":"🎮","tokens_out":5156,"duration_ms":46784,"temperature":0.7,"pith_summary":"The paper aims to solve the inverse differential game problem for nonlinear multi-player systems: from observed state and control trajectories that constitute a feedback Nash equilibrium, recover the cost functions of all players. Its central claim is that an offline HJB-based method computes the full set of equivalent cost function parameters that reproduce the observed trajectories, and that an online gradient-descent counterpart is guaranteed to converge to one element of that set. A separate approximation analysis shows that errors in value-function approximation produce bounded trajectory errors, whereas errors in cost-function approximation generally produce unbounded or unquantifiable errors unless cost and value structures are aligned so the coupled Hamilton-Jacobi-Bellman equations can be fulfilled. A two-player numerical example illustrates the results under both known and violated structural assumptions.","feed_headline":"Recover every cost function consistent with observed play","feed_subtitle":"Offline and online HJB methods compute all equivalent cost parameters from Nash trajectories, with bounded approximation errors.","key_machinery":"The central machinery is the reformulation of the coupled HJB equations into N decoupled linear-in-parameters equations after replacing the true Nash strategies with identified ones. Solving these equations as least-squares problems gives parameter sets parametrized by the null-space vectors of the stacked regression matrices, displayed as equations (25) and (26). The online method uses the same regression structure in gradient-descent laws, with persistent excitation assumptions guaranteeing exponential convergence. The approximation analysis uses the Stone-Weierstrass theorem to represent value and cost functions as basis expansions plus bounded residuals, and compares how the two types of residuals propagate.","core_discovery":"The core discovery is that the coupled HJB equations, which characterize a feedback Nash equilibrium, can be decoupled after first identifying the equilibrium strategies from data, and then solved as convex quadratic programs whose null spaces describe all equivalent cost parameters. When the value functions are known up to parameters, the offline method returns the complete solution set of equivalent parameters; the online method, under persistent excitation, converges to one element of this set. When value-function structures are approximated, the identified strategies stay within a bounded error of the true Nash strategies, but this only transfers to cost parameters if an additional existence assumption holds. When cost functions are approximated, the residual approximation error biases the parameter estimates and the resulting Nash equilibrium generally differs from the observed one, with no guaranteed bound.","pith_inferences":["The paper leaves open the problem of constructively characterizing the free null-space vectors that make the decoupled optimal control problem well-posed; a practical extension would be to project the solution set onto parameters satisfying positive definiteness or other regularity constraints.","The asymmetry between value and cost approximation suggests a design rule: choose value-function basis functions first, then construct cost basis functions via a converse HJB argument so the coupled equations are exactly solvable; the paper hints at this in Remark 2 but does not turn it into an algorithm.","Because the online estimator converges to an arbitrary element of the solution set, the converged parameters are not individually identifiable; set-valued tracking or additional selection criteria would be needed for interpretations in applications like human-robot interaction.","A concrete testable extension for linear-quadratic games would be to check whether the implicit free-vector condition reduces to positive-definiteness constraints, which would give a fully constructive version of Theorem 1 in that setting."],"forward_implications":["For a given dataset, all cost parameters consistent with the data can be enumerated or sampled from the solution set, making non-uniqueness explicit rather than hidden by a scaling assumption.","The online method is the first nonlinear multi-player inverse differential game algorithm with a convergence guarantee to the offline solution set, enabling real-time identification from streaming trajectories.","Value-function approximation errors lead to bounded errors in the identified Nash strategies and, under an additional existence assumption, in the final trajectories.","Cost-function approximation errors can bias parameters and break trajectory matching, so the paper's alignment condition on cost and value basis functions is necessary for bounded final errors.","When the structural assumptions hold, the estimated cost parameters produce a forward Nash equilibrium whose trajectories match the observed ones, as demonstrated by the numerical example."],"supporting_citations":[{"why":"Supplies the necessary and sufficient coupled Hamilton-Jacobi-Bellman condition for a feedback Nash equilibrium that underlies all identification equations.","marker":"[21]"},{"why":"Provides the single-player optimal control conditions that justify the decoupling logic when reducing to one player.","marker":"[22]"},{"why":"Establishes the prior solution-set result for linear-quadratic inverse differential games that this paper generalizes to nonlinear settings.","marker":"[5]"},{"why":"Gives the policy iteration algorithm used to compute the forward Nash equilibrium from estimated cost parameters for verification.","marker":"[23]"},{"why":"Supplies the Stone-Weierstrass approximation theorem used to represent approximated value and cost functions as basis expansions plus bounded residuals.","marker":"[25]"},{"why":"Provides the persistent-excitation stability theorem used to prove exponential convergence of the online parameter adaptation.","marker":"[20]"},{"why":"Defines exponential stability and provides convergence results used in the online error-dynamics analysis.","marker":"[27]"},{"why":"Contributes the technical lemma used in Lemma 7 to establish convergence to a residual set under bounded disturbances.","marker":"[28]"}],"fun_headline_variants":["Inverse differential games: pinpoint all cost functions from player paths","Offline and online methods recover every cost function that fits observed play","Decouple Hamilton-Jacobi equations to infer all cost parameters from data","From observed trajectories to all equivalent cost functions in nonlinear games","Online inverse game solver converges to costs matching observed Nash play"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that for every parameter in the computed sets one can choose the free null-space vector so that the decoupled optimal control problem has a unique solution, and that cost parameters exist making approximated value functions exact; neither choice is given constructively.","fun_headline_variants_meta":{"raw":{"variants":["Inverse differential games: pinpoint all cost functions from player paths","Offline and online methods recover every cost function that fits observed play","Decouple Hamilton-Jacobi equations to infer all cost parameters from data","From observed trajectories to all equivalent cost functions in nonlinear games","Online inverse game solver converges to costs matching observed Nash play"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000181,"raw_usage":{"total_tokens":1265,"prompt_tokens":860,"completion_tokens":405,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":319}},"tokens_in":476,"tokens_out":405,"duration_ms":4705,"temperature":1.0,"reasoning_tokens":319,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:46:30.717478+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the solution set for a nonlinear game with two distinct free-vector choices that both satisfy the paper's conditions, then solve the forward differential game for each parameter pair and compare trajectories to the observed ones; if any parameter inside the set produces a trajectory that deviates from the observations, the claim that the set contains all equivalent parameters fails. Alternatively, construct a game with value-function approximation error where no parameterized cost function can realize the identified strategy, which would show that Assumption 7 and Lemma 4 do not apply.","supporting_citations":[{"cited_title":"Bas ¸ar and G","cited_arxiv_id":null,"evidence_quote":"Supplies the necessary and sufficient coupled Hamilton-Jacobi-Bellman condition for a feedback Nash equilibrium that underlies all identification equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the single-player optimal control conditions that justify the decoupling logic when reducing to one player."},{"cited_title":"Solution sets for inverse non-cooperative linear-quadratic differential games,","cited_arxiv_id":null,"evidence_quote":"Establishes the prior solution-set result for linear-quadratic inverse differential games that this paper generalizes to nonlinear settings."},{"cited_title":"Excitation for adaptive optimal control of nonlinear systems in differential games,","cited_arxiv_id":null,"evidence_quote":"Gives the policy iteration algorithm used to compute the forward Nash equilibrium from estimated cost parameters for verification."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Stone-Weierstrass approximation theorem used to represent approximated value and cost functions as basis expansions plus bounded residuals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the persistent-excitation stability theorem used to prove exponential convergence of the online parameter adaptation."},{"cited_title":"Convergence properties of adaptive systems and the definition of exponential stability,","cited_arxiv_id":null,"evidence_quote":"Defines exponential stability and provides convergence results used in the online error-dynamics analysis."},{"cited_title":"Online actor-critic algorithm to solve the continuous-time infinite horizon optimal control problem,","cited_arxiv_id":null,"evidence_quote":"Contributes the technical lemma used in Lemma 7 to establish convergence to a residual set under bounded disturbances."}],"review_version":1}