{"id":"2a9d512c-8a44-40d1-ac16-8726b7de36b0","arxiv_id":"2507.02203","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"Two Koopman-based solution approaches, resolvent feedback and EDMD-MCP, reproduce the analytical turret-defense equilibrium for most initial states but visibly fail on singular surfaces and lack error metrics.","lead":"Two new solvers for two-player zero-sum differential games, one using the Koopman operator resolvent to produce a global feedback policy and one using a lifted data-driven model with complementarity to produce open-loop policies, are demonstrated on a turret defense game. The methods reproduce the known optimal behavior for most initial conditions, but the paper's own results show they miss the optimal solution on singular surfaces and need more careful validation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The resolvent method optimizes the spatially aggregated payoff ∫_X J(x) dx (Eqs. 23, 49), not the pointwise value function; no proof shows the aggregate-optimal feedback policy is a pointwise saddle policy, and Section 6.1 concedes the method is 'not directly solving for the Nash equilibrium.'","rationale":"The reader's weakest assumption identifies exactly the same load-bearing gap: the resolvent method minimizes an aggregate payoff rather than the pointwise value function, and no equivalence is proven. The full text strengthens this concern: Section 6.1 explicitly states that the resolvent method is 'not directly solving for the Nash equilibrium,' and the numerical results show the expected failures (Universal Line, smooth approximation of sharp discontinuities). The concern is not merely a disagreement with the existing analytical solution; it is a correctness risk internal to the method's formulation. The EDMD-MCP component is less affected because it solves one initial condition at a time via optimality conditions. The appropriate remedy is a proof or explicit limitation statement connecting the aggregate optimum to the pointwise saddle property, plus a pointwise verification of the saddle inequalities. This is addressable in revision, so the reader's CONDITIONAL verdict remains appropriate; no change to the verdict is needed.","tokens_in":28209,"tokens_out":4946,"duration_ms":65768,"concrete_test":"Construct a two-point variant of the turret game with x_A=(0.9, 2.0) and x_B=(0.1, 0.5), using a single shared constant feedback policy (u0, v0). Brute-force compute (i) the saddle policy for x_A alone, (ii) the saddle policy for x_B alone, and (iii) the maximin of J(x_A)+J(x_B) over (u0, v0). If the maximin of the sum is not simultaneously a saddle policy for both points (i.e., saddle inequalities (65) fail at either x_A or x_B), the aggregate objective is not equivalent to pointwise equilibrium. Then repeat with the published RBF basis and a finer grid of sampled initial states: evaluate the published resolvent policy's pointwise saddle inequalities at 20 states by computing one-sided control deviations while holding the opponent's policy fixed. If violations persist as the basis is refined, the concern is confirmed as a structural issue rather than an approximation artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the global feedback policies u*(x), v*(x) produced by the resolvent method are equilibrium policies for the differential game. However, Eq. 49 optimizes only the sampled aggregate Σ_i J(x_i,T) from Eq. 23. Stationarity of this aggregate with respect to a control coefficient u_k is Σ_i ∂J(x_i)/∂u_k = 0, which does not imply the pointwise stationarity ∂J(x_i)/∂u_k = 0 needed for the saddle inequalities (65) at each initial state. The paper supplies no argument — e.g., via dynamic programming or the Isaacs/HJI equation — that the minimax of the aggregate equals the pointwise value function. This is not merely a numerical-fidelity issue: even with an exact finite basis, a policy that is optimal on average over the sampled domain need not be a saddle policy at any particular initial condition. The small-r region is particularly vulnerable, since the running and terminal costs are proportional to r and therefore contribute little to the aggregate objective. The paper's own numerical results show the expected symptoms: the fixed smooth basis misses the sharp Universal Line behavior, and the agent policy cannot represent the time-dependent bifurcation that a true feedback saddle policy would require. Section 6.1 explicitly concedes that the alternating-optimization scheme is 'not directly solving for the Nash equilibrium.' The EDMD-MCP method does not share this specific gap because it solves trajectory-by-trajectory open-loop equilibria; the concern is targeted at the resolvent method's global-feedback claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two operator-theoretic approaches to solving two-player zero-sum differential games and demonstrates them on a turret defense game from the literature. The first approach approximates the Koopman generator on a fixed RBF basis and uses a resolvent contour-integral representation of the cost functional to compute a continuous-time global feedback policy by alternating optimization. The second approach builds a discrete-time EDMD-with-control model using random Fourier features, lifts the agent's controls into parallel and perpendicular components, and solves the resulting mixed complementarity problem trajectory by trajectory for open-loop policies. The methods are benchmarked against the analytical solution of Von Moll et al. for the turret defense game. The EDMD-MCP method matches the analytic trajectories closely for most of the approximately 1500 initial conditions tested, while the resolvent method shows qualitative agreement but visible deviations near the Universal Line and around alpha = 3 pi / 4, as the authors acknowledge. The paper also compares computational costs and discusses extensions of both methods.","tokens_in":28649,"tokens_out":4993,"duration_ms":67175,"significance":"The paper is a useful computational demonstration of how Koopman-operator ideas can be combined with complementarity and resolvent methods for differential games, and it provides a comparatively honest and transparent account of where the methods work and where they do not. Its strengths include the use of an external analytical benchmark, validation over a large set of initial conditions, and explicit discussion of the limitations of fixed-basis approximations. However, the central theoretical claim that both methods yield equilibrium solutions is only fully established for the EDMD-MCP method. For the resolvent method, the objective being optimized is an aggregate over initial states, and no proof or verification is supplied that the resulting global feedback policy satisfies the pointwise saddle-point inequalities of the original game. Because of this gap and the acknowledged deviations in the numerical solution, the paper's main claim is stronger than what is demonstrated. The methods are of interest to the computational differential games and Koopman control communities, but the theoretical gap must be closed or the claims substantially qualified.","major_comments":[{"comment":"The resolvent method optimizes the aggregate payoff J(T) = \\int_X [g(F_T(x)) + \\int_0^T h(F_t(x)) dt] dx (Eq. 23), not the pointwise value function V(x). For a feedback policy to be a saddle-point equilibrium, the inequalities in Eq. (65) must hold at every initial state. Stationarity of the aggregate objective with respect to a control coefficient u_k gives only \\sum_i \\partial J(x_i)/\\partial u_k = 0, which does not imply the pointwise stationarity needed for Eq. (65). No argument via dynamic programming, the Isaacs equation, or the HJI equation is provided. This is not a numerical-fidelity issue: even with an exact finite basis, a policy that is optimal on average over the sampled domain need not be a saddle policy at any particular initial condition, and the small-r region is especially vulnerable because the running and terminal costs are proportional to r. The paper should either prove the equivalence for the considered class of games or verify the pointwise saddle inequalities on a dense grid and restate the result as an approximate global feedback policy.","section":"Section 3.2, Eq. (49)"},{"comment":"The paper's abstract and Section 1.1 claim that the approaches \"yield the equilibrium solutions\" and \"replicate the behavior of the analytical solution,\" but Section 6.1 concedes that the resolvent method is \"not directly solving for the Nash equilibrium\" and instead relies on alternating optimization. This is not merely a wording issue: it means the reported resolvent-based feedback policies have not been shown to satisfy the defining equilibrium property of the game. The authors should either supply a convergence or equivalence argument for the alternating scheme, or explicitly frame the resolvent results as approximate solutions to a related aggregate objective, not as verified equilibrium policies.","section":"Section 6.1"},{"comment":"The reported resolvent results show that the smooth fixed basis cannot produce the sharp discontinuity at alpha = 0 (the Universal Line) and that the agent leaves the r = 1 surface near alpha = 3 pi / 4, a region the analytic solution treats as constrained. These are not peripheral artifacts: they are exactly the singular and constrained regimes that define the equilibrium structure of the game. The paper acknowledges these limitations, but the central claim that the method replicates the analytical solution needs to be qualified with a quantitative statement of where in the domain the approximation is valid, or the method needs a basis or representation able to capture the discontinuities.","section":"Section 5.2, Figs. 3 and 4"},{"comment":"The lower bound v^2 + v_perp^2 >= 0.9 v_A^2 is introduced because the finite-basis approximation sometimes makes the agent's speed too small, especially at small r. This changes the admissible control set relative to the original game, in which the agent's speed is fixed at v_A. The assertion in Section 4.2 that the relaxed inequality v^2 + v_perp^2 <= v_A^2 is tight at the optimum does not apply to the resolvent solution, since the paper explicitly states that the speed is sometimes well below v_A. The equilibrium status of a policy whose speed is 0.9 v_A in part of the domain is unclear; the authors should either project the final policy onto the feasible set or provide evidence that the lower bound is inactive at the reported solution.","section":"Section 4.3.2, Eq. (92)"}],"minor_comments":[{"comment":"There is a typographical double period in \"provided in the literature..\".","section":"Abstract"},{"comment":"The inequality in Eq. (72) appears to have the direction reversed: the intended constraint is v^2 + v_perp^2 <= v_A^2, not v_A^2 <= v^2 + v_perp^2.","section":"Section 4.2, Eq. (72)"},{"comment":"The sentence \"However, in this particular case, these was a far simpler alternative approach\" contains a grammatical error and should be rephrased.","section":"Section 4.3.2"},{"comment":"The phrase \"the constraint ( r = 1)\" should read \"the constraint r <= 1\" for consistency with the problem statement in Section 4.1.","section":"Section 5.1"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle to acceptance is the unresolved relationship between the resolvent method's aggregate objective and the pointwise Nash equilibrium condition. If the authors can add either a proof of equivalence for the relevant game class or a dense-grid verification of the saddle inequalities, the paper would be a solid computational contribution. The paper is best viewed as a methods demonstration rather than a theoretical advance; the editor may wish to weigh the novelty of the resolvent-based scheme accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper genuinely introduces something: using the resolvent of a control-dependent Koopman generator to solve a zero-sum differential game with global feedback policies. I checked the cited literature as best I can, and that particular move is new. Second, the paper's stronger practical result is the EDMD-MCP pipeline, which solves trajectory-by-trajectory open-loop equilibria via KKT conditions and PATH, and it reproduces the analytic turret-defense solutions for most of 1500 initial conditions. That part holds up reasonably well.\n\nWhat is good: the authors use an external analytical solution, not a fitted target. The cost matrix Q is specified a priori, and no step in the derivation takes the target result as input. They explicitly discuss singular surfaces, state constraints, lifted control variables, and computational tradeoffs. The EDMD-MCP method is a useful synthesis of EDMDc and MCP; the lifted controls (nu = r^2 v, nu_perp = r v_perp) are a nice domain-knowledge trick. They are also honest about limitations, including the basis-function failure to capture discontinuities near the Universal Line.\n\nThe soft spots are real but uneven. The resolvent method's objective is J(T) = integral over X of J(x,0) dx, an aggregate over sampled initial states. Stationarity of that aggregate with respect to control coefficients does not imply pointwise stationarity, so the resulting feedback policy need not be a saddle policy for any given initial condition. The paper never proves that the minimax of the aggregate equals the pointwise value function, and Section 6.1 explicitly concedes it is 'not directly solving for the Nash equilibrium.' This is not a minor numerical fidelity issue; it is a gap between the claim and the method. The numerical results show the symptom: the resolvent policy misses the Universal Line and the alpha ~ 3 pi / 4 boundary. The EDMD-MCP method does not share this flaw because it optimizes each trajectory separately.\n\nA second, smaller issue: there are no quantitative error metrics. The text claims 'strong quantitative agreement' but gives only trajectory plots and value heat maps. A table of errors would make the comparison reproducible.\n\nWho this is for: people working on Koopman-based control who want to see a serious attempt at differential games, and differential-game people who want to see whether data-driven lifting can avoid hand-constructed singular surfaces. The EDMD-MCP part deserves citation; the resolvent part needs more work before it can be trusted as a global-feedback equilibrium solver.\n\nIf you review it, send it out. The paper is original, clearly written, and honest; the resolvent gap is addressable with a proof or a sharply bounded limitation statement, plus error metrics. I would recommend major revision, not rejection.","headline":"New and honest Koopman differential-game pipeline, but the resolvent method's aggregate-objective gap undermines the global-feedback equilibrium claim.","tokens_in":29131,"tokens_out":2883,"would_cite":true,"duration_ms":35457,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49N70","47A10","37N35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that two Koopman-based pipelines — a resolvent-based global feedback search and a data-driven complementarity approach — both reproduce the analytic equilibrium of a zero-sum turret defense game.","keywords":["differential games","Koopman operator","Extended Dynamic Mode Decomposition","Mixed Complementarity Problems","resolvent operator","zero-sum game","turret defense","feedback control"],"falsifier":"Take a zero-sum game with a known pointwise equilibrium solution and a basis rich enough to represent the true value. Minimize the aggregate payoff $J(T)=\\int_X J(x,T)\\ dx$ with the resolvent method, then test the resulting feedback policy against the pointwise saddle inequalities $J(x,u,v^*)\\le V(x)\\le J(x,u^*,v)$ at a state where the aggregate optimum compromises one initial condition. If the policy violates either inequality, the method has not found an equilibrium of the original game despite having a perfect basis.","tokens_in":27991,"feed_emoji":"🎯","tokens_out":9547,"duration_ms":107535,"temperature":0.7,"pith_summary":"Two-player zero-sum differential games are normally solved by hand, because nonlinear dynamics produce singular surfaces that resist general numerical treatment. This paper claims that the Koopman operator — a linear operator that evolves functions of the state instead of the state itself — can automate the solution. The first proposed method uses the resolvent of the Koopman generator, $(zI-L)^{-1}$, to propagate terminal and running costs over the entire state domain and solves for a global feedback policy by alternating optimization. The second learns a finite-dimensional Koopman model with control from data and solves each trajectory's equilibrium as a mixed complementarity problem. On a turret defense game with a known analytical solution, both methods reproduce the equilibrium trajectories and the value of the game, with the per-trajectory data-driven method matching the analytic solution more closely and the resolvent method supplying a single feedback law valid over the whole domain.","feed_headline":"Two Koopman methods reproduce a differential game's equilibrium","feed_subtitle":"A resolvent-based global policy and a data-driven trajectory-wise solver both match the analytic solution.","key_machinery":"The load-bearing object is the Koopman generator $L=f\\cdot\\nabla_x$ and its resolvent $R_L(z)=(zI-L)^{-1}$, used through the contour-integral identity $K_t=\\frac{1}{2\\pi i}\\int_\\gamma e^{zt}R_L(z)\\ dz$ that propagates any observable forward in time. The paper folds a running cost into this identity with the factor $\\frac{1}{z}(1-e^{-zT})$, so a single resolvent calculation delivers the full cost functional of the game. Both methods then replace the infinite-dimensional operator by a finite dictionary: random Fourier features for the EDMD model, radial basis functions for the resolvent model, with the control policies expanded in the same basis. The EDMD route additionally assumes control-affine lifted dynamics, which lets the players' optimality conditions be assembled into a mixed complementarity problem; the resolvent route avoids that assumption and instead carries the control dependence inside $L$, at the price of nonlinear, alternating optimizations.","core_discovery":"The central claim is that Koopman-based reformulations can yield the equilibrium solutions of a zero-sum differential game, at least for the finite approximations of the dynamics that the methods use. In continuous time, the paper expresses the game value as a contour integral of the resolvent, $J(x_0,T)=\\frac{1}{2\\pi i}\\int_\\gamma e^{zT}(zI-L)^{-1}\\left(g(x_0)+\\frac{1}{z}(1-e^{-zT})h(x_0)\\right)dz$, and then makes the generator and both feedback policies finite-dimensional through a fixed basis, turning the saddle-point search into a finite-dimensional max-min problem solved by alternating optimizations. In discrete time, it fits an EDMD-with-control model $\\Psi(x_{t+1})=K\\Psi(x_t)+K_u u_t+K_v v_t$ with quadratic cost surrogates in lifted states, derives the two players' KKT conditions, and solves the resulting mixed complementarity problem trajectory by trajectory. On the turret defense game, the EDMD-MCP solutions are close to the analytic equilibrium for regular, constrained, and universal-line trajectories, while the resolvent-based feedback policy reproduces the structure of the analytic solution with the expected loss of fidelity near the $\\alpha=0$ discontinuity.","pith_inferences":["A natural reading is that the resolvent method is solving a population-level min-max problem over the sampled domain rather than the pointwise game; whether its feedback law is also a pointwise saddle policy is a testable question, and if not, the method should be presented as an average-case synthesis tool.","The poor fidelity near the universal line points to basis resolution rather than method failure; a multi-resolution or discontinuous basis could sharpen the feedback law, though any state-feedback policy cannot represent the time-of-entry-dependent bifurcations that appear in the analytic solution.","The success of lifted controls like $\\nu=r^2v$ and $\\nu_\\perp=rv_\\perp$ suggests that choosing control coordinates from the structure of the generator $L=\\sum_i f_i(x,u)\\partial_{x_i}$ could be a general recipe for making EDMD-based game solvers work when naive additive control fails.","Combining the MCP read-out with learned dictionaries such as autoencoder-based DMD is a plausible path to higher-dimensional games, while the resolvent approach will likely remain limited by exponential growth in basis size and quadrature nodes."],"forward_implications":["If the methods hold, differential games with nonlinear dynamics can be solved numerically without first characterizing their singular surfaces by hand.","The resolvent method produces a global feedback policy after one expensive computation, so evaluating it in real time is cheap; this matters for low-latency applications.","The EDMD-MCP method computes open-loop trajectories quickly from data and can enforce state and control constraints directly, making it suitable for high-dimensional, data-rich settings.","Both methods are approximations of the dynamics, so their equilibrium solutions are equilibria of the approximate model, not necessarily of the original game.","The two approaches are complementary: the resolvent method handles control-nonlinear dynamics without control lifting but is limited by smooth feedback bases, while the EDMD-MCP method captures discontinuities and constraints but needs lifted controls or domain knowledge."],"supporting_citations":[{"why":"supplies the analytical solution of the constrained turret defense game that both new methods are compared against.","marker":"[44]"},{"why":"provides the high-order contour-integral quadrature with error bounds that the resolvent-based method uses to propagate costs.","marker":"[24]"},{"why":"provides the complementarity solver used to find the Nash equilibrium of the MCP in the data-driven method.","marker":"[12]"},{"why":"provides the random Fourier features used to build the EDMD dictionary for the second method.","marker":"[46]"},{"why":"introduces the zero-sum turret defense differential game treated in the demonstration.","marker":"[6]"},{"why":"supplies the interior-point optimizer used in the alternating optimizations of the resolvent-based method.","marker":"[42]"}],"fun_headline_variants":["Koopman resolvent and EDMD reproduce differential game equilibrium","Two Koopman methods match analytic turret defense solution","Resolvent and data-driven Koopman replicate game equilibrium","Koopman-based policies solve zero-sum differential game"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a feedback policy minimizing the payoff averaged over sampled initial states is also the saddle-point policy for each individual initial condition; the paper uses this equivalence without proving it.","fun_headline_variants_meta":{"raw":{"variants":["Koopman resolvent and EDMD reproduce differential game equilibrium","Two Koopman methods match analytic turret defense solution","Resolvent and data-driven Koopman replicate game equilibrium","Koopman-based policies solve zero-sum differential game"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000345,"raw_usage":{"total_tokens":1890,"prompt_tokens":942,"completion_tokens":948,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":880}},"tokens_in":558,"tokens_out":948,"duration_ms":9773,"temperature":1.0,"reasoning_tokens":880,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:35:17.843203+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a zero-sum game with a known pointwise equilibrium solution and a basis rich enough to represent the true value. Minimize the aggregate payoff $J(T)=\\int_X J(x,T)\\ dx$ with the resolvent method, then test the resulting feedback policy against the pointwise saddle inequalities $J(x,u,v^*)\\le V(x)\\le J(x,u^*,v)$ at a state where the aggregate optimum compromises one initial condition. If the policy violates either inequality, the method has not found an equilibrium of the original game despite having a perfect basis.","supporting_citations":[{"cited_title":"In: Modeling, Estimation and Control Conference","cited_arxiv_id":null,"evidence_quote":"supplies the analytical solution of the constrained turret defense game that both new methods are compared against."},{"cited_title":"arXiv (2024)","cited_arxiv_id":null,"evidence_quote":"provides the high-order contour-integral quadrature with error bounds that the resolvent-based method uses to propagate costs."},{"cited_title":"Optimization methods and software 5(2), 123–156 (1995)","cited_arxiv_id":null,"evidence_quote":"provides the complementarity solver used to find the Nash equilibrium of the MCP in the data-driven method."},{"cited_title":"In: 2017 IEEE Conference on Control Technology and Applications (CCTA), pp","cited_arxiv_id":null,"evidence_quote":"introduces the zero-sum turret defense differential game treated in the demonstration."},{"cited_title":"Mathematical programming 106(1), 25–57 (2006)","cited_arxiv_id":null,"evidence_quote":"supplies the interior-point optimizer used in the alternating optimizations of the resolvent-based method."}],"review_version":1}