{"id":"f51bad60-0a0c-4e92-af2d-ac372fae2d86","arxiv_id":"2507.17014","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Open-loop and closed-loop Nash equilibria of N-player Mean Field Games of Controls converge to the mean field equilibrium at explicit rates under displacement monotonicity, without assuming separable Hamiltonians.","lead":"This paper proves quantitative convergence rates for N-player stochastic games with interactions through controls toward their mean field limit, under displacement monotonicity. It removes a restrictive separability assumption on the Hamiltonian, which matters for economic and financial models where agents interact through both states and controls.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Closed-loop convergence (Theorem 1.8) is conditional on an unproved existence assumption: no admissible classical solution to the N-player Nash system (1.11) with bounded second derivatives is known under the quadratic growth allowed by Assumption 2.1, so the abstract's claim to 'conclude'…","rationale":"The reader's weakest-assumption analysis is accurate. I found no additional internal error in the open-loop argument: the fixed-point construction (Lemmas 3.1–3.6) is coherent modulo standard Lipschitz/Young estimates, and the stability bound (Proposition 4.1) is the right tool for the open-loop theorem. The closed-loop proof is internally consistent but entirely dependent on the existence of admissible solutions to (1.11); the paper does not pretend otherwise, yet the abstract overstates the situation by saying the paper 'conclude[s] the convergence of closed-loop equilibria.' The concrete check I propose is therefore to settle the well-posedness question, either by a continuation argument using the paper's own Section 6 estimates or by a counterexample. If no counterexample is found and the continuation closes, the paper's result becomes unconditional; if not, the published version should state Theorem 1.8 as conditional on the existence of admissible Nash-system solutions. This does not change the reader's CONDITIONAL verdict.","tokens_in":49898,"tokens_out":16521,"duration_ms":169893,"concrete_test":"Prove or disprove well-posedness of the Nash system (1.11) for large N under Assumptions 2.1 and 2.3 by attempting a Leray–Schauder/continuation argument on the space of functions with bounded second derivatives, using the uniform bound on the quantity (1.27) from Proposition 6.5 as the essential a priori estimate. Concretely, check whether the a priori bounds can be upgraded to C^{2,α} controls on uN that allow passage to the limit as the truncation of the Hamiltonian is removed; if the argument cannot be closed (e.g., quadratic growth of H produces blow-up of second derivatives), then the admissibility assumption in Definition 1.2 is not a harmless hypothesis and Theorem 1.8 should remain explicitly conditional. A positive closedness argument would make Theorem 1.8 unconditional.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of the paper has two parts. The open-loop result (Theorem 1.6) is well supported: it rests on the existence of the fixed-point maps Φ and aN (proved in Section 3) and on the stability estimate Proposition 4.1, and it is explicitly conditional only on the existence of open-loop equilibria and a mean-field equilibrium, which are natural hypotheses. The closed-loop result (Theorem 1.8) carries a much heavier and unproved assumption: Definition 1.2 requires an admissible classical solution uN to the N-player Nash system (1.11), i.e., a global classical solution with bounded second spatial derivatives and exchangeability. The authors state in Section 1.2 that no well-posedness result for (1.11) is known under the quadratic growth permitted by Assumption 2.1, and Remark 1.3 admits that exchangeability is not automatic without a uniqueness result. Thus, if such admissible solutions fail to exist for some N, Theorem 1.8 is vacuous. This is not an internal inconsistency, but it is a genuine gap between the abstract's claim to 'conclude the convergence of closed-loop equilibria' and the theorem's conditional hypothesis. The a priori estimates in Sections 6 and 7 (notably Proposition 6.5) are conditional on the existence of an admissible solution as well, so they do not by themselves provide existence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies N-player stochastic differential games with interactions through both state and control variables, in the displacement-monotone regime. It establishes quantitative convergence of open-loop Nash equilibria (Theorem 1.6) and of closed-loop Nash equilibria (Theorem 1.8) to the corresponding mean field game equilibrium, with explicit rates r_{d,p}(N). The proof is built on a detailed analysis of the fixed-point maps Phi and a_N (Section 3), a uniform-in-N stability estimate for the Pontryagin system (Proposition 4.1), and a bootstrap argument for the N-player Nash system (Proposition 6.5). The closed-loop result is explicitly conditional on the existence of admissible classical solutions to the Nash system (Definition 1.2), a point the authors state clearly in Section 1.2.","tokens_in":50130,"tokens_out":3780,"duration_ms":39132,"significance":"If correct, the open-loop result is a significant advance: it removes the separability assumption on the Hamiltonian and prior regularity assumptions on the fixed-point maps, and it provides a global-in-time quantitative rate under displacement monotonicity. The structural analysis of the fixed-point maps Phi and a_N (Lemmas 3.3, 3.5, 3.6) is a genuine contribution, as is the uniform stability estimate Proposition 4.1. The bootstrap argument in Proposition 6.5 is intricate and appears internally coherent. The main caveat is that Theorem 1.8 rests on an unproved existence hypothesis for the Nash system; this limits the scope of the closed-loop claim but does not undermine the open-loop part.","major_comments":[{"comment":"The closed-loop convergence result is conditional on an unproved existence assumption: the authors state in Section 1.2 that no well-posedness result for the N-player Nash system (1.11) is known under the quadratic growth permitted by Assumption 2.1. Since Definition 1.2 requires an admissible classical solution with bounded second spatial derivatives and exchangeability, Theorem 1.8 is vacuous for any N for which such a solution fails to exist. This is not an internal inconsistency, but it means the abstract's claim to 'conclude the convergence of closed-loop equilibria' overstates what is proved. The open-loop Theorem 1.6 is not affected by this issue.","section":"Section 1.2, Definition 1.2, Theorem 1.8"},{"comment":"The uniform a priori estimate in Proposition 6.5 is derived under the standing assumption that an admissible solution u_N to (1.11) exists. Consequently, the bootstrap leading to (6.20) does not by itself provide existence of such solutions. If the authors wish to present Theorem 1.8 as an unconditional theorem, they need either a well-posedness result for (1.11) under Assumption 2.1 (perhaps using their a priori estimates) or an explicit reformulation of Theorem 1.8 as a conditional statement whose hypothesis is highlighted in the abstract as well as in the theorem.","section":"Section 6, opening paragraph and Proposition 6.5"},{"comment":"The exchangeability condition in Definition 1.2 is not automatic without a uniqueness result for the Nash system, as the authors note in Remark 1.3. This is load-bearing for Theorem 1.8 because Proposition 7.1 explicitly uses the assumed exchangeability of u_N to pass from a sum over players to the per-player bound in (7.2). Thus even a classical solution with bounded second derivatives would not suffice for the closed-loop proof unless exchangeability is guaranteed separately.","section":"Remark 1.3 and Proposition 7.1"}],"minor_comments":[{"comment":"There is a typo: 'inital state' should be 'initial state'.","section":"Definition 1.1"},{"comment":"The word 'equlibirum' appears in the definition of a closed-loop Nash equilibrium; it should be 'equilibrium'.","section":"Section 1.2, closed-loop game paragraph"},{"comment":"The word 'Lipchitz' appears in the discussion of the fixed-point maps; it should be 'Lipschitz'.","section":"Section 1.5, proof strategy"},{"comment":"The extension of Phi from measures with bounded support to all of P_2 is only sketched; a few lines verifying that the extended map still satisfies the fixed-point equation (3.1) would improve readability.","section":"Lemma 3.3"}],"recommendation":"major_revision","confidential_remarks":"The open-loop part of the paper is solid and would alone be a substantial contribution. The closed-loop theorem is conditional in a way that should be prominently flagged if the authors choose not to prove existence for the Nash system. I would encourage the authors to either supply a well-posedness argument or adjust the abstract and introduction to avoid overclaiming."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it delivers the first quantitative convergence rates for mean field games of controls without the separability assumption on the Hamiltonian. The fixed-point map on measures is genuinely nontrivial here, and the paper proves its Lipschitz regularity under displacement monotonicity rather than assuming it. That is a real advance. Second, the open-loop result (Theorem 1.6) is on solid ground; the closed-loop result (Theorem 1.8) is honest but conditional, and the abstract oversells it slightly.\n\nThe open-loop argument is clean. The authors prove uniform-in-N Lipschitz properties for the finite-dimensional fixed-point maps, then use a synchronous coupling and the stability estimate in Proposition 4.1 to get the rate r_{d,p}(N). The dependence on the empirical measure bound from Fournier-Guillin is standard. I checked the structure of the proof and it hangs together. The a priori estimates for the Nash system in Sections 6 and 7 are detailed, and the bootstrap in Proposition 6.5 is coherent. The paper also handles common noise without extra pain.\n\nThe soft spot is exactly where the reader's report puts it: Theorem 1.8 assumes, for all large N, the existence of an admissible classical solution to the N-player Nash system (1.11) with bounded second spatial derivatives and exchangeability. The authors state plainly in Section 1.2 that no well-posedness result is known under the quadratic growth their Assumption 2.1 allows. Remark 1.3 admits exchangeability is not automatic. So the closed-loop theorem is conditional on a hypothesis that may be true but is not proved. The abstract's phrase 'conclude the convergence of closed-loop equilibria' goes beyond what the theorem establishes. This is not a fatal flaw, because the authors are transparent about the assumption, and the open-loop result does not need it. But it is a genuine gap between the advertising and the mathematics, and a referee should make them fix the wording.\n\nThere is no circularity and no fitted parameters. The citation pattern looks fair: they cite the two prior quantitative works [LT22, PT25] and correctly identify that those require separability or small-time/dissipative conditions. Self-citation is limited and appropriate.\n\nWho is this for? People working on propagation of chaos and convergence for MFGs with control interactions, and anyone who needs rates without separability. It deserves a serious referee, not a desk reject. My recommendation: send it out, ask the referee to scrutinize Sections 6-7 and to require the authors to either prove existence of admissible Nash-system solutions under their assumptions or else restate Theorem 1.8 and the abstract as explicitly conditional.","headline":"First quantitative convergence for MFGC without separability; open-loop theorem is solid, closed-loop result is conditional on an unproved Nash-system existence assumption.","tokens_in":50681,"tokens_out":1625,"would_cite":true,"duration_ms":20298,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A16","60H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves explicit convergence rates for open- and closed-loop Nash equilibria in mean field games of control under displacement monotonicity, with common noise allowed.","keywords":["mean field games of controls","displacement monotonicity","quantitative convergence","Nash equilibria","open-loop and closed-loop convergence","Nash system","fixed point on Wasserstein space","common noise"],"falsifier":"Solve the one-dimensional linear-quadratic version of this model, with quadratic $L$ and $G$ satisfying Assumptions 2.1 and 2.3, explicitly for both the mean-field and N-player closed-loop equilibria; if the mean-square gap between the representative path and the mean-field path decays slower than $r_{1,p}(N)=N^{-1/2}+N^{-(p-2)/p}$, the claimed rate is wrong.","tokens_in":49615,"feed_emoji":"🎮","tokens_out":8738,"duration_ms":89362,"temperature":0.7,"pith_summary":"This paper proves that in a broad class of N-player stochastic differential games where agents interact through both their states and their controls, Nash equilibria converge to the unique mean-field equilibrium at explicit algebraic rates as N grows. The main results give bounds of order $r_{d,p}(N)$, roughly $N^{-1/2}$ in low dimension, for both open-loop and closed-loop equilibria, under a displacement-monotonicity condition that allows non-separable Hamiltonians and common noise. Prior quantitative results for mean field games of control required either separability of the Hamiltonian, short-time or dissipative assumptions, or unverified regularity of a fixed-point map; here those hypotheses are replaced by one structural condition, and the fixed-point map is analyzed rather than assumed. The open-loop result is unconditional given existence of equilibria, while the closed-loop counterpart assumes, as a tractable but unproved hypothesis, that the N-player Nash system has a classical solution with bounded second derivatives.","feed_headline":"Quantitative convergence proven for mean-field games of control","feed_subtitle":"Open-loop rates need only equilibrium existence; closed-loop rates assume a regular Nash-system solution.","key_machinery":"The central object is the fixed-point map $\\Phi:\\mathcal{P}_2(\\mathbb{R}^d\\times\\mathbb{R}^d)\\to\\mathcal{P}_2(\\mathbb{R}^d\\times\\mathbb{R}^d)$ defined implicitly by $\\Phi(\\mathcal{L}(X,Y))=\\mathcal{L}(X,-D_p H(X,Y,\\Phi(\\mathcal{L}(X,Y))))$, which encodes the self-referential nature of equilibrium when controls enter the interaction; a finite-dimensional counterpart $a^N$ plays the same role for the N-player game. Displacement monotonicity, a quantitative convexity-type condition on $L$ and $G$, is shown to imply existence, uniqueness, and Lipschitz regularity of $\\Phi$, and decay of the distance between $\\Phi$ and $a^N$ at order $1/N$. The proof then rests on a dimension-free stability estimate for the Pontryagin forward-backward system and, for closed loops, on differentiating the Nash system to compare $D_{\\mathrm{diag}}u^N$ with the Pontryagin vector field $v^N$; a bootstrap that turns a hypothetical uniform bound on the second-derivative matrix $A^N$ into itself with better constants yields the uniform bound needed to control the open-loop/closed-loop gap.","core_discovery":"For any initial distribution with finite $p$-th moment, $p>2$, the paper establishes that solutions of the N-player Pontryagin system, which characterize open-loop Nash equilibria, stay within distance $C\\,r_{d,p}(N)$ of i.i.d. copies of the mean-field equilibrium, measured in the mean-square path error and in the control energy. The same quantitative bound is then transferred to closed-loop equilibria, provided the closed-loop value functions solve the Nash system in the admissible sense of Definition 1.2. The rate $r_{d,p}(N)$ is the Wasserstein empirical-measure rate: $N^{-1/2} + N^{-(p-2)/p}$ for $d<4$, $N^{-1/2}\\log(1+N) + N^{-(p-2)/p}$ for $d=4$, and $N^{-2/d} + N^{-(p-2)/p}$ for $d>4$. As a corollary, the empirical state and joint state-control distributions of the N-player equilibrium converge to the mean-field flow at the same rate in squared 2-Wasserstein distance, with common noise allowed throughout.","pith_inferences":["The rate $r_{d,p}(N)$ is plausibly sharp in the $d<4$ regime, since the $N^{-1/2}$ term is exactly the fluctuation scale of empirical measures; the paper does not address optimality.","The a-priori estimates of Section 6 are a natural starting point for a well-posedness theorem for the Nash system under quadratic growth, which would make Theorem 1.8 unconditional.","The structural analysis of $\\Phi$ may serve a separate purpose: establishing higher regularity of $\\Phi$ would feed directly into master-equation approaches for mean field games of control, which currently require smoothness of $\\Phi$ that is not verified in the displacement-monotone setting.","A natural testable extension is mean field games of control with controlled volatility, combining the present control interactions with the controlled-diffusion techniques used in earlier displacement-monotone convergence results."],"forward_implications":["Open-loop N-player Nash equilibria approach the mean-field equilibrium at rate $r_{d,p}(N)$ in mean-square path and control error, whenever both equilibria exist.","Empirical state and joint state-control measures of the N-player equilibrium converge to the mean-field flow at the same rate in squared 2-Wasserstein distance.","The same rates hold for closed-loop feedback equilibria as soon as an admissible solution of the N-player Nash system exists for all large N.","Neither separability of the Hamiltonian nor short horizons are needed; displacement monotonicity alone carries the global-in-time argument, and common noise is incorporated without additional rates."],"supporting_citations":[{"why":"Supplies the Wasserstein empirical-measure rate $N^{-1/2}$, $N^{-1/2}\\log(1+N)$, or $N^{-2/d}$ that defines $r_{d,p}(N)$.","marker":"[FG15]"},{"why":"Provides the displacement-monotone synchronous-coupling approach that the open-loop convergence proof adapts to mean field games of control.","marker":"[JT24]"},{"why":"Shows how to quantify the gap between open- and closed-loop equilibria in N-player games, a strategy extended here to control interactions.","marker":"[CR24]"},{"why":"Provides the bootstrap philosophy for comparing open- and closed-loop equilibria via a priori estimates on the Nash system.","marker":"[CJR25]"},{"why":"Supplies the method of continuation and BSDE stability results used to prove well-posedness of the mean-field Pontryagin system in Appendix A.","marker":"[Zha17]"},{"why":"Supplies the four-step scheme connecting forward-backward SDE solutions to the associated PDE system, linking open- and closed-loop formulations.","marker":"[Del02]"},{"why":"Supplies the Itô–Krylov formula used when differentiating the Nash system to obtain equations for the second-derivative matrix $A^N$.","marker":"[Kry80]"},{"why":"Supplies the parabolic Calderón–Zygmund estimates that justify differentiability of the Nash-system solution in the closed-loop analysis.","marker":"[LSU68]"}],"fun_headline_variants":["MFG of control: quantitative convergence for open and closed-loop equilibria","Quantitative convergence for displacement monotone MFG of control","Open-loop and closed-loop rates for mean-field games of control","Mean-field games of control: convergence rates for Nash equilibria","Quantitative rates for MFG of control with common noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The closed-loop convergence statement assumes that for every large N the N-player Nash system has a classical solution whose second spatial derivatives are uniformly bounded, and the paper does not prove such solutions exist under its own assumptions; if they fail to exist, that theorem has no content.","fun_headline_variants_meta":{"raw":{"variants":["MFG of control: quantitative convergence for open and closed-loop equilibria","Quantitative convergence for displacement monotone MFG of control","Open-loop and closed-loop rates for mean-field games of control","Mean-field games of control: convergence rates for Nash equilibria","Quantitative rates for MFG of control with common noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001591,"raw_usage":{"total_tokens":6364,"prompt_tokens":986,"completion_tokens":5378,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":5292}},"tokens_in":602,"tokens_out":5378,"duration_ms":41443,"temperature":1.0,"reasoning_tokens":5292,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:58:55.071161+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the one-dimensional linear-quadratic version of this model, with quadratic $L$ and $G$ satisfying Assumptions 2.1 and 2.3, explicitly for both the mean-field and N-player closed-loop equilibria; if the mean-square gap between the representative path and the mean-field path decays slower than $r_{1,p}(N)=N^{-1/2}+N^{-(p-2)/p}$, the claimed rate is wrong.","supporting_citations":[],"review_version":1}