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Quantitative convergence for displacement monotone Mean Field Games of control

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict First quantitative convergence for MFGC without separability; open-loop theorem is solid, closed-loop result is conditional on an unproved Nash-system existence assumption. read the letter →

arxiv 2507.17014 v1 pith:JAFU6JAP submitted 2025-07-22 math.PR math.APmath.OC

Joe Jackson , Alpár R. Mészáros This is my paper · ORCID
classification math.PRmath.APmath.OC MSC 91A1660H10
keywords meanfieldgamesofcontrolsdisplacementmonotonicityquantitativeconvergenceNashequilibriaopen-loopandclosed-loopsystemfixedpointonWassersteinspacecommonnoise
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section 1.2, Definition 1.2, Theorem 1.8] 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.
  2. [Section 6, opening paragraph and Proposition 6.5] 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.
  3. [Remark 1.3 and Proposition 7.1] 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.
minor comments (4)
  1. [Definition 1.1] There is a typo: 'inital state' should be 'initial state'.
  2. [Section 1.2, closed-loop game paragraph] The word 'equlibirum' appears in the definition of a closed-loop Nash equilibrium; it should be 'equilibrium'.
  3. [Section 1.5, proof strategy] The word 'Lipchitz' appears in the discussion of the fixed-point maps; it should be 'Lipschitz'.
  4. [Lemma 3.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems follow from explicit structural assumptions and internally proved estimates; the closed-loop result is conditional on an openly stated existence hypothesis, which is a limitation but not a circular reduction.

full rationale

The paper does not fit parameters to data, rename a known result, or invoke a uniqueness theorem from the authors' prior work as a substitute for proof. The central open-loop result, Theorem 1.6, is proved from Assumptions 2.1 and 2.3 via the internally established fixed-point analysis of Section 3 (Lemmas 3.3, 3.5, 3.6), the stability estimate Proposition 4.1, and an external empirical-measure rate result [FG15]. The closed-loop result, Theorem 1.8, is conditional on the existence of an admissible classical solution to the N-player Nash system (1.11), as stated in Definition 1.2 and Section 1.2: the paper explicitly says 'we are not aware of any results which obtain even local in time existence results for (1.11) when H and G can grow quadratically in x' and 'we choose to focus here on the convergence problem, and simply assume the existence of a sufficiently nice solution to (1.11).' This is an honest hypothesis, not a consequence of the theorem being proved, so it does not make the derivation circular. The a priori estimates in Sections 6 and 7 are conditional on that same hypothesis, but they are used only to prove the conditional convergence statement, not to manufacture existence. Self-citations to [JT24] and [CJR25] are used for proof strategy ('similar to the one carried out in [JT24]') and for philosophical roadmapping; the load-bearing inequalities are proved in this paper, and the cited mean-field FBSDE well-posedness is established in Appendix A via continuation, with [Zha17] used only as a standard external tool. No step reduces, by the paper's own equations, to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted and no new physical entities are postulated. The fixed point map Phi is not an invented entity: it is defined by the equilibrium condition and its existence is proved. The only unproved input that is load-bearing for the closed-loop theorem is the existence of an admissible Nash system solution.

assumptions (6)
  • domain assumption Assumption 2.1: L and G are C2 with uniformly bounded second derivatives, L is uniformly strictly convex in a, with coercive growth (2.1)-(2.2), and the measure derivatives of DxL, DaL, DxG are uniformly Lipschitz.
    Used throughout; it implies the Hamiltonian H has the growth and regularity in Remark 2.2 and makes all fixed point maps and FBSDEs well behaved.
  • domain assumption Assumption 2.3: displacement monotonicity (2.4)-(2.5) with C_disp = CL,a - T CG - T^2/2 CL,x > 0.
    This is the structural condition driving uniqueness of fixed points, uniform stability of the Pontryagin system, and the a priori Nash system estimates.
  • domain assumption The initial condition m0 lies in P_p(R^d) for some p > 2, with p not in {4, d/(d-2)}.
    Needed for the moment bound (5.1) and for the empirical measure convergence rate r_{d,p}(N) from [FG15].
  • domain assumption Existence of a mean field equilibrium and open-loop Nash equilibria for each N, as assumed in Theorem 1.6.
    The convergence statements are conditional on these equilibria; Remark 1.7 notes they are automatic under convexity (1.7), for example when CL,x = CG = 0.
  • ad hoc to paper Existence of an admissible solution uN to the Nash system (1.11) with bounded second derivatives for all large N, as in Definition 1.2 and Theorem 1.8.
    This is not proved in the paper; the text says no well-posedness result for (1.11) is known under quadratic growth. It is the main open gap for the closed-loop part.
  • standard math Standard background results: Pontryagin maximum principle, four step scheme for FBSDEs, Ito-Krylov formula, continuation method for mean field FBSDEs (Appendix A), and empirical measure convergence rates of [FG15].
    Invoked with references and standard in the literature; none are machine-checked.

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Cite this review

Pith. "Pith review of Quantitative convergence for displacement monotone Mean Field Games of control." pith.science (2026). https://pith.science/paper/JAFU6JAP

@misc{pith2026250717014,
  author       = {Pith},
  title        = {Pith review of: Quantitative convergence for displacement monotone Mean Field Games of control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAFU6JAP}},
  note         = {Machine review of arXiv:2507.17014}
}
read the original abstract

In this paper we establish quantitative convergence results for both open and closed-loop Nash equilibria of N-player stochastic differential games in the setting of Mean Field Games of Controls (MFGC), a class of models where interactions among agents occur through both states and controls. Our analysis covers a general class of non-separable Hamiltonians satisfying a displacement monotonicity condition, along with mild regularity and growth conditions at infinity. A major novelty of our work is the rigorous treatment of a nontrivial fixed-point problem on a space of measures, which arises naturally in the MFGC formulation. Unlike prior works that either restrict to separable Hamiltonians - rendering the fixed-point map trivial - or assume convergence or regularity properties of the fixed point map, we develop a detailed structural analysis of this equation and its N-player analogue. This leads to new regularity results for the fixed-point maps and, in turn, to quantitative convergence of open-loop equilibria. We further derive sharp a priori estimates for the N-player Nash system, enabling us to control the discrepancy between open and closed-loop strategies, and thus to conclude the convergence of closed-loop equilibria. Our framework also accommodates common noise in a natural way.

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Reference graph

Works this paper leans on

60 extracted references · 58 canonical work pages

  1. [1]

    u rich. Birkh\

    L. Ambrosio, N. Gigli, and G. Savar\' e . Gradient flows in metric spaces and in the space of probability measures . Lectures in Mathematics ETH Z\" u rich. Birkh\" a user Verlag, Basel, second edition, 2008

    work page 2008

  2. [2]

    S. Ahuja. Wellposedness of mean field games with common noise under a weak monotonicity condition. SIAM J. Control Optim. , 54(1):30--48, 2016

    work page 2016

  3. [3]

    Achdou and Z

    Y. Achdou and Z. Kobeissi. Mean field games of controls: finite difference approximations. Math. Eng. , 3(3):Paper No. 024, 35, 2021

    work page 2021

  4. [4]

    Ambrose and A.R

    D.M. Ambrose and A.R. M\'esz\'aros. Well-posedness of mean field games master equations involving non-separable local hamiltonians. Trans. Amer. Math. Soc. , 376(4):2481--2523, 2023

    work page 2023

  5. [5]

    Bonnans, J

    J.F. Bonnans, J. Gianatti, and L. Pfeiffer. A L agrangian approach for aggregative mean field games of controls with mixed and final constraints. SIAM J. Control Optim. , 61(1):105--134, 2023

    work page 2023

  6. [6]

    Bansil and A.R

    M. Bansil and A.R. M\'esz\'aros. Hidden monotonicity and canonical transformations for mean field games and master equations. arXiv:2403.05426 , 2024

    work page arXiv 2024

  7. [7]

    Bansil, A.R

    M. Bansil, A.R. M\'esz\'aros, and C. Mou. Global well-posedness of displacement monotone degenerate mean field games master equations. SIAM J. Control Optim. , 63(2):993--1021, 2025

    work page 2025

  8. [8]

    Chassagneux, D

    J.-F. Chassagneux, D. Crisan, and F. Delarue. A probabilistic approach to classical solutions of the master equation for large population equilibria. Mem. Amer. Math. Soc. , 280(1379):v+123, 2022

    work page 2022

Show all 60 references
  1. [9]

    Cardaliaguet, M

    P. Cardaliaguet, M. Cirant, and A. Porretta. Splitting methods and short time existence for the master equations in mean field games. J. Eur. Math. Soc. (JEMS) , 25(5):1823--1918, 2023

    1918

  2. [10]

    Carmona and F

    R. Carmona and F. Delarue. Probabilistic theory of mean field games with applications. I . M ean field FBSDE s, control, and games , volume 83 of Probability Theory and Stochastic Modelling . Springer, Cham, 2018

    2018

  3. [11]

    Carmona and F

    R. Carmona and F. Delarue. Probabilistic theory of mean field games with applications II . Mean field games with common noise and master equations , volume 84 of Probab. Theory Stoch. Model. Springer, Cham, 2018

    2018

  4. [12]

    Cardaliaguet, F

    P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions. The master equation and the convergence problem in mean field games , volume 201 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 2019

    2019

  5. [13]

    Cirant, J

    M. Cirant, J. Jackson, and D.F. Redaelli. A non-asymptotic approach to stochastic differential games with many players under semi-monotonicity. arXiv:2505.01526 , 2025

    2025 arXiv

  6. [14]

    Carmona and D

    R. Carmona and D. Lacker. A probabilistic weak formulation of mean field games and applications. Ann. Appl. Probab. , 25(3):1189--1231, 2015

    2015

  7. [15]

    Cardaliaguet and Charles-A

    P. Cardaliaguet and Charles-A. Lehalle. Mean field game of controls and an application to trade crowding. Math. Financ. Econ. , 12(3):335--363, 2018

    2018

  8. [16]

    Cirant and D.F

    M. Cirant and D.F. Redaelli. A priori estimates and large population limits for some nonsymmetric nash systems with semimonotonicity. arXiv:2406.10822 , 2024

    2024 arXiv

  9. [17]

    F. Delarue. On the existence and uniqueness of solutions to FBSDE s in a non-degenerate case. Stochastic Process. Appl. , 99(2):209--286, 2002

    2002

  10. [18]

    M.F. Djete. Extended mean field control problem: a propagation of chaos result. Electron. J. Probab. , 27:Paper No. 20, 53, 2022

    2022

  11. [19]

    M.F. Djete. Large population games with interactions through controls and common noise: convergence results and equivalence between open-loop and closed-loop controls. ESAIM Control Optim. Calc. Var. , 29:Paper No. 39, 42, 2023

    2023

  12. [20]

    M.F. Djete. Mean field games of controls: on the convergence of N ash equilibria. Ann. Appl. Probab. , 33(4):2824--2862, 2023

    2023

  13. [21]

    Djete, D

    M.F. Djete, D. Possama\"i, and X. Tan. Mc K ean- V lasov optimal control: the dynamic programming principle. Ann. Probab. , 50(2):791--833, 2022

    2022

  14. [22]

    Fournier and A

    N. Fournier and A. Guillin. On the rate of convergence in W asserstein distance of the empirical measure. Probab. Theory Related Fields , 162(3-4):707--738, 2015

    2015

  15. [23]

    M. Fischer. On the connection between symmetric N -player games and mean field games. Ann. Appl. Probab. , 27(2):757--810, 2017

    2017

  16. [24]

    Gangbo and A.R

    W. Gangbo and A.R. M\'esz\'aros. Global well-posedness of master equations for deterministic displacement convex potential mean field games. Comm. Pure Appl. Math. , 75(12):2685--2801, 2022

    2022

  17. [25]

    Graber and A.R

    P.J. Graber and A.R. M\'esz\'aros. On monotonicity conditions for mean field games. J. Funct. Anal. , 285(9):Paper No. 110095, 45, 2023

    2023

  18. [26]

    Graber and E

    P.J. Graber and E. Matter. New uniqueness results for a mean field game of controls. arXiv:2410.13950 , 2024

    2024 arXiv

  19. [27]

    Graber and A.R

    P.J. Graber and A.R. M\'esz\'aros. On some mean field games and master equations through the lens of conservation laws. Math. Ann. , 390(3):4497--4533, 2024

    2024

  20. [28]

    Gangbo, A.R

    W. Gangbo, A.R. M\'esz\'aros, C. Mou, and J. Zhang. Mean field games master equations with nonseparable H amiltonians and displacement monotonicity. Ann. Probab. , 50(6):2178--2217, 2022

    2022

  21. [29]

    Graber, A

    P.J. Graber, A. Mullenix, and L. Pfeiffer. Weak solutions for potential mean field games of controls. NoDEA Nonlinear Differential Equations Appl. , 28(5):Paper No. 50, 34, 2021

    2021

  22. [30]

    Gomes, S

    D.A. Gomes, S. Patrizi, and V. Voskanyan. On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. , 99:49--79, 2014

    2014

  23. [31]

    Gangbo and A

    W. Gangbo and A. \' S wiech. Existence of a solution to an equation arising from the theory of mean field games. J. Differential Equations , 259(11):6573--6643, 2015

    2015

  24. [32]

    Graber and R

    P.J. Graber and R. Sircar. Master equation for C ournot mean field games of control with absorption. J. Differential Equations , 343:816--909, 2023

    2023

  25. [33]

    Gangbo and A

    W. Gangbo and A. Tudorascu. On differentiability in the W asserstein space and well-posedness for H amilton- J acobi equations. J. Math. Pures Appl. (9) , 125:119--174, 2019

    2019

  26. [34]

    Gomes and V.K

    D.A. Gomes and V.K. Voskanyan. Extended mean field games. Izv. Nats. Akad. Nauk Armenii Mat. , 48(2):63--76, 2013

    2013

  27. [35]

    Gomes and V.K

    D.A. Gomes and V.K. Voskanyan. Extended deterministic mean-field games. SIAM J. Control Optim. , 54(2):1030--1055, 2016

    2016

  28. [36]

    Huang, R.P

    M. Huang, R.P. Malham\'e, and P.E. Caines. Large population stochastic dynamic games: closed-loop M c K ean- V lasov systems and the N ash certainty equivalence principle. Commun. Inf. Syst. , 6(3):221--251, 2006

    2006

  29. [37]

    Jakobsen and A

    E.R. Jakobsen and A. Rutkowski. The master equation for mean field game systems with fractional and nonlocal diffusions. J. Eur. Math. Soc. (JEMS) , to appear, 2025+

    2025

  30. [38]

    Jackson and L

    J. Jackson and L. Tangpi. Quantitative convergence for displacement monotone mean field games with controlled volatility. Math. Oper. Res. , 49(4):2527--2564, 2024

    2024

  31. [39]

    Kobeissi

    Z. Kobeissi. Mean field games with monotonous interactions through the law of states and controls of the agents. NoDEA Nonlinear Differential Equations Appl. , 29(5):Paper No. 52, 33, 2022

    2022

  32. [40]

    Kobeissi

    Z. Kobeissi. On classical solutions to the mean field game system of controls. Comm. Partial Differential Equations , 47(3):453--488, 2022

    2022

  33. [41]

    Nicolai V. Krylov. Controlled Diffusion Processes . Springer-Verlag Berlin Heidelberg, 1980

    1980

  34. [42]

    D. Lacker. A general characterization of the mean field limit for stochastic differential games. Probab. Theory Related Fields , 165(3-4):581--648, 2016

    2016

  35. [43]

    D. Lacker. On the convergence of closed-loop N ash equilibria to the mean field game limit. Ann. Appl. Probab. , 30(4):1693--1761, 2020

    2020

  36. [44]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions. Jeux \`a champ moyen. I . L e cas stationnaire. C. R. Math. Acad. Sci. Paris , 343(9):619--625, 2006

    2006

  37. [45]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions. Jeux \`a champ moyen. II . H orizon fini et contr\^ole optimal. C. R. Math. Acad. Sci. Paris , 343(10):679--684, 2006

    2006

  38. [46]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions. Mean field games. Jpn. J. Math. , 2(1):229--260, 2007

    2007

  39. [47]

    Lacker and L

    D. Lacker and L. Le Flem. Closed-loop convergence for mean field games with common noise. Ann. Appl. Probab. , 33(4):2681--2733, 2023

    2023

  40. [48]

    Liao and C

    H. Liao and C. Mou. A particle system approach towards the global well-posedness of master equations for potential mean field games of control. arXiv:2412.11742 , 2024

    2024 arXiv

  41. [49]

    Lady z enskaja, V.A

    O.A. Lady z enskaja, V.A. Solonnikov, and N.N. Ural'ceva. Linear and Q uasi-linear E quations of P arabolic T ype . American Mathematical Society, 1968

    1968

  42. [50]

    Lauri\`ere and L

    M. Lauri\`ere and L. Tangpi. Convergence of large population games to mean field games with interaction through the controls. SIAM J. Math. Anal. , 54(3):3535--3574, 2022

    2022

  43. [51]

    S. Mayorga. Short time solution to the master equation of a first order mean field game. J. Differential Equations , 268(10):6251--6318, 2020

    2020

  44. [52]

    R.J. McCann. A convexity principle for interacting gases. Adv. Math. , 128(1):153--179, 1997

    1997

  45. [53]

    M\'esz\'aros and C

    A.R. M\'esz\'aros and C. Mou. Mean field games systems under displacement monotonicity. SIAM J. Math. Anal. , 56(1):529--553, 2024

    2024

  46. [54]

    J. Ma, P. Protter, and J.M. Yong. Solving forward-backward stochastic differential equations explicitly---a four step scheme. Probab. Theory Related Fields , 98(3):339--359, 1994

    1994

  47. [55]

    Mou and J

    C. Mou and J. Zhang. Mean field games of controls: propagation of monotonicities. Probab. Uncertain. Quant. Risk , 7(3):247--274, 2022

    2022

  48. [56]

    Mou and J

    C. Mou and J. Zhang. Wellposedness of second order master equations for mean field games with nonsmooth data. Mem. Amer. Math. Soc. , 302(1515):v+86, 2024

    2024

  49. [57]

    G. Parker. Some convexity criteria for differentiable functions on the 2- W asserstein space. Bull. Lond. Math. Soc. , 56(5):1839--1858, 2024

    2024

  50. [58]

    Possama\"i and L

    D. Possama\"i and L. Tangpi. Non-asymptotic convergence rates for mean-field games: weak formulation and M c K ean- V lasov BSDE s. Appl. Math. Optim. , 91(3):Paper No. 58, 73, 2025

    2025

  51. [59]

    Santambrogio and W

    F. Santambrogio and W. Shim. A C ucker- S male inspired deterministic mean field game with velocity interactions. SIAM J. Control Optim. , 59(6):4155--4187, 2021

    2021

  52. [60]

    J. Zhang. Backward stochastic differential equations. From linear to fully nonlinear theory , volume 86 of Probability Theory and Stochastic Modelling . Springer, New York, 2017

    2017

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Reviewed August 6, 2026 · model on record in the stance chip above.