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arxiv: 2606.26853 · v1 · pith:3N7XNEL6new · submitted 2026-06-25 · 🧮 math.NA · cs.NA· math.AP· math.OC

Numerical analysis of first-order mean field games under displacement monotonicity

Alp\'ar R. M\'esz\'aros , Yohance A. P. Osborne This is my paper

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classification 🧮 math.NA cs.NAmath.APmath.OC
keywords mean field gamesparticle methodnumerical approximationdisplacement monotonicityWasserstein distancePontryagin systemfirst-order systemsimplicit Euler
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The pith

A particle method converges in L^infty(W_2) for distributions and L^infty(L^2) for value gradients in first-order mean field games with non-separable displacement-monotone Hamiltonians.

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

The paper introduces a particle method that discretizes the characteristic system of a first-order mean field game via implicit Euler steps in time and spatial sampling of agents. It proves that the resulting approximations of the player distribution converge to the true solution in the L^infty of the Wasserstein-2 metric and the gradients of the value function converge in the L^infty of the L^2 norm, jointly as the number of particles tends to infinity and the time step vanishes. The result holds for non-separable Hamiltonians over arbitrary time horizons and for possibly singular initial distributions in the space of probability measures with finite second moment. A sympathetic reader would care because mean field games describe the collective behavior of large populations of rational agents, and this supplies the first rigorous convergence theory that covers the non-separable case with long horizons.

Core claim

We introduce a particle method for the numerical approximation of time-dependent first-order Mean Field Games systems with non-separable, displacement monotone Hamiltonians and terminal costs, for arbitrary time-horizons and possibly singular initial player distributions in P_2(R^d). The numerical scheme is based on an implicit Euler discretization in time and sampling in space of the characteristic Hamiltonian/Pontryagin system. We prove convergence of the approximations of the player distribution in the L^infty(W_2)-metric and the approximations for the gradient of the value function along optimal trajectories in the L^infty(L^2)-norm as the number of spatial samples tends to infinity join

What carries the argument

Implicit Euler discretization in time and spatial sampling of the characteristic Hamiltonian/Pontryagin system associated with the continuous MFG system.

If this is right

  • The scheme applies directly to singular initial agent distributions.
  • Convergence holds uniformly for arbitrarily long time horizons.
  • The error bound yields explicit rates once a specific spatial sampling method is chosen.
  • Under additional local Lipschitz continuity of the Lagrangian and terminal costs, the value function itself converges in L^infty(L^1).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same particle discretization might be combined with other monotonicity notions to cover a wider class of Hamiltonians.
  • Because the method works in any dimension and avoids grids, it could be used to simulate economic or traffic models whose state spaces are too large for finite-difference schemes.
  • If displacement monotonicity is relaxed, one would expect to need a different proof technique or a modified scheme to recover comparable guarantees.

Load-bearing premise

The Hamiltonians and terminal costs satisfy displacement monotonicity.

What would settle it

Numerical evidence that the L^infty(W_2) distance between the particle approximation and a reference solution stays bounded away from zero as the number of particles tends to 10,000 and the time step tends to 0.001, for a system that meets every other hypothesis of the theorem.

Figures

Figures reproduced from arXiv: 2606.26853 by Alp\'ar R. M\'esz\'aros, Yohance A. P. Osborne.

Figure 1
Figure 1. Figure 1: Depiction of j-th iteration of Algorithm B – The direction of the blue arrows indicate downward induction along the temporal subintervals {[tn, tn+1]} Mk−1 n=0 starting from the right-most grey block. Each grey block corresponds to performing standard Picard iteration by Algorithm A to approximate the dis￾crete Hamiltonian system posed on the n-th subinterval [tn, tn+1] with initialization being the value … view at source ↗
Figure 2
Figure 2. Figure 2: First experiment – Illustration of reference solution ( [PITH_FULL_IMAGE:figures/full_fig_p051_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: First experiment: L∞(L 2 )-Error plots for approximations of the reference (Y (t))t∈[0,T] curve and the reference trajectories of the players (X(t))t∈[0,T] for T = 1 using Algorithm A and Algorithm B. For k ∈ {2, 3, · · · , 9}, we apply both the standard Algorithm A and the local Algorithm B in MATLAB to solve the discrete Hamiltonian system (3.6), where for each algorithm we initialized by setting Y k,N i… view at source ↗
Figure 4
Figure 4. Figure 4: First experiment: L∞(L 2 )-Error plots for approximations of the reference flow (Y (t))t∈[0,T] and the reference trajectories of the players (X(t))t∈[0,T] for T ∈ {1, 2, 4, 8, 16, 32} using Algorithm B for mesh counts k ∈ {1, 2, 3, 4, 5, 6, 7, 8} Second experiment. In this experiment we test the performance of the numerical scheme (3.6) for a MFG system with an exact solution. We consider the continuous MF… view at source ↗
Figure 5
Figure 5. Figure 5: Second experiment: L∞(L 2 )-Error plots for approximations of the exact flow (−Dxu(t, X(t)))t∈[0,T] and the exact trajectories of the players (X(t))t∈[0,T] for T = 1 using Algorithm A and Algorithm B. X k,Nk init ∈ V − k ([0, T]; R2(ANk , ONk )) where X k,Nk init |(0,T] := 0 with X k,Nk init (0) := X Nk 0 . For both algorithms, we set the global tolerance to be tolglobal = 10−8 with the local error toleran… view at source ↗
Figure 6
Figure 6. Figure 6: Second experiment: L∞(L 2 )-Error plots for approximations of the exact flow (−Dxu(t, X(t)))t∈[0,T] and the exact trajectories of the players (X(t))t∈[0,T] for T ∈ {1, 2, 4, 8, 16, 32} using Algorithm B for mesh counts k ∈ {1, 2, 3, 4, 5, 6, 7, 8} In the second part of this experiment, we assess the robustness of the numerical scheme (3.6) for longer time horizons T ∈ {2, 4, 8, 16, 32} by using Algorithm B… view at source ↗
Figure 7
Figure 7. Figure 7: Third experiment: L∞(L 2 )-Error plots for approximations of the exact flow (−Dxu(t, X(t)))t∈[0,T] and the exact trajectories of the players (X(t))t∈[0,T] for d ∈ {1, 2, 3, 4, 5, 6} using Algorithm A for mesh counts k ∈ {1, 2, 3, 4, 5, 6, 7} (3.5) that Z tn tm E [PITH_FULL_IMAGE:figures/full_fig_p056_7.png] view at source ↗
read the original abstract

We introduce a particle method for the numerical approximation of time-dependent first-order Mean Field Games (MFGs) systems with non-separable, displacement monotone Hamiltonians and terminal costs, for arbitrary time-horizons and (possibly) singular initial player distributions in $\mathcal{P}_2(\mathbb{R}^d)$. The numerical scheme is based on an implicit Euler discretization in time and sampling in space of the characteristic Hamiltonian/Pontryagin system associated with the continuous MFGs system. We prove convergence of the approximations of the player distribution in the $L^{\infty}(\mathcal{W}_2)$-metric and the approximations for the gradient of the value function along optimal trajectories in the $L^{\infty}{(L^2)}$-norm as the number of spatial samples tends to infinity jointly with the temporal time-step vanishing. The error bound that we establish for this convergence further implies rates of convergence of the scheme for a range of spatial sampling techniques. Provided that the Lagrangian and terminal costs are additionally locally Lipschitz continuous, we also establish an asymptotic error bound in the $L^{\infty}(L^1)$-norm for the approximations of the value function along optimal trajectories. This is the first work in the literature on rigorous numerical approximation and analysis of first-order MFG systems that handles non-separable Hamiltonians and potentially singular initial agent distributions for arbitrary long time horizons. We illustrate the performance of the scheme in numerical experiments for a range of initial agent distributions, time horizons and space dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces a particle method for first-order time-dependent MFG systems with non-separable displacement-monotone Hamiltonians and terminal costs, allowing singular initial measures in P2(R^d) and arbitrary finite time horizons. The scheme applies implicit Euler discretization to the Pontryagin characteristic system, followed by spatial sampling. Under displacement monotonicity, it establishes L^∞(W2) convergence of the empirical player distribution and L^∞(L2) convergence of value-function gradients along trajectories as N, Δt → 0 jointly; an additional local Lipschitz assumption yields an L^∞(L1) bound on the value function. Error bounds imply rates for various sampling methods. Numerical experiments are shown for different initial distributions, horizons, and dimensions. The work claims to be the first rigorous numerical analysis of this generality.

Significance. If the stated convergence theorems hold, the contribution is significant: it supplies the first rigorous particle-scheme analysis for first-order MFGs that accommodates non-separable Hamiltonians, singular initial data, and long horizons under the displacement-monotonicity hypothesis. The explicit error bounds and the derivation of convergence rates from the joint limit constitute concrete strengths that advance the numerical analysis of MFGs beyond separable or short-horizon settings.

minor comments (3)
  1. [§2.2] §2.2, Definition 2.3: the precise statement of displacement monotonicity for the pair (H, g) should be recalled verbatim before its use in the stability estimate, to make the dependence on the monotonicity constant explicit.
  2. [Theorem 3.5] Theorem 3.5: the constant C in the L^∞(W2) bound appears to depend on T; an explicit statement of this dependence (or uniformity in T) would strengthen the claim of arbitrary horizons.
  3. [§5] §5, numerical experiments: the reported L^∞(W2) errors are tabulated only for one sampling method; adding a short table or plot comparing at least two sampling techniques would directly illustrate the rate statements in Corollary 3.7.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for recommending minor revision. The report correctly summarizes the scope and contributions of the paper.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a particle scheme via implicit Euler discretization of the Pontryagin system associated to the continuous first-order MFG and derives L^∞(W₂) and L^∞(L²) convergence directly from displacement monotonicity of the Hamiltonian and terminal cost. All error bounds and rates follow from standard stability estimates under the stated assumptions without any reduction of a claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation chain. The scheme is explicitly derived from the continuous system rather than being equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on the displacement monotonicity assumption, which is a domain condition rather than a derived property; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Hamiltonians and terminal costs are displacement monotone
    This property is required to prove convergence of the particle approximations in the L^infty(W_2) metric and the gradient approximations in L^infty(L^2).

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