The 2011 lecture-notes chapter of Guéant, Lasry and Lions, 'Mean Field Games and Applications', introduced an 'eductive' relaxation for stationary mean field games: the coupled stationary Hamilton–Jacobi–Bellman / Kolmogorov system is rewritten as a fixed point (via the change of variables β=eu/σ2\beta=e^{u/\sigma^2} and inversion of the elliptic operators), and an artificial forward time θ\theta is added so that the pair (u,m)(u,m) relaxes toward the fixed point; the stationary points of this relaxation are exactly the mean field game equilibria. Only local stability is known: the linearization around the explicit Gaussian equilibria of the logarithmic-coupling model decays in every Hermite mode when ρσ2<2\rho\sigma^2<2, and discretized (Fourier-truncated) versions converge in the chapter's numerical experiments. Whether the relaxation converges globally — from arbitrary initial data, far from equilibrium, and through its computable discretizations — has neither been established nor refuted: no subsequent treatment of the eductive scheme is known in the literature, and the numerical survey of Achdou and Laurière (2020) reports that no general proof of convergence is known for iterative mean field game solvers. The problem asks for a global convergence theory (or a rigorous counterexample) for this forward relaxation, together with convergence proofs and error estimates for its discretizations, on the stationary classes to which the source applies it — principally the non-monotone logarithmic-coupling class where only local stability is known.

Background

A mean field game (MFG) describes the limit of a symmetric differential game with many identical agents: each agent optimizes against the distribution of the others' states, and equilibria are characterized by a system of two coupled partial differential equations — a backward Hamilton–Jacobi–Bellman (HJB) equation, the dynamic-programming equation of the agent's stochastic optimal control problem, for the value function uu; and a forward Kolmogorov (Fokker–Planck) equation for the population density mm. The theory was introduced by Lasry and Lions (2007). For stationary equilibria of discounted problems one seeks a time-independent pair (u,m)(u,m); in the reference model used below (idiosyncratic diffusion σ>0\sigma>0, discount ρ\rho, quadratic control cost, logarithmic coupling), the system on Rn\mathbb{R}^n reads

ρu(x)=σ22Δu(x)+12∣∇u(x)∣2+ln⁡m(x),σ22Δm(x)=∇⋅(m(x)∇u(x)),\rho u(x)=\frac{\sigma^2}{2}\Delta u(x)+\frac{1}{2}|\nabla u(x)|^2+\ln m(x), \qquad \frac{\sigma^2}{2}\Delta m(x)=\nabla\cdot\big(m(x)\nabla u(x)\big),

with m>0m>0 a probability density. Uniqueness for such forward–backward systems is known essentially only for small time horizons or for couplings that are monotone in the sense of Lasry–Lions, meaning that the agents' running cost increases with the population density. The logarithmic coupling here is an agglomeration externality of the opposite sign: the running payoff ln⁡m\ln m increases with the density, so the associated cost coupling −ln⁡m-\ln m decreases with it — an anti-monotone coupling — and the model indeed admits a continuum of equilibria.

The source of this problem is the chapter 'Mean Field Games and Applications' by Guéant, Lasry and Lions (2011), the written version of a 2009 lecture course. Beyond collecting the authors' applied models, the chapter proposes a distinctive way to analyze and compute stationary equilibria. The stationary system is rewritten as a fixed point — informally: given mm, solve the HJB equation for uu (which becomes linear after the change of variables β=eu/σ2\beta=e^{u/\sigma^2}); given uu, solve the Kolmogorov equation for mm; each step is an inversion of an elliptic operator, performed numerically after a Fourier/harmonic truncation and a small regularization ϵ>0\epsilon>0. One then introduces an artificial forward time θ\theta and the relaxation dynamics that moves (u,m)(u,m) (equivalently (β,m)(\beta,m)) toward the value produced by these solved equations; stationary points of the relaxation are exactly the MFG equilibria. The word 'eductive' is borrowed from Guesnerie's analysis of rational-expectations equilibria in economics (1992): an equilibrium is eductively stable if it can be reached by iterated reasoning of the agents from common-knowledge primitives, and the forward evolution in θ\theta is a continuous-time model of such iterated reasoning.

For the logarithmic-coupling reference model the chapter exhibits the explicit family of Gaussian equilibria m∗=N(μ,s2In)m^*=\mathcal{N}(\mu,s^2 I_n), u∗=−η∣x−μ∣2+ωu^*=-\eta|x-\mu|^2+\omega with s2=σ4/(4−2ρσ2)s^2=\sigma^4/(4-2\rho\sigma^2), η=1/σ2−ρ/2\eta=1/\sigma^2-\rho/2, centered at arbitrary μ∈Rn\mu\in\mathbb{R}^n, and proves local eductive stability: linearizing the relaxation around a Gaussian equilibrium in one dimension gives

∂θφ=−Lφ−ρφ+ψ,∂θψ=−Lψ+2σ2Lφ,Lf=−σ22f′′+2ηxf′,\partial_\theta\varphi=-\mathcal{L}\varphi-\rho\varphi+\psi, \qquad \partial_\theta\psi=-\mathcal{L}\psi+\frac{2}{\sigma^2}\mathcal{L}\varphi, \qquad \mathcal{L}f=-\frac{\sigma^2}{2}f''+2\eta x f',

and the Hermite polynomials (the orthogonal polynomials with respect to the Gaussian equilibrium density) diagonalize L\mathcal{L} with eigenvalues 2ηn2\eta n, so that for ρσ2<2\rho\sigma^2<2 every mode decays exponentially in θ\theta, up to the neutral direction generated by translations of the equilibrium center (which reflect the translation freedom of the equilibrium family). The chapter also implements the scheme numerically and reports that it works in practice, for example for the benchmark coupling g(x,m)=m−δx2g(x,m)=\sqrt{m}-\delta x^2 (the term gg playing the role of the density-dependent coupling, in place of ln⁡m\ln m) with σ2=0.4\sigma^2=0.4, ρ=0.4\rho=0.4, δ=0.5\delta=0.5.

What is missing — and acknowledged as missing in the source's own summary of its limitations — is everything beyond the linearized regime: global convergence of the relaxation for arbitrary initial data, justification of the scheme far from equilibrium, and convergence proofs with error estimates for the discretized (truncated, regularized) iterations that are actually run in practice.

Problem Statement

Consider the stationary mean field game system on Rn\mathbb{R}^n with idiosyncratic diffusion σ>0\sigma>0, discount rate ρ\rho satisfying ρσ2<2\rho\sigma^2<2, quadratic control cost, and logarithmic coupling (the source's reference model for spatial population distribution with an agglomeration externality, i.e. a running payoff that increases with the local population density):

ρu(x)=σ22Δu(x)+12∣∇u(x)∣2+ln⁡m(x)(stationary HJB),\rho u(x)=\frac{\sigma^2}{2}\Delta u(x)+\frac{1}{2}|\nabla u(x)|^2+\ln m(x) \quad\text{(stationary HJB)}, σ22Δm(x)=∇⋅(m(x)∇u(x))(stationary Kolmogorov / Fokker–Planck),\frac{\sigma^2}{2}\Delta m(x)=\nabla\cdot\big(m(x)\nabla u(x)\big) \quad\text{(stationary Kolmogorov / Fokker–Planck)},

for a pair (value function uu, probability density m>0m>0). Its equilibria include the explicit family of Gaussians m∗=N(μ,s2In)m^*=\mathcal{N}(\mu,s^2 I_n), u∗=−η∣x−μ∣2+ωu^*=-\eta|x-\mu|^2+\omega with s2=σ4/(4−2ρσ2)s^2=\sigma^4/(4-2\rho\sigma^2), η=1/σ2−ρ/2>0\eta=1/\sigma^2-\rho/2>0, centered at arbitrary μ∈Rn\mu\in\mathbb{R}^n.

The source introduces an eductive relaxation of this system, in the sense of Guesnerie's eductive stability (an equilibrium is called eductively stable if it can be reached by iterated reasoning of the agents starting from common-knowledge primitives — Guesnerie's model, within the rational-expectations program in economics, of when such iterated reasoning converges to an equilibrium): writing the stationary system as a fixed point — given mm, solve the HJB equation for uu (which is linear in the variable β=eu/σ2\beta=e^{u/\sigma^2}), then given uu, solve the Kolmogorov equation for mm, each step being the inversion of an elliptic operator, performed with a small regularization ϵ>0\epsilon>0 and, numerically, after a Fourier/harmonic truncation — one introduces an artificial forward time θ\theta and lets the pair (u,m)(u,m) evolve at unit speed toward the value produced by these solved equations. Explicitly, in the variables (β,m)(\beta,m) with β=eu/σ2\beta=e^{u/\sigma^2}, with the elliptic inversions taken on Rn\mathbb{R}^n and with the regularization ϵ>0\epsilon>0, and with g(x,m)=ln⁡mg(x,m)=\ln m for the reference class, the continuous relaxation is

∂θβ=−β+[σ22Δ−ϵ Id]−1 ⁣(β[ρln⁡β−1σ2g(x,m)−ϵ]),∂θm=−m+[σ22Δ]−1 ⁣(σ2∇⋅(m∇ββ)),\partial_\theta\beta=-\beta+\Big[\tfrac{\sigma^2}{2}\Delta-\epsilon\,\mathrm{Id}\Big]^{-1}\!\Big(\beta\big[\rho\ln\beta-\tfrac{1}{\sigma^2}g(x,m)-\epsilon\big]\Big), \qquad \partial_\theta m=-m+\Big[\tfrac{\sigma^2}{2}\Delta\Big]^{-1}\!\Big(\sigma^2\nabla\cdot\big(m\tfrac{\nabla\beta}{\beta}\big)\Big),

whose stationary points are exactly the mean field game equilibria (with ∂θu=σ2∂θβ/β\partial_\theta u=\sigma^2\partial_\theta\beta/\beta). This continuous relaxation is the object of part 1; its Fourier/harmonic truncation and operator-inversion discretization are the iterations treated in part 3. Around the Gaussian equilibria, in one dimension, the linearized relaxation is

∂θφ=−Lφ−ρφ+ψ,∂θψ=−Lψ+2σ2Lφ,Lf=−σ22f′′+2ηxf′,\partial_\theta\varphi=-\mathcal{L}\varphi-\rho\varphi+\psi, \qquad \partial_\theta\psi=-\mathcal{L}\psi+\frac{2}{\sigma^2}\mathcal{L}\varphi, \qquad \mathcal{L}f=-\frac{\sigma^2}{2}f''+2\eta x f',

and the Hermite polynomials diagonalize L\mathcal{L} with eigenvalues 2ηn2\eta n, so that for ρσ2<2\rho\sigma^2<2 every mode decays exponentially in θ\theta (up to the neutral translation freedom of the equilibrium family) — this is the source's local eductive stability result, and it is the only stability information available.

The problem is to establish, or rigorously refute, the global reliability of this scheme:

  1. Global convergence of the continuous relaxation: for arbitrary admissible initial data (a positive density and a positive β\beta, not necessarily close to an equilibrium), does the relaxation trajectory (u(θ),m(θ))(u(\theta),m(\theta)) converge as θ→∞\theta\to\infty to the set of stationary equilibria (or to a selected equilibrium)? Or can it fail — convergence to a non-equilibrium attractor, sustained oscillation, or loss of regularity or mass?
  2. Justification far from equilibrium: whenever convergence holds, quantify it — rates, and basins of attraction that provably extend beyond the linearized regime; in particular resolve the role of the neutral translation modes and of the anti-monotone coupling (running payoff ln⁡m\ln m increasing in the density, equivalently a running cost −ln⁡m-\ln m decreasing in it — the sign opposite to the Lasry–Lions monotonicity under which the standard uniqueness and contraction machinery works).
  3. Justification of the computable discretizations: prove convergence of the iterations actually used — the Fourier/harmonic truncations and the operator-inversion discretizations with the ϵ\epsilon-regularization — and provide error estimates with respect to the truncation and regularization parameters, on the stationary classes to which the source applies the scheme (the logarithmic-coupling reference class above is the required class, with the source's benchmark couplings such as g(x,m)=m−δx2g(x,m)=\sqrt{m}-\delta x^2 with σ2=0.4\sigma^2=0.4, ρ=0.4\rho=0.4, δ=0.5\delta=0.5 as optional illustrations).

These parts form one objective: a rigorous foundation for the eductive scheme as a method for computing stationary mean field game equilibria — part 1 is the analytical core, parts 2 and 3 its completion for the scheme's intended use.

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