The founding note of mean field game theory, Lasry and Lions' 'Jeux à champ moyen. I – Le cas stationnaire' (2006), derives the stationary (ergodic) mean field system on the torus as the N→∞N\to\infty limit of Nash equilibria, proves uniqueness under a strict L2L^2-monotonicity condition on the interaction operator V\mathcal{V}, and exhibits examples showing uniqueness fails in general. The deferred proofs, extensions, broader strategy classes, and multi-population or Vlasov-type generalizations announced in that note have been substantially developed by the subsequent literature. What survives as a precise open core is: determine verifiable conditions on V\mathcal{V} (and the Hamiltonian HH) under which the stationary system has a unique solution beyond the monotonicity condition, in the strongest form a characterization delimiting the well-posedness boundary, and, where uniqueness fails, establish a principled selection of the relevant equilibrium, clarifying how selection interacts with the three limits N→∞N\to\infty, regularization ε→0\varepsilon\to0, and vanishing noise ν→0\nu\to0, which the source proved do not commute.

Background

A differential game with a large number NN of identical players, each controlling a stochastic dynamics dXti=σ dWti−αti dtdX_t^i=\sigma\,dW_t^i-\alpha_t^i\,dt and each evaluated by a long-time-average (ergodic) cost, leads to Nash equilibria, i.e. profiles of strategies such that no single player can lower its own cost by unilaterally changing strategy. The idea of a population of individually negligible players, each rational and crediting the others with the same rationality, goes back to Aumann's model of markets with a continuum of traders, which the source note itself cites as the starting point of this rational-expectations line. Lasry and Lions (2006) rigorously passed to the limit N→∞N\to\infty and obtained the stationary mean field game (MFG) system: an unknown triple (λ,v,m)(\lambda, v, m), with λ\lambda a real number (the effective ergodic cost), vv a value function and mm a probability density, solving on the torus Q=[0,1]dQ=[0,1]^d with periodic boundary conditions

−νΔv+H(x,∇v)+λ=V[m],∫Qv dx=0,-\nu\Delta v + H(x,\nabla v) + \lambda = \mathcal V[m], \qquad \int_Q v\,dx = 0, −νΔm−div(∂H∂p(x,∇v) m)=0,∫Qm dx=1,m>0.-\nu\Delta m - \mathrm{div}\Big(\frac{\partial H}{\partial p}(x,\nabla v)\,m\Big) = 0, \qquad \int_Q m\,dx = 1,\quad m>0.

Here ν>0\nu>0 is the diffusion intensity (half the noise variance), H(x,p)=sup⁡α(p⋅α−L(x,α))H(x,p)=\sup_\alpha(p\cdot\alpha - L(x,\alpha)) is the convex Hamiltonian obtained by Legendre transform of the running cost LL (the usual Hamilton-Jacobi-Bellman structure of ergodic control), and V\mathcal V is the interaction operator: it maps the population distribution mm to the running cost density V[m]\mathcal V[m] felt by a representative player, e.g. V[m](x)=F(V∗m(x),x)\mathcal V[m](x)=F(V*m(x),x) with V∗mV*m a convolution against a kernel VV. The first equation is of Hamilton-Jacobi-Bellman type (backward, for the value), the second of Fokker-Planck (Kolmogorov) type (forward, for the population); the coupling v↔mv\leftrightarrow m through V[m]\mathcal V[m] encodes rational expectations: the players' aggregated behavior shapes the cost that guides each of them. When H=12∣p∣2H=\frac12|p|^2 the system reduces, after the exponential change of variable φ∝e−v/2ν\varphi\propto e^{-v/2\nu}, to the Hartree equation of quantum mechanics (a nonlinear Schrödinger equation describing a particle interacting with its own charge density), whose solutions are well known to be non-unique in general; this is one of the mechanisms of multiplicity here. Lasry and Lions proved (their Theorem 4.1) that if V\mathcal V is strictly monotone in L2L^2, i.e.

∫Q(V[m1]−V[m2])(m1−m2) dx≤0 ⟹ V[m1]≡V[m2],\int_Q \big(\mathcal V[m_1]-\mathcal V[m_2]\big)(m_1-m_2)\,dx \le 0 \ \Longrightarrow\ \mathcal V[m_1]\equiv \mathcal V[m_2],

(a condition now called Lasry-Lions monotonicity; the sign convention makes congestion-type costs --- costs that increase where the population is dense --- monotone), then the stationary system has at most one solution. Without it, uniqueness fails: the note gives explicit stationary counterexamples, including a local focusing-type (attractive) operator V[φ2]=−cφ\mathcal V[\varphi^2]=-c\varphi. The note also proves that the deterministic limit ν→0\nu\to0 selects an explicit solution, but shows by an explicit example that the three limits appearing in the theory --- the number of players N→∞N\to\infty, the regularization parameter ε→0\varepsilon\to0 used to approximate a singular or local operator V\mathcal V by smoothing V[m∗ρε]\mathcal V[m*\rho_\varepsilon], and the vanishing noise ν→0\nu\to0 --- need not commute: taken in one order they yield effective cost λ=0\lambda=0, in another a strictly positive λ\lambda. Since that note, mean field games have become a large field at the interface of stochastic control, analysis of PDEs, and economics; two standard references are the survey of Lasry and Lions (2007) and the lecture notes of Cardaliaguet and Porretta (2020). Uniqueness of MFG solutions is essential to the convergence of NN-player equilibria to the mean field limit (the convergence program of Cardaliaguet, Delarue, Lasry and Lions (2019) requires monotonicity or must handle multiplicity through the master equation, an equation in the space of probability measures that describes the value as a function of the population law), and to the convergence of the standard numerical schemes for MFG systems. Beyond the monotonicity condition, later work has multiplied sufficient conditions for uniqueness: displacement monotonicity, introduced for nonseparable Hamiltonians, tests monotonicity along optimal-transport plans between two population measures rather than pointwise between densities and covers pairwise-interaction couplings that Lasry-Lions monotonicity does not (Gangbo, Mészáros, Mou and Zhang 2022; Mészáros and Mou 2024), and further newer monotonicity conditions have been introduced, each in partial overlap with Lasry-Lions monotonicity (Graber and Mészáros 2023). Partial answers of this kind, and partial answers on selection, are so far confined to specific model subclasses of the stationary system.

Problem Statement

For the stationary (ergodic) mean field game system of Lasry and Lions (2006) on the torus Q=[0,1]dQ=[0,1]^d with periodic data, ν>0\nu>0 and convex Hamiltonian HH:

−νΔv+H(x,∇v)+λ=V[m],∫Qv dx=0,-\nu\Delta v + H(x,\nabla v) + \lambda = \mathcal V[m], \qquad \int_Q v\,dx = 0, −νΔm−div(∂H∂p(x,∇v) m)=0,∫Qm dx=1,m>0 in Q,-\nu\Delta m - \mathrm{div}\Big(\frac{\partial H}{\partial p}(x,\nabla v)\,m\Big) = 0, \qquad \int_Q m\,dx = 1,\quad m>0 \text{ in } Q,

where V\mathcal V is a continuous operator on probability measures (the interaction/cost operator, e.g. V[m](x)=F(V∗m(x),x)\mathcal V[m](x)=F(V*m(x),x)), Lasry and Lions prove that uniqueness of the solution triple (λ,v,m)(\lambda,v,m) holds under strict L2L^2-monotonicity

∫Q(V[m1]−V[m2])(m1−m2) dx≤0 ⇒ V[m1]≡V[m2],\int_Q \big(\mathcal V[m_1]-\mathcal V[m_2]\big)(m_1-m_2)\,dx \le 0 \ \Rightarrow\ \mathcal V[m_1]\equiv\mathcal V[m_2],

and prove by explicit examples (including the Hartree reduction and a focusing-type operator) that uniqueness fails in general. The open problem is: (a) determine verifiable conditions on V\mathcal V and HH, going beyond the monotonicity condition above and not restricted to specific model subclasses (linear-quadratic dynamics and costs, potential systems deriving from a single energy functional, finite state spaces, first-order vanishing-noise systems), under which this stationary system has a unique solution, and in the strongest form characterize which interaction operators yield a unique stationary equilibrium, so that the monotonicity condition's relation to the true uniqueness boundary (sufficient but not necessary, or close to sharp) is settled; and (b) since uniqueness genuinely fails for natural operators, establish a principled selection, in general (non-subclass) form, of the relevant stationary equilibrium when several coexist, clarifying in particular how the selection interacts with the three limits that appear in the derivation --- the number of players N→∞N\to\infty, the regularization parameter ε→0\varepsilon\to0 used to approximate singular or local operators by smoothing, and the vanishing noise ν→0\nu\to0 --- across the orderings in which they are defined, so that how the selected equilibrium depends on that ordering is settled; Lasry and Lions demonstrated the noncommutation of these limits by an example in which one order of limits gives effective cost λ=0\lambda=0 and another gives λ>0\lambda>0.

References

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