In the finite-horizon mean field game framework introduced by Lasry and Lions (2006), the value functions of the N-player stochastic differential game satisfy a coupled system of N Hamilton–Jacobi equations (the Nash system), and its expected limit is a forward–backward mean field game system coupling a backward Hamilton–Jacobi–Bellman equation to a forward Kolmogorov–Fokker–Planck equation. The source note explicitly flags the rigorous justification of the passage N→∞ as open, having been carried out only in special cases (e.g. H = 0), and it remarks that without monotone couplings the limit system may be ill-posed or admit multiple equilibria. The subsequent literature resolved the convergence problem under the Lasry–Lions monotonicity condition for arbitrary horizon, and for general data over short horizons, through the master equation program of Cardaliaguet, Delarue, Lasry and Lions (2015/2019) and the probabilistic theory of Carmona and Delarue (2018). The open problem addressed here is the general non-monotone regime over arbitrary horizons in the continuous-state diffusion setting: proving or disproving convergence of N-player Nash equilibria, and characterizing the set of limit points and the selection among multiple mean field solutions when uniqueness fails.

Background

Mean field games (MFG) model stochastic differential games with a very large population of identical, individually negligible players; the theory was introduced in 2006 by Lasry and Lions and, independently, by Huang, Malhamé and Caines. In the finite-horizon setting of the source note, N symmetric players move on the torus Q=[0,1]dQ=[0,1]^d with periodic boundary conditions, with independent stochastic dynamics dXsi=σ dWsi−αsi dsdX^i_s=\sigma\,dW^i_s-\alpha^i_s\,ds where σ=2ν\sigma=\sqrt{2\nu}, ν>0\nu>0, the WiW^i are independent Brownian motions, and αi\alpha^i is the control of player ii. Each player minimizes a criterion that couples to the other players only through the empirical distribution of the remaining ones, schematically E∫tT(L(Xsi,αsi)+V[1N−1∑j≠iδXsj]) ds+v0[1N−1∑j≠iδXTj]E\int_t^T\big(L(X^i_s,\alpha^i_s)+V\big[\tfrac{1}{N-1}\sum_{j\ne i}\delta_{X^j_s}\big]\big)\,ds+v_0\big[\tfrac{1}{N-1}\sum_{j\ne i}\delta_{X^j_T}\big], with running Lagrangian LL, coupling operators VV and v0v_0 acting on probability measures, and Hamiltonian H(x,p)=sup⁡α[α⋅p−L(x,α)]H(x,p)=\sup_\alpha[\alpha\cdot p-L(x,\alpha)]. An equilibrium in which no player can lower its cost by unilaterally changing its control is a Nash equilibrium. Propagation of chaos — the statistical-mechanical phenomenon by which weakly interacting particles become asymptotically independent and their empirical measure converges to a deterministic flow — suggests that as N→∞N\to\infty a player's value function depends only on its own state and on a limiting density mm, so that the equilibrium is described by the forward–backward mean field game system (written in the time convention of the source)

∂v∂t−νΔv+H(x,∇v)=V[m],v∣t=0=v0[m(0)],\frac{\partial v}{\partial t}-\nu\Delta v+H(x,\nabla v)=V[m],\qquad v\big|_{t=0}=v_0[m(0)], ∂m∂t+νΔm+div(∂H∂p(x,∇v) m)=0,m∣t=T=m0,\frac{\partial m}{\partial t}+\nu\Delta m+\mathrm{div}\Big(\frac{\partial H}{\partial p}(x,\nabla v)\,m\Big)=0,\qquad m\big|_{t=T}=m_0,

on Q×[0,T]Q\times[0,T]: the backward equation is a Hamilton–Jacobi–Bellman equation for the typical player's value function (the equation solved by an optimally controlled diffusion), and the forward equation is a Kolmogorov (Fokker–Planck) equation for the population density mm. The source note proves existence of smooth solutions for smoothing operators VV, v0v_0 and of weak solutions for local couplings V[m](x)=F(x,m(x))V[m](x)=F(x,m(x)) under growth conditions, and uniqueness when VV and v0v_0 are strictly monotone in the Lasry–Lions sense: ∫Q(A[m1]−A[m2])(m1−m2) dx≥0\int_Q(A[m_1]-A[m_2])(m_1-m_2)\,dx\ge 0 for all m1,m2m_1,m_2 forces A[m1]=A[m2]A[m_1]=A[m_2] (economically, players dislike congestion). Without monotonicity the system may be ill-posed or have multiple equilibria: linearizing around m≡1m\equiv 1 for F(x,λ)=−cλF(x,\lambda)=-c\lambda yields a forward–backward system that is well posed only for small horizons T<ν/cT<\nu/c, explicit non-uniqueness examples exist (e.g. V≡0V\equiv 0, v0[m]=−ψ(m)v_0[m]=-\psi(m) with ψ\psi increasing), and in the first-order case the system reduces to the compressible Euler equations, hyperbolic (with formation of discontinuities in finite time) exactly when the coupling is anti-monotone and elliptic exactly when VV is monotone. The authors state that rigorously justifying the N→∞N\to\infty passage for these finite-horizon Nash equilibria is 'extremely delicate', that they had completed it only in special cases (such as H=0H=0), and that they would return to it in future publications; they also flag 1/N1/N asymptotic expansions of the N-player equilibria and multi-population systems as research directions. The subsequent literature introduced the master equation — an infinite-dimensional partial differential equation on the space of probability measures, whose solution U(t,x,m)U(t,x,m) gives the typical player's value given the population distribution mm — as the object that directly describes the expected limit of the N-player Nash system. Under monotonicity its well-posedness was established by Cardaliaguet, Delarue, Lasry and Lions, who thereby proved convergence of the N-player equilibria with propagation of chaos, even with common noise (noise acting simultaneously on all players); monotonicity can be dropped only for small horizons. Without monotonicity, the master equation develops singularities: in a two-state anti-monotone model it becomes a scalar conservation law whose entropy solution (the vanishing-viscosity-type limit selected by admissibility) is the limit of the N-player value functions, while the mean field system has exactly three solutions; and in general all limit points of closed-loop (feedback) N-player equilibria can so far be characterized only as weak, possibly randomized, mean field equilibria — equilibria in a relaxed sense in which the population's collective state may carry extraneous randomness beyond the players' individual noises. Which mean field solutions the finite games select, and for which data regimes the limit problem is well posed and the convergence holds, remain unsettled in the continuous-state setting.

Problem Statement

Consider the finite-horizon, continuous-state mean field game framework of Lasry and Lions (2006): NN symmetric players with independent diffusion dynamics dXsi=σ dWsi−αsi dsdX^i_s=\sigma\,dW^i_s-\alpha^i_s\,ds (σ=2ν\sigma=\sqrt{2\nu}, ν>0\nu>0) on the periodic domain Q=[0,1]dQ=[0,1]^d, where the WiW^i are independent Brownian motions, αi\alpha^i is the control of player ii, and each player minimizes a cost coupled to the other players only through the empirical distribution of the remaining players, via coupling operators VV and v0v_0 acting on probability measures, with Hamiltonian H(x,p)=sup⁡α[α⋅p−L(x,α)]H(x,p)=\sup_\alpha[\alpha\cdot p-L(x,\alpha)] where LL is the running cost density. The formal limit of the NN-player Nash equilibria as N→∞N\to\infty is the forward–backward mean field game system on Q×[0,T]Q\times[0,T], written in the time convention of the source:

∂v∂t−νΔv+H(x,∇v)=V[m],v∣t=0=v0[m(0)],\frac{\partial v}{\partial t}-\nu\Delta v+H(x,\nabla v)=V[m],\qquad v\big|_{t=0}=v_0[m(0)], ∂m∂t+νΔm+div(∂H∂p(x,∇v) m)=0,m∣t=T=m0,\frac{\partial m}{\partial t}+\nu\Delta m+\mathrm{div}\Big(\frac{\partial H}{\partial p}(x,\nabla v)\,m\Big)=0,\qquad m\big|_{t=T}=m_0,

where m0m_0 is the initial population distribution: the backward equation is a Hamilton–Jacobi–Bellman equation (the equation solved by an optimally controlled diffusion) for the typical player's value function vv, and the forward equation is a Kolmogorov (Fokker–Planck) equation for the population density mm. Determine, in the generality in which that system is posed — general coupling operators VV, v0v_0 not assumed monotone in the Lasry–Lions sense (that is, no congestion-type hypothesis such as ∫Q(V[m1]−V[m2])(m1−m2) dx≥0\int_Q(V[m_1]-V[m_2])(m_1-m_2)\,dx\ge 0 is imposed), non-zero Hamiltonian HH, and arbitrary horizon TT (the monotone case for arbitrary TT and the general case for small TT being already settled) — the following question, which makes precise the rigorous passage to the limit announced as open in the source:

Convergence and selection. Prove, or disprove by an explicit counterexample within this framework, that the Nash equilibria of the NN-player game — value functions and optimal trajectories in both open-loop form (controls depending only on time) and closed-loop form (controls depending on the current state), together with the closed-loop feedback controls — converge as N→∞N\to\infty to solutions of the mean field game system above, where the admissible limit class is fixed explicitly: a classical (strong) solution of the displayed system wherever such a solution exists, and otherwise a weak (possibly randomized) mean field equilibrium in the relaxed sense recalled in the Background. When the limit system admits multiple solutions or is not uniquely solvable, characterize the set of limit points of the NN-player equilibria and identify which mean field solutions are selected.

The convergence assertion and the limit-point and selection characterization are complementary parts of a single question. The convergence claim is universal over the objects and formulations listed above — the value functions and optimal trajectories in each of the open-loop and closed-loop forms, and the closed-loop feedback controls — so a single explicit counterexample, within this framework, to the convergence of any one of the listed objects (in any formulation in which that object is defined) refutes it, since a universal claim is disproved by one valid counterexample. Whenever the limit system admits multiple solutions or is not uniquely solvable, the limit-point and selection characterization is part of the same question.

References

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