Multi-Population Mean Field Games

Updated 11 November 2025

Multi-population mean field games are mathematical models that analyze interactions among diverse agent groups using coupled McKean–Vlasov dynamics and PDE formulations.
They encompass non-cooperative, cooperative, and mixed regimes, employing forward–backward SDEs and Hamilton–Jacobi–Bellman equations to characterize equilibrium concepts.
The framework supports finite-agent approximations and practical numerical methods, offering rigorous error bounds and applications in economics, energy systems, and networked markets.

Multi-population mean field games (MP-MFGs) are mathematical models for analyzing the behavior of large-scale systems consisting of multiple interacting populations of rational agents, where each population may differ in terms of its dynamics, cost functions, information structure, or cooperative behavior. In the asymptotic regime where each population contains infinitely many agents, the impact of a single agent on the aggregate evolution becomes negligible, allowing tractable formulations in terms of McKean–Vlasov stochastic differential equations (SDEs), forward-backward SDEs (FBSDEs), partial differential equations (PDEs) of the Hamilton–Jacobi–Bellman (HJB) and Fokker–Planck (FP) type, and variational and entropic formulations. The MP-MFG paradigm encompasses a variety of agent interactions—including competition (“games”), collaboration (cooperative control), and mixed regimes—and admits a rich array of mathematical techniques for the paper of existence, uniqueness, numerics, and approximation to finite-agent settings.

1. Mathematical Formulation of Multi-population Mean Field Games

Consider $M$ populations indexed by $k=1,\dots,M$ . Each population $k$ consists of a continuum of agents whose states $X^k_t$ evolve under McKean–Vlasov SDEs driven by stochastic processes and controlled to optimize an objective. The general SDE for a representative agent in population $k$ takes the form

$dX^k_t = b_k(t, X^k_t, \mu^1_t, \dots, \mu^M_t, \alpha^k_t)\,dt + \sigma_k(t, X^k_t, \mu^1_t, \dots, \mu^M_t)\,dW^k_t,$

where each $\mu^k_t = \text{Law}(X^k_t)$ , $\alpha^k_t$ is an admissible control, and $W^k_t$ is a Brownian motion. The agent’s cost functional is

$J^k(\alpha^1,\dots,\alpha^M) = \mathbb{E}\left[ \int_0^T f_k(t, X^k_t, \mu^1_t, \dots, \mu^M_t, \alpha^k_t)\,dt + g_k(X^k_T, \mu^1_T, \dots, \mu^M_T) \right].$

All drift, volatility, and running cost coefficients may depend on all populations via the measure flows, yielding highly coupled dynamics.

The main objects studied are:

Hamiltonian: For each $k$ , $H_k(t,x,\mu^1,\dots,\mu^M, y, z, \alpha)$ defines agent $k$ ’s pointwise optimization, capturing drift, volatility, and running cost contributions.
Pontryagin Stochastic Maximum Principle (SMP): The optimal feedback $\hat{\alpha}_k$ is characterized via minimization of the Hamiltonian with respect to controls—requiring differentiability and convexity assumptions.
Forward–backward SDE system: The optimal state-adjoint (FBSDE) system for each population is fully coupled through measures and possibly their derivatives.

Key distinctions in MP-MFGs arise from the specification of cooperative or competitive behaviors within and across populations, the dependence structure of coefficients, and the type of solution concepts (Nash equilibrium, social optimum, team optimum) employed.

2. Interaction Regimes and Equilibrium Concepts

Three canonical multi-population interactions are distinguished:

(i) Non-cooperative Mean Field Game regime (MFG–MFG):

Each agent in every population acts selfishly to minimize its own cost, treating the measure flows as given. The equilibrium is characterized by a system of coupled FBSDEs—one for each population—without mean-derivative terms in the backward equation. Existence is obtained under affine and convexity assumptions ((MFG-a), (MFG-b)) on model coefficients, or for small time horizons/small coupling (Fujii, 2019).

(ii) Cooperative Mean Field Type Control regime (MFTC–MFTC):

Agents of each population cooperate (e.g., under a team planner), so each population shares a common feedback policy. The resulting McKean–Vlasov control problems for each population require an additional mean-derivative term (L-derivative or Lions derivative) in the backward equation, reflecting anticipation of the collective impact on the mean-field (Fujii, 2019). Existence relies on differentiability and convexity in measures (MFTC-a).

(iii) Mixed cooperation (“MFTC–MFG”):

Some populations cooperate (as above), while others act non-cooperatively. The resulting FBSDE system mixes both types: backward components for cooperative populations include L-derivative terms, and others do not. Existence follows under appropriate hybrid regularity and convexity assumptions (MFTC–MFG-a,b).

For all regimes, a mean field equilibrium (MFE) is a tuple $(\mu^1,\dots,\mu^M)$ such that, for each $k$ , the controlled state process under the optimal feedback produces the assumed measure flow: $\mu^k_t = \text{Law}(X^{k,\mu}_t)$ for all $t$ .

3. Existence and Characterization of Equilibria

Existence of equilibria is established in (Fujii, 2019) using fixed-point arguments adapted to the structure of the MP-MFG FBSDEs—specifically, continuation/Schauder fixed-point techniques originally due to Carmona–Delarue for McKean–Vlasov FBSDEs. The main existence results are:

Regime	Assumptions	Existence Theorem	FBSDE Structure
MFG–MFG	(MFG-a), (MFG-b)	Thm 2.6, 2.9	No mean-derivative in backward SDE
MFTC–MFTC	(MFTC-a), (MFTC-b)	Thm 3.4, 3.7	Includes mean-derivative (L-derivative)
MFTC–MFG	hybrid	Thm 4.3, 4.8	Mixed: some with, some without L-deriv.

The precise assumptions include:

Affine structure in $(x,\alpha)$ ,
Convexity and Lipschitz continuity in measure flows,
Quadratic growth for cost and terminal data,
L-differentiability for measure derivatives (MFTC-a).

Unique solvability under small time and/or coupling conditions is verified using contractivity of the decoupling fields.

The equilibrium can also be interpreted in Martingale Optimality Principle terms, via the stochastic maximum principle.

4. Finite-population Approximation and Propagation of Chaos

A crucial property of MP-MFGs is that the solution of the mean-field model approximates equilibria for the corresponding finite-agent game with many agents in each population. Given a Nash equilibrium $(\hat{\alpha}_1,\dots,\hat{\alpha}_M)$ of the limiting system, an approximate equilibrium for the finite game is constructed by assigning each agent $p$ in population $k$ its corresponding mean-field feedback and propagating independent copies $X^{k,p}_t$ . The empirical measures $\mu_t^{k,N_k}$ (based on the $N_k$ agents of population $k$ ) converge to $\mu_t^k$ in $W_2$ at a rate $O(N_k^{-2/\max(d,4)})$ . The cost for any unilateral deviation from the mean-field strategy is penalized quadratically, leading to an $\epsilon$ –Nash equilibrium property: $J^{N}_i(\text{deviator}) \geq J^N_i(\text{MFE feedback}) - C\sum_k N_k^{-2/\max(d,4)},$ with $\epsilon = O(\sum_k N_k^{-2/\max(d,4)})$ (Fujii, 2019). The key ingredients in the proof are stability estimates for the controlled processes with respect to empirical measures, propagation of chaos, and penalization by convexity.

5. Extensions: Discrete Time, Variational, and Strong Solution Approaches

MP-MFGs have been extensively generalized beyond the continuous semi-martingale regime.

Discrete-time and general state spaces: Existence and equilibrium concepts for multi-population discrete-time MFGs (with both discounted and total payoff criteria) are established under general continuity and compactness assumptions (Więcek, 2023). The approach combines Bellman contraction, invariant-measure construction, and Kakutani fixed-point arguments, covering Markov and stationary equilibria in general Polish state spaces.
Variational and entropic formulations: For systems with quadratic Hamiltonians and non-local interactions, Eulerian and Lagrangian variational formulations allow minimization over trajectory distributions subject to mass conservation (Pascale et al., 6 Aug 2024). Despite non-convexity induced by interaction terms, weak solutions exist owing to separate convexity and can be computed via a Sinkhorn-like scheme.
Strong solution and weakly regular coefficients: The existence of strong (pathwise) solutions for mean-field FBSDEs in the multi-population context has been demonstrated under minimal regularity—allowing measurable and only continuous-in-measure coefficients—using Pontryagin’s maximum principle and martingale arguments (Nam et al., 2 Feb 2024).

6. Applications, Numerical Methods, and Quantitative Approximations

MP-MFGs are applied in economic models (oligopoly, regulation, market design), energy systems (carbon emission regulation with major/minor populations), crowd dynamics (multiple populations with congestion), and networked markets.

Notable computational methods include:

Online Mirror Descent (OMD) for computing Nash equilibria in large-scale finite-state multi-population MFGs, with provable convergence under monotonicity and empirically superior performance over Fictitious Play (FP), scaling to models with billions of state-action pairs (Perolat et al., 2021).
Variational Sinkhorn-like block coordinate algorithms for weakly-coupled nonlocal interactions (Pascale et al., 6 Aug 2024).
Mixed-integer convex programming for optimal partitioning of heterogeneous agent systems into near-homogeneous sub-populations, enabling sharp $\epsilon$ –Nash quantitative homogenization (Cont et al., 17 Feb 2025).

Analytical results provide explicit, non-asymptotic error bounds for finite-agent approximations, revealing precise trade-offs between the number of populations, group sizes, and inter-group heterogeneity (Cont et al., 17 Feb 2025).

7. Structural and Theoretical Generalizations

MP-MFGs admit extensions to models with fully asymmetric data, teams/coalitions with arbitrary partition structure, and major–minor player hierarchies (Subramanian et al., 2023 Dayanikli et al., 2023). The analysis of second-order PDEs with critical exponents reveals detailed existence and blow-up profiles for ergodic MP-MFG ground states, including explicit classification of existence regimes in terms of intra- and inter-population coupling coefficients (Kong et al., 27 Jan 2025). For two or more populations engaged in Nash-type competition, the solution of a fully coupled system of master equations is required, especially when anticipating the reactions of other (possibly cooperative) groups (Bensoussan et al., 2018).

The variational framework and master equations support treatment of geometric, nonlocal, or singular interactions and boundary effects, and allow for generalization to time-dependent, spatially inhomogeneous, and infinite-horizon models.

The mathematical analysis of multi-population mean field games integrates stochastic optimal control, infinite-dimensional analysis, PDE and FBSDE theory, and optimization. The outcomes provide a rigorous and flexible toolset for describing and approximating equilibria in large-scale interacting agent systems with complex population structures, permitting precise error control and facilitating computation at unprecedented scale (Fujii, 2019 Więcek, 2023 Perolat et al., 2021 Pascale et al., 6 Aug 2024 Cont et al., 17 Feb 2025 Subramanian et al., 2023 Dayanikli et al., 2023 Nam et al., 2 Feb 2024 Kong et al., 27 Jan 2025 Bensoussan et al., 2018).