Mean Field Games of Controls

• Mean Field Games of Controls are models where numerous agents interact through the joint distribution of states and controls rather than states alone.
• The framework is rigorously formulated using forward–backward PDE systems, McKean–Vlasov FBSDEs, and measure-valued methods to achieve fixed-point consistency.
• This approach underpins practical applications in economics, crowd dynamics, resource management, and facilitates advanced numerical and theoretical developments.

Mean field games of controls are mean field games in which a continuum of negligible agents interact through the joint distribution of their states and controls, rather than through the state distribution alone. In this setting, a representative player minimizes a cost functional that may depend on the law of state–control pairs, and equilibrium requires a fixed-point consistency between optimal feedback and the induced joint law. The literature treats this structure through forward–backward PDE systems, McKean–Vlasov FBSDEs, relaxed and measure-valued formulations, and numerical schemes adapted to nonlocal and often nonmonotone couplings (Graber et al., 29 Aug 2025, Achdou et al., 2020).

1. Historical emergence and defining feature

The general theory of mean field games studies deterministic or stochastic differential games in the limit as the number of agents tends to infinity. In the formulation emphasized in the numerical survey of control-interaction models, mean field games were introduced in the pioneering works of J.-M. Lasry and P.-L. Lions, and independently in the engineering literature by M. Y. Huang, P. E. Caines, and R. Malhamé (Achdou et al., 2020).

The distinguishing feature of a mean field game of controls, often abbreviated MFGC, is that interaction occurs through the joint law of states and controls. One standard formulation uses a family of probability measures

$\mu_t \in \mathcal P(\Omega\times\mathbb R^n),$

with state marginal

$$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$

so that the conditional law of the control given the state is encoded in $\mu_t$ (Graber et al., 29 Aug 2025). Other formulations express the same idea through a local average control field $V(t,x)$, a price variable determined by aggregate control, or a conditional joint law under common noise (Achdou et al., 2020, Bonnans et al., 2019, Mou et al., 2022).

This extension is not merely formal. It allows models in which agents react to average speed in a neighborhood, to aggregate production or price, or to the empirical distribution of strategies themselves. The data include crowd-motion models, debt refinance dynamics, pedestrian evacuation, exhaustible-resource production, price-interaction models, and flocking or synchronization examples (Achdou et al., 2020, Bongini et al., 2021, Camilli et al., 2024, Höfer et al., 2024).

2. Canonical mathematical formulations

A representative-agent formulation on a bounded domain $\Omega\subset\mathbb R^n$ with control set $A=\mathbb R^n$ is given by

$$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$

together with the cost

$$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$

The Hamiltonian is the $p$-Legendre transform

$$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$

and equilibrium is described by the forward–backward system

$$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$0

This is the basic PDE realization of MFGC on bounded domains (Graber et al., 29 Aug 2025).

A closely related Dirichlet formulation on a closed domain writes the dynamics as

$$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$1

stopped upon exit, with Hamiltonian

$$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$2

and fixed-point condition

$$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$3

The induced PDE system couples HJB and Fokker–Planck equations under absorbing boundary conditions (Bongini et al., 2021).

The probabilistic side is equally prominent. In the Pontryagin formulation of large-population games with interaction through the empirical distribution of states and controls, equilibria satisfy McKean–Vlasov FBSDEs. In common-noise settings the master equation is posed on the joint law of state and control, and the population is represented by a random probability measure $$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$4 with state marginal $$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$5 (Laurière et al., 2020, Mou et al., 2022). In a Hilbert-space formulation, the equilibrium control is characterized as the zero of a monotone variational inequality built from a coupled mean-field FBSDE (Meynard, 16 Feb 2026).

This suggests that MFGC is less a single model than a family of equivalent equilibrium formalisms—PDE, FBSDE, martingale problem, and variational inequality—linked by the same fixed-point constraint on the joint law of state and control.

3. Boundary-value problems and constrained-state variants

Boundary conditions are a central structural feature in recent MFGC analysis. On bounded domains, two cases are treated in parallel: Dirichlet boundary conditions, interpreted as absorption or exhaustion, and Neumann boundary conditions, interpreted as reflection (Graber et al., 29 Aug 2025).

For the system above, the boundary conditions on $$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$6 are

• Dirichlet:

$$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$7

• Neumann:

$$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$8

In the Dirichlet case, the condition $$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$9 models absorption at exit, and mass is not conserved. This lack of mass conservation is the principal technical novelty in the Dirichlet analysis of Bongini and Salvarani, which develops a priori estimates specifically adapted to the fact that standard $\mu_t$0 and maximum-principle arguments cannot be used directly (Bongini et al., 2021). By contrast, the Neumann condition yields mass preservation and a no-flux boundary relation, and in the numerical reflected-diffusion formulation it arises from Itô’s formula applied to

$\mu_t$1

with reflection on $\mu_t$2 (Achdou et al., 2020).

Reflection can also be imposed against a stochastic boundary process. In the reflected-state model with an exogenous continuous boundary $\mu_t$3, the state solves

$\mu_t$4

where $\mu_t$5 is continuous, nondecreasing, $\mu_t$6, and

$\mu_t$7

The corresponding dynamic Skorokhod problem yields a Lipschitz Skorokhod map $\mu_t$8 and permits a weak relaxed equilibrium theory on an enlarged canonical space (Bo et al., 5 Mar 2025).

State constraints also appear in application-specific forms. In the Cournot exhaustible-resource model, the state is an inventory level on $\mu_t$9 with reflection at $V(t,x)$0 and absorption at $V(t,x)$1; the HJB uses $V(t,x)$2 and $V(t,x)$3, while the Fokker–Planck equation uses $V(t,x)$4 and a reflecting boundary relation at $V(t,x)$5 (Camilli et al., 2024). Across these examples, boundary conditions are not secondary regularity choices; they encode exit, depletion, preservation of mass, or exogenous state reflection.

4. Existence, uniqueness, monotonicity, and nonuniqueness

Two main well-posedness mechanisms are explicit in the recent PDE theory on bounded domains. The first is a small-coupling theory without monotonicity. Under coercivity and controlled growth of $V(t,x)$6 and $V(t,x)$7, Lipschitz and Hölder dependence on the measure argument, uniformly bounded terminal cost $V(t,x)$8 in $V(t,x)$9, and $\Omega\subset\mathbb R^n$0, a quantitative smallness condition on the constants in the growth and convexity bounds yields a unique classical solution $\Omega\subset\mathbb R^n$1 for both Dirichlet and Neumann boundary conditions (Graber et al., 29 Aug 2025).

The second is a monotone-coupling theory in the Lasry–Lions sense. If $\Omega\subset\mathbb R^n$2 is strictly convex in $\Omega\subset\mathbb R^n$3, $\Omega\subset\mathbb R^n$4 in $\Omega\subset\mathbb R^n$5, and satisfies

$\Omega\subset\mathbb R^n$6

while the state couplings $\Omega\subset\mathbb R^n$7 and $\Omega\subset\mathbb R^n$8 are monotone in $\Omega\subset\mathbb R^n$9, then for arbitrary $A=\mathbb R^n$0 there exists a unique strong solution $A=\mathbb R^n$1 with either Dirichlet or Neumann boundary conditions (Graber et al., 29 Aug 2025). A related theory on $A=\mathbb R^n$2 and on the torus assumes power-type growth of the Hamiltonian and monotonicity in the law of states and controls, and proves existence and uniqueness through a priori estimates and Leray–Schauder (Kobeissi, 2020).

The Dirichlet theory of Bongini and Salvarani employs a double fixed-point procedure: first solve the fixed point

$A=\mathbb R^n$3

then solve linear HJB and Fokker–Planck equations, and finally close the map by Leray–Schauder. Much of the argument is devoted to positivity and mass estimates for $A=\mathbb R^n$4 despite absorbing boundaries (Bongini et al., 2021). In the 2025 boundary-value treatment, the proof strategy combines Leray–Schauder, a priori $A=\mathbb R^n$5 and $A=\mathbb R^n$6 estimates for $A=\mathbb R^n$7 and $A=\mathbb R^n$8, a Bernstein-type boundary-adapted argument for $A=\mathbb R^n$9, and Moser iteration or classical parabolic theory for $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$0 (Graber et al., 29 Aug 2025).

Monotonicity is also studied at the master-equation level. Mou and Zhang analyze Lasry–Lions monotonicity, displacement monotonicity, anti-monotonicity, and displacement $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$1-monotonicity or semi-monotonicity for MFGC with common noise, and prove that these properties propagate along classical master flows. They describe this as the first step toward a global well-posedness theory for master equations of mean field games of controls (Mou et al., 2022).

A frequent misunderstanding is that interaction through controls admits either complete uniqueness or complete instability. The literature is more differentiated. The finite-difference study stresses that in models where agents try to adjust speed to an average speed, the monotonicity assumptions frequently made in MFG theory do not hold and uniqueness cannot be expected in general; its symmetric numerical example exhibits at least three distinct solutions (Achdou et al., 2020). At the same time, uniqueness is recovered under monotone couplings (Graber et al., 29 Aug 2025), under quantitative smallness assumptions (Graber et al., 29 Aug 2025), and even without monotonicity in a linear–quadratic setting with common noise, where the Nash certainty equivalence system and the associated master equation are solvable over an arbitrary time horizon (Li et al., 2022).

5. Finite-player limits, relaxed equilibria, and singular controls

A substantial part of the MFGC literature concerns the relation between finite $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$2-player games and the mean-field limit. In the open-loop framework with interaction through the empirical distribution of states and controls, Laurière and Tangpi derive convergence of Nash equilibria to the unique mean field equilibrium by developing propagation of chaos for forward and backward weakly interacting particles. Their results include moment and concentration bounds, and in linear–quadratic examples the rate is $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$3 (Laurière et al., 2020).

A complementary route uses controlled Fokker–Planck equations and measure-valued equilibria. In that framework, $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$4-Nash equilibria in $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$5-player games have limits as $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$6, and each limit is a measure-valued solution of the mean field game of controls; conversely, any measure-valued solution can be obtained as the limit of a sequence of $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$7-Nash equilibria. The same work also proves existence of measure-valued solutions in the case without common noise (Djete, 2020).

For reflected states and joint-law dependence, Bo, Wang, and Yu establish a relaxed mean field equilibrium on an enlarged canonical space built around the dynamic Skorokhod mapping. They further prove two directional limit results: any weak limit of empirical measures induced by $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$8-Nash equilibria in $$dX_t = \alpha_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_0=x_0,$$9-player games is supported on the set of relaxed mean field equilibria, and any Markovian mean field equilibrium in the weak sense can be approximated by a sequence of constructed $$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$0-Nash equilibria as $$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$1 (Bo et al., 5 Mar 2025).

Singular control enters the MFGC literature through relaxed formulations and compactness in the Skorokhod weak $$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$2 topology. In the extended mean-field game with interaction through both states and controls, relaxed regular controls are combined with singular finite-variation controls, and existence is proved by first smoothing singular controls into continuous controls, establishing compactness and closed-graph properties of the best-response correspondence, and then passing to the singular limit in weak $$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$3 topology (Fu, 2019). An approximation result for $$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$4-player stochastic games with singular controls shows that the optimal control of an MFG with bounded velocity $$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$5 is an $$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$6-Nash equilibrium of the bounded-velocity $$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$7-player game with

$$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$8

and an $$J^{x_0}(\alpha(\cdot);\mu) = \mathbb E\Big[ \int_0^T L(t,X_t,\alpha_t,\mu_t)\,dt +\int_0^T f(t,X_t,m_t)\,dt +g(X_T,m_T) \Big].$$9-Nash equilibrium of the finite-variation game, with $p$0 as $p$1 (Cao et al., 2022).

6. Potential structures, linear–quadratic classes, and nonlocal extensions

One major subclass of MFGC is potential. On the torus, a market-price model with endogenous price

$p$2

leads to the coupled system

$p$3

Here an incomplete potential and, under an additional monotonicity-of-$p$4 assumption, a full potential functional $p$5 are available, and existence of a unique classical solution follows from Leray–Schauder together with Schauder estimates (Bonnans et al., 2019).

A broader non-Markovian perspective appears in the mean field control approach of Höfer and Soner. They show that minimizers of a mean field optimal control problem with common noise and jumps are Nash equilibria of an associated mean field game of controls, and that these associated games are necessarily potential. Their examples include a mean field game of controls with interactions through a price variable, and mean field Cucker–Smale flocking and Kuramoto models (Höfer et al., 2024).

The linear–quadratic class is especially explicit. In the common-noise model with state

$p$6

the stochastic maximum principle yields a feedback control

$p$7

the mean field equilibrium is determined by a Nash certainty equivalence system, and the master equation reduces to a finite-dimensional second-order parabolic equation. The paper proves global existence and uniqueness of the master equation on arbitrary time horizons without monotonicity conditions, and quantitative convergence of the $p$8-player game to the mean field game with $p$9 estimates (Li et al., 2022).

Nonlocal diffusion has also entered the theory. In the fractional MFGC on $$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$0 with $$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$1, the Brownian Laplacian is replaced by the generator of a $$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$2-stable pure-jump process,

$$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$3

and existence is proved under Lasry–Lions monotonicity by combining moment estimates on $$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$4, abstract fractional-parabolic estimates, and time-regularity estimates for the joint law derived from the associated Lévy process (Graber et al., 4 Sep 2025).

7. Computation, learning, and model behavior

The computational literature reflects the dual difficulty of MFGC systems: they are forward–backward and nonlocal, and they may be nonmonotone. A finite-difference approximation on bounded domains discretizes the HJB backward in time and the Fokker–Planck equation forward in time, uses a monotone Hamiltonian built from an upwind Godunov flux, enforces discrete Neumann conditions through ghost nodes, and preserves nonnegativity and total mass at the discrete level in the reflecting case (Achdou et al., 2020).

The associated nonlinear solver combines continuation, Newton iterations, and an inner bigradient-like loop. In the reported experiments, Newton has local quadratic convergence, in practice $$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$5–$$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$6 outer steps suffice, BiCGStab takes on average $$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$7–$$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$8 inner iterations, and the method captures both “gathering–kissing–splitting” in a square and smooth queue formation in a corridor (Achdou et al., 2020). The same study emphasizes multiplicity in symmetric nonmonotone regimes, which is consistent with the absence of a general uniqueness theory outside monotone or specially structured classes (Achdou et al., 2020).

For Cournot mean field games of controls, a learning-based numerical method is built around Smoothed Policy Iteration. Given a smooth policy guess, one alternates Fokker–Planck policy evaluation, price update, HJB policy evaluation, best response, and smoothing

$$H(t,x,p,\mu):=\sup_{\alpha\in\mathbb R^n}[p\cdot\alpha-L(t,x,\alpha,\mu)],$$9

The paper proves uniqueness of the equilibrium under general assumptions on the price function and proves convergence of the learning algorithm; numerically it reproduces depletion dynamics and the Hubbert peak in oil-production models (Camilli et al., 2024).

A different numerical perspective treats MFGC equilibria as zeros of a monotone variational inequality in a Hilbert space. The extragradient iteration

$$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$00

is shown to converge with $$m_t := \mathrm{proj}_x{}_\#\mu_t \in \mathcal P(\Omega),$$01 averaged rate under strong monotonicity and Lipschitz assumptions, and exponentially fast for the last iterate under stronger assumptions. The paper explicitly relates this construction to fictitious play and extends it to general mean field type FBSDEs that do not necessarily come from optimal control (Meynard, 16 Feb 2026).

Taken together, these methods indicate that computation in mean field games of controls is not tied to a single numerical paradigm. Finite differences, policy iteration, continuation–Newton methods, and monotone-operator algorithms all appear, and the choice depends on whether the target problem is boundary-driven, potential, monotone, nonmonotone, or formulated primarily through PDEs or FBSDEs.