Mean Field Games of Controls (MFGC)

• Mean Field Games of Controls (MFGC) are models where agents’ costs depend on the joint distribution of states and controls, requiring a fixed-point equilibrium formulation.
• They integrate PDE systems, Hamilton–Jacobi–Bellman equations, and Fokker–Planck dynamics to link optimal control strategies with population interactions.
• MFGC frameworks connect finite-player games with continuum limits and inspire computational approaches including reinforcement learning and numerical schemes.

Searching arXiv for recent and foundational papers on Mean Field Games of Controls to ground the article. arXiv search query: Mean Field Games of Controls joint distribution state and control Nash equilibrium Fokker-Planck Hamilton-Jacobi-Bellman Mean Field Games of Controls (MFGC) are mean field games in which the interaction is through the joint law of agents’ states and controls, rather than only through the law of the states. In this setting, a representative agent minimizes a cost that depends on the joint distribution of players’ states and controls, and equilibrium requires a fixed-point relation between the optimal feedback and the induced state–control law. Across the literature, MFGC appears in PDE, variational, probabilistic, ergodic, and learning-based formulations, with recurrent structural ingredients given by a Hamilton–Jacobi–Bellman equation, a Fokker–Planck equation, and a consistency condition for the joint measure of states and controls (Graber et al., 8 Apr 2026, Mou et al., 2022, Djete, 2020).

1. Definitional scope and distinguishing features

In a standard mean field game, each agent controls a state process and incurs a running cost depending on the state and the current population distribution $m_t$ of states. The MFG coupling is through $m_t$ only. In a mean field game of controls, the cost depends on the joint distribution of states and controls. At each time one considers a probability measure on a state–control space, such as $$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$ or, in discrete-time formulations, a state–action distribution $\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$. Analytically, this adds a fixed-point condition: given a candidate joint measure, the agent solves a control problem, obtains an optimal feedback, and at equilibrium the candidate measure must coincide with the law of the optimally controlled state–control pair (Graber et al., 8 Apr 2026, Angiuli et al., 2022).

Several papers in adjacent areas are explicitly distinguished from full MFGC. “Meanfield games and model predictive control” studies couplings through the state distribution only (Degond et al., 2014). “Mean Field Game and Control for Switching Hybrid Systems” states that there is no explicit dependence on the distribution of controls or on joint state–control measures such as $\mu_t(dx,d\alpha)$ (C. et al., 2024). “Approximation of N-player stochastic games with singular controls by mean field games” has interaction through the empirical distribution of states only (Cao et al., 2022). “A Mean Field Game System and a Related Deterministic Optimal Control Problem” is presented as “very close in spirit to MFG of controls,” but its coupling is local in the density (Anita, 20 Mar 2025). A common misconception is therefore to identify any mean field model with control in the drift or in the cost as an MFGC; the stricter usage in this literature requires dependence on the distribution of controls or on the joint law of states and controls.

A second structural distinction concerns the equilibrium notion. Some works use feedback Nash equilibria characterized by PDE systems; some use open-loop formulations and controlled Fokker–Planck equations; some use relaxed controls and measure-valued solutions; and some organize the interaction as competition between groups that cooperate internally. This suggests that “MFGC” names a class of interaction structures rather than a single canonical formalism.

2. Canonical probabilistic and PDE formulations

A representative PDE form of MFGC is the stationary second-order system

$\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$

Here $u$ is the value function, $\rho$ the ergodic cost, $m$ the invariant density of states, and $\mu$ the joint distribution of states and controls. The third line is the MFGC-specific consistency relation: under $m_t$0, almost surely the control is the optimal feedback $m_t$1 when the state is $m_t$2 (Graber et al., 8 Apr 2026).

A time-dependent potential MFGC can be written as

$m_t$3

with initial-terminal data for $m_t$4 and $m_t$5. In this formulation the state coupling $m_t$6 is supplemented by a control coupling through a time-dependent “price” $m_t$7, determined by the aggregate feedback $m_t$8 (Graber et al., 2020).

Probabilistic formulations place the same interaction in a McKean–Vlasov control problem. With common noise, the state equation may be

$m_t$9

where $$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$0 and $$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$1. In this setup the mean field is the joint conditional law of state and control, and the paper introduces measure-valued control rules and a notion of measure-valued MFG equilibrium through controlled Fokker–Planck equations (Djete, 2020).

These formulations differ in regularity and information structure, but they share the same equilibrium architecture: a control problem for a representative agent, a forward law equation, and a fixed point for the joint state–control distribution.

3. Variational structure, monotonicity, and uniqueness mechanisms

A major line of work treats MFGC as a potential system. In the potential second-order case on $$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$2, the MFGC system is the Euler–Lagrange system of a pair of convex optimization problems in duality. After the Benamou–Brenier substitution $$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$3, the weak formulation involves the HJB side

$$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$4

the Fokker–Planck equation

$$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$5

the coupling

$$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$6

and the constitutive identity

$$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$7

The main result is existence and uniqueness of weak solutions, with uniqueness of $$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$8 and of $$\mu_t\in\mathcal P(\overline{\Omega}\times\mathbb R^n)$$9 on $\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$0, obtained by exploiting potentiality and convexity (Graber et al., 2020).

A related density-control formulation, explicitly described as “very close in spirit to MFG of controls,” identifies an MFG system as the Euler–Lagrange system for an optimal control problem on densities driven by a controlled Fokker–Planck equation. The cost functional

$\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$1

produces an optimality system with a forward density equation, an adjoint equation, and a pointwise relation

$\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$2

This shows how MFG-type PDE systems can arise as optimality systems for deterministic control on the space of densities (Anita, 20 Mar 2025).

Monotonicity is the central uniqueness mechanism. In MFGC master equation theory, Lasry–Lions monotonicity, displacement monotonicity, and anti-monotonicity are all studied at the level of the Hamiltonian and shown to propagate along classical solutions. The paper extends displacement monotonicity to semi-monotonicity and proves its propagation; this propagation result is stated to be new even for standard mean field games. The work is presented as the first step towards the global wellposedness theory for master equations of Mean Field Games of Controls (Mou et al., 2022).

In the ergodic state-constrained regime, well-posedness is proved under monotone coupling and Hamiltonians with at most quadratic growth. There the monotone coupling extends Lasry–Lions monotonicity to the joint state–control setting through both $\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$3 and $\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$4, and yields uniqueness up to the additive constant in $\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$5 (Graber et al., 8 Apr 2026).

4. Boundary conditions, reflections, and control constraints

Boundary phenomena are not peripheral in MFGC; they change the equilibrium class. With Dirichlet boundary conditions on a bounded domain, agents may exit the domain, the mass is not conserved, and the PDE system becomes

$\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$6

with

$\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$7

The main theorem establishes existence of a $\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$8-classical solution by combining a fixed point for $\nu_t\in\Delta^{|\mathcal X\times\mathcal A|}$9 with Leray–Schauder on $\mu_t(dx,d\alpha)$0. Much of the analysis is devoted to a priori estimates needed to circumvent the fact that the mass is not conserved (Bongini et al., 2021).

State constraints lead to a different boundary mechanism. In the ergodic second-order MFGC with state constraints, the value function blows up at the boundary and the density vanishes at a commensurate rate. For $\mu_t(dx,d\alpha)$1,

$\mu_t(dx,d\alpha)$2

while for $\mu_t(dx,d\alpha)$3,

$\mu_t(dx,d\alpha)$4

The invariant density satisfies

$\mu_t(dx,d\alpha)$5

so the singular inward drift generated by $\mu_t(dx,d\alpha)$6 keeps the state inside the domain (Graber et al., 8 Apr 2026).

Reflection yields a third regime. In “Mean Field Game of Controls with State Reflections: Existence and Limit Theory,” the state process is reflected along an exogenous stochastic reflection boundary. The paper introduces a customized relaxed formulation, an enlarged canonical space, the dynamic Skorokhod mapping, and an extension transformation technique to handle the joint measure flow of the state and the relaxed control when continuity may fail (Bo et al., 5 Mar 2025).

A nearby but distinct control-centric regime is singular control. The singular-control approximation paper does not study a genuine MFGC, because the interaction is through the empirical distribution of states only, but it shows that bounded-velocity singular controls induce $\mu_t(dx,d\alpha)$7-Nash equilibria with

$\mu_t(dx,d\alpha)$8

and that finite-variation singular controls can be approximated with an additional $\mu_t(dx,d\alpha)$9 term (Cao et al., 2022). This suggests that singular-control extensions of MFGC are technically plausible, but that implication is inferential.

5. Finite-player games, distributed equilibria, and large-$\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$0 limits

A defining achievement of recent MFGC theory is the bidirectional connection between finite-player equilibria and mean field limits. In a probabilistic open-loop framework with controlled volatility, the $\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$1-player game is driven by empirical distributions

$\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$2

and the paper proves that $\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$3-Nash equilibria have subsequential limits, each limit being 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 $\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$4-Nash equilibria. The same paper also shows that measure-valued solutions are the accumulating points of $\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$5-strong solutions when $\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$6 (Djete, 2020).

A different finite-player route uses distributed equilibria. For possibly nonsymmetric $\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$7-player differential games with interaction through controls and nonseparable running costs, existence and uniqueness of distributed equilibria are established under displacement semimonotonicity assumptions. In the symmetric setting, the paper proves quantitative convergence toward the corresponding MFGC as $\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$8. For i.i.d. initial data, the convergence rate is expressed through

$\begin{cases} -\sigma\Delta u + H(D_x u,\mu) + \rho = F(\mu,x), & x\in\Omega,\[0.3em] \sigma\Delta m + \nabla\cdot\bigl(m\,D_pH(D_x u,\mu)\bigr) = 0, & x\in\Omega,\[0.3em] \mu = (I, -D_pH(D_x u,\mu))_\# m,\[0.3em] m \ge 0,\quad \int_\Omega m(x)\,dx=1,\quad \lim_{d(x)\to 0} u(x) = +\infty. \end{cases}$9

and the paper proves both convergence of state/control trajectories and

$u$0

It also states that, for deterministic models, distributed equilibria correspond to open loop equilibria (Lam et al., 31 Mar 2026).

These results position MFGC as more than an asymptotic heuristic. They show that, under structural monotonicity and compactness hypotheses, the mean field model and the finite-player game determine each other in the large-population limit.

6. Reinforcement learning, numerical schemes, and adjacent computational frameworks

Algorithmic work on MFGC divides sharply between genuine state–control mean fields and adjacent state-only models. In mixed mean field control games, which are directly relevant to MFGC because they allow mean field interactions through both states and controls, a three-timescale reinforcement learning algorithm $u$1 is proposed. In the extended finite-horizon formulation, the mean field is a state–action distribution $u$2, the transition kernel is $u$3, and the algorithm updates the local distribution, the Q-function, and the global distribution on separate timescales. The paper explicitly states that the algorithm can be adapted for solving problems when both the distributions of states and controls are involved (Angiuli et al., 2022).

A deep RL line targets continuous-state MFCG/MFGC-type models by reformulating the representative-agent problem as an MDP coupled with a learned mean field. “Efficient and Scalable Deep Reinforcement Learning for Mean Field Control Games” uses actor–critic methods together with parallel sample collection, mini-batching, target network, proximal policy optimization (PPO), generalized advantage estimation (GAE), and entropy regularization. On a linear-quadratic benchmark with an analytical equilibrium, the paper reports that some versions of the proposed approach achieve faster convergence and closely approximate the theoretical optimum, outperforming the baseline algorithm by an order of magnitude in sample efficiency (Peng et al., 2024).

By contrast, several computational papers remain outside strict MFGC but provide methods that are technically adjacent. Degond–Herty–Liu connect mean field games and model predictive control, showing that short-horizon optimization leads to best-reply dynamics in a purely state-based mean field model (Degond et al., 2014). The switching-hybrid paper develops a finite-difference Newton scheme for MFG and MFC systems with state-distribution coupling and time-dependent boundary data, but states explicitly that there is no explicit dependence on the distribution of controls (C. et al., 2024). A plausible implication is that many numerical devices used there—Godunov discretization, global Newton iterations across switching intervals, and continuity constraints at event times—may transfer to genuine MFGC once a joint state–control PDE characterization is available.

Across these computational approaches, the recurring difficulty is not merely dimensionality. It is the need to represent, update, and enforce consistency for a measure on state–control space, often under boundary effects, control constraints, or weak solution concepts. That requirement is the computational signature of MFGC.