Distributed Equilibria for $N$-Player Differential Games with Interaction through Controls: Existence, Uniqueness and Large $N$ Limit

Abstract: We establish the existence and uniqueness of distributed equilibria to possibly nonsymmetric $N$ player differential games with interactions through controls under displacement semimonotonicity assumptions. Surprisingly, the nonseparable framework of the running cost combined with the character of distributed equilibria leads to a set of consistency relations different in nature from the ones for open and closed loop equilibria investigated in a recent work of Jackson and the second author. In the symmetric setting, we establish quantitative convergence results for the $N$ player games toward the corresponding Mean Field Games of Controls (MFGC), when $N\to+\infty$. Our approach applies to both idiosyncratic noise driven models and fully deterministic frameworks. In particular, for deterministic models distributed equilibria correspond to open loop equilibria, and our work seems to be the first in the literature to provide existence and uniqueness of these equilibria and prove the large $N$ convergence in the MFGC setting. The sharpness of the imposed assumptions is discussed via a set of explicitly computable examples in the linear quadratic setting.

• The paper establishes existence and uniqueness of distributed Nash equilibria under displacement semimonotonicity in both stochastic and deterministic settings.
• It uses a novel fixed point approach in the admissible control space to manage nonseparable running costs without relying on traditional measure-based methods.
• The analysis provides quantitative convergence rates to the mean field limit, supported by explicit linear-quadratic examples demonstrating sharp technical conditions.

Distributed Nash Equilibria in $N$-Player Differential Games with Control Coupling

Problem Formulation and Scope

The paper rigorously formulates and analyzes the existence, uniqueness, and asymptotic behavior of distributed Nash equilibria in $N$-player differential games where the agent interactions occur through their controls and not only through their states. The evolution of each agent's state variable is governed by controlled stochastic or deterministic dynamics, and the cost functional for each player features coupling via the controls of the other players—a framework substantially more general and technically subtle than classical state-only interactions.

The authors address both nonsymmetric and symmetric games, accommodate full nonseparability of the running cost with respect to the other players’ controls, and handle both parabolic (stochastic) and first-order (deterministic) regimes. Crucially, the distributed feedback (Markovian) setup is investigated, in contrast to the open- or closed-loop strategies more commonly explored in related mean field games of control (MFGCs).

The main theoretical contributions are: (a) existence and uniqueness of distributed equilibria for the $N$-player problem under displacement semimonotonicity assumptions, (b) explicit analysis of the consistency/fixed point structure resulting from nonseparability, (c) rigorous mean field limit as $N\to\infty$ with quantitative convergence results, and (d) explicit characterization in linear quadratic (LQ) scenarios to illustrate the sharpness of the assumptions.

PDE Characterization and Consistency Relations

The optimal responses for each player are characterized by a coupled system of Hamilton-Jacobi-Bellman (HJB) and Kolmogorov-Fokker-Planck (KFP) equations. The nonseparable nature of the running costs—especially the dependence on the joint distribution of other players' controls—forces a novel form of consistency: the feedback law for each agent emerges from a fixed point equation involving functional derivatives of the cost with respect to both the agent's control and the control distribution of the population. This departs from the correspondences found in separable cost frameworks, and its formal structure is distinct from that of open- or closed-loop equilibria (Jackson et al., 22 Jul 2025).

The deterministic limit further simplifies the analysis but motivates distinct technical tools, as the equivalence between distributed (Markov) and open-loop feedback ceases to hold except in very specific circumstances.

Existence and Regularity Theory

The core analytical difficulty lies in the high-dimensional, nonlinear, and non-locally coupled PDE system that defines distributed Nash equilibria for the system. The paper introduces a technique based on Schauder-type compactness arguments on the space of admissible feedback controls, circumventing the need to analyze the fixed point mapping on the (infinite-dimensional) measure flows directly. This is essential since nonseparable running costs break the commutation between the Legendre transform and the “lifting” operation, which is exploited in the open-loop analysis of (Jackson et al., 22 Jul 2025). The iteration occurs over control laws, not on measures, yielding robust existence theory for both finite $N$ and the mean field limit.

The authors establish the following technical results:

• For arbitrary $N$, existence of classical solutions to the distributed Nash system under mild coercivity and regularity assumptions on the data.
• Stability and uniqueness under a quantified displacement semimonotonicity, a condition known to be minimal in the analysis of monotone mean field games [56, (Cirant et al., 2 May 2025)].
• Well-posedness (existence, uniqueness, regularity) of the associated MFGC system in the limit $N \rightarrow \infty$ using similar analytic techniques.

Quantitative Convergence to the Mean Field Game

In the symmetric setting, it is shown that, as $N\to\infty$, the distributed Nash equilibria of the $N$-player games converge quantitatively to the solution of the corresponding Mean Field Game of Controls (MFGC). Let $X^i$ and $N$0 denote the state and control processes in the $N$1-player system, and let $N$2 and $N$3 denote their mean field analogues from independent copies of the MFGC system. The paper establishes the convergence rate:

$N$4

for an explicit function $N$5 quantifying deviation of initial data and a statistical error rate $N$6 in Wasserstein distance depending on the dimension and moment bounds, with forms compatible with classical limit theorems (sub-Gaussian or poly-logarithmic rates depending on $N$7 and $N$8). The rate for value function convergence is provably sharper, with individual (rather than averaged) convergence in uniform norm.

These results apply in both stochastic and deterministic cases. Notably, the convergence of gradients of value functions is resolved in the deterministic regime—addressing a technically delicate open question in nonseparable settings.

Structural and Technical Novelty

The primary technical innovation is the management of the analytic challenges posed by full nonseparability of the running cost via controls—a scenario that, in the distributed feedback framework, produces a fixed point equation that cannot be directly reduced or solved by approaches from the open/closed-loop literature (Jackson et al., 22 Jul 2025, Cirant et al., 2 May 2025). The “lifting” to infinite-dimensional function spaces prevents the use of the Legendre transform in the usual way, and the fixed-point construction operates within the admissible control space rather than the space of distributions.

A further strength of the approach is that existence and convergence results do not require the number of players $N$9 to be large (unconditional on $N$0), and apply in both stochastic and deterministic regimes. The analysis is sharp, as demonstrated via explicit LQ counterexamples in which violation of the displacement semimonotonicity or related contractivity assumptions leads either to loss of uniqueness or absence of (quadratic) solution structure.

Explicit Linear-Quadratic Examples

In LQ scenarios, explicit construction of equilibria validates the theory and elucidates the impact of the displacement semimonotonicity and contractivity conditions. When these are not met, the system is shown to admit either multiple equilibria or none, depending on parameter regimes. The feedback structure and convergence to the MFGC can be explicitly computed and analyzed, confirming the theoretical rates and limitations.

Implications and Future Directions

The established well-posedness and quantitative convergence for distributed equilibria in settings with control coupling significantly expands the mathematical foundation of MFGCs, which are relevant for decentralized engineering controls, economics (e.g., mean field models of Cournot-Bertrand competition), and large-scale learning systems. The techniques introduced here open several avenues:

• Direct analysis of more general interaction forms (non-Markovian, common noise, controlled volatility).
• Extension to weak or relaxed equilibria beyond classical settings, as studied probabilistically in [Dje:23-aap, PosTan].
• Numerical and computational methods for high-dimensional coupled PDE systems with control-distribution feedback.
• Asymptotics in the presence of additional sources of heterogeneity or further nonconvexity.

Quantitative propagation of chaos for value functions provides new benchmarks for algorithms and for understanding equilibrium stability in large populations.

Conclusion

This work presents a comprehensive PDE and probabilistic analysis of distributed Nash equilibria in $N$1-player games with interaction through controls, extending mean field limits to cases of full nonseparability in costs. Existence, uniqueness, and quantitative convergence are all established unconditionally on $N$2, and the analysis in both stochastic and deterministic settings is rigorous and sharp. By clarifying model-dependent subtleties and providing new analytic methodologies, the paper marks notable progress in the mathematical theory of mean field games of controls and opens further research directions in the analysis of high-dimensional, non-locally coupled game-theoretic systems (2603.29707).