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

TL;DR

This paper proves the existence, uniqueness, and convergence of distributed equilibria in N-player differential games with control interactions, extending to mean field limits and deterministic models.

Contribution

It introduces a novel framework for distributed equilibria in nonsymmetric N-player games, establishing key theoretical results and convergence to mean field games of controls.

Findings

Existence and uniqueness of distributed equilibria under displacement semimonotonicity.

Quantitative convergence of N-player games to mean field games of controls as N→∞.

Applicability to both stochastic and deterministic models, including linear quadratic examples.

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.