Mean Field Game and Control for Switching Hybrid Systems
Tejaswi K. C., Taeyoung Lee
TL;DR
The paper tackles coordinating large populations in switching hybrid environments by formulating mean field game (MFG) and mean field control (MFC) problems that couple continuous dynamics with discrete transitions. It develops a forward-backward PDE framework (HJB and FPK) for both MFG and MFC, and extends these to switching systems by discretizing continuous flow with a finite-difference scheme and enforcing matching conditions at switching times. A Newton-based solver with a continuation strategy in viscosity is proposed to compute solutions across multiple intervals, including a specialized root-finding formulation $\,\Phi(Z)=0$ that handles interval coupling. The approach is demonstrated on an emergency evacuation scenario with environment changes, showing how MFC achieves near-optimal social performance (PoA $\approx 1.0181$) and how switching events shape congestion and path selection, highlighting a scalable framework for decentralized decision-making in large switching systems.
Abstract
Mean field games and controls involve guiding the behavior of large populations of interacting agents, where each individual's influence on the group is negligible but collectively impacts overall dynamics. Hybrid systems integrate continuous dynamics with discrete transitions, effectively modeling the complex interplay between continuous flows and instantaneous jumps in a unified framework. This paper formulates mean field game and control strategies for switching hybrid systems and proposes computational methods to solve the resulting integro-partial differential equation. This approach enables scalable, decentralized decision-making in large-scale switching systems, which is illustrated through numerical examples in an emergency evacuation scenario with sudden changes in the surrounding environment.