$N$-Player Stochastic Differential Games with Regime Switching and Mean Field Convergence
Mingrui Wang, Prakash Chakraborty
TL;DR
This work addresses finite-horizon $N$-player stochastic differential games with regime-switching driven by a finite-state Markov chain. It develops both a PDE-based analysis for the finite $N$-player Nash system and a probabilistic regime-switching McKean–Vlasov FBSDE framework for the mean-field limit, establishing existence and uniqueness of Nash equilibria and the unique solvability of the limiting FBSDE. By proving propagation of chaos and providing explicit convergence rates, the paper shows that strategies derived from the mean-field limit yield an $\varepsilon_N$-Nash equilibrium for large $N$, with $\varepsilon_N\to0$ as $N\to\infty$. The results extend mean-field game theory to regime-switching common noise, offering a rigorous bridge between finite-player equilibria and their limiting counterparts in complex, regime-dependent environments.
Abstract
In this study, we investigate $N$-player stochastic differential games with regime switching, where the player dynamics are modulated by a finite-state Markov chain. We analyze the associated Nash system, which consists of a system of coupled nonlinear partial differential equations, and establish the existence and uniqueness of solutions to this system, thereby proving the existence of a unique Nash equilibrium. Additionally, we examine the mean field game problem under the same regime-switching framework. We derive a connection between the Nash equilibrium of the MFG and forward-backward stochastic differential equation with jumps, and demonstrate the unique solvability of this equation. Finally, we explore the propagation of chaos and show that the optimal control obtained from the MFG serves as an approximate Nash equilibrium for the $N$-player problem.