Approximately Solving Continuous-Time Mean Field Games with Finite State Spaces

TL;DR

This paper develops computational methods for approximating Nash equilibria in continuous-time mean field games with finite state spaces, extending discrete-time algorithms to improve tractability.

Contribution

It introduces regularized equilibria for continuous-time MFGs and adapts fixed-point and fictitious play algorithms for these equilibria, filling a gap in computational approaches.

Findings

Regularized equilibria enable tractable computation.

Extended algorithms show effectiveness in numerical examples.

Approach bridges theory and practical computation for continuous-time MFGs.

Abstract

Mean field games (MFGs) offer a powerful framework for modeling large-scale multi-agent systems. This paper addresses MFGs formulated in continuous time with discrete state spaces, where agents' dynamics are governed by continuous-time Markov chains -- relevant to applications like population dynamics and queueing networks. While prior research has largely focused on theoretical aspects of continuous-time discrete-state MFGs, efficient computational methods for determining equilibria remain underdeveloped. Inspired by discrete-time approaches, we approximate the classical Nash equilibria by regularization methods, enabling more computationally tractable solution algorithms. Specifically, we define regularized equilibria for continuous-time MFGs and extend the classical fixed-point iteration and fictitious play algorithm to these equilibria. We validate the effectiveness and practicality of our approach via illustrative numerical examples.