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

Yannick Eich, Christian Fabian, Kai Cui, Heinz Koeppl

TL;DR

This paper defines regularized equilibria for continuous-time MFGs and extends the classical fixed-point iteration and fictitious play algorithm to these equilibria, approximate the classical Nash equilibria by regularization methods, enabling more computationally tractable solution algorithms.

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.

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

TL;DR

This paper defines regularized equilibria for continuous-time MFGs and extends the classical fixed-point iteration and fictitious play algorithm to these equilibria, approximate the classical Nash equilibria by regularization methods, enabling more computationally tractable solution algorithms.

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.
Paper Structure (19 sections, 3 theorems, 40 equations, 3 figures, 2 algorithms)

This paper contains 19 sections, 3 theorems, 40 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. MFGs in Continuous Time with finite state spaces
  3. Finite Game
  4. Mean Field Game
  5. Equilibria in continuous-time finite-state MFGs
  6. Nash equilibria
  7. Regularized equilibria
  8. Learning Non-Cooperative Mean Field Equilibria
  9. Fixed Point Iteration.
  10. Fictitious Play.
  11. Experiments
  12. Conclusion and Discussion
  13. Proofs
  14. Lipschitz $\Gamma_{\mathcal{M}}$.
  15. Lipschitz $\Gamma^\alpha_{\Pi} \circ \Gamma_{ Q}$.
  16. ...and 4 more sections

Key Result

Theorem 1

For any $\alpha > 0$, a RE exists under Assm. ass:ccont.

Figures (3)

  • Figure 1: Convergence of FP and FPI on the LR problem.
  • Figure 2: Convergence of FP on a randomly generated MFG.
  • Figure 3: Fraction of infectious agents in SIS RE over temperatures $\alpha$.

Theorems & Definitions (6)

  • Definition 1: Mean field NE
  • Definition 2: RE
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Definition 3: Distance to equilibria