Computational Mean-field Games on Manifolds

TL;DR

This paper extends mean-field game theory to Riemannian manifolds, formulating Nash equilibria and developing numerical methods, demonstrating effectiveness through experiments on various manifolds.

Contribution

It introduces a framework for mean-field games on manifolds, including PDE formulation, variational equivalence, and a proximal gradient method for numerical solutions.

Findings

Effective numerical methods for mean-field games on manifolds

Validation through experiments on different manifold geometries

Establishment of PDE and variational equivalence on manifolds

Abstract

Conventional Mean-field games/control study the behavior of a large number of rational agents moving in the Euclidean spaces. In this work, we explore the mean-field games on Riemannian manifolds. We formulate the mean-field game Nash Equilibrium on manifolds. We also establish the equivalence between the PDE system and the optimality conditions of the associated variational form on manifolds. Based on the triangular mesh representation of two-dimensional manifolds, we design a proximal gradient method for variational mean-field games. Our comprehensive numerical experiments on various manifolds illustrate the effectiveness and flexibility of the proposed model and numerical methods.