Discrete Mean Field Games on Finite Graphs as Initial Value Optimization

TL;DR

This paper introduces an initial value formulation for discrete mean field games on finite graphs and proposes a neural network method to efficiently solve the resulting constrained optimization problem.

Contribution

It reformulates potential mean field games on finite graphs as a reduced-order initial value problem and develops a neural network approach for its solution.

Findings

Reformulation as a finite-dimensional optimization problem reduces complexity.

Neural network approach effectively solves the initial value formulation.

Avoids time-discretization, leading to computational efficiency.

Abstract

In this paper, we propose an initial value fomulation of the discrete mean field games on finite graphs (Graph MFG), and design a neural network based approach to solve it. Graph MFG describes infinite, non-cooperative and interactive homogeneous agents move on node states through the edges to optimize their own goals. Nash Equilibrium of the Graph MFG is characterized by a coupled ordinary differential equations (ODE) system, including the discrete forward continuity equation and the discrete backward Hamilton-Jacobi equation. In this paper, we mainly focus on the potential mean field games (Potential MFG) on finite graphs, which has an infinite-dimensional constrained optimization structure. We reformulate Potential MFG as an initial value finite-dimentional optimization problem with dynamics constrains, names Graph MFG-IV. Specifically, the initial condition of the Hamilton-Jacobi equation is regarded as the unique variable, constrained by the coupled Hamilton-Jacobi and continuity equation system as the ODE integrator. This formulation is a reduced-order model, which avoids time-discretization of the infinite-dimensional path and has a much smaller searching space than the general path-wise problem setting. We design a neural network-based approach to solve the Graph MFG-IV problem.