Beyond separability: convergence rate of vanishing viscosity approximations to mean field games via FBSDE stability

TL;DR

This paper establishes the convergence rate of vanishing viscosity approximations in mean field games with nonlocal Hamiltonians, using probabilistic and analytical methods, and discusses applications to various related problems.

Contribution

It provides the first rigorous proof of the convergence rate for vanishing viscosity in MFGs with non-separable Hamiltonians, extending classical results.

Findings

Value function converges at rate O(β)

Distribution of players approximations also converge at the same rate

Applications to N-player games, mean field control, and policy iteration

Abstract

This paper studies the vanishing viscosity approximation to mean field games (MFGs) in $\mathbb{R}^d$ with a nonlocal and possibly non-separable Hamiltonian. We prove that the value function converges at a rate of $\mathcal{O}(\beta)$, where $\beta^2$ is the diffusivity constant, which matches the classical convergence rate of vanishing viscosity for Hamilton-Jacobi (HJ) equations. The same rate is also obtained for the approximation of the distribution of players as well as for the gradient of the value function. The proof is a combination of probabilistic and analytical arguments by first analyzing the forward-backward stochastic differential equation associated with the MFG, and then applying a general stability result for HJ equations. Applications of our result to $N$-player games, mean field control, and policy iteration for solving MFGs are also presented.