This work introduces a new general approach for the numerical analysis of stable equilibria to second order mean field games systems in cases where the uniqueness of solutions may fail. For the sake of simplicity, we focus on a simple stationary case. We propose an abstract framework to study these solutions by reformulating the mean field game system as an abstract equation in a Banach space. In this context, stable equilibria turn out to be regular solutions to this equation, meaning that the linearized system is well-posed. We provide three applications of this property: we study the sensitivity analysis of stable solutions, establish error estimates for their finite element approximations, and prove the local converge of Newton's method in infinite dimensions.

翻译：本文提出一种新的通用方法，用于在解可能不唯一的情况下对二阶平均场博弈系统的稳定平衡进行数值分析。为简化问题，我们聚焦于一个简单平稳情形。通过将平均场博弈系统重述为巴拿赫空间中的抽象方程，我们构建了一个研究这些解的抽象框架。在此框架下，稳定平衡恰为该方程的正则解，即线性化系统是适定的。我们提供了该性质的三个应用：研究稳定解的灵敏度分析、建立其有限元逼近的误差估计，以及证明无穷维牛顿方法的局部收敛性。