Connections between Mean-Field Game and Social Welfare Optimization

TL;DR

This paper establishes a link between mean-field games and social welfare optimization, showing that under certain conditions, the equilibrium of the game aligns with the social optimum, even with non-convex costs.

Contribution

It demonstrates that mean-field game equilibria can be characterized as solutions to social welfare problems, providing insights into their efficiency and computation.

Findings

Equilibrium coincides with social welfare optimum under mild conditions

Results hold even for non-convex individual cost functions

Numerical validation confirms theoretical insights

Abstract

This paper studies the connection between a class of mean-field games and a social welfare optimization problem. We consider a mean-field game in function spaces with a large population of agents, and each agent seeks to minimize an individual cost function. The cost functions of different agents are coupled through a mean-field term that depends on the mean of the population states. We show that although the mean-field game is not a potential game, under some mild condition the $\epsilon$-Nash equilibrium of the mean-field game coincides with the optimal solution to a social welfare optimization problem, and this is true even when the individual cost functions are non-convex. The connection enables us to evaluate and promote the efficiency of the mean-field equilibrium. In addition, it also leads to several important implications on the existence, uniqueness, and computation of the mean-field equilibrium. Numerical results are presented to validate the solution, and examples are provided to show the applicability of the proposed approach.