On Social Optima of Non-Cooperative Mean Field Games

TL;DR

This paper establishes a link between mean field games and social welfare optimization, showing that equilibria can be found via convex optimization, enabling efficient decentralized computation.

Contribution

It proves that under mild conditions, mean field game equilibria align with solutions to convex social welfare problems, facilitating practical computation methods.

Findings

Equilibrium solutions coincide with convex social welfare optima.

Decentralized algorithms efficiently compute mean field equilibria.

Numerical simulations validate the proposed approach.

Abstract

This paper studies the connections between mean-field games and the social welfare optimization problems. We consider a mean field game in functional spaces with a large population of agents, each of which 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 under some mild conditions any $\epsilon$-Nash equilibrium of the mean field game coincides with the optimal solution to a convex social welfare optimization problem. The results are proved based on a general formulation in the functional spaces and can be applied to a variety of mean field games studied in the literature. Our result also implies that the computation of the mean field equilibrium can be cast as a convex optimization problem, which can be efficiently solved by a decentralized primal dual algorithm. Numerical simulations are presented to demonstrate the effectiveness of the proposed approach.