  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.
