Optimization Perspectives of Mean Field Games

This paper studies the connections between meanfield games and an auxiliary optimization problem, and compares the auxiliary optimization with potential functions. We formulate a large-population game in function space. The cost functions of all agents are weakly coupled though the mean of the population states/controls. We show that under some conditions, the -Nash equilibrium of the mean-field game is the optimal solution to an auxiliary optimization problem, and this is true even when the optimization problem is non-convex. The result enables us to evaluate the mean-field equilibrium, and also has some interesting implications on the existence, uniqueness and computation of the mean-field equilibrium. While the auxiliary optimization is similar to the potential function in potential games, we show that in general, the meanfield game considered in this paper is not a potential game. We compare the auxiliary optimization problem with potential function minimization, and discuss their differences in terms of solution concept and computation complexity.