Connections between mean-field game and social welfare optimization

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 -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.

Mean-Field Transmission Power Control in Dense Networks, Part II -- Social Welfare Evaluation

“…However, in an optimization problem, the change of optimal value of the objective function under different constraints can be evaluated based on the Lagrangian dual method [24]. As motivated by Li et al [13], where the connection between a class of mean-field games and optimization problems is established, it is desirable to conduct the social welfare comparison from the perspective of optimization. In a mean-field game with a large number of players, we define the social welfare as the average utility achieved by all players.…”

“…Later, both Nourian et al [12] considered mean-field games for synchronization of a large number of oscillators, and characterized the efficiency loss at the game equilibrium. In order to perform computation and analysis on mean-field equilibrium more conveniently, Li et al [13] proposed an equivalent formulation of a class of mean-field games as optimization problems, in which the LQG mean-field game is an example. For modeling the evolution of the game equilibria when the number of players in a game approaches infinity, Lacker and Ramanan [14] adopted a probabilistic model to characterize the speed for some rare equilibria to vanish.…”

Achieving Social Optimum in Non-convex Cooperative Aggregative Games: A Distributed Stochastic Annealing Approach

“…In fact, social optimum, origned from [8] and of which Pareto optimum is a useful necessary condition [9], seeks global optimum of the whole network. Still, for intrinsic interest or means of measuring the efficiency of different Nash equilibria, the social optimum has been widely studied in various situations, including resource allocation games [10]- [12], meanfield games [13]- [15] and so on. Different from Nash equilibria, the social optimum offers cooperation mechanism for networked games.…”

Linear quadratic mean field social optimization: Asymptotic solvability

“…For these control problems, the dynamic programming approach is developed in [5,43]. Both mean field games and mean field social optima are analyzed and compared in [11,35]. The bounds for their efficiency difference are provided in [11] while [35] shows that the mean field equilibrium may be interpreted as the solution of a modified social optimization problem.…”

“…Both mean field games and mean field social optima are analyzed and compared in [11,35]. The bounds for their efficiency difference are provided in [11] while [35] shows that the mean field equilibrium may be interpreted as the solution of a modified social optimization problem. Performance comparisons of the two approaches are presented in [54] for a static mean field model arising in dense wireless networks.…”