Mean Field Game Theory: A Potential Game Changer in the Decentralized Control of Complex Systems

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Mean Field Games Theory (MFG) is perceived by numerous control theorists as probably the most important control theoretic development during the past decade. Canada's Caines, Huang and Malhamé acted as pioneers in the field (initial conference paper 2003), on this side of the Atlantic , while Lasry and Lions (a Fields medalist) did so in France, both independently with journal publications in the years 2006-2007. MFG is a control theory of large scale multi agent systems in a game situation, i.e. involving multiple and potentially conflicting optimizers. Such agents either have an intrinsic existence (e.g. an economy where individuals seek economic self fulfillment), or they may be deliberately created in a "divide to conquer" effort. This latter pattern is found in many large management or engineering systems where the major decision maker does not own the sensing, computation or communication capabilities, required to centrally control the system. Instead, the system is deliberately broken up into a set of local decision centers defined as agents, which may not have access to the same information sets, and which are assigned local performance functions. Provided both these functions and needed agent coordination signals are adequately engineered, the loss of optimality resulting from system breakup could be largely compensated by the corresponding decision making efficiency gains and lower communication costs. This is the essence of game theoretic based decentralized control of complex systems. However, computing different types of "equilibria" (generalizing notions of optimality) in games , particularly dynamic games, is notoriously difficult, with a difficulty generally compounded by the number of agents. The breakthrough made possible by MFG's is that of realizing that if agents share a lot of similarity, and the weight of an individual in the global welfare of the group vanishes as the number of agents increases without bound (e.g. an isolated individual's driving habits influence on the price of gasoline), the mass behavior (mean field) could become deterministic in the limit, even if individuals remain stochastic, as a result of pairs of individuals becoming gradually more independent, and the law of large numbers. It is a similar effect which is exploited in Statistical Mechanics analyses and indeed MFG's have emerged thanks to a blend of large interacting particle systems ideas, with the theory of Dynamic Games. Simply put, the virtual infinite agents game turns out to be much easier to analyze than its real large but finite game counterpart, and is used as a device to compute approximate equilibria. We propose a two pronged research programme, theory / applications of MFGs. Two important applications are delineated: (i) Peak load shaving and improved renewables integration in smart grids; (ii) Biologically inspired collective decision making and navigation schemes in robotic systems.