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

平均场博弈理论（MFG）被许多控制理论家认为可能是过去十年中最重要的控制理论发展。在大西洋的这一边，加拿大的Caines、Huang和malham<e:1>是这一领域的先驱（2003年的首次会议论文），而法国的Lasry和Lions（菲尔兹奖得主）在2006-2007年独立发表了期刊。MFG是一种大规模多智能体系统在博弈中的控制理论，即涉及多个潜在冲突的优化器。这些代理人要么具有内在的存在（例如，个人寻求经济自我实现的经济），要么可能是在“分而治之”的努力中故意创造出来的。后一种模式出现在许多大型管理或工程系统中，其中主要决策者不拥有集中控制系统所需的传感、计算或通信能力。相反，系统被故意分解成一组定义为代理的本地决策中心，这些代理可能无法访问相同的信息集，并被分配了本地性能函数。如果这些功能和所需的代理协调信号都得到了充分的设计，则系统崩溃导致的最优性损失可以通过相应的决策效率提高和较低的通信成本来弥补。这是基于博弈论的复杂系统分散控制的本质。然而，在游戏中，特别是在动态游戏中，计算不同类型的“平衡”（优化的泛化概念）是非常困难的，这种困难通常与代理的数量有关。MFG实现的突破在于，它认识到，如果智能体有很多相似之处，并且个体在群体整体福利中的权重随着智能体数量的无限制增加而消失（例如，一个孤立个体的驾驶习惯对汽油价格的影响），即使个体仍然是随机的，由于成对的个体逐渐变得更加独立，群体行为（平均场）在极限下可能变得确定性。还有大数定律。统计力学分析中也利用了类似的效应，事实上，MFG的出现要归功于大型相互作用粒子系统思想与动态博弈理论的融合。简单地说，虚拟的无限代理博弈比实际的大而有限的博弈更容易分析，并被用作计算近似均衡的工具。**我们提出了一个双管齐下的研究计划，理论和应用的mfg。描述了两个重要的应用：(i)峰值负荷调整和改进智能电网中的可再生能源整合；机器人系统中受生物学启发的集体决策和导航方案。**

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. **