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是过去十年中最重要的控制理论发展。加拿大的Caines，Huang和Malhamé在大西洋的这一边成为该领域的先驱（2003年的初始会议论文），而Lasry和Lions（菲尔兹奖获得者）在法国也是如此，两者都在2006-2007年独立发表期刊论文。 MFG是一种大规模多智能体系统在博弈情况下的控制理论，即涉及多个和潜在冲突的优化器。 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），要么他们可能是在“分而治之”的努力中故意创造的。后一种模式存在于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被定义为代理的本地决策中心，这些决策中心可能无法访问相同的信息集，并且被分配了本地性能函数。 只要这些功能和所需的代理协调信号都得到充分的设计，系统崩溃导致的最优性损失可以在很大程度上通过相应的决策效率增益和较低的通信成本来补偿。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，特别是动态游戏中，计算不同类型的“均衡”（最优性的广义概念）是非常困难的，通常由于代理的数量而变得更加困难。MFG的突破是实现了如果代理人共享很多相似性，并且随着代理人数量的无限增加，个体在群体整体福利中的权重消失（例如，一个孤立的个人的驾驶习惯对汽油价格的影响），大众行为由于成对的个体逐渐变得更加独立，以及大数定律，即使个体保持随机性，平均场（mean field）也可能在极限中变得确定性。 这是一个类似的效果，这是利用统计力学分析和实际MFG的出现，由于大的相互作用的粒子系统的想法，与动态游戏的理论相结合。 简单地说，虚拟的无限代理人博弈比真实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/应用MFG。描述了两个重要的应用：（一）调峰和改进的可再生能源集成在智能电网中;（二）生物启发的集体决策和机器人系统中的导航方案。**

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. **