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近十年来最重要的控制理论发展。加拿大的Caines、Huang和Malhamé是该领域的先驱(最初的会议论文是2003年)，在大西洋的这一边，而Lasry和Lions(菲尔兹奖牌获得者)在法国这样做，两人都在2006-2007年独立发表了期刊文章。MFG是一种大规模多智能体系统在博弈情况下的控制理论，即涉及多个潜在冲突的优化器。这样的代理人要么有内在的存在(例如，在一个个人寻求经济自我实现的经济中)，要么它们可能是在“分而治之”的努力中故意创造出来的。后一种模式存在于许多大型管理或工程系统中，在这些系统中，主要决策者并不拥有集中控制系统所需的传感、计算或通信能力。相反，系统被故意分解为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，并且被分配了本地执行功能。如果这些功能和所需的代理协调信号都得到了充分的工程设计，那么系统崩溃造成的最优性损失可以通过相应的决策效率收益和较低的通信成本得到很大程度的补偿。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”(推广最优概念)是出了名的困难，困难通常因代理人的数量而复杂化。MFG的突破是认识到，如果代理人有很多相似之处，并且随着代理人数量的增加，个体在群体整体福利中的权重消失(例如，孤立的个体的驾驶习惯对汽油价格的影响)，集体行为(平均场)可能在极限中变得确定，即使个体保持随机，这是由于成对的个体逐渐变得更加独立，以及大数定律的结果。这是一种类似的效应，在统计力学分析中被利用，事实上，MFG的出现要归功于大型相互作用的粒子系统思想与动态博弈理论的融合。简单地说，虚拟无限代理博弈比真正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MFGs的双管齐下的研究方案、理论和应用。描述了两个重要的应用：(I)智能电网中的高峰负荷削减和改进的可再生能源集成；(Ii)机器人系统中受生物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.