Cooperative and non-cooperative mean field control: road to taming complexity

自从平均场博弈(MFG)理论在过去十年中诞生以来(Caines，Huang和Malhame 2003,2006,2007；Lasry and Lions，2006,2007)，这个领域已经发展成为一个重要的科学社区，具有密集的研究活动，跨越学科边界；参见(Caines，Huang和Malhame，2017)中的概述。它为解决大型动力学决策问题中臭名昭著的维难问题提供了有力工具。MFGS的基本理论建立在两个基本方法之上。第一种方法被称为直接方法，它从求解一个大规模博弈开始，并在种群规模趋于无穷大时推导出一组极限方程(Lary和Lions，2007)。第二种方法应用平均场近似，并将具有代表性的球员的不动点问题形式化(Huang等人，2006,2007)。通过考虑主要球员(Huang，2010；Nourian和Caines，2013；Bensoussan等人，2015；Carmona和朱，2016)或共同噪声(Cadaliaguet等人，2015；Carmona和Delarue，2018)，MFG理论进一步丰富了MFG理论，这导致了随机平均场而不是确定性平均场。另一个重要的扩展是社会优化，即大量代理合作优化他们的总成本(Huang等人，2012；Lacker，2017)。虽然在过去的十年中，平均场控制理论及其应用经历了惊人的发展，形成了一个核心研究共同体，但由于其绝对的丰富性，这一领域仍在迅速发展。这项研究计划将调查这一领域的重要前沿课题。首先，一个基本问题是关于最惠国集团理论的两种方法之间的关系。最近，黄和周(2018b)将渐近可解性概念形式化，作为直接方法的一个例子，并给出了一个齐次代理人的LQ平均场对策的完整答案。这是通过发展多尺度方法和重新尺度程序来实现的。我们的目标是将这种方法扩展到更大范围的建模。不同结构的LQ模型将继续在我们的一些发展中发挥重要作用，因为它们在系统和控制理论中占据核心地位，并且在平均场控制中很重要(Bardi，2012；Huang等人，2007；Yong，2013)。另一个重要方向是混合参与者的社会优化，这不仅是数学上的兴趣，而且有实际背景(Chen，Busic等人，2017)，例如运营电网为大量个人用户(如住宅单位)服务。我们将进一步研究在具有限制的密集图的背景下具有空间交互的MFG；将结构化解作为降低计算和实现复杂性的手段的马尔可夫决策模型；以及当人类代理在平均场决策问题中交互时的主观性建模。

Since the inception of mean field game (MFG) theory in last decade (Caines, Huang, and Malhame 2003, 2006, 2007; Lasry and Lions, 2006, 2007), this area has evolved into a major scientific community with intense research activities, crossing the border of disciplines; see an overview in (Caines, Huang and Malhame, 2017). It provides a powerful tool to tackle the notorious dimensionality difficulty in large dynamics decision problems. The basic theory of MFGs has been built upon two fundamental approaches. The first, called the direct approach, starts by solving a large-scale game and derives a set of limiting equations as the population size tends to infinity (Lary and Lions, 2007). The second approach applies mean field approximations and formalizes a fixed point problem for a representative player (Huang et al, 2006, 2007). MFG theory has been further enriched by considering major players (Huang, 2010; Nourian and Caines, 2013; Bensoussan et al, 2015; Carmona and Zhu, 2016) or common noise (Cadaliaguet et al, 2015; Carmona and Delarue, 2018), which leads to stochastic rather than deterministic mean field. Another significant extension is social optimization where a large number of agents cooperatively optimize their aggregate cost (Huang et al, 2012; Lacker, 2017). Although in the past decade mean field control theory together with applications has undergone a phenomenal growth leading to the formation of a core research community, this area is still quickly evolving due to its sheer richness. This research program will investigate important frontier topics of this area. First, a fundamental question is about the relation of the two approaches of MFG theory. Recently Huang and Zhou (2018b) formalizes an asymptotic solvability notion as an instance of the direct approach and gives a complete answer for an LQ mean field game of homogeneous agents. This is accomplished by developing a multi-scale method and a re-scaling procedure. We aim to extend this approach to much larger scopes of modeling. LQ models of different structures will continue to have an important role in some of our developments since they occupy a central place in systems and control theory and are important in mean field control (Bardi, 2012; Huang et al, 2007; Yong, 2013). Another important direction is social optimization with mixed players, which not only are of mathematical interest but have practical backgrounds (Chen, Busic, et al, 2017) such as operating the power grid to service a large number of individual users (like residential units). We will further investigate MFGs with spatial interactions in the context of dense graphs with limits called graphons; Markov decision models with structured solutions as a means to reduce computational and implementational complexity; and subjectivity modeling when human agents interact in mean field decision problems.