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

自过去十年中平均场博弈（MFG）理论诞生以来（Caines，Huang和Malhame 2003，2006，2007; Lasry和Lions，2006，2007），这一领域已发展成为一个主要的科学社区，具有激烈的研究活动，跨越学科边界;见概述（Caines，Huang和Malhame，2017）。它提供了一个强大的工具，以解决臭名昭著的维数困难的大型动态决策问题。MFG的基本理论建立在两种基本方法之上。第一种称为直接方法，首先解决一个大规模的游戏，并推导出一组极限方程，因为人口规模趋于无穷大（拉里和狮子，2007年）。第二种方法应用平均场近似并形式化代表性玩家的不动点问题（Huang et al，2006，2007）。通过考虑主要参与者（Huang，2010; Nourian和Caines，2013; Bensoussan等人，2015; Carmona和Zhu，2016）或共同噪声（Cadaliaguet等人，2015; Carmona和Delarue，2018），MFG理论得到了进一步丰富，这导致了随机而不是确定性平均场。另一个重要的扩展是社会优化，其中大量代理人合作优化其总成本（Huang et al，2012; Lacker，2017）。虽然在过去的十年中，平均场控制理论与应用一起经历了一个惊人的增长，导致核心研究社区的形成，这一领域仍然是迅速发展，由于其纯粹的丰富性。本研究计划将探讨这一领域的重要前沿课题。首先，一个基本问题是关于MFG理论的两种方法的关系。最近，Huang和Zhou（2018 b）将渐近可解性概念形式化为直接方法的一个实例，并给出了齐次代理的LQ平均场博弈的完整答案。这是通过开发一个多尺度的方法和重新缩放的程序。我们的目标是将这种方法扩展到更大的建模范围。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中心地位，并且在平均场控制中很重要（巴尔迪，2012;黄等人，2007;勇，2013）。另一个重要的方向是混合参与者的社会优化，这不仅是数学上的兴趣，而且具有实际背景（Chen，Busic等人，2017），例如运营电网以服务大量个人用户（如住宅单元）。我们将进一步研究MFG与空间相互作用的背景下，密集的图与限制称为graphons;马尔可夫决策模型与结构化的解决方案作为一种手段，以减少计算和实现的复杂性;和主观性建模时，人类代理人在平均场决策问题。

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.