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

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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.**