Equilibria in Large Populations: Asymmetric Mean Field Games and Optimal Control

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Since its inception about a decade ago, the theory of mean field games has developed into an important source of progress in the study of the dynamical and equilibrium behavior of large systems. The introduction of ideas from statistical physics in the search for approximate equilibria for sizable dynamic systems created new interest in the study of large populations of competitive individuals with "mean field" interactions. The search for equilibria in a mean field game is an attempt to identify and compute global statistics (macroscopic behavior) of large populations of rational individuals who optimize their own cost/reward trade-offs (microscopic behavior). The challenge is to understand and quantify the way the behavior of individuals affects the overall conduct of the population. Propagation of chaos and rational irrationality are terms that have been used to describe the possible macroscopic impact of individual choices at the microscopic level. This broad-brush description has its parallel in the structure of the mathematical models and analytical tools used to describe these phenomena. This project aims at a better understanding of the mathematics underpinning the models. Special emphasis is put on realistic models characterized by the presence of a small number of major players and a large number of minor players interacting in a mean field manner. These models have important applications in population biology (bee swarming, bird flocking, and crowd motion), biology (circadian rhythm synchronization), sociology, behavioral economics, and finance.The project focuses on a class of equilibrium problems for which symmetry is broken by the presence of two groups of individuals having conflicting interests. These models can be cast as mean field games with major and minor players. Special attention will be given to potential models that can be solved as the optimal control of McKean-Vlasov stochastic systems. One goal of the project is the development of numerical algorithms for the computation of Nash equilibria, hence the focus on models with finite state spaces. The second part of the project is devoted to the analysis of the recently-introduced mean field games of timing and again, the extension to models with major and minor players is a significant part of the research agenda. Finally, the project will compare the body of work on mean field games with the strand of economic literature concerning games with a continuum of players.