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

平均场博弈理论自10多年前诞生以来，已经发展成为研究大系统动力学和平衡行为的一个重要来源。从统计物理学中引入的思想在寻找近似平衡的大型动态系统创造了新的兴趣在研究大人口的竞争性个人与“平均场”的相互作用。在平均场博弈中寻找均衡是试图识别和计算大量理性个体的全局统计（宏观行为），这些个体优化了自己的成本/回报权衡（微观行为）。挑战在于理解和量化个人行为影响人口整体行为的方式。混沌的传播和理性的非理性是用来描述个人选择在微观层面上可能产生的宏观影响的术语。这种粗略的描述在用于描述这些现象的数学模型和分析工具的结构中也有其平行之处。该项目旨在更好地理解支撑模型的数学。 特别强调的是现实的模型，其特点是存在少量的主要球员和大量的未成年人的球员在平均场的方式进行交互。这些模型在种群生物学（蜜蜂群集、鸟类群集和群体运动）、生物学（昼夜节律同步）、社会学、行为经济学和金融学中有重要的应用。该项目关注一类均衡问题，其中对称性被两组具有冲突利益的个体打破。这些模型可以被铸造成与主要和次要玩家的平均场游戏。将特别注意潜在的模型，可以解决的McKean-Vlasov随机系统的最优控制。该项目的一个目标是发展用于计算纳什均衡的数值算法，因此重点关注有限状态空间的模型。该项目的第二部分是专门分析最近推出的平均场游戏的时间和再次，扩展到模型与主要和次要的球员是研究议程的重要组成部分。最后，该项目将比较机构的平均场游戏的工作与经济学文献链有关的游戏与连续的球员。

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.