CAREER: Mean Field Games with Economics Applications: New Techniques in Partial Differential Equations

平均场博弈是理性主体之间大规模相互作用的数学模型。它们可以用来解释经济中的复杂现象，如商业周期和不平等，但我们并不完全理解它们背后的数学理论。该项目将在微分方程领域开发新的数学技术，以确定经济学中出现的平均场博弈的理论性质。这项研究将通过确定哪些数学模型是可行的，以及它们的解决方案如何运行，从而有助于对经济学以及公共政策的理性讨论。此外，这项研究还与跨学科教育和外展计划相结合，通过展示数学和社会动态之间的联系来吸引公众。该项目还将在经济学和政治学教师的合作下，开发课程和研讨会，利用经过测试的做法将数学教学与社会科学相结合。该项目的研究和教育方面都将为研究生和本科生提供培训，从而有助于培养一支多元化的、具有全球竞争力的STEM劳动力队伍。平均场博弈理论利用偏微分方程组为经济学提供了有用的数学模型，但由于非局部和随机相互作用的存在，使得分析变得困难。这个项目将开发偏微分方程的新技术，以确定这些模型是否适定。第一个目标是证明描述宏观经济现象的非线性积分-微分方程组的前向-后向耦合系统的新结果。第二个目标是证明模拟经济冲击的无限维方程的新结果。为了实现这些目标，首席研究人员将基于非局部方程、向前-向后系统和平均场游戏的主方程的最新进展开发新的分析技术。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Mean field games are mathematical models of large-scale interactions between rational agents. They can be used to explain complex phenomena in the economy, such as business cycles and inequality, but we do not fully understand the mathematical theory behind them. The project will develop new mathematical techniques in the field of differential equations to determine the theoretical properties of mean field games that arise in economics. This research will contribute to a rational discussion of economics, and by extension public policy, by determining which mathematical models are viable and how their solutions behave. Moreover, the research is integrated with an interdisciplinary education and outreach plan, engaging the public by showing the connection between mathematics and social dynamics. The project will also develop courses and seminars that use tested practices to integrate mathematics instruction with the social sciences, with the cooperation of instructors in economics and political science. Both research and education aspects of the project will provide training to students at graduate and undergraduate levels, thus contributing to the development of a diverse, globally competitive STEM workforce. Mean field game theory provides useful mathematical models in economics using coupled systems of partial differential equations, but their analysis is rendered difficult by the presence of nonlocal and stochastic interactions. This project will develop new techniques for partial differential equations to determine whether these models are well-posed. The first objective is to prove new results for forward-backward coupled systems of nonlinear, integro-differential equations that describe macroeconomic phenomena. The second objective is to prove new results for infinite dimensional equations that model economic shocks. To achieve these objectives, the principal investigator will develop new analytical techniques based on recent advances in nonlocal equations, forward-backward systems, and the Master Equation for mean field games.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.