Interacting agent systems in optimal control, game theory, and beyond

In an increasingly connected world in which individuals have access to unprecedented amounts of data, models of large, complex systems of interacting agents are ubiquitous, with examples including economic systems, telecommunication grids, social networks, and generative artificial intelligence (AI) models. Traditional approaches to studying such systems, which tend to exhibit statistical uncertainty and high dimensionality, break down as the number of agents grows large, rendering the relevant tasks of decision makers numerically intractable. The investigator will develop a thorough mathematical understanding of the equations arising in these settings by developing innovative approaches to optimal control and game theory. The findings will be disseminated widely across scientific communities and the general public, and the techniques developed in the project will inform applications in a broad range of scientific, technological, and economic disciplines. This project will generate opportunities to train and mentor graduate and undergraduate students and lead to community outreach activities involving mathematical and technological literacy. A quantitative understanding of large interacting agent systems will be achieved through the use of mean field models, which allow for the study of a relatively smaller collection of equations that nevertheless require a sophisticated mathematical treatment. The first main purpose of the project is to employ mean field techniques in order to predict the behavior of large player populations in games in which several game equilibria can be witnessed, which, analytically, translates into singularities in the relevant equations. This thrust of the project will therefore lead to the development of advanced theories of partial differential equations and stochastic analysis, as well as machine learning algorithms. The second category of activity will involve the use of the methodologies developed in the first part of the project to further the understanding of mean field optimal control problems with one or more central planners. Quantitative estimates with a high degree of precision will be developed, imparting practical justification to the application of mean field models in decision-making contexts. The results will also allow for the study of related optimization problems arising in material science and statistical physics. 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.