2018 AFOSR MURIInnovations in Mean-Field Game Theory for Scalable Computation and Diverse ApplicationsPO: Dr. Fariba Fahroo, Computational Mathematics: comp.math@us.af.mil Lead PI: Stanley Osher (UCLA, Mathematics)Website: TBD
Mean field games (MFGs) study strategic decision making in large populations where the individual players interact via certain mean-field quantities. MFGs are studied by taking the limit of infinitely many individual players and replacing individual interactions by an average or effective interaction. Each agent in an MFG decides his/her strategies continuously in time to minimize his/her cost-to-go. Their strategies influence the controlled dynamics, which may be perturbed by the noise of both the individuals and the population. The simplest MFG is a (stochastic) optimal control problem in which the payoff function and the system dynamics depend on mean-field variables, which are the averages (density) of all individual quantities.
The Nash equilibrium (NE) of an MFG refers to a consensual state in which no player has any incentive to change his/her strategy unilaterally. The NE is mathematically described by a pair of backward-forward equations. The (forward) Fokker-Planck equations represents the population's strategy, and the (backward) Hamilton-Jacobi (HJ) equations describe their velocities (strategy changing rate).
MFGs have recently gained enonnous popularity. They provide powerful tools ranging from the trace of quantum mechanics (via the Schrodinger equation) to biodiversity ecology and also have a great impact on the evolution of social cooperation, macroeconomics, stock markets, and wealth distribution, and on the evolution of biological systems.
Despite successful applications of the MFG theory in some areas, its theory, modeling, and computation are far from being mature. Current challenges include, but are not limited to, certain monotonicity assumptions in the theory, the strategy sets commonly being assumed to be continuous, and the numerical difficulty due to nonconvexity, nonsmoothness, and the curse of dimensionality.
This MURI team consists of four universities: UCLA (lead), U. Houston, U. Maryland, and Princeton U. The team believes real progress on MFGs requires a coordinated effort based on an overarching mathematical framework, accurate modeling in diverse applications, and a set of algorithmic tools that support fast simulation, prediction, and inverse design. Therefore, this project assembles a multidisciplinary team of eight experts from pure and applied mathematics, computer science, psychology, and electrical engineering who have rich experience in social and economic topics and will work systematically and in close collaboration. The previous works of this team are well connected to these challenges. Subsets of the (co)Pis have co-authored multiple papers and books on related topics. In this project, their work includes the following list of key components: (i) A finite-state (e.g., player strategies form a discrete/graph set) MFG framework;
(ii) MFGs with non-physical interactions, multiple scales, and multiple populations; (iii) Explanation and prediction of long-time population behaviors and non-equilibrium behavior; (iv) MFG inverse problems; (v) Fast, robust, and scalable numerical schemes for both MFGs and MFG inverse problems; (vi) data collection and experimental validation.
Their efforts will result in accurate modeling of knowledge evolution and opinion formation in social networks, social and culture norm dynamics, socio-economical dynamics in energy consumption and crime modeling, election modeling, and other psycho-/socio mathematical models of rational and irrational agents.