Mean Field Models with Heterogeneous Agents: Extensions and Learning

Abstract

We first study the extensions of mean field game and control models with heterogenous players (i.e., major players) with applications to socioeconomic problems. Later, in order to solve Stackelberg mean field game and graphon game models that incorporate realistic features, we propose novel numerical approaches. In the first part, first, we introduce two different equilibrium notions for the mean field games with multiple major players. We compare these equilibrium notions and show the existence and uniqueness results in an application to one-shot advertisement competition in a duopoly where a mean field population of consumers are also included in the model. Second, we look at mean field game and control models with a regulator where mean field population of agents choose both time dependent and time independent controls. In our motivating example, the mean field population corresponds to the electricity producers and later, we add a regulator to find the optimal carbon tax levels. We propose nonstandard forward backward stochastic differential equation systems that characterize the equilibrium and social optimum in the mean field population, and we show the existence and uniqueness of the solutions. In the second part, we focus on learning the optimal policies and equilibrium behavior in large population games with the motivation of mitigating the epidemics. First, we develop the theory for a Stackelberg extended mean field game between a principal (i.e., the government) and a mean field population of identical agents evolving on a finite state space that gives the possible health conditions. The agents play a noncooperative game in which they control their transition rates between states to minimize their individual costs by choosing their socialization levels. The principal influences the Nash equilibrium in the population through policies to optimize its own objective. Second, we consider a graphon game to add heterogeneities among agents and show the existence of the solution of a continuum of forward backward ordinary differential equations that characterize the finite state graphon game equilibrium. We propose numerical approaches that are based on machine learning and Monte Carlo simulation to solve both Stackelberg mean field game and graphon game models.