Mathematics

In this talk, we consider epidemic models with a continuum of agents where the evolution of the epidemics is represented by an ODE system. In contrast with most of the existing literature, we allow the agents to make decisions and we incorporate game theoretical ideas in the model such as the notion of Nash equilibrium. When the population is homogeneous, this leads to a continuous time, finite state mean field game. We consider, from a mathematical viewpoint, mainly two questions: (1) How to find optimal public policies to reduce the impact of the epidemics while taking into account the agents' rational choices? (2) How to handle heterogeneities among the population while keeping a continuum of agents? For the first point, we use a Stackelberg mean field game model, while for the second point, we rely on the framework of graphon games. In each case, we develop numerical methods based on machine learning tools to efficiently compute approximately optimal solutions.

Numerous applications of mean-field games theory assume nonlocal interactions between agents. Although somewhat simpler from a mathematical analysis perspective, nonlocal models are often challenging for numerical solutions. Indeed, direct discretizations of mean-field interaction terms yield dense systems that are not economical from computational and memory perspectives. In this talk, I will discuss several options to mitigate the challenges above by importing methods from Fourier analysis and kernel methods in machine learning.

To go back to part 1 of the talk, click here: https://mathtube.org/lecture/video/nonlocal-interactions-mean-field-game...

Graphon control (CDC 17-18-19, IEEE TAC 20, Gao and Caines) and graphon mean field games (CDC18, CDC19, Caines and Huang) were used to address decision problems on very large-scale networks by employing graphons to represent arbitrary size graphs, from, respectively centralized and decentralized perspectives. Graphon couplings may be considered as a generalization of mean-field couplings with network heterogeneity. Such couplings may appear in states, controls and cost, and may be represented by different graphons in each case. In this talk, I will present the use of graphon spectral decomposition in graphon control and graphon mean field games in a linear quadratic setting. The complexity of the method does not directly depend on the number of agents or number of nodes, instead, it depends on the dimension of the characterizing graphon invariant subspace shared by the coupling operators.

n this paper, we study the optimal control problem arising from the mean-field game formulation of the collective decision-making in honeybee swarms. A population of homogeneous players (the honeybees) has to reach consensus on one of two options. We consider three states: the first two represent the available options (or strategies), and the third one represents the uncommitted state. We formulate the continuous-time discrete-state mean-field game model. The contributions of this paper are the following: i) we propose an optimal control model where players have to control their transition rates to minimize a running cost and a terminal cost, in the presence of an adversarial disturbance; ii) we develop a formulation of the micro-macro model in the form of an initial-terminal value problem (ITVP) with switched dynamics; iii) we study the existence of stationary solutions and the mean-field Nash equilibrium for the resulting switched system; iv) we show that under certain assumptions on the parameters, the game may admit periodic solutions; and v) we analyze the resulting microscopic dynamics in a structured environment where a finite number of players interact through a network topology.

The Graphon Mean Field Game equations consist of a collection of parameterized Hamilton-Jacobi-Bellman equations, and a collection of parameterized Fokker-Planck-Kolmogorov equations coupled through a given graphon. In this talk, we will discuss the sensitivity of the gradient of HJB solutions with respect to the coefficients, which can be used for the solvability of Graphon Mean Field Game equation. It's based on the joint work with Peter Caines, Daniel Ho, Minyi Huang, and Jiamin Jian, see https://arxiv.org/pdf/2009.12144.pdf.