Mathematics

The existence of Nash equilibria in the Mean Field Game (MFG) theory of large non-cooperative populations of stochastic dynamical agents is established by passing to the infinite population limit. Individual agent feedback strategies are obtained via the MFG equations consisting of (i) a McKean-Vlasov-Hamilton-Jacobi-Bellman equation generating the Nash values and the best response control actions, and (ii) a McKean-Vlasov-Fokker-Planck-Kolmogorov equation for the probability distribution of the state of a generic agent in the population, otherwise known as the mean field. The applications of MFG theory now extend from economics and finance to epidemiology and physics.

In current work, MFG and MF Control theory is extended to Graphon Mean Field Game (GMFG) and Graphon Mean Field Control (GMFC) theory. Very large scale networks linking dynamical agents are now ubiquitous, with examples being given by electrical power grids, the internet, financial networks and epidemiological and social networks. In this setting, the emergence of the graphon theory of infinite networks has enabled the formulation of the GMFG equations for which we have established the existence and uniqueness of solutions. Applications of GMFG and GMFC theory to systems on particular networks of interest are being investigated and computational methods developed. As in the case of MFG theory, it is the simplicity of the infinite population GMFG and GMFC strategies which, in principle, permits their application to otherwise intractable problems involving large populations on complex networks. Work with Minyi Huang

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.