Mean Field Games (May 5-7-12-13 2015)
Abstract
Mean-field games were introduced by Lasry and Lions and by Huang, Caines and Malhamé ten years ago. Their purpose is to describe asymptotic Nash equilibria within large population of controlled agents interacting with one another through the empirical distribution of the system. When players are driven by similar dynamics and subject to similar cost functionals, asymptotic equilibria are expected to obey some propagation of chaos, limiting the analysis of the whole population to the analysis of one single player and thus reducing the complexity.
The general program of the lectures is twofold. A first objective is to address the existence and possibly uniqueness of solutions to mean-field games or, equivalently, of equilibria to the asymptotic formulation. A second one is to justify the convergence of the equilibria of the finite-player-games to the solution (if unique) of the corresponding mean-field game, which, quite surprisingly, turns out to be a much more challenging question than it might seem at first sight.
Lecture 1. Forward-Backward Stochastic Differential Equations (PDF)
(May 5, 10:00-12:00, CDT Lecture Room 3)
In this first lecture, I will present forward-backward stochastic differential equations and the connection with optimal stochastic control. I will focus on two examples: (i) first, the interpretation of Hamilton-Jacobi-Bellman equation satisfied by the value function of the optimal control problem; (ii) second, the stochastic Pontryagin principle. I will discuss the main strategies for solving such a kind of equations.
Lecture 2. Introduction to Mean-Field Games: Examples (PDF)
(May 7, 10:00-12:00, CDT Lecture Room 3)
In this second lecture, I will present the concept of Mean-Field Games (MFG). I will start with Nash equilibria among a finite population of agents. Then I will explain how to define a consensus in infinite population of agents interacting with one another in a mean-field way. For that purpose, I will recall what the propagation of chaos is. Then, I will detail some simple examples coming from economy or finance where the consensus can be explicitly found.
Lecture 3. Solvability of Mean-Field Games (PDF)
(May 12, 10:00-12:00, CDT Lecture Room 3)
In the third lecture, I will address the question of solvability of MFG. I will interpret the solution of a MFG as the solution of a FBSDE of McKean-Vlasov type. I will show how to prove existence by means of Schauder's theorem and present the Lasry Lions monotonicity conditions for uniqueness. Then, I will switch to the more difficult case when the players are subject to a common source of noise (in addition to idiosyncratic noises) and revisit existence and uniqueness in that extended framework.
Lecture 4. Master Equation (PDF)
(May 13, 14:00-16:00, CDT Lecture Room 3)
To finish with, I will show that MFG can be characterized by means of a PDE (called master equation), set on the space of probability measures. I will start with simple examples and then expose a more general strategy for solving the master equation. I will discuss extensions to the case when the players are subject to a common noise and also show that the master equation is a very useful tool for investigating approximated equilibria.
References
- RENÉ CARMONA, FRANÇOIS DELARUE. Mean Field Forward-Backward Stochastic Differential Equations
- RENÉ CARMONA, FRANÇOIS DELARUE. Probabilistic Analysis of Mean-Field Games
- J.-F. CHASSAGNEAUX, D. CRISAN, F. DELARUE. Classical solutions to the master equation for large population equilibria. arXiv:1411.3009
- M. HUANG, P.E. CAINES, and R.P. MALHAME ́. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Communications in Information and Systems, 6:221–252, 2006.
- J.-M. LASRY and P.-L. LIONS. Mean field games. Japanese Journal of Mathematics, 2(1), Mar. 2007.
Contact us
CFM-Imperial Institute of Quantitative Finance
Department of Mathematics,
Imperial College
London
SW7 1NE
Email: iqf-events@imperial.ac.uk