The theory of Mean Field Games, introduced among others by Lasry and Lions a little bit more than 10 years ago, is an emerging topic in applied mathematics, and has connections with game theory, fluid mechanics, PDEs, calculus of variations and optimal transport. It describes the evolution of a population, where each agent has to choose the strategy (i.e., a path) which best fits his preferences, but is affected by the others through a global mean field. A differential game is considered, with a continuum of players, all indistinguishable and all negligible. In the most typical case we face a congestion game (agents try to avoid the regions with high concentrations), and we look for a Nash equilibrium, which can be translated into a system of PDEs.
Contents:
Introduction to congestion games Notion of Nash equilibria, Nash equilibria for non-atomic games, potential games, price of anarchy, examples, traffic congestion, Wardrop equilibria on networks and on domains.
Introduction to Mean Field Games Optimal control, value funcion, Hamilton-Jacobi-Bellmann equation, continuity equation, definitions of equilibria in terms of optimal trajectories and MFG system.
Existence of equilibria for regular MFG The case of non-local dependence in the density, approach via measures on the space of trajectory, proof of existence via the Kakutani fixed-point theorem, recovering the MFG system.
Some tools from optimal transport The Monge and Kantorovich problems, Kantorovich duality, Brenier theorem, Wassersein distances, curves in the Wasserstein spaces, Benamou-Brenier formulation.
Variational MFG - the MFG system Formal derivation of the MFG system as a primal-dual condition, proof of duality, conditions for the existence of a dual solution.
Variational MFG - trajectory equilibria Optimization in terms of measures on trajectories, optimality conditions, how to obtain optimality on each trajectory, the role of the maximal function
Regularity via duality in variational MFG. Examples of Sobolev regularity results via convex duality. Local H^1 regularity in time and space in variational MFG.
$L^\infty$ estimates for quadratic MFG Tools from optimal transport: geodesic convexity and flow interchange. Time discretization of a quadratic variational MFG. $L^p$ and $L^\infty$ estimates.
Uniqueness Conditions to guarantee uniqueness of the equilibria: monotonicity, strict convexity.
ODEs and PDEs in the space of measures Interpretation of the MFG system as a Pontryagin principle. Value functions in differential games. The Master equation. Convergence of the N-players game to the MFG as $N\to \infty$.
Variant - density-constrained MFG Variational and non-variational models, connections with incompressible fluid mechanics and crowd motion. Regularity of the pressure.
Variant - minimal-time MFG A non-variational variant of exit in minimal time, relations with the Hughes model for pedestrian motion. Existence of an equilibrium and estimates.
ECTS Credits:
Prerequisites:
MA1001 Analysis 1, MA1002 Analysis 2, MA2003 Measure and Integration, MA3001 Functional Analysis. Suggested optional: MA2504 Convex Optimization, MA3005 Partial Differential Equations, MA3504 Convex Analysis. The parallel course by G.Carlier complements this lecture nicely, attendance to that parallel course is recommended but not necessary for successful completion of this course.