Mean-Field Games: Theory, Numerics and Applications

Abstract

In dynamical systems with a large number of agents, competitive, and cooperative phenomenaoccur in a broad range of designed and natural settings. Such as communications, environmental,biological, transportation, trading, and energy systems, and they underlie mucheconomic and financial behavior. Analysis of such systems is intractable using the classicalfinite N-players game theoretic methods is often intractable. The mean-field games (MFG)framework was developed to study these large systems, modeling them as a continuum ofrational agents that interact in a non-cooperative way.In this thesis, we address some theoretical aspects and propose a definition of relaxedsolution for MFG that allows establishing uniqueness under minimal regularity hypothesis.We also propose a price impact model, that is a modification of the Merton’s portfolio problemwhere we consider that assets’ transactions influence their prices.We also study numerical methods for continuous time finite-state MFG that satisfy amonotonicity condition, and for time-dependent first-order nonlocal MFG. MFG is determinedby a system of differential equations with initial and terminal boundary conditions. Thesenon-standard conditions make the numerical approximation of MFG difficult. Using themonotonicity condition, we build a flow that is a contraction and whose fixed points solveboth for stationary and time-dependent MFG.We also develop Fourier approximation methods for the solutions of first-order nonlocalmean-field games (MFG) systems. Using Fourier expansion techniques, we approximate agiven MFG system by a simpler one that is equivalent to a convex optimization problem overa finite-dimensional subspace of continuous curves. We solve this problem using a variant ofa primal-dual hybrid gradient method.Finally, we introduce a price-formation model where a large number of small players canstore and trade electricity. Our model is a constrained MFG where the price is a Lagrangemultiplier for the supply versus demand balance condition. We establish the existence of aunique solution using a fixed-point argument. Then, we study linear-quadratic models thathold specific solutions, and we find that the dynamic price depends linearly on the instantaggregated consumption.