Stochastic Differential Mean Field Game Theory

Abstract

Mean field game (MFG) theory generalizes classical models of interacting particle systems by replacing the particles with rational agents, making the theory more applicable in economics and other social sciences. Intuitively, (stochastic differential) MFGs are infinite-population or continuum limits of large-population stochastic differential games of a certain symmetric type, and a solution of an MFG is analogous to a Nash equilibrium. This thesis tackles several fundamental problems in MFG theory. First, if (approximate) equilibria exist in the large-population games, to what limits (if any) do they converge as the population size tends to infinity? Second, can the limiting system be used to construct approximate equilibria for the finite-population games? Finally, what can be said about existence and uniqueness of equilibria, for the finite- or infinite-population models?

This thesis presents a complete picture of the limiting behavior of the large-population

systems, both with and without common noise, under modest assumptions on the model

inputs. Approximate Nash equilibria in the n-player games admit certain weak limits as n

tends to innity, and every limit is a weak solution of the MFG. Conversely, every weak

MFG solution can be obtained as the limit of a sequence of approximate Nash equilibria

in the n-player games. Even in the setting without common noise, a new solution concept

is needed in order to capture all of the possible limits. Interestingly, and in contrast with

well known results on related interacting particle systems, empirical state distributions often admit stochastic limits which are not simply randomizations among the deterministic solutions.

With the limit theory in mind, the thesis then develops new existence and uniqueness

results. Using controlled martingale problems together with relaxed controls, a general

existence theorem is derived by means of Kakutani's xed point theorem. In the common

noise case, a natural notion of weak solution is introduced, and the existence and uniqueness theory is designed in perfect analogy with weak solutions of stochastic differential equations. An existence theorem for weak solutions is proven by a discretization procedure, and a Yamada-Watanabe result is presented and illustrated under some stronger assumptions which ensure pathwise uniqueness.