Asymptotics for mean field games of market competition

The goal of this thesis is to analyze the limiting behavior of solutions to a system of mean field games developed by Chan and Sircar to model Bertrand and Cournot competition. We first provide a basic introduction to control theory, game theory, and ultimately mean field game theory. With these preliminaries out of the way, we then introduce the model first proposed by Chan and Sircar, namely a coupled system of two nonlinear partial differential equations. This model contains a parameter ε that measures the degree of interaction between players; we are interested in the regime ε goes to 0. We then prove a collection of theorems which give estimates on the limiting behavior of solutions as ε goes to 0 and ultimately obtain recursive growth bounds of polynomial approximations to solutions. Finally, we state some open questions for further research.