A Finite Dimensional Approximation of a Density Dependent Mean Field Game

The goal of this thesis is to establish the existence and uniqueness of a Nash equilibrium of a density dependent mean field game and approximate the solution with numerical methods. We first briefly introduce both mean field game theory and measure theory. Next, we define a game in which the final cost is the density of the equilibrium measure. Then we prove a unique solution exists by using the Browder-Minty Theorem. To conclude, we will show how Newton’s method can be used to approximate a solution and look at some specific examples of this approximation in action.