A Mean Field Game (MFG) is the limit of an N-player noncooperative differential game with mean field interaction. The players are assumed to be symmetric and statistically homogeneous. The latter means that any individual differences are negligible when N gets large. In addition, it is typically assumed that no player can see the unique private states of its competitors. Instead, each player experiences only the average of these states at each point in time t, which is often modeled using empirical probability measures. This probability measure, which is nothing more than the weighted sum of Dirac measures centered at the individual players‘ states, is called the mean field interaction.

Due to the nonlinear interaction between the players, these N-player games become numerically intractable for large N. Sometimes we are able to derive a limiting game, which can be characterized by the interaction of a single representative agent with a transport equation whose solution represents a macroscopic approximation of the original mean field interaction term. For large N, the solution of the individual agent problems using the macroscopic mean field term, in the form of an absolute continuous curve of probability measures, yields optimal open loop controls that are close to the original one. This property motivates mean field games.

Mean field games were introduced by Lasry and Lions in 2007 and independently by Huang, Malhamé, and Caines. They have a wide range of applications, i.e., evolutionary biology, macroeconomics, crowd behavior, and swarm robotics.

The goal of this project is to analyze „constrained“ Mean Field Games. In contrast to most of the literature, we add some constraints on the N-player differential game, in particular on the decisions spaces and the states themselves. This is very natural, if you consider that birds cannot accelerate infinitely quickly nor can crowds pass through walls. However, constraints add a significant source of difficulty to the problem, as the resulting mean field limit can longer be represented by coupled partial differential equations. Instead one or several variational inequalities may arise.

The goals of the project include both theoretical as well as numerical questions. On the one hand, we seek to rigorously derive a „constrained“ MFG for various types of constraints typical for optimal control. Furthermore, we want to show the existence of a solution for both the MFG and the N-player differential game. On the other hand, we suggest several numerical schemes to efficiently solve these problems. Additional generalizations will be considered. For example, nonsmooth objective functions, sparse control, and ideally stochastic dynamics.