Mean field games

Abstract

[en] In recent years, at the interface of game theory, control theory and statistical mechanics, a new baby of applied mathematics was given birth. Now named mean-field game theory, this new model represents a new active field of research with a huge range of applications.

Inspired by ideas from statistical particle physics particles are replace by agents with strategic interactions.

Consider an N-player stochastic dynamic game, thus we can consider that is a MFG when $N \rightarrow \infty$. It comes into a system of coupled equations:

Fokker-Plank

Hamilton-Jacobi-Bellman.

This ungraduate thesis tryes to introduce the tools that MFG need to be understood, it is given mostly whith analysis tools, it does not go deep into functional analysis but it is enogh to introduce the main ideas from this field.

It will begin from the derivation of HJB equation from the Dynamic Programing Principle of Bellman and then adapt it into the stochastic case. The following chapters will derive the Fokker - Plank equation and give an heuristic interpretation. Finally it will study a very simple model and some variation to illustrate the the main idea from MFG.