• aggregation
• coupled PDE
• growth model
• Mean field games
• rationality

Mean field games is a branch of game theory that deals with large symmetrical games.

Born in 2006, it is now a lively research topic that poses tough mathematical challenges and already features a broad range of applications.
Among the challenges, a system of coupled PDE plays the major role, while current mean field games models deal with crowds behaviour, traffic flows, macroeconomics dynamics and cancer evolution.

Our work has a dual aim: to highlight the core intuitions of this approach and to present some of its prospects as an applied tool. To do this, we merge the Mean Field Games analytic backbone by P. Cardaliaguet with some of the applications by O. Guéant, J-M. Lasry and P-L. Lions.

The first part of our work introduces the main intuitions behind the mean field game solution, such as the limit game definition, the forward-backward reasoning and the fixed point nature of the mean field. We then solve a {single-shot} game via the mean field approach and present a numerical simulation of the results.

The second part shows how the same steps can be employed in the continuous time case to obtain the coupled PDE formulation. In this general problem setting, we prove an important existence and uniqueness theorem for the mean field solution and characterize this latter as an epsilon-Nash equilibrium.

In the last part we focus on the prospects for this recent theory, presenting some economics related applications. After a brief model of imitative preferences, we discuss an heterogeneous agents, human capital based growth model, highlighting how easily real-like scenarios can give birth to new unconventional PDE systems.