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This volume provides an introduction to the theory of Mean Field Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a mean-field model for Nash equilibria in the strategic interaction of a large number of agents. Besides giving an accessible presentation of the main features of mean-field game theory, the volume offers an overview of recent developments which explore several important directions: from partial differential equations to stochastic analysis, from the calculus of variations to modeling and aspects related to numerical methods. Arising from the CIME Summer School "Mean Field Games" held in Cetraro in 2019, this book collects together lecture notes prepared by Y. Achdou (with M. Laurière), P. Cardaliaguet, F. Delarue, A. Porretta and F. Santambrogio. These notes will be valuable for researchers and advanced graduate students who wish to approach this theory and explore its connections with several different fields in mathematics.

This volume is based on lectures delivered at the 2020 AMS Short Course "Mean Field Games: Agent Based Models to Nash Equilibria," held January 13-14, 2020, in Denver, Colorado.Mean field game theory offers a robust methodology for studying large systems of interacting rational agents. It has been extraordinarily successful and has continued to develop since its inception. The six chapters that make up this volume provide an overview of the subject, from the foundations of the theory to applications in economics and finance, including computational aspects.The reader will find a pedagogical introduction to the main ingredients, from the forward-backward mean field game system to the master equation. Also included are two detailed chapters on the connection between finite games and mean field games, with a pedestrian description of the different methods available to solve the convergence problem. The volume concludes with two contributions on applications of mean field games and on existing numerical methods, with an opening to machine learning techniques.

ABSTRACTThis paper analyses the consequences of young researchers' scientific choice on the dynamics of sciences. We develop a simple two state mean field game model to analyse the competition between two paradigms based on Kuhn's theory of scientific revolution. The dynamics of the model are driven by the scientific choice of young researchers at the beginning of their career. Despite the possibility of multiple equilibria, the model exhibits at least one stable solution in which both para digms coexist. The occurrence of shocks on the parameters may induce the shift from one paradigm to the other. During this shift, researchers' choice is proved to be having a great impact on the evolution of sciences.