Mathematics Think Tank

Mean Field Games and Applications

Olivier Guéant, Jean-Michel Lasry, Pierre-Louis Lions at Paris-Princeton Lectures on Mathematical Finance, 2011

This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three au- thors and upon unpublished materials developed by the authors. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.

The content of this text is therefore far more important than the actual “Cours Bachelier” conferences, though the guiding principle is the same and consists in a progressive introduction of the concepts, methodologies and mathematical tools of mean field games theory.

Mean field games theory was created in 2006 by Jean-Michel Lasry and Pierre- Louis Lions and the first results and developments are given in the publications [34, 35, 36]: structures, concepts, definitions of equilibria, forward-backward Hamilton-Jacobi-Bellman/Kolmogorov equation systems, existence theorems in static and dynamic cases, links with Nash equilibria and dynamics in n-player games theory when n tends to infinity, variational principle for decentralization, etc. A number of developments were then implemented by Jean-Michel Lasry and Pierre-Louis Lions, several of them in collaboration with Olivier Gu ́eant: notions of stability of solutions, specific numerical methods, numerical educ- tive algorithms, and developments in 1/n for a better approximation to n-player games. These developments were presented in three successive courses at the Coll`ege de France [38], in a Bachelier course, in various publications [23, 24] and in Olivier Gu ́eant’s PhD thesis [23]. Various applications, notably on the economics of scarce resources, were implemented or are ongoing (in collabo- ration: Pierre No ̈el Giraud, Olivier Gu ́eant, Jean-Michel Lasry, Pierre-Louis Lions). Advances in population dynamics were made by Olivier Gu ́eant [23]. Since 2008, several other authors have made further contributions, or are work- ing on new applications and/or properties of MFG models [33, 21].

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