## Research Papers on Mean Field Games

**Existence and uniqueness result for mean field games with congestion effect on graphs, Applied Mathematics and Optimization**,

**Volume 72, Issue 2**,

**October 2015**

*Abstract:*

*This paper presents a general existence and uniqueness result for mean field games equations on graphs. In particular, our setting allows to take into account congestion effects of almost any form. These general congestion effects are particularly relevant in graphs in which the cost to move from one node to another may for instance depend on the proportion of players in both the source node and the target node. Existence is proved using a priori estimates and a fixed point argument à la Schauder. We propose a new criterion to ensure uniqueness in the case of Hamiltonian functions with a complex (non-local) structure. This result generalizes the discrete counterpart of existing uniqueness results.*

**New numerical methods for mean field games with quadratic costs,**

**Networks and Heterogenous Media, Volume 7, Number 2, June 2012***Abstract: Mean field games have been introduced by J.-M. Lasry and P.-L. Lions as the limit case of stochastic dierential games when the number of players goes to infinity. In the case of quadratic costs, we present two changes of variables that allow to transform the mean field games (MFG) equations into two simpler systems of equations. The first change of variables, already introduced in a preceding paper, leads to two heat equations with nonlinear source terms. The second change of variables, which is introduced for the first time in this paper, leads to two Hamilton-Jacobi-Bellman equations. Numerical methods based on these equations are presented and numerical experiments are carried out.*

Mean field games equations with quadratic Hamiltonian: a specific approach, Mathematical Models and Methods in Applied Sciences (M3AS), Volume 22, Issue 9, September 2012

*Abstract:*

*Mean field games models describing the limit of a large class of stochastic differential*

*games, as the number of players goes to infinity, have been introduced by J.-M. Lasry and P.-L. Lions. We use a change of variables to transform the mean field games (MFG) equations into a system of simpler coupled partial differential equations, in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the MFG equations. Effective numerical methods based on this constructive scheme are presented and numerical experiments are carried out.*

**Mean Field Games with a Quadratic Hamiltonian: A Constructive Scheme,**

*in***Advances in Dynamic Games, Annals of the International Society of Dynamic Games, 2013.**

*Abstract: Mean field games models describing the limit case of a large class of stochastic differential games, as the number of players goes to infinity, were introduced by Lasry and Lions. We use a change of variables to transform the mean field games equations into a system of simpler coupled partial differential equations in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the mean field games equations.*

**Mean Field Games and Applications, with J.-M. Lasry and P.-L. Lions,**

*in*Paris-Princeton Lectures on Mathematical Finance 2010, Ed. Springer, January 2011*Abstract: 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 authors 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 and Oil Production, with J.-M. Lasry and P.-L. Lions,**

*in*The Economics of Sustainable Development, Ed. Economica, 2010*Abstract: In this paper we study the evolution of oil production in the long run. A first optimization model is presented, that can be solved using Euler-Lagrange tools. Because these classical tools are not the best suited to the model, we adopt a mean field games approach based on two partial differential equations. An extended model is then presented to analyze the influence of new competitors which might enter the market with energy from renewable sources. The usefulness of a subsidy to potential entrants is discussed.*

**A reference case for mean field games,**

**Journal de Mathématiques Pures et Appliquées, Volume 92, Issue 3, September 2009***Abstract: In this article, we present a reference case of mean field games. This case can be seen as a reference for two main reasons. First, the case is simple enough to allow for explicit resolution: Bellman functions are quadratic, stationary measures are normal and stability can be dealt with explicitly using Hermite polynomials. Second, despite its simplicity, the case is rich enough in terms of mathematics to be generalized and to inspire the study of more complex models that may not be as tractable as this one.*

**Tournament-induced risk-shifting: A mean field games approach**

**, Risk and Decision Analysis**

**, Volume 4, Issue 2, 2013***Abstract:*

**The agency problem between an investor and his mutual funds managers has long been studied in the economic literature. Because the very business of mutual funds managers is not only to manage money but also, and rather, to increase the money under management, one of the numerous agency problems is the implicit incentive induced by the relationship between inflows and performance. If the consequences of incentives, be they implicit or explicit – as for compensation schemes of individual asset managers – are well known in terms of risk-shifting when the incentives are linked to a benchmark, the very fact that the mutual fund market is a tournament does not seem to be modeled properly in the literature. In this paper, we propose a mean field games model to quantify the risk-shifting induced by a tournament-like competition between mutual funds.**