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thors and upon unpublished materials developed by the authors.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 eld games theory.
Mean eld games theory was created in 2006 by Jean-Michel Lasry and Pierre-
Louis Lions and the rst results and developments are given in the publications
[34, 35, 36]: structures, concepts, de nitions 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 in nity, 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, speci c 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 Noel 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].
1 Introduction to mean eld games
Mean eld games theory is a branch of game theory. It is therefore a set of con-
cepts, mathematical tools, theorems, simulation methods and algorithms, which
like all game theory is intended to help specialists model situations of agents
who take decisions in a context of strategic interactions. These specialists, as
in other areas of game theory, will probably be economists, micro- or macro-
economists and, given the speci cities of mean eld games theory, possibly also
sociologists, engineers and even architects or urban planners. In any case, this
view of the eld of application emerges, we feel, from the panorama created by
the rst \toy models" presented in this text.
We choose the term \toy models" to indicate the particular status of game the-
ory and of many \examples" of it. Consider the famous \prisoner's dilemma".
Nobody thinks of taking the story literally, nor that this example was created
to be applied to the real-life situations it is supposed to evoke. In fact it is a
fable intended to introduce an archetype of strategic interaction: an archetype
that can thus be recognized in many negotiation situations in business life and
elsewhere. Many of our examples have a similar status. \What time does the
meeting start?" or the \Mexican wave equation" should not be taken literally,
as a desire to scienti cally model these situations in social life. Even if there is
clearly an element of truth in our models for these two examples, we believe that
the interest for the reader is primarily in the structure that is indicated through
these \toy models". The Mexican wave equation, for example, shows how a
sophisticated propagation phenomenon in social space can be constructed from
non-cooperative individual behaviors in a rational expectation context, once a
certain taste for imitation is present in agents' utility function.
Introducing mean eld games through these \toy models" is also a way of lead-
ing the reader to progressively discover the concepts and the mathematics of
mean eld games theory.