Table Of ContentProbability Theory and Stochastic Modelling 84
René Carmona
François Delarue
Probabilistic
Theory of Mean
Field Games with
Applications II
Mean Field Games with Common Noise
and Master Equations
Probability Theory and Stochastic Modelling
Volume 84
Editors-in-chief
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AndreasE. Kyprianou,Bath,UK
YvesLeJan,Orsay,France
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GeorgePapanicolaou,Stanford,CA,USA
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EdwinPerkins,Vancouver,Canada
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René Carmona • François Delarue
Probabilistic Theory of
Mean Field Games
with Applications II
Mean Field Games with Common Noise and
Master Equations
123
RenéCarmona FrançoisDelarue
ORFEDepartment InstitutUniversitairedeFrance
PrograminAppliedand &LaboratoireJ.A.Dieudonné
ComputationalMathematics UniversitéNiceSophiaAntipolis
PrincetonUniversity Nice,France
Princeton,NJ,USA
ISSN2199-3130 ISSN2199-3149 (electronic)
ProbabilityTheoryandStochasticModelling
ISBN978-3-319-56435-7 ISBN978-3-319-56436-4 (eBook)
DOI10.1007/978-3-319-56436-4
LibraryofCongressControlNumber:2017940847
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Foreword
Sinceitsinceptionaboutadecadeago,thetheoryofMeanFieldGameshasrapidly
developed into one of the most significant and exciting sources of progress in the
studyofthedynamicalandequilibriumbehavioroflargesystems.Theintroduction
of ideas from statistical physics to identify approximate equilibria for sizeable
dynamic games created a new wave of interest in the study of large populations
of competitive individuals with “mean field” interactions. This two-volume book
grew out of series of lectures and short courses given by the authors over the last
fewyearsonthemathematicaltheoryofMeanFieldGamesandtheirapplications
in social sciences, economics, engineering, and finance. While this is indeed the
object of the book, by taste, background, and expertise, we chose to focus on the
probabilisticapproachtothesegamemodels.
Inatrailblazingcontribution,LasryandLionsproposedin2006amethodology
to produce approximate Nash equilibria for stochastic differential games with
symmetric interactions and a large number of players. In their models, a given
playerfeelsthepresenceandthebehavioroftheotherplayersthroughtheempirical
distribution of their private states. This type of interaction was studied in the
statistical physics literature under the name of mean field interaction, hence the
terminology Mean Field Game coined by Lasry and Lions. The theory of these
new game models was developed in lectures given by Pierre-Louis Lions at the
Collège de France which were video-taped and made available on the internet.
Simultaneously,Caines,Huang,andMalhaméproposedasimilarapproachtolarge
games under the name of Nash Certainty Equivalence principle. This terminology
fellfromgraceandthestandardreferencetothesegamemodelsisnowMeanField
Games.
While slow to pick up momentum, the subject has seen a renewed wave of
interest over the last seven years. The mean field game paradigm has evolved
fromitsseminalprinciplesintoafull-fledgedfieldattractingtheoreticallyinclined
investigators as well as applied mathematicians, engineers, and social scientists.
The number of lectures, workshops, conferences, and publications devoted to the
subjecthasgrownexponentially,andwethoughtitwastimetoprovidetheapplied
mathematics community interested in the subject with a textbook presenting the
stateoftheart,asweseeit.Becauseofourpersonaltaste,wechosetofocusonwhat
v
vi Foreword
we like to call the probabilistic approach to mean field games. While a significant
portionofthetextisbasedonoriginalresearchbytheauthors,greatcarewastaken
toincludemodelsandresultscontributedbyothers,whetherornottheywereaware
ofthefacttheywereworkingwithmeanfieldgames.Sothebookshouldfeeland
readlikeatextbook,notaresearchmonograph.
Mostof thematerial and examples found in thetext appear for the firsttimein
book form. In fact, a good part of the presentation is original, and the lion’s share
oftheargumentsusedinthetexthavebeendesignedespeciallyforthepurposeof
thebook.Ourconcernforpedagogyjustifies(oratleastexplains)whywechoseto
dividethematerialintwovolumesandpresentmeanfieldgameswithoutacommon
noisefirst.Weeasetheintroductionofthetechnicalitiesneededtotreatmodelswith
a common noise in a crescendo of sophistication in the complexity of the models.
Also,weincludedattheendofeachvolumefourextensiveindexes(authorindex,
notationindex,subjectindex,andassumptionindex)tomakenavigationthroughout
thebookseamless.
Acknowledgments
First and foremost, we want to thank our wives Debbie and Mélanie for their
understandingandunwaveringsupport.Theintensityoftheresearchcollaboration
whichledtothistwo-volumebookincreaseddramaticallyovertheyears,invading
our academic lives as well as our social lives, pushing us to the brink of sanity at
times.Weshallneverbeabletothankthemenoughfortheirpatienceandtolerance.
This book project would not have been possible without them: our gratitude is
limitless.
Next we would like to thank Pierre-Louis Lions, Jean-Michel Lasry, Peter
Caines, Minyi Huang, and Roland Malhamé for their incredible insight in intro-
ducing the concept of mean field games. Working independently on both sides of
the pond, their original contributions broke the grounds for an entirely new and
fertile field of research. Next in line is Pierre Cardaliaguet, not only for numerous
private conversations on game theory but also for the invaluable service provided
by the notes he wrote from Pierre-Louis Lions’ lectures at the Collège de France.
Althoughtheywereneverpublishedinprintedform,thesenoteshadatremendous
impactonthemathematicalcommunitytryingtolearnaboutthesubject,especially
atatimewhenwritingsonmeanfieldgameswerefewandfarbetween.
Wealsoexpressourgratitudetotheorganizersofthe2013and2015conferences
onmeanfieldgamesinPadovaandParis:YvesAchdou,PierreCardaliaguet,Italo
Capuzzo-Dolcetta,PaoloDaiPra,andJean-MichelLasry.
While we like to cast ourselves as proponents of the probabilistic approach to
mean field games, it is fair to say that we were far from being the only ones
followingthispath.Infact,someofourpaperswerepostedessentiallyatthesame
timeaspapersofBensoussan,Frehse,andYam,addressingsimilarquestions,with
thesametypeofmethods.Webenefittedgreatlyfromthisstimulatingandhealthy
competition.
Foreword vii
Wealsothankourcoauthors,especiallyJean-FrançoisChasagneux,DanCrisan,
Jean-Pierre Fouque, Daniel Lacker, Peiqi Wang, and Geoffrey Zhu. We used our
jointworksasthebasisforpartsofthetextwhichtheywillrecognizeeasily.
Also, we would like to express our gratitude to the many colleagues and
studentswhogracefullytoleratedourrelentlesspromotionofthisemergingfieldof
researchthroughcourses,seminar,andlectureseries.Inparticular,wewouldliketo
thankJean-FrançoisChasagneux,RamaCont,DanCrisan,RomualdElie,Josselin
Garnier,MarcelNutz,HuyenPham,andNizarTouziforgivingustheopportunity
todojustthat.
Finally, we would like to thank Paul-Eric Chaudru de Raynal, Mark Cerenzia,
Christy Graves, Dan Lacker, Victor Marx, and Peiqi Wang for their help in
proofreadinganearlierversionofthemanuscript.
The work of R. Carmona was partially supported by NSF DMS-1211928 and
DMS-1716673, and by ARO W911NF-17-1-0578. The work of F. Delarue was
partially supported by the Institut Universitaire de France and by the Agence
NationaledelaRecherche(projectMFG,ANR-16-CE40-0015-01).
Princeton,NJ,USA RenéCarmona
Nice,France FrançoisDelarue
July29,2016
Preface of Volume II
While the first volume of the book only addressed mean field games where the
sourcesofrandomshockswereidiosyncratictotheindividualplayers,thissecond
volumetacklestheanalysisofmeanfieldgamesinwhichtheplayersaresubjectto
a common source of random shocks. We call these models games with a common
noise.Generalsolvabilityresults,aswellasuniquenessconditions,areestablished
in Part I. Part II is devoted to the study of the so-called master equation. In the
finalchapterofthispart,weconnectthevariousasymptoticmodelsinvestigatedin
thebook,startingfromgameswithfinitelymanyplayers.Evenformodelswithouta
commonnoise,thistypeofanalysisheavilyreliesonthetoolsdevelopedthroughout
thissecondvolume.LikeinthecaseofVolumeI,VolumeIIendswithanepilogue.
There, we discuss several extensions which fit naturally the framework of models
withacommonnoise.Theyincludegameswithmajorandminorplayers,andgames
oftiming.
Thereaderwillfindonpagexiiibelowadiagramsummarizingtheinterconnec-
tionsbetweenthedifferentchaptersandpartsofthebook.
Chapter 1 is the cornerstone of the second volume. It must be seen as a
preparationfortheanalysisofmeanfieldgameswithacommonnoise.Asequilibria
become random in the presence of a common noise, we need to revisit the tools
introducedanddevelopedinthefirstvolumeforthesolutionofoptimalstochastic
control problems and establish a similar technology for stochastic dynamics and
cost depending on an additional source of randomness. To that effect, we provide
a general introduction to forward-backward systems in random environment and
possiblynon-Brownianfiltrations.Akeypointinouranalysisistoallowtherandom
environment not to be adapted to the Brownian motions driving the controlled
dynamics. This forces us to impose compatibility conditions constraining the
correlation between the noise carrying the random environment and the general
filtration to which the solution of the forward-backward system is required to be
adapted.Althoughthesecompatibilityconditionsareratherdifficulttohandle,they
turnouttobeabsolutelycrucialastheyplayafundamentalrolethroughoutthetext,
inparticularforthenotionofweakequilibriadefinedinthesubsequentChapter2.
Asfarasweknow,thismaterialdoesnotexistinbookform.
ix
x PrefaceofVolumeII
Thenotionofsolutionformeanfieldgameswithacommonnoiseendsupbeing
more subtle than one could expect. In contrast with the case addressed in Volume
I, the fixed point problem underpinning the definition of an equilibrium cannot
be tackled by means of Schauder’s fixed point theorem. In order to account for
the dependence of the equilibria upon the realization of the common noise, it is
indeed necessary to enlarge the space in which the fixed point has to be sought.
Unfortunately, proceeding in this way increases dramatically the complexity of
the problem, as the new space of solutions becomes so big that it becomes very
difficult to identify tractable compactness criteria to use with Schauder’s theorem.
ThepurposesofChapters2and3ispreciselytoovercomethisissue.Thefirststep
of our strategy is to discretize the realization of the common noise entering the
definition of an equilibrium in such a way that the equilibrium can take at most
a finite number of outcomes as the realization of the common noise varies. This
makesthesizeofthespaceofsolutionsmuchmorereasonable.Thesecondstepis
topasstothelimitalongdiscretizedsolutions.Thesuccessofthisapproachcomes
atthepriceofweakeningthenotionofequilibriuminthesensethattheequilibrium
is only adapted to a filtration which is larger than the filtration generated by the
commonnoise.ThisnotionofweaksolutionsistherationalefortheCompatibility
ConditionintroducedinChapter1.Theconceptsofweakandstrongsolutionsare
explained in Chapter 2. In analogy with strong solutions of stochastic differential
equations,strongequilibriaarerequiredtobeadaptedtothefiltrationgeneratedby
thecommonnoise.Naturally,weestablishaformofYamada-Watanabetheoryfor
Nash equilibria. Quite expectedly, it says that weak solutions are strong provided
that uniqueness holds in the strong sense. In analogy with the theory developed
in the first volume, we prove that strong and weak equilibria may be represented
by means of forward-backward systems of the conditional McKean-Vlasov type.
Duetothepresenceofthecommonnoise,conditioningappearsintheformulation.
To wit, we first develop, in the first section of the chapter, a theory of conditional
propagation of chaos for particle systems with mean field interaction driven by a
commonnoise.
The construction of weak solutions is addressed in Chapter 3. There, we
implement the aforementioned discretization approach. It consists in forcing the
total number of realizations of an equilibrium to be finite. The passage to the
limit along discretized solutions is achieved in the weak sense: we consider the
asymptoticbehaviorofthejointlawoftheequilibriaandtheforwardandbackward
processes characterizing the best response of the representative agent under the
discretizedenvironment.Whiletightnessoftheforwardcomponentisinvestigated
for the standard uniform topology on the space of continuous paths, tightness of
thebackwardcomponentisestablishedfortheMeyer-Zhengtopologyonthespace
ofright-continuouspathswithleftlimits.TheresultsoftheMeyer-Zhengtopology
needed for this purpose are recalled in the first section of Chapter 3. In the last
sections,weaddressthequestionofuniqueness.Likeformeanfieldgameswithout
acommonnoise,meanfieldgameswithacommonnoiseareshowntohaveatmost
