Table Of ContentEquilibrium Problems and Applications
Mathematics in Science and
Engineering
Equilibrium Problems
and Applications
Gábor Kassay
FacultyofMathematicsandComputerScience
Babes-BolyaiUniversity
Cluj-Napoca,Romania
Vicent¸iu D. Ra˘dulescu
FacultyofAppliedMathematics
AGHUniversityofScienceandTechnology
Kraków,Poland
InstituteofMathematics“SimionStoilow”oftheRomanianAcademy
Bucharest,Romania
Series Editor
Goong Chen
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This book is dedicated to the memory of my beloved mother,
Erzsébet Kassay(1932-1992).
GáborKassay
With Gratitude to my parents, Professor Dumitru Ra˘dulescu
(1914–1982) and Ana Ra˘dulescu (1923–2011). They are loved
beyondwords andmissedbeyondmeasure...
Vicen¸tiuD.Ra˘dulescu
About the Authors
Gábor Kassay received his Ph.D. thesis at the Babes-Bolyai University in
Cluj-Napoca, Romania, under the supervision of József Kolumbán in 1994.
He is a Professor in Mathematics at the same University, with more than 75
published research papers, several books, and book-chapters in the larger area
of nonlinear analysis, and more than 1500 citations. Gábor Kassay delivered
many invited and plenary talks, was session organizer and guest of honor at
prestigious international conferences. He has more than 35 coauthors and col-
laboratorsfromallovertheworld.HeiscurrentlythesupervisoroftheResearch
GroupofAnalysisandOptimizationaccreditedbytheFacultyofMathematics
andComputerScienceoftheBabes-BolyaiUniversity.Between2002and2004
he was an associate professor of Eastern Mediterranean University in Fama-
gusta,Cyprus.
Vicent¸iu D. Ra˘dulescu received his Ph.D. at the Université Pierre et Marie
Curie (Paris 6) in 1995 under the supervision of Haim Brezis. In 2003 he de-
fendedhisHabilitationMémoireatthesameuniversity.Ra˘dulescuisProfessor
attheAGHUniversityofScienceandTechnologyinKraków,ProfessorialFel-
lowatthe“SimionStoilow”MathematicsInstituteoftheRomanianAcademy,
andProfessorofMathematicsattheUniversityofCraiova.Heistheauthorof
morethan300researchpapersinnonlinearanalysisandseveralbooks,includ-
ingVariationalandNonvariationalMethodsinNonlinearAnalysisandBound-
aryValueProblems(Kluwer,2003),SingularEllipticProblems:Bifurcationand
AsymptoticAnalysis(OxfordUniversityPress,2008),ProblemsinRealAnaly-
sis: Advanced Calculus on the Real Axis (Springer, 2009), Variational Princi-
ples in MathematicalPhysics, GeometryandEconomics: QualitativeAnalysis
ofNonlinearEquationsandUnilateralProblems(CambridgeUniversityPress,
2010),NonlinearPDEs:MathematicalModelsinBiology,ChemistryandPop-
ulationGenetics(Springer,2012),PartialDifferentialEquationswithVariable
Exponents: Variational Methods and Qualitative Analysis (CRC Press, 2015),
VariationalMethodsforNonlocalFractionalProblems(CambridgeUniversity
Press,2016).HewasaHighlyCitedResearcher(2014).Hewaselectedtothe
AccademiaPeloritanadei Pericolanti(2014) and the Accademiadelle Scienze
dell’Umbria(2017).
xiii
Preface
Tothinkfreelyisgreat,buttothinkrightlyisgreater.
ThomasThorild(1759–1808),engravedingoldenletterstotheGrand
Auditorium,UppsalaUniversity
TheremaybenoNobelinmathematics,butthatneednotstopmanymathe-
maticianswinningtheNobelPrizeforEconomics1.Indeed,manyofthewinners
ofthisprizeeitherweremathematiciansorhadconductednotableresearchusing
mathematical models. For instance, Leonid Kantorovich and Tjalling Koop-
mans received the Nobel Prize in Economic Sciences “for their contributions
tothetheoryofoptimumallocationofresources”.BothKantorovichandKoop-
mansacknowledgedthatGeorgeB.DantzigdeservedtosharetheirNobelPrize
forlinearprogramming.Economistswhoconductedresearchinnonlinearpro-
gramming also have won the Nobel prize, notably Ragnar Frisch (1969). The
1994SverigesRiksbankPrizeinEconomicSciencesinMemoryofAlfredNo-
belwasawardedtoJohnForbesNashJr.,2 JohnHarsanyi,andReinhardSelten
“for their pioneering analysis of equilibria in the theory of non-cooperative
games”.ManysuchexamplescanbeprovidedbutwereferonlytoLloydShap-
leywhowontheNobelPrizeforEconomicSciencesin2012“forthetheoryof
stableallocationsandthepracticeofmarketdesign”.LloydShapleydescribed
himself in an Associated Press interview: “I consider myself a mathematician
andtheawardisforeconomics.Inever,neverinmylifetookacourseineco-
nomics.”
This monograph is intended to fill a gap in an interdisciplinary field at the
interplaybetweenappliedmathematics,optimization,equilibria,andeconomic
mathematics. Our analysis deeply relies on concrete models in the real world.
Models play crucial roles in appliedmathematicsand economics, from identi-
fyingnonstandardbehaviorofmathematicalmodelsineconomytoforecasting
howeconomicswillevolve.Yetmajorchangesareafootinequilibriumtheory,
triggeredbyglobaleconomicproblemsandthehigherandhigherimpactofbig
datasets.
Thisvolumeisanattempttostudyinarigorousmannerqualitativeproblems
arisinginappliedsciences.Moreprecisely,thismonographlooksatmodelingin
thesefieldsthroughthreelenses.ThefirstisthatofNashequilibriumandman-
agement,whicharefundamentalissuesinexperimentaleconomics.Thesecond
xv
xvi Preface
is through the variational analysis of equilibrium problems, which has been
challenged by the events of the macro-economy. The third looks at the opti-
mality and stability of the models of economic analysis at the interface with
powerfulandefficientmathematicaltheories.
Thismonographisasystematicexpositionoftheauthors’researchongen-
eral equilibrium models arising in optimization, economics, and applied sci-
ences. It is intended to serve both as a graduate text on aspects of general
equilibrium theory and as an introduction, for economists and mathematicians
working in mathematical economics, to current research in a frontier area of
general equilibrium theory. This book presents a systematic approach to prob-
lems in economic equilibrium based on fixed-point arguments and rigorous
variationalanalysismethods.Itdescribesthehighest-levelresearchontheclas-
sicaltheme,fixedpointsandeconomicequilibria,inthetheoryofmathematical
economics, and also presents basic results in this area, especially in the gen-
eralequilibriumtheoryandnoncooperativegametheory.Convexitytheoryand
topologyhavebeenthecentraltoolsfortherigorousaxiomatictreatmentofeco-
nomic theory since the 1950s. In this book, the notion of convexity is used to
describeideaswithinamixtureofalternativechoices,amoderateviewamong
extremes, and especially to ensure the existence of equilibrium depending on
suchstableactionsasafixedpointforamathematicalmodelofsociety.
In this monograph we aim to show how a special mathematical method
(a tool for thinking) can be utilized for constructing or developing part of an
economictheory. The arguments also containdistinguishable developmentsof
the main theme in the homology theory for general topological spaces, in the
modeltheoryandmathematicallogic,andinthemethodologyandphilosophy
ofsocialsciences.
Many of the theorems contained in this book are technical extensions of
fixed-point arguments, variational analysis methods, and tools for economic
equilibriumresults.Themainconcernofthisvolumeisnotonlytoshowabun-
dantwaystoapplysuchextensions,butalsotolisttheminimallogical,analytic,
oralgebraicrequirementsfortheconstructionofasolideconomicequilibrium
theory. Accordingly, we use in this monograph many highly abstract settings
(e.g., fixed-pointargumentsbasedonalgebraicsettings,actionswithoutconti-
nuityconditionsorconvexityassumptions,orspaceswithoutlinearstructures)
while basing our arguments on topics that are quite usual. Among others, the
concept of convex combination, approximate and iterative methods, and ar-
guments based on mathematical logic form the distinguishing features of this
book’smathematicalarguments.
Acentralroleinthismonographisplayedbythestudyofsomefundamen-
talaspectsrelatedtotheNashequilibrium.Thisconceptstartedtodevelopfrom
theearlyinsightsofÉmileBorel3andJohnvonNeumann.4Acoupleofdecades
later, at the beginning of the 1950s, those insights were developed and gener-
alized by John F. Nash Jr. His name, quite appropriately, became attached to
the equilibrium state characterized by the condition that all possible unilateral
Preface xvii
actions of any actor in that state lead to states that are no better for the devia-
torthantheoriginalone.ThefundamentalresultofNashwasthatallgames—
(with constant-sum or nonconstant sum) with a finite number of players each
endowedwith a finite set of strategies and anykind of goal function(egoistic,
altruistic,egalitarian)haveanequilibriuminpureormixed(probabilistic)strate-
gies.Ingametheory,aNashequilibriumisanarrayofstrategies,oneforeach
player,suchthatnoplayercanobtainahigherpayoffbyswitchingtoadifferent
strategy while the strategies of all other players are held fixed. As pointed out
in[161],“ifChrysler,Ford,andGMchooseproductionlevelsforpickuptrucks,
acommoditywhosemarketpricedependsonaggregateproduction,anequilib-
riumisanarrayofproductionlevels,oneforeachfirm,suchthatnonecanraise
itsprofitsbymakingadifferentchoice”.Aspointedoutin[1,p.495],“ingame
theory,thesinglemostimportanttoolhasproventobeNashequilibrium”.
This volume can serve as a graduate-level textbook on mathematical eco-
nomicsaswellasanadvancedmonographforstudentsandresearcherswhoare
concernedaboutrigorousmathematicaltreatmentinthesocialsciences.
Our vision throughout this volume is closely inspired by the following
propheticwordsofJohnF.NashJr.,inaninterviewgiveninOsloonMay18,
2015,thedaybeforetheAbelPrizeceremony:
“Ihadachievedmyproofoftheequilibriumtheoremforgametheoryusing
the Brouwer fixed-point theorem, while von Neumann and Morgenstern used
other things in their book. But when I got to von Neumann, and I was at the
blackboard, he asked: “Did you use the fixed-point theorem?” “Yes,” I said.
“IusedBrouwer’sfixed-pointtheorem.”
I had already, for some time, realized that there was a proof version using
Kakutani’s fixed-point theorem, which is convenient in applications in eco-
nomicssincethemappingisnotrequiredtobequitecontinuous.Ithascertain
continuityproperties,so-calledgeneralizedcontinuityproperties,andthereisa
fixed-pointtheoreminthatcaseaswell.IdidnotrealizethatKakutaniproved
that after being inspired by von Neumann, who was using a fixed-point theo-
rem approach to an economic problem with interacting parties in an economy
(however,hewasnotusingitingametheory)”.
GáborKassayandVicen¸tiuD.Ra˘dulescu
Cluj-NapocaandCraiova
May11,2018
NOTES
1. TheSverigesRiksbankPrizeinEconomicSciencesinMemoryofAlfredNobelwasestablished
in 1968 by the Bank of Sweden, and it was first awarded in 1969, more than 60 years after
thedistributionofthefirstNobelPrizes.AlthoughnottechnicallyaNobelPrize,thePrizein
EconomicSciencesisidentifiedwiththeaward;itswinnersareannouncedwiththeNobelPrize
recipients,anditispresentedattheNobelPrizeAwardCeremony.ItisconferredbytheRoyal
SwedishAcademyofSciencesinStockholm.
