Equilibrium problems and applications PDF
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Equilibrium 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 AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1650,SanDiego,CA92101,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom Copyright©2019ElsevierInc.Allrightsreserved. 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LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-811029-4 ForinformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:CandiceJanco AcquisitionEditor:J.ScottBentley EditorialProjectManager:SusanIkeda ProductionProjectManager:JoyChristelNeumarinHonestThangiah Designer:AlanStudholme TypesetbyVTeX 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.
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