文獻綜述：

博弈論在供應鏈管理中的應用數9艾松1博弈論在供應鏈管理中的應用現在還處于探索的階段,所用的博弈論理論還比較淺；更多的是用博弈論中的概念、已有的結論等，最常用的就是Nash均衡，Game的模型，Stackelberg模型等；部分模型用顯示原理、Nash均衡的存在性定理來求解均衡結果。2文獻綜述Huang,Z.M.,S.X.Li.2001.Co-opadvertisingmodelsinmanufacturer-retailersupplychains:Agametheoryapproach.EuropeanJournalofOperationalResearch135,527-544.Li,S.X.,Z.M.Huang,J.Zhu,P.Y.K.Chau.2002.Cooperativeadvertising,gametheoryandmanufacturer-retailersupplychains.Omega30,347–357.3Huang,Z.M.,S.X.Li.2001.

Co-opadvertisingmodelsinmanufacturer-retailersupplychains:Agametheoryapproach.

EuropeanJournalofOperationalResearch135,527-544.

Keyword:Decisionanalysis;Gametheory;Co-opadvertising;Equilibrium;Coordination;Bargainingproblems;Utilities.41.IntroductionMoststudiestodateonverticalco-opadvertisinghavefocusedonarelationshipwherethemanufacturerisaleaderandtheretailerisafollower.Thispaperisintendedtodiscusstherelationshipbetweenco-opadvertisingandefficiencyofmanufacturer-retailertransactions.61.IntroductionThreeco-opadvertisingmodel:1.aleader-followernoncooperativegame:manufacturerisaleader;2.anoncooperativesimultaneousmovegame;3.acooperativegame.72.AssumptionsOne-periodsalesresponsevolumefunction:Expectedsalesresponsevolume:92.AssumptionsThemanufacturer’s,retailer’s,system’sexpectedprofitfunctionsareasfollows:Note:“cq”shouldbe“q”103.StackelbergequilibriumWemodeltherelationshipbetweenthemanufacturerandtheretailerasasequentialnoncooperativegamewiththemanufacturerastheleaderandtheretailerasthefollower.113.StackelbergequilibriumWefirstsolveforthereactionfunctioninthesecondstageofthegame:isaconcavefunctionofSettingthefirstderivativeofwithrespecttotobezero:

ThenwehaveEq(5):123.StackelbergequilibriumWecanobservethat:Sothemanufacturercanusehisco-opadvertisingpolicyandhisnationalbrandnameinvestmenttoinducetheretailertoincreaseordecreaselocaladvertisingexpenditureatalevelheexpects.133.StackelbergequilibriumNexttheoptimalvalueofandaredeterminedbymaximizingthemanufacturer’sprofitsubjecttotheconstraintimposedbyEq(5).Hence,themanufacturer’sproblemcanbeformulatedas143.StackelbergequilibriumSolvingEq(9),andsubstitutingtheoutcomeintoEq(5),wehavetheuniqueequilibriumpointofthetwo-stagegame:163.StackelbergequilibriumProposition1:If(1)themanufacturerofferspositiveadvertisingallowancetotheretailer,otherwisehewilloffernothing;(2)173.Stackelbergequilibrium(2)thehigher(thelower)theretailer’s(manufacturer’s)marginalprofit,thelowerthemanufacturer’sadvertisingallowancefortheretailer;(3)theincreaseofsuchthatwillcauseanincreaseinthesalesandthenwillgivetheretailerincentivetodolocaladvertisingwithoutmanufacturer’sfinancialhelp.193.StackelbergequilibriumInthisgame,themanufacturerholdsextremepowerandhasalmostcompletecontroloverthebehavioroftheretailer.Therelationshipisthatofanemployerandanemployee!204.NashequilibriumRecentstudiesinmarketinghavedemonstratedthatinmanyindustriesretailershaveincreasedtheirpowerrelativetomanufacturersoverthepasttwodecades.Especially,fordurablegoodssuchasappliancesandautomobiles,theretailerhasmoreinfluenceontheconsumer’spurchasedecision.214.NashequilibriumInthissection,werelaxtheleader-followerrelationshipandassumeasymmetricrelationshipbetweenthemanufacturerandtheretailer.Themanufacturerandtheretailersimultaneouslyandnoncooperativelymaximizetheirprofitswithrespecttoanypossiblestrategiessetbytheothermember.224.NashequilibriumHence,themanufacturer’soptimalproblemis:Theretailer’soptimalproblemis:234.NashequilibriumItisobviousthatthemanufacturer’soptimalfractionlevel,,iszero,becauseofitsnegativecoefficientintheobjective.ANashequilibriumadvertisingschemecanbeobtainedbysimultaneouslysolvingthefollowingconditions:244.NashequilibriumThreeimplications:(1)sincethemanufacturer’sallowancepoliciesdoesnotinfluencethesalesresponsevolumefunction,independentactionstakenbybothmemberssimultaneouslymakenoimpactofthesharingpoliciesonthedeterminationoftheretailer’slocaladvertisinglevel;264.Nashequilibrium(2)(3)274.Nashequilibrium(b)Iftheretailerprefersthesimultaneousmovegamestructure,otherwisehepreferstheleader-followergamestructure.294.NashequilibriumProposition3:(a)Themanufacturer’sbrandnameinvestmentishigheratNashthanatStackelberg;(c)Themanufacturer’sadvertisingallowanceforretaileriszero.304.Nashequilibrium(b)Iftheretailer’slocaladvertisingexpenditureishigheratNashthanatStackelberg,otherwiseitisloweratNashthanatStackelberg.315.Anefficiencyco-opadvertisingmodelInthissectionwewillretaintheassumptionofthesymmetricrelationshipbetweenthemanufacturerandtheretailer.Wewilldiscusstheefficiencyofmanufacturerandretailertransactionsinverticalco-opadvertisingagreements.325.Anefficiencyco-opadvertisingmodelWeconsiderParetoefficientadvertisingschemesinourco-opadvertisingarrangements.AschemeiscalledParetoefficientifonecannotfindanyotherscheme(a,t,q)suchthatneitherthemanufacturer’snortheretailer’sprofitislessat(a,t,q)butatleastoneofthemanufacturer’sandretailer’sprofitsishigherat(a,t,q)thanat.335.Anefficiencyco-opadvertisingmodelSinceandarequasi-concave,thesetofParetoefficientschemesconsistsofthosepointswherethemanufacturer’sandtheretailer’siso-profitsurfacesaretangenttoeachother,i.e.,forsome>=0345.Anefficiencyco-opadvertisingmodelThisleadstothefollowingproposition

355.Anefficiencyco-opadvertisingmodelThistheoremtellsusthatallParetoefficientschemesareassociatedwithasinglelocaladvertisingexpenditure

andasinglemanufacturer’sbrandnameinvestment

andwiththefractiontofthemanufacturer’sshareofthelocaladvertisingexpendituresbetween0and1.Thelocusoftangencyliesonaverticallinesegmentatin(a,t,q)space.365.Anefficiencyco-opadvertisingmodelProposition5:AnadvertisingschemeisParetoefficientifandonlyifitisanoptimalsolutionofthejointsystemprofitmaximizationproblem.

Thistheoremtellsusthat,amongallpossibleadvertisingschemes,thesystemprofit(i.e.,thesumofthemanufacturer’sandtheretailer’sprofits)ismaximizedforEveryParetoefficientscheme,butnotforanyotherschemes.375.Anefficiencyco-opadvertisingmodelProposition6:(a)ThesystemprofitishigheratanyParetoefficientschemethanatbothnoncooperativeequilibriums;(c)ThelocaladvertisingexpenditureishigheratanyParetoefficientschemethanatbothnoncooperativeequilibriums;385.Anefficiencyco-opadvertisingmodel(b)Ifthenthemanufacturer’sbrandnameinvestmentishigheratanyParetoefficientschemethanatbothnoncooperativeequilibriums,otherwisethemanufacturer’sbrandnameinvestmentatanyParetoefficientschemeishigherthanatStackelbergequilibriumandislowerthanatNashequilibrium.395.Anefficiencyco-opadvertisingmodelProposition6leadstothepossibilitythatboththemanufacturerandtheretailercangainmoreprofitscomparedwithStackelbergequilibriums.ButitshouldbenotedthatnotallParetoefficientschemesarefeasibletoboththemanufacturerandtheretailer.Neitherthemanufacturernortheretailerwouldbewillingtoacceptlessprofitsatfullcooperationthanwithnoncooperation.405.Anefficiencyco-opadvertisingmodelAnadvertisingschemeiscalledfeasibleParetoefficientif415.Anefficiencyco-opadvertisingmodelthefeasibleParetoefficientsetofadvertisingschemes.Sinceonlyschemessatisfying(24)and(25)areacceptableforboththemanufacturerandtheretailerwhentheydocoordinate,wethencall425.Anefficiencyco-opadvertisingmodelReferringtoProposition2,weknowthat:(1)(2)Ifthenotherwise435.Anefficiencyco-opadvertisingmodelThereforeForthepurposeofsimplicity,weassumethat445.Anefficiencyco-opadvertisingmodelHencerelationshipsinEq(24)and(25)canberewrittenas455.Anefficiencyco-opadvertisingmodelLetHereweassume465.Anefficiencyco-opadvertisingmodelThenandZcanbesimplifiedas475.Anefficiencyco-opadvertisingmodelItcanbeshownthatTherefore,foranygivent

whichsatisfieswehaveThissimplyimpliesthatthereexistParetoefficientadvertisingschemessuchthatboththemanufacturerandtheretailerarebetteroffatfullcoordinationthanatnoncooperativeequilibrium.485.Anefficiencyco-opadvertisingmodelWeareinterestedinfindinganadvertisingschemeinZwhichwillbeagreeabletoboththemanufacturerandtheretailer.AccordingtoProposition6,foranyParetoschemewhereisapositiveconstant.495.Anefficiencyco-opadvertisingmodelThispropertyimpliesthatthemorethemanufacturer’sshareofthesystemprofitgain,thelesstheretailer’sshareofthesystemprofitgain,andviceversa.Sothemanufacturerandtheretailerwillagreetochangethelocaladvertisingexpenditurestoandthebrandnameinvestmentsto.However,theywillnegotiateoverthemanufacturer’sshareofthelocaladvertisingexpenditures.506.BargainingresultsAssumethatthemanufacturerandtheretaileragreetochangelocaladvertisingexpenditurestoandbrandnameinvestmentstofromand,respectively,andengageinbargainingforthedeterminationofreimbursementpercentagetodividethesystemprofitgain.516.BargainingresultsAfractionclosertoispreferredbytheretailer,andafractionclosertoispreferredbythemanufacturer.Todeterminethedivisionofthesystemprofitgain,wemustgivesomefurtherassumptions.526.BargainingresultsSincethereisanenvironmentuncertaintyinsalesvolume,bothmembersareassumedtobeuncertainaboutthesystemprofitgain,.ForeachParetoefficientadvertisingschemes,theuncertaintyisrepresentedintermsofaprobabilitydistributionfor.Weassumethatbothmembersagreeontheprobabilitydistributionsofinterest.536.BargainingresultsSupposeboththemanufacturerandtheretailerhavepreferencesfortheamountofsharesofthesystemprofitgain,whichpreferencesarerepresentedbyeachsystemmember’svonNeumann–Morgensterncardinalutilityfunctionfor.Themanufacturer’sandtheretailer’sutilityfunctionsaredenotedbyand,respectively.546.BargainingresultsWeassumetheutilityfunctionsareadditive,thatistosayitcanbewrittenintheform

whereistheconditionalutilityfunctionofmemberi(i=m,r)for(j=m,r).556.BargainingresultsIthasbeenalsoshownthat,foradditiveindividualutilityfunctions,thesystemutilityfunction,,isalsoadditiveunderthelinearaggregationrule.Theformofusisasfollows:whereisthevectorofaggregationweightsand.566.BargainingresultsInordertoincorporatethemanufacturer’sandtheretailer’sriskattitudeintoouranalysis,wedefinethePratt–Arrowriskaversionfunctionasfollows:istheriskaversionfunctionofmemberi(i=m,r)totheshareofthejthmember(j=m,r).576.BargainingresultsHerewepresenttheNash(1950)bargainingmodeldeterminingthebargainingreimbursementfractionoverthelinesegmentofParetoefficientsolutionsdescribedbyThebargainingoutcomeisobtainedbymaximizingtheproductindividualmarginalutilitiesoverParetoefficientlocus.586.BargainingresultsTodemonstratethisapproach,weconsidertwodegeneratedexponentialutilityfunctionsforthemanufacturerandtheretailerasfollows:whereandarepositiveconstant.596.BargainingresultsEqs.(37)and(38)implythatboththemanufacturerandtheretailerhaveconstantriskaversionfunctionswithand606.BargainingresultsSincetheproductofandcanberewrittenastheformintermsof:616.BargainingresultsTakingthefirstderivativeofwithrespecttoandsettingittobezero:626.BargainingresultsNowweconsiderseveralspecialcases.First,assumethatboththemanufacturerandtheretailerhavethesamedegreeofriskaversionmeasures,i.e.Thensolving(40),wehave636.BargainingresultsTherefore,thebestParetoadvertisingreimbursementisSoifthemanufacturerandtheretailerhavethesamedegreeofriskaversionmeasures,themodelsuggeststhatthemembersshouldequallysharethesystemprofitgain.646.BargainingresultsSecond,assumethatthemanufacturerhasahigherdegreeofriskaversionmeasuresthantheretailerandThensolving(40),wehave656.Bargainingresults666.BargainingresultsTherefore,thebestParetoadvertisingreimbursementis676.BargainingresultsSowhenthemanufacturer’sdegreeofriskaversionishigherthantheretailer’s,hereceivesalowershareofthesystemprofitgain,whichisconsistentwiththeresultinthecaseofnegotiationwithbargainingpower.Asimilaranalysiscanbeaccomplishedforthecasewheretheretailer’sdegreeofriskaversionishigherthanthemanufacturer’s.687.ConcludingremarksThispaperattemptstoinvestigatetheefficiencyoftransactionforthesystemofmanufacturer–retailerco-opadvertisinginthecontextofgametheory.697.ConcludingremarksTherearethreepossibleavenuesforfutureresearch:First,thesinglemanufacture–retailersystemassumptioncanberelaxedtoaduopolysituationofmanufacturerswhoselltheirproductsthroughacommonmonopolisticretailerwhosellsmultiplecompetingbrandswithvaryingdegreesofsubstitutability.707.ConcludingremarksSecond,inouranalysisweemployednonlinearsalesresponsefunctiontosatisfythesaturationrequirement.Theuseofalinearsalesresponsefunctionmayyielddifferentandinterestingresultsintheanalysisforverticalco-opadvertisingagreements.717.ConcludingremarksFinally,inourstudythemanufacturer’sspendingonlocaladvertisingischaracterizedonlybyitsreimbursementpolicy.Wecanincorporateanotherfactor,accrualrate,intoourmodeltoyieldsomeinterestingresults.72Li,S.X.,Z.M.Huang,J.Zhu,P.Y.K.Chau.2002.

Cooperativeadvertising,gametheoryandmanufacturer-retailersupplychains.

Omega30,347–357.Keyword:

Co-opadvertising;Supplychains;Leader–followerrelationship;Paretoefficiencies;Bargainingmodel.731.Introduction

2.Interactivetwo-stageco-op

advertisingmodel

3.HigherorderStackelberg

equilibrium

4.Fullycoordinatedco-op

advertisingmodel

5.Concludingremarks743.HigherorderStackelbergequilibriumNowletusconsiderboththemanufacturer’sandtheretailer’sprofitfunctionsagain:G