the bargaining problem

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TheBargainingProblemJohnF.Nash,Jr.Econometrica,Vol.18,No.2.(Apr.,1950),pp.155-162.StableURL:http://links.jstor.org/sici?sici=0012-9682%28195004%2918%3A2%3C155%3ATBP%3E2.0.CO%3B2-HEconometricaiscurrentlypublishedbyTheEconometricSociety.YouruseoftheJSTORarchiveindicatesyouracceptanceofJSTOR'sTermsandConditionsofUse,availableathttp://www.jstor.org/about/terms.html.JSTOR'sTermsandConditionsofUseprovides,inpart,thatunlessyouhaveobtainedpriorpermission,youmaynotdownloadanentireissueofajournalormultiplecopiesofarticles,andyoumayusecontentintheJSTORarchiveonlyforyourpersonal,non-commercialuse.Pleasecontactthepublisherregardinganyfurtheruseofthiswork.Publishercontactinformationmaybeobtainedathttp://www.jstor.org/journals/econosoc.html.EachcopyofanypartofaJSTORtransmissionmustcontainthesamecopyrightnoticethatappearsonthescreenorprintedpageofsuchtransmission.TheJSTORArchiveisatrusteddigitalrepositoryprovidingforlong-termpreservationandaccesstoleadingacademicjournalsandscholarlyliteraturefromaroundtheworld.TheArchiveissupportedbylibraries,scholarlysocieties,publishers,andfoundations.ItisaninitiativeofJSTOR,anot-for-profitorganizationwithamissiontohelpthescholarlycommunitytakeadvantageofadvancesintechnology.FormoreinformationregardingJSTOR,pleasecontactsupport@jstor.org.http://www.jstor.orgFriOct2618:13:052007 THEBARGAININGPROBLERI1Anewtreatmentispresentedofaclassicaleconomicproblem,onerhichoccursinmanyforms,asbargaining,bilateralmonopoly,etc.Itmayalsoberegardedasanonzero-sumt~o-persongame.Inthistreatmentafewgeneralassumptionsaremadeconcerningthebehaviorofasingleindividualandofagroupoft~voindividualsincertaineco-nomicenvironments.Fromthese,thesolution(inthesenseofthispaper)oftheclassicalproblemmaybeobtained.Inthetermsofgametheory,valuesarefoundforthegame.INTRODUCTIONATWO-PERSONbargainingsituationinvolvest~~oindividualswhohavetheopportunitytocollaborateformutualbenefitinmorethanoneway.Inthesimplercase,whichistheoneconsideredinthispaper,noactiontakenbyoneoftheindividualswithouttheconsentoftheothercanaffecttheI-ell-beingoftheotherone.Theeconomicsituationsofmonopolyversusmonopsony,ofstatetradingbetweentwonations,andofnegotiationbetweenemployerandlaborunionmayberegardedasbargainingproblems.Itisthepurposeofthispapertogiveatheoreticaldiscussionofthisproblemandtoobtainadefinite"solutionn-making,ofcourse,certainidealizationsinordertodoso.A"solution"heremeansadeterminationoftheamountofsatisfactioneachindividualshouldexpecttogetfromthesituation,or,rather,adeterminationofhowmuchitshouldbeworthtoeachoftheseindividualstoharethisopportunitytobargain.Thisistheclassicalproblemofexchangeand,morespecifically,ofbilateralmonopolyastreatedbyCournot,Bou-ley,Tintner,Fellner,andothers.AdifferentapproachissuggestedbyvonYeumannandRIorgensterninTheoryofGamesandEcono~nicBehavior2whichpermitstheidentificationofthistypicalexchangesituationwithanonzerosumtwo-persongame.Ingeneralterms,weidealizethebargainingproblembyassumingthatthetwoindividualsarehighlyrational,thateachcanaccuratelycomparehisdesiresforvariousthings,thattheyareequalinbargainingskill,andthateachhasfulllino~vledgeofthetastesandpreferencesoftheother.1Theauthorwishestoacknon-ledgetheassistanceofProfessorsvonNeu-mannandMorgenstern~vhoreadtheoriginalformofthepaperandgavehelpfuladviceastothepresentation.2JohnvonSeumannandOskarlIorgenstern,TheoryofGan:esandEconomicBeha1,ior.Princeton:PrincetonUniversityPress,1944(SecondEdition,1917),pp.15-31.155 156JOHNF.NASH,JR.Inordertogiveatheoreticaltreatmentofbargainingsituationsweabstractfromthesituationtoformamathematicalmodelintermsofwhichtodevelopthetheory.InmakingourtreatmentofbargainingTeemployanumericalutility,ofthetypedevelopedinTheoryofGames,toexpressthepreferences,ortastes,ofeachindividualengagedinbargaining.Bythismeanswebringintothemathematicalmodelthedesireofeachindividualtomaximizehisgaininbargaining.Weshallbrieflyreviewthistheoryintheterminologyusedinthispaper.UTILITYTHEORYOFTHEINDIVIDUALTheconceptofan"anticipation"isimportantinthistheory.Thisconceptwillbeexplainedpartlybyillustration.SupposeMr.SmithknowshewillbegivenanewBuicktomorrow.WemaysaythathehasaBuickanticipation.Similarly,hemighthaveaCadillacanticipation.Ifheknewthattomorron*acoinwouldbetossedtodecidewhetherhewouldgetaBuickoraCadillac,weshouldsaythathehadaBuick,4Cadillacanticipation.Thusananticipationofanindividualisastateofexpectationwhichmayinvolvethecertaintyofsomecontingenciesandvariousprobabilitiesofothercontingencies.Asanotherexample,Mr.SmithmightknowthatheTillgetaBuicktomorrowandthinkthathehashalfachanceofgettingaCadillactoo.TheBuick,3Cadillacanticipationmentionedaboveillustratesthefollowingimportantprop-ertyofanticipations:if00.Lettingcapitallettersrepre-sentanticipationsandsmallonesrealnumbers,suchautilityfunctionwillsatisfythefollowingproperties:(a)u(A)>u(B)isequivalenttoAismoredesirablethanB,etc.(b)If0ul(cr)andu2(p)>u~(Q(),thena#c(S).7.IfthesetTcontainsthesetSandc(T)isinS,thenc(T)=c(S).WesaythatasetSissymmetricifthereexistutilityoperatorsulanduzsuchthatwhen(a,b)iscontainedinS,(b,a)isalsocontainedinS;thatis,suchthatthegraphbecomessymmetricalwithrespecttothelineul=uz.8.IfSissymmetricandulandu2displaythis,thenc(S)isapointoftheform(a,a),thatis,apointonthelineul=uz.Thefirstassumptionaboveexpressestheideathateachindividualwishestomaximizetheutilitytohimselfoftheultimatebargain.Thethirdexpressesequalityofbargainingskill.Thesecondismorecompli-cated.Thefollowinginterpretationmayhelptoshowthenaturalnessofthisassumption:Iftworationalindividualsm~ouldagreethatc(T)wouldbeafairbargainifTwerethesetofpossiblebargains,thentheyshouldbewillingtomakeanagreement,oflesserrestrictiveness,nottoattempttoarriveatanybargainsrepresentedbypointsoutsideofthesetSifScontainedc(T).IfSwerecontainedinTthiswouldreducetheirsitua-tiontoonewithSasthesetofpossibilities.Hencec(S)shouldequalc(T).Wenowshowthattheseconditionsrequirethatthesolutionbethepointofthesetinthefirstquadrantwhereuluzismaximized.Weknowsomesuchpointexistsfromthecompactness.Convexitymakesitunique.Letusnowchoosetheutilityfunctionssothattheabove-mentionedpointistransformedintothepoint(1,1).Sincethisinvolvesthemulti-plicationoftheutilitiesbyconstants,(1,1)willnowbethepointofmaximumuluz.Fornopointsofthesetwillul+uz>2,now,sinceiftherewereapointofthesetwithu1+u2>2atsomepointonthelinesegmentbetween(1,1)andthatpoint,therem~ouldbeavalueofuluzgreaterthanone(seeFigure1).Wemaynowconstructasquareintheregionul+uz<2whichissymmetricalinthelineul=uz,whichhasonesideonthelineul+uz=2,andwhichcompletelyenclosesthesetofalternatives.Consideringthesquareregionformedasthesetofalternatives,insteadoftheolderset,itisclearthat(1,1)istheonlypointsatisfyingassumptions(6)and(8).Xon-usingassumption(7)wemayconcludethat(1,1)mustalsobethesolutionpointwhenouroriginal(transformed)setisthesetofalternatives.Thisestablishestheassertion.Weshallnowgiveafewexamplesoftheapplicationofthistheory. 160JOHNF.XASH,JR.EX4MPLESLetussupposethattn-ointelligentindividuals,BillandJack,areinapositionwheretheymaybartergoodsbuthavenomoneywithwhichtofacilitateexchange.Further,letusassumeforsimplicitythattheutilitytoeitherindividualofaportionofthetotalnumberofgoodsin-volvedisthesumoftheutilitiestohimoftheindividualgoodsinthatportion.Wegivebelon-atableofgoodspossessedbyeachindividualwiththeutilityofeachtoeachindividual.Theutilityfunctionsusedforthet~oindividualsare,ofcourse,toberegardedasarbitrary. THEBARGA~INGPROBLEM161Bill'sUtilityUtilitygoodstoBilltoJackbookwhipballbatboxJack'sgoodsPentoyknifehatThegraphforthisbargainingsit,uationisincludedasanillustration(Figure2).Itturnsouttobeaconvexpolygoninwhichthepointwheretheproductoftheutilitygainsismaximizedisatavertexandwherethereisbutonecorrespondinganticipation.Thisis:BillgivesJack:book,whip,ball,andbat,JackgivesBill:pen,toy,andknife.Whenthebargainershaveacommonmediumofexchangetheproblemmaytakeonanespeciallysimpleform.Inmanycasesthemoneyequiva-JACK'SUTILITYtb'UIFIGURE2FIGURE3FIGURE2-Thesolutionpointisonarectangularhyperbolalyinginthefirstquadrantandtouchingthesetofalternativesatbutonepoint.FIGURE3-Theinnerarearepresentsthebargainspossiblewithouttheuseofmoney.Theareabetweenparallellinesrepresentsthepossibilitiesallowingtheuseofmoney.Utilityandgainmeasuredbymoneyarehereequatedforsmallamountsofmoney.Thesolutionmustbeformedusingabarter-typebargainforwhichul+u,isatamaximumandusingalsoanexchangeofmoney. 162JOHNF.NASH,JR.lentofagoodwillserveasasatisfactoryapproximateutilityfunction.(Bythemoneyequivalentismeanttheamountofmoneywhichisjustasdesirableasthegoodtotheindividualwithwhomweareconcerned.)Thisoccurswhentheutilityofanamountofmoneyisapproximatelyalinearfunctionoftheamountintherangeofamountsconcernedinthesituation.Whenwemayuseacommonmediumofexchangefortheutilityfunctionforeachindividualthesetofpointsinthegraphissuchthatthatportionofitinthefirstquadrantformsanisoscelesrighttri-angle.Hencethesolutionhaseachbargainergettingthesamemoneyprofit(seeFigure3).PrincetonUniversity