Time-varying fairness concerns, delay, and disagreement in bargaining

ContentslistsavailableatScienceDirect

Journal

of

Economic

Behavior

&

Organization

jo u r n al ho me p ag e :ww w . e l s e v i e r . c o m / l o c a t e / j e b o

Time-varying

fairness

concerns,

delay,

and

disagreement

in

bargaining

Emin

Karagözo˘glu

a,b,∗

,

Kerim

Keskin

c

a_{BilkentUniversity,Turkey} b_{CESifo-Munich,Germany} c_{KadirHasUniversity,Turkey}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received18September2017 Receivedinrevisedform 27November2017 Accepted3January2018 Availableonline9January2018 JELclassification: C72 C78 D03 Keywords: Bargaining Deadlineeffect Delay Disagreement Fairnessconcerns Justicesensitivity

a

b

s

t

r

a

c

t

Westudyanalternating-offers,bilateralbargaininggamewhereplayersmayderive disu-tilityfromacceptingsharesbelowwhattheydeemasfair.Moreover,weassumethatthe valuestheyattachtofairness(i.e.,theirsensitivitytoviolationsoftheirfairnessjudgments) decreaseovertime,asthedeadlineapproaches.Ourresultsofferanewexplanationto delaysanddisagreementsindynamicnegotiations.Weshowthatevenmutually compati-blefairnessjudgmentsdonotguaranteeanimmediateagreement.Wepartiallycharacterize conditionsfordelayanddisagreement,andstudythechangesinthelengthofdelayin responsetochangesinthemodelparameters.

©2018ElsevierB.V.Allrightsreserved.

1. Introduction

Delays in reaching an agreement and stalemates are ubiquitous features of real life bargaining encounters. Labora-tory experiments on bargaining games report similar observations (see Roth et al., 1988; Babcock et al., 1995; Babcock and Loewenstein, 1997; Gächter and Riedl, 2005; Karagözo˘glu and Riedl, 2015; Karagözo˘glu and Kocher, 2016; among others). Roth et al. (1988), who observed subjects’tendency to reach agreements towards the deadline, labeled this phe-nomenon as the deadline effect.1 _{Babcock et al. (1995), Babcock and Loewenstein (1997), Gächter and Riedl (2005), and}

Karagözo˘gluandRiedl(2015)observedthatdelaysinreachingagreementsanddisagreementsarepositivelycorrelatedwiththe incompatibilitybetweenthebargainers’fairnessjudgments.Basedontheseobservations,theyinformallyarguedthat fairmind-edness, combined withbiasedandincompatiblefairness judgments,couldbeone ofthereasons behind delays,last-minute agreements,andevendisagreements.Standardbargainingmodelswithcommonknowledgeofrationality,complete/perfect

infor-∗ Correspondingauthorat:BilkentUniversity,DepartmentofEconomics,06800Bilkent,Ankara,Turkey. E-mailaddresses:karagozoglu@bilkent.edu.tr(E.Karagözo˘glu),kerim.keskin@khas.edu.tr(K.Keskin).

1_{Thisisalsofrequentlyobservedoutsidethelab.Forinstance,laborunionsandemployersusuallyreachagreementsjustbeforeastrikestarts,a}

phenomenonknownasthe“eleventhhourdeal”.Settlementsinpre-trialnegotiationsareusuallyreachedjustbeforethecourtdate.Similarly,in2011and 2013,DemocratsandRepublicansreachedanagreementinthenegotiationsabouttheU.S.debtceiling,justbeforethedeadline.

https://doi.org/10.1016/j.jebo.2018.01.002 0167-2681/©2018ElsevierB.V.Allrightsreserved.

mation,anddiscountingfailtocapturethesephenomena.Inthosemodels(seeRubinstein,1982),theplayerwhomakesthefirst offerusesalltherelevantinformationandmakesanofferthattheotherplayercannotreject.Thisresultsinanimmediateagreement. Delaysanddisagreementsinnegotiationscanhaveimportanteconomicimplications.Forinstance,strikesorlockoutsdueto disagreementsbetweenlaborandmanagement(e.g.,theNBAlockoutinthe2011–2012season,theUSAirwaysunionworkers’ strikein2014),continuingconflictsbetweencountries(e.g.,theCyprusconflict,theIsraeli-Palestinianconflict),postponeddecisions duetostalematesinpoliticalnegotiations(e.g.,delaysingovernmentformationduetolengthycoalitionnegotiationsinBelgiumin 2011andintheNetherlandsin2017),mergerandacquisitionnegotiations(e.g.,theVerizon–Yahoomerger),orlitigationnegotiations (e.g.,thePennzoilvs.Texacocase)implygreatwelfarecostsnotonlyforthenegotiatingpartiesinvolvedbutalsoforthethirdparties whohaveinterestsconnectedtothenegotiation(e.g.,fans,tvbroadcastingcompanies,sportsbrandsintheNBAlockoutexample). Therefore,understandingthefactorsthatcausebargainingdelaysanddisagreementsisofutmostimportance.Overthelastthree decades,bargainingscholarsofferedvarioustheoreticalexplanationsfordelayanddisagreements.2_{Asymmetricinformationorlack}

ofcommonpriorsonimportantbargainingcharacteristics(e.g.,discountfactors,outsideoptions),stochasticallyevolvingmodel parameters(e.g.,stakesize),oruncertaintyregardingplayertypesandreputationalconcernsarethemainreasonsforequilibrium delayanddisagreementinthosepapers.

Inthecurrentpaper,westudyanalternating-offersbargaininggamewithaknowndeadlinebetweentwoplayerswhereplayers mayderivedisutilityfromacceptingsharesbelowwhattheydeemasfair.Moreprecisely,eachplayerhassomesubjectivejudgment regardingthefairdivisionofthepie,whichiscommonknowledge;andifshereceivesasharebelowwhatshedeemsasfair,then shederivesdisutilityfromthat.Anaturalexampleaddressedinvariousbargainingexperimentsisasfollows(seeGantneretal., 2001;BirkelandandTungodden,2014;Karagözo˘gluandRiedl,2015;BoltonandKaragözo˘glu,2016):Supposethattwoplayers,Alan andBetty,exertedeffortstojointlyproducethepietheyarebargainingover.Alan’seffortproduced70%ofthepie,whereasBetty’s effortproduced30%ofthepie.Now,Alanbelievesthattheappropriatejusticenormisequityandhencethepieshouldbedivided inproportiontotheircontributions(i.e.,70–30division);whereasBettybelievesthattheappropriatejusticenormisequalityand hencethepieshouldbedividedequally(i.e.,50–50division).Eachplayerwouldexperiencedisutilityfromreceivingsharesbelow theirfairshares(70vs.50,respectively).

Importantly,weallowtheweightstheplayersattachtofairnessconcerns(i.e.,theirsensitivitytoviolationsoftheirfairness judgments)todiminishovertime.Accordingly,asthedeadlineapproaches,theplayerscarelessandlessaboutfairnessandbecome moreandmorematerial-gain-oriented.Thismodelingchoiceisinspiredbytherecentexperimentalfindings,whichhighlightthe primacyofeconomicconcernsoverfairnessconcerns—especiallyundertime/cognitivepressure(seeMooreandLoewenstein,2004; Knochetal.,2006;KnochandFehr,2007;Halalietal.,2013;Hochmanetal.,2015;amongothers)andthetemporalinstability ofjusticesensitivity(seeFortinetal.,2016foranexcellentreview).Furthermore,itisalsoinlinewiththehabituationandcostly self-controlargumentsinthetheoreticalliterature,whichwouldimplythatanagentwhorepetitivelyreceivesastimulithatis cognitivelydisturbingorcostlytohandlestartstobecomelessresponsive(seeKaragözo˘glu,2014forhabituation;andFudenberg andLevine,2006;Dreberetal.,2016forcostlyself-control).

We,first,analyticallyshowthatdelayand/ordisagreementmayexistintheequilibriumofthisbargaininggame.Weprovide necessaryandsufficientconditionsfortheexistenceofdelayanddisagreement.Ourresultsshowthatwhetherplayerswillbeableto reachanagreementandifsowhendependonmultiplefactorssuchasplayers’fairnessjudgments,thepaceofthedecreaseintheir weightsforfairness,theidentitiesofplayers(i.e.,proposerorresponder)indifferentperiods,andthehorizonlength.Furthermore, oursimulationresultsshowthatourmodelcanproduceempiricallyrelevantoutcomepatternssuchasU-shapedorJ-shaped distributionofagreementtimes,whichareusuallyobservedindynamicbargainingexperiments(seeRothetal.,1988;Gächterand Riedl,2005;Karagözo˘gluandRiedl,2015;Sullivan,2016).Ourcomparativestaticanalysesrevealnon-trivialinteractionsbetween themodelparametersandbargainingdelay.Forinstance,moredemandingfairnessjudgmentsorstrongerconcernsforfairness mayleadtofasteragreements.Finallyandsomewhatsurprisingly,weshowthatmutuallycompatiblefairnessjudgmentsdonot guaranteeanimmediateagreement.

Itisworthwhileemphasizingthatalltherelevantinformationiscommonknowledgeinourmodel.Thereisnoincompleteor imperfectinformation.Playersareperfectlyforward-looking.3_{Ourmodelisthefirsttoexplainbothdelayanddisagreementina}

finite-horizon,alternatingoffersbargainingframeworkwithoutresortingtousualincompleteorimperfectinformation assump-tions.Moreover,itisthefirstmodeltoformallystudytime-varyingfairnessconcerns.Tothebestofourknowledge,Birkelandand Tungodden(2014)presentthecloseststudytooursinthattheyalsoutilizefairnessconcerns.TheseauthorsworkonastaticNash bargainingmodel,withoutanytime-varyingcomponentinfairnessjudgments.Naturally,theirmodelcanexplaindisagreements, butnotdelay.

Theorganizationofthepaperisasfollows.InSection2,weintroduceourmodel.InSection3,wefirstpresentouranalytical resultsontheexistenceofequilibriuminvolvingdelayanddisagreement.Later,wepresentsimulationresultsontherelationships betweenthemodelparametersandthelengthofdelay.Section4concludes.

2. Themodel

Weinvestigateatwo-playerfinite-horizonalternatingoffersbargaininggamealaRubinstein(seeRubinstein,1982).LetN={1, 2}bethesetofplayerswhobargainoveradivisiblepie(withanormalizedsizeof1)forafinitenumberofperiodsT.Inperiodt=1, player1proposesadivisionofthepiechoosingastrategy(anoffer)from[0,1].Observingplayer1’soffer,player2chooseswhether toacceptorreject:{a,r}.Ifsheacceptstheoffer,thentheproposeddivisionisimplementedandthegameends.Incasesherejects

2 _{SeeRubinstein(1985),AdmatiandPerry(1987),GulandSonnenschein(1988),Cho(1990),Cramton(1991),FershtmanandSeidmann(1993),Maand}

Manove(1993),JehielandMoldovanu(1995),MerloandWilson(1998),AbreuandGul(2000),Bac(2000),Yildiz(2004),FeinbergandSkrzypacz(2005), Ali(2006),SimsekandYildiz(2014),Romm(2016),Fanning(2016)amongothers.

theoffer,thegameproceedstothenextperiod.Inperiodt=2,itisplayer2’sturntomakeanoffer,choosingastrategyfrom[0, 1],afterwhichplayer1istodecidewhethertoaccept(a)orreject(r)theoffer.Ifplayer1accepts,thentheproposeddivisionis implementedandthegameends;ifsherejects,thenthegameproceedstoperiod3inwhicheverythingfollowsasinthefirstperiod. ThisbargainingprocedurecontinuesuntiloneoftheplayersacceptsanofferoruntilperiodTends.Incasetheformerhappens,each playergetssomeutilitybasedontheagreeddivision,whereasifthelatterhappens,thenthegameendswithadisagreementwhich yieldsautilityof0tobothplayers.

Players have fairness concerns. More precisely, each player i_∈N believes that it is fair for her to receive a share ϕi∈[0,1]of thepie.If playerigetsashare xi<ϕi,thensheexperiences adecreaseintheutilityderivedfromthematerial

gain of having xi∈[0, 1].4 This type of preferences is commonly represented with an additively separable utility

func-tion in the literature (see Bolton, 1991; Fehr and Schmidt, 1999; Bolton and Ockenfels, 2000). The novel feature of our model isthe introduction of time-varying fairnessconcerns. Accordingly,each utilityfunction has different weightsfor the material gain and thedisutility comingfrom an unfair division, and theweight (relative importance) for fairness concerns weakly decreases through time. We normalize the weight of the material gain inboth utility functions to 1, and we let ˛i:{1,...,T}→[0,∞)beanon-increasingfunctionofplayeri’srelativeweightsforherfairnessconcernsateachperiod.Ifplayers

agreeonanallocation(x,1−x)atsomeperiodt∈{1,...,T},theirutilitiesfromsuchanagreementaregivenby: u1((x,1−x),t)=x−˛1(t)·max{ϕ1−x,0}

u2((x,1−x),t)=1−x−˛2(t)·max{ϕ2+x−1,0}

Playersdiscountfuturepayoffswithıi∈(0,1).Thismeansthattheutilitiesgivenabovearediscountedbyarateofıit−1when

evaluatedinthefirstperiod.Alltherelevantinformation(e.g.,fairnessconcerns,theinitialvaluesandtheevolutionoftheweights attachedtofairness,thediscountfactors,functionalforms,etc.)iscommonknowledge.Wedenotethisgameby_T_.

3. Theresults

Inwhatfollows,wefirstpresenttheanalyticalresultsontheexistenceofequilibrium,delay,anddisagreements.Later,wepresent simulationresultsonthelengthofdelay.

3.1. Analyticalresults

Throughoutthepaper,anequilibriummeans(purestrategy)subgameperfectNashequilibrium.Wedenotetheoffermadeby player1atsomeperiodtbyxtandtheoffermadebyplayer2atsomeperiodtbyyt.Asaconvention,nomatterwhotheproposeris, anofferalwaysindicatestheshareofplayer1.Toputitdifferently,player1proposesthesharehewantstoreceive,whereasplayer 2proposestheshareshewantstogive.Ourfirstpropositionconcernstheexistenceofequilibrium.

Proposition1. LetT_{beaT-period,bilateralbargaininggamewithtime-varyingfairnessconcerns.Anequilibriumof}T_exists. Proof. T_{isafinite-horizon,completeinformation,extensiveformgamewithcontinuousutilityfunctionsandcompactstrategy}

setsateachdecisionnode.Theexistenceresultaccordinglyfollows._

Itisworthmentioningthattheremayexistmultipleequilibriaandthatallequilibriaarepayoffequivalent:utilizingbackward induction,onecanseethatifaplayerpreferstomakeanofferthatcannotberejected,thenthereexistsauniqueoptimaloffer;but ifaplayerpreferstomakeanofferthatwillberejected,thenthereexistsinfinitelymanyoptimaloffers.Thelatterwouldleadtoa multiplicityofequilibria,whereastheformerguaranteesthatallequilibriaarepayoffequivalent.Notethattheplayersmaynotbe abletosimplyagreeontheequilibriumpayoffsinthefirstperiodsincethosepayoffsmaynotbefeasibleinthefirstperioddueto higherweightstheyattachtofairnessatthebeginningofthegame.

Oursecondpropositionconcernstheexistenceofdisagreementsanddelayedagreementsontheequilibriumpath.Forthat purpose,itisenoughtoanalyzeatwo-periodbargaininggame.

Proposition2. Let2_{beatwo-period,bilateralbargaininggamewithtime-varyingfairnessconcerns.Then,(i)thereexistsanequilibrium}

of_2_{withadisagreementand(ii)thereexistsanequilibriumof}2_{withadelayedagreement(i.e.,anagreementinthesecondperiod).} Proof. SeeAppendixA.

ThecorollarybelowshowsthatthesameresultsfollowforanyT-periodbargaininggame.

Corollary1. LetT_{beaT-period,bilateralbargaininggamewithtime-varyingfairnessconcerns.Then,(i)thereexistsanequilibriumof}

T_{withadisagreementand(ii)thereexistsanequilibriumof}T_{withadelayedagreement(i.e.,anagreementsometimeafterthefirst}

period).

Proof. TheproofsimplyfollowsfromProposition2sinceonecanaddarbitrarynumberofperiodstothetwo-periodgameanalyzed inProposition2.

4_{Notethatplayers’fairnessjudgmentsareself-servinginthattheydonotexperiencedisutilityiftheyreceiveasharelargerthantheirfairshare.}

Introducingthecounterpartforxi>ϕiisneitheressentialforourmodelnorchangesourresultsqualitatively.Hence,weoptforsimplicity.Satisfaction

withadvantageousinequityincognitivelydemandingsituationsissupportedbyexperimentalresearch,aswell(seeFalkandFischbacher,2006;Vanden Bosetal.,2006).

TheproofofProposition2maygivetheimpressionthattheincompatibilitybetweenplayers’fairnessjudgmentsnecessitatesa delayedagreement,ifnotadisagreement.Bycontrast,Proposition3showsthatevenwhenthereisnosuchconflictbetweenfairness judgments,itispossibletohavedelayedagreements.Thepropositionfurtherindicatesthatdisagreementisimpossibleinthatcase. Proposition3. Let2_{beatwo-period,bilateralbargaininggamewithtime-varyingfairnessconcerns.Letϕ}

1+ϕ2<1,i.e.,thereexists

anallocationwhichsatisfiesbothplayers’fairnessconcerns.Then,(i)theremaystillexistanequilibriumwithadelayedagreementbut(ii) thereexistsnoequilibriumwithdisagreement.

Proof. SeeAppendixA.

Noticethattheformerstatementisaboutapossibility.Whenthereisnoconflictbetweenplayers’fairnessjudgments,an immedi-ateagreementisgenerallyreached;however,theoppositemightalsobetrueinsomecases.Forinstance,incasethebestacceptable offerinthefirstperiodisunfairtooneoftheplayers5_{andtheweightsattachedtofairnessdecreasedramaticallyinthenextperiod,}

thenthatplayermayprefertoproceedtothenextperiodinwhichshemayfindsuchanunfairoffermoreacceptableduetohaving alowerweightforfairnessconcerns.Thelatterstatementinthepropositionisbasedonthefactthatifplayers’fairnessjudgments arecompatible,eithertheyimmediatelyagreeoriftheydonot,theremustbeabetteralternativeinthefuture,whichcannotbea stalemate.

Proposition4presentsanecessaryandsufficientconditionfortheexistenceofadisagreementinequilibrium.Infact,itboils downtogeneralizingthecondition(*)givenintheproofofProposition2.

Proposition4. LetT_{beaT-period,bilateralbargaininggamewithtime-varyingfairnessconcerns.Anequilibriumof}T _involves

disagreementifandonlyif 1+˛2(T)·(1−ϕ2)

1+˛2(T) <

˛1(T)·ϕ1

1+˛1(T). Proof. SeeAppendixA.

Theleft-hand-sideoftheinequalityinthepropositionistheshareofthepiewhichyieldsautilityof0toplayer2inperiodT, therebymakingplayer2indifferentbetweenagreementanddisagreement,whereastheright-hand-sideistheshareofthepiewhich yieldsautilityof0toplayer1inthesameperiod.Accordingly,theleft-hand-siderepresentsthehighestofferplayer2iswillingto acceptinperiodTandtheright-hand-siderepresentsthelowestofferplayer1iswillingtomakeinthesameperiod.Incasethe formerislessthanthelatter,thereexistsnoofferthatisacceptabletobothplayers.Thisimpliesadisagreement.Notethatifsucha conditionissatisfiedinperiodT,similarconditionswouldbesatisfiedinanyperiodt<T.

3.1.1. TheLengthofDelay:APartialCharacterization

Anaturalquestionatthispointis:Canweidentifythelengthofdelay(orequivalently,theagreementperiod)?Itturnsoutthat writingthelengthofdelayasafunctionofthemodelparametersisfarfromtrivial;andhence,isleftforfutureresearch.Herewe takeanintermediatestep.Inordertopartiallycharacterizetheagreementperiod,wemakeuseofthefollowinginterpretation: Considerafinitehorizon,alternatingoffersbargaininggamewithnofairnessconcerns(i.e.,withstandardutilityfunctions).Inthe firstperiod,playersbargainoverasetwithalinearfrontierpassingfromtheutilitypairs(0,1)and(1,0).6_{Asamatteroffact,a}

similarinterpretationfollowsforeachperiodwiththedifferencethatthebargainingsetinperiodtisalwayslargerthanthatin periodt+1foreveryt∈Nduetothediscountfactorıi∈(0,1)(seeFig.1).Thisisthereasonwhythereisanimmediateagreement

inthestandardmodel:supposethatplayersagreeinsomeperiodt>1andgetdiscountedutilitiesof ¯u1and ¯u2,wecaneasilysee

thatplayer1couldoffer ¯_u2toplayer2inthefirstperiodandreceiveahigherutilitythan ¯u1;acontradiction.

Inourmodelwithtime-varyingfairnessconcerns,thereisanumberofpossibilitiesregardinghowtheabove-describedbargaining setschangeacrossperiods.Forinstance,ifthereisaconflictbetweenplayers’fairnessconcernsandif˛i(t)issufficientlyhighforeach

player,thenitmaybethatthebargainingsetinperiodtdoesnotincludeautilitypairgreaterthanorequalto(0,0),invectorterms (seeS1_{inFig.2).Insuchaperiod,weknowforsurethatthereexistsnoagreement.Ontheotherhand,when˛}

i(t)issufficientlylow

foreachplayer,thenthereexistallocationsyieldingpositiveutilitiestobothplayers(seeS2_{inFig.2).Notethatifthisisthecasein}

someperiodt,thenforeveryperiodt>t,therealwaysexistallocationsyieldingpositiveutilitiestobothplayers.Furthermore,note thatduetotime-varyingfairnessconcerns,wecannotsimplysaythatthebargainingsetinperiodtisalwayssmallerorlargerthan thebargainingsetinperiodt_{forsomeperiodst}_,t_{>t.Thereasonisthatalthoughthediscountfactorshrinksthebargainingsetin}

thenextperiod,non-increasingfairnessconcernsexpandthebargainingset.Andtheseparametersmaybesuchthatthebargaining setsinperiodt_andt_{donotcontainoneanother(seeS}3_,S4_,andS5_inFig.2).

Becauseofthesedifferentformseachbargainingsetmaytake,itisaverydifficulttasktoidentifytheagreementperiodinclosed form.Thisisthereasonwhyweturntoapartialcharacterization.Westartwiththefollowingremark.

Remark1. LetSt_{denotethebargainingsetinperiodt.IfS}1_{doesnotincludeautilitypairgreaterthanorequalto(0,0),i.e.,iffor}

everyx∈[0,1],

x−˛1(1)·(ϕ1−x)<0 and 1−x−˛2(1)·(ϕ2+x−1)<0,

5 _{Althoughthereexistallocationswhichbothpartiesdeemasfair,an‘optimal’offermightbeunfairtooneofthepartiesduetotheasymmetrycreated}

bytheidentityoftheproposerinthefinalperiod.

6 _{Althoughthemodelrestrictsplayerstochooseautilitypaironthefrontier,wecanalsoassumethatplayersareabletochoosefromthewholebargaining}

Fig.1.Apossiblecollectionofbargainingsetsinthestandardmodel.

thenthereexistsdelayordisagreement.

Fromnowon,weavoidtrivialcasesandonlyconsidertheonesinwhichS1_{includesatleastoneutilitypairgreaterthan(0,0),}

invectorterms.Inthefollowing,wedefinetwoconceptsaboutthecorrespondingbargainingsetsineachperiodandusethem toprovideadditionalconditionsforhavingadelayedagreement.First,wedefinedominationinbargainingsetsanddescribethe conditionsforSt

⊂St_.

Definition1. LetT_{beaT-period,bilateralbargaininggamewithtime-varyingfairnessconcerns.Aperiodtdominatesperiodt} _/=t

inbargainingsetsif_St ⊂St_.

Thefollowingobservationsummarizestheconditionsforthebargainingsetinperiodttodominatethebargainingsetinperiod t /=t.

Observation1. LetT_{beaT-period,bilateralbargaininggamewithtime-varyingfairnessconcerns.Considertwoperiodstandt}_satisfying

u1



1−˛2(t)·ϕ2 1+˛2(t), ˛2(t)·ϕ2 1+˛2(t)



,t



>ıt_−t 1 ·u1



1−˛2(t)·ϕ2 1+˛2(t), ˛2(t)·ϕ2 1+˛2(t)



,t



u2



_˛ 1(t)·ϕ1 1+˛1(t), 1−˛1(t)·ϕ1 1+˛1(t)



,t



>ıt_−t 2 ·u2



_˛ 1(t)·ϕ1 1+˛1(t), 1−˛1(t)·ϕ1 1+˛1(t)



,t



_. If 1−˛2(t)·ϕ2 1+˛2(t)<ϕ1 and 1−˛1(t)·ϕ1 1+˛1(t)>ϕ2,

thenperiodtdominatesperiodtinbargainingsets;andif 1−˛2(t)·ϕ2

1+˛2(t)>ϕ1

or1−˛1(t)·ϕ1

1+˛1(t) >ϕ2,

thenperiodtdominatesperiodtinbargainingsetsonlywhent<t.

Thisobservationisbasedonthefactthatdominationinbargainingsetscanbeidentifiedusingtheendpointsofthecorresponding bargainingfrontiers.Thelatterpartoftheobservationindicatesthatwhenthereexistsnostrongconflict(i.e.,whenthereexistsa utilitypairwhichsatisfiesthefairnessconcernofoneplayerandyieldsapositiveutilityfortheotherplayer),onlyanearlierperiod candominateaparticularperiodinbargainingsets.7_{Utilizingthisobservation,wepresentthefollowingnecessaryconditionfora}

delayedagreement.

Remark2. Ifforeveryi∈{1,2}andeveryt∈{1,...,T}, ˛i(t)

˛i(1)>ı t−1 i ,

7_{RemembertheresultinProposition3:evenwhenthereisnoconflictinplayers’fairnessconcerns,t=1doesnotnecessarilydominatethefollowing}

Fig.2. Apossiblecollectionofbargainingsetsinourmodel.

thent=1dominatesalltheotherperiodsinbargainingsets.Accordingly,thereisanimmediateagreementinanyequilibrium of_T_{.Inordertohaveadelayedagreementinequilibrium,theinequalityabovemustbeviolatedforsomei∈{1,2}andsome}

t∈{1,...,T}.

Inotherwords,iftheinequalityinRemark2issatisfiedforbothplayersandforanyperiod,thenthediscountfactorsaresufficiently lowsuchthatdiscountingshrinksthebargainingsetsinthefollowingperiodsmorethannon-increasingfairnessconcernsexpands thebargainingsetintherespectiveperiod.Thisimpliesthatthefirstperioddominatesalltheotherperiodsinbargainingsets,so thatplayerswouldbebetteroffagreeinginthefirstperiod.

Below,weprovidethedefinitionofdominationthroughtheunionofbargainingsets,whichwillbeusefulin(partially)identifying theagreementperiod.

Definition2. Let_T_{beaT-period,bilateralbargaininggamewithtime-varyingfairnessconcerns.Acollectionofperiodst}1_,...,tk

dominatesanotherperiodtthroughtheunionofbargainingsetsif St_⊂



i∈{1,...,k}S ti

.

Furthermore,aperiodtisundominated,ifnocollectionofperiodsdominatesperiodtthroughtheunionofbargainingsets. Now,wearereadytodescribeapropertyoftheagreementperiodandanothersufficientconditionfordelay.

Remark3. InaT-periodbargaininggame,theagreementperiodt*_{isundominated.}8_{Thus,ifthefirstperiodisdominatedbysome}

collectionofperiods,thenthereexistsdelayordisagreement;andifthereexistsauniqueperiod ˆtwhichisundominated,then ˆtis theagreementperiod.

Asmentionedabove,theremaybemultipleundominatedperiodsinthepresenceoftime-varyingfairnessconcerns.Inthatcase, inwhichperiodplayerswillagreedependsonmultiplefactorssuchastheidentityoftheproposersintheundominatedperiods, theshapeofbargainingsetsintheundominatedperiods,theendpointsofbargainingfrontiersintheundominatedperiods,etc.For suchsituations,wearguethatsolvingfortheequilibriumistheonlywaytodeterminetheagreementperiod.

3.1.2. TheLengthofDelay:AnIllustrativeExample

Wenowprovideathree-periodexampleinwhichwecharacterizethesubgameperfectNashequilibrium.Thisistounderstand howtheequilibriumconditionsinourmodelappearandhowthevaluesofmodelparametersdeterminewhethertherewillbe anagreementandifsowhen.Notethat(i)anyT-periodgamecanbeanalyzedinasimilarmannerforanyoddT ∈Nand(ii)the equilibriumanalysisbelowcanbeextendedtothecaseswhereTiseven.

Example.Considerthebargaininggame_3_{andassumethatthedisagreementconditionpresentedinProposition4isnotsatisfied.}

AlthoughT=3,westicktogeneralnotationinthefollowinganalysis.

ConsiderperiodTinwhichplayer1istheproposer.Theanalysisforheroptimalofferhasalreadybeenmadeintheproofof Proposition4.Giventheassumptionof“nodisagreement”specifiedabove,player1offers

x∗ T=

1+˛2(T)·(1−ϕ2)

1+˛2(T)

whichwillbeacceptedbyplayer2.

ConsiderperiodT−1inwhichplayer2istheproposer.Sheanticipatesthatifherofferisrejected,thenshewillreceiveautility of0inperiodT,whereasplayer1willreceiveautilityofı1·u1((x∗_T,1−x∗_T),T).Accordingly,player2makesanoffersoastomake

player1indifferentbetweenaandr,ifsuchanofferyieldsplayer2autilitynotlessthan0.Wedivideourequilibriumanalysisinto sixcases:

Forthesecases,wedefine mT−1=ı1x ∗ T+˛1(T−1)·ϕ1 1+˛1(T−1) and nT−1=ı1· [x∗ T−˛1(T)·(ϕ1−x∗_T)]+˛1(T−1)·ϕ1 1+˛1(T−1) . CaseA[ϕ1≤ı1x∗_T≤x∗_Tandu2((ı1x_T∗,1−ı1x∗_T),T−1)≥0]:

Inthiscase,player1’sofferint=Tisfairforher.Suchanofferyieldsherautilityofx∗

T.Player2makesthefollowingoffer

y∗

T−1=ı1x∗_T.

CaseB[ϕ1≤ı1x∗_T≤x∗_Tandu2((ı1x∗_T,1−ı1x∗_T),T−1)<0]:

Itturnsoutthatanofferthatwillbeacceptedbyplayer1yieldsplayer2anegativeutility.Hence,player2canmakeanyofferless thanı1x∗_T,andanysuchofferwillberejectedbyplayer1.

CaseC[ı1x∗_T≤ϕ1≤x∗_Tandu2((mT−1,1−mT−1),T−1)≥0]:

Inthiscase,player1’sofferint=Tisfairforher.Suchanofferyieldsherautilityofx∗

T.Player2makesthefollowingoffer

y∗

T−1−˛1(T−1)·(ϕ1−y∗_T−1)=ı1x∗_T

y∗

T−1=mT−1.

CaseD[ı1x∗_T≤ϕ1≤x∗_Tandu2((mT−1,1−mT−1),T−1)<0]:

Itturnsoutthatanofferthatwillbeacceptedbyplayer1yieldsplayer2anegativeutility.Hence,player2canmakeanyofferless thanmT−1,andanysuchofferwillberejectedbyplayer1.

CaseE[ϕ1≥x∗_Tandu2((nT−1,1−nT−1),T−1)≥0]:

Inthiscase,player1’sofferint=Tisunfairforher.Suchanofferyieldsherautilityofx∗

T−˛1(T)·(ϕ1−x∗_T).Player2makesthe

followingoffer y∗ T−1−˛1(T−1)·(ϕ1−y∗_T−1)=ı1·[x∗_T−˛1(T)·(ϕ1−x∗_T)] y∗ T−1=nT−1. CaseF[ϕ1≥x∗_Tandu2((nT−1,1−nT−1),T−1)<0]:

Itturnsoutthatanofferthatwillbeacceptedbyplayer1yieldsplayer2anegativeutility.Hence,player2canmakeanyofferless thannT−1,andanysuchofferwillberejectedbyplayer1.

Tosumup,player2’sbestofferisdescribedbythefollowingpiecewiseset-valuedfunction:9

y∗ T−1=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

ı1x∗_T ifCaseA mT−1 ifCaseC nT−1 ifCaseE [0,ı1x∗_T) ifCaseB [0,mT−1) ifCaseD [0,nT−1) ifCaseF

ConsiderperiodT₋2inwhichplayer1istheproposer.IncasesA,C,andEabove:player1anticipatesthatifherofferisrejected, thenshewillreceiveautilityof

UT−1≡ı1·u1((y∗T−1,1−y∗T−1),T−1),

whereasplayer2willreceiveautilityof ı2·u2((y∗T−1,1−y∗T−1),T−1).

Accordingly,player1makesanoffersoastomakeplayer2indifferentbetweenaandr,ifsuchanofferyieldsplayer1autilitynot lessthanU_T−1.Wedivideourequilibriumanalysisintosixcases:

Forthesecases,wedefine kT−2=ı2·(1−y∗_T−1), mT−2= 1+˛2(T−2)·(1−ϕ2)−ı2·(1−y∗_T−1) 1+˛2(T−2) and nT−2= 1+˛2(T−2)·(1−ϕ2)−ı2·[1−y∗_T−1−˛2(T−1)·(ϕ2+y∗_T−1−1)] 1_+˛2(T−2) . Case1A[ϕ2≤kT−2≤1−y∗T−1andu1((kT−2,1−kT−2),T−2)≥UT−1]:

Inthiscase,player2’sofferint=T−1isfairforher.Suchanofferyieldsherautilityof1−y∗

T−1.Player1makesthefollowingoffer

x∗

T−2=kT−2.

Case1B[ϕ2≤kT−2≤1−y∗T−1andu1((kT−2,1−kT−2),T−2)<UT−1]:

Itturnsoutthatanofferthatwillbeacceptedbyplayer2yieldsplayer1anundesirableutility.Hence,player1canmakeanyoffer higherthankT−2,andanysuchofferwillberejectedbyplayer2.

Case1C[kT−2≤ϕ2≤1−y∗_T−1andu1((mT−2,1−mT−2),T−2)≥UT−1]:

Inthiscase,player2’sofferint=T−1isfairforher.Suchanofferyieldsherautilityof1−y∗

T−1.Player1makesthefollowingoffer

1−x∗

T−2−˛2(T−2)·(ϕ2+x∗_T−2−1)=ı2·(1−y_T−1∗ )

x∗

T−2=mT−2.

Case1D[kT−2≤ϕ2≤1−y∗_T−1andu1((mT−2,1−mT−2),T−2)<UT−1]:

Itturnsoutthatanofferthatwillbeacceptedbyplayer2yieldsplayer1anundesirableutility.Hence,player1canmakeanyoffer higherthanmT−2,andanysuchofferwillberejectedbyplayer2.

Case1E[ϕ2≥1−y∗_T−1andu1((nT−2,1−nT−2),T−2)≥UT−1]:

Inthiscase,player2’sofferint=T−1isunfairforher.Suchanofferyieldsherautilityof1−y∗

T−1−˛2(T−1)·(ϕ2+y∗_T−1−1).

Player1makesthefollowingoffer 1−x∗

T−2−˛2(T−2)·(ϕ2+x∗_T−2−1)=ı2·[1−y∗_T−1−˛2(T−1)·(ϕ2+y∗_T−1−1)]

x∗

T−2=nT−2.

Case1F[ϕ2≥1−y∗_T−1andu1((nT−2,1−nT−2),T−2)<UT−1]:

Itturnsoutthatanofferthatwillbeacceptedbyplayer2yieldsplayer1anundesirableutility.Hence,player1canmakeanyoffer higherthannT−2,andanysuchofferwillberejectedbyplayer2.

IncasesB,D,andFabove:player1anticipatesthatifherofferisrejected,thenshewillreceiveautilityof VT−1≡ı21·u1((x∗_T,1−x∗_T),T),

whereasplayer2willreceiveautilityof0.Accordingly,player1makesanoffersoastomakeplayer2indifferentbetweenaandr,if suchanofferyieldsplayer1autilitynotlessthanVT−1.Wedivideourequilibriumanalysisintotwocases:

Forthesecases,wedefine pT−2=1+˛₁2(₊T_˛−2)·(1−ϕ2)

2(T−2) . Case2A[u1((pT−2,1−pT−2),T−2)≥VT−1]:

Inthiscase,player2’sofferint=T−1isirrelevantandplayer1’sofferinperiodTyieldsplayer2autilityof0.Player1makesthe followingoffer

x∗

T−2=pT−2.

Case2B[u1((pT−2,1−pT−2),T−2)<VT−1]:

Itturnsoutthatanofferthatwillbeacceptedbyplayer2yieldsplayer1anundesirableutility.Hence,player1canmakeanyoffer higherthanpT−2,andanysuchofferwillberejectedbyplayer2.

Tosumup,player1’sbestofferisdescribedbythefollowingpiecewiseset-valuedfunction: x∗ T−2=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

kT−2 ifCase1A mT−2 ifCase1C nT−2 ifCase1E pT−2 ifCase2A (kT−2,1] ifCase1B (mT−2,1] ifCase1D (_nT−2,1] ifCase1F (pT−2,1] ifCase2B

Thiscompletesouranalysisofequilibriuminclosedform.Now,weconsiderdifferentsetsofparametervaluesinordertoillustrate thatagreementinanyperiodispossible.Wefirstfix

ϕ1=ϕ2=0.8 and ı1=ı2=0.8

If˛1(T)=˛2(T)=2,thenitfollowsbyProposition4thatwehaveadisagreement.Instead,assumingthat

˛1(T) =˛2(T)=1

˛1(T−1) =˛2(T−1)=2

wehaveadelayedagreementinperiodT=3.Assumingthat ˛1(T) =˛2(T)=2/3

˛1(T−1) =˛2(T−1)=3/4

˛1(T−2) =˛2(T−2)=2

wehaveadelayedagreementinperiodT−1=2.Finally,consideringstandardpreferenceswithnofairnessconcerns,wewouldhave animmediateagreementinperiodT₋2=1._♦

3.2. Simulations

Inthissubsection,wereportsimulationresultsidentifyingtheagreementperiodandanalyzehowitrespondstochangesinthe modelparameters.10_{Forthesakeoftractability,weconsiderthefollowingfunctionalformfor˛}

i(·):

˛i(t)=iati

foreveryi_∈{1,2_},andusethefollowingvaluesforthemodelparameters T ∈ {20,21}

ıi ∈ {0.95}

ϕi ∈ {0.1,0.2,...,0.9,1}

i ∈ {6,24}

ai ∈ {0.1,0.2,...,0.8,0.9}

foreveryi∈{1,2},whichcorrespondstoatotalof64,800numericalresults.11

ThenumberofperiodsTischoseninordertocapturesufficientamountofvariabilityintheagreementperiods.The func-tionalformfor˛i andthecorrespondingparametervaluescoverafairlywiderangeof interestingsituations.For instance,if

ai=0.1and i=6,thenplayer i alwaysvalues materialgain morethanshe valuesfairness; ifai=0.9and i=24,thenplayer

i cares aboutfairnessat the beginningof thegame so much thather materialgain never becomes relativelymore impor-tantthroughoutthegame;andinalloftheremainingcases,fairnessisrelativelymorevaluableforplayeriatthebeginning, butitbecomesrelativelylessvaluablesomewhereduringthegameasthedeadlineapproaches.Theswitchingperiod signif-icantly depends ontheactualvaluesof these parameters:ai governs thespeed of decreaseinplayer i’sweight for fairness

concerns and i determines whethersuch a speed is enough to make thematerial gain relativelymore important before

theend.

10_{ThesimulationsareconductedinVisualStudiousingC++astheprogramminglanguage.}

11_{Duetospacelimitations,wereportasmallsubsetoftheseresultsinFigs.3and5.Allequilibriumagreementperiodscanbefoundintheonlineappendix}

Fig.3.TheagreementperiodswhenT=20.(Boldfacenumbersinrowsandcolumnsarethevaluesforϕ1andϕ2,respectively.)

Toprovideaclearunderstanding,herewerestrictthesetofparametervaluesandreporttheresultsinFig.3.12_{Asitcanbeseen}

inthisfigure,weobserveawiderangeofagreementperiods;withanimmediateagreementwhenplayers’beliefabouttheirfair sharesaresufficientlylow,andwiththeagreementperiodgenerallyincreasingasϕi,ai,oriincreasereachingamaximumvalueof

19whenϕ1=ϕ2=0.8,a1=a2=0.8,and1=1=24.

Below,wepresenttwoobservationsfromoursimulationresults.Thefirstobservationisaboutacommonlyreportedfindingin theexperimental/empiricalliteratureonbargaining:thedistributionofagreementperiodsisU-orJ-shaped,meaningthatplayers mostlyagreeeitherclosetothebeginningorclosetothedeadline,withlatterbeingobservedmorefrequently,buttheyrarelyagree inthemiddleofthegame(seeRothetal.,1988;Kessler,1996;GächterandRiedl,2005;Karagözo˘gluandRiedl,2015;Sullivan,2016; VassermanandYildiz,2016,amongothers).Thisobservationcanbesupportedbyourtheoreticalmodelforsomevaluesofmodel parameters.Forexample,considera1=a2=0.9and1=2=6whenT=20andı1=ı2=0.95.Settingperiods1–7asthebeginningpart

andperiods15–20astheendpartofthegame,andassumingthateachplayer’sbeliefaboutherfairshareisuniformlydistributed over_{0.2,...,0.9_},13_{wecanarguethatthedistributionofagreementperiodsisU-shaped(seethechartontheleftinFig.4).Asfor}

anotherexample,considera1=a2=0.8,1=2=24,andϕ1,ϕ2∈{0.3,...,0.9}keepingtheotherparametervaluesunchanged.Now,

thedistributionofagreementperiodsbecomesJ-shaped(seethechartontherightinFig.4).

Thesecondobservationisthattheagreementperiodisnotnecessarilymonotonicinaplayer’sbeliefaboutherfairshareϕior

inherfairnessweightparameterai.Thatis,asoneoftheseparametersincreasesforaplayer,itmayturnoutthatplayersreachan

agreementmorequickly.Forexample,fixinga1=a2=0.8and1=2=6,playersagreeinthe15thperiodwhenϕ1=0.8andϕ2=0.5;

whereastheyagreeinthe13thperiodwhenϕ1remainstobe0.8andϕ2increasesto0.6.Althoughsuchcasesarequiteuncommon

inthedomaincoveredbytheparametervaluesweused,theydoexist,preventingustoreachageneralmonotonicityresult. Theintuitionbehindthissomewhatcounterintuitiveresultisasfollows.Considerthenumericalexampleabove.Wecanseevia backwardinductionthatplayerswouldagreeinthe17thperiodyieldinganexpectedutilityof0.3399toplayer2inbothcases.From thatagreementplayer1’sexpectedutilityisequalto0.2727incase(i)ϕ2=0.5andto0.2667incase(ii)ϕ2=0.6.Simply,inthelatter

case,player1hastoofferahighersharetoplayer2inordertomakeherindifferentbetweenacceptingandrejectingtheoffer,so thatplayer1’sownexpectedutilityturnsouttobelower.Now,incase(i),player1hasachancetoofferanamountyielding0.3399 toplayer2inthe15thperiodincreasingherownexpectedutilityto0.2910.Thus,playersagreeinthe15thperiod.Ontheother hand,incase(ii),player1doesnotwishtoofferanexpectedutilityof0.3399toplayer2inthe15thperiod.Thereasonisthatplayer 2’sfairamountisnowhighersothatanunfairdivisionwouldgiveherahigherdisutility(incomparisontotheformercase),sothat makingplayer2indifferentbetweenacceptingandrejectingturnsoutnottobebeneficialforplayer1.Thus,theydonotagreeinthe 15thperiod.Finally,inthe13thperiodofbothcases(i)and(ii),player1istoofferanexpectedutilityof0.3399toplayer2inorderto makeherindifferentbetweenacceptingandrejectingtheoffer.However,incase(i),player1wouldmaketheofferonlyifityields anexpectedutilitynolessthan0.2910toherself;whereasincase(ii),shewouldmaketheofferonlyifityieldsanexpectedutility

12 _{ForT=21,forwhichthelastproposerisplayer1,theresultsarerelegatedtotheAppendixA.} 13 _{Arguably,casesinwhichϕ}

Fig.4.U-orJ-shapedagreements.

nolessthan0.2667toherself.Giventhattheofferthatmakesplayer2indifferentyieldsanexpectedutilityof0.2854toplayer1, theofferismadeincase(ii)leadingtoanagreementinthe13thperiod,butnotmadeincase(i).Asimilarresultaswellasasimilar interpretationarevalidforai.14

4. Conclusion

Westudyafinitehorizonbargaininggamebetweentwoplayerswhovaluenotonlytheabsoluteamounttheyreceivebutalsothe relativeshare.Thenoveltyofourmodelisthattheweightsplayersattachtofairnessdecreaseovertimeasthedeadlineapproaches. Thus,asthepushcomestoshove(i.e.,asthedeadlineapproachesandlosingthemoneyonthetablebecomesmoresalient),players becomemorefocusedonmaterialgainsandlessfocusedonfairness.Thismodelingassumptionisinspiredbytherecentexperimental studiesonhotvs.coldpsychologicalstates,therelativeprimacyofeconomicvs.socialconcerns,timepressure,andthetemporal instabilityofjusticesensitivity.

Ourresultsshedlightontheinfluenceoffairnessjudgmentsondelays,last-minuteagreements,anddisagreementsindynamic bargainingsituations.We,first,analyticallyshowthatdelayordisagreementmayexistintheequilibriumpath.Then,weprovide necessaryandsufficientconditionsfordisagreementanddelay.Wealsofindthatmutuallycompatiblefairnessjudgmentsdonot guaranteeanimmediateagreement.Ouranalyticalresultsshowthatwhetherplayerswillbeabletoreachanagreementandifso whendependsonmultiplefactorssuchasplayers’fairnessjudgments,thepaceofthedecreaseintheirweightsforfairness,the iden-titiesofplayers(i.e.,proposerorresponder)indifferentperiods,andthehorizonlength.Ourmodelproducesempiricallyrelevant outcomepatternssuchasU-shapedorJ-shapeddistributionofagreementtimes,whicharefrequentlyobservedindynamic bargain-ingexperiments.Finally,oursimulationresultsrevealsomeinterestingcomparativestaticresults.Forinstance,moredemanding fairnessjudgmentsorstrongerconcernsforfairnessmayleadtofasteragreements.

Herewepresentedanempiricallyinspiredmodelofbargaining,whichsuccessfullymatchesmultipleregularitiesinthe experi-mental/empiricaldata.Tothebestofourknowledge,thisisthefirstfinite-horizonalternating-offersbargainingmodelthatproduces delayanddisagreementinequilibriumwithoutresortingtousualincompleteorimperfectinformationassumptions.15_Obviously,

tobetterunderstandtherelationshipbetweentime-varyingfairnessconcernsanddelay/disagreementinbargaining,experiments particularlydesignedtotestthepredictionsofourmodelshouldbeconducted.

Thisisafirststepinincorporatingtime-varyingfairnessjudgmentsintobargainingmodels.Afurtherstepwouldbetomodelthe micro-foundationsoftime-varyingfairnessjudgments.Toachievethisgoal,webelievethatexperimentsinvestigatingthechangesin brainactivityintheregionsthatreacttounfairness—asaproxyforchangesinthevaluepeopleattachtofairness—duringnegotiations withfMRIstudiesoralternativemethodssuchasskinconductancewouldbenaturalcomplements.

Acknowledgments

Wewouldliketothanktheeditor,twoanonymousreviewers,andGaryCharnessforusefulcommentsandsuggestions.Weare gratefultoHamideTuranforherhelpinwritingthecodeusedforthesimulationresults.Wealsothankaudiencesatthe“Bargaining: TheoryandExperiments”workshopatBilkentUniversityandthe2ndTurkishExperimentalandBehavioralEconomicsWorkshop at ˙IstanbulBilgiUniversity.EminKaragözo˘gluthanksTÜB˙ITAK(TheScientificandTechnologicalResearchCouncilofTurkey)forthe post-doctoralresearchfellowship,andMassachusettsInstituteofTechnology,DepartmentofEconomicsfortheirhospitality.The usualdisclaimersapply.

AppendixA.

ProofofProposition2. Considert=2inwhichplayer2istheproposer.Shemakesanoffer,whichmakesplayer1indifferent betweenplayingaandr.Notingthatplayingrwouldyieldplayer1autilityof0,wefindplayer2’soptimalofferasfollows:

14_{Asamatteroffact,onecanfindnumericalexamplesforthenon-monotonicityina}

iintherestrictedsetofexampleswehavereportedinFigs.3and5. 15_{Incorporatinguncertaintyintotheevolutionofplayers’fairnesspreferencesovertimemaybeareasonableextension.Nevertheless,itwould}

y∗ 2−˛1(2)·(ϕ1−y∗2)=0 y∗ 2= ˛1(2)·ϕ1 1+˛1(2)

Noticethatplayer2wouldnotprefertomakethisoffer,iftheremainingpieamountyieldsheranegativeutility;i.e.,if16

1−y∗

2−˛2(2)·(ϕ2+y∗2−1)<0. (*)

Ifthisisthecase,player2makessomeofferlessthany∗

2,whichwillthenberejectedbyplayer1.So,therewillnotbeanyagreement

int=2.Ifthisisnotthecase,therewillbeanagreementontheallocation(y∗ 2,1−y2∗).

Now,tocompletetheproof,wemustshowthatthebargainingproceedstot=2.Considert=1.Notethatplayer1istheproposer andthatsheanticipatestheoutcomeoft=2.Shemakesanofferthatmakesplayer2indifferentbetweenplayingaandr.Notingthat playingrwouldyieldplayer2autilityof

u2((y∗2,1−y∗2),2)

int=2,wedividetheproblemintotwocases: (i) ϕ2≤1−y∗2 and (ii) ϕ2>1−y∗2.

Incase(i),player2’sutilityis1−y∗

2.Dividethiscaseintotwosubcases:

(i.a) ϕ2≤ı2·(1−y∗2) and (i.b) ϕ2>ı2·(1−y∗2).

Incase(i.a),player1’soptimalofferis: x∗

1=1−ı2·(1−y∗2)

Incase(i.b),player1’soptimaloffercanbefoundbythefollowing: 1−x∗ 1−˛2(1)·(ϕ2+x1∗−1)=ı2·(1−y∗2) x∗ 1= 1+˛2(1)·(1−ϕ2)−ı2·(1−y∗₂) 1+˛2(1)

Incase(ii),player2’sutilityis1−y∗

2−˛2(2)·(ϕ2+y∗2−1).Player1’soptimaloffer—whichshouldbeunfairforplayer2—canbe

foundbythefollowing: 1−x∗ 1−˛2(1)·(ϕ2+x∗1−1)=ı2·[1−y∗2−˛2(2)·(ϕ2+y∗2−1)] x∗ 1= 1+˛2(1)·(1−ϕ2)−ı2·[1−y∗2−˛2(2)·(ϕ2+y∗2−1)] 1+˛2(1) Tosumup,ifx∗

1yieldsanegativeutilitytoplayer1int=1foroneoftheabovecases,thenforthatparticularcase,thereis

noagreementinthefirstperiod.Notethatthisispossible,if˛1(1)issufficientlyhigh.Ifthishappensincase(i),thenthereisan

agreementint=2.Ifthishappensincase(ii),thenthereisanagreementint=2iftheequation(*)isnotsatisfied,andthereis disagreementiftheequation(*)issatisfied.

ProofofProposition3. Fortheformer,weprovideanexampleinwhichthereexistsdelay.Consideratwo-periodmodelwith ı1 =ı2=0.9,

ϕ1 =ϕ2=0.4,

˛1(2) =˛2(2)=0.25,and

˛1(1) =˛2(1)=1.

Considert=2inwhichplayer2istheproposer.Shemakesanoffersoastomakeplayer1indifferentbetweenaandr.Thisoffer is_y∗

2=0.08,whichyieldsautilityof0toplayer1andadiscountedutilityof0.828toplayer2.Now,considert=1.Anticipatingthe

outcomeoft=2,player1’soffercouldbetotake0.172forherself,andtherebymakingplayer2indifferentbetweenplayingaandr. However,suchanofferwouldyieldautilityof0.172−0.228=−0.056<0.Then,player1wouldprefermakinganofferwhichplayer 2rejects.Hence,inequilibrium,agreementisreachedwithdelay.

Forthelatter,supposethatthereexistsdisagreement.ThisalsomeansthatplayersdidnotagreeuntilperiodT(i.e.,theyreached thefinalperiod).ConsiderperiodT.Withoutlossofgenerality,assumethatplayer1istheproposer.Inthisperiodthereexistsanoffer forplayer1whichyieldsapositiveutilitytoherselfandautilityof0toplayer2,sinceϕ1+ϕ2<1byassumption.Sincedisagreement

16 _{Thisisnotnecessarilyplayer2’sutility,becauseif1−y}∗

2isfairforher,thenherutilitywouldbe1−y∗2.Insuchacase,player2getsapositiveutilityfor

yieldsautilityof0tobothplayers,player1woulddeviatetosuchanofferandplayer2wouldaccept.Thisisacontradiction.Hence, thereisalwaysanagreement.

ProofofProposition4. AssumethatTisodd(forevenT,similarargumentsfollow).ConsiderperiodTinwhichplayer1isthe proposer.Shemakesanofferthatmakesplayer2indifferentbetweenplayingaandr.Notingthatplayingrwouldyieldplayer2a utilityof0,wefindplayer1’soptimalofferasfollows:

1−x∗ T−˛2(T)·(ϕ2+xT∗−1)=0 x∗ T= 1+˛2(T)·(1−ϕ2) 1+˛2(T)

Now,ifsuchanofferisnotbeneficialforplayer1,i.e.,if x∗ T−˛1(T)·(ϕ1−x∗_T)<0 x∗ T<˛1 (T)·ϕ1 1+˛1(T)

thenplayer1wouldoffersomethinghigherthanx∗

Tandplayer2wouldreject.

Tosumup,if 1+˛2(T)·(1−ϕ2)

1+˛2(T) <

˛1(T)·ϕ1

1+˛1(T),

thereisnoagreementinthefinalperiod.Noticethatifthisisthecase,thentherewouldnotbeanagreementinthepreviousperiods either.Tobemoreprecise,ifthereisnoallocationthatyieldsautilitypairgreaterthan(0,0)inperiodT,thenthereisnoallocation thatyieldssuchautilitypairinthepreviousperiods.Thisisbecause_˛i(t)≥˛i(T)foreveryi∈{1,2}andt<T.Thiscompletesthe

proof.

SimulationresultswhenT=21:

Fig.5. TheagreementperiodswhenT=21.(Boldfacenumbersinrowsandcolumnsarethevaluesforϕ1andϕ2,respectively.)

References

Abreu,D.,Gul,F.,2000.Bargainingandreputation.Econometrica68(1),85–87.

Admati,A.D.,Perry,M.,1987.Strategicdelayinbargaining.Rev.Econ.Stud.54(3),345–364.

Babcock,L.,Loewenstein,G.,1997.Explainingbargainingimpasse:theroleofself-servingbiases.J.Econ.Perspect.11(1),109–126. Babcock,L.,Loewenstein,G.,Issacharoff,S.,Camerer,C.,1995.Biasedjudgmentsoffairnessinbargaining.Am.Econ.Rev.85(5),1337–1343. Bac,M.,2000.Signalingbargainingpower:strategicdelayversusrestrictedoffers.Econ.Theory16(1),227–237.

Birkeland,S.,Tungodden,B.,2014.Fairnessmotivationinbargaining:amatterofprinciple.TheoryDecis.77(1),125–151.

Bolton,G.E.,Karagözo˘glu,E.,2016.Ontheinfluenceofhardleverageinasoftleveragebargaininggame:theimportanceofcredibleclaims.GamesEcon. Behav.99,164–179.

Bolton,G.E.,Ockenfels,A.,2000.ERC:atheoryofequity,reciprocity,andcompetition.Am.Econ.Rev.90(1),166–193. Bolton,G.E.,1991.Acomparativemodelofbargaining:theoryandevidence.Am.Econ.Rev.81(5),1096–1136. Cho,I.K.,1990.Uncertaintyanddelayinbargaining.Rev.Econ.Stud.57(4),575–595.

Cramton,P.,1991.Dynamicbargainingwithtransactioncosts.Manage.Sci.37(10),1221–1233.

Dreber,A.,Fudenberg,D.,Levine,D.K.,Rand,D.,2016.Self-control,socialpreferencesandtheeffectofdelayedPayments.WorkingPaper.MIT. Falk,A.,Fischbacher,U.,2006.Atheoryofreciprocity.GamesEcon.Behav.54,293–315.

Fanning,J.,2016.Reputationalbargaininganddeadlines.Econometrica84(3),1131–1179.

Fehr,E.,Schmidt,K.M.,1999.Atheoryoffairness,competition,andcooperation.Q.J.Econ.114(3),817–868. Feinberg,Y.,Skrzypacz,A.,2005.Uncertaintyaboutuncertaintyanddelayinbargaining.Econometrica73(1),63–91.

Fershtman,C.,Seidmann,D.,1993.Deadlineeffectsandinefficientdelayinbargainingwithendogenouscommitment.J.Econ.Theory60(2),306–321. Fortin,M.,Cojuharenco,I.,Patient,D.,German,H.,2016.Itistimeforjustice:howtimechangeswhatweknowaboutjusticejudgmentsandjustice

effects.J.Organ.Behav.37,S30–S56.

Fudenberg,D.,Levine,D.K.,2006.Adualselfmodelofimpulsecontrol.Am.Econ.Rev.96(5),1449–1476. Gächter,S.,Riedl,A.,2005.Moralpropertyrightsinbargainingwithinfeasibleclaims.Manage.Sci.51(2),249–263.

Gantner,A.,Güth,W.,Königstein,M.,2001.Equitablechoicesinbargaininggameswithjointproduction.J.Econ.Behav.Organ.46(2),209–225. Gul,F.,Sonnenschein,H.,1988.Ondelayinbargainingwithone-sideduncertainty.Econometrica56(3),601–611.

Halali,E.,Bereby-Meyer,Y.,Ockenfels,A.,2013.Isitallabouttheself?Theeffectofself-controldepletiononultimatumgameproposers.Front.Hum. Neurosci.7,240.

Hochman,G.,Ayal,S.,Ariely,D.,2015.Fairnessrequiresdeliberation:theprimacyofeconomicoversocialconsiderations.Front.Psychol.6,747. Jehiel,P.,Moldovanu,B.,1995.Negativeexternalitiesmaycausedelayinnegotiation.Econometrica63(6),1321–1335.

Karagözo˘glu,E.,Kocher,M.,2016.Bargainingundertimepressure,CESifoWorkingPaper#5685.

Karagözo˘glu,E.,Riedl,A.,2015.Performanceinformation,productionuncertainty,andsubjectiveentitlementsinbargaining.Manage.Sci.61(11), 2611–2626.

Karagözo˘glu,E.,2014.Adynamicmodelofdissonancereductioninamodularmind.J.Neurosci.Psychol.Econ.7(1),15–32. Kessler,D.,1996.Institutionalcausesofdelayinthesettlementoflegaldisputes.J.LawEcon.Organ.12(2),432–460.

Knoch,D.,Fehr,E.,2007.Resistingthepoweroftemptations:therightprefrontalcortexandself-control.Ann.N.Y.Acad.Sci.1104,123–134. Knoch,D.,Pasqual-Leone,A.,Meyer,K.,Treyer,V.,Fehr,E.,2006.Diminishingreciprocalfairnessbydisruptingtherightprefrontalcortex.Science314,

829–832.

Ma,C.A.,Manove,M.,1993.Bargainingwithdeadlinesandimperfectplayercontrol.Econometrica61(6),1313–1339. Merlo,A.,Wilson,R.,1998.Efficientdelaysinastochasticmodelofbargaining.Econ.Theory11(1),39–55.

Moore,D.,Loewenstein,G.,2004.Self-interest,automaticity,andthepsychologyofconflictofinterest.Soc.JusticeRes.17,189–202. Romm,A.,2016.Buildingreputationattheedgeofacliff.WorkingPaper.HebrewUniversityofJerusalem.

Roth,A.E.,Murnighan,J.K.,Schoumaker,F.,1988.Thedeadlineeffectinbargaining:someexperimentalevidence.Am.Econ.Rev.78(4),806–823. Rubinstein,A.,1982.Perfectequilibriuminabargainingmodel.Econometrica50(1),97–110.

Rubinstein,A.,1985.ABargainingmodelwithincompleteinformationabouttimepreferences.Econometrica53(5),1151–1172. Simsek,A.,Yildiz,M.,2014.Durability,Deadline,andelectioneffectsinbargaining.WorkingPaper.MIT.

Sullivan,S.P.,2016.Whywaittosettle?Anexperimentaltestoftheasymmetric-informationhypothesis.J.LawEcon.59(3),497–525.

VandenBos,K.,Peters,S.L.,Bobocel,D.R.,Ybema,J.F.,2006.Onpreferencesanddoingtherightthing:Satisfactionwithadvantageousinequitywhen cognitiveprocessingislimited.J.Exp.Soc.Psychol.42(3),273–289.

Vasserman,S.,Yildiz,M.,2016.Pretrialnegotiationsunderoptimism.WorkingPaper.MIT. Yildiz,M.,2004.Waitingtopersuade.Q.J.Econ.119(1),223–248.