On the influence of hard leverage in a soft leverage bargaining game: the importance of credible claims

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Contents lists available atScienceDirect

Games

and

Economic

Behavior

www.elsevier.com/locate/geb

On

the

inﬂuence

of

hard

leverage

in

a

soft

leverage

bargaining

game:

The

importance

of

credible

claims

Gary

E. Bolton

a

,

Emin Karagözo˘glu

b

,

∗

a_Naveen_Jindal_School_of_Management,_Department_of_Managerial_Economics,_University_of_Texas_at_Dallas,_Richardson,_TX_75080,

United States

b_Department_of_Economics,_Bilkent_University,_Bilkent_06800,_Ankara,_Turkey

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received15January2015 Availableonline26August2016 JELclassiﬁcation: C91 D33 D63 Keywords: Bargaining Credibility Experiments Hardleverage Softleverage

What makes a bargaining proposal credible? We study how hard leverage (binding commitment) influences soft leverage (appealing to a focal point) in a rich-context bargaininggameknowntoexhibitcompetingclaimstofocality.Inthreetreatments,our experimentvaries onebargainer’sability tocommit, holdingthesoftleverage condition fixed. As in previous studies, we observe that opening offers are consistent with the availablesoftleverage.Theinfluenceofhardleverageismostevidentintheconcessionary stage.Hardandsoftleverageinteracttoinfluenceoutcomes:Hardleverageadvantagesits holder,yetsettlementslargelystaybetweenthetwofocalpoints.Wepositthatfocalpoints induce mutual expectations of bargainer social preferences that, when combined with the Nashbargainingsolution, imply the comparative statics concerning the settlements observed.Asimilarly modifiedversion oftheZeuthen–Harsanyimodelofthebargaining process,agreeswiththecomparativestaticsonopeningoffersandconcessions.

[E]achpartywillmakeaconcessionatagivenstageofthenegotiationifandonlyifhethinkshehasatleastasmuch”reason”as hisopponenthastoyieldgroundatthatpoint.

[JohnC.Harsanyi(1956)]

[I]toftenlooksliketheultimatefocusofagreementdidnotjustreﬂectthebalanceofbargainerpowerbutprovidedbargaining advantagetoonesideortheother.

. . .

Onehastohaveareasonforstandingﬁrmlyonaposition.

[ThomasC.Schelling(1960)]

1. Introduction

What makes a bargaining proposal credible? Where will a bargainer make a stand and where will he back down?

Both Harsanyi andSchelling argued that the answer hasto do withreasons.Harsanyi argued that thereasons could be deduced from the mathematical game formand bargainer rationality. From this view, we derive hardleverage strategies for establishing credibility, strategies such asbinding commitments. Schellingargued that focal points –context speciﬁc variables extraneous to themathematical deﬁnitionof thegame form– can providebargaining advantageas well.Focal points shaped by “analogy, precedent

. . .

and who the partiesare and what they know abouteach other” amongother

*

Correspondingauthor.

E-mailaddresses:gbolton@utdallas.edu(G.E. Bolton),

karagozoglu@bilkent.edu.tr

(E. Karagözo˘glu). http://dx.doi.org/10.1016/j.geb.2016.08.005

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variables(Schelling,p.57).1 Focalpointsgiverisetosoftleverage strategiesforestablishingcredibility,softinthesensethat theyrelyonmutual perceptionstocoordinatebargainerexpectations.2

Intheeconomicsliterature, hardandsoftleveragelivelargely separatelives.Softleveragestrategieshavebeen inves-tigated mostly in bargaininggames with symmetric strategy spaces (ex.,Roth, 1995).3 Hardleverage strategies play the central role in asymmetric bargaininggames, particularlythose wherecommitment power is distributed unequally (ex.,

Tadelis,2013,chapter11).Onereasontowonder whetherthisdivisionoflabor characterizesthe entirerelationship hard andsoft leverageshare is that manyif not mostasymmetric bargainingencounters take place in a rich context. A‘rich context’isonethatadmitsappealtoanalogy,precedent,etc.Inarichcontext,evenifonesideinthenegotiationhasahard leverageadvantage,bargainerscanﬁndgristtofashionsoftleveragestrategiesaswell.

Infact,arich contexttypicallyadmitscompetingclaimstofocality.Togivean examplethatwillbe relevanthere, one bargainer might arguefor a 50–50split whileanother mightargue foran asymmetric split based onprecedent. Studies suggest that competing focal points can act to coordinate expectations.In assessing a series of bargaining experiments conductedwithcolleagues, Roth (1985) conjectured that,“[The] bargainers sought toidentify initial bargainingpositions thathadsomespecialreasonforbeingcredible,andthatthesecrediblebargainingpositionsthenservedasfocalpoints that inﬂuencedthesubsequentconductofnegotiations.”Someofthegamesendedinasettlementatoneortheotherfocalpoint, whileother settlementsappeared ascompromisesbetweenfocalpoints. GächterandRiedl (2005)studyinga richcontext bargaininggameusinganovelexperimentaldesign,foundthatmostobservedsettlementsappearascompromisesbetween focalclaims.GächterandRiedl (2006) foundthatvaryingtheasymmetricfocalpointleadstoashiftinthedistributionof settlements.

Themajorgoalofourexperimentistogeneratedatathatinformsusofwhatatheorythat accountsforbothsoftand hard leverageneedexplain. Wefashion ourguidinghypothesesfromtherobust patternsofbehaviorpreviously observed insymmetricandasymmetricbargaininggameexperiments.Thebaselinetreatmentisasymmetricbargaininggame. Bar-gainersare givena fixedamountoftime tocommunicate andreach amutualagreementina freeformsetting.Wethen createan asymmetricbargaining game by givingthe highperformer the option,which hecan exercise at his discretion duringthebargaining,tomakeafinalofferthatcannotberevoked(hencethehighperformerhasfullcommitmentpower). Athirdtreatmentthendrawsthebargaininggameclosertoonethat isfullyasymmetric:Thehighperformerisrestricted toafinalofferthatthelowperformermusteitheracceptorreject,withnofurthercommunication,eliminatingtheability ofthelowerperformer toaskforconcessions.Weexamine howtreatmentmanipulationsaffectboth settlementsandthe processofthenegotiations.Tomorefullygaugethecoordinatinginfluenceofthefocalpointsandcommitmentoption,we askbargainerspriortobargaining,whattheythinkaneutralarbitratorwouldjudgetobeafairsettlement.

From the data,theobserved influence ofhard leverageon softleverage runsasfollows:The critical influenceofsoft leverageis onfairnessjudgmentsandonthe openingoffersmadeby bargainers.Theinfluence ofhardleverage isfound in concession behavior. Abargainer facing a bargainer withthe final offer option is theone morelikely to bendin the concessionprocess.Hardleverageholdersdobestinthefinaloffertreatment(withmodaloffersandrejectionratesbelow thoseinatypicalultimatumgame).

None ofour original hypotheses provides a totally satisfactory explanationfor what we observe.Instead, we propose asimplemodelinwhichfocal pointsinduce mutualexpectationsofthe socialpreferencesofthebargainers.Thesesocial preferencesareageneralizationofthoseusedtoexplain bargainingoutcomesinlowcontextbargaininggames.Modifying theNashbargainingsolution(Nash,1950) toreﬂectthesepreferencesimpliesthecomparativestaticsweobserveregarding settlements.ModifyingtheZeuthen–Harsanyibargainingmodel(closelyassociatedwiththeNashsolutionintheliterature) inasimilarmannerimpliesthecomparativestaticsweobservewithregardtoopeningoffersandconcessions.

To the bestof our knowledge, our work is the first to studyboth hard and softleverage in rich context bargaining. Ourworkcomplementsotherrecentstudiesoftheinfluencesonbargainingsettlements. Birkeland (2013),inabargaining gamewithproductionandarbitration,showsthatsettlementscanbeapproximatedbythemodalfocaloutcomes bargain-ersattribute tothe arbitratorpool.Anbarcı andFeltovich(2013,2014) studybehaviorinbargaininggames that differby disagreementoutcomes;andfindthatsettlementsaresubstantiallylesssensitivetothelevelofdisagreementpayoffsthan standard theory suggests. Our resultsshare a similar flavor in that we find that hard leverage isless effectivethan one would expect.Rode andLe Menestrel (2011), using dictatorand repeatedNash demand games, findthat distributionof powerinfluencesbothfairnessjudgmentsanddistributivedecisions.

The organization of the paper is as follows: Section 2 describes the experimental design and hypotheses. Section 3

reports theresults. Section 4 presents themodiﬁed Zeuthen–Harsanyimodel anddiscusses how it ﬁts with ourresults. Section5presentsconcludingremarks.

1 _In_the_negotiation_literature_(ex.,

_Fisher

_et_al.,₂₀₁₁_),_common_focal_points_include_market_value,_ethical_standards,_scientiﬁc_judgment,_professional

codes,eﬃciency,cost,legalprecedent,tradition,reciprocityandfairprocedures.

2 _See

_{Innocenti (2008)}

_for_{a historical}_account_of_Harsanyi’s_and_{Schelling’s}_contrasting_views_on_bargaining.

3 _Focal_points_are_also_central_to_the_study_of_tacit_bargaining;_ex.,

_Isoni

_et_{al. (2013)}_and

_Embrey

_et_{al. (2014)}_._Here_we_focus_on_direct_(or_explicit)

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2. Experimentaldesignandhypotheses

Theexperimentisdesignedtogaugetheinﬂuenceofhardleverageonsoftleveragethroughthephasesofthenegotiation process.

2.1. Experimentaldesign

Following Gächter and Riedl (2005), we createa bargaining environment with two competingclaims to focality: the 75–25splitbasedonprecedent;andthe50–50split.4,5 _At_the_beginning_of_the_experiment,_each_subject_is_informed_that

she is one oftwo department heads workingfor a (hypothetical) company. In the past, a salary budget of4480 points was dividedbetweendepartmentheads.Highandlowperformingdepartmentheadsreceived,respectively,3360and1120 points: a75–25split. Now,duetothe economicdownturn, thesalarybudget willbe smaller, takingoneof threevalues depending onthejointperformance ofthetwo departmentheadsinarealeffort task.6 _Another_difference _from_previous

years isthattop management willnotdictatea salarydistribution.Instead,thetwo departmentheads willnegotiatethe distribution.Thenegotiationprocedureisvariedoverthreetreatments:

•

Treatment 1 (T1) Soft Leverage

◦

Unstructuredbargaining.

◦

Maximum10minutes.

◦

Incaseofadisagreement,bothsubjectsreceivezeropoints.

◦

Communicationthroughwrittenverbalmessagesisallowed.

•

Treatment 2 (T2) Combined Leverage

◦

Treatment1plus:

◦

The high-performing departmenthead is granted theoption to makean ultimatum proposalatany point intime within10minutes.7

◦

Onceandifthehigh-performingdepartmentheadexercisestheultimatumproposaloption,thereisnoturningback totheunstructuredbargaining.

◦

Incasethelow-performingdepartmentheadrejectstheultimatumproposal,bothsubjectsreceivezeropoints.

◦

Communication through written verbal messages is allowed in the unstructured bargaining phase but not in the ultimatumphase.

•

Treatment 3 (T3) Hard Leverage

◦

Ultimatumbargaining.

◦

Thehigh-performingdepartmentheadisgrantedthe“proposer”role.

◦

In casethe low-performing department head accepts the other’s proposal, they each receive their corresponding shares.

◦

Incaseofarejection,bothsubjectsreceivezeropoints.

◦

Communicationisnotallowed.8

Table 1summarizesthemainelements oftheexperimentinthesequencethey werepresentedtothesubjects.Below, weexplaineachpartindetail.

Performanceandoutcomedetermination. Afterreadinginstructionsaloud,departmentheads’performancesaredeterminedby a generalknowledge quiz(Hoffman etal., 1994; GächterandRiedl,2005; Karagözo˘gluandRiedl,2015).The quizconsists of36multiplechoicequestionstakenfromﬁeldsofknowledgesuchaspolitics,music,arts,religion,astronomy,geography. Eachquestion hasﬁvepossibleanswers,andexactlyoneiscorrect(unansweredquestionsarescored incorrect).Each

par-4 _See

_Güth

_and_{Tietz (1986)}_,

_Hoffman

_and_{Spitzer (1985)}_,

_Hoffman

_et_{al. (1994)}_,

_Gantner

_et_{al. (2001)}_,

_Cherry

_et_al.₍₂₀₀₂₎_,_and

_Gächter

_and_{Riedl (2005)}

forrelatedmethodsofcreatingasymmetricfocalpointsinbargaininganddistributiongames.

5 _The_focal_points_in_our_study_are_payoff_relevant._See_the_conclusion_section_for_discussion_of_the_{applicability}_of_our_ﬁndings_when_focal_points_are

payoffirrelevant.

6 _We_chose_salary_budgets_that_are_lower_than_what_was_available_in_the_past_to_create_a_conﬂicting_claims_environment._Historical_claims_(i.e.,₃₃₆₀_and

1120)arenotfeasibleunderanyofthepossiblebudgetlevels.

7 _To_be_as_neutral_as_possible,_this_option_is_labeled_as_“ﬁnal_proposal_option”,_not_as_“ultimatum_proposal_option”_in_the_experimental_{instructions.}

Moreover,weusetheultimatumgametovarythehardleverageacrosstreatmentssinceusingthedictatorgamewouldpracticallyturnthewholegame inT2toadictatorgameandusingthealternatingoffersbargaininggamewouldunnecessarilyaddmorecomplexitywithoutbringingmuchadvantage

(e.g.,alternatingoffersbargaininggamesexhibitresultsconsistentwithultimatumgamewithregardtosocialpreferences(see

Bolton,

1991; Roth,1995; De BruynandBolton,2008)).

8 _We_stick_to_the_standard_ultimatum_game_setup_in_T

3tohaveacompletelyasymmetric,hard-leveragebargaininggame,andsonocommunication.This

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Table 1

Sequenceofevents. 1. Reading of instructions 2. Performance determination 3. Outcome determination

4. Elicitation of beliefs on performances 5. Relative performance information 6. Fairness judgments

7. Bargaining

8. Post-experimental questionnaire

ticipant receives thesameset ofquestionsinthe sameorderandhas20 secondstoanswer each question.All ofthisis

commonknowledge.9

Thesizeofthesalarybudgetisdeterminedbythejointperformanceofdepartmentheadsinapairandexplainedtothe subjectsinthefollowingway:

•

Ifthetotalnumberofcorrectanswersinapairisfrom0 to24,thenthesalarybudgetwillbe1510points.

•

Intheexperiment,100pointsareworthUS$0.75.Themediumsalarybudgetof2420pointsthencorrespondstoroughly $32.

Elicitationofbeliefsonperformances. We ask each subjectto report her beliefs abouther own numberof correctanswers as well as her beliefs aboutthe number of correct answers ofthe other department head.These belief elicitations are incentivized:foreachestimationwith0/1/2error(s),asubjectearns100/50/25points;estimateswithlargererrorsearnno points.

Relativeperformanceinformation. Once therealeffort taskis complete,each subjectistoldwho inher pair was the‘high performer’or‘lowperformer’,dependingonthenumberofcorrectanswersgivenonthegeneralknowledgequiz.Incaseof atie,highandlowperformertitlesaredeterminedatrandom.Tokeepthingssimple,wedidnotprovideexacttaskscores tothesubjects.Earlierstudiesshowthattheinformationprovidedisenoughtosigniﬁcantlyinﬂuencesubjects’bargaining behavior(ex.,GächterandRiedl,2005,2006;Karagözo˘gluandRiedl,2015; Karagözo˘gluandKocher,2015).

Fairnessjudgments. We asked bargainers to report what they thought an impartial arbitrator would judge to be a ‘fair’ distributionofsalarybudgetusingaquestion,adaptedfromBabcocketal. (1995):“Accordingtoyouropinion,whatwouldbea ‘fair’distributionofthesalarybudgetfromthevantagepointofanon-involved neutral arbitrator?”Whatweseektogaugeishow fairnessjudgmentschangewiththenegotiatingenvironmentandtowhatextenttheactions andsettlementstakenreﬂect theseperceptions.Subjects’fairnessjudgmentsarereportedprivatelytoavoidanystrategicvaluethereportingmighthave.

GächterandRiedl (2005)compareaskingforfairnessjudgmentsbeforeandafterrolespecificinformation(e.g.,highvs.low performerinformation)andfindnodifference.Franco-Watkinsetal. (2013)alsofindthataskingaboutfairnessjudgments did not influencebargaining behavior. There isoverwhelming evidence that bargainingsubjects think aboutwhat isfair even absent asking(ex., Güth etal., 1982; Kahnemanetal., 1986; FehrandSchmidt, 2006, essaysinBoltonandCroson, 2012).

Bargaining. In T1 and T2, bargaining is (almost) unstructured. Unstructured bargaining avoids exogenous first-mover ef-fectsandgivessubjects asmuch bargainingfreedom aspossible(e.g.,in thetiming, sequence,numberofproposals, and communicationetc.). Subjectsexchangeproposals thatconsist ofan amountforthemselvesandan amountforthe other departmenthead.Asubjectcansendoneverbalmessageperproposal(eliminatesconfusionaboutwhichproposalisbeing discussed).10Ifsubjectsreachanagreementwithintheallotted10minutes,payoffsareawardedaccordingly.Iftheydonot, eachearnszeropoints.InT2 thehighperformerhastheoption,exercisableatanytimeinthe10minutes,toconvertthe bargainingto an ultimatum game(conversion is irrevocable). Once exercised, thehigh performer makes a final proposal that thelow performermusteitheraccept orreject.Ifheaccepts, bothget theircorresponding shares; ifherejects both getzero.InT3,subjectsplay theultimatumgame(highperformeristhe proposer).Toisolatetheinfluenceofconcession behavior,noverbalcommunicationisallowedinT2 orT3ultimatumgames.

9 _See

_{Karagözo˘glu (2012)}

_for_a_survey_of_bargaining_and_distribution_games_with_joint_production._See

_{Birkeland (2013)}

_,

_Gantner

_and_{Kerschbamer (2016)}_, Gantneretal. (2013),

Luhan

etal. (2013),

Anbarcı

andFeltovich(2014),and

Fischbacher

etal. (2014)forsomerecentexamples.

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Post-experimentalquestionnaire. After the experiment, subjectsare asked to report their (i) level of satisfaction with the bargainingoutcome,(ii)opinionaboutthelegitimacyofthequizasameasureofgeneralknowledge,(iii)perceivedfairness of the bargainingprocedure,and (iv)explanations fortheir bargaining behavior. Theyare alsopresented the Machiavelli personality test (MACH IVscale ofChristie, 1970) to measurepersonality dimensionssuch astoughness,self-orientation, competitiveness, etc. that could be important in bargaining, and a risk attitude questionnaire(Dohmen etal., 2011). Fi-nally, subjectsanswer questions abouttheir personal background.We report on all of theseindividual characteristics in

Appendix A.

The experiment was computerized and programmed in z-Tree (Fischbacher, 2007). We conducted the experiment in LEMA (Laboratory forEconomics, Management, and Auctions) atThe Pennsylvania State University. Intotal 172 subjects participated in seven randomized experimental sessions. We had 29 pairs for T1, 29 pairs for T2 and 28 pairs for T3. Most ofoursubjects wereundergraduates in economics, management,accounting, marketing, supplychain management, andﬁnance.Atypicalsessionlasted60–75minutes(dependingonthetreatment).The averageearningsper subjectwere approximately$22.70(includingashow-upfeeof$5).

2.2. Hypotheses

Let AgreeHigh

(T

j

)

be theaverage agreed sharethe highperforming bargainer receives inTreatment j. We studythree simplebenchmarkhypothesesonhowhardandsoftleveragemayinﬂuencesettlements.Eachismotivatedbytheexisting literatureoneithersoftorhardleverage.

Softleveragehypothesis (null).Bargainingadvantageisdeterminedbythesoftleverageinthebargainingenvironment. In a de-sign similarto T1, GächterandRiedl (2006)demonstratethat the inducedfocal point hascleareffectson settlementsin symmetricbargaining.Underthenullhypothesisthispatternextendstoallthreetreatments:

0

.

50

<

Agree_High

(

T1

)

=

AgreeHigh

(

T2

)

=

AgreeHigh

(

T3

) <

0

.

75

.

Whatweknowofhardleveragefromstudiesofasymmetricbargaininggamesdoesnotspeakdirectlytowhatwillhappen insymmetricgames. Forthisreason,the hardleveragehypotheseswe considerareconﬁnedtopredictionsforT2 andT3. Thehypothesesdon’tmakeanydistinctionbetweenT2 andT3sincehardleverageisequallyavailabletothehighperformer inboth.

Hardleveragehypothesis1.Thehardleverageadvantageisequivalenttothatoftheﬁrstmoverinthestandardultimatumgame.In thestandard ultimatumgame,theaveragesettlement hasﬁrstmovers receivingan averagesomewhatmorethan50%,up to60%ofthepie(Roth,1995).

0

.

50

≤

AgreeHigh

(

T2

)

=

AgreeHigh

(

T3

)

≤

0

.

60

.

The 0.60 cutoff isan approximation butwe think a defensible one;forexample, ina widely cited study, Roth et al. (1991),forproposersina$10ultimatumgame,themodal(andmostgenerous)offerwas typically50–50withanaverage offer of 61–39 split in their favor with a 27% rejection rate (lower offers to responder being more likely rejected). So settlementswereintherangeofthehypothesis(alsoseeSlonimandRoth,1998).11

Hardleveragehypothesis2.Hardleverageenablesthehighperformertoobtain75%ormoreofthepie. The high performer in our experimentearnedhis role. ListandCherry (2000), studyingan ultimatumgamewherethe pieis fullyproducedby the proposer, report a higher number of asymmetric offers than is typical in a standard ultimatum game. The average demand was roughly 68%, with27% ofall offers demanding more than 75%. Rejection rates,however, were substantial, with demands ofmore than 50% rejectedatan averagerate of31% to 44% depending on the treatment (rejection rates increased asoffers decreased).12 These ﬁgures suggest average settlements substantially lessasymmetric than 75–25; in fact, themodalsettlement intheexperimentwas 50–50.Thehard leveragehypothesis2conjectureisthat thecontextof the presentexperiment–inducinga focalpoint at75–25– lends75–25a legitimacy notfound intheearlierultimatum gameexperiments.

Agree_High

(

T2

)

=

AgreeHigh

(

T3

)

≥

0

.

75

.

Weusethesehypothesesasguidepostsforourinvestigation.

11 _Statistics_cited_are_averaged_over₄_treatments._There_were_some_{statistically}_signiﬁcant_differences_in_play_across_the_treatments_but_the_differences

wererathersmallineconomicterms;see

De Bruyn

andBolton (2008).Rothetal.alsoconducted$30ultimatumgameandfoundsimilarresultstothe$10 games.

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Fig. 1. (a) Agreements in T1. (b) Agreements in T2. (c) Agreements in T3.

3. Results

Inthissectionwereviewresultsonsettlements,fairnessjudgments,openingoffers,andconcessionbehavior.The inves-tigationwillleadustoasetofobservedregularitieswhichwecheck againstthehypotheseswe beganwith.Throughout, we expresssettlementsintermsofsharestothe high-performersandusenonparametric tests.13 _In_the_tables, _we_report

averagesandstandarddeviations(inparentheses).

The overall averagenumber ofcorrectquiz answers was 20.9out of a possible36.High-performers (low-performers) averaged 23.8(18.0) correct.In estimatingtheir own (bargaining partner’s)number ofcorrectanswers, subjects’average estimationmistakewas1.77(1.12).Moreover,subjects’beliefswereinlinewiththeiractualrelativeperformances:the av-eragehigh-performerpredictedthathehadmorecorrectanswersthanhispartnerandtheaveragelow-performerpredicted thathehadlesscorrectanswersthanhispartner.Answerstodebrieﬁngquestionsshowthatsubjectsperceivedthequizas alegitimatemeasureofgeneralknowledge.14Answersalsoshowthatgrantingtherighttoexerciseanultimatumoptionto thehighperformingdepartmentheadinT2wasjudgedlegitimate.15Thethreebudgetlevels1560,2420,and3510occurred 2,70,and14times,respectively.Thereare nosigniﬁcantdifferencesinvariablesofinterestacrossdifferentbudgetlevels. Hence,wepoolallsalarybudgetstogetherinouranalyses.

3.1. Settlementsandcomparisonwiththethreehypotheses

Tostartwith,disagreementratesinourexperimentwerelow.OnebargaininggameendedindisagreementinT1

(3.4%),

noneinT2

(0%),

andtwoinT3

(7.0%).

Wereturntothedisagreementratesbelow.

13 _We_check_for_treatment_effects_on_averages_and_{distributions}_using_{Mann–Whitney}_and_{Kolmogorov–Smirnov}_tests_{respectively.}_We_report_only_the

Mann–Whitneytestresultswhenbothtestsgivesigniﬁcantresults.

14 _On_a_7-point_Likert_scale_where_“7₌_agree_very_much”,_average_(median)_degree_of_agreement_with_the_statement_“In_my_view_the_knowledge_questions

havebeendiﬃcult”is4.5(5).Average(median)degreeofagreementwiththestatement“Theonewiththebettergeneralknowledgeisabletoanswer morequestionscorrectly”is5.58(6).

15 ₂₅_out_of₂₉_{high-performers}_and₁₉_out_of₂₉_{low-performers}_in_T

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Table 2

Averageagreedsharesforhigh-performers.

T1 T2a T3

0.55 0.60 0.64

(0.06) (0.09) (0.08)

a _The_average_agreed_share_of_high_performers_in_the_three_pairs_where

theﬁnalofferoptionisexercisedis0.69.Intheremainingpairs,itis0.58. Finally,highperformers’expectedpayoffsacrossT2andT3(i.e.,factoringin

rejections/disagreements)areidentical.

With regard to settlements: Figs. 1(a), (b),and (c) show the distribution of high-performers’ agreed shares foreach treatment. Thedistributions differsubstantially andsystematically:Withone exception,T1 settlementsrangefrom50–50 toabout60–40,withpeaksthatroughlycorrespondtothetwoendsoftherange.ForT2,thereisstillapeakat50–50but nowtherangehaswidenedtoinclude75–25withlessclusteringat60–40.Onlythreehigh-performersexercisedtheﬁnal offeroptioninT2(correspondingproposalswere0.63,0.70,and0.74).T3 showsafurthershifttotheright;nowthereisa newpeakat70–30,the60–40peakremainsbutthe50–50peakisgone.

Average settlements, reported in Table 2, show the rise in average agreed share across treatments, conditional on agreement (results arequalitatively thesameifwe includedisagreements).16_The _{Kruskal–Wallis}_test _strongly_rejects _the

hypothesis that agreed shares are identicalacross treatments (p

=

0.0004). Pairwise testresults are alsoalong the same lines:AgreeHigh

(T

1

)

issigniﬁcantlylessthanAgreeHigh

(T

2

)

(one-sided,p

=

0.0438)andAgreeHigh

(T

2

)

issigniﬁcantlylessthan AgreeHigh

(T

3

)

(one-sided,p

=

0.0136).

The shift in average settlements across treatments clearly rejects the soft leverage hypothesis. That said, the hy-pothesis does get the range of settlements correct. In particular, no single focal point makes a satisfactory account of the data: AgreeHigh

(T

1

)

=

0.55 is signiﬁcantly different from 0.5(Wilcoxon signed-rank test, two-sided, p

=

0.0002) and AgreeHigh

(T

3

)

=

0.64 issigniﬁcantlydifferentfrom0.75(Wilcoxon signed-ranktest,two-sided, p

<

0.0001).Below,we will see thatcertainfeatures ofthenegotiationprocessare alsoﬁxedacrosstreatments asthe softleveragehypothesis would leadustoexpect.

Hardleveragehypothesis 1(basedonstandardultimatumgamebehavior)anticipatestheclustering insettlements ob-served near50–50inT2and(maybelessso)inT3.Butaveragesettlementsarehigherthananticipatedwithasigniﬁcant numberofsettlementsabove60–40sothatthedistributionsofsettlementsisnotskewedtowards50–50asiscommonin standardultimatumgames(GüthandKocher,2014).Itisclear,particularlyfromtheT3distribution,that75–25settlements haveaviabilitythehypothesisdoesnotanticipate.

The hardleverage2hypothesisdoesanticipatetheacceptabilityofa75–25offer.Butactualaverageoffers(and settle-ments) are lower than 75%.Moreover, the 50–50focal point still appears relevant toexplaining settlements, particularly for T2. Other patterns differ substantially from the ultimatum game evidence that the hypothesis is based on (see sec-tion 2.2). The rejectionrates of 0% and7% observed in T2 andT3, respectively, are farlower than the 31% to 44% rates reportedbyListandCherry (2000)ultimatumgameexperimentwithjointproductionandearnedroles.Also, thereisonly one demandin T3 (3.5%)above 75%versus 27% inListandCherry (theirexperimentdidnot inducean asymmetricfocal point).

Observe thatneither ofthe hard leveragehypotheses predicts thedifference in averageagreements across T2 andT3. Onehypothesisisthatthedifferencehastodowiththepotentialtoinduceconcessionbehavior(communication).Thiscan explain whyhard leverage ismore effectivein the final offertreatment than in theinteraction treatment. But itcannot explain theinfluenceofhard leveragewhenmoving fromT1 andT2,sincecommunicationisheld constant.Likewise, the fact that thefinaloffer mechanismis optionalcan explainthe differencesbetweenT2 andT3 butdoesnotexplain much abouttheothersettlementregularitiesweobserve.

To summarize, none of the hypotheses provide a clean explanation for the shifts we observe in settlements across treatments.Tobetterunderstandthepatternofsettlements,wenextinvestigatebehaviorduringtheprocessofnegotiation.

3.2. Fairnessjudgments

Afterproducingthebargainingpiebutpriortonegotiatingweaskedbargainerstoreportwhattheythoughtanimpartial arbitrator wouldjudgetobea ‘fair’distribution ofthesalarybudget.The resultingdatasheds lighton howandtowhat extentthecompetingfocalpointsshapebargainers’expectationsofabargainingsettlement.Table 3shows averagefairness judgmentsofhighandlow-performersinallthreetreatments.

One of the robust ﬁndings on fairness judgments in bargaining is that they exhibit a self-serving (egocentric) bias (MessickandSentis,1979; Bazerman,1985; Babcocketal.,1995; BabcockandLoewenstein,1997; GächterandRiedl,2005, andKaragözo˘glu andRiedl,2015). Ourdataexhibitsself-serving biasesaswell.Writing the averagefairnessjudgment of

16 _Median_agreed_shares_for_{high-performers}_are_similar:_0.55_in_T

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Table 3

Averagefairnessjudgmentsofhighandlowperformers.

T1 T2 T3

High-performers 0.65 0.66 0.67

(0.13) (0.09) (0.10)

Low-performers 0.56 0.59 0.57

(0.07) (0.08) (0.06)

Fig. 2. (a) Fairness judgments in T1. (b) Fairness judgments in T2. (c) Fairness judgments in T3.

bargainer i in Treatment j asFairi

(T

j

),

Fairhigh

(T

j

)

>

Fairlow

(T

j

)

(two-sided, p

<

0.0025 forall j

=

1, 2,3Mann–Whitney test).

From Fig. 2,observethatthefocal pointslargely boundtheself-serving biasinsettlement expectations.Only5outof 86low performers’fairnessjudgmentsarelower than50%andonly2areabove 75%.Only11outof86highperformers’ fairnessjudgmentsarehigherthan75%andnonearebelow50%.Alsoobservethatthemajorityofbothlowandhigh per-formersreportjudgmentsthatimplyhighperformersshouldgetmorethanlowperformers17_;_{in fact,}_Wilcoxon_signed-rank

testsshowthat Fairhigh

(T

j

)

>

1/2 andFairlow

(T

j

)

>

1/2 forall, j

=

1,2,3(two-sided, forhigh-performers all p

<

0.0001; andforlow-performersall p

<

0.0003).18

Howdoeshardleverage inﬂuencefairnessjudgments?LernerandMiller (1978)argue thatpeople haveatendencyto believe that theworld they are livingin isa just one(known asthe beliefinajustworld hypothesis). Animplication for ourexperimentisthatthemorehardleverageabargainerhasthehighershareofthesurplushejustlydeserves.Thedata wehave,however,doesnotfavorthishypothesis.AsTable 3reports,bothhighandlowperformers’fairnessjudgmentsare constantacrosstreatments.Kruskal–WallistestsfailtorejectbothFairhigh

(T

1

)

=

Fairhigh

(T

2

)

=

Fairhigh

(T

3

)

(p

=

0.8002)and Fairlow

(T

1

)

=

Fairlow

(T

2

)

=

Fairlow

(T

3

)

(p

=

0.1103).Binarycomparisonsyieldanidenticalresult:Fairhigh

(T

1

)

andFairhigh

(T

2

)

17 _Observe_that_agreements_in_all_treatments_are_between_high_and_low_{performers’}_fairness_judgments._In_T

1 andT2,agreementsarecloserto

low-performers’ fairnessjudgments(0.55≤0.55<0.66 and0.59<0.60<0.66, respectively), whereasinT3 they areclosertohigh-performers’fairness

judgments(0.57<0.64<0.67).

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Table 4

Averageopeningoffersofhighandlowperformersa_.

T1 T2

High-performers’ opening offer 0.72 0.73 (0.13) (0.13) Low-performers’ opening offer 0.46 0.46

(0.15) (0.11)

a _In_T

3,thereisnounstructuredbargainingand theproposerisﬁxed.Hence,wedonot

includetheproposalsmadeinT3here.NotethattheaverageultimatumofferinT3is0.65.

Fig. 3. Distributions of high and low performers’ opening offers.

are not different (two-sided, p

=

0.6912). Fairhigh

(T

2

)

and Fairhigh

(T

3

)

are not different either (two-sided, p

=

0.7368). Fairlow

(T

1

)

andFairlow

(T

2

)

are statistically not differentaccording to Kolmogorov–Smirnov test (corrected p

=

0.279) but different accordingtoMann–Whitney test (two-sided, p

=

0.0435). Finally,Fairlow

(T

2

)

and Fairlow

(T

3

)

arestatisticallynot different(two-sided,p

=

0.2431).

Tosumup,thepresenceofhardleveragehaslittleeffectonfairnessjudgments.Thecompetingfocalpointscoordinate settlement expectationsin the sense that bargainerslargely agreethat a settlement deemed fair by a neutralarbitrator would be one thatfalls withinthe compromise interval betweenthe two focalpoints. Withinthisinterval, however,we observeself-servingbiasesacrosshighandlowperformers.

3.3. Openingoffers

The openingofferinabargainingpairistheveryﬁrstoffermade. Themainreasonwefocusonopeningoffersisthat theyareknowntobehighlypredictiveoftheremainderofthebargainingprocessandeventualsettlements(Chertkoffand Conley,1967; Yukl,1974; Wilsonetal.,1996; Parketal.,2010;Galinskyetal.,2009;Karagözo˘gluandRiedl,2015).

In T1, high and low performers make the opening offer in roughly equal numbers (14 and 15, respectively). In T2, 18 (11)ofthesubjectswhomaketheopeningofferwerehigh(low)performers(two-sampletestofproportions,two-sided, p

=

0.066).Table 4reportstheaveragesofopeningoffersinpairsinT1 andT2,Fig. 3showsthedistributions.

Observe that averageopeningoffers for thetwo bargainingrolesare nearidentical acrosstreatments. Denote the av-erage openingoffer ofbargainer i inTreatment j by Firsti

(T

j

).

Firsthigh

(T

1

)

andFirsthigh

(T

2

)

are statisticallynot different (two-sided, p

=

0.8195),nor areFirstlow

(T

1

)

andFirstlow

(T

2

)

statisticallydifferent (two-sided, p

=

0.6941).The important implicationisthattheintroductionofhardleveragehaslittleinﬂuenceonopeningofferbehavior.19

Fig. 4shows that thelargestfractionofhighperformers’ﬁrstoffers centeraround75–25division,whereasthe largest fractionoflowperformers’ﬁrstofferscenteraround50–50divisioninT1andT2.20

Allinall,highperformeropeningoffersweremoreaggressiveinT1andT2 thanultimatumproposalsinT3.Thereisno evidencethatthepresence(T2)orabsence(T1)ofahardleverageoptioninfluencesubjects’openingoffers.Openingoffers reflect thefairnessbias identifiedinsection3.2,withtheaveragehighperformer offeringclosertoa75–25splitandthe averagelowperformerofferingcloserto50–50.

19 _There_is_no_signiﬁcant_correlation_between_opening_proposals_and_{counter-proposals}_in_either_T 1orT2. 20 _Note_that_transposed_ﬁgures_would_represent_cumulative_{distributions.}

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Fig. 4. High performers’ ﬁrst offers and low performers’ opening offers in T1and T2.

Table 5

Settlements,openingoffers,counter-offersandyielding(inhighperformershares).

T1 T2 p-value High_performer_agreed_share 0.55 0.60 0.043∗∗ High_performer_ﬁrst_offer 0.68 0.69 0.616 Low_performer_ﬁrst_offer 0.45 0.51 0.075∗ High_performer_opening_offer 0.72 0.73 0.820 Low_performer_opening _offer 0.46 0.46 0.659 High_performer_counter_offer 0.64 0.64 0.770 Low_performer_counter_offer 0.45 0.54 0.008∗∗∗ 3.4. Concessionbehavior

Sofar,wehavedetectednodifferenceinbargainingbehavioracrossT1andT2.Yetwesawthatsettlementsacrossthese treatments do infact differ. It turns out that the difference hasto do withconcessions implicit ininitial counter-offers.

Table 5 displays highandlow performers’ first offers conditional on agreements, inT1 andT2. First,a clarification: ifa bargainer was the first toput a division on thetable, her first offerqualifies as an openingoffer. Ifa bargainer did not initiatethenegotiation,her firstofferisacounter-offertotheopeningoffer.FromTable 5weseethatthemorefavorable agreementsforhighperformersinT2thaninT1arenotduetodifferencesinfirstoffers,butratherreflectdifferenceinlow performercounter-offers(0.54comparedto0.45).

Neither openingoffers norcounter-offersofhighperformersdifferacross treatments,nor dolow performers’opening offers.But lowperformers’ counter-offersdo differ:facingthe sameaverageopening offerasinT1,low performer coun-teroffersin T2 concedealmost 10% point moreto highperformers. Ahigherpercentageof openingoffers were madeby highperformerinT2 thanT1(section3.3),compoundingtheeffectofthelowperformercounter-offerdifferenceon settle-ments.FurtherinvestigationusingregressionanalysisyieldssimilarresultstothosereportedinTable 5;seeTablesA.3–A.6 inAppendix A.

3.5. Asummaryoftheobservedregularities

1. Settlements fallwith few exceptionsbetweenthe two focalpoints, 50–50 and75–25. The highperformer capturesa highershare,onaverage,whenendowedwithhardleverageinT2 andT3,althoughthesizeoftheeffectissigniﬁcantly smallerinT2.

2. Judgmentsofwhataneutralarbitratorwoulddeemfair are inall cases betweenthe two focal points and donot differ statisticallyacrosstreatments(varyinghardleveragedoesnotinﬂuencefairnessassessments).Highandlowperformer assessmentsexhibitself-servingbiases.

3. Openingoffers inT1andT2aresigniﬁcantlymoreaggressivethanfairnessassessments:Theaverageopeningofferofhigh (low)performersapproximates75–25(50–50).InT3 highperformers’ openingoffersapproximatefairnessjudgments, andaresigniﬁcantlylowerthanhighperformeropeningoffersintheothertreatments.

4. Concessionbehavior. Low performer counter-offers (made in response to an opening offer) concede more inT2 than in T1.

4. AmodiﬁedNashbargainingsolution

Howmightwe explaintheregularitiesstatedinsection3.5?Insection 4.1we sketchamodelpositingthat competing focalpointscreatemutualexpectationsoftherange ofacceptablesettlements,thecharacterizationoftheunderlyingsocial preferencesbeinganextensionofthoseusedtoexplainultimatumgamebehavior.TheNashbargainingsolutioncanexplain

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thecomparativestaticsofobservedsettlements.Insection4.2wesimilarlymodifyamodelofthebargainingprocessthat

Harsanyi (1956,1977)usedtomotivate theNashbargainingsolution.Themodiﬁedmodelcanexplainwhyopeningoffers clusteraroundthefocalpointsandwhyhardleveragehasitsmaininﬂuencethroughconcessions.

We hastentonote thatoursisnottheonlypotential candidateforexplainingtheseregularities. Anotherapproach,for example, isa model withendogenousreferencepoints that shiftswith thestructure or contextof the bargaininggame. The theory we sketch borrows from the social preference models of behavior in ultimatum and associated alternating offer bargaining games (ex., De Bruyn and Bolton, 2008). Subgame perfect equilibriumfrom these models is consistent with observed ultimatumgame settlements clustering near 50–50withthe first mover having a modest advantage. But this characterizationdoes not fitour data:For example,the clustering around 50–50inT1 andT2 disappears in T3, the treatmentmostliketheultimatumgame(Fig. 1).Shiftingthemodelreferencepointfrom50–50to75–25helpsexplainthe low rejectionrates inT2 to T3,butalso makes itdifficultto explain whythereare settlementsbelowthenewreference point.Themodelwesketchtakesexplicitaccountofthecompetingfocalpoints.

4.1. SocialpreferencemodiﬁcationoftheNashbargainingsolution

The key innovationwe introduce totheNash bargainingsolution istosuppose that thefocal points deﬁnethe limits of potential compromise. We capturethisthrough a socialpreference characterization.Consider a negotiationbetween a high (h) and low (l) performer. Bargainers are risk averse withutility uh

(x

h

)

and ul

(x

l

)

for a settlement of x

= (

xh,xl

),

xh

+

xl

=

100.Let fi

= (

fii

,

f

i

∼i

)

bethefocalpointfavoredbybargaineri.Weassumethat f i i

>

f∼

i

i ;thatis,eachbargainer favorsthefocalpointthatgiveshimthelargershareofthebargainingpie.Weconsidertwotypesofbargainer:

Compromisingbargainer. Let Xi

_{= {}

_x

_|

_x

i

≥

fi∼i

}

. Then for all x,y

∈

Xi such that xi

>

yi

,

ui

(x

i

)

>

ui

(y

i

)

>

ui

(0).

For all x

∈

/

Xi

_,

_u

i

(x

i

)

<

ui

(0).

That is, a compromising bargainer has standard self-interested preferences over the domain in which his share ofa settlementisatleastashighasthesharehewouldreceivefromthefocalpointtheother bargainerfavors.Forsharesbelow thispoint,heprefersthe(0,0)outcome.

Non-compromisingbargainer. Let X_ni

= {

x

|

xi

≥

f_ii

}

.Thenforall x,y

∈

Xni such thatall xi

>

yi

,

ui

(x

i

)

>

ui

(

yi

)

>

ui

(0).

Forall x

∈

/

X_ni

,

ui

(x

i

)

<

ui

(0).

Hence non-compromisingbargainerspreferthe (0,0) outcometo acceptinga sharelessthan whatthey wouldreceive bytheirfavoredfocalpoint.

Forpurposesofillustration,wesupposethehighperformingbargainerisacompromisingbargainer.Thelowperforming bargainer isa compromisingbargainer withprobability 1

−

ε

andanon-compromiserwithprobability

ε

.Weassume the highperformerknows(only)theprobabilityofhisbargainingpartnerbeingacompromiser.21

Combining this preferencespeciﬁcation with the Nash bargainingsolution, we can explain the difference in average settlements we observed across treatments. To do so we need suppose that

ε

is small in that for the high performer bargainer

uh

z_h

= (

1

−

ε

)

uh

(

75

)

+

εu

h

(

0

)

for zh

>

50

.

(1)

TheNashbargainingsolutionpredictsthesettlementofthegametobe

x∗

=

argmax x=(xh,xl)

uh

(

xh

)

−

uh

(

dh

)

ul

(

xl

)

−

ul

(

dl

)

(2)

where d

= (

dh

,

dl

)

deﬁnesthe smallest settlement shares that each bargainer ﬁnds acceptable to agree on.Consider the bargaininggameinT1:Ifthehighperformerismatchedwithacompromisinglowbargainer,thesetofacceptabledivisions xh

−

xl arebetween50–50and75–25,implyingd

= (

25,50).Letx∗1betheT1bargaininggamesettlementpredictedby(2). The value of x∗1 will be a compromise between the two extremes 50

<

x∗_h1

<

75,the exact compromise depending on thecomparative riskaversionofthebargainers(Roth, 1979).Ifthelowbargainer isanon-compromiser,thereisonlyone feasiblesettlement,50–50.TheexpectedsettlementforT1 istherefore50

ε

+ (

1

−

ε

)x

∗_h1.

FortreatmentsT2 andT3,dl remainsat25butdh increaseswiththeintroductionofthefinalofferoption.Toseethis, observe that condition (1) impliestheoptimal highperformer final offer is75–25. So inthe T2 unstructuredbargaining phase, the highperformer will agree to a settlement no smaller than z_h (as definedin (1)); that is,dh

=

zh.If thelow performer isa non-compromiser,thegameendswithnosettlement–bothunstructuredandﬁnal offerbargainingendin disagreement. But ifthe low performer is a compromiser, the increase indh will resultis a Nash bargainingsettlement x_h∗2

>

x∗_h1 (Thomson,1987)where z_h

<

x∗_h2

<

75.SotheexpectedsettlementforT2 isgreaterthanthatforT1.ForT3,there

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isno unstructuredbargainingphase andthehighperformer ﬁnal offerwillbe 75–25whichis acceptedwithprobability 1

−

ε

.Since75

>

x∗2

h ,theexpectedsettlementinT3isgreaterthanthatinT2.

Thissimple modelcapturessome features of thevariation insettlementsbutfailsinregard to others.ForT1 andT2, the model predicts settlements should range from 50–50to 75–25, a close approximation to what is observed (Fig. 1). But neither therange in settlementsin T3 nor the clustering (sometimes) observed near focal points are accounted for. A morereﬁnedmodel,onethatallowsformorenuancedcompromisingtolerancethanfoundinthetwo-typemodel,might explainatleastsomeofthemissingvariation.Thefairnessassessmentsgivenbybargainers(Fig. 2)aresuggestiveofawide variation incompromise tolerance.Some low performers, forexample, mightbe willing tocompromise up to 67–33but not all the wayto 75–25. Anotherfactor that might explain themissing variation isdifferences innegotiating skill (see section4.2).

Notes:Thecompromiser/non-compromiserformulationsuggestsameasurementofafocalpoint’scredibility,withhigher recognition(lower

ε

) making forgreatercredibility. Amodel that characterizesthe value of

ε

morefullymightlink low focalpointcredibilitytodisagreementanddelayinbargaining,aswellasexplainsomeofthevariationinoutcomes.

Finally,inthe symmetricversion ofthe modelall bargainersfavor thesamefocal point,andso both bargainertypes, compromisingandnon-compromising, requireatleast50%toacceptasettlement.Thepredictedsettlementisthen50–50, consistentwiththewell-knownregularity offree formsymmetricbargainingthat settlementscluster tightlyabout50–50 (Roth,1995).

4.2. Asimplemodelofthebargainingprocess

Harsanyi (1956,1977) extendedandgeneralizedZeuthen’s(1930) modelof thebargainingconcessionprocess by way ofprovidingafoundationfortheNash bargainingsolution.Wecanusethesamemodel,modiﬁed bythesocialpreference characterizationprovided above,to explainwhyopening offerscluster atthefocal points,andwhytheinﬂuence ofhard leverageisthroughtheconcessionprocess(regularities3and4,section3.5).22

AppliedtothebargaininggameinT1,theZeuthen–Harsanyimodelrunsasfollows:Bargainingisbrokenintoperiods.In each period,high(h)andlow (l)performingbargainers simultaneouslymakeaproposalforsplittingthepie,respectively h

= (

hh

,

hl

)

andl

= (

lh

,

ll

)

where ij is theshare i offers to j.If the proposalsare compatiblein that hh

≤

lh andll

≤

hl, thenthe gameendswitheach bargainer gettingthe shareheproposed forhimself.Otherwisethegamecontinuestothe next period,where eachbargainer (simultaneously)hastheoptiontoeithermake aconcession(decrease herown payoff andincrease the payoff of theother bargainer) or makeno concession. The gameends inthe first periodproposals are compatible,orinthefirstperiodinwhichneitherbargainermakesaconcession.Inthelattercase,payoffsaregivenbythe disagreementoutcome.Toensurethegameisfinite,weassumeanyconcessiontobeatleastafixedminimumamountas specifiedbyHarsanyi(1977,p.152).

TheZeuthen–Harsanyimodelfocusesonthequestionofwhichbargainerwillmakeaconcessionatagivenperiodwith givenproposalsonthetable. Theanswer,known astheZeuthen Principle,assertsthatthebargainer leastwilling totake a risk thatthe bargainingwillend inconflictis theone that concedes. Herewe modify theprinciple toaccount forthe influenceoffocalpointsonperceivedcredibility.Wefirstcalculatetherisklimit foreachbargainer.Supposethath andl are thetwo(incompatible)proposalscurrentlyonthetable.Thehighperformer’srisklimit,statedasaprobability,isrh.When 50

≤

lh

≤

75,rh isdeﬁnedby

uh

(

lh

)

= (

1

−

rh

)

·

uh

(

hh

)

+

rh

·

uh

(

50

).

(3)

Thatis,therisklimitisthebreakevenprobabilitysuchthatthehighperformerisindifferentbetweenacceptingthepresent lowperformerofferandtakingtheriskofstandingonhispresentofferontheassumptionthattheworstpossibleoutcome is 50–50, theleast favorable, mutuallyacceptable division. Mathematically, the calculation of therisk limit in(3) differs fromZeuthen–Harsanyi injustone respect,the substitution of50for thedisagreement payoff.23 Ifthe low performeris offeringthehighperformerlessthan50,rh

=

1,thehighestpossiblerisklimit.Intuitively,thehighperformerdoesnotﬁnd itcredible thatthelow performerwill standonademand formore than50%.Bysimilarreasoning,ifthelow performer offersmorethan75,rh

=

0,thelowestpossiblerisklimitsincethisismorethanthehighperformercouldcrediblydemand forhimself.Puttingthistogetherwehave

rh

=

⎧

⎨

⎩

1

,

if lh

<

50 uh(hh)−uh(lh) uh(hh)−uh(50)

,

if 50

≤

lh

≤

75

.

0

,

if lh

>

75 (4)

Therisklimitforthelowperformerismeasuredinananalogousway.

22 _See

_Birkeland

_and_{Tungodden (2014)}_for_a_recent_paper_that_studies_the_effect_of_fairness_motivations_on_the_bargaining_outcome_in_Nash_bargaining. 23 _The_original_{Zeuthen–Harsanyi}_has_been_criticized_for_being_too_pessimistic,_omitting_the_possibility_of_compromise_from_the_calculation._The_modiﬁed

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rl

=

⎧

⎨

⎩

1

,

if hl

<

25 ul(ll)−ul(hl) ul(ll)−ul(25)

,

if 25

≤

hl

≤

50

.

0

,

if hl

>

50

We then apply a version of the Zeuthen Principle, we call it the credibilityprinciple because ofthe modiﬁcations we made: Bargaineri makesa concessionifshehasthelowest risk limit:ri

≤

rj.The intuitionissimilar tothat behind the ZeuthenPrinciple:Deadlockedwithincompatibledemands,andwithtimepassing,theprobabilityofconﬂictrisesuntilone bargainer reacheshisrisklimitandbreaksthedeadlockbymakinga concession.24 Inthecaseofthecredibilityprinciple, the concession isbased on mutualexpectations ofwhat is acceptable,with theﬁnal fallback position forthe conceding bargainerbeingtheleastfavorablefocalpoint.

The credibility principleconstitutesa formofrisk dominance inthat thebargainer running thehighestrisk concedes groundtotheotherbargainer(Harsanyi,1977).InZeuthen/Harsanyi’streatment,risklimitsaredeﬁnedasgivenabovesave that thelast terminthe denominatoris adisagreementpayoff insteadofthe low-endfocalpoint thatthe bargainer can achieve.Asaresult, allofthetheoremsprovedfortheZeuthen–Harsanyigameapplyhere,treating thefocalpoints asthe disagreementpayments.

Note thatthere isno reasonforeitherbargainer todemandmore thantheir preferredfocal pointsince doingso will simplyleadthemtomakeunreciprocatedconcessions.

Proposition1.Ifbargainersoffertheirpreferredfocalpointinperiod1andthenfollowthecredibilityprinciple,thebargainingends intheNashbargainingsolutionwithdisagreementoutcomed

= (

dhdl

).

Proof(followssection 4,Harsanyi,1956). Bargainer i makesa concessionanytimeri

≤

rj making i the concedingbargainer. Let’ssupposei

=

h (thechoiceisarbitrary).Thensubstitutingtheformulasforriandrj,anddoingsomealgebra,wehave:

π

(

H

)

=

uh

(

hh

)

−

uh

(

dh

)

ul

(

lh

)

−

ul

(

dl

)

≤

ul

(

lh

)

−

uh

(

dh

)

∗

ul

(

ll

)

−

ul

(

dl

)

=

π

(

L

)

where

π

(X)

istheNashproduct ofbargainer X ’sproposal.Inwords,the concedingbargainer istheonewiththelowest Nash product. Solongasthe Nashproduct isbelowits optimum value,one ortheother bargainer will needconcede to increaseit.AsettlementisreachedonceproposalsconvergeontheNashproduct.

2

Whatdoesthemodelexplainabouttheexperiment?Weobservedthatthemodelpredictsthattheopeningoffersshould approximatetheofferers’favoredfocalpoint(75–25or50–50)inbothT1 andT2,consistentwithregularity3.Inaddition, fromProposition1itisclearthattheextentofconcessionsmadeisrelatedtothevalues(dh

,

dl).Speciﬁcally,anincreasein dh ofthesortimpliedinmoving fromT1 toT2 (seesection 4.1) ispredictedtoleadthelowperformer tomakerelatively greater concessions than thehighperformer, consistentwithobservation(regularity 4, section3.5). Asbefore, themodel doesnotaccountforthefullvariationinbehaviorthatweobserve.Forexample,someopeningoffersarebetweenthetwo focal points (Fig. 3). Onehypothesis isthat thisvariation isdue toa lackof skillonthe partofsome bargainers –they concede too muchtoo soon.Thismight alsoexplain some ofthe aforementionedvariation insettlementsmissed bythe modiﬁedNashbargainingmodel(section4.1).

Proposition 1exploresthedynamicpropertiesofagamebutisnotanequilibriumanalysis.Harsanyi(1977,p.158) con-sidersa“compressedZeuthen game”withjusttwoperiods.Inperiod1,bargainersmakeopeningoffers.Inperiod 2,they each eitheraccept theotherbargainer’sproposalorstandﬁrmontheir ownproposal.Harsanyi showsthattheonly equi-libriumsettlementthat isconsistentwiththeZeuthenPrincipleistheNashbargainingsolution.Substitutingthemodiﬁed Nash bargaininggameconsideredinsection 4.1,thesameproofshowsthat theonlysubgameperfectequilibriumforthe above gamethatis consistentwiththecredibilityprincipleis oneinwhichbargainersbothpropose andaccept theNash solution.

5. Conclusions

We investigatedtherelative credibilityofhard leverage(commitment actions)andsoftleverage (focalpoints) for bar-gaineradvantageinarichcontextbargaininggamewheretherearetwocompetingfocalpoints.Themostimportantfindings havetodowithwherehardandsoftleverageenterthebargainingprocess. Theinfluenceofsoftleverageismostevident on bargainers’fairnessjudgmentsandopeningoffers.The influenceofhardleverageismostevident intheconcessionary stage. Hard andsoftleverageinteractto influenceoutcomes:Hard leverageadvantages its holder,yetsettlementslargely staybetweenthetwofocalpoints.

The comparative staticswe observe (section3.5) fit,to a firstapproximation, withthe Nash andassociatedZeuthen– Harsanyi bargainingmodels, modified to account forthe role that competing focalpoints play incoordinating bargainer expectations.Themodelisafirsteffortinanarea(richcontextbargaining)wherelittleisknown.No doubtothermodels

24 _Thus_the_prediction_of_who_will_back_down_does_not_assume_complete_information_about_risk_preferences._See

_{Roth (1979)’s}

_discussion_of_the_intuition

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couldbe constructedtoﬁtthedata.Thevalueofexplainingtheresultswitha simplemodel,we think,isthatthemodel impliesuntestedhypothesestoguideadditionalresearch,datafromwhichmightthenleadtoareﬁnementofthepresent modeloradifferentmodelaltogether.Forexample,themodelimpliesthatshiftingtheasymmetricfocalpointfrom75–25 (in either direction) should shift bargaining outcomes in all three treatments inthe same direction, withthe compara-tivestaticsacrosstreatments thatwefound stillholding.Datafromsuch anexperimentcouldgive usgreaterinsightinto bargainerwillingnesstocompromise,somethingthesimplemodelwepresentapproximatesonlycoarsely.

Animportantcaveatto ourresultsisthatthe focalpoints arepayoff relevant.Focal points canbenon-payoff relevant aswell;see forexample,Sugden (1995).Potentially,bargaininginvolvingnon-payoff relevantfocalpoints mightdisplaya differentpatternthan weobservehere.Note,however,thatthecharacterizationoftheroleoffocalpoints presentedhere rests not with thepayoff relevance of the focal point butrather on whetherbargainers recognize one another claim to focalityas credible.So we conjecture that ourresults will generalize tonon-payoff relevantfocal points so long asthey meetthecredibilitycriterion.

What elsecould changeourresult? Onemighthypothesize thathavingmore thantwo focalpoints attenuatesthe in-fluenceofanyonefocalpoint.Butwhatappearstobeimportantaboutfocalpoints isthatthey arestartingplacesforthe compromise process.Solongaseachbargainer canidentifyamostfavoredfocalpoint –sufficientlycredibletotheother side – thisshould be sufficient to orient thenegotiation. Morelikely to change things isa situation whereone or both bargainersclaimtofocalityisnotconsideredsufficientlycrediblebytheotherside.Forinstance,iftheexogenously imple-mentedreferencepointintheexperimentwasratherextreme(e.g.,99–1),itmightnotserveasacrediblereferencepoint. Thesecircumstances,mightforexample,leadtomoredisagreementoutcomesthanthecaseexaminedhereoragreements thatarelesspredictablebythereferencepointoutcome.25Theassumptionsonthecredibilityparameterinourmodelcould beadjustedtoaccommodatethesecircumstances.

Whatmightmakehardleveragemoreinﬂuentialinthesenseoftrumpingthefocalstructureofthenegotiation?Perhaps higherstakessuchthatoneside’scommitmentthreatenstheothersidewithverysubstantiallosses.Butthisisnotobvious: Mostexperimentsincreasingthestakesinultimatumgamesﬁndatmost,modestshiftsinbehavior;theexceptionsinvolving veryhighstakes(seeKaragözo˘gluandUrhan,2016forasurvey;thereviewinCooperandKagel,forthcoming;Andersenet al.,2011foranexception).

Another potential caveat to our results is that the bargaining game is played only once. Whether our results would surviveifsubjectshadmoreexperienceplayingthegameisanopen question.Also, movingfromT2 toT3,boththehard leverageandtheavailabilityofcommunicationchange.Researchshowsthatcommunicationmayhelpbargainersinarguing for their favored focal points (see Karagözo˘glu and Kocher, 2015 fora recent example). Hence, if we hadan ultimatum game treatment withpre-play communication, depending on the persuasiveness and risk attitudes (since possiblysome messages would employ threats) of bargainers we could have observed different outcomes. Nevertheless, we conjecture that the results from such a treatment would be qualitatively similar (to the ones from T3) since the limited pre-play communicationwouldpreservetheeffectsofincreasedhardleverage.

Inclosing, wepointtooneareaofinvestigationinparticularthatmightleadtolargegainsinunderstanding:Whilewe were ableto offera preliminary explanationforthe comparativestatic regularitiesoftheexperiment, observed behavior varied more widely thanthe explanations imply. Atleast some ofthis variationmight be explainedby a more nuanced understandingofbargainerwillingnesstomakeconcessionstowardsthefocalpointthatfavorstheotherbargainer.Finding a reliable quantitative measure for the ‘credibility’ of a focal point might be the key here. The model we sketched in section 4.1offers asimple measurethat mightbe takenasa starting pointbutsurely other candidates are possible;the idealwouldbeamodelinwhichreferencepointsemergeendogenously.

Acknowledgments

Gary Bolton thanks the Pennsylvania State University forthe grant used in thisresearch. We thank two anonymous referees andan associateeditor for their comments that improved the paper. We also thankAlexander Cappelen, Anita Gantner, Werner Güth, Hakan Holm, Rudi Kerschbamer, Theo Offerman, AndersPoulsen, ArnoRiedl, Bettina Rockenbach, ArthurSchram,RobertSugden,TedTurocy,andRobertoWeberforvaluablecomments.

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefoundonlineathttp://dx.doi.org/10.1016/j.geb.2016.08.005.

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