NBERWORKINGPAPERSERIES

INFORMATIONFAVORITISMANDSCORINGBIASINCONTESTS

ShanglyuDeng

HanmingFang

QiangFu

ZenanWu

WorkingPaper31036

/papers/w31036

NATIONALBUREAUOFECONOMICRESEARCH

1050MassachusettsAvenue

Cambridge,MA02138

March2023

WethankAngusC.Chu,KevinHe,JiangtaoLi,AllenVong,RakeshVohra,MingYang,andseminarparticipantsattheUniversityofPennsylvania,UniversityofMacau,SingaporeManagementUniversity,andJinanUniversityforhelpfuldiscussions,suggestions,andcomments.FuthankstheSingaporeMinistryofEducationTier-1AcademicResearchFund(R-313-000-139-115)forfinancialsupport.WuthankstheNationalNaturalScienceFoundationofChina(Nos.72222002,72173002,and71803003),theWuJiapeiFoundationoftheChinaInformationEconomicsSociety(No.E21100383),andtheseedfundoftheSchoolofEconomics,PekingUniversity,forfinancialsupport.Anyremainingerrorsareourown.TheviewsexpressedhereinarethoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheNationalBureauof

EconomicResearch.

NBERworkingpapersarecirculatedfordiscussionandcommentpurposes.Theyhavenotbeenpeer-reviewedorbeensubjecttothereviewbytheNBERBoardofDirectorsthataccompaniesofficialNBERpublications.

?2023byShanglyuDeng,HanmingFang,QiangFu,andZenanWu.Allrightsreserved.Shortsectionsoftext,nottoexceedtwoparagraphs,maybequotedwithoutexplicitpermissionprovidedthatfullcredit,including?notice,isgiventothesource.

InformationFavoritismandScoringBiasinContests

ShanglyuDeng,HanmingFang,QiangFu,andZenanWu

NBERWorkingPaperNo.31036

March2023

JELNo.C72,D44,D82

ABSTRACT

Twopotentiallyasymmetricplayerscompeteforaprizeofcommonvalue,whichisinitiallyunknown,byexertingefforts.Adesignerhastwoinstrumentsforcontestdesign.First,shedecideswhetherandhowtodiscloseaninformativesignaloftheprizevaluetoplayers.Second,shesetsthescoringruleofthecontest,whichvariestherelativecompetitivenessoftheplayers.Weshowthattheoptimumdependsonthedesigner’sobjective.Abilateralsymmetriccontest—inwhichinformationissymmetricallydistributedandthescoringbiasissettooffsettheinitialasymmetrybetweenplayers—alwaysmaximizestheexpectedtotaleffort.However,theoptimalcontestmaydeliberatelycreatebilateralasymmetry—whichdisclosesthesignalprivatelytooneplayer,whilefavoringtheotherintermsofthescoringrule—whenthedesignerisconcernedabouttheexpectedwinner’seffort.Thetwoinstrumentsthusexhibitcomplementarity,inthattheoptimumcanbemadeasymmetricinbothdimensionseveniftheplayersareexantesymmetric.Ourresultsarequalitativelyrobustto(i)affiliatedsignalsand(ii)endogenousinformationstructure.Weshowthatinformationfavoritismcanplayausefulroleinaddressingaffirmativeactionobjectives.

ShanglyuDeng

DepartmentofEconomics

UniversityofMaryland

3114TydingsHall

7343PreinkertDr.

CollegePark,MD20742

sdeng1@

HanmingFang

DepartmentofEconomics

RonaldO.PerelmanCenter

forPoliticalScienceandEconomics

UniversityofPennsylvania

133South36thStreet,Suite150

Philadelphia,PA19104

andNBER

hanming.fang@

QiangFu

DepartmentofStrategyandPolicy

NUSBusinessSchool

15KentRidgeDrive

Singapore

bizfq@.sg

ZenanWu

SchoolofEconomics

PekingUniversity

Beijing,China

zenan@

2

1Introduction

Contestisawidelyadoptedmechanismthatmobilizesfocusede?ortstoachievestatedgoals.Inacontest,theorganizersetsalimitednumberofprizesandinvitesentries;con-tenderssinkcostlyandnonrefundablee?orts,andonlythefrontrunnersarerewarded.Gov-ernments,?rms,nonpro?torganizations,andevenwealthyindividualsoftensponsorR&Dconteststosolicitnoveltechnologicalsolutions,procureinnovativeproducts,orencouragescienti?cbreakthroughs(

Taylor

,

1995

;

FullertonandMcAfee

,

1999

;

CheandGale

,

2003

).

1

TheU.S.DepartmentofDefense(DoD)famouslyoperatestheSmallBusinessInnovationResearchprogram,whichpromotesprivateR&De?ortsonmilitarytechnology,andawardsprocurementcontractstooutstandinginnovators(

Bhattacharya

,

2021

).Theinternallabormarketsinside?rmsprovideanotheranalogy(

LazearandRosen

,

1981

;

GreenandStokey

,

1983

;

Nalebu?andStiglitz

,

1983

;

Rosen

,

1986

).Firmsinduceworkers’e?ortsbyscarcelysuppliedbonuspackages,promotions,andopportunitiesforcareeradvancement,aswellasthethreatoflayo?s.In2013,30%ofFortune500companieswereestimatedtorewardorpunishtheiremployeesbasedonrankingsystemsinspiredbyJackWelch’spracticeofthe“vitalitycurve.”

Theubiquityofcontest-likecompetitiveactivitieshassparkedextensiveandcontinuouse?orts,inbothpracticeandacademicresearch,toidentifyfeasibleande?cientmeansofadministeringsuchmechanisms(see,e.g.,

FuandWu

,

2019

,

2020

;

Fang,Noe,andStrack

,

2020

;

LemusandMarshall

,

2021

;

Hofstetter,Dahl,Aryobsei,andHerrmann

,

2021

).Twophenomenaarebroadlyobservedthatinspirecontestdesign.

First,discriminatorymeasuresareoftenadoptedtomanipulatethecompetitivebalanceofacompetition,whicheitherhandicaporfavorcertaincontenders.Forinstance,favoritehorsescanberequestedtocarryextraweightsinhorseracing(

Chowdhury,Esteve-Gonz′alez,

andMukherjee

,

2023

).In?rms’successionselectionprocesses,incumbentCEOsandboardmembersmaydeliberatelydevotetheire?ortstogroomingpreferredcandidates,withthecelebratedexamplesofGEandIdeaXerox(

FuandWu

,

2022

).GovernmentsinOECDcountriesstrivetosupportsmallandmedium-sizedenterprises(SMEs)throughpreferentialaccesstopublicprocurementopportunities(OECD,2018).

2

Second,contestantscanbesubjecttouncertaintysurroundingthenatureandenviron-

1Similarly,inSingapore,theMinistryofDefence(MINDEF)andSingaporeArmedForcessponsortheannualDefenseInnovationChallenge,whichprovideswinnerswithgrantsandtheopportunitytoco-developtheirsolutionswithMINDEF.TheEuropeanInnovationCounciladministersEuropeanSocialInnovationCompetitionschemestoincentivizeentrepreneurialsolutionsthatenhancesocietalwell-being,andtheAri-zonaStateUniversityInnovationOpenencouragesstudent-ledventurestoundertakehard-techR&D.

2ThereaderisreferredtoOECD(2018),SMEsinPublicProcurement:PracticesandStrategiesforSharedBene?ts,OECDPublicGovernanceReviews,OECDPublishing,Paris,

/10.1787/9789264307476-en

.

3

mentofthecontest—e.g.,theexactvalueoftheprize.Imaginethefollowingscenarios.

(i)Privatemilitarycontractors,wheninvestingintheirprototypestocompeteforDoD’sprocurementcontract,maynotbefullyinformedoftheexactcoststobeincurredwhenexecutingthedeliverycontract—e.g.,thee?ortsto?ne-tuneandmanufacturetheproductincompliancewithDoD’sregulationsandprovisions—whichcausesun-certaintyabouttheeventualpro?tabilityofthecontract.

(ii)Workersinsidea?rm,whencompetingforavacancy,areoftenunawareofthenatureandchallengeofthenewpost—e.g.,thescopeofdutiesandtheresources,andsupportavailablefromseniormanagement—anditsimplicationsforfuturecareeradvancement.

(iii)TheArizonaStateUniversityInnovationOpenencouragesstudent-ledventurestoundertakehard-techR&D.Itawardsthewinnernotonlycashprizesbutalsoaccesstotheuniversity’sincubatorandacceleratorprogramandopportunitytocollaboratewiththesponsoringcompanies,whichareestablished?rmsinrelevantareas.However,theoptionvaluesofthesenonmonetaryrewardscannotbeascertainedwithoutsu?cientdetails—e.g.,theresourcesavailablefromtheincubatorandtheextentofinvolvementandcommitmentofthesponsoringcompanies.

Thesephenomenasparknaturalquestionsforcontestdesign.First,howshouldacontestdesigneroptimallysetbiasesincontestsfore?cientincentiveprovision—i.e.,whichcontes-tantsaretobetreatedpreferentially,andtowhatextent?Second,howshouldthedesignerdistributeherinformation—i.e.,shouldshediscloseherinformationand,ifso,towhom?

Thispaperexploresoptimalcontestdesigntoaddresstheabovequestions.Incontrasttothevastmajorityofpreviousstudiesofcontestdesign,weallowthecontestdesignertodeploytwoinstrumentsandchooseanoptimalcombination:(i)apotentiallyselectiveinformationdisclosureschemethatmaydiscloseinformationtooneplayerwhileconcealingitfromtheotherand(ii)ascoringbiasthatcaneitherdiscountorin?ateacontestant’sperceivedoutputintheevaluationprocessrelativetoothers.

ASnapshotoftheModelWeconsideracontestinwhichtwopotentiallyasymmetricplayerscompeteforaprizeofacommonvalue.Theprizevalueisunknownandcanbeeitherhighorlow,withapubliclyknowndistribution.Playerssimultaneouslycommittotheire?ortstoviefortheprize—e.g.,privatemilitarycontractors’e?ortstodeveloptheirprototypes—whilethee?ortincursaconstantmarginalcost.Thewinnerisselectedthroughanall-payauction(see,e.g.,

HillmanandRiley

,

1989

;

Baye,Kovenock,andDeVries

,

1993

,

1996

;

CheandGale

,

1998

).Thedesignerevaluatesplayers’entries;eachplayer’se?ortisconvertedintoascore,andthehigherscorerwins.Theasymmetryoftheplayersissuch

4

thatplayer1bearsaweaklyhighermarginale?ortcostc1thanplayer2—i.e.,c1>c2—sothelatteristhefavoriteinthecontest.

Thedesignerconductsaninvestigationandacquiresaninformativebinarysignalabouttheunderlyingprizevalue,whichenablesamorepreciseposteriorthroughBayesianupdat-ing.Priortothecompetition,thedesignercommitstothecontestrule,whichconsistsoftwoelements.First,adisclosureschemespeci?eswhetherthesignalistobedisclosedandwhichplayeristoreceiveit.Thedisclosureschemeisasymmetricwhenthedesignerconveysthesignaltoonlyoneplayerwhileconcealingitfromtheother,inwhichcasetherecipientisawardedinformationfavoritism.Forinstance,theorganizerofabusinesspitchingcom-petitionmaybriefpreferredentrepreneursmoreelaboratelyonthefundingopportunitiesavailabletowinningprojects.Second,acoe?cientisimposedoneachplayer’se?orttogeneratehisscore.Wenormalizethecoe?cientfortheunderdog—i.e.,player1—tooneandthatforplayer2to6>0,whichiscalledascoringbias.Thebiascanbeinterpretedasanominaljudgingrule,aswellasmeasuresthatelevateordiscountone’s(perceived)output.Forinstance,theleadingcandidateforcorporatesuccessionisoftenappointedthepresidentorCOO,whichendowsthemaccesstoadditionalcorporateresourcesandimprovestheirvisibilitytotheboard.

Weconsidertwoobjectivesforcontestdesign.The?rstistheusualmaximizationofexpectedtotale?ort(see,e.g.,

MoldovanuandSela

,

2001

;

Moldovanu,Sela,andShi

,

2007

).Forinstance,thegovernmentoranonpro?torganizationmayuseanR&Dchallengetofuelthepublic’sinterestinacertainareaofscienti?cortechnologicalresearch;e.g.,cleanenergy.Thesecondisthemaximizationoftheexpectedwinner’se?ort,whichhasattractedincreasinginterestinrecentliterature(see,e.g.,

MoldovanuandSela

,

2006

;

FuandWu

,

2022

).Consider,forinstance,acontestsponsoredbyapharmaceuticalcompanytoprocureaninnovativeingredient.Onlythequalityofthewinningsolutionaccruestothebene?tofthesponsor.Asweshowbelow,theoptimalcontestdesigncruciallydependsonwhetherthedesigneraimstomaximizetheexpectedtotale?ortortheexpectedwinner’s.Intuitively,thedi?erenceisdrivenpartlybythefactthattheexpectedtotale?ortisthesumofthemeans,whiletheexpectedwinner’se?ortisthemodi?ed?rst-orderstatistic,ofthe(random)e?ortvariablesbythetwoplayers.

3

SummaryofResultsandImplicationsThecontestgamecanbeviewedasanall-payauctionwithinterdependentvaluationsanddiscretesignalspaces.

Siegel

(

2014

)providedthetechniqueforthecasewithaneutralscoringrule6=1.Weextendtheanalysistocharacterizethemixed-strategyequilibrium,whileallowingforabiasedscoringrule61,

3Itisnoteworthythatinourcontext,theexpectedwinner’se?ortisnotthesimplehigheste?ortexceptforthecaseof6=1.Underabiasedscoringrule(61),thewinnermaynotbetheonewhoexertsthehigheste?ort.Wethuscalltheexpectedwinner’se?ortamodi?ed?rst-orderstatistictore?ectthenuance.

5

whichpavesthewayfortheoptimalcontestdesign.

Resultsforthemaximizationofexpectedtotale?orta?rmtheconventionalwisdomthatalevelplaying?eldfuelscompetition.Theoptimumisachievedbyanexpostsymmetriccontest,whichsetsthescoringbiasto6=c2/c1,togetherwithasymmetricdisclosurescheme(fulldisclosureorfullconcealment).Neitherplayerisawardedinformationfavoritism.Thebias6=c2/c1,calledthefairbiasofthecontest,preciselyo?setstheinitialadvantageofplayer2intermsofthemarginalcostofe?ort.

Theoptimalcontestdesigndepartsfromtheconventionalwisdomwhenthedesigneraimstomaximizetheexpectedwinner’se?ort.Inacomplete-informationall-payauction,thefairbiasmaximizestheexpectedtotale?ortandtheexpectedwinner’se?ortsimultaneously(see,e.g.,

Fu

,

2006

).Inourcontext,thedesignermaypreferatilting-and-relevelingcontest,whichcreatesexpostbilateralasymmetrybetweenplayers.Speci?cally,thedesignermayawardoneplayerinformationfavoritism—i.e.,disclosinghersignalexclusively—whilerelevelingtheplaying?eldbylettingthescoringbiasdeviatefromthefairleveltofavortheother.Threeobservationsarenoteworthy.

(i)Thetilting-and-relevelingcontestcanbeoptimalevenifplayersareexantesymmetric.

(ii)Whenplayersareexanteasymmetric,thetilting-and-relevelingcontest,wheneveroptimal,awardstheunderdog,player1,informationfavoritism.

(iii)Thetwoinstruments—i.e.,disclosureschemeandscoringbias—arecomplementarytoeachother:Asymmetryneveremergesintheoptimuminasingledimension;theoptimalcontestiseitherexpostsymmetric(symmetricdisclosuretogetherwithfairbias)orexpostasymmetricintermsofbothinformationdisclosureandscoringrule.

Atilting-and-relevelingcontestcouldenableanupwardshiftofthewinner’se?ortdistri-butionundercertaincircumstances.Weelaborateonthelogicofourresultsinthissimplebutcounterintuitivecasewithsymmetricplayers.Speci?cally,wecompareabilaterallyasymmetriccontest—whichdisclosesthesignaltoplayer1andsetsafavorablescoringbias6>1forplayer2—withasymmetricone,inwhichthesignalisconcealedfrombothplayersand6=1.

Assumeaunitymarginale?ortcost.Inthesymmetriccontest,itisstraightforwardtoinferthattheire?ortsareboundedfromabovethecommonexpectedprizevalue.Wenowletthedesigner“tilt”thecontestbydisclosingthesignaltoplayer1—whichallowshimtoupdatehisbeliefabouttheprizevalue—whilemaintainingtheneutralscoringrule.Itiscrucialtonotethatplayer1’smixedstrategywouldbetype-dependent:Thebiddingsupportsforlowandhightypes—i.e.,player1whenreceivinglowandhighsignals,respectively—donotoverlap.Player2istheuninformed,sohecontinuestobidaccordingtotheprior,

6

althoughhetakesintoaccountthefactthatplayer1receivesasignal.Theexpectedwinner’se?ortinthiscontestfallsshortoftheinitiallysymmetricone.Thelowtypeisdiscouragedbyhislowvaluation.Despitethehighvaluation,thehightype,whoisnowanexpostfavoriterelativetotheuninformedplayer2,hasnoincentivetoraisehismaximume?ort—i.e.,theuppersupportofhisbiddingstrategy—sincehisopponent’se?ortscontinuetobeboundedbytheprior.

Nowweletthedesigner“relevel”thecontestbylifting6above1.Thescoringrulefavorsplayer2,whicha?ectsthelowandhightypesofplayer1di?erently.Thebiasfurthersqueezesthelowtype,whilediminishingtheadvantageofthehightype.Thelatteristhusforcedtostepuphise?ort.Asaresult,thehightype’sbiddingsupportisstretchedupward.Thereexistsauniquescoringbias—calledtherelevelingbias—underwhich(i)thelowtyperemainsinactivewithprobabilityone,(ii)boththehightypeandtheuninformedplayer2remainactivewithprobabilityone,and(iii)thehightype’smaximume?ortrisestohisupdatedprizevaluation.

Insummary,thetilting-and-relevelingcontestmayoutperformthesymmetriconewhenahighsignalisrealized,inwhichcasethehigh-typeplayer1maycontributeahigherwinner’se?ortthatisotherwiseimpossible.Theformercontestunderperformsthelatterwhenalowsignalappears,becausethelowtypewouldexitthecompetition.However,thelossispartlymadeupforbyplayer2:Hebidsactivelyregardlessoftherealizedsignal.Weidentifyaconditionthatsummarizesthetrade-o?anddeterminestheoptimum.Thesametrade-o?governsthecontestdesignwithasymmetricplayersandexplainstheallocationofinformationfavoritismandfavorablescoringbiasbetweentheweakandstrongcontenders.Section

3.2.2

delvesintothelogicindepth.

Weextendourmodeltothreealternativesettings.Inthe?rst,thedesignerdecideswhethertoprovideplayerswithrelevantinformation,andplayerscanacquireinterdependentsignalsfromtheinformation(

MilgromandWeber

,

1982

).Thesecondallowsthedesignertochoosetheinformationstructureofherinvestigation,whichcorrespondstotheconceptoftheBayesianpersuasionapproachpioneeredby

KamenicaandGentzkow

(

2011

).Thethirdextensionrequiresthattheweakerplayerwinwithaminimumprobability,whichmirrorsthepracticeofa?rmativeaction(

CoateandLoury

,

1993

).Weshowthatourmainresultsarequalitativelyrobusttothesemodelingvariationswithadditionalinsights.

RelatedLiteratureInthispaper,weconsidertheoptimaldesignofacontestmodeledasanall-payauction.Thecontestmodelcanbeviewedasavariantofthefamilyofall-payauctionswithinterdependentvaluations,includingthoseof

KrishnaandMorgan

(

1997

);

LizzeriandPersico

(

2000

);

Siegel

(

2014

);

RentschlerandTurocy

(

2016

);

LuandParreiras

(

2017

);and

Chi,Murto,andV¨alim¨aki

(

2019

).Ourstudyisprimarilylinkedtotwostrands

7

oftheliteratureoncontestdesign:(i)(identity-dependent)di?erentialtreatmentsand(ii)informationdisclosure.Tothebestofourknowledge,wearethe?rsttoallowthedesignerto?ne-tunethetwoinstrumentssimultaneouslyandchoosetheiroptimalcombination.

Anenormousamountofscholarlye?orthasbeenexpendedontheoptimalwaytobiasacontestbyimposingdi?erentiatedtreatments(

MealemandNitzan

,

2016

).Theliteraturehasconventionallyespousedthemeritsofalevelplaying?eldforfuelingcompetitionandsug-gestedhandicappingthefavoritetoo?setinitialasymmetrybetweenplayers—e.g.,

Epstein,

Mealem,andNitzan

(

2011

);

Franke,Kanzow,Leininger,andSchwartz

(

2013

,

2014

);

Franke,

Leininger,andWasser

(

2018

).Ahandfulofrecentstudies—e.g.,

DrugovandRyvkin

(

2017

);

FuandWu

(

2020

);

BarbieriandSerena

(

2022

);

WasserandZhang

(

2023

);

EcheniqueandLi

(

2022

)—identifythecontextsinwhichtheoptimalcontestmightinsteadrequireupsettingtheinitialbalanceoftheplaying?eld.However,thisstrandofstudiesmainlyfocusesontheoptimalconstructionofidentity-dependenttreatmentsanddoesnotinvolveinformationdisclosure.

Theliteraturehasincreasinglyrecognizedinformationdisclosureasavaluableadditiontoacontestdesigner’stoolkit,e.g.,

Halac,Kartik,andLiu

(

2017

)and

Ely,Georgiadis,

Khorasani,andRayo

(

2022

).

4

Lu,Ma,andWang

(

2018

)and

Serena

(

2022

)exploreoptimaldisclosureincontestswithcontestantsofindependentandprivatetypes.Thesestudiesrequiresymmetricdisclosurerules,suchthatthesignalfromthedesignermustbepublic.

W¨arneryd

(

2012

)considersacommon-valuecontestandallowsonlyaportionoftheplayerstoknowtheprizevalue.Heshowsthatselectivelyinformingplayersoftheprizevalueissuboptimalwhenthedesignermaximizestotale?ort.TheBayesianpersuasionapproachhasbeenappliedinahandfulofstudiestoexploreoptimalinformationdisclosureincontests;e.g.,

ZhangandZhou

(

2016

);

ChenandChen

(

2022

);

MeloPonce

(

2021

);and

Antsyginaand

Teteryatnikova

(

2022

).However,thesestudiesfocusexclusivelyoninformationdisclosureincontests.

Therestofthepaperproceedsasfollows.Section

2

setsupthemodelandSection

3

carriesouttheanalysis.Section

4

presentsthreeextensionsandSection

5

concludes.Appendix

A

collectstheanalyticalresultsforSection

4.1

.ProofsofourmainresultsarecollectedinAppendix

B

.

2TheModel

Tworisk-neutralplayers,indexedbyieN={1,2},competeforaprizeofacommonvalueve{vH,vL},withvH>vL>0.ThehighvaluevHisrealizedwithaprobability

4Bothstudiesfocusoninformationdisclosureinthecourseofdynamiccontests.Thisstrandofliteraturealsoincludes

Yildirim

(

2005

);

Aoyagi

(

2010

);

Ederer

(

2010

);and

GoltsmanandMukherjee

(

2011

).

8

Pr(v=vH)=:μe(0,1),withthelowvaluevLtoberealizedwiththecomplementaryprobability.Playersareinitiallyuninformedabouttheprizevalue,whileitsdistributioniscommonknowledge.Theysimultaneouslyexerttheire?ortsxi>0towintheprize.Oneplayer’se?ortincursaconstantmarginale?ortcostci>0.Withoutlossofgenerality,weassumethatplayer2isthestrongercontender;i.e.,c1>c2.

Winner-selectionMechanismandScoringBiasThecontestdesignerimposesascor-ingbiasδi>0oneachplayeri’se?ortentryxi,whichgenerateshisscoreδixi.Wenormalizeδ1to1andsetδ2=δ>0.Fixingasetofe?ortentriesz:=(x1,x2)eR,player1’sprob-abilityofwinningthecontest—i.e.,thecontestsuccessfunction(CSF)—isgivenby

(0,

1,p1(x1,x2)=,

ifx1>δx2,

ifx1=δx2,

ifx1<δx2,

andplayer2winswithaprobabilityp2(x1,x2)=1-p1(x1,x2).Thatis,aplayerwinsthecontestwithcertaintyifhisscoreexceedsthatofhisopponent.Thewinnerispickedrandomlyintheeventofatieinscores.

DisclosureSchemesThedesignerconductsaninvestigation