Competitive balance: Information disclosure and discrimination in an asymmetric contest

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Journal of Economic Behavior and Organization

journalhomepage:www.elsevier.com/locate/jebo

Competitive balance: Information disclosure and discrimination in an asymmetric contest ^R

Derek J. Clark

^a

, Tapas Kundu

^b,∗

aSchool of Business and Economics, UiT the Arctic University of Norway, Norway

bOslo Business School, Oslo Metropolitan University, Norway

a rt i c l e i nf o

Article history:

Received 18 August 2020 Revised 21 December 2020 Accepted 30 January 2021

JEL classification:

D02 D72 D82 Keywords:

Asymmetric contest Information design Discrimination

a b s t ra c t

Westudyadesignproblemforaneffort-maximizingprincipalinatwo-playercontestwith twodimensionsofasymmetry.Playershavedifferentskilllevelsandaninformationgap exists,asonlyoneplayerknowstheskilldifference.Theprincipalhastwopolicyinstru- mentstoredressthelackofcompetitivebalanceduetoasymmetry;shecancommittoan information-disclosingmechanism,andshecandiscriminateoneoftheplayersbybiasing hiseffort.Wecharacterizetheoptimallevelofdiscriminationtomaximizeaggregateeffort, showing howthisinextricablydeterminesthe choiceofinformation disclosure.Applica- tionsarefoundinnewcomer-incumbentsituationsinaninternallabormarket,sales-force management,andresearchcontests.

© 2021TheAuthor(s).PublishedbyElsevierB.V.

ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Competitioninsocial,politicalandeconomicspheresisoftenanalyzedasacontestinwhichresourcesaresunkinorder to win a prize. Numerous applications of these frameworkscan be found in the literature relatingto conflict andwar- fare, lobbying,elections,internal andexternallabor marketsorvarious typesofresearch competition.¹ Acommontheme inmuch oftheexisting workishowthecharacteristics ofthe competitorsandstructure ofthecontestaffecttheamount ofresources oreffortusedinthe competition,andhowa contestdesignermayattempttoinfluencethis;themostusual assumptionisthatthedesignerwishestomaximizetheresourcesexpended.²Weconsideracontestbetweenanincumbent andanewcomerforafixedprize.Suchsituationsareoftencharacterizedbythenewcomerhavingbetterinformationthan boththerivalandthecontestdesigneraboutattributessuchasownabilitythatarerelevanttotheplayingofthecontest.

The differenceinthisattributemaybe largeorsmall,andit isnotcertain thattheincumbentisthe superiorcontestant.

Designingthe contesttomaximize effortinthissituationisnot atrivialexerciseforthe principalsinceit involvesnego- tiatingtwodimensionsofheterogeneity.First,hiddeninformationmakes thereturntoeffortuncertainfortheuninformed

R We would like to thank two anonymous referees for insightful comments. Any remaining errors are our own.

∗Corresponding author.

E-mail addresses: [email protected] (D.J. Clark), [email protected] (T. Kundu).

1See Konrad et al. (2009) and the references therein.

2Other aims are possible. In some contexts achieving a close contest may be the objective ( Runkel, 2006 ), or maximizing participation ( Azmat and Möller, 2009 ), or securing the highest quality winner ( Serena, 2017b ).

https://doi.org/10.1016/j.jebo.2021.01.034

0167-2681/© 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

player,discouraging effort.³ Second,heterogeneity,ascapturedbyarelativeskilldisparity,isgenerallyacknowledgedtobe onefactorthatlimitsresourceuseincontestsettings(seeChowdhuryetal.,2020).Aplayerwithalargerelativeability,or alargerelativevaluationofwinning,mayintimidateopponentsintosubmittinglowefforts,andcanhencereduce hisown effortsandstillwinwithalargeprobability.⁴

Inthispaper,we setupa simplemodelthateffectivelycapturesthe incumbent-newcomerscenario,andinwhichthe principalhastwopolicyinstrumentsatherdisposal.Shecancommittoasignalingmechanismwhichmayreveal-atleast partially-thehiddeninformation;furthermore,shecanuseapolicywhichtreatsoneoftheplayerspreferentiallybybiasing positively hiseffortlevelinthecontest.⁵ We demonstratethatthere isaninteresting interplaybetweenthesetwopolicy instruments, andthatthe optimallevel anddirectionofdiscrimination inextricablydetermines the choiceof information disclosure. Furthermore,weshowthattheprincipalmustbemindfulofthefactthat herchoicesmayaffecttheincentives ofthelessskilledtoexerteffortinthecontest.⁶

Ourcontributionistwofold.First,weanalyzeamodelinwhichthereisasymmetricinformationabouttheabilitiesofthe contestantsandshowhowtheskilldifferentialandtheexistingdiscriminationpolicyaffectstheincentivesoftheprincipal torevealinformation;secondweallowtheprincipaltochoosetheoptimallevelofdiscrimination,inwhichshehastotake accountofhowthisaffectstheoptimaldisclosureofinformation.Ourmodelfeaturesuncertaintyaboutthecharacteristicsof thecontestants,ratherthanthevalueofthecontestedprize.Specifically,thereisaskilldifferentialbetweenthetwoplayers incarryingoutthecontesttask,therelativevalueofwhichisknownonlytooneplayer,thenewcomer;theskilldifferential canbelargeorsmall,andmaypositivelyfavoreitherplayer.Ourmodelingassumptionisthat aplayer’sskillismultiplied by his effortto give his effective level of contest effortwhich directly affects the success probabilities. The uninformed incumbentandprincipalhaveapriordistributionovertwopossiblevaluesoftherelativeskill.Inseekingtomaximize the expected effortfrom thecontest,the designercan committo aset ofsignalsthat are sent afterthestate is determined.

Furthermore,weintroduceadiscriminationparameterwhichchangestherelativeproductivityofthecontestants’effortin determiningtheprobabilityofsuccess.Thediscriminationparameteraffectseffectivecontesteffortmultiplicatively,scaling a contestant’s effortup or down depending on the directionof the discrimination;given that skill also affects effective contesteffortmultiplicatively,weshowthatamultiplicativediscriminationparameterisabetterpolicyinstrumentforthe principalthanforexampleanadditiveone.

When the discrimination parameter is fixed, thedesigner can influence effortonly through thesignaling mechanism.

We calculatetheoptimalsignalssetbythe principal,dependingon thefixeddiscrimination policy.Furthermore,wesub- sequentlyallowher tochoose thisparameter inordertoachieve themaximalamountofeffortpossible.Insodoing, she mustbe mindfulofthefact thatthechoiceofthemagnitudeanddirectionofdiscrimination affectstheoptimalpolicy of informationdisclosure.Weshowthatthedesignerwillnotwanttoimplementalevelofdiscriminationthatinvolvespartial disclosureofthehiddeninformation.Shechoosesoptimallybetweenvaluesofthediscrimination parameterforwhichno disclosureorfulldisclosureisoptimal,andweshowhowthisisconnectedtothepriorbeliefsoftheuninformedincumbent.

When itisthoughtthattheinformednewcomerisverylikelytobe skill-inferiorthenthedesignerdoesnotbenefitfrom revealingthistotheuninformedopponent,andshechoosestodiscriminateinfavoroftheinformed(butlikelylow-skilled) player;weshowfurtherhowthemagnitudeofthediscriminationdependsontheskilllevel.Ontheother hand,whenthe uninformedincumbentthinksthatitislikelythattheopponentwillbehighlyskilled,thedesignermustalleviateriskiness ofeffortforthisplayerbyrevealingthetruestate;shewillalsodiscriminateagainsttheinformedplayertoencourageeffort byboth.

Onecan imagineseveralapplicationsinwhichanewcomercompetesagainstan incumbentforaprize,andtheability oftheincumbentisknownbyall,butthenewcomerhashiddentalent.⁷Aninternallabormarkethasfeaturesincommon with ourframework in whichan insiderand an outsiderto a firm compete fora position ora promotion. The outsider knows his ownskill level,butthe insideris uncertainof thequality ofthe rival. Inthe context ofa tournamentmodel, Chan(1996)analyses thepreferentialtreatmentofonetypeofcandidatedependingontheunknownskilllevelofexternal workers. If they are expected to be highly skilled, the performance of the firm can be improved by giving preferential treatment to internal candidates; ifexternalcandidates are oflittle threat interms ofskilllevel,then they can be given an advantageinthe promotioncontestto incentivizetheinternal candidates.Thisisasimplemechanismforlevelingthe playingfield.Ourmodeldevelopsthisapproachbymaking manipulationoftheinformationstructureapolicy instrument in thecontest design.Inthiscase, thesignaling mechanismthatwe consider canbe likenedto theuseofaptitude tests

3Asymmetry in what the players know about the structure of the contest can generally lead to a low effort level ( Wärneryd, 2003 ).

4In a dynamic race setting, Konrad and Kovenock (2009) investigate the discouragement effect that arises as one competitor nears the finishing line, causing opponents to simply give up.

5In this way we combine incentive manipulation with information provision, which Kamenica (2019 , footnote 2) regards as a promising area for future research.

6Here, the principal chooses whether and how to disclose information. Denter et al. (2020) consider a situation where the agents themselves can choose to reveal information. See also Epstein and Mealem (2013) .

7Denter et al. (2020) also consider this framework, analyzing the information disclosure decisions made by the newcomer himself by choosing to show off his talent, or lie low.

by firms toglean some informationabouttheir skilllevel.⁸ Theinformativeness of thesignal can be chosen by thefirm accordingtothetestdesign.Afullyinformativesignalaboutanagent’sabilitycouldrevealmanydimensionsofskillthrough testsofcognitiveability,problem-solving,criticalthinkinganddecisionmaking.Alessinformativesignalmayonlyconsider oneofthesedimensions.Insales-forcemanagement,anestablishedsellermayfacecompetitionfromanoutsidechallenger andthefirm maychoosetodiscloseinformationaboutpastsalesperformance ofthechallenger;thisinformationmaybe coarse(numberofsalesorrevenueachieved)orfiner(givingdetailsofsalesterritories,productinformation,andarelative comparisonwithpeers).Discriminationinthiscasecan bethoughtofaslessadministrativeduties,betteraccesstoback- office resources, moretraining, andbetterterritories; see,e.g., SkieraandAlbers (1998), FarrellandHakstian(2001),and Krishnamoorthyetal.(2005).

Researchcontestsbetweencompetingteamsalsofitourframework.Theeffortmadebyeach applicantcanbe thought ofasbuildingupthequalityoftheteamthatiscompetingforaresearchgrant;oneteammayalreadybewellestablished whereas achallengerisup-and-comingandnottested. Inthiscase,theresearchsponsorcould grantthenewteamapre- projecttogather informationaboutitsrelativeskilllevel.⁹Discriminationinthiscasecouldinvolvepreferentialtreatment ofyoungresearchersoraidinwritingagoodproposal.Manylargecorporationsruninternalinnovationcompetitionswhich see employees(or teamsofemployees) compete witheach other inorderto achievefurther fundingfortheir projects.¹⁰ Rathi (2014)documents thatThompson Reuters,theUS DepartmentofHealthandHumanServices,ReedElsevier andTC Transcontinental(the largestprintingcompany inCanada)usedifferentformsofinnovationcontest,releasinginformation aboutpreviouscontestsandfindings,andusinginternalmentorsasawayofgivinganadvantageinthecompetition.Sim- ilarly,one maythinkofprocurementcontractingasfittingourmodel,inwhichan incumbententrepreneuriswell known, whereasachallengerhashiddenqualities.Instructingthenewcomertodevelopaprototype,andrevealingtheresultsofits testing,isawayinwhichinformationcanpotentiallybegarneredandreleased.Furthermechanismsofinformationdisclo- suremaybe legallyspecified.ZhangandZhou(2016)note that politicalcandidatesintheUSare requiredbytheFederal Election Campaign Act to reveal campaigncontributions and expenditures; this provides information aboutthe financial supportbaseofthecandidates.

RelatedLiterature

We draw together three strands of literature in this paper. One relates to discrimination in contests, the second to asymmetricinformationstructuresincontests,andthethirdtotheuseofsignalingmechanismstorevealhiddeninforma- tion,otherwiseknownasBayesianpersuasion(KamenicaandGentzkow,2011).¹¹ Chowdhuryetal.(2020)andMealemand Nitzan (2016) discussdifferentformsof discriminationaimed atleveling theplayingfield to achieve competitivebalance in asymmetriccontests. Instruments atthe disposalofan effort-maximizing principalinclude exclusionofstrong players (Bayeetal.,1993),caps onefforts (CheandGale,1997)orvarious formsofdiscriminationsuch asdifferentialtaxation of the prize,orgivinghead startsorhandicapstosome players(see the survey byMealemandNitzan, 2016). Inthe latter case, one mayaffect the structure ofthe contest environment through the probability of success by giving a headstart or biasingthe efforts ofone ormoreplayers. Forthetwo-player case, Franke(2012) showshow biasingthe efforts opti- mally inrelationto competitors’ costof effort(orequivalentlyprize valuation)leadstomaximal contest effort.¹² Fu and Wu (2020) show that multiplicativebiasesoutperform additive heads startsin ann-player lotterycontestwithheteroge- neouscontestantsdifferingintheirprizevaluations.Theprincipalpursuesabroadrangeofobjectives,includingtotaleffort maximization.

Leaving the framework of complete information, Hurley and Shogren (1998a), Hurley and Shogren (1998b) and Wärneryd (2003)considerhow asymmetriesin informationcan affectcontestbehavior. Particularly relevantto ourwork is their focuson caseswithone-sided informationalasymmetry,whereone player hasbetter informationthanthe com- petitor.¹³ Supposethat ina two-player framework,one knowsthe exactvalue oftheprize buttheother knowsonly the underlyingdistribution.TheeffortoftheuninformedplayeristhenlikenedbyHurleyandShogren(1998a)toariskyinput which tends todecreaseeffortin equilibrium, afinding that isreinforcedby Wärneryd(2003)who showsthat thetwo- playerlotterycontestwithasymmetricinformationyieldslowerequilibriumeffortthanwhentheplayersaresymmetrically informedoruninformed.

OurpaperisclosesttothatofEpsteinandMealem(2013)whoattempttobridgethegapbetweenone-sidedincomplete informationon theonehand,andinformationdisclosureon theother.Weusetheir basiccontest framework,andextend it in twoimportant directions.First, we allow forexogenous andendogenousdiscrimination ofthe contestants, showing

8Several large firms use aptitude tests in hiring such as Apple, Samsung, Microsoft and Nike. See https://www.aptitude-test.com/blog/articles/

10- major- companies- that- use- aptitude- testing/ . SHL’s Global Assessment Trends Report in 2018 documents that 76% of organizations with over 100 em- ployees use aptitude and personality tests in hiring decisions. See https://www.shl.com/en/assessments/trends/ .

9Serena (2017a) mentions several research contests in which there is an initial stage in which information on the rivals may be gathered and revealed, before final proposals are made. These include the Horizon 2020 submissions to the European Research Council and design competitions run by the Royal Australian Institute of Architects.

10See Adamczyk et al. (2012) for a review of research on innovation contests, and Höber (2017) for internal contests.

11Our focus is on a lottery contest, originating in Tullock (1980) , rather than an all-pay auction ( Hillman and Riley, 1989 ). See Lu et al. (2018) for an analysis of information disclosure in an all-pay auction.

12See also Epstein et al. (2013) .

13Serena (2017a) in contrast considers a model of information disclosure in a lottery contest with two-sided private information.

how thisaffectscontestincentives;second, weconsideroptimalinformationdisclosurebyaneffort-maximizingprincipal.

EpsteinandMealem(2013)considertheincentivesoftheinformedplayertorevealowntype beforeitisknown,inorder toincreasehisownexpectedpayoff fromthecontest.Whilsttheyconsiderinformationdisclosureasabinarydecision,our analysis coversa much widerspecificationof possibilitiesinwhichthe principal canreveal all orno informationattwo extremes,butcanalso chooseasetofsignalsto imparta particularposteriorbelieftotheuninformed type whenthisis optimal.Furthermore,weanalyzetheinterplaybetweenthisinformationdisclosureandtheoptimaldiscriminationpolicy.

Manipulationoftheinformationstructureisaninstrumentthatcanbeemployedtoachieveaspecificgoal.Kamenicaand Gentzkow (2011) have operationalized a method of Bayesian persuasion in which a principal can commit to a state- conditional distributionof signalsbeforerealizationofthe state;thishasbeenappliedto alottery contestby Zhangand Zhou(2016),FengandLu(2016)andFuetal.(2016).Thelatterpapersconsiderinformationdisclosureaboutanunknown numberofcompetitors,whilstZhangandZhou(2016)ismorerelevantforouranalysissinceitisatwo-playercontest.The asymmetricinformationrelatestothevalueoftheprize,andtheeffort-maximizingdesignermustrevealthestateoptimally bycommittingtoasignalingmechanism.¹⁴KamenicaandGentzkow(2011)showgenerallythatfulldisclosureisanoptimal policy ifthepayoff ofthesender(principal)asafunction ofthebeliefofthe receiver(uninformedcontestant)isglobally convex,whilstnodisclosureisbestwhenitisgloballyconcave;ifthepayoff functionofthesenderhasconcaveandconvex portions,thenpartialdisclosureisoptimal.ZhangandZhou(2016)considerfirstastructureinwhichthehiddenprizevalue isbinary,whichyieldsanexpectedeffortfunctionthatisgloballyconvexorconcavedependingonthevaluationbythein- formedplayerandthetwopossiblevaluationsoftheuninformed;hence,asignalisoptimalthatgiveseitherfulldisclosure ofthehiddenstate,ornodisclosure.Onlywhentherearemorethantwopossiblevaluationscanpartialdisclosureappear, inwhichthe signalrevealsthetruevalueoftheprizeimperfectly totheuninformed player.Ourfindings showthat even for abinarydistribution of theskilldifferential, partial disclosurecan be optimalforsome givenlevel ofdiscrimination, althoughtheprincipalwillnotimplementsuchadiscriminationlevelifshecanchooseit.¹⁵

The paper isorganizedas follows.Section 2 sets up thebasic contest andframework forinformation disclosure, and Section 3 solves forequilibrium effortlevels underdifferent informationalassumptions. Section 4 considers the optimal informationdisclosurepolicyforagivenlevelofdiscrimination,andtheoptimaldirectionandmagnitudeofdiscrimination iscalculatedinSection5.WediscussourmainmodelingassumptionsinSection6,andSection7concludes.TheAppendix containsproofsofourresults.¹⁶

2. Model

Tworisk-neutral agents, N (newcomer)andI (incumbent),compete fora fixed prizeof value 1ina contest designed by theprincipalP.Thecontest scoreachievedby playerIissimplygivenbyhis efforte_I,andthescoreofthenewcomer is a multipleof his effort:

α

^seN,where

α

>0ands>0. The parameter

α

^{measures thedegree of} discrimination:

α

>1 (0<

α

<1)impliesthat N is positively(negatively) discriminated. Thisparameteris chosen optimallyby theprincipal in theanalysisbelow.Theparametersisoneoftheprimitivesofthemodel,andmeasuresN’srelativeskill:s>1(0<s<1) impliesthat N issuperior(inferior)inskill.Thesuccess probabilitiesofN andI,givenan effortprofile(^eN,e_I),e_N,e_I>0, aredeterminedbyaplayer’sscorerelativetothetotalscoreinthecontest:

ρ

^N=

α

^seN

α

^seN+eI,

ρ

^I=

α

^seNeI+eI.

Thiscontestsuccessfunctioniscommonlyused,andhasbeenaxiomatizedbyClarkandRiis(1998).¹⁷SchallerandSkaper- das(2020) suggest themultiplicativeapproachtakenhereasa generalwayofcapturingasymmetry incontests.The skill differential implies that the newcomeris a certain percentage better orinferior than the incumbentat carryingout the contesttask,anditwouldthenappearnaturalthattheinstrumentoftheprincipalshouldalsobemultiplicative.Thisisalso inlinewiththeapproachtomodelingdiscriminationtakenbyEpsteinetal.(2011).¹⁸

Weassumethattherelativeskillsisthesourceofinformationasymmetryatthebeginningofthegameandisreferred to asthe stateofthegame.PlayerN knowsthe state.PlayerIandtheprincipaldo notknowthestate, butknowthat N is fullyinformed.Forsimplicity,we assume that thereare two possiblestates,one inwhich N issuperior andtheother

14Kamenica and Gentzkow (2011 , p. 2599) discuss the plausibility of committing to a signaling mechanism, noting that: “Firms often commit to the information they will seek out in performance reviews. Academic departments follow fixed rules about the information they will solicit for midterm or tenure reviews”.

15A major difference between our model and Zhang and Zhou (2016) is the fact that our use of the discrimination parameter renders the relative asymmetry between the players a continuous variable, even though the skill differential takes one of two values. For some of these values partial disclosure is optimal. In Zhang and Zhou (2016) , it is the relative valuation of the two players that captures the asymmetry, and this takes a finite number of values;

either full disclosure or no disclosure is optimal for all of these values in their model.

16Appendix B gives supplementary material relating to the analysis in Section 6 .

17If αse A+ e B= 0 , we assume that the prize is not awarded. This does not occur in equilibrium, however.

18“The bias is represented by the different values of a unit of effort made by the contestants”, Epstein and Mealem (2013 , p. 89). Modeling the skill differential as an additive head start is also a possibility. However, it is widely acknowledged that head starts tend to dampen contest effort in two-player contests ( Franke et al., 2013 ). With several players they can, however, be a useful tool for encouraging the participation of desirable players (with a high prize valuation for example), and excluding others.

inwhichNisinferior.Specifically,scantakeonlytwovaluesinS:=

{

^x,y

}

,y>1>x>0,withpriorprobabilitiesq∈(⁰,1) and(¹−q),respectively.ThiscontestspecificationisthesameasEpsteinandMealem(2013)whoanalyzethespecialcase

α

=1.Inordertobeabletoisolatetheeffectofthediscriminationpolicyonbehavior,weassumethaty= ¹x andx∈(⁰,1); implicationsofrelaxingthisassumptionarediscussedinSection6.¹⁹SinceS=

{

^x,¹_x

}

^{containsonlytwovalues,a}distribution overthestatespacecanbeexpressedwithascaler p∈[0,1]suchthatp=Pr[s=x].Wewillfollowthisconvention,unless statedotherwise.Inaddition,weusethenotationptodenoteagenericdistributionwhereverneeded,whileqalwaysrefers totheprior,whichisaparameterofthemodel.

2.1. Informationdisclosure

Beforethestateisrealized,theprincipalcommitstoandpubliclydisclosesapairofstate-conditionalsignaldistributions

π

(·

|

^x),

π

·

|

¹x

suchthat

π

(·

|

^s)∈(^M),whereMisafinitesetofsignalsand(^M)^{isthesetofall}probabilitydistri- butionsoverM.²⁰OncethestatesisrealizedandrevealedtoN,naturedrawsasignalm∈Mfromthedistribution

π

(·

|

^s)^. BothagentsobservethesignalandtheuninformedagentIupdateshisbelief.WeletqmdenoteI’sposteriorbeliefthatNis inferior,afterobservingasignalm,i.e.,q_m=Pr[s=x

|

^{m].NotethatsinceNalsoobservesthesignal,Ncaninferq}m.

The contest then takesplaces withagents exerting effortsimultaneously. The cost ofeffortis linearandidentical for eachagent,withaconstantmarginalcostofone.Theagenti∈

{

^N,I

}

^{choosesefforte}i≥0tomaximizehisexpectedpayoff, denotedby

v

_i.Theprincipal’spayoff isgivenbythetotalexpectedeffortsandisdenotedbyV_P.

Thetimingofthegameisasfollows:

1. Pchoosesthedegreeofdiscrimination

α

>0,whichiscommonknowledge;

2. P commitstoandpubliclydisclosesasetofdistributions

π

(·

|

^x),

π

·

|

¹x

where

π

(·

|

^s)∈(^M),thesetofallprob- abilitydistributionsoverthesignalspaceM;

3. Nature draws a realizationof the state s∈S, which is revealedto player N. Next,nature draws a signal m∈M from

π

(·

|

^s)^{.Thesignalmispubliclyobserved,leadingtoaposteriorbelief}distributionforI;

4. ThecontesttakesplacewithNandIchoosingeffortsimultaneously.

WestudytheperfectBayesianequilibriumofthegame.

2.2. Discussionofthemodelassumptions

Wemakecertainassumptions inourmodelforanalyticaltractability.First,weconsideronlytwopossiblestates.When therearemorethantwostates,theanalysisofoptimalinformationdisclosurerequiresanalyzingtheconvexitypropertyof afunction withamulti-dimensionaldomain.Sincewe characterizetheinformationdisclosurepolicy forallpossiblelevels ofdiscrimination,thisanalysisbecomesquiteintractable.Inordertofocusonthekeyissueoftheoptimaldiscrimination, we restrictthestate space.Further,ifbothstate valuesareabove(below)one,thenthe principalcaninferthattheunin- formedplayerIisalwaysinferior(superior),andtheresultonthedirectionofdiscriminationisstraightforward.Themore interesting caseisthe oneinwhich theuninformed playercan beeithersuperior orinferiorto theinformedplayer. Our assumption that thetwo statevaluesare reciprocaltoeach other allowsusto analyzethisinteresting caseinastraight- forwardmanner.InSection 6,we showthat thereducedmodelthatwe analyzeisrelatedtothegeneraloneby asimple isomorphicmapping.Ourmainresultsremainintact.

A finalassumption worthyofcommentisthat theprincipal chooses thediscrimination policyatthe beginningofthe game,sothatitisnotconditionedonanyinformationthat mightberevealedaboutthestate.Knowingthepossiblestates oftheworld,theprincipalsetsthepolicyinadvance,regulatingeffortthroughthesignalingmechanism;theprincipalonly observesthestateifthesignalisfullyrevealing,andweshowbelowthatthisisnotalwaysanoptimalchoice.Thedegreeof discriminationandthecommitmenttothesignaldistributionscanbechosensimultaneouslywithoutchangingourresults.

Theimportantpointisthatthesearedecidedbeforeanyinformationisrevealedaboutthestate.Hence,theprincipalmakes optimaldecisionsbaseduponthestructureoftheenvironmentinwhichsheoperates,ofwhichuncertaintyovernewcomer skillisan inherentfeature.Wethink ofthepolicychoice asbeingan impersonalandoverarchingprinciplethat doesnot changeaccordingtotheskilloftheplayerthatcompetes.Thisfitsmanycompetitiveenvironmentsinwhichrulesaremade withoutknowingtheexactsetofparticipants,includingresearch contests,sportingevents,litigation,politicalcompetition, andinternallabormarkets.

3. Contest

Webeginatstage4wherethecontestcantakeplaceundertwopossibleinformationstructures:(i)fullinformationand (ii)asymmetricinformation.

19See also the discussion of the main assumptions in Section 2.2 .

20Kamenica (2019 , p. 251) notes that “The key object in Bayesian persuasion models is the thing that Sender chooses, which goes by many names, including signal, signal structure, information structure, experiment, Blackwell experiment, or data-generating process.” In line with Kamenica (2019) , we refer to this as signal.

3.1. Fullinformation

Suppose both agents know the value of s. Agent i chooses effort to maximize his expected payoff

v

i=

ρ

i−e_i. Lemma 1documents the principal’s expectedpayoff in theNash equilibrium ofthe game.We use thenotationEp[f(^s)^] todenotetheexpectedvalueof f(^s)^{giventhat p}=Pr[s=x];therefore,Ep[f(^s)^]:=p f(^x)+(¹−p)^f

₁

x

.

Lemma1. TheNashequilibriumofthefullinformationcontestissymmetric,andtheprincipal’sexanteexpectedpayoff is V_P^F

( α

^,q

)

^=Eq

2

α

^s

(

¹⁺

α

^s

)

²

. (1)

It isreadilyestablishedthat thelow-skillednewcomer(x)hasmore(less) effortthan thebetterskilledrival (¹_x) when

α

>(<)^1;sincetheequilibriuminthiscaseissymmetric,theincumbentfightsharderagainstthelow-skilledopponentin theformercase.Inthecaseofnodiscrimination(

α

=1),theincumbentandbothtypesofnewcomerexertthesameeffort inthecontest.

3.2. Asymmetricinformation

Suppose agentsare asymmetricallyinformed. AgentN knowsthetrue value ofs.Agent Iknows thatthe opponent is fullyinformed aboutthestate, butdoesnot knowits value himself.SupposeI’s beliefisgivenby some p∈[0,1],where p=Pr[s=x].Agentschooseefforttomaximizetheirexpectedpayoffs:

v

N=

ρ

N−e_N,and

v

I=Ep(

ρ

I)−e_I.Asinthemodel of EpsteinandMealem(2013),two possibilitieshaveto be considered,one inwhich both skilltypesare active,andone in whichthelow abilitynewcomerdoesnot exerteffortinthecontest.Intuitively,when thelow-skilled newcomerfaces negativediscriminationintheformofalow

α

,theeffectofhiseffortontheprobability ofsuccessislimited,andhewill prefertoexertnoeffortandsavecost.

Lemma2. Letq˜=1−₁^α_−x.IntheBayes–Nashequilibriumoftheasymmetricinformationcontest,theprincipal’sexpectedpayoff is

V_P^A

( α

,p

)

⁼

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

2

α

Ep

√1s

α+Ep[¹s]

²

if

α

^>1−x∪

{

^1−x>

α

^∩p∈

(

^q˜,1]

}

2

α

₁

x

_1−p

α

(

^1x

)

^+(1−p)

2

if1−x>

α

^{∩p∈[0,q˜}

)

.

(2)

Thefirstlinein(2)representsthecaseinwhichbothnewcomertypesexerteffortinthecontest;thesecondlineoccurs when only themostskilled newcomer participates,making expectedeffortexerted a function ofonly his skilllevel(¹_x). Notethatwhenthediscriminationparameter

α

≥1,thenbothnewcomertypesexerteffort,andthisisalsothecasefor

α

belowbutsufficientlycloseto1.Whenfinally decidingontheoptimalvalueofthediscriminationparameter,theprincipal must bear inmind the differentpatternsof behavior that canensue asdocumented by Lemma2. However, itturns out thattheprincipalhasbetteroptionsinher policysettingthanexcludingonetypeofnewcomerfromexertingeffort.Write theamountofeffortexertedbytheincumbentunderfullinformationwhenfacingnewcomersase^F_I[s],andtheamountof effortunderasymmetricinformationase^A_I.Whenbothtypesofnewcomerareactive,Lemmas1and2implythefollowing:

Corollary1. When

α

>1,thene^F_I[x]>e^A_I >e^F_I

₁

x

.For1>

α

>1−x,thene^F_I

₁

x

>e^A_I >e^F_I[x].

When theprincipalfavorsthenewcomer, theincumbentexertsmoreeffortagainstaknownlow-skilled opponentand is discouraged whenfacing the advantagedsuperior player. The reverseis thecasewhen thenewcomer isdiscriminated against; theincumbentfightsintenselyagainst thesuperiortype, withlower effortagainst thealready-disadvantagedlow skillrival.Ineachcase,theexpectedeffortoftheincumbentunderasymmetricinformationliesinbetweenthefullinfor- mationeffortlevels.

4. Informationdisclosurewithdiscrimination 4.1. Theposteriorbelief

Atstage3,naturedrawsarealizationofthestates∈Sandsubsequently,asignal m∈Mfrom

π

(·

|

^s),whichleadsIto updatehisbelieffromthepriorqtoaposteriorqm∈[0,1]:

qm=Pr[s=x

|

^m]=Pr[m

|

^s=x]Pr[s=x]

s

π (

^m

|

^s

)

^Pr

(

^s

)

⁼

π (

^m

|

^x

)

^q

π (

^m

|

^x

)

^q+

π

m

|

¹x

(

^1−q

)

^{. (3)}

Observethataposteriorqmisarandomdrawfromadistributionof

π

(^m

|

^s)^{givensomestates},whichisstochastically drawnfromabinarydistributionoverthestatespaceS=

x,¹x

.Therefore,anysetofsignaldistributions

π

(·

|

^x),

π

·

|

¹x

generatesadistributionofposteriors

{

^qm

}

m∈Mwithprobabilities

s∈S

π

(^m

|

^s)^Pr(^s)^.Theprincipal’sexpectedpayoff,given thesignaldistributions

π

(·

|

^x),

π

·

|

¹x

,is

s∈S

m∈M

V_P^A

( α

,qm

) π (

^m

|

^s

)

^Pr

(

^s

)

^{. (4)}

4.2. Optimalinformationdisclosure

Atstage2,theprincipal’sinformationdisclosurepolicythereforesolvesthefollowingproblem:

π(·|^x)∈(^Mmax),π

(

^·|1x

)

^∈(M)

s∈S

m∈M

V_P^A

( α

^,qm

) π (

^m

|

^s

)

^Pr

(

^s

)

^subjectto

(

³

)

. (5)

Following Kamenica and Gentzkow(2011), we reformulate (5) as a constrained optimization problem of choice over posteriorsandderivethesignaldistributionsfromtheoptimalposteriors.Whilethistechniqueisquitegeneral,weillustrate itwithanabridgedproof,byconstructingthesignaldistributionsspecifictoourcontext,anditisincludedintheAppendix.

Lemma3. Theindirectvaluefunctionof(5)isthesameastheindirectvaluefunctionofthefollowingoptimizationproblem:

{^qm∈[0,1]max,β^m∈[0,1]}m∈M

m∈M

β

^mVP^A

( α

^,qm

)

⁽⁶⁾

subjectto

m∈M

β

^m=1and

m∈M

β

^mqm=q.

KamenicaandGentzkow(2011)establishthattheindirectvaluefunctionof(6)iswelldefinedandgivenbytheconcave closureoftheprincipal’s expectedpayoff underasymmetricinformation.The concaveclosuredenotedbyCa

v

₍

α

,q)^isthe smallestconcavefunctionthat iseverywhereweaklygreaterthanV_P^A(

α

,q)^{.21 WhetherornotV}P^A(

α

,q)^{isstrictly lessthan} Ca

v

₍

α

,q)^forq∈(⁰,1),hasimplicationforhowinformationisdisclosed.IfV_P^A(

α

,q)=Ca

v

₍

α

,q),thenthemaximumpayoff theprincipalcanachievebycommittingtosomestate-conditionalsignaldistributionsisexactlythesameaswhatshegets withnoinformationdisclosure.IfinsteadV_P^A(

α

,q)<Ca

v

₍

α

,q),thentheprincipalcanmanipulateI’sbeliefbychoosingthe signaldistributionssuitablyandimproveherexpectedpayoff fromV_P^A(

α

,q)^toCa

v

₍

α

,q)^.Proposition1documentstheabove findings.Theproofisomitted;seeKamenicaandGentzkow(2011,Proposition1,Corollaries1and2)forageneralanalysis.

Proposition1 (KamenicaandGentzkow 2011). Theindirect valueof (6),as a functionofthepriorq,is givenbyCa

v

₍

α

,q), theconcaveclosureofV_P^A(

α

,q)^{.TheprincipalbenefitsfromadjustingI’sbeliefbydisclosing}informationifandonlyifV_P^A(

α

,q)<

Ca

v

₍

α

,q)^.

Wedeterminetheprincipal’spreferredinformationdisclosurepolicyfromtheshapeofV_P^A(

α

,q)^{withrespecttoq},which issummedupinLemma4.

Lemma4. Define

α

^:=3+2xand

α

¯^:=3+(²/x)^{.Thefollowing}characterizestheshapeofV_P^A(

α

,q)^:

(a) If0<

α

<1,thenV_P^A(

α

,q)^{is decreasinginq.For1}>

α

>min

{

^2x,1−x

}

,V_P^A(

α

,q)^{is convexinq;for1}>min

{

^2x,1−x

}

>

α

>0,V_P^A(

α

,q) ^{is concave in q forq}∈

0,min

˜ q,q

, and piecewise convex forq∈

min

˜ q,q

,1

, where q˜=1−₁^α_−x, q =1−₂^α_x.

(b) If

α

=1,thenV_P^A(

α

,q)^isindependentofq.

(c)If1<

α

≤

α

,thenV_P^A(

α

,q)^{isincreasingandconcaveinq.} (d) If

α

<

α

<

α

¯,thenV_P^A(

α

,q)^{isincreasingandconvexinqforq}∈

0,qˆ

,andincreasingandconcaveinqforq∈

ˆ q,1

where qˆ:=(

α

−

α

)/(

α

^¯−

α

)^.

(e) If

α

¯≤

α

,thenV_P^A(

α

,q)^{isincreasingandconvexinq.}

Fig.1plotsV_P^A(

α

,q) ^{againstq forvarious valuesof}

α

^.For

α

>1−x,V_P^A(

α

,q)^{isgivenby thefirstlineof(2)andboth} typesofnewcomerareactiveattheconteststage.For

α

∈(⁰,1−x),therearefourpossibleshapesofV_P^A(

α

,q)^inpart(a)of Lemma4;theoneillustrated inFig.1holdsforx∈[¹₃,1),andnewcomertype xisinactiveforq<q˜andactiveotherwise.

Fig.A.1intheAppendixillustratesthepossibleshapesofV_P^A(

α

,q)^forx∈

0,¹3

.Inallofthesecases,V_P^A(

α

,q)^isdecreasing inq,andthereisacutoff valueofthepriorq˜∈(⁰,1)^{suchthatnewcomertypexisinactiveforpriorsbelowthislevel,and} active above.²² Theexpectedpayoff of theprincipal isfallinginq for1>

α

, andincreasing for

α

>1.Recall Corollary 1; in theformer case, theincumbentfights mostintensely against thehighlyskilled newcomer underfull information,and increasingtheprioronthelowskilledtypehencegivesmoreandmoreweighttothelowereffortthatisexpendedagainst thatrival.When

α

>1,effortsarehigheragainsttheleastskillednewcomer,andraisingthepriorbeliefonthistypehence increasestotalexpectedeffort.

21The concave closure is formally defined as follows. Fix α^{. Let co} V _P^A(α^{, q} )

be the convex hull of the graph of V _P^A(α^{, q} ) as a function of q . Then, the concave closure is given by Ca v₍α, q )= sup

p |(p, q )∈ co V _P^A(α, q )

.

22We show later that the principal will not choose αrelating to cases where one type of newcomer exerts zero effort.

Fig. 1. Plot of V _P^A(α, q ) against q for various values of α.

Depending on the curvature ofV_P^A(

α

,q), it follows fromLemma 4 that the principal may choose one of the follow- ing three information disclosurepolicies in equilibrium- (i) Full informationdisclosure, (ii) Noinformationdisclosure, and (iii)Partialinformationdisclosure.Belowwedescribethesignaldistributionsassociatedwithvariousinformationdisclosure regimes.

First,considerLemma4for

α

>1−x;cases(^a)^and(^e)^{aredrawninthesecondandfifthpanelsofFig.1.SinceV}P^A(

α

,q) is convex for all values of q∈(⁰,1),Ca

v

₍

α

,q) ^{is a straight linejoining V}P^A(

α

,0) ^{and V}P^A(

α

,1) ^andVP^A(

α

,q)<Ca

v

₍

α

,q) except forq=

{

⁰,1

}

^{. The principal benefits fromfull} information disclosurefor anyq∈(⁰,1), generatesa unique signal in eachstate so that Iperfectlyinfers thestate fromobserving thesignal. Fixa pairofsignals m₁,m₂∈M,m₁=m₂; the followingsignaldistributionsgenerateposteriorsqm₁=1andqm₂=0:

π (

^m

|

^x

)

⁼

1 ifm=m₁

0 ifm=m1 and

π

m

|

¹_x

=

1 ifm=m₂

0 ifm=m2. (7)

Lemma4,case(^c),isdrawninthethirdpanelofFig.1.SinceV_P^A(

α

,q)^{isconcaveforallvaluesofq}∈(⁰,1),V_P^A(

α

,q)= Ca

v

₍

α

,q)^forallq.Noinformationdisclosureisoptimalandtheprincipalimplementsthisbygeneratingoneandthesame signal ineverystate sothatIfindsthesignal uninformative.Fix somem₁∈M;thefollowingsignaldistributions generate posteriorq_m1=q:

π (

^m

|

^x

)

⁼

1 ifm=m1

0 ifm=m1 and

π

m

|

¹_x

=

1 ifm=m1

0 ifm=m1. (8)

Lemma 4,case (^d), is depictedin the fourthpanel of Fig.1, whereV_P^A(

α

,q) ^{is partlyconvex andpartlyconcave for} q∈(⁰,1)^{. There exists a posterior}

μ

R∈

ˆ q,1

such that V_P^A(

α

,q)<Ca

v

₍

α

,q) ^{for q}∈(⁰,

μ

R) ^andVP^A(

α

,q)=Ca

v

₍

α

,q) ^for q∈(

μ

R,1)^{.23Formally,forany}

α

>

α

,

μ

R(

α

)^isdefinedas

μ

^R

( α )

^:=

p∈

(

^0,1

)

^{:VPA(α,p)−}p^VPA(α,0)=^dVPA_dp^(α,p) ifV_P^A

( α

^,1

)

^−VP^A

( α

^,0

)

^{> dVPA}dp^(α,1)

1 ifV_P^A

( α

^,1

)

^−VP^A

( α

^,0

)

^≤dVPAdp^(α,1)

. (9)

Fig.1,panelfour,illustratesthattheprincipalimplementspartialinformationdisclosureforq∈(⁰,

μ

R(

α

))^{andnoinforma-} tiondisclosureforq∈(

μ

R(

α

),1)^.Forq∈(⁰,

μ

R)^{theprincipalgeneratesacommonsignalinbothstates,while}randomizing withanother signalin oneofthe twostates.Fix apair ofsignalsm₁,m₂∈M,m₁=m₂; thefollowing signaldistributions generateposteriorsqm₁=

μ

R andqm₂=0:

π (

^m

|

^x

)

⁼

1 ifm=m1

0 ifm=m₁ and

π

m

|

¹_x

=

⎧ ⎨

⎩

(¹−μ^R(α))^q

μR(α)(¹−q) ifm=m1

1−⁽_μ¹_R⁻₍^μ_α)^R₍⁽₁^α)₋⁾_q^q₎ ifm=m₂ 0 ifm∈/

{

^m1,m2

}

. (10)

23It is possible that μR= 1 so that V _P^A(α, q ) is always strictly less than Ca v(α, q ), except for q = {0 , 1 }, so that full information disclosure is optimal.