Creating balance in dynamic competitions

ContentslistsavailableatScienceDirect

International Journal of Industrial Organization

journalhomepage:www.elsevier.com/locate/ijio

Creating balance in dynamic competitions ^R

Derek J. Clark

^a,∗

, Tore Nilssen

^b

aSchool of Business and Economics, UiT - The Arctic University of Norway, Tromsø NO-9037, Norway

bDepartment of Economics, University of Oslo, P.O. Box 1095, Blindern, Oslo NO-0317, Norway

a r t i c l e i n f o

Article history:

Received 25 April 2019 Revised 16 October 2019 Accepted 27 December 2019 Available online 2 January 2020 JEL classification:

D74 D72 Keywords:

Balanced competition All-pay auction Momentum Multiple rounds

a b s t r a c t

Weconsiderincentivesfororganizingcompetitionsinmultiplerounds,focusingonsitua- tionswherethereisheterogeneityamongthecontestantsexante,whichdiscourageseffort inasinglecontest.Heterogeneityevolvesacrossroundsdependingupontheoutcomesof previousrounds.Wepresentconditionsunderwhichbalanceinsuchacompetitioncanbe created,bydeterminingthenumberofroundsanddividingtheprizefundcarefullyacross them,sothatfullrent dissipationentails.Inthemodel,eachroundisanall-payauction wherecontestantsdifferintheirabilitiestogainamomentumfromwinning.Wealsodis- cussthecasewhennegativeprizesarefeasible,demonstratingthatthisstrengthensthe fulldissipationresult;andweconsideracasewherethesizeofthewinner’smomentum isrelatedtothesizeoftheprizeattained,showingthatthestrongerthislinkage,theless oftheprizeisawardedearlyon.

© 2020TheAuthors.PublishedbyElsevierB.V.

ThisisanopenaccessarticleundertheCCBYlicense.

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

1. Introduction

Manycompetitionsare organizedinmultiplerounds.A challengewiththispracticeisthe difficultyofkeepingup the efforts ofthe playersasthecompetition advances insituations wherean early lossdiscouragesa player fromputtingin effortinlaterrounds (Konrad,2012).When thisso-calleddiscouragementeffectisstrong,theremaybecauseto organize the competitionas one grand single-stage contest.When multiple-roundcompetitions are so prevalent, thismay be for reasonssuchasconvexityofper-roundeffortcosts;budgetconstraintsonthepartofcontestants,organizer,orboth;orthe needfortheorganizertokeepthesuspense,whichislikelyacrucialargumentfororganizersofsportcompetitionsliving off oftheaudiencethatthecompetitionsattract.¹

Inthispaper,we point tostill anotherreasonfororganizing a multi-roundcompetition: Whencontestants cometo a competitionwithex-antedifferencesinabilities,thesedifferencesinthemselvesmaybediscouraging.²Wepointtocircum- stances where,by organizinga competitionin multiplerounds, theorganizer can notonly attenuate thediscouragement

R We would like to thank Yongmin Chen (editor) and two referees, as well as participants at the Annual Meeting of the Norwegian Association of Economists, and at seminars at the Norwegian School of Economics and Oslo Metropolitan University, for helpful comments.

∗ Corresponding author.

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

1On convexity of costs in contests see Moldovanu and Sela (2006) ; for budget constraints in sequential contests refer to Megidish and Sela (2014) ; for the design of sports contests, see Szymanski (2003) .

2See the survey by Chowdhury et al. (2019) on the effect of contestant heterogeneity on competition effort.

https://doi.org/10.1016/j.ijindorg.2019.102578

0167-7187/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license.

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

effectstemmingfromsuchdifferencesbutevenliftoveralleffortssomuchthatfullrentdissipationentails.Thismayhap- penwhentheex-antelaggardwouldgetahigherboostinhisskillsfromanearlywinthantheex-anteleaderwould.This secondasymmetrybetweenthecontestantscreatesascopefortheorganizertomakeuseoftwoormoreroundsandsplit theprizefundacrosstheseroundsinsuchawaythattheinitialroundbecomesabalancedcontest.

In order to discussthis, we presenta model of severalrounds of all-pay auctions between two contestants who are different along two dimensions:the productivityof their efforts at the outset, and the ability to obtain a boost in this productivityatlaterstagesfromanearlywin.Theprincipal,whoorganizesthecompetition,hasavailableafixedprizefund todistributeacrossthesequenceofcontestswiththeaimofmaximizingtotaloverallexpectedefforts,subjecttoanoverall budgetconstraintand(initially)anon-negativityconstraintoneachround’sprize.

Wefindthattheprincipalisabletoobtainfullrentdissipation– withtotalexpectedeffortsamongtheplayersequalto theprizefund– whentheex-anteheterogeneityinproductivityissmallenoughandtheex-antedisadvantagedcontestant issufficientlybetteratgainingaboostfromwinning.Whenthisisthecase,theprincipalisabletodistributetheprizefund such thatthevariousskillsofthetwo contestantsareexactlybalancedinthefirstcontest inthe sequenceandtherewill be fullrentdissipation,withtotalexpectedefforts amongtheplayersequalto theprizefund.Moreover,we findthatthe extenttowhichfullrentdissipationisachievableisincreasedwhenitispossibletoorganizethecompetitionoveralonger sequenceofcontests.

Thecentralnotionforthepossibilityoffullrentdissipationisthatofabalancedcontest.Ofcourse,asymmetriccontest isbalanced.Inordertoobtainbalanceinthefirstcontestwhenthereareex-anteasymmetriesthatarenoteasilycorrected, theunderdogmusthaveagreatergainfromwinningthatcontest,sothathecancatchupwithandleapfrogthebetterrival.

Whenthisisthecase,thereisascopeforsplittingtheprizefundacrossthecontests suchthatabalanced firstcontestis created.

Forthetwo-contestcase,weofferinadditionacompletecharacterizationoftheoptimalorganizationofthecontest.For an intermediate rangeof theex-ante heterogeneity wherefull rentdissipation is not possible,we find thatthe principal shouldputalltheprizefundinthesecondcontest,whileforlargesuchheterogeneity,thebestistohavealltheprizefund in thefirst contest,essentially closingdown the second one andrun a singlecontest.We alsodiscuss extensionsofour modeltothecasewheretheboostgainedfromwinningdependsonthesizeoftheprizewon,aswellasthecasewherea prizemaybenegative.

Ouranalysisisofrelevancetomanykindsofcompetitionwheredesignersfaceheterogeneouscontestants.Consider,in particular,competitionsforresearchgrants,commonlyorganizedbyresearchcouncilsandsimilarentities.Ithaslongbeen recognized,atleastsinceMerton(1968)coinedtheconcept oftheMattheweffect,that therearewin effectsinthecom- petitionforresearchgrants.³ Asnotedby GalliniandScotchmer(2002,p.54),“futuregrants arecontingentuponprevious success. The linkagebetween previous success andfuture fundingseems even morespecific in the caseof the National ScienceCouncil”.Thosewhosucceedinobtainingresearchgrantsmayexperienceanincreaseinstatus,winningagrantto fundcurrentworkandbuildupacompetentresearch team,whichagainimprovestheir chanceofwinningfurthergrants.

Losingteamsmustusetimeandresourcesinseekingpresumablyinferiorformsoffunding;insum,thisgivesanadvantage infutureroundsofcompetitionforscarceresearchfunding.OuranalysispointstoascopeforattenuatingtheMatthewef- fect,andevenobtainfullrentdissipation,bycarryingoutaresearchprogramacrossseveralroundsofcontestsforfundsand carefullyspreadingout thetotalresearch budgetacross theserounds.Takingthisview mightleadto adistributionofre- searchfundsthatislesssusceptibletodiscouragementamonginitiallaggards,bothforresearchcouncilsandfororganizers ofrelatedcompetitions,suchasinnovationcontests.⁴

Anotherareawhereouranalysishasapotentialtocontributeisthatofinternallabourmarkets,inparticularsales-force management.⁵There,itisnotuncommonforthemoresuccessfulagentstobegivenlessadministrativeduties,betteraccess to back-officeresources, more trainingthan the lesssuccessful, and betterterritories; see,e.g.,Skiera andAlbers (1998), FarrellandHakstian(2001),andKrishnamoorthyetal.(2005).Thisis,again,amechanisminwhichwinningcreateswinners, andusingseveralroundsofcontestsamongthesalesforcemaybewhatisneededinordertocreateabalancedcontestin ordertokeeptotaleffortshigh.⁶ Ourmodelmayalsobeappliedtofranchising,whichissuggestedbyGillisetal.(2011)to resembleadynamictournamentsetting.Franchiseescompete witheachotherinordertogainmoreunitsinthefranchise, asaprizeforgoodrelativeperformanceovertime.Initially,onefranchiseemayhaveanadvantageoverrivalsduetolocation orother factors,andthefranchisorcanrewardthe highperformer withanotherfranchise unit.Thisallowsthewinnerto buildabusinessandenjoyeconomiesofscale,scopeand/ormanagementinfuturecontests.Thefranchisormustdetermine when, and how large a franchise to award. In our analysis, we show that several reward schemes can induce maximal

3Win effects are also well documented in biology, where the male of a species experiences an increase in testosterone level following a win in a contest, while the loser has a reduced level of testosterone ( Chase et al., 1994 ). Hence, the winner is in better physiological shape to compete in the next contest.

A similar winner effect has been demonstrated in male judo competitions by Cohen-Zada et al. (2017) and in male tennis competitions by Gauriot and Page (2018) , while batters in baseball are found to perceive the ball as bigger when they have had recent success in hitting ( Witt and Proffitt, 2005 ). See also Iso-Ahola and Dotson (2014) for a discussion of the occurrence of psychological momentum in sports and other contexts.

4See Adamczyk et al. (2012) for a discussion of innovation contests.

5But also, and related to the previous paragraph, our analysis may be relevant to the organization of internal innovation contests; see ( Höber, 2017 ).

6Our analysis can also find application in political competition, where the winner of a first election may gain additional media attention and funding from campaign contributors which helps to build future momentum (see Strumpf, 2002 ).

expectedeffort;some oftheseinvolveperiods ofeffortin whichnodirect prize(franchise) isawarded, butwhererivals seektobuildtheirclaimsforthenextprize.

ThepaperclosesttooursisClarkandNilssen(2019),whichanalyzesasimilarsettingtothepresentone,exceptthatthe heterogeneitiestherearewithrespecttoadditiveheadstartsratherthanmultiplicativeproductivitybiases.Inparticular,one playerhasanex-anteheadstart,whiletheplayersalsodifferintheirgainsfromanearlywin.Alsothere,fullrentdissipation canbeobtainedthroughanoptimumdistributionoftheprizebudgetiftheex-anteheterogeneityissmallenough.Although resultsare similarforthe two-contest case,the mechanismsdiffer. Withheadstarts, cases mayoccur wheretheleading playerhassuch abiglead thathecanwin withoutexertingeffort, simplytrustinghisheadstart, andthe principalmust takesuchcases,thatwouldentaillowtotalexpectedefforts,intoaccountwhendesigningthecompetition.Thisissuedoes notshowupinthepresentcaseofproductivitybiases.Oneeffectofthisdifferenceisthatwehereareabletogetoutsome resultsforlongercompetitions,that is,sequences ofthreeormorecontests,thus beingabletodiscussthemeritsofsuch longcompetitions,whiletheotherpaperislimitedtotheanalysisofthetwo-contestcase.

Aprecursorto boththesepapers isClark andNilssen(2018), whichanalyzes asequence ofall-payauctions withwin advantages,butwhereprizesareconstantacrosstimeandplayersareidentical.Inparticular,playersobtainthesamehead start and/or productivity boost fromwinning a contest.In that paper,the design issue that we focus on presently does notappear,sinceidenticalplayersimplythatthefirstcontestisalwaysbalancedandthereforethatthereisalwaysfullrent dissipationinequilibrium.Instead,thefocusisontheextenttowhichaninitiallaggardwillstayinthegameandeventually havehigherexpectedeffortsthantheleader.⁷ClarkandNilssen(2018)findthatheterogeneitybetweentherivalsoccurring throughoutthecompetitionaswinsandlossesarerecordedcanreducetheireffortssincetheweakerplayerreducesefforts duetoaperceivedincreaseintheprobabilityoflosing,andthestrongerplayerreduceseffortasaresponse.⁸Counteracting thismechanismthroughthecontestdesignisakeyissueinthecurrentpaper.

Other papers that discuss design inheterogeneous all-pay auctions have focused on how the principal can optimally set, or reset, biases in order to maximize total expected efforts; see, e.g., Epstein et al. (2011), Li and Yu (2012), and Frankeetal.(2018).⁹ Thisapproachdiffers fromours, inthat we take theview that biasesare fixed andnotpossible to adjustdirectly andrather explore how, in dynamiccompetitions, distributingthe prize fundacross time can affect total expectedefforts.A complicating– butrealistic – factorinouranalysisisthat rivalshavedifferentproductivitiesofeffort initially,andthat theseevolve atdifferentratesastheseriesofcontests progresses.Contestantsimprovetheir effortpro- ductivityovertime inrelationtothe patternoflossesandwins, andthisopensup forthepossibilitythat aproductivity advantagemaybeenhanced,neutralizedoroverturnedinthecourseoftheplay.AsRigney(2010,p.1)putsit,“i]nitialad- vantagedoesnotalwaysleadtofurtheradvantage,andinitialdisadvantagedoesnotalwaysleadtofurtherdisadvantage”.¹⁰ OurworkalsorelatestoFudenbergetal.(1983).Inoneversionoftheirpatentrace,competitorsmustprogressthrough discretestagesinordertosecurethefinalinvention.Theyassumethatalaggardmayprobabilisticallycompleteanecessary stageintheracebeforethefirmwithex-antehigherexpectedvalue,givingitanadvantagesince itcanstartworkonthe nextstageintheprocess.Insomecompetitions,successfulplayersmaygainaccesstomaterialgoodsthatmakecompeting easier.Similarly,Konrad andKovenock(2010)showthat thediscouragementeffectinsequentialcontestscanbe mitigated ifcontestants’abilitiesarenotconstant,andrathertheresultofastochasticprocess.Thisensuresthattherearesituations inwhichanunderdogmaybemoreablethanthefavoriteonagivenday,leadingtolesspronounceddiscouragement.

Formostofouranalysis, weassumethat thesizeofthemomentum gainedby acontestwinnerisindependentofthe sizeof theprize actually won.Möller (2012),Beviá andCorchón (2013),andLuo andXie (2018)presenttwo interlinked Tullockcontests in whichthe size ofthe prizeattained inthe firstaffects the probability ofwinning inthe second. We showthat ourresultofattaining contestbalanceandfullrentdissipationalsoholds fora seriesofall-payauctions when thesizeofthewinner’smomentumisdirectlyrelatedtotheearlyprizewon.

A further extension that we explore is that of a negative prize in the first contest witha fixed budget constraint, a notionthatissimilartotheideapursued byMealemandNitzan(2016)involvingtaxation ofcontestprizes.Clearly,there aremanycircumstancesinwhich negativeprizes arenotfeasible,includingtheapplications mentionedaboveofresearch competitionsandinternallabourmarkets. Still,itisofvalue tonoteourfindingthat allowingnegativeprizesmaygreatly expandtherangeofparametersforwhichfullrentdissipationoccurs.

Ourpaperalsojoinsagrowingliteraturethatdiscussessequentialcompetition,andstandsoutsinceittacklesthedesign issueofhowtodistributeaprizemassoverasequenceofcontests,wherealsothenumberofcontestsisadesignvariable.

Severalpapersassume a structure inwhich a certain numberofrounds (often termed battles)mustbe won inorder to achieveanoverallprize(KlumppandPolborn,2006; KonradandKovenock,2009;Sela,2011).Takingadifferentapproach, FengandLu(2018) allowtheprizeachievedto beanon-decreasingfunction ofthenumberofcomponentbattleswonin

7Clark et al. (2019) discuss a sequence of two Tullock contests in which the winner of the first has a lower cost of exerting effort in the second or a higher win probability compared to ex ante symmetric rivals. In that model, with no discounting of future payoffs, the optimum for the principal is to put the prize fund into the second contest.

8A similar mechanism is noted in the innovation tournament of Terwiesch and Xu (2008) .

9See also the surveys by Mealem and Nitzan (2016) and Chowdhury et al. (2019) .

10 When the direction and magnitude of the contestants’ heterogenity evolves, the principal must continually rebias contest efforts if this is the instrument being used. This may be seen as a rather erratic policy in which the favoured contestant constantly changes. Our approach in this setting is simple, involving only a division of the prize mass over contest rounds.

athree-stagecontest.Theyfindthatintermediate prizesincomponentbattlesmaybe awardedasawayofmitigatingthe discouragementeffect;seealsoKonradandKovenock(2009).

Similarlytous,FuandLu(2012)allowa principaltochoosethenumberofcontestsandhowtodividetheprizemass betweenthem; since they consider an elimination tournament, theprincipal can also decide thenumber ofparticipants remainingateachstage.ThecomponentbattleisaTullockcontest,andtheresultsechothoseofFengandLu(2018):low discriminatorypowerinthecontestsuccessfunctiontendstoleadtoasinglecontestbeingoptimal,whereashigherlevels ofdiscriminatorypowermakethemulti-contestenvironmentmoreefficientatelicitingeffort.

Thepaperisorganizedasfollows.Inthenextsection,weofferapreliminaryanalysisofthestagegame.InSection3,we presentourmodelofthetwo-contestcompetitionandprovideacompletesolutionfortheprincipal’soptimumdistribution ofthe prizefund. Section4 extendstheanalysisto morethantwo contests, limitingthediscussionto findingconditions such that full rentdissipation isfeasible.Section 5 discussestwo extensionsofour analysis;one allowing negativestage prizes,andone allowingthemomentum thatthe first-contestwinnerachievesto dependonthesize oftheprizeatthat stage.Section6offerssomeconcludingremarks,whileproofsarerelegatedtoanAppendix.

2. Preliminaries

Therearetworisk-neutralplayers,sandw,whocompeteforaprizethattheyvalueatvs>0andvw>0,respectively, bymakingirreversibleeffortsxs≥0andxw≥0.Theprobabilitythatplayerswinstheprizeis

ps

(

^xs,xw

)

=

⎧ ⎨

⎩

1if

α

^sxs>

α

^wxw;

1

2 if

α

^sxs=

α

^wxw; 0if

α

^sxs<

α

^wxw;

where

α

i>0isabiasparameterinfavourofplayeri∈{s,w},andtheprobabilitythat wwinsis pw=1−ps.Weassume that

α

^svs≥

α

^{wvw,implyingthatplayersisthestrongerone.Theexpectedpayoffsofthetwoplayersaregivenby}

π

^s

(

^xs,xw

)

^{= ps}

v

s−xs;

π

^w

(

^xs,xw

)

^{= pw}

v

w−xw.

Thisgamehasauniqueequilibrium, whichisdescribedinLemma1intheAppendix.Inthisequilibrium, theexpected effortsoftheplayersare

x^∗_s=

α

^w

v

w

2

α

^{s , andx∗w=}

α

^w

v

²_w

2

α

^s

v

s; (1)

expectedpayoffsare

π

s^∗=

v

s−

α

^w

α

s

v

w, and

π

w^∗=0; (2)

andprobabilitiesofwinningare p^∗_s=1−

α

^w

v

w

2

α

^s

v

s, and p^∗_w=

α

^w

v

w

2

α

^s

v

s. (3)

From(1),wehavethatthetotalexpectedeffortis x^∗_s+x^∗_w=

α

w

v

w

α

s

v

s

( v

s+

v

w

)

2 . (4)

Wesaythecontestisbalancedwhen

α

^s

v

s=

α

^w

v

w. (5)

Itfollowsfromtheabovethat,inabalancedcontest, x^∗_s=

v

s

2; x^∗_w=

v

w

2 ; x^∗_s+x^∗_w=

v

s+

v

w

2 ; (6)

π

s^∗=

π

w^∗=0; andp^∗_s=p^∗_w=1 2.

AsdiscussedintheIntroduction,thenotionofabalancedcontestiscrucialfortheprincipal’ssearchforfullrentdissipation.

Note that, ina biased contest,total expected effort(4) isa fraction ofthe average valuation of the prize.Balancing the contestyieldsexpectedeffortsequaltotheplayers’averagevaluation.Wewillreturntothisbelow.

Theabovesimpleall-payauctionwithbiasescomprisesthestagegameofouranalysisinthenextsections.

3. Thetwo-contestmodel

Westartouranalysisbycompletelysolvingthemodelforthecaseoftwocontests,identifyingconditionsunderwhich theexpectedtotaleffortsareequaltothevalueofthetotalprizemass.Therearetworisk-neutral players,i∈{1,2},who competeintwosuccessivecontests,t∈{1,2},bymakingirreversibleefforts,x_i,t.Thetwoplayersdifferintworespects:in thebiasestheyhavebeforethegamestarts,andinthebiastheycanobtainbeforecontesttwobywinningcontestone.A principalhasaprizemassofsizeonetodividebetweenthetwocontests,making(¹−

v

₎availableinthefirst,andvinthe second;fornowweassumenon-negativeprizes,v∈[0,1].

Onlyeffortsinthecurrentcontestaffecttheprobabilityofwinning,butdosoaccordingtoabiasedversionoftheall-pay auction.Oneofthe players– player 1,withoutloss ofgenerality– hasa positivebias incontestone,so that thecontest successfunctionofplayer1is

ρ

¹,1

(

^x1,1,x2,1

)

⁼

⎧ ⎨

⎩

1ifbx1,1>x2,1;

1

2 ifbx1,1=x2,1; 0ifbx1,1<x2,1;

(7)

whereb>1isthebiasinfavourofplayer1incontestone;thecontestsuccessfunctionofplayer2,hereandthroughout, is

ρ

²,1(^x1,1,x2,1)=1−

ρ

¹,1(^x1,1,x2,1)^{.Incontesttwo,thebiasdevelopsaccordingtowhohaswonthefirstcontest.Should} thealreadyadvantaged player 1 winthe firstcontest, then hisbias parameter isincreasedby a factorofa₁ > 1 toa₁b. Shouldtheinitiallaggard,player2,winthefirstcontest,thenhehasabiasparameterofa₂ >1incontesttwo,andplayer 1retainshisbiasofb.Hencetheprobabilitythatplayer1winsthesecondcontest,havingwonthefirst,is

ρ

1,2

(

^x1,2,x2,2;1

)

=

⎧ ⎨

⎩

1ifa1bx1,2>x2,2;

1

2 ifa1bx1,2=x2,2; 0ifa1bx1,2<x2,2;

whilsttheprobabilitythatplayer1winsthesecondcontestaftertheopponenthaswonthefirstis

ρ

¹,2

(

^x1,2,x2,2;2

)

⁼

⎧ ⎨

⎩

1ifbx1,2>a2x2,2;

1

2 ifbx1,2=a2x2,2; 0ifbx1,2<a2x2,2.

Denoteby

π

i^∗,2(ⁱ)^{thepayoff ofplayeriinthesecond contesthavingwonthefirst,and}

π

i^∗,2(^j)^thecorresponding payoff ifilostthefirstcontest.¹¹ Seenfromcontestone,theexpectedpayoff functionsofplayericanbewrittenas

π

i,1 =

ρ

i,1

1−

v

+

π

_i,^∗2

(

ⁱ

)

+

(

¹−

ρ

i,1

) π

_i,^∗2

(

^j

)

−xi,1

=

π

_i,^∗2

(

^j

)

+

ρ

i,1

1−

v

+

π

_i,^∗2

(

ⁱ

)

−

π

_i,^∗2

(

^j

)

−xi,1

=

π

_i,^∗2

(

^j

)

+

ρ

i,1Vi,1−xi,1.

Winningthefirst contestgivesthe currentprize1−

v

andtheexpectedpayoff inthe secondcontest havingwon the first;losingthefirstcontestgivesonlythecontinuation valueofproceedingtothesecond contestastheloserofthefirst.

ThevalueV_i,1isthetotalvaluethatplayerifightsforinthefirstcontest,consistingofthefirstcontestprize,andthepayoff difference inthe second contestbetweenwinning andlosingthe first one.The modelis solved by backwardsinduction, startinginthesecond contest.Proposition1characterizestheoptimalprizesplitbetweenthetwo contestsaswellasthe totalexpectedefforts.

Proposition1. Inthetwo-contestmodel,theoptimalsettingofv,andthecorrespondingrealizedtotalexpectedefforts,areas follows:

(i)If ^a_a^2(1+a1)

1(1+a₂)≥b>1,then

v

^∗=

v

,withtotalexpectedefforts1,where

v

:=a1a2

(

^b−1

)

a₂−a₁b . (8)

(ii)If ^a_a²

1 >b>^a_a^2(1+a1)

1(¹+a₂),then

v

^∗=1,withtotalexpectedefforts ¹_b+^a_a^2−a1b

1a₂b ∈

₁

b,1

. (iii)Ifb=^aa²₁,thenv^∗∈[0,1],withtotalexpectedefforts ¹_b= ^aa¹₂.

(iv)Ifb>^a_a²₁,then

v

^∗=0,withtotalexpectedefforts ¹_b.

Fig.1 gives an illustrationof the results,depending on thevalue ofb.When the initial bias in favour ofplayer 1 is sufficiently small, it is possible to use the division of the prize mass to ensure balance in the first contest so that the principalcanachieveexpectedeffortequaltothetotalprizemass.Forintermediatevaluesofb,thisisnotattainable,but

11 Closed expressions for these payoffs are provided in the proof of Proposition 1 in the Appendix.

2(1 + 1)

1(1 + 2)

2 1

1 2

∗= , ^∗= 1 ^∗= 1, ^∗=1

+ ²− 1 1 2

∗= 0, ^∗=1

Fig. 1. Illustration of Proposition 1.

b v*

2(1 + 1)

1(1 + 2)

2 1

1 1

0

∗= ^{1 2}( − 1)

2− 1

Fig. 2. Optimal second contest prize.

savingthewholeprizeforcontesttwoyieldsthemostexpectedeffort.Whentheleadofplayer1attheoutsetistoolarge, theprincipalcandonobetterthantorunasingle(biased)contest.

Key to achievingfull dissipation ofthe prize ismaking the firstcontest balanced. This requiresdividing up theprize masssothat bV_1,1=V_2,1,giving

v

asthesecondcontestprize.Theparameterrestrictioninpart(i)ofProposition1derives fromthefactthat

v

∈[0,1].¹²Fig.2indicateshowthesecond-contest prizedependsontheinitialbiasinfavourofplayer 1,increasinginthisbiasuntil

v

^∗₌1isreached,andthenfallingtozerowhenthebiasistoolarge.

Whenthefirstcontestisbalanced,eachplayerhasanequalprobabilityofbeingthevictorthere.Inthesecondcontest, each player hasvaluation v ofwinning, andfrom(1),one can seethat each playerhas thesame expectedeffortinthat contest.¹³ That second-contest effortdepends on who isthe winner ofthe first, being _a^v

1b if player 1wins, and ^b_a^v

2 if2 wins.IntheAppendix,weshowthatexpectedeffortincontestoneis1−^v2

1

a₁b+a^b₂ ,whichisdecreasinginthesecond- period balancingprize.Expected effortin contesttwo is ^v₂

1

a₁b+a^b₂ , whichexactly neutralizesthe effectthat

v

hason first-contesteffort,leavinganexpectedeffortof1.

Withthefirstcontestbalanced,theplayerscompeteawaythefullvalueofthetotalprize,eachendingupwithanoverall payoff ofzero.Toseethis,notethat,inthebalancedcontest,

ρ

_i,^∗₁=¹2 andx^∗_i,1= ^V2^i,1,sothat

π

_i,^∗₁= ^Vi,2¹−^V2^i,1 =0.Thisoccurs forsufficientlysmallvaluesoftheex-anteheterogeneityb.Whenb> ^a_a²₁⁽₍¹₁₊^+a_a¹₂⁾₎,itisnolonger possibletobalancethefirst contest,sincedoingsowouldrequire

v

>1,whichisruledoutbyassumption.Itiswellknownthatdistributingthewhole prizeincontestone willgive anexpectedeffortof ¹_b.Forintermediate valuesofb,part(ii)ofPropositionshowsthat itis possibletogetsome benefitfromthe possiblecatching-upby player2by awardingthewholeprizemassincontesttwo.

Thisworksaslongasthecatching-upparameterofplayer2issufficientlylarge(a₂ >a₁b).WeshowintheAppendixthat, whenthefirstcontestisnotbalanced,thetotalexpectedeffortinthetwocontestsis

v ₍

a2−a1b

)

a1a2b +1

b,

12 In Section 5.1 we consider the case when this restriction is not imposed and the first-contest prize may be negative.

13From (1) , we have that _x^x∗w^∗s = _v^v_w^s.

b X*

2(1 + 1)

1(1 + ₂)

2 1

1 1

0

1 2

∗=1

+ ²− 1 1 2

∗=1

Fig. 3. Maximum expected effort.

whichis linearin v.Thus, the optimum decisionforthe principal dependson thesign ofa₂−a₁b.When a₁b > a₂,the principalcandonobetterthansetting

v

=0,andgettingtheexpectedeffortfromasinglecontest, ¹_b.Whena₂>a₁b,

v

=1 givesmoreexpectedeffortthan ¹_b.

Fig.3indicatesthetotalexpectedeffortsachieved,whichareX^∗=1forsufficientlylowvaluesofb,andfallinginbafter this.

4. Morethantwocontests

WhilethepreviousSectionoffersacompletesolutionofthetwo-contestcase,thequestionremainswhetheritwouldbe intheinterestoftheprincipaltosplitthecompetitionintoevenmoreroundsthantwo.InthepresentSection,weprovide ananswertothis.Inparticular,wedelineatecaseswhereitisintheinterestoftheprincipaltohavemorethantworounds ofcontestsinordertoenlargethescopeforfullrentdissipation.

Inconstructinga seriesofmorethantwo contests,we assume thatthe momentum/wineffectis multiplicative;when contestt is aboutto be played, and player 1has won m ofthe previous t−1,the bias foreach competitor is a^m₁b and a^t₂^−1−m,respectively.¹⁴ Inourextendedmodel,weretaintheassumptionfromthetwo-contestcasethattheinitiallaggard can catch up andsurpass the leader.The key to full rentdissipation is againbalancing the first contest.As in the two- contestcase,whenthefirstcontestisbalanced,thevalue ofthegametoeachplayeriszero,sincetheycompeteawaythe fullvalueoftheprizemassinexpectation.Wepresentbelowaconditionsuchthat,inthegeneralcase,thefirstcontestis balancedandthereisfullrentdissipationacrossthesequenceofcontests.

Proposition2. Supposethatb≤^aa²₁,thatthesequenceconsistsofT≥2contests,andthattheprincipalallocateshertotalprize fundof1overtheTcontestssuchthattheprizeincontestt∈

{

¹,...,T

}

^isvt≥0,andT

t=1

v

t=1.Thefirstcontestintheseries isbalanced,andthetotalexpectedeffortequalsthetotalprizefund,when

v

1

(

^b−1

)

⁼ T t=2

φ

t

v

t, where (9)

φ

t = a^t₁−1 a^t₁⁻¹

(

^a1−1

)

^−b

a^t₂−1

a^t₂⁻¹

(

^a2−1

)

^{,t=2,....,T. (10)}

Sincetheleft-hand-side of(9)is atleastzero,balancerequiresthat theright-hand-sidebenon-negative.WhenT=2, wesee,from(10),that

φ

2 > 0for ^a_a^2(1+a1)

1(¹+a₂)≥b,consistentwithpart(i) ofProposition 1.If, ontheotherhand,T=2,and

2(1 + 1)

1(1 + 2)

2 1

1

2> 0 2< 0, 3> 0 ²< 0, 3< 0, 4> 0

22( 1+ 12+ 1)

12( 2+ 22+ 1) ²

3( 1+ 12+ 13+ 1)

13( 2+ 22+ 23+ 1)

Fig. 4. Balance with T contests.

a₂(¹+a₁)

a₁(¹+a₂)<b,then

φ

2 <0,andaseriesoftwocontests cannotbebalanced. Fig.4depictshowthesignofthe

φ

t depends upontheinitialbiasparameterb.

Thisfigureshowsthat,for

a2

(

^a1+1

)

a1

(

^a2+1

)

^<b≤

a²₂

a₁+a²₁+1

a²₁

a2+a²₂+1

,

theprincipalcanobtainabalanced firstcontestbyusingthreecontests,T=3.Increasingboutsideofthisintervalmeans that

φ

2,

φ

3 <0, andafourthcontest mustbe added toachieve balance, andsoon forfurtherincreases inb. Notethat thecriticalvalueofbmaking

φ

t positivecanbeexpressedastheratiobetweentwogeometricalseries.Asweincreasethe numberofcontests,therestrictiongetsweakerandweaker,modulotheotherrestrictionofb<^a_a²₁.From(10),wehavethat

φ

^t >0for b<

a1−_at−1¹

1

(

^a2−1

)

a2−_at−1¹

2

(

^a1−1

)

^:=

θ

^t.

Hence, theseriesof criticalvaluesfortheinitial biasparameter

θ

t that makes

φ

t positiveincreasesaseach newcontest isadded.¹⁵However,itmaynotbepossibletoachieve fullrentdissipationforall b∈

1,^aa²₁ ,asdetailedinthefollowing corollary.

Corollary1. Consideraprizestructurevt≥0,t=1,2,....,T.Fullrentdissipationcanbeachievedforb∈

1,^a_a²₁ iff _a^a1

1−1≥a₂. Ifa2>a₁^a−¹1,thenfullrentdissipationcanbeachievedforb∈

1,^aa¹₂⁽(^aa21⁻¹−1⁾)

,butnotforb∈

a₁(a₂−1) a₂(^a1−1),^aa²₁ .

Thereare threetypesofasymmetrycapturedbythemodel: theinitialbiasinfavourofplayer1,andtheratesofmo- mentumfollowingacontestwin.Achievingfullrentdissipationinvolvesaddingcontestsuntilthefirstoneisbalanced.In ordertodothis,theasymmetry cannotbetoolarge,andthisistheessenceofCorollary 1.Iftheinitialasymmetry istoo large (b>^aa²₁), then the firstcontest cannot be balanced by adding moresince player 2cannot catchup the lead; ifthe momentumoftheinitiallydisadvantaged player2istoolarge(a₂>_a1^a₋₁¹ ),then player1cannotcatchup iftherival pulls ahead.

Arewardschemeforachievingfullrentdissipationconsistsofanycombinationofnon-negativeprizesthatsumto1and solve(9);whentherearemorethantwoprizesonoffer,therewillhencenotbeauniquesolution.Asanexample,consider a₁=1.2,a₂=2, b=1.3. Thisimplies

φ

2=−0.11667, sothat two contests cannot be usedto createbalance,anda third onemustbeadded.Wecancalculate

φ

3=0.25278,sothatbalanceinthefirstcontestrequires

0.3

v

1=−0.11667

v

2+0.25278

v

3;and (11)

v

1+

v

2+

v

3=1, (12)

where(11)followsfrominsertionsinto(9).

Fig. 5depictscombinations offirst-contest andthird-contest prizesin (v₃,v₁) spacefor thisexample.The v₁(v₃) line in the figure is the locus of combinations of the two prizes that satisfy the two Eqs. (11) and (12), in addition to the non-negativity constraintsv_t ≥0,t ∈{1,2,3}.At pointy inthefigure,thefirst-contest prizeisatzeroandwehavethe prizevector (

v

1=0,

v

2=0.68,

v

3=0.32)^{.At pointz,itisthe}second-contest prizethatisatzero,andtheprizevectoris (

v

1=0.46,

v

2=0,

v

3=0.54)^{.Inbetweenthetwoextremes,the}third-contestprizemovesintherange[0.32,0.54].

14This is not the only way of modelling this parameter over time but is in line with the approach taken by de Roos and Sarafidis (2018) for modelling momentum in continuous-time dynamic contests.

15Also, ^∂θ_∂a^t₁< 0 , and ^∂θ_∂a^t₂> 0 .

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0

0.1 0.2 0.3 0.4 0.5 0.6 10.7

3 1

(

₃

) .

.

z

y

Fig. 5. Reward Schemes for T = 3 , b = 1 . 3 , a 1= 1 . 2 , a 2= 2 .

4.1. First-and-lastprizeschemes

Withonly two equations to tie down the profile of T rewards, the exact prize structure cannot be determined. It is possibletodeterminethesystembyspecifyingthattheprincipalwilljustusetwoprizes,oneinthefirstcontestandone inthelast;thisisrepresentedinFig.5bythepointz.WehavethefollowingCorollary toProposition2forthis“first-and- last” prizescheme:

Corollary2. WithaseriesofTcontests,theprincipalobtainsfullrent dissipationbydistributingtheprizefundstrictly between thetwocontests1andT,keepingv₂through

v

T−1 atzero,aslongas

b≤min

⎧ ⎨

⎩

a2

a1,

a1−_aT−1¹

1

(

^a2−1

)

a₂−_aT−1¹

2

(

^a1−1

)

⎫ ⎬

⎭

^{. (13)}

with

v

T = b−1

b−1+

φ

T; (14)

v

1=

φ

T

b−1+

φ

T

;

v

2,...,

v

T−1=0; where

φ

Tisdefinedin(10).

Note,from(10),that ^∂φ_∂t^t >0.Byinsertingfrom(10)into(14),wecanderivethefollowingcomparative-staticsproperties ofthisfinal-contestprize:

∂ v

T

∂

^{b >0;}

∂v

T

∂

^a1 >0;

∂v

T

∂

^a2 <0;

∂ v

T

∂

^{T <0.}

Iftheinitialbiasinfavorofplayer1increases,and/orifthegaintothisplayerfromwinningincreases,thenthefinalcontest prizeshouldbeincreased,aslongas(13)isstill satisfied.Savingtheprizemassuntillaterencouragesthelaggardtostay inthe game,fighting forthepossibility ofwinninga large rewardat theend oftheseries. Themore thedisadvantaged playergainsfromwinningacontest,themoreoftheprizemassitisoptimaltohaveearly.Thisgivestheinitiallaggarda largeincentivetowinearly,catchingupandsurpassingtheinitialleader.Asthenumberofcontestsintheseriesbecomes larger,theprincipalshouldgivealargershareofthespoilsearlytobalancethecontest.Thisiseasily seenfrom(9),since

φ

^{Tisincreasing inTandall prizesfromv}2 through

v

T−1 are zerointhisparticularrewardscheme.Theprincipalinduces

mosteffortinthefirstcontest,sincethefollowingcontests aresimplyforposition,withamodestprizeintheend. Note, however,thatthefinalprizeisalwayspositive.

There is an interesting interplay betweenthe comparative-statics effects noted above. Ceteris paribus, increasing the initialbiasbmakesitoptimaltoshiftprizemasstolateintheseries.However,thisalsoincreasesthenumberofcontests thatmustbeusedinordertoachieve fullrentdissipation,whichlowerstheoptimalfinalprize.Thiscanbeillustratedby recalling the numericalexamplein whicha₁=1.2,anda₂=2,andwhere we varythe initialbias b andthe numberof contestsT;atb=1.3andT=3,thisexampleisidenticaltotheoneusedinconjunctionwithFig.5above.Inthefollowing table,foreachrow,a“+” indicatesthecontestinwhich

φ

^{tturnspositive.}

b φ2 φ3 φ4 φ5 vT

1.1 + 0.35

1.3 − + 0.54

1.5 − − + 0.63

1.6 − − + 0.85

1.66 − − − + 0.64

Twocontests can be used fora low value of the initialbias (b=1.1), withmost ofthe prizemass givenin the first contest.Withabiasofb=1.3,threecontestsareutilizedandmorethanhalfoftheprizeisdistributedinthefinalcontest.

Fourcontestsareusedforbiasesof1.5and1.6,withthefinal prizeincreasing inb.Whentheinitialbiasisat1.66, which is close to its top level at ^a_a²

1 =1².2= ⁵3, five contests are necessary, and here we see that the amount of prize that is awarded latefallsfrom0.85withb=1.6 to0.64withthehigherbias parameter.As discussedabove,the increaseinthe biasparametertendstoincreasethelateprize,whereasthefactthatitisawardedonecontestlaterreducesthelateprize.

Asnoted,thesimplerewardschemeinCorollary2isnottheuniqueonethatbalancesthefirstcontest.Butthereisno otherrewardschemethatachievesthisgoalbyusingfewercontestsintheseries.Toseethis,consider(9).Iftheleft-hand sideispositive(i.e. v₁ >0),thentheright-handsidemustalsobe.HencecontestsmustbeaddeduntilwefindTsuchthat

φ

T>0,justasinthesimpleschemeabove.If

v

1=0,thentheright-hand-sidemustsumtozero,sothattheearlynegative valuesof

φ

tmustbecanceledoutbythefirstpositiveone,justasabove.Hence,wecanstatethefollowing:

Corollary3. Arewardschemeinwhichv₁>0,

v

T>0,

v

1+

v

T=1,and

v

2,....,

v

T−1=0,andwhere(9)issatisfied,achieves balanceincontestonewiththefewestnumberofcontestspossible,whichisthelowestTsuchthat

φ

T>0.

ItisnotdifficulttoconstructexamplesinwhicharewardschemeusesmoreconteststhanthatinCorollary3.Suppose forinstancethattheprincipalwantstodividetheprizemassasequallyaspossibleacrosscontests,whilststillmaintaining balance, sothat all contestprizes afterthefirst are ofequalvalue v,whilst the prizeincontest one isv₁ > 0,and

v

1+ (^T−1)

v

₌1.Theconditionforbalanceinthiscaseis

v

1

(

^b−1

)

⁼

v

T

t=2

φ

^t,

requiring that T

t=2

φ

t>0. As an illustration, return to the numerical example above, and put b=1.5. The condition T

t=2

φ

^t>0isnotfulfilledforfourcontests,sotheprincipalmustusefiveinthisinstance,onemorethandiscussedabove.

Contestoneis balancedinthiscaseforT=5,

v

1=0.188,and

v

2=

v

3=

v

4=

v

5=0.203.Amoreeven distributionofthe prizemassthusrequiresmorecontestsinordertopreservebalance.

5. Extensions

Heterogeneitytendstoreduceexpectedeffortinasinglecontest,andtheprevioussectionshavehighlightedthecircum- stancesunderwhichaprincipalcanuseasequenceofcontestsinordertoelicitfullprizedissipation.Addingcontests has beenshowntofacilitatethisforlargerandlargervaluesoftheinitialdegreeofheterogeneity.InthisSection,wereturnto asequenceoftwocontests,relaxingsomecoreassumptions.First,weconsiderhowthescopeforbalanceischangedwhen theprincipalcanofferanegativefirst-contestprize;¹⁶ thenwerelaxtheassumptionthatthemomentumfromwinningis constant,allowingittodependonthesizeoftheprizewoninthefirstcontest.

5.1. Negativeprizes

Intheanalysisabove,werestrictprizestobenon-negative.Thisisanaturalrestrictiontoimposeinmanycircumstances, whichis whywehavemaintained itinour mainanalysis.Butthe samesetofcontestants competingseveraltimesdoes openforupthepossibilityofmakingsome prizesnegative.Withtwocontests,thesecondcontestprizewouldhavetobe positiveinordertogive theplayersan incentivetocompete atthatstage;thefirstcontestprizecanbenegative, butthe

16We would like to thank a referee for suggesting this.