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Increasingtheshotataqualitydraft-decision–

ABayesianapproachtoimprovepredictingthree-point-accuracytranslationintheNBADraft

TobiasBerger|Friedrich-Schiller-UniversitätJena|tobias.berger@gmx.netFrankDaumann|Friedrich-Schiller-UniversitätJena

1. IntroductionThe NBA Draft is a mechanism to regulate competitive balance of a sports league. Itsambition-togivetheworstfranchisesachancetoacquireyoungprospects,improvelong-termandclosetheon-fieldtalentgaptobetterteams-isclearandnoble.IntheorythepolicystrengthenstheNBA’sproductinmanyways.It issupposedtodistributeleague-enteringplayer capital equally, and by this dynamic improve the quality of play [14, 32] whileincreasing the competitive balance among all competitors by raising the uncertainty ofoutcomeonagameandseason-level.Thishasbeendeemedanimportantgoalforsportsleagues for a long time [63]. The potential benefits of the mechanism should lead toincreasingattractivenessoftheentertainmentproducttheNBAoffers.The crux of these theoretical outcomes is the nature of the mechanism itself. For thedescribeddynamicstowork,allNBAmanagersneedtoconstantlyseizetheopportunitytheNBADraftprovidesthemwith.Unfortunately,thisisanincrediblycomplexexpectationtohavefortheactingdecision-makers.Inthedraftenvironment,choicesarenotmadewithinaknownvariance,butamongwidelyunknownoutcomes.Thepolicycanthereforebetreatedas a proxy for decisions under uncertainty [51]. Within this complex process decision-makersultimatelydonotonlyhavetojudgewhothebestprospectisatthemomentofthedraft,butalsowhowilldevelopintothemostvaluableplayerinthefuture,whileadditionally(amongconsideringotherexternalfactors)forecastingthedirectioninwhichthesportitselfismovinginthelong-term.Thisprovidesmuchroomforindividualfailuresduetopersonalmisjudgmentsormisconceptionsbythemanagers,andleavesthedooropenforsystematicerrorsonaleaguewidelevelcausedbycollectivedecision-makingbiases[64].Overthepastfewdecadesmanyeffortshavebeenmadetoimprovedraft-decision-making-qualityandreachthegoalsthedraftmechanismintendstofulfill.Still,themechanismdoesnotproducesuchpositiveoutcomesconstantly,eventhoughitwouldbebeneficialfortheentire league.Researchhas shown that it doesnot pay off to solely build up a franchisethrough the draft [11, 54]. Meanwhile, the NBA historically has performed as the most

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imbalancedofalltheNorthAmericansportsleagues[70].Atleasttosomeextendthisstateof the association canbebasedon the ratherpoordraft-decision-making-quality next tosalarycap-inducedfactors[74].Weidentifiedtheincreaseofmanagerialdecision-making-qualitywithinthedynamicsasacrucialcomponenttofulfilltheaimsofthefundamentalpolicyproposal.Initialjudgementsintalentevaluationandfuturepredictionsofperformanceforthepotentialdrafteesandtheenvironmenttheywillplayinneedtobeimprovedintheiraccuracyandreliabilitytoreduceuncertaintyinsidethecomplexchoicemechanismandthus,provokebetteroverallresults.Thispaperaspirestoreachthisgoalbyexaminingoneofthekeycomponentsofthesportmoreclosely–shooting thebasketball fromdistance.Wewillexplore league-wide three-pointscoringtendenciesandinvestigateinwhichdirectionthesporthasdevelopedandwillprogress.Afterwardswestriveforimprovingthepredictionofshooting-translationfromthecollegetotheprofessionallevelbyimplementingaBayesianapproachtopre-draftmetricsandincludingaplay-by-play-basedmetricintomodeling.Beingabletoprojectanimportantbasketballskillmoreaccuratelyshouldimprovetheindividualmanagersplayerevaluationsand,intheend,increasethedraft-decision-making-qualityonaleaguewidelevel.Theresultswill provide anewbaseline for the judgementof draft prospectswithinoneof themostimportantperformancefacetsofthesport.2. Backgroundandrelatedliterature2.1.TheNBADraftpolicyanditsshortcomingsThedraftasaregulatingsportsmechanismhasbeenanintegralpartofallNorthAmericanmajorleaguesandisinvestigatedinmanyenvironments.Researchersmostlyanalyzedthestructureaswellasthedecision-makingwithintheprocessorlookforinefficienciesfromaneconomicstandpoint.Papersonthedraft,canbefoundforhockey[e.g.24,73],baseball[e.g.20,69],football[e.g.34,49]andbasketballintheWNBA[e.g.31,33].ThedraftpolicyoftheNBAwasfirstlydiscussedfromastrictlylegalperspective.Overhalfacenturyago,themechanismwasdebatedfocusingonantitrustlaws,laborrightsanditsgeneralvalidity [e.g.2,18,21]. It tookawhileuntilearlystructural criticismwasraised,because it failed to deliver the intended league-wide outcomes concerning competitivebalancethroughequaltalentdistribution.AsBurkowandSlaughter(1981)pointedoutteamsuccesswasmoredependenton themanagerialdecision-makingqualities inother areassuchaspersonnelhiringandplayertrading[19].Draftingdidnotseemtoleveltheplayingfieldasmuchasitwassupposedto,sincemostofthedecidingorganizationsseemtolackpropertalentevaluationabilities[12].Strongerteamstendtoprofitmorefromthepolicythanbadones,whichiscompletelycontrarytoitsintentions[11].Consequently,researchevenadvicesagainstthedraftpolicyasleague-widetooltostrengthencompetitivebalance.Historicalresultsshow,therearenotenoughcompetentdecision-makerstoimplementthe

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mechanismwell enough to produce those favorable outcomes for the entire associationconstantly[55].2.2.TheNBADraftdynamicasaproxyfordecisionsunderuncertaintyThereasonsforthesepolicyshortcomingsoftheNBADraftaremanifoldbutcanbemostlybe attributed to the ‘human judgement-factor’ inmany parts of this particular decision-makingequation.Thedecisionproblemof theentiremechanismisclear.Every franchisestrivestooptimizetheopportunityitscurrentdraftpositionprovidesthemwithbyselectingthebesttalentavailable.Talentinthisenvironmentcanbeseenasamixtureofmostlyon-courtbutalsooff-courtbenefitstheprospectwillgenerateforhisnewemployer.Thisvalueallowsthedraftingorganizationtomaximizefinancialandsportiveutilityfollowingclassicaleconomictheory[25,41].Theprocesstowardssolvingthisproblemisimmenselycomplexduetothehurdlesthedecision-makersmustmaster.Withinthisstraight-forwardsetupadraftchoicerequiresNBAfrontofficestogivetheirjudgementintwomainareaswhichbothpresenthugelevelsofuncertainty,asprobabilitiesofdecisionoutcomesinthisframeworkcanonlyberoughlyestimatedandaremostlyunknowable[78].One the one hand, amanager needs to evaluate the talent level of all the players at themomentofthedraft,whilefactoringtheirpotentialdevelopmentinthefuture.Ontheotherhand,thedirectionoftheentiresports,asanunderlyingenvironmenttheathleteneedstoperformin,mustbecorrectlyanticipatedaswell.TheNBAhascomealongwayoverthepasttwodecadesbyinstallingmoresophisticatedprocessesaroundtheseinterwovendynamics.Especiallywiththerecentemergenceofsportsanalyticshugestepshavebeentakentowardsamorescientificapproachofthesporttobetterunderstandtheentiredisciplineandreduceuncertainties [1, 46]. The evaluationof player talent along the triangulationof eyes (e.g.scouting,personalworkouts),ears(e.g.personal interviews,medicalrecordsexploration,furtherbackgroundresearch)andnumbers(e.g.anthropometricmeasurements,highschoolandcollegeperformancemetrics)hasbecomemuchmoresophisticated[8],whileleague-wideplayingtrendscanbediscoveredandfollowedmorecloselythaneverbefore([68].However,judgingindividualsisextremelydifficultbecausebasketballasasporthasnotbeen‘solved’ yet. It is hard to isolate the personal impact of one player in a dynamic teamdiscipline[48].Andevenifmanagersknewwhatexactdatatolookat,howtoweighalltheinformationcorrectlytoanticipatecareerarcsofplayersperfectly,therewouldprobablystillbe discussions about career values of prospects arguing about peak versus averageperformance-level,relativevalueorlongevityaspotentiallydrivingutilitydimensions[e.g.7,72].Atsomepointtheanalysisofthosefactorssimplycomesdowntotaste,preferences,andthedecision-maker’sphilosophyonhowthesportsofbasketballshouldbeplayed.Theseclearaspectsofjudgement,whicharedefinedas‘thesetofevaluativeandinferentialprocesses

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thatpeoplehaveattheirdisposalandcandrawonintheprocessofmakingdecisions’[43,p.XV]openuptheentireprocesstotherealmofclassicaldecision-making-quality-loweringfallaciesandbiases.ManyofthosehavebeeninvestigatedintheNBADraftenvironmenttoeitherreduceuncertaintyoruncoverhiddensystematicerrors.Basketballdraftresearchhascovered nationality bias [54] as well as recency and availability bias [12, 17, 38].Additionally,itwasshownthatageandfavorablepositionallengthasproxiesforpotential[4,13,29]getmisjudgedconstantly,while certainboxscore statisticsdonot translateassmoothlyfromtheamateurtotheprofessionallevelasonewouldexpect[e.g.22,31,65].2.3.TheNBAthree-pointrevolutionArguably the most influential rule change in the history of basketball was theimplementation of the three point-line. After much experimentation in other basketballenvironmentsthethree-pointlinewasintroducedtotheNBAin1979.Thisrulechangewasa clear reaction to the dominance of the bigmen in early decades of the sport. It wasintentionallydesignedtogivesmallerplayersbetteropportunitiestoscore,whilemakingthegamemoreenjoyabletowatchforthefans[30].

Figure1.LeaguewideAverageof3PApGovertimeinNBAandNCAA.

Atfirsttheleaguewasslowtoreacttothischange.Onlyafewplayerspossessedtheshootingskillstousethisnewoffensiveweaponconstantly,whilemanagersandcoacheshadnotfullygrasped theadvantages theadditionalwayof scoringdoesprovide.Butasmoreathleteswere brought up with the three-point line being present for their entire basketballupbringingshooting-competenceandtheacceptanceofthethree-pointerroseintheNBA[27]. In a trickle-down fashion this change affected college basketball as well, whichintroducedthelinein1986.

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Figure2.LeaguewideAverage3P%overtimeinNBA.

With the general skill level of the players catching up, the era of sports analytics led toshootingfromdistancebecomingevenmorefashionable.Buildingonthe‘Moneyball’-ideasearlyintroducedbytheOaklandAsinbaseball[45],whichcenteraroundtheuseofdatatofindexploitable inefficiencieswithinthesport[e.g.1,46], thethree-pointerasastrategicoptionwas finally recognizedasanexcellentdriver forefficientbasketballoffenses [57].Backedupbynumbers,managersandplayersnotonlystartedtorealizethehugebenefitthepotentialextra-pointtheshotgrants.Theyalsorecognizedthepositiveeffectthethreatofsuchattemptsaloneproduced,forcingopposingdefensestospaceout,openinguproomforotherbasketballactionsaswell.Theentiregeometryofthesportchanged[e.g.27,67].

Figure3.LeaguewideAverageOffensiveRatingovertimeinNBA.

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The strategic and stylistic innovation the extensive use of the three-pointer has causedwithin sport is enormous. The increased application of sports analytics only fuels thisdevelopmenttothisday.Currently,someofthearguablymostinfluentialplayersofthesportmakemorethree-pointersoveraseasonthanentireteamsdidoverayearjustthreedecadesago[6].Thistrendshouldlast,sinceithasledtoavastincreaseinoffensiveefficiency,whichisonekeyfactorinwinningbasketballgames[57].Withthe2010sGoldenStateWarriorsandClevelandCavaliersdisprovingtheoldNBAsentiment,thatteamsrelyingonthejumpshotwillneverwinchampionships[76]andnodiminishingrule-changesonthehorizon,thethree-point-movementisheretostay.Thus,theleague-widedemandforcompetentthree-pointshootersisincreasing.

Figure4.Leaguewide3PAtrendsbyplayerheightinNBA[23].

Thisdevelopmentbecomesespeciallyobviousbylookingatfunctionaldemandsfordifferentpositions.ThetallestofplayersoftheNBAusedtohavethesimplejobofconvertingscoringopportunitiesclosetothebasket.Sincethestartoftheanalyticsera,mostteamswantalltheirplayerstobeabletoshootthethree,makingtheabilitytohitopentriplesoneofthemostvaluableindividualbasketballtraitsnomattertheheightorposition[23].Aswecanseeeverybodyissupposedtoshootmore.Curiously,NBAfrontofficesseemtobeslowtoreacttothistrend.Theyarguablystilldraftmorenon-shootingbigmeninhigherspotsthantheyshould[58],whichpotentiallyillustratesrigidunderlyingmanagementstructuresandapotentialstatus-quo-bias.2.4.Three-pointshootingasanNBADrafttraitDuetothecurrentleagueenvironmentandgeneralplaystyle,wehaveidentifiedthree-pointshooting as a valuable skillset-element for every NBA player. Experts forecast thisassociation-wideshooting-trendtolastsinceanincreasingnumberofathletesandteamsis

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adapting[e.g.16,27,](e.g.Nasru,2017).Consequently,thistraitbecomesanincreasinglyinterestingpre-draft ability toevaluate.Especially at themoment, it isoftendeemed theswing skill for players that can make or break successful careers [35]. Identified andcorrectlyprojected, theaccurate judgementofaprospectsthree-pointshootingskillscanresultinbetterdraftdecision-making.Ontheonehand,draftingunderratedelitelong-rangeshooterscansetupateamforadecadeofsuccess[e.g.52,53].Ontheotherhand,simplyavoidingplayerswhofailtomeettheirshootingprojectionandturnouttobelessofatalentasanticipatedforthatmatterindirectlyhelpsafranchiseaswell[e.g.44,47].Eitherway,theprojectionofthisabilityiskeytobeabletoseizetheopportunitiestheNBAdraftprovidesforthefranchises.However, predicting the translation of pre-draft three-point shooting towards NBAperformanceinthiscategoryisrathercomplexanddifficultbecausemanyfactorsneedtobeconsidered. From a player’s internal perspective, managers need to judge the shootingtechnique[e.g.37,42]whileconsideringtheprospect’smentalandphysicalcapabilities[e.g.3,59].ExternalvariableslikeadaptiontothegreaterNBAshootingdistance1,differinglevelofplayaroundtheplayer,varyingrolewithintheteamsystem,changeofplayerpositionaswell as potentially shifting degrees of difficulty of the types of three-point attempts aprospect is taking, complicate a clean projection.Due to differing circumstances and theuncertaintysuchexternalfactorsproduce,evensmartforecastsforNBAperformancebasedonpre-draftindictorscanneverbefullyaccurate[56].Additionally, within these pre-draft components several biases can cloud judgement.Recency, availability, or small sample size-biases cannegatively influenceevaluators, if aprospecthasahotshootingstreakinthemarchmadnesstournament[12].Overconfidencemightalsoleadtoinaccurateprojectionsonthree-pointability,asmanagersfalselybelievethat they will teach a non-shooting player how to hit more threes, or on the contrary,irrationallydisregardasolidshooterbecauseoftheirunorthodoxthrowingmotion[64].Toavoidsomeofthesebiases,analyticswouldsuggestblockingoutasmuchsubjectivenoiseaspossible by projecting shooting and its potential development using a model based ontangible,historicdata.3. Methodology3.1.Data

Thewebsite‘Hoop-Math’wasusedasaprimarydataresource[36].TheirdatabaseallowedustocapturetheshootingstatisticsofeveryNCAAathleteandtheirrespectiveschoolsfrom2012to2020(N=47408).Weisolatedthedraftedplayersfromthissampleandwereable

1TheNBAline(7.24m)isfartherbackthantheoneincollegeandFIBAbasketball(both6.75m)[77].

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tomatchfurther,missingshootingdatafromthestats-portal‘Barttorvik’aswellasscrapingandaddingtheNBAperformancedataofalldrafteesfrom‘Basketball-Reference’[5,6].

Consequently, theconstructeddataset includespre-draft information(e.g.name,college,position, age, years of pre-draft-experience), detailed pre-draft shooting data (e.g. shotattemptsandmakesfromseveraldistances)basedonplay-by-playinformationanddetailedpost-draftperformancedata (e.g. shot attempts andmakes fromseveraldistances)of allcollege playerswhowere drafted between 2012 and 2020 (N = 386). 2012 became thenaturalstartingpointforourrecordssincecollegebasketballplay-by-playdataandresultingdeeperstatisticsforfulldraftclasseswereonlyavailabletousfromthispointon.

Theongoingnatureofthedatasetisanimportantlimitationwewanttopointout.Whiletheplayers pre-draft statistics cannot change anymore, their post-draft numbers are stilldevelopingasmostoftheplayersarecurrentlyactiveintheleague.ThedatawasscrapedinJune2020anddoesnot include themostrecentcontinuationof theNBAseasonafteranintermissionofthe2019/20season,inducedbytheCOVID-19pandemic.

Weusedthe ‘R’-softwaretoprepareandanalyzethedata.ForempiricalBayesbinominalestimations,wereliedonthe“ebbr”-packagewithinthisprogram[62].

3.2.Modelingthree-pointaccuracytranslationbyusingdifferentshootingmeasures

Predicting the translation of three-point accuracy from the college to the NBA level isdifficult.Therearemanycontextvariables(e.g.pre-draftteamtacticalsystem,teammates,competition) thatwouldneed tobe included into amoreholistic approach.We chose todisregardthesefactorscompletelytosimplifyourapproachasmuchaspossibleandreducetheamountofpotentialstatisticalnoisearoundthetwointroducedmetrics.Consequently,weapproachedshooting-abilitysimplyasameasureofhowmanyofthethree-pointersaplayerattempted(3PA)heconverted(3PM).

Introducing shooting as this basicmeasure of volume and efficiency alreadyposes somedifficulties.Researchhasshownthatthree-pointpercentage(3P%)asametricstabilizesataround750attempts.Hence,itneedsacertainkindofvolumeuntilthisstatisticbecomesareliablemeasure for aplayer’s shooting competence [15].This creates adraftprojectiondilemma,becausealmostnopotentialdrafteeplaysenoughofficialgamestobeabletooffersuchadependablepre-draftsample-size.Especiallysincedraftedplayershavebeengettingalotyoungeroverthepastfewdecades[79].

Currentpubliclyavailablemodelstrytoaccountforthiscomplicationbyfactoringinotherpotentiallyusefuldimensionsthatcontributetomeasuringshootingability.Regularthree-pointmetricsinthesemodelsareusuallyconsideredwithrawpercentagesandashootingvolumeestimatebasedonthree-pointmakesorattemptsona‘per-40-minutes-basis’tobeabletocomparetalentswithdifferentplayingtimes.Pre-draftfree-throw-percentage(FT%)isapopularadditioninsuchdesignsaswell.Ithasbeenshownthatincludingfree-throwpercentageinmodelssignificantlyimprovesprojectionsofNBAthree-pointaccuracy,even

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thoughtheshotsseemtodifferalotintermsofdistanceorgamecircumstance[e.g.28,39,40,71].Despitethedifferences,itseemslogical,thatfree-throwpercentagecontributestotheprojectionofthree-pointskills.Initscoreitisonlycapturinganotherthrowing-activityonthecourtandisthereforejustanadditional,interestingmeasureof‘shootingtouch’,whileincreasingthedatabasisthesystemcangathermeaningfulinformationfrom.

Inourlatermodelingwewilladdtwo-point-jumperaccuracy(2PJ%)asanothershootingmetrictoourpredictionattemptwiththesamelogic.Weassumetwo-pointjumpersrequireadrafteetoapplymanyofthesamebiomechanicalcapabilitiesthatareneededtohitathree-pointerorafree-throw.Eventhoughthegamecontextdiffersalotinthegivensituations,allscoringattemptsareusuallyexecutedwithasimilarshootingtechniqueorstyleandrequirealevelof‘shootingtouch’tosuccessfullyputtheballintothebasket[80].Hence,two-point-jump shooting could also be a reliable measure of overall shooting-ability and shouldthereforecontributetothepredictionofthree-pointcompetenceintheNBA.Thishypothesiswillbetested.

3.3.AddingaBayesianapproachtoidentifytruepre-draftshootingcompetencemoreaccurately

To‘applyaBayesianapproach’usuallymeanstousestatisticaltechniquesrelyingon‘Bayestheorem’,whichassignsarathersubjectivetreatmenttoprobabilitiesandhandlesunknowninformation probabilistically [10]. Taking a Bayesian perspective when dealing withstatistical problems has become popular in many fields because it provides severaladvantages. For the discipline of sports science and analytics particularly - facing a datainflux rapidly growing in volume and complexity - it proves to be beneficial especiallybecauseitallowsto:“

•incorporateexpertinformationorpriorbelieves,[…]•useBayesianlearningwherethecurrentposteriordistributionbecomesthepriorforfuturedata,[…]•modelcomplexproblems,[…]•dealmoreeffectivelywithsmalldatasetusingpriorinformationtoimprovetheparameterestimates,[…]•makepredictionstakingintoconsiderationuncertainty”[66,p.2]

among other listed benefits. Various applications of Bayesian methods for basketballshooting problems and modeling can be found [9, 26, 60, 75]. In our opinion, such anapproachshouldbefavorableforpre-draftthree-pointshootingaswellbecauseitallowstoaddmorecontextandpotentiallyassignmoremeaningtothismeasurebyusinganempiricalBayesianshrinkagetowardsaBetapriorbasedonhistoricaldata.Thistechniquehasbeenapplied successfully to produce a more well-informed assessment of baseball’s battingaverage, an estimate of a player’s hitting ability [61]. Statistically speaking, the logic ofbatting average is very similar to three-point-shooting inmanyways. This approachhasthereforealreadybeenappliedtopre-draftthree-pointpercentageandprovidedinteresting

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results in thebasketball realmaswell [50].Withouranalysis,wewill buildon this firstmodelingattemptandhopeforevenmoreaccurateresultsbyaddingmoreinformeddata,factoringintwo-point-jumperaccuracy.

4. Analysis4.1.EmpiricalBayesestimationofpre-draftthree-pointpercentageThree-point-percentageisastraight-forwardstatistic.Itiscalculatedbydividingthenumberofthree-point-makesbythenumberofattempts:

3𝑃% =3𝑃𝑀3𝑃𝐴

Asthetermshows,three-point-percentagemeasuresshootingaccuracyandwillalwaysbeavaluesomewherebetween0and1.Carelesslyjustgoingby3P%amanagerlookingforagood shooting prospect would have to draft a talent that went 1/2 over a player whoconverted 40/100 of his triples as .5 is a lot greater than .4. But when evaluating andprojectingshootingabilityandnotonlyaccuracythisapproachseemssilly.Weintuitivelyknowthatthenumberofattemptsaplayerneededforhisresultsshouldbefactoredinsinceeventhoughtheypresentthesameaccuracyonthesurfaceneither1/2and50/100nor0/1and0/100shouldbetreatedequally.Hence,toproduceamoreusefuloreven‘true’estimateofshootingabilities,morecontextneedstobeaddedtothesimplepercentage.Oftenbasicfilteringissufficient.Asdescribedearlier,aftercrossingacertainthresholdinsample-sizeplainpercentagesbecomereliablestatistics.Forthree-pointshootingthispointhasproventobearound750attempts[15],meaning we can use simple 3P% as a tool to compare shooting capabilities of talentssomewhatconfidentlyifalloftheevaluatedplayershavehitthismark.However,thisexcludesalotofprospects.AswewanttobeabletoprojectallpotentialNBAplayersintheirshootingtranslation,wechoseempiricalBayesestimationasourmethodtoworkwithpre-draftthree-point-percentage-agoodexampleforabinominaldistribution.With this technique, context is added by fitting a beta distribution to all available data,gaining a prior that adds information based on the investigated environment. Later, thisprioris individuallyupdatedwiththeevidenceeachobservationgaveandconvertedtoaposteriordistribution[61].Toproduceourprior,wefirstneedtofitaregularbetadistributiontothehistogramofallthethree-point-percentagesofourdatabase,usingthismodel:

𝑋~𝐵𝑒𝑡𝑎(𝛼!, 𝛽!)

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Thisestimationgivesusα≈114andβ≈201forallplayerswhoattemptedatleastonethree-pointer,usingthemostdatapointsfromourdatasetpossible.

Figure5.DensityplotofPre-Draft-3P%withbetadistributionfitcurve.

Thedensityplotofthebetadistributionfitwithmaximumlikelihoodoverahistogramofthree-point percentage seems to match our data satisfyingly well. Hence, we can applyBayesianinference,adaptthefoundvaluesasourpriorsandupdateeachcasebasedonthisfoundationusingthefollowingformula:

𝐵𝑒𝑡𝑎(𝛼! +𝑚𝑎𝑘𝑒𝑠, 𝛽! +𝑚𝑖𝑠𝑠𝑒𝑠)

Thismeansthattheexpectedmeanvalueofthebetadistributionis

𝛼!𝛼! + 𝛽!

𝑜𝑟 ≈ 0.362

Thefoundprioristheheadstartbasicallyeveryplayergetsassignedwithbeforeevaluatingtheiractualperformance.Dependingonthethree-pointmakesandmissesatalentshowedthefoundpriorisupdated.Thegreaterthesamplesizeaprospectproduced,thegreaterthechancetoshiftthepriorinonedirectionortheother.Simplyapplyingtheexamplesfromearlier,wecanseethatthisapproachwouldestimatethatanathletehitting40/100ismorelikelytobeabettershooterthanacompetitormaking1/2despitehavingalowerthree-pointpercentage:

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1 + 𝛼!2 + 𝛼! + 𝛽!

=1 + 114

2 + 114 + 201 ≈ 0.36340 + 𝛼!

100 + 𝛼! + 𝛽!=

40 + 114100 + 114 + 201 ≈ 0.371

Applyingthismethod,weupdatedeachobservationusingourfoundprior.Orasthisfigureshows,webasicallyshrunkeachvaluetowardstheexpectedmeanvalue.Thesmallerthesample-size a player put up the more reason the model has to move the three-point-percentage towards the general expectation of average based on the investigatedenvironment.

Figure6.RawPre-Draft-3P%versusShrunken-3P%

Inourfirststep,weupdatedallthree-point-percentagesofourdatabasethiswayandgotalittle closer to a more insightful metric to use for our shooting translation projection.Extremecasesofeitherextremelyhighorlowaccuracygotcorrectediftheywerebasedonlittleevidence.4.2.Betabinomialregressionforpre-draftthree-pointpercentageShrinkingthree-pointpercentagestowardsanexpectedmeanseemstohaveimprovedthemetricalready.Allplayersgettingluckybyhittingmostattemptsofaverylownumberoftriesfrombehindthethree-pointlinearenotlistedastheplayerspotentiallypossessingthemost shooting accuracy anymore. But our method did not account for one importantdynamicwithin the sport of basketball:Weak shooters generally tend to shoot less thanbetterlong-rangethreatsduetoe.g.alackofself-confidenceorcoachingstaffsrestrictingprospectsfromtakingtriples.Ontheflipsidestrongshootersusuallygetthegreenlightintheirteam’soffense,eveniftheyhitacold-streakthatcouldalreadyinfluenceasmallcollege-

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sampleimmensely,becausetheyshowedtheirtalentsinpractices.Aswecansee,wedidnotaccountforthiscomplicationinourfirstestimation.Playersshootingverylittletendtogetoverratedbyonlyusingaone-fit-all-prior.

Figure7.ComparisonRaw3P%versusShrunken3P%byCBB3PA.

Factoringinthiscomplication,forcesustoadjustourfoundestimatesagain,sincewecanseethatourEmpiricalBayesestimatetowardsourprior(indicatedbythereddottedline)systematically overvalues shooters that took very little shots. Theblue line, showing themedianaverageforeveryattemptrange,bearsoutthatsentiment.Removingthisbiaswithinourestimates,weneedtoadjustourmodel.Wewantourpriorstobeinfluencedbythenumberofthreesaprospecthastaken/wasallowedtotake.Robinson(2017) suggests using beta-binomial regression and update the model in a generativeprocesstothefollowing,withpibeingdefinedasthetrueshootingpercentageofplayeriandletting3PAibeknownandfixedperplayer[61]:

𝑝" ~𝐵𝑒𝑡𝑎(𝛼!, 𝛽!)

3𝑃𝑀" ~𝐵𝑖𝑛𝑜𝑚(3𝑃𝐴" , 𝑝")

The“ebbr”-packageinRletsuscalculateindividualpriorsforourshootingestimation,basedonthenumberofattempts.Theirdistributionslooklikethis:

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Figure8.Overviewovernewpriorsbasedonbeta-binomialregression.

Ournewfoundpriorsstillstateafairlygreatamountofuncertainty.Attempting750threesincollegestillleavesopenabroadcorridorofa‘true’shooting-percentageof.31to.46foraprospectastherangeofprobableresults.Butwiththerangedependingon3PA,itcanbeassumedthatatalentthatonlyattempted10threesisalmostcertainlyaworseshooterthanaplayertaking750triples,assumedtheyhadequalplayingopportunities.

Figure9.Shrunken-3P%versus3P%estimatewithregression.

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Withthenewpriorsmostofthepercentageestimatesforplayerswithlittleattemptsdropquiteabitandcorrectforourformersystematicovervaluationofshooting.

Figure10.OverviewofRaw3P%,Shrunken3P%and3P%estimatewithregressionbyCBB3PA.

Investigating theeffectsofour last step,wereachedour intendedresults.Ournewbasicshooting-estimates lookmore context driven and thereforemore accurate than our rawpercentages. We eliminated unreasonably high or low percentages produced by (bad)shootingluckinsmallsample-sizecircumstances,usingadataeducatedprior.Afterwardswecorrectedourshrinkagevaluesbyintroducingindividualpriorsforeveryplayerbasedontheirnumberofthree-pointattemptsasbasketballlogicsuggeststhatthisfactorshouldplayaroleintheestimationofshootingability.Thenewfoundstatisticisstillproblematicasvariousexternalfactors(e.g.tacticalsystem,injury,suspensions,competition)caninfluencethenumberofthree-pointattemptsaplayeristakingdespitehisassumedshootingabilityasoursimpleapproachputstotheforefronthere.However,ourvaluesnowbetter lineupwith themedianaverageacertainattemptthresholddictates,whichispromising.Additionally,thefoundestimatesareonlyonepieceofthethree-point-translation-modelpresentedlater.4.3.Modelingthree-point-percentagetranslationaddingtwo-point-jumperaccuracyFinally,wecanputtogetherallthepreparedcomponentsandcombinethemintoamodelingapproach. Based on our theoretical work we assume the pre-draft factors free-throwpercentage, two-point-jumper percentage and our estimation of three-point-percentage(afteranempiricalBayesestimationwhileaccordingforthree-point-attemptsinourpriors)tobeafruitfulfoundationfortheprojectedtranslationofNBAthree-point-shootingbased

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on pre-draft-data. Hence, the formula of ourmultiple regressionmodel (M1)with post-shooting-accuracyasourdependentvariableshallbe:

𝑀1:𝑃𝑜𝑠𝑡𝐷𝑟𝑎𝑓𝑡3𝑃% =𝛽! +𝛽"𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡𝐹𝑇%+𝛽#𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡2𝑃𝐽%+𝛽$𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡3𝑃%𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒

Wewillcomparetheresultsofthisapproachwithtwootherpopularmethodswefound[39,40], using pre-draft three-point-accuracy-data without a shrinkage concept. While thesecondmodel(M2)addssimplepre-draftthree-point-percentageandthree-pointattempt-rateper40minutes,thethirdmodel(M3)factorsinacombinationofthetwovariablesbycallingonthree-pointersmadeper40minutes:

𝑀2:𝑃𝑜𝑠𝑡𝐷𝑟𝑎𝑓𝑡3𝑃% =𝛽! +𝛽"𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡𝐹𝑇%+𝛽#𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡3𝑃%+𝛽$𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡3𝑃𝐴𝑝40

𝑀3:𝑃𝑜𝑠𝑡𝐷𝑟𝑎𝑓𝑡3𝑃% =𝛽! +𝛽"𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡𝐹𝑇%+𝛽#𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡3𝑃𝑀𝑝40

Wetestedallmodelingideasonourdatasetforallathleteswhoattemptedatleastonethreeincollege(tobeabletocalculateapre-draftthree-point-percentageestimate)andplayedatleast50gamesintheNBAalready(N=255).Wechosethisthresholdtobeabletoincludeasmanyplayersaspossible,whileworkingwithareasonablesample-sizeofgames.Evenrookiesfromthe2019/20seasonhadthechancetoqualifyiftheyplayedenoughuntiltheCOVID-19intermission.M1provedtobeahighlysignificantgeneralmodelwithanoverallexplanatorypowerofanadjustedR²of0.231.WecanrejecttheH0ofthesefactorsnothavinganinfluenceonthedependentvariable.Allindividualcomponentswerehighlysignificantaswellandhavinganeffectintheexpecteddirection.Goodpre-draftperformancetranslatesintoaprojectionofhigherpost-draftskills:Table1

CoefficientOverviewModel1

Model1

UnstandardizedCoefficients StandardizedCoefficients t Sig.

B Std.Error Beta

1 (Constant) -.134 .052 -2.576 .011

Pre-Draft-FT% .153 .062 .169 2.456 .015

Pre-Draft-2PJ% .202 .081 .145 2.502 .013

Pre-Draft-3P%-Estimate .756 .146 .343 5.177 .000

a.DependentVariable:Post-Draft-3P%M2alsopresentsahighlysignificantmodelwithandanadjustedR²of0.213.ContrarytoM1,itdoesnotonlyconsistofsignificantparts.Curiously,Pre-Draft-3P%doesnotimprovetheprediction of the design. Though, removing the variable only improves the explainedvarianceby0.002.

17

Table2

CoefficientOverviewModel2

Model2

UnstandardizedCoefficients StandardizedCoefficients t Sig.

B Std.Error Beta

1 (Constant) .126 .040 3.137 .002

Pre-Draft-FT% .215 .059 .238 3.645 .000

Pre-Draft-3P% -.020 .042 -.028 -.468 .640

Pre-Draft-3PAp40 .009 .002 .316 4.722 .000

a.DependentVariable:Post-Draft-3P%M3issimilartoM2initsresults.Presentingahighlysignificantmodelingconceptagain,M3hasanadjustedR²of0.216. Its individualvariablesareallhighlysignificantandhaveaneffectintheexpecteddirection:Table3

CoefficientOverviewModel3

Model3

UnstandardizedCoefficients StandardizedCoefficients t Sig.

B Std.Error Beta

1 (Constant) .123 .040 3.106 .002

Pre-Draft-FT% .212 .058 .233 3.621 .000

Pre-Draft-3PMp40 .009 .002 .308 4.771 .000

a.DependentVariable:Post-Draft-3P%Comparingtheexplainedvarianceofallthedesigns,M1faresthebest.Itspredictionofpost-draft three-point-percentage based on pre-draft-statistics is superior to the exploredpublicly available methods. Examining the standardized beta coefficients, the Pre-Draft-3P%-Estimatehasthemostinfluenceontheprojectionofthedependentvariable.Buttakingtwo-point-jump shoot-accuracy into account also contributes to the overall modelperformance.Withoutposingahugeincreaseinexplainedvariance,ourdesignstilldeliversaslightimprovementovertheknowntechnique.Tofurtherevaluatethemodels,wecalculatedthemeanabsolutedeviationforalldesigns.M1performedbestagainwithageneralerrorsizeof0.0447comparedtoM2(0.0466)andM3(0.0463).Oncemore,thedifferencesaremarginal,butstillrelevant.

18

Figure11.AbsoluteErrorsofModel1byNBAGames.

Figure12.AbsoluteErrorsofModel1byNBA3PA.

Finally, we used absolute error terms to evaluate M1 individually. With an increasingnumber of NBA games and three-point attempts the size of the errors made decreasesrapidly and seems to stabilize. Curiously,we can also show that predicting the shooting

19

accuracyofbigmenoffersthemostproblemstoourdesignandcanclearlybepickedoutasanareaofimprovement,eventhoughpositionvariablesdidnotprovetobesignificantinourinitialmodeling.4.4.Predictingpost-draftthree-point-attemptrateShootingability intheNBAissurelyabouthowaccurateplayersareathittingthebasketwhen takingshots fromdistance.Therefore,we investigatedPost-Draft-3P%asametric,managersshouldbeeagertoprojectaccuratelyinadraftee.However,prospectsalsoneedtobewillingtotakethesekindsofshotsatareasonableratetomakeenoughuseoftheirown competence andkeepdefenses honest.Only a combinationof accuracy and volumerepresentsshooting-abilitycorrectly.Thatiswhyweincorporatedthree-point-attemptsintoourestimationsof‘true’pre-draft-three-point-percentage.Consequently,inourquestofpredictingshooting-translationfromtheNCAAtotheNBA,wealso need to look at the three-point-attempt-rate at the professional level and suggest amodelthatallowstoprojectthismetricasaccurateaspossibletoaiddraftdecision-makingevenfurther.Thus,wechosearatherexploratoryapproachsincewecouldnotderivearegressiondesignsolely from theory. We selected NBA three-point-attempt rate per 40 minutes as ourdependentvariable.Fortheindependentcomponentsofthemodel,wetookallsixavailablefactorsfromthethreealreadyexploredideasforthree-point-percentageasallofthesecanbearguedasmeasuresforshootingability(allpre-draftmeasures:3P%,3PAp40,3PMp40,FT%,2PJ%,3P%-estimate).Afterwards,weappliedastepwisemultipleregressionmethodtocheckifasignificantapproachwasthereandwhichcombinationoftheindividualpartswouldofferthemostexplanatorypower.Again,weusedtheentiredataset,filteringforonecollegethreeattemptand50playedNBAgames.Table4

ModelSummariesofViableDesignswithStepwiseAdditionofVariables

Model R RSquare AdjustedRSquare

Std.Errorofthe

Estimate

ChangeStatistics

RSquareChange FChange df1 df2

Sig.F

Change

1 .750a .562 .560 1.531 .562 324.692 1 253 .000

2 .759b .575 .572 1.511 .013 7.962 1 252 .005

3 .767c .588 .583 1.491 .012 7.547 1 251 .006

a.Predictors:(Constant),Pre-Draft-3PMp40

b.Predictors:(Constant),Pre-Draft-3PMp40,Pre-Draft-2PJ%

c.Predictors:(Constant),Pre-Draft-3PMp40,Pre-Draft-2PJ%,Pre-Draft-3P%-Estimate

d.DependentVariable:Post-Draft-3PAp40

20

Table5

CoefficientOverviewStepwiseModel

Model

Unstandardized

Coefficients

Standardized

Coefficients t Sig. CollinearityStatistics

B Std.Error Beta Tolerance VIF

1 (Constant) 1.696 .181 9.396 .000

Pre-Draft-3PMp40 .685 .038 .750 18.019 .000 1.000 1.000

2 (Constant) -.203 .696 -.292 .770

Pre-Draft-3PMp40 .687 .038 .752 18.320 .000 1.000 1.000

Pre-Draft-2PJ% 4.977 1.764 .116 2.822 .005 1.000 1.000

3 (Constant) -3.830 1.488 -2.573 .011

Pre-Draft-3PMp40 .563 .058 .616 9.642 .000 .402 2.488

Pre-Draft-2PJ% 4.990 1.741 .116 2.865 .005 1.000 1.000

Pre-Draft-3P%-

Estimate

11.955 4.352 .176 2.747 .006 .402 2.488

a.DependentVariable:Post-Draft-3PAp40Thebestmodelappearstoincludepre-draftthree-pointer-makerateper40minutes,pre-draft-two-point-jumper-percentage,andournewlyintroducedestimateofpre-draft-three-point-percentage. The model itself is highly significant as well as all its individualcomponents.TheadjustedR²of0.588isfairlystrong,whileallcomponentcoefficientshaveaneffectintheanticipateddirection.Thisgivesusreasontobelieve,thatthefoundmodel(M4)𝑀4:𝑃𝑜𝑠𝑡𝐷𝑟𝑎𝑓𝑡3𝑃𝐴𝑝40 = 𝛽! +𝛽"𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡3𝑃𝑀𝑝40 +𝛽#𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡2𝑃𝐽%+𝛽$𝑃𝑟𝑒𝐷𝑟𝑎𝑓𝑡3𝑃%𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒

should be useful in the projection of shooting competence translation, while our newlyintroducedcomponentscontributetothisprediction.Again,weevaluatedthemodelbyinvestigatingitserrorterms.Themeanabsolutedeviationofthedesignis1.2,whichseemsreasonable,whenprojecting3PAp40.

21

Figure13.AverageErrorsofModel4byNBAGames.

Figure14.AverageErrorsofModel4byNBA3PA.

Again,itcanbeobservedthatthemodel’sprojectionsbecomemoreaccuratewithincreasingNBA games played as well as threes taken, and then stabilize. The distinction between

22

bigmen and non-bigs appears to be important again, even though bigmen-attempt-ratesseemtobemorepredictablebythevariablesusedthanthree-pointaccuracy.5. ConclusionandOutlookInthispaperweinvestigatedwhetheranempiricalBayesianderived‘true’pre-draftthree-pointpercentageandpre-drafttwo-point-jumper-percentagecouldcontributepositivelytothequestofprojectingtheshootingtranslationofNBAdraftprospects.Ouranalysisshowedthatinadditiontopre-draftfree-throw-shootingbothstatisticsprovidevalueasabasisforpredictive modeling. The newfound three-point-percentage estimate based on data-introducedBayesianpriors, factoring in thenumberofattemptsasan indicator forbasicshooting-quality lends more context to the former simple percentages. The two-pointjumper-percentage also behaves as expected. Our notion of this metric being anotherindicatorofgeneral‘shootingtouch’,justlikefree-throwpercentage,turnedouttoberightandshouldbeconsideredforshootingtranslationideasfromnowon.Interestingly, this statistic also contributes positively to the projection of three-point-attempt-rate,whichshoulddrawsomeattention.Theshowedconnectionindicatesthatthepopularbasketballnarrative, implyingthathittingmidrange-shotseffectivelycouldbeanindicatorforaplayerhavingshooting-rangewhichmightbeexpandabletobehindthethree-pointline,couldbetrue.The general model we found, including the components in question, presents animprovementtootherpubliclyavailabledesignsofshooting-translation.Wewereabletoshowthatcomparedtosystemsbasedonsimplermetrics,ourmodelwasable toprojectthree-pointaccuracywithlesserrorsthanformerapproaches.Additionally,wewerepleasedtoseethatourintroducedvariablesofferanevenmorepromisingbasisfortheprojectionofpost-draftthree-pointshooting-volume,addingevenmoreinformationtotheevaluationofa prospects shooting-competence. Such contributions could ultimately improve draftdecision-making league-wide and therefore move the results of the policy towards theintendedoutcomes.However,thesimplemodelingpresentedhere,cannotbeusedreliablyonitsown.Aswecanseewiththesharesofvariancethatbothmodelsforshootingtranslationfailtoexplain,manyimportantdynamicsarenotcapturedbythesimplemetricsincludingtheBayesianinformedthree-pointpercentageandtwo-pointjump-shooting.Thedesignhasdifficultiespickingupone.g.conservativecollegeschemessuppressingshootingaspirationsoftalentedbigmenorpossiblenon-linear futuredevelopmentofplayersdue to improvedcoachingor shootingmechanic change. Hence, it should rather be used as a valuable decision-informing toolduring the draft process, triggering conversation and possibly closer investigations ofprospects,ratherthanbeingthemainargumentforachoice.

23

Toproduceamodelpowerfulenoughtobeusedinthisway,moreresearch,time,anddataisneeded.Ourapproachregardingtwo-point-jumperswasbasedonpre-draftplay-by-playdatathathasbeenavailableforNCAAprospectsforadecadeonly.Suchasample-sizeisnotsatisfying,butwillautomaticallyexpandinthecomingyears,maybeevenwiththeadditionofinformationoninternationalprospects,toallowmoreaccurateresults.Besides this natural progression, more sophisticated future approaches should considercapturingmoreof the external factors thatpotentially go into theprojectionof shootingprogression.Havingdataonsystem-basedshootingsuppression/enhancementofaformerteamoramoredetailedoverviewofthetypesofthreesaplayertook,wouldfurtherinformestimations of a ‘true’ pre-draft three-point percentage. Analysis of biomechanical orpsychological attributes could educate estimates of development curves as well, by e.g.showing an obvious, but easy to fix flaw in the shooting motion, a lack of throwingconsistencyduetoweakconditioningoralackofconfidenceintheownabilitiesandthenindicatingahighlikelihoodofimprovementintheseareasbyshowingagoodwork-ethicandhigh coachability with psychological profiling. Picking up on such signals, which shouldinformshooting translationeven further,wouldallow to improvedraftdecision-making-qualityevenmore.References[1] Alamar,B.C.(2013).SportsAnalytics.ColumbiaUniversityPress.

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