APageRankModelforPlayerPerformanceAssessment
inBasketball,SoccerandHockey
ShaelBrown
Basketball
Paper#1494
Abstract
Inthesportsofsoccer,hockeyandbasketballthemostcommonlyusedstatisticsforplayer
performanceassessmentaredividedintotwocategories:offensivestatisticsanddefensive
statistics.However,qualitativeassessmentsofplaymaking(forexample,making“smart”passes)
aredifficulttoquantify.Itwouldbeadvantageoustohaveavailableasinglestatisticthatcan
emphasizetheflowofagame,rewardingthoseplayerswhoinitiateandcontributetosuccessful
playsmore.InthispaperwewillexamineamodelbasedonGoogle'sPageRank.Otherpapershave
exploredrankingteams,coaches,andcaptainsbuthereweconstructratingsandrankingsfor
individualmembersonbothteamsinagamethatemphasizeinitiatingandpartakinginsuccessful
playsandforcingdefensiveturnovers.
Forasoccer/hockey/basketballgame,ourmodelassignsanodeforeachofthenplayerswhoplay
inthegameanda“goalnode”.Arcsbetweenplayernodesindicateapassinthereverseorder
(turnoversaredealtwithseparately).Everysport-specificsituation(fouls,out-of-bounds,playstoppages,turnovers,missedshots,defensiveplays)isaddressed,tailoredforeachsport.Aswell,
someadditionalarcsareaddedintoensurethattheassociatedMarkovchaintransitionmatrixis
primitive(somepowerofthematrixhasallpositiveentries)andhencethereisauniquePageRank
vector,whichisusedtorateandranktheplayersofthegame.
Toillustratethemodel,datawastakenfromnineNBAgamesplayedbetween2014and2016.The
applicationshowsthatthismodeldoesindeedprovidethetypeofcomprehensivestatistic
describedintheintroductoryparagraph.Manyofthetop-rankedplayers(inthemodel)inagiven
gamehavesomeofthemostimpressivetraditionalstat-lines.However,fromthemodelthereare
surpriseswheresomeplayerswhohaveimpressivestat-lineshavelowerranks,andothers,who
havelessimpressivestat-lineshavehigherranks.
Overall,themodelprovidesanalternatetoolforplayerassessmentinsoccer,basketballand
hockey.Themodel'srankingandratingsreflectmoretheflowofthegamecomparedtotraditional
sportsstatistics.
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1. Background
GooglePageRankwascreatedasthebackbonetowhatisnowthemostinfluentialsearchengine
evercreated[7].Itspurposeistoranktheimportanceofwebpageswhenausermakesaquery.
ThefoundationsofPageRanklieinMarkovchaintheory(see,forexample[9]):givenafinitesetof
states𝑆 = 𝑠$ , ⋯ , 𝑠' ,let𝑡),* betheprobabilityofmovingfromstate𝑠) tostate𝑠* fromtimekto𝑘 +
1(notethat𝑡),* isindependentofk).Let𝑇 = [𝑡),* ]denotethetransitionmatrixoftheMarkovchain.
ProvidedthatthematrixTisprimitive(i.e.𝑇 1 > 0forsomepositiveintegerm),thereisaunique
stationaryvectorv,suchthat𝒗6 𝟏 = 1and𝑻6 𝒗 = 𝒗,where𝟏isthevectorinℝ' consistingofall1's
(thatis,visaneigenvectorof𝑻6 witheigenvalue1whosenonnegativeentriessumto1).Wecall
suchavectorthePageRankvectoroftheMarkovchain.ThelackofprimitivityinGoogle'sMarkov
modelingeneralrequiressomealterationtothetransitionmatrixinthatcase.Eachcomponentof
thePageRankvectoristhoughtofastherankofthatstate(andanorderingofthestatesisderived
fromthesevalues).
ThereisanaturalwaytoconstructaMarkovchainfroma(finite)directedgraphanditsadjacency
matrix.Thestatesofthechainarethenodesofthegraph.Ifthereare𝑛),* (= 𝐴),* )arcsfromnodei
tonodejandnodeihasatotalof𝑛) outgoingarcs,then𝑇),* =
'<,=
'<
.ForaMarkovchainderivedfrom
adirectedgraph,primitivityoftheMarkovchaincorrespondstotheexistenceofapositiveintegerk
suchthatthereisawalkoflengthkbetweenanytwonodes.Insuchacasethereisaunique
stationaryvectorfortheassociatedMarkovchain,whichcanbecalculatedfromalinearsystem
(see[7]formoredetails).WeremarkthatithasbeenobservedthatthePageRankvectorisfairly
insensitivetosmallchangesinthenetworkinvolvingonlylowlyrankednodes[4,5].
PreviousapplicationsofPageRanktosportsmetricsusuallyaddressrankingeitherteams[1,2,10],
coaches[8],orindividualplayersonvariousteams[10,11].Anumberofmodelshavereliedon
underlyinggraphnetworksofgamesintheirrespectivesports.In[12]adirectedgraph
representingthepassesbetweenthestartingplayersonanindividualsoccerteamwasconstructed
andaPageRankvectorwascomputedtohighlightwhowerethemostimportantplayersonthe
teambasedonballreception;thenetworkaswellwasanalyzedtodeterminethestrategiesand
weakpointsofeachteam.In[6]everyteamhasacorrespondingweighteddirectedgraph
representingpassesandtwoadditionalnodesrepresentingtheotherteam'sgoalandmissedshots.
Cricketbatsmenandbowlersthatfaceeachotheronseparateteamsarecomparedusingabasic
modelmuchliketheonethatcomparesteamsinthe``win-loss''PageRankmethodforranking
teams[11].Aweighteddirectedgraphforplayersacrossmanybasketballteamsiscreatedin[13]
wherearcsexistonlybetweenplayerswhoplayedonthecourttogetheronthesameteamatsome
pointandtheweightofthesearcscorrespondstohoweffectivetheywereinplayingtogether.
Finally,in[3]aPageRanknetworkiscreatedforeachindividualsoccerteam,wherearcsindicate
passesbetweenplayers.
Noneofthesemodelsallowfortheeffectivecomparisonofanytwoplayersplayinginthesame
gametogether(orevenindifferentgames),possiblyofdifferentpositionsorteams,usinga
PageRankmethod.Moresignificantly,theydonotemphasizeplaymakingability,asopposedto
pureoffensivestatistics,andthatisexactlywhatweplantodo.
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2. TheModel
Wecreateadirectedgraph(whichweabbreviateasadigraph)representingtheprogressionofplay
duringaparticulargame–whethersoccer,basketballorhockey.Foreachofthenplayersinthe
gamethereisanode,andthedigraphcontainsoneadditionalgoalnode(whileourimplementation
ofagoalnodeisnotunique,theuseofa“missedshot”nodein[6]isredundantwhenwehaveboth
teamsthatarecompetingagainsteachotherinagamerepresentedonthesamedigraph).Sincewe
desireamodelthatvaluesplaymaking,wemustreversethedirectionofmostarcsthatwouldresult
fromastraightforwardprogressionofplay(suchanideawasraisedbutnotdeeplypursuedina
multi-teamsettingin[3]).Forplayeriandplayerj(representedbynodeiandnodej,respectively)
onthesameteam,wheneverplayeripassestoplayerjwedrawanarcfromnodejtonodei.
However,ifplayersiandjareonseparateteamsandplayerilosestheball/pucktoplayerjwe
drawanarcfromnodeitonodej.Ifplayeriscores,thesportspecificvalueofthescorewillbethe
numberofarcsdrawnfromthegoalnodetonodei(forexampleanNBA3-pointerwouldresultin
threearcs).Allofourchoicesforarcdirectionensurestheflowofrankrewardsplaymaking.There
willbemoregamespecificarcs,tobediscussedbelow.
Weinitializethedigraphforagivengameasfollows.Wedrawarcsinbothdirectionsbetweeneach
playernodeandthegoalnode,andinaddition,wedrawaloopfromthegoalnodetoitself.Outside
ofgoalscoring,nootherarcswillbeaddedtoorfromthegoalnode.Ifteam1has𝑛$ playersand
team2has𝑛> players(𝑛$ + 𝑛> = 𝑛),theninthecorrespondinggameadjacencymatrix,A,𝐴 ∈
𝑀'A$,'A$ (ℝ),thefirst𝑛$ columns(androws)ofTrepresentplayersonteam1,columns(androws)
𝑛> + 1tonrepresentingplayersonteam2,andthe(𝑛 + 1)thcolumnandrowrepresentingthegoal
node.Thus,beforethegamestarts,𝐴'A$,* = 𝐴),'A$ = 1for1 ≤ 𝑖, 𝑗 ≤ 𝑛 + 1and𝐴),* = 0
otherwise.Themethodofconstructionoftheinitialdigraphensuresthatanytwonodesare
connectedbyapathoflengthexactlytwo,andhencethecorrespondingMarkovchaintransition
matrix,T,willhaveeachentryof𝑇 > nonnegative,makingTprimitive,andthereforetheMarkov
chainassociatedwiththedigraphhasauniquePageRankvector.
Welet𝒓 = (𝑟$ , ⋯ , 𝑟' , 𝑟I )bethePageRankvectorofthegametransitionmatrix.Foranumberof
reasons,tobelisted,weshallrescalethevaluesinthePageRankvector(suchaprocessdoesnot
changetheinducedorderingoftheplayers’ranks).Oneimmediateissuewiththemodelisthefact
thatthegoalnodemayhavedifferentrankineachgamedependingonthenumberofplayersinthe
game,thusmakingthecomparisonofranksofplayersindifferentgamesdependentontherankof
thegoalnode.Wecanscalethecomputedrankvectorrbyanyscalar,asrisaneigenvectorof𝑇 6 .
Westandardizerbydefiningtherelativerankofplayeriinthegameby
𝑟𝑒𝑙) = 50𝑛 ∙
𝑟)
'
*N$ 𝑟*
=
50𝑛 ∙ 𝑟)
1 − 𝑟I
Thechoiceofscalingisasfollows:thedenominatorremovestheeffectofthegoalnode’srank,and
thenumeratorensures(a)thattherelativerankisinsensitivetothenumberofplayersinthegame
and(b)providesvaluesonareasonablescale,between0and50𝑛(itisnothardtoseethatthe
averagerelativeranksofallplayersinagameis50).Therelativerankscanthusbeusedfor
meaningfulcomparisonofplayersindifferentgames.
Wenowreturntohowthedigraphitselfisbuiltupinthethreesportspecificsituations.We
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illustratetheprocesswithbasketball(therulesforsoccerandhockeycanbefoundinthe
Appendix).Eachbulletpointisaplay“event”,withitssubsequentdescriptorthecorresponding
arc(s)toaddtothedigraph.
Basketballrulesofimplementation:
• Passfromplayeritoplayerj.
−anarcfromnodejtonodei
• Playeridispossessesplayerj.[Thiscouldincludecaseswhereplayeridoesnotgainpossession
oftheballafterdispossessingplayerj–forexampleadefensivetouchleadingtotheballbeing
outofplayordeflectingapassstillintoplaybutawayfromitsintendedtarget.]
− anarcfromnodejtonodeI
• PlayerIscoresnpointswhere1 ≤ 𝑛 ≤ 4.
− narcsfromthegoalnodetonodeI
• PlayerIshootswhenbeingcontestedanddefendedbyplayerjandmissesthenet.[Sameas
playerjdispossessingplayerI,playresumingwiththerebounding/inboundingplayer.Thiscase
includesthesituationwhereplayerjblocksplayeri.]
−anarcfromnodeitonodej
• Playerishootsandmissesthenetundernopressureandtheballisreboundedbyplayerj.[Same
asplayerjdispossessingplayeri.]
−anarcfromnodeitoplayerj
• Playerifoulsplayerjandplayerjmakesatleastonefreethrow.
−ifplayerjmakesn>0freethrowsthennarcsarecreatedfrom
thegoalnodetonodej
• Playerifoulsplayerjandplayerjmakeszerofreethrows.[Sameasplayerjdispossessingplayer
i–itwasa“smart”foul.]
−anarcfromnodejtonodei
• Anystoppageofplaythatdoesnothavetodowiththegame(i.e.a
technicalfoul,fan
interference,injury,altercationetc.).[Playisdead.]
−noarcdrawn
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• Playeriinterceptsapassfromplayerj.[Sameasplayeridispossessingplayerj.]
−anarcfromnodejtonodei
• Playeritouchestheballwithouthavingpossession(forexamplethe
ballhitsplayeri,ora
“pinball”play).
−noarcdrawn
• Anyunforcedturnoverbyplayeri.
−noarcsdrawn
Weillustratetheprocesswithasmallexample.Supposethattwobasketballteams,theRedsand
theBlues,areplayingagainsteachotherina“3-on-3”match(whereallbasketsareworthone
point).WewilldenotetheplayersontheRedsbyA,BandC,andthoseontheBluesbyD,EandF.
BeforethegamebeginswehavethesetupofthedigraphshowninFigure1(whereGstandsforthe
goalnode).Inthisandsubsequentdiagramsofdigraphs,ifthereexistsmorethanonearcfrom
nodextonodeywewilldrawonearcfromnodextonodeybutlabelitwiththenumberofarcs
thatexistbetweenthetwonodes.
A
F
G
E
B
C
D
Figure1:Smallexample’sinitialdigraph(beforeplay).
Nowsupposewehavethefollowingsequenceofplaysinthegame:(where“→”representsthe
movementoftheballbetweennodes,and“0”istheendofplaysequencesymbol):
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A→B→A→F→G
D→F→E→F→D→C→B→C→A→C→B→A→G
D→C→A→C→B→A→G
D→F→0→B→C→A→G
D→F→E→F→D→G
A→B→F→G
Afterthesesequencesofplaysourupdatednetworkbecomes:
A
3
2 3
F
2
2
G
B
E
2
2
3
C
3
D
2
Figure2:Smallexampleupdated
Theadjacencymatrixofthedigraph(whose(𝑖, 𝑗)–thentryisthenumberofarcsinthedigraphfrom
nodeitonodej)is
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⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
2
2
0
0
0
4
3
0
2
0
0
0
1
3
3
0
2
0
0
1
0
0
0
0
0
3
2
0
0
0
0
0
2
1
1
1
0
2
2
0
3
1
1
1
1
1
1
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
andsothetransitionmatrixforthisgameis
⎡
⎢
⎢
⎢
T =⎢
⎢
⎢
⎢
⎢
⎣
0
3/8
3/8
0
0
1/ 8
1/ 8
2/7 2/5
0
0
0
0
2/5
0
0
0
3/ 7
0
2/5
0
0
0
0
0
0 1/ 2
0
0
0
0 1/ 3
1/ 7
0
2/5 2/3 0
1/ 7 1/ 5 1/ 5 1/ 3 1/ 6
4 /13
1 /13
1 /13
2 /13
1 /13
3 /13
1 /13
⎤
⎥
⎥
⎥
⎥ ⎥
⎥
⎥
⎥
⎦
Thecalculatedrelativeranksoftheplayersinthesystemareasfollows:
Player
Team
RelativeRank
C
Reds
64.66
F
Blues
60.38
A
Reds
58.79
B
Reds
52.39
D
Blues
39.17
E
Blues
24.61
WhileplayerAscoredthemostgoalsinthegameandplayerFhadthesamenumberoftotalgoals
andassists,weseethatinfactplayerC,whohadonly1assistandnogoals,hasthehighestrank.
However,acursoryexaminationoftheplaysclearlyshowshowintegralplayerCwasinthegame
asaplaymaker.Theexampleshowshowaplayerwhosecontributiontothegamemightbeignored
undertheusualstatlinesreceivestheirwell-deservedacknowledgementundertheproposed
model.
Beforewecontinuewithsomeexperimentalresultsitisnaturaltoaskhowthismodelfitswithour
intuitionofevaluatingtheperformanceofathletes.Weobservefirstthatthelowestpossible
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relativerankofanyplayerinagameofnplayersisnomorethan50,sinceifthesmallestrelative
rankofanyplayerinthegameismorethan50,sinceiftherelativerankofeveryplayerwerelarger
than50,wewouldcontradictthefactthattheaveragerelativerankofallplayersinthegameis50.
Thefollowingpropositionsconsiderthespacingofrelativeranksinagame.
Proposition2.1.Iftherearekstarters(outofnplayers)inagivengamewithacumulativestarter
'(QRSTU)
relativerankofR,thentheremustexistabenchplayerwitharankthatisatmost
distance
U('RU)
fromtherankofsomestarter.
ST'RQ
Proof.Wefirstnotethattheaveragebenchplayerrelativerankmustbe
asthereare𝑛 − 𝑘
'RU
benchplayersandthetotalsumofalltherelativeranksis50𝑛.Clearlytheworst-casescenario
Q
ST'RQ occursifallstartershavearankof andallbenchplayershavearankof
(asotherwisethere
U
ST'RQ
'RU
wouldhavetobeonebenchplayerwithrelativerankgreaterthan
oronestarterwithrank
'RU
Q
lessthan ).Thus,inthisworst-casescenariothedistancebetweenanystarterandanybench
U
Q
ST'RQ
U
'RU
playeris −
=
Q'RQURST'UAQU
U 'RU
=
'(QRSTU)
U('RU)
.n
Proposition2.2.Iftherearenplayersinagamethentheremustexistatleasttwoplayerswhohave
>S'
relativerankswithin ofeachother.
'R>
Proof.Clearly0isalowerboundontherelativerankofaplayerinagivengameandanupper
boundis50𝑛(thesumofalltherelativeranks).Thus,wemaypartitionthisrangeinto𝑛 − 2equal
ST'
lengthintervals,eachoflength using𝑛 − 1ofthentotalplayers.Wemusthavethatatleast
'R>
threeplayersmustbewithin(orontheboundaryof)oneofthe𝑛 − 2intervals,meaningthattwo
oftheirrelativeranksmustbeinthesamehalfinterval,makingthedifferenceoftheirrelative
>S' ranksnomorethan
.n
'R>
Theabovetworesultsshowthattherehastobesomebenchplayerofnon-negligibleimportanceto
thegame,andthatsomeplayershavetohaveranksthatare“somewhat”closetoeachother.The
firstresultfitswiththewidelyacceptednotionthatbenchplayisacomponenttothesuccessofa
basketballteam.
Finally,wemaybeinterestedinhowthenumberofgoalsscoredbyplayersondifferentteams
affectstheirrelativeranks.Asanillustration,supposewehaveonlytwoplayers,𝑃$ and𝑃> ,inthe
system,eachonseparateteams,withnointeractionbetweenthemand𝑃$ scores𝑔$ goalsand𝑃$ scores𝑔> goals.Thenif𝑔$ < 𝑔> thentherelativerankof𝑃$ isgreaterthanthatof𝑃> .Thisfollowsas
ST'Z[ (I\ A$) itisclearthattherelativerankof𝑃$ is
andtherelativerankof𝑃> is
ST'Z[ (I] A$)
(I\ AI] A>)($RZ[ )
(I\ AI] A>)($RZ[ )
where𝑟I istherankofthegoalnode.
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3. SampleDataAnalysis
Wenowapplyourmodeltosomereal-lifebasketballgames.Thespecificgamesusedwere
•ChicagoBullsvs.SanAntonioSpurs(November30th2015),
•GoldenStateWarriorsvs.ClevelandCavaliers(January18th2016),
•SanAntonioSpursvs.BrooklynNets(December3rd2014),
•ChicagoBullsvs.CharlotteHornets(December3rd2014),
•LosAngelesLakersvs.GoldenStateWarriors(November1st2014),
•TorontoRaptorsvs.ClevelandCavaliers(December9th2014),
•GoldenStateWarriorsvs.ClevelandCavaliers(December25th2015),
•ChicagoBullsvs.OklahomaCityThunder(December25th2015),and
•WashingtonWizardsvs.ClevelandCavaliers(November26th2014).
Thegamesareidentifiedbytheteamsplaying,date,scoreandwinningteam.Ineachgame,plays
weremanuallyanalyzedandtranscribed,andtheresultingtransitionmatricesandPageRank
vectorswerecalculated.Theninetables,indecreasingorder,therelativeranksofeachofthe
players(roundedtothenearesthundredth)inallnineNBAbasketballgamesfromwhichdatawas
taken,followedbyeachplayerspoints(P),assists(A),rebounds(R),steals(S),turnovers(T)and
fieldgoalpercentage(FG%),roundedtothenearestwholenumber,(inthatorder)allaccessed
fromnba.com.
ChicagoBullsvs.SanAntonioSpurs,November30th2015(Bullswin92-89)
Playername
Team
RelativeRank
P A R S T FG%
Parker
Rose
Duncan
Gasol
Leonard
Aldridge
Noah
Green
Butler
Mirotic
Ginobli
Diaw
Mills
Moore
West
Snell
McDermott
Gibson
Anderson
Spurs
Bulls
Spurs
Bulls
Spurs
Spurs
Bulls
Spurs
Bulls
Bulls
Spurs
Spurs
Spurs
Bulls
Spurs
Bulls
Bulls
Bulls
Spurs
103.66
83.68
74.05
72.30
63.62
59.99
58.32
57.69
49.23
46.55
44.89
44.82
40.31
33.42
31.02
27.48
24.97
20.86
13.13
13
11
6
18
25
21
8
9
14
8
4
5
4
6
2
11
12
4
0
9
6
3
4
3
0
7
1
3
2
2
0
1
1
1
1
0
1
0
1
4
12
13
8
12
11
4
3
5
1
6
0
2
3
6
3
4
0
0
1
0
1
2
0
0
2
1
0
1
0
0
0
0
0
0
1
0
0
1
2
1
2
2
0
1
5
2
1
1
0
1
0
0
0
0
0
50
29
43
33
77
55
67
30
55
38
25
40
25
50
20
80
42
40
0
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GoldenStateWarriorsvs.ClevelandCavaliers,January18th2016(Warriorswin132-98)
Playername
Team
RelativeRank
P A R S T FG%
Curry
Green
Dellavedova
James
Irving
Barnes
Livingston
Love
Iguodala
Bogut
Varejao
Shumpert
Barbosa
Thompson
Clark
Ezeli
Mozgov
Smith
Thompson
JThompson
Cunningham
Jefferson
Jones
Speights
Rush
Warriors
Warriors
Cavaliers
Cavaliers
Cavaliers
Warriors
Warriors
Cavaliers
Warriors
Warriors
Cavaliers
Cavaliers
Warriors
Warriors
Warriors
Warriors
Cavaliers
Cavaliers
Cavaliers
Warriors
Cavaliers
Cavaliers
Cavaliers
Warriors
Warriors
122.91
109.34
109.26
91.43
78.75
66.94
62.80
60.54
60.24
58.61
50.63
39.84
39.52
39.05
38.24
29.79
28.84
26.99
23.73
21.89
20.39
18.47
17.43
17.22
17.17
35
16
11
16
8
12
4
3
20
4
5
10
8
15
6
4
6
14
2
1
9
6
8
4
3
4
10
6
5
3
0
0
2
5
0
3
0
4
2
2
0
3
1
0
1
1
0
1
0
1
5
7
1
5
5
2
2
6
3
6
4
2
1
1
2
2
0
2
2
3
3
3
0
1
3
3
0
0
1
0
0
0
0
0
0
1
0
2
1
0
0
0
1
0
0
0
0
0
0
0
1
1
1
3
2
2
2
1
1
0
1
3
0
1
0
2
1
1
0
0
0
1
1
0
0
67
50
50
44
27
50
67
20
88
67
50
67
50
45
29
67
50
67
0
1
60
100
60
25
33
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SanAntonioSpursvs.BrooklynNets,December3rd2014(Netswin95-93)
Playername
Team
RelativeRank
P A R S T FG%
Williams
Teletovic
Duncan
Parker
Ginobli
Green
Lopez
Leonard
Joseph
Johnson
Jack
Diaw
Bonner
Bogdanovic
Baynes
Anderson
Belinelli
Jordan
Plumlee
Nets
Nets
Spurs
Spurs
Spurs
Spurs
Nets
Spurs
Spurs
Nets
Nets
Spurs
Spurs
Nets
Spurs
Nets
Spurs
Nets
Nets
111.40
90.99
81.11
79.92
68.23
65.77
62.01
56.59
55.32
53.48
47.11
45.72
35.93
32.72
17.78
14.49
11.89
9.83
9.73
17
26
14
9
15
20
16
12
7
8
8
0
7
14
4
2
5
2
2
9
2
1
6
5
2
3
1
3
2
3
3
0
0
1
1
1
0
0
3
15
17
1
6
10
16
13
3
5
1
2
1
8
4
1
1
2
3
0
0
0
0
1
2
0
1
0
1
0
0
0
0
1
0
0
1
0
2
0
2
3
0
0
0
0
0
1
2
2
0
2
0
2
1
0
1
40
69
28
50
46
50
35
25
38
25
40
0
30
50
40
20
67
100
33
ChicagoBullsvs.CharlotteHornets,December3rd2014(Bullswin102-95)
Playername
Team
RelativeRank
P A R S T FG%
Walker
Rose
Gasol
Noah
Mirotic
Stephenson
Zeller
Williams
Butler
Brooks
Hinrich
Roberts
Jefferson
Dunleavy
Henderson
Hairston
Snell
Biyombo
Hornets
Bulls
Bulls
Bulls
Bulls
Hornets
Hornets
Hornets
Bulls
Bulls
Bulls
Hornets
Hornets
Bulls
Hornets
Hornets
Bulls
Hornets
110.08
85.77
84.94
77.80
73.59
64.26
63.21
59.65
52.88
51.90
48.90
37.93
35.36
32.20
30.55
14.67
13.54
12.64
23
15
19
14
11
20
12
6
15
7
12
3
13
9
10
4
0
4
4
5
3
7
1
4
2
0
5
3
2
3
2
0
1
1
1
0
5
2
15
10
2
8
8
3
2
3
3
1
7
1
4
2
1
4
1
0
0
1
0
1
0
1
2
0
0
0
0
1
0
2
0
0
0
2
2
2
1
4
0
0
1
2
1
0
0
0
1
0
0
0
39
42
37
67
50
50
45
40
45
43
44
13
38
60
50
14
0
50
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Pargo
0.14
0 0 0 0 0 0
LosAngelesLakersvs.GoldenStateWarriors,November1st2014(Warriorswin127-104)
Playername
Team
RelativeRank
P A R S T FG%
Curry
Lin
Green
Iguodala
Bogut
Bryant
Hill
Price
Davis
Barnes
Thompson
Livingston
Boozer
Ezeli
Barbosa
Johnson
Ellington
Speights
Sacre
Clarkson
Henry
Holiday
Hornets
Warriors
Lakers
Warriors
Warriors
Warriors
Lakes
Lakers
Lakers
Lakers
Warriors
Warriors
Warriors
Lakers
Warriors
Warriors
Lakers
Lakers
Warriors
Lakers
Lakers
Lakers
Warriors
132.74
94.24
89.61
84.83
79.29
76.87
74.27
64.48
59.39
53.12
50.27
42.72
38.93
37.05
34.37
33.40
20.53
14.58
7.69
5.52
4.95
1.18
31
6
9
9
6
28
23
1
13
15
41
2
9
3
9
15
2
2
4
3
0
0
10
6
1
6
3
1
4
6
2
3
2
1
1
1
3
0
1
0
0
0
0
0
5
4
5
4
10
6
5
4
6
4
5
2
4
4
1
4
4
3
1
1
0
0
3
1
1
2
1
2
0
2
1
1
0
1
0
0
1
0
1
0
0
2
0
0
2
5
1
4
5
7
2
2
1
1
1
1
0
2
3
1
1
0
2
1
0
0
53
0
33
50
30
43
71
0
71
83
78
50
44
100
50
67
50
50
50
20
0
0
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TorontoRaptorsvs.ClevelandCavaliers,December9th2014(Cavalierswin105-101)
Playername
Team
RelativeRank
P A R S T FG%
Lowry
Irving
James
Love
Valanciunas
Dellavedova
Patterson
Thompson
Williams
A.Johnson
Ross
Vasquez
Fields
Varejao
Waiters
J.Johnson
Marion
Jones
Hayes
Raptors
Cavaliers
Cavaliers
Cavaliers
Raptors
Cavaliers
Raptors
Cavaliers
Raptors
Raptors
Raptors
Raptors
Raptors
Cavaliers
Cavaliers
Raptors
Cavaliers
Cavaliers
Raptors
123.55
118.30
95.39
70.70
65.34
59.32
47.36
44.27
43.97
41.76
37.68
33.11
32.38
32.15
29.92
24.98
23.12
20.41
6.30
16
13
35
17
18
6
12
8
6
10
18
3
4
8
18
12
0
0
2
14
10
4
4
0
5
1
0
4
2
1
2
2
1
2
0
0
1
0
4
1
2
9
15
3
4
8
1
2
3
0
1
6
1
4
1
0
1
1
2
2
0
0
0
0
0
0
0
0
0
2
0
0
1
0
0
0
0
2
2
3
3
0
1
1
1
2
5
1
1
1
1
1
1
0
0
33
42
57
40
86
50
71
60
25
50
62
33
100
40
70
46
0
0
100
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GoldenStateWarriorsvs.ClevelandCavaliers,December25th2015(Warriorswin89-83)
Playername
Team
RelativeRank
P A R S T FG%
Green
Curry
Love
Dellavedova
James
Iguodala
Irving
Thompson
Livingston
Thompson
Bogut
Ezeli
Shumpert
Smith
Rush
Mozgov
Clark
Barbosa
McAdoo
Speights
Jones
Williams
Warriors
Warriors
Cavaliers
Cavaliers
Cavaliers
Warriors
Cavaliers
Cavaliers
Warriors
Warriors
Warriors
Warriors
Cavaliers
Cavaliers
Warriors
Cavaliers
Warriors
Warriors
Warriors
Warriors
Cavaliers
Cavaliers
140.89
117.29
106.85
98.70
97.45
74.47
65.18
57.09
54.35
47.08
47.03
36.77
30.38
28.77
18.84
15.40
14.20
14.04
13.33
10.75
5.78
5.35
22
19
10
10
25
7
13
8
16
18
4
3
0
14
0
0
0
0
0
0
0
3
7
7
4
1
2
3
2
1
2
1
1
0
1
0
0
0
0
0
0
0
0
1
15
7
18
5
9
2
3
10
3
6
7
4
4
1
3
3
0
1
1
0
2
0
0
2
0
1
1
1
1
1
1
0
0
0
1
1
1
0
1
0
0
1
0
0
4
3
1
1
4
0
2
0
4
1
0
2
0
2
1
1
0
0
0
1
0
0
47
40
31
36
38
17
27
50
89
38
100
25
0
44
0
0
0
0
0
0
0
0
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ChicagoBullsvs.OklahomaCityThunder,December25th2015(Bullswin105-96)*last36.4
secondsofsecondquarterandfirst17secondsof3rdquarterwerenotabletobeseenfromthe
source.
Playername
Team
Westbrook
Gasol
Butler
Durant
Rose
Kanter
Gibson
Ibaka
Portis
Hinrich
Adams
Brooks
Mirotic
Augustin
Roberson
McDermott
Morrow
Snell
Waiters
Collison
Thunder
Bulls
Bulls
Thunder
Bulls
Thunder
Bulls
Thunder
Bulls
Bulls
Thunder
Bulls
Bulls
Thunder
Thunder
Bulls
Thunder
Bulls
Thunder
Thunder
RelativeRank
P
A R
S T FG%
125.96
104.06
94.85
79.36
79.34
70.19
62.38
58.25
49.98
41.15
38.62
35.67
32.04
28.25
23.22
19.31
18.27
17.27
12.41
9.42
26
21
23
29
19
14
13
6
7
2
3
6
6
3
2
5
9
3
2
2
8
6
4
7
1
1
2
0
3
2
0
1
2
1
1
1
0
0
2
0
6
0
4
1
0
0
1
2
1
0
0
0
1
1
0
1
1
0
1
0
7
13
6
9
4
13
10
7
5
0
4
4
7
1
4
3
1
1
0
2
6
4
3
2
4
0
1
2
1
0
0
0
1
2
0
1
0
1
1
0
39
50
45
52
39
50
75
25
38
50
25
50
20
25
17
29
50
25
17
50
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WashingtonWizardsvs.ClevelandCavaliers,November26th2014(Cavalierswin113-87)
Playername
Team
RelativeRank
P A R S T FG%
James
Irving
Wall
Beal
Love
Gortat
Thompson
Waiters
Miller
Seraphin
Varejao
Humphries
Pierce
Marion
PorterJr.
Cherry
Blair
Butler
Amundson
Gooden
Harris
Cavaliers
Cavaliers
Wizards
Wizards
Cavaliers
Wizards
Cavliers
Cavaliers
Wizards
Wizards
Cavaliers
Wizards
Wizards
Cavaliers
Wizards
Cavaliers
Wizards
Wizards
Cavaliers
Wizards
Cavaliers
117.45
117.00
113.96
62.07
61.40
52.63
50.89
49.51
48.21
47.26
47.03
45.24
42.62
40.00
30.72
28.63
23.79
23.11
20.52
16.66
11.30
29
18
6
10
21
12
10
15
7
7
10
3
15
6
2
2
0
23
0
2
2
8
5
7
2
0
1
0
6
6
3
0
1
3
2
1
0
0
0
1
0
0
10
1
4
2
5
2
1
3
2
3
7
3
3
4
2
0
1
1
1
3
2
3
3
0
3
0
1
0
2
0
0
0
0
0
2
0
2
0
1
0
0
0
4
1
5
1
2
3
0
1
0
2
1
1
3
0
0
0
1
2
0
0
0
50
47
33
40
70
50
100
35
75
38
100
14
80
25
25
0
0
60
0
50
50
Weobserveseveralgeneraltrendsfromthedatathatintuitivelymatchwhatwewouldexpect.For
^_
example,outofalltheplayerswithrelativeranksunder30, ≈ 87%ofthemhavetheirsumof
SS
points,assists,reboundsandstealsnogreaterthan10,correspondingtoa“small”statline.Onthe
>_
otherhand,outofallplayerswhohaverelativeranksgreaterthan70, ≈ 64%havetheirsumof
^^
points,assists,reboundsandstealsbeatleast25,correspondingtoa“large”statline.Intermsof
highestrankedpositions,sevenofthegameshadapointguardrankedthehighest;however,a
forwardwasrankedinthetopfiveranksinallninegamesaswell.Onaverage,thestartersofboth
teamsownedapproximately7outofthefirst10highestrelativeranks,whichfitswithour
knowledgethatatleastonebenchplayermusthavesomesignificantimportancetothegame.Out
oftheninegamessampled,thehighestrankedplayerwasonthelosingteamfourtimes.
Nowletusconsidersomebasictrendsincertainaveragesoftherelativeranksintheninegames.In
thetablebelowallvalueswereroundedtothenearesthundredth,WTstandsforwinningteam,LT
standsforlosingteamandARRstandsforaveragerelativerank.
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Game
WTARR
LTARR
WTstarterARR
LTstarterARR
1
2
3
4
5
6
7
8
9
46.31
52.59
47.97
57.95
56.34
54.84
49.09
53.60
54.37
53.32
47.19
51.82
42.85
43.66
45.64
51.10
46.40
46.03
55.85
79.37
70.11
66.72
81.00
67.93
74.23
74.53
76.58
71.80
57.31
65.82
60.69
63.54
60.14
62.73
65.08
63.31
Outoftheninegamessampled,thewinningteamhadahigheraveragerelativerankthanthatofthe
losingteamsixtimes.However,theaverageranksofstartersonthewinningteamwasgreaterthan
thatforthelosingteamineightoutoftheninegames.
Inmanycasesourintuitionofhowastrongoverallperformancebyaplayershouldberanked
agreedwiththegeneratedrelativeranks.Forexample,considerthecasesofLeBronJames
(ClevelandCavaliers),StephenCurry(GoldenStateWarriors)andPauGasol(ChicagoBulls).Each
arerecognizedasbeingverytalentedplayersintheNBAandineachgamesurveyedinwhichthey
playedtheyeachhada“strong”statline,andalsoaveryhighrelativerank(correspondingtothem
beingexcellentplaymakersaswell).However,therewerecertainlyalsosomesurpriseswhere
playerswith“strong”statlinesdidnothavelargerelativeranks.IntheLakersvs.Warriorsgame,
KlayThompsonscored41pointsyethadaveryaveragerelativerankof50.27.IntheWizardsvs.
Cavaliersgame,Butlerhad23pointsandarankofabout23.
Ageneraltrendthatweobserveisthatoutofalltheplayerswhohadatleast15pointsandhavea
relativeranklessthan50,89%hadthesumoftheirassists,reboundsandstealsbeunder10,i.e.
scoringalonewasnotgenerallyhighlyvaluedbythemodel.Incontrast,outofthe30playersinall
gameswhohadatleast5assists,28ofthemhadarelativerankofatleast50(about93%),showing
ageneraltrendofrewardingpassingcomparedtoscoring.Therewereofcoursealsosome
overachieversinthesampledata;ofthetop10ranksineachgame,onaverage40%hadasumof
theirpoints,assists,reboundsandstealsbeatmost20,correspondingtowhatonecouldcallanat
most“standard/average”statline.Specificexamplesofhighrankingplayerswithlowstatlinesin
thismodelwereJohnWall(Wizardsvs.Cavaliers),RonniePrice(Lakersvs.Warriors),JeremyLin
(Lakersvs.Warriors),TonyParker(Spursvs.Nets)andMatthewDellavedova(Cavaliersvs.
WarriorsandCavaliersvs.Raptors).
4. Conclusion
Whiletherelativeranksofplayersarecomparablebetweengames,infutureworkwemay
consider,overawholeseasoninonesportsleague,havingallplayersandteams
representedbyonelargedigraphwithonegoalnode.Thislarge-scalemodelcouldbeused
asanothermeasureofperformancebetweenanyathletesinthesameleague.Notethat
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adjustingthismodelfordealingwitheventssuchasplayertradeswouldnotbedifficult.In
anycase,automatingtheprocessingofinputisanecessarystepinscalingthemodel.
Moreover,adjustingthemodeltoencompassmoresportsthatfollowsimilarmodelsto
soccer,basketballandhockey,forinstancevolleyballorwaterpolo,couldfinduseful
applicationsaswell.
ThePageRank-basedmodel,whichwehaveconstructed,appearstobethefirstofitskind
togiveaquantifiablemeasureinwhichplayersondifferentteamsandevendifferent
gamescanberankedandcompared,inclusiveoftheiroffensiveanddefensiveskills.While
themodelcouldserveasausefulmainstreamstatisticforscouts,coaches,managersand
fans,moredataanalysisisrequiredtoseeifthemodelcanprovideaccurateoutcome
predictionsforgames(forexample,comparingtheaveragerelativerankofthestartersof
eachteam,priortothatgame).Thestatisticsfromourproposedmodelcouldbeusedin
conjunctionwiththestandardplayerperformancemetricsineachsporttohelpdeepenour
understandingofwhoisreallyinfluencingthegamethemost.
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References
[1] Balreira, Eduardo C., Brain K. Miceli, and Thomas Tegtmeyer. "An Oracle Method to
PredictNFLGames."JournalofQuantitativeAnalysisinSports10(2014):183-196.
[2]Barrow,Daniel,IanDrayer,PeterElliott,GarrenGaut,andBraxtonOsting.”Ranking
rankings:anempiricalcomparisonofthepredictivepowerofsportsrankingmethods.”
JournalofQuantitativeAnalysisinSports9(2013):187-202.
[3]Brandt,Markus,andUlfBrefeld.”Graph-basedApproachesforAnalyzingTeam
InteractionontheExampleofSoccer.”ProceedingsoftheECML/PKDDWorkshopon
MachineLearningandDataMiningforSportsAnalytics8(2015).
[4]Chartier,TimothyP.,ErichKreutzer,AmyN.Langville,andKathrynE.Pedings.
”Sensitivityandstabilityofrankingvectors.”SiamJournalofScientificComputing33
(2011):1077-1102.
[5]deKerchove,Cristobald,LauraNinove,andPaulVanDooren.”MaximizingPageRankvia
outlinks.”LinearAlgebraanditsApplications429(2008):1254-1276.
[6]Duch,Jordi,JoshuaS.Waitzman,andLuisA.NunesAmaral.”Quantifyingthe
PerformanceofIndividualPlayersinaTeamActivity.”PLoSONE5(2010).
[7] Glelch,DavidF.”PageRankBeyondtheWeb.”SIAMReview57(2015):321-363.
[8]Hu,Zhi,JingZhou,MengZhang,andYangZhao.”Methodsforrankingcollegesports
coachesbasedondataenvelopmentanalysisandPageRank.”TheJournalofKnowledge
Engineering32(2015):652-73.
[9]Langville,AmyN.,andCarlD.Meyer.Google’sPageRankandBeyond:TheScienceof
SearchEngineRankings.N.p.:PrincetonUniversityPress,2011.
[10]Mukherjee,Satyam.”Identifyingthegreatestteamandcaptain-Acomplexnetwork
approachtocricketmatches.”PhysicaA:StatisticalMechanicsanditsApplication391
(2012):6066-6076.
[11]Mukherjee,Satyam.”QuantifyingindividualperformanceinCricket-Anetwork
analysisofbatsmenandbowlers.”PhysicaA:StatisticalMechanicsanditsApplication393
(2013):624-637.
[12]Pea,JavierL.,andHugoTouchette.”Anetworktheoryanalysisoffootballstrategies.”
Proc.2012EuromechPhsycisofSportsConference(2013)517-528.
[13]Piette,James,LisaPham,andSathyanarayanAnand.”EvaluatingBasketballPlayer
PerformanceviaStatisticalNetworkModeling.”MITSloanSportsConference(2011).
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Appendix
Soccerrulesofimplementation:
•Passfromplayeritoplayerj.
–anarcfromnodejtonodei
•Playeridispossessesplayerj.[Thiscouldincludecaseswhereplayeridoesnotgain
possessionoftheballafterdispossessingplayerj;forexampleadefensivetackleleading
totheballbeingoutofplayordeflectingapassstillintoplaybutawayfromitsintended
target.]
–anarcfromnodejtonodei
•Playeriscores.
–anarcfromthegoalnodetonodei
•Playerishootswhenbeingpressuredbyplayerjandmissesthenet.[Sameasplayerj
dispossessingplayeri.Thiscaseincludesthesituationwhereplayerjblocksplayeri.]
–anarcfromnodeitonodej
•Playerishootsandmissesthenetundernopressure.[Playisdead.]
–noarcsdrawn
•Playerishootsandshotissavedbythegoalkeeper,playerj.[Sameasplayerj
dispossessingplayeri.]
–anarcfromnodeitonodej
•Playerifoulsplayerj,notresultinginagoal.[Playisdead.]
–noarcsdrawn
•Playerifoulsplayerj,resultinginapenaltyoragoalfromafreekick.[Smartdrawingofa
foulbyplayerj;scoringfromafreekickcouldincludeadirectshot,aheaderorvolley
fromthefreekickorareboundinsidetheboxfollowingthefreekick.]
–anarcfromnodeitonodej
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•Anystoppageofplaythatdoesnothavetodowiththegame(i.e.weather,fan
interference,injury,altercationetc.).[Playisdead.]
–noarcsdrawn
•Playeriinterceptsapassfromplayerj.[Sameasplayeridispossessingplayerj.]
–anarcfromnodejtonodei
•Playeritouchestheballwithouthavingpossession,forexample,theballhitsplayeri,or
a“pinball”play.[Playerididnothavepossession.]
–noarcsdrawn
•Anyunforcedturnoverbyplayeri.
–noarcsdrawn
•Playeriisoffsidewhenplayerjpassestheball.[Playerjpassestheballtoplayeriwhois
inanoffsideposition,sotheplayendsatplayeri.]
–anarcfromnodeitonodej
Hockeyrulesofimplementation:
• Passfromplayeritoplayerj.
–anarcfromnodejtonodei
• Playeridispossessesplayerj.[Thiscouldincludecaseswhereplayeridoesnotgainpossession
ofthepuckafterdispossessingplayerj,forexampleadefensivetouchleadingtothepuckbeing
outofplayordeflectingapassstillintoplaybutawayfromitsintendedtarget.]
–anarcfromnodejtonodei
• Playeriscores.
–anarcfromthegoalnodetonodei
• Playerishootswhenbeingdefendedbyplayerjandmissesthenet.[Sameasplayerj
dispossessingplayeri;playresumeswiththeplayerthatcollectsthepuckaftertheshot.This
caseincludesthesituationwhereplayerjblockstheshotofplayeri.]
–anarcfromnodeitonodej
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• Playerishootsandtheshotissavedbythegoalkeeper,playerj.[Sameasplayerjdispossessing
playeri.]
–anarcfromnodeitonodej
• Playerishootsandmissesthenetundernopressure.[Playisdead.]
–noarcsdrawn
• Playeridrawsapenaltyfromplayerjduringwhichnopower-playgoalisscored.[A“smart”
penalty.]
–anarcfromnodeitonodej
• Playeridrawsapenaltyfromplayerjduringwhichapower-playgoalisscored.[A“smart”
drawingofapenalty.]
–anarcfromnodejtonodei
• Anystoppageofplaythatdoesnothavetodowiththegame(i.e.apenalty,faninterference,
injury,altercationetc.).[Playisdead.]
–noarcsdrawn
• Playeriinterceptsapassfromplayerj.[Sameasplayeridispossessingplayerj.]
–anarcfromnodejtonodei
• Playeritouchesthepuckwithouthavingpossession(forexamplethepuckhitsplayeri,ora
“pinball”play).[Playerididnothavepossession.]
–noarcsdrawn
• Anyunforcedturnoverbyplayeri.
–noarcsdrawn
• Playeriisoffsidewhenplayerjpassesthepuck.[Playerjstillpassedplayerithepuck,theplay
endedbyplayeribeinginanoffsideposition.]
–anarcfromnodeitonodej
• Playeriicesthepuckwhichistouchedbyplayerj.[Sameasplayeriturningthepuckoverto
playerj.]
–anarcfromnodeitonodej
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