MixedMembershipMartialArts:
Data-DrivenAnalysisofWinningMartialArtsStyles
SeanR.HackettandJohnD.Storey
Lewis-SiglerInstituteforIntegrativeGenomics
PrincetonUniversity,Princeton,NJ08544,USA
Abstract
AmajoranalyticschallengeinMixedMartialArts(MMA)iscapturingthedifferencesbetween
fightersthatareessentialforbothestablishingmatchupsandfacilitatingfanunderstanding.
Here, we model ~18,000 fighters as mixtures of 10 data-defined prototypical martial arts
styles,eachwithcharacteristicwaysofwinning.Bybalancingfighter-leveldatawithbroader
trends in MMA, fighter behavior can be predicted even for inexperienced fighters. Beyond
providinganinformativesummaryofafighter'sperformance,styleisamajordeterminantof
successinMMA.Thisisreflectedbythefactthatchampionsofthesportconformtoanarrow
subsetofsuccessfulstyles.
1. Introduction
EarlyeventsinMixedMartialArts(MMA)weretoutedasachancetodeterminewhichstylesofpure
martialartsweremosteffectivefordefeatinganopponent[1,2].Theseevents,epitomizedbythe
mid-1990seventsoftheUltimateFightingChampionship(UFC)andValeTudo,featuredmatchups
ofstylisticallydiversemartialartistsrepresentingsportssuchasBoxing,JudoandKarate[1,3].As
thesportofMMAhasevolved,fightershavebecomeincreasinglywell-rounded[1,2].ModernMMA
fightersneedtobeeffectivestrikerswhodictatewherethefightoccurs,byeithertakingdowntheir
opponentsorforcingthemtofightstanding.
Each MMA fighter’s skills are drawn from a mix of multiple pure martial arts that define his/her
personalizedfightingstyle.WhilemodernMMAfightersaremorestylisticallymixedthantheirpure
martialartistforbearers,themixturesofindividualfightersdiffer.Somefighterswouldbereferred
toasstrikersandothersasgrapplerswhospecializeinmartialartssuchasBrazilianJiu-Jitsu(BJJ)
[4].Thesestylisticcharacterizationsareimportantwhenestablishingmatchups;buttodate,thereis
noquantitativeprocedureforcharacterizingthestyleofMMAfighters.Furthermore,becauseofthe
absenceofsuchanapproach,theimpactofstyleonsuccessinMMAhasneverbeenquantitatively
investigated.
Every sport is enriched by the behaviors that distinguish its athletes and teams. Whether one is
concernedwithapitcher’sarsenalofpitches,theplaysafootballteamemploysorthelocationsfrom
which a basketball player likes to shoot, style is an important (albeit often nebulous) concept in
sports.Tomakestylemoreaccessible,data-drivenmodelingapproacheswillbeinvaluable.These
models have enormous potential; they can be used to identify prospects who are similar to
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superstars,buildabalancedteamandguidevisualizationsthatinformcasualfansofeachathlete’s
strengths.Todate,approachesthathaveaimedtodefineathletes'styleshavegenerallyattemptedto
identifylatentfactorsthatcouldserveasarepresentationofstyle[5,6,7].Theseapproacheshave
beenrestrictedtospatialsummariesofathletes(e.g.,wheretheyscorepoints).Consequently,their
latentfactorsreflectcommonspatialpatterns,likethree-pointshootingordunkinginbasketball[6].
Whilelatentvariablebasedapproacheshavebeensuccessfullyappliedtosportsforwhichalarge
amountofathlete-specificperformancedataisavailable,suchmethodstendtooverfitpatternswhen
littledataisavailableorwhencomplexpatternsaresought(forexample,ifstylewereallowedto
vary between seasons or between games). Most sports are not immune to such overfitting; for
instance,thecapacityforoverfittingobserveddatahasbeenwelldemonstratedwiththedata-rich
applicationofbaseballbattingaverages[8,9].Wewouldintuitivelyexpectabatterwith300hitsout
of 1000 at-bats to perform better in the future than a batter with 5 hits out of 10 at-bats, yet the
secondbatterwouldhaveahigherbattingaverageusingtheMaximumLikelihoodEstimate(MLE)of
battingaverages.Indeed,itwasdemonstratedthattheMLEwaslessaccuratethantheJames-Stein
estimate of batting averages, which shrinks observed batting averages towards the mean of all
playersbasedontheamountofindividual-specificdata[8].Bayesianapproachesofthissorthavethe
capacitytobalanceathlete-specificdatawithoverallbehaviorinordertoimproveprediction.
CharacterizingthestylesofMMAfightersisanunprecedentedchallengebecauseofboththeamount
andtypeoffighter-leveldataavailable.Mostprofessionalathletescompeteinupwardsof100games
inasingleseason.Incontrast,asuccessfulMMAfightermayretireafteronly20bouts.Withsuch
limiteddata,afighter'sto-datebehavioraloneisinsufficienttopredicthis/herfutureperformance.
Additionally,whilethemannerbywhichafighterfinishesafight(e.g.,finishessuchaspunches,kicks
orchokes)isthepurestdepictionofhis/herstyle[3],thesemetricsarecategorical.Consequently,
methods for defining spatially-oriented styles from other sports are not directly applicable. We
addresstheabovetwochallengesbyapplyinglatentDirichletallocation(LDA)[10,11],aBayesian
mixed membership model that naturally accommodates categorical data and shares information
across fighters, thereby balancing fighter-specific behavior with broader trends. The goal of this
model is to estimate two mixtures that define a plausible underlying structure of fighters' styles
(Figure1):(i)Distinctmartialarts(prototypes)havecharacteristicfinishesand(ii)Fightersarea
mixtureofprototypes(withindividualizedweightssummingtoone).
Figure 1: Identifying
fighter styles based on
performance. We assume
thatanMMAfighter'swins
generally reflect his/her
training, either as a pure
martial artist or as a
practitioner of multiple
prototypicalstyles.Eachof
these
fighting
styles
utilizesasubsetofpossible
finishes at characteristic
frequencies.
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Using data from >250,000 fights, we applied the mixed membership martial arts (3MA) model to
identify10prototypicalmartialartsstyles,eachcomposedofasetofcharacteristicfinishes.Thestyle
of each of ~18,000 fighters was defined based on his/her individualized combination of these
prototypes.Thesestylisticmakeupsgreatlyimprovedthepredictionofheldoutresults.Beyondtheir
valueasanimplicitreadoutofafighter’sabilities,fighterstylesareamajordeterminantofsuccess
inthecage.Inparticular,winningfightsintheupperechelonsofMMArequiresdynamicstrikingand
theabilityto“gothedistance”bywinningfightsthroughjudges’decision.DefendingUFCchampions,
themostsuccessfulathletesinthesport,generallyconformtothesestyles,providinginsightsinto
howfightingstrategyaffectssuccessinMMA.
2. Dataset
UsingtheSherdogFightFinder[12],weobtainedrawsummariesof257,582previousamateurand
professionalboutsamong151,746fightersthroughOctober29,2016.Eachboutwassummarized
basedonthepairoffighters,theresultofthebout(win,loss,draw,nocontest)andthewayinwhich
the bout was finished (e.g. by punches or unanimous decision). Rare finishes were manually
combined with the most similar high-frequency finish to generate 50 distinct high-frequency
categories (shown in Figure 2). Only fights that ended in a win for one fighter were considered,
leaving230,126boutsforthisanalysis.
3. Identifyingpatternsinfinishusage
Attheirmostbasiclevel,martialartsaresetsoftechniquesthatareusedtowinfights.Eachfightis
wonbyusingasingletechnique,butifwelookatafighter'scareer,patternsmayemergewherein
setsoffinishesrecuracrossafighter'svictories.Ifthesepatternsareduetoamartialartthatisshared
amongfighters,wewouldexpectsetsoffinishestoco-occurnotonlyinasinglefighterbutalsoina
recurring manner across many individuals. For example, Muay Thai fighters use both elbows and
knees,soweexpectthatfighterswhowinwithelbowstendtowinbyknees.Thus,elbowsandknees
shouldco-occur.
To determine whether any subsets of the 50 finishes are frequently used in conjunction, we
quantifiedhowofteneachpairoffinishesco-occurs(i.e.,isusedtowinfightsbythesamefighter),
aggregatingacrossallfighters.Toanswerthisquestion,wecomparedobservedco-occurrencesof
eachfinishpair(numberofpairsofafinishxandyinafighter'srecord,summedoverfighters)with
the expectation under independence. Then, using permutation testing, we determined how often
such large deviations were expected. Out of 1,225 pairs of finishes, 289 pairs co-occur more
frequentlythanexpected,and292pairsco-occurlessfrequentlyatafalsediscoveryrateof0.05[13].
Pairs of finishes that significantly co-occur are far from random; rather, they are organized into
cliques of generally mutually correlated finishes (Figure 2). For example, kicks/knees form one
tightlyconnectedcliqueandlegsubmissions,another.Fromthisanalysis,wealsoseethatpatterns
aremorestronglyinfluencedbythepositionfromwhichasubmissionisapplied,ratherthanbywhat
typeofsubmissionisapplied,perse.Forexample,theanacondachoke,appliedfromafrontheadlock
position,isindependentfromthetrianglechoke,whichisappliedfromguard.Incontrast,triangle
chokesandarmbarsarefrequentlyappliedfromguardandthustendtoco-occur.
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Figure 2: Structure of
MMA finishes based on
pairwiseco-occurrence.
289 pairs of finishes
significantly co-occurred
amongfighters.Thethree
strongest connections to
each finish (dark edges)
were used to construct a
layout; other significant
edges are shown as faint
backgroundlines.
4. Determiningfighter-specificstyles
By aggregating data over all fighters, we demonstrated that finishes exist in co-occurring groups.
Directly applying this information at the fighter-level is challenging, however. Although we
collectivelyhavealargeamountofinformationaboutfighters’performances,wehaveamoremodest
amountoffighter-specificdata.Nofighterhaswonafightwitheachofthe50possiblefinishes.In
fact,thevastmajorityofexperiencedfightersdonotevenhave50winstodate.Becauseweonlyhave
anincompletereadoutofafighter’scapabilities,weneedtobalancewhatweknowaboutafighter
with what we do not, hedging the uncertainty in fighter-specific data by using broader patterns
acrossfighters.Specifically,weusethebehaviorofsimilarfighterstodefinecommonfinishpatterns.
Wethencharacterizeindividualfightersinthecontextofhowfrequentlytheyhavefoughtandthe
classesoffinishestheyhaveemployed.
4.1. Themixedmembershipmartialarts(3MA)modeloffighterstyles
Increatingamodeloffighterstyles,ourultimategoalistodeterminethefinishprobabilitiesforeach
fighter. Statistically, this objective amounts to estimating fighter-specific finish probabilities:
Pr(Finishj|Fighteri)foreachfinishj=1,2,…,50andeachfighteri=1,2,…,17,778includedinour
dataset.(Werestrictedouranalysistofighterswithfourormorewinstodate.)Ourdataiscollected
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intoanI=17,778byJ=50matrixX,wherewenotethatthe(i,j)entryisthetotalnumberofwins
forfighterioffinishtypej.Wewilldenotewij=Pr(Finishj|Fighteri),whicharethevalueswewishto
estimate,andwecollectthesevaluesintotheIxJmatrixW.
ThemostobviousestimateofthewijvaluesisthemultinomialMLE:
𝑥"#
*
𝑥
# + ,- "#)
MLE
𝜔"#
=
However,whenusedonaper-fighterbasis,theMLEissusceptibletooverfittingtheobserveddata
(particularly for fighters with a smaller number of observations), and the MLE estimates that
unobserved finishes have zero probability. It also does not utilize information shared across the
manyfighters’observeddata.
Instead,inlinewithotherlatentvariablebasedapproaches(suchasprincipalcomponentanalysis
and factor analysis) [14], we assume a lower dimensional structure to W. By doing so, we are
assumingthatfighters’individualizedstylescanbeattributedtoasetofprototypicalwinningstyles
whichwillbesmallerthanthenumberofJ=50finishtypes.Wewilldenotethenumberofprototypes
by K, where K < J, and we will estimate the value of K from the data. We assume the following
decompositionofW:
Ω/×* = Θ/×3 Φ3×* 3
𝜔"# =
(1)
𝜃"6 𝜙6# 6,-
where 3
6,- 𝜃"6 = 1foralliandwhere fkjÎ[0,1]forallkandj.Theinterpretationofthismodelis
thatthe“prototypes”arecapturedby fkj=Pr(Finishj|Prototypek).Theseprototypesaremappedto
each individual fighter i through the values qik where qik = Pr(Prototypek|Fighteri), where 𝜔"# =
3
6,- 𝜃"6 𝜙6# . For fighter i, the vector (qi1, qi2, …, qiK) yields his/her mixture of prototypes, i.e., the
fighter’smixtureofmartialartswinningstyles.Thismodeldecompositionisoftenreferredtoasa
“mixedmembershipmodel”[10]inthateachindividualiscomposedofanindividual-specificmixture
ofmembershipstodifferentclasses.Here,thedifferentclassesaretheprototypes.
ThisapproachresultsinthegroupingofJ=50finishesintoasmallernumberofprototypesK,and
thus provides increased generalizability when predicting previously unobserved finishes. For
example,submissionsthattargetthelegs(e.g.,kneebarandanklelock,asshowninFigure2)should
be associated with the same prototype. Because of this, a fighter with only kneebar submissions
wouldhavesomeweightinthis“leg-submission”prototype.Therefore,wewouldpredictthathe/she
alsohasthecapacitytoapplyanklelocks.Onelimitationofthisapproach,however,isthatifafighter
hadneverusedlegsubmissions,he/shewouldhavezeroweightinthe“leg-submission”prototype.
Asaresult,wewouldpredictthatthereisprobabilityzeroforhim/hertoperformanylegsubmission.
Inordertoallowforfinishescontainedinunrepresentedprototypesandtoalsolimittheinfluence
ofunrepresentativepatternsoffinishes,wecanplaceaprioronmixturecomponents,inlinewiththe
earlierexampleofapplyingshrinkagetobattingaveragesinbaseball[8].LDAisapopularBayesian
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modelingapproachforincorporatingsuchapriorintothecategoricalmatrixfactorizationshownin
model(1),andfurthermore,forfittingthismodel(1)todatathroughatechniquecalledvariational
inference[10].InLDA,Dirichletpriorswithparameters a=(a1, a2,…, aK)and b=(b1, b2,…, bJ)are
placedonQandF,respectively.LDAthenutilizesvariationalinferencetoestimatethemaximuma
posteriori probability (MAP) of the posterior distributions of Q and F, Pr(Q|X) and Pr(F|X). The
estimates𝜃"6 and𝜙6# aresetastheseMAPestimates,andwefinallyform𝜔"# = 3
6,- 𝜃"6 𝜙6# .
ThevaluesofbdonotgreatlyaffectinferenceonFbecauseFisarelativelysmallmatrix(KxJ)that
isestimatedbypoolingdataacrossallfighters[15].Incontrast,choiceofagreatlyimpactsestimation
of Q [15]. Each component of a, denoted by ak, can be interpreted as the number of effective
observations (pseudocounts) [16] of a prototype k that supplements each fighter's history. The
overall magnitude of a, which we denote by 𝛼 S = 𝛼6 , reflects the degree to which fighters’
prototypes will be determined by their own records (low aS) versus the average of prototype
frequencies across all fighters (high aS). Treating a as asymmetric (meaning the ak values across
prototypes may differ) is another important consideration because this allows larger and smaller
prototypestoexistratherthanarbitrarilyenforcingthatprototypesbeequallyprevalent.
Toapproximateoptimalvaluesof aSandK(sinceboththedistributionof aandthevaluesof bcan
beadaptivelyestimated),possiblecombinationsofaSandKwerecomparedtodeterminewhichset
generatedanestimateofWthatbestpredictedheldoutfinishes.Quantitatively,foreachpairof aS
andKvalues,weused20-foldcrossvalidationtoestimateWforeachof20subsetsofthefulldataset
using the software MALLET [17]. We then calculated the log-likelihood of observations that were
heldoutfromeachdataset:
IJ
𝒮<
log 𝐿 =
𝑥"# log 𝜔"# (2)
K,- "∈𝒮< #∈𝒮<
𝒮<
where 𝒮 faretheindicesofthetestsetforfoldfand𝜔"# istheestimatebasedonthetrainingsetof
foldf.
ParametersetswithhighKgenerallyresultedinheldoutlog(L)similarinmagnitudetothosewith
moderate K, but high K models were less interpretable. Accordingly, we adopted a parsimonious
approach to choosing a set of parameter values (small K, small aS) among those with good
performance (³ 95% of the difference between maxS (log(𝐿)) and a multinomial likelihood model
3,A
containingnofighter-specificinformation).ThisparsimoniousapproachresultedinsettingK=10
prototypes(twopairsofprototypeswerecombinedbecausetheycontainedthesamemajorityfinish)
withaS=16.
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4.2.Performanceandgeneralityof3MA:winsvs.lossesandamateursvs.theelite
To evaluate the performance of 3MA and assess its broader applicability, we applied the above
approach both to elite subsets of fighters and to datasets constructed from fighters' losses rather
thantheirwins(Table1).
Dataset
Type
InclusionCriteria
Nfighters
Nbouts
K
a S
AllData
allfighters
wins
4+wins
SubsetsofData
topfighters
wins
4+wins,8+fights,1+fightsin"toppromotions"
topvs.top
allfighters
wins
losses
4+wins,8+fightsagainsttopfighters
4+losses
topfighters
losses
4+losses,8+fights,1+fightsin"toppromotions"
topvs.top
losses
4+losses,8+fightsagainsttopfighters
17,778
142,775
10
16
2,973
40,244
11
14
408
14,690
3,673
97,569
14
4
22
4
2,201
18,043
4
10
485
3,824
3
8
Table 1: 3MA was applied to six datasets constructed based on the fighters included and
whetherwinsorlosseswereinvestigated.Weconsidered“toppromotions”tobeUFC,Bellator,
WorldSeriesofFighting,PRIDE,DREAMandStrikeforce.
Our estimates of K and aS are
similarregardlessofwhetherall
fightersoronlyelitefightersare
investigated. This suggests that
regardless of the tier of
competition, similar move sets
are employed, and accordingly
amateur fighters may be
important to better understand
the elite. To quantitatively
evaluate this assertion, for each
of the six datasets described in
Table1,predictionaccuracywas Figure3:Performanceof3MAmodelrelativetomultinomial
calculated
for
both modelwhereallfightersareassumedtousethesamestyle
inexperienced and experienced (1prototype).
fighters(Figure3).
Inallcases,3MAgreatlyoutperformsamultinomial,1prototypemodel(wherefinishprobabilities
foreachfighterareproportionaltofinishfrequenciesacrossallfighters).Ingeneral,itiseasierto
predicthowafighterwinsthanhowhe/sheloses.(Losingissomethingoverwhichafighterhasmuch
lesscontrol.)Additionally,thefinishesofexperiencedfighters(thosewith>20wins/losses)canbe
moreaccuratelypredictedwhenamateurfightersareincludedratherthanusingasmallerdataset
composed exclusively of elite fighters. This suggests that patterns of MMA prototypes are nearly
universal;thus,informationfromamateurscanimprovetheaccuracyofwithwhichwedefinethe
stylesofchampions.Becausethecompletedatasetresultsinthebestpredictionoffinishaccuracy
regardlessoffighterexperience,subsequentanalyseswillutilizeall~18,000fighters.
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WefoundK=3orK=4forlossescomparedtoK=10forwins,indicatingthatfighterstendtolose
inmoregenericwaysthanhowtheywin.Losingprototypesaregroupedintothelargecategoriesof
strikes, submissions, and decisions. Because a fighter's weaknesses can be faithfully read from
his/herrecordwhilewinningprototypesareconstructedfromnuancedcombinationsoffinishes,we
focusourremainingattentiononwinningstylesderivedfromanalyzingallfighters.Thesestylesare
reflected in the MALLET software estimates of both the fighter-specific prototypes (Θ) and
prototype-specificfinishes(Φ).
4.3.Prototypesasmixturesoffinishes
By studying the estimated values of fkj = Pr(Finishj|Prototypek), we can characterize which finish
types make up each prototype. We can also calculate the marginal probability Pr(Prototypek) =
/
",- 𝜃"6 /𝐼 as a summary of each prototype's overall relevance in the data. Summaries of
Pr(Prototypek)andPr(Finishj|Prototypek)asestimatedfromthedatacanbefoundinTable2.
Prototype
Description
Pr(Prototype)
P1
Punches(TKO)/Punches(KO)
0.22
P2
UnanimousDecision
0.18
P3
Rear-NakedChoke
0.11
P4
UnanimousDecision/SplitDecision
0.09
P5
Armbar/LegSub
0.09
P6
Kicks/Knees/Punches(TKO)
0.08
P7
TriangleChoke/Armbar
0.07
P8
GuillotineChoke/Rear-NakedChoke
0.07
P9
Rear-NakedChoke/Kimura
0.06
P10
Punches(Submission)/Choke
0.05
FinishBreakdown,Pr(Finish|Prototype)
76%Punches(TKO),20%Punches(KO),3%
Punches(Submission)
82%UnanimousDecision,8%SplitDecision,7%
MajorityDecision
100%Rear-NakedChoke
48%UnanimousDecision,26%SplitDecision,
16%Punches(TKO),11%Other
72%Armbar,11%HeelHook,5%Kneebar,12%
Other
27%Punches(TKO),17%Punches(KO),14%
Knees,42%Other
56%TriangleChoke,38%Armbar,3%RearNakedChoke
84%GuillotineChoke,15%Rear-NakedChoke
33%Rear-NakedChoke,25%Kimura,25%ArmTriangleChoke,17%Other
35%Punches(Submission),19%Choke,13%
Strikes,33%Other
Table 2: Summary of LDA-derived MMA prototypes. Pr(Prototype) is given by the overall
estimatedfrequencyofeachprototypeinfighters( /",- 𝜃"6 /𝐼).Pr(Finish|Prototype)isdetermined
fromthevaluesinΦ.Thetopfinishesforeachprototypeareshown.
Theprototypecharacterizationbasedon3MAreproducesmuchofthestructurefoundinFigure2
butbreaksthisstructureintodiscreteprototypes(Figure4).Thesediscreteprototypesappearto
capturemeaningfulstylesusedbyMMAfighters:
•
•
ThemajorstrikingprototypesareP1andP6.P1containsTKOandKOwinsbypunches,andthus
representsboxing.P6containspunchesaswellaselbows,kicksandotherdiversestrikesthat
areheavilyfeaturedinMuayThai,Karate,TaeKwonDoandkickboxing.
PrototypesP2andP4containwinsbyjudges’decision.P2primarilyreflectsunanimousdecision,
i.e.casesinwhichafighterdominatedhis/heropponentforthelengthofthefightandalljudges
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•
•
agreedthathe/shewon.P4hassomeweightinunanimousdecisionbutalsostronglyrepresents
splitdecisions:fightsinwhichjudgesdisagreedaboutwhowon.
BJJandsubmissiongrapplingarerepresentedbyfivegenerallysmallprobabilityprototypes.P5
primarily contains the armbar and submissions that target the legs. P7 are submissions from
guard. P9 has some contribution from the rear-naked choke but is primarily composed of
submissionsappliedfromsidecontrolandfrontheadlock.P8andP3eachlargelycontainasingle
submission:theguillotinechokeandrear-nakedchoke,respectively.Thesesubmissions,while
traditionallyapartofBJJ,areknownandpracticedbymostMMAfighters.
Thefinalandsmallestprototype,P10,isahodgepodgeofunusualstoppages,generictermsand
easilydefendedsubmissionsthatwererelativelycommoninMMAcirca2000buthavedecreased
infrequencybyover10-foldsince.Thus,P10isaprototypethatdatesafightertotheformative
yearsofMMA.
This analysis does not
includewrestling,judoand
othermartialartsthatare
specialized in achieving
takedowns. Because these
martial arts are not
directly tied to MMA
finishes, wins by their
practitioners
could
manifest in many ways,
such as through decision
(P2) or submissions from
strong control positions
(P3,P8orP9).
Figure4:3MAidentifiessets
offinishesusedtogetherby
fighters.Finishesfromthe
networkfoundinFigure2are
coloredaccordingto
Pr(Prototype|Finish)where
viaBayes’theoremwehave:
Pr(Prototype|Finish)=
Pr(Prototype)´
Pr(Finish|Prototype).
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4.4.Fightersasmixturesofprototypes
Prototypesexhibitedabroadrangeofaverage
frequencies, ranging from 5% to 22%.
Individual fighters vary greatly about these
averages,
reflecting
either
fighter
specializationinorneglectofagivenprototype
anditsassociatedfinishes(Figure5).Because
all pairs of prototype loadings are negatively
correlated,themixturesofprototypesusedby
fightersarerelativelyindependent;theweight
in any one prototype does not necessarily
entail higher weight in another prototype.
While this trend would be partially expected
duetotradeoffsbetweenmixturecomponents,
the absence of strong associations between
prototypes (as could be dealt with using Figure5:Violinplotshowingthedistributions
correlatedtopicmodels[18])suggeststhatthe offighterprototypeabundances.
structure of finishes is appropriately
discretizedusingLDA.
5. Notallprototypesarecreatedequal:winningstylesinMMA
MMAfightersemployavarietyofapproachestodefeattheiropponents,butitisunclearwhetherall
stylesareequallyviableorifsomehavedisproportionatesuccessinthecage.Thesuccessofastyle
mayalsobecontextual.MMAfansfrequentlytalkaboutanopponentwhoisa“goodmatchup”fora
fighter,reflectingthatafighter’soddsofwinningmightbeinfluencedbybothhis/herownstyleas
well as an opponent’s style. By creating a quantitative representation of style, we systematically
assess whether some styles are superior to others and whether this success depends upon an
opponent’sstyleaswell.
5.1.Theroleofmatchups:prototypesformastrictdominancehierarchy
IfsomeprototypesandtheirassociatedfinishesaremoreimportantforwinningMMAfightsthan
others, we would expect that fighters who favor these finishes would win more frequently. The
relativestrengthofaprototypemayalsodependonanopponent'sstyle.Insuchacase,therelative
strengthofaprototypemaybecontextual,resultinginintransitivepropertieslikethoseseeninrockpaper-scissors.
Toinvestigatehowtheprototypesperformagainsteachother,wecanconsiderwhichprototypes
tend to win when all pairs of prototypes are matched against each other. Since every fighter is
representedbyaprototypemixtureratherthanbyapureprototype,wecannotdirectlycountwhich
prototypewinsduringmatchups.Instead,weconsiderthesetsofboutsℬ={(b1,b2)}whereb1isthe
winningfighterandb2isthelosingfighter(sob1,b2Î{1,2,…,I}),andcharacterizetheirprototype
mixturesforeachpairofprototypesm(inwinningfighters)andn(inlosingfighters)wherem,nÎ
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{1,2,…,K}andm¹n.Totheextentthat𝜃OP Q and𝜃OR S arelarge,prototypemhasdefactodefeated
prototypen.Moregenerally,weweighteachmatchbasedon𝜃OP Q 𝜃OR S ,reflectingtheextenttowhich
prototypemhasdefeatedprototypen.Thenthescoreforwinnermandlosern(smn)canbecompared
tothescoreforwinnernandloserm(snm)toseeifoneprototypesystematicallydominatesanother:
𝑠QS =
𝜃OP Q 𝜃OR S OP ,OR ∈ℬ
𝑑QS =
𝑠QS − 𝑠SQ
P
R
WXY ZWYX
(3)
To determine whether these dominance scores (dmn) were significant, the observed value of each
score was compared to 10,000 null values where winner and loser labels were permuted, and
significantdominator-dominatedprototypepairswerefoundatafalsediscoveryrateof0.05[13].
Visualizingthedominancesbetweenallpairsofprototypes(Figure6)suggeststhatintransitivities
are a relatively minor phenomenon. Instead, prototypes occupy five tiers of dominance, wherein
prototypes assigned to higher tiers dominate all prototypes in tiers beneath them. The top, most
successful,tierofprototypescontainsonlyP2(unanimousdecision),whiletheleastsuccessfultier
includesP5(armbarandlegsubmissions),P8(guillotinechoke)andP10(circa2000submissions).
Figure6:Prototypesform
tiersofeffectiveness.
Networkedgesthatindicate
asignificantdominance
scorelinkpairsof
prototypes,pointingfroma
dominatedprototype(light)
toitsdominator(dark).
Groupsofprototypeswith
mutuallyweakor
insignificantdominance
scoresweregroupedinto
tiers,andtierswere
arrangedfromthebest
performing(1)tothe
weakestperforming
prototypes(5).
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5.2.Effectofprototypemixturesonoverallsuccess
Becauseprototypesareeithereffectiveorineffectivewithoutregardtoanopponent’sstrategy,each
fighter'sperformancecanbeexplainedbasedonhis/herprototypemixture,settingasideattributes
of his/her opponent. To investigate prototype performance from the perspective of individual
fighters,wesoughttoevaluatethedegreetowhichfighterwinpercentagescouldbeexplainedby
prototype frequencies, i.e., Pr(win|𝜃). One challenge with this strategy is that, by construction,
fighterswithextremevaluesofanyprototypecomponentareveterans;theyhavewonenoughfights
thattheirobservedfinisheshaveoverwhelmedtheinfluenceoftheprior.Bycontrast,fighterswhose
prototypeproportionsarenearthepriorproportionsgenerallyhaveveryfewfights.Therefore,their
performanceismorerepresentativeofan“averagefighter.”Becauseprototypevaluesarepartially
conflatedwithexperience,weneedtoteaseoutthesetwoeffectsinordertodeterminethedirect
contributionoffighterstylesonperformance.
In order to separate the influence of experience and fighter styles (Θ), using generalized additive
models (GAMs) with a logistic link function, the win percentage of each fighter i, Pr(win)i, was
modeledbasedonthenumberoffightswon(binnedintointervalswhichincludeasimilarnumber
of fighters) and the 10 estimated prototype mixture components (log 𝜃" ). Logistic regression was
usedbecausePr(win)canbetreatedasBinomialsuccesses(wins)andfailures(losses);accordingly,
inferenceisbestperformedonthelog-oddsofwinsversuslosses[19,20].Theuseofgeneralized
additive models provides flexibility, allowing for the identification of nonlinear relationships
betweeneachexplanatoryvariableandthelog-oddsofwinning[21].Fightswerebinnedandmixture
components were log-transformed in order to approximate Normal distributed values so that
predictionwasnotundulyinfluencedbyaminorityoffighters.
Fromthefittedmodel(Figure7A),wecanseethattheoddsofwinningincreasesmonotonicallywith
thenumberofwins,asexpected.Bycontrast,theinfluencesofprototypeabundancesontheoddsof
winning,thoughhighlysignificant(p<10-200;ANOVAversusamodelexcludingprototypes)aremore
complicated. In Figure 6, we noted that prototypes form tiers of dominance. Here, we see that a
fighter'ssuccesswithaprototypegreatlydependsonthedegreetowhichthefighterhasinvestedin
it. Based on dominance, P2 is the most important prototype for success overall, but its benefits
actually peak once a fighter possesses 25% of this style and its effectiveness decreases at higher
values.AsidefromP2,athighvaluesofotherprototypes,therankingofprototypeeffectsontheodds
ofwinningissimilartothetiersofdominance(Figure6).PrototypesP5,P8andP10stillnegatively
impacttheoddsofwinning,whiletheremainingprototypesareallgenerallypositive.Thesepositive
prototypeseitherhavestrongeffectsbutimpactfewfighters(e.g.,extremevaluesofP4,P6andP9)
orrelativelyweakeffectsthatimpactmanyfighters(e.g.,intermediatevaluesofP2andhighvalues
ofP1).Toprovideasenseofthemagnitudeoftheseeffects,consideringstylealone,a50%:50%P6:P9
fighteranda50%:50%P5:P7fighterwouldtranslatetoanexpectedwinpercentageof60%and43%,
respectively.
Together, experience and prototypes strongly predict fighters’ true win frequencies; but, the
predictionisstillquiteremarkablewiththeroleofexperienceremoved(generatingamodelwith
experienceandthencorrectingforthisinfluenceinthefit)(Figure7B).
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Figure7:Predicting
winsbasedon
fighters’experience
andprototypes.A)
Theimpactof
prototypemagnitude
andnumberoffights
onfighters’oddsof
winning(versus
losing)wasestimated
usingageneralized
additivemodel.B)The
fittedvaluesof
fighters’Pr(win)is
comparedtofighters’
truewinpercentage.
Theimpactofnumber
ofwinswas
subtractedfrom
fighters’Pr(win)to
generatean
experience-corrected
estimatebasedon
prototypesonly.
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Becauseprototypespecializationisanimportantpredictorofhowfrequentlyeachfighterwins,an
interesting question is whether the most successful MMA fighters, UFC champions who have
successfully defended their title at least once, utilize these effective prototypes. Looking at the
predicted win frequencies of fighters (corrected for experience), defending UFC champions favor
effectiveprototypemixtures(Figure8A).Visualizingthemostsuccessfulofthesechampions(those
withthemosttitledefenses),strongdynamicstriking(representedbyP6andtoalesserextentP1)
andtheabilityto“gothedistance”bywinningthroughdecision(P2)arethemajorattributesthat
distinguish the best of the best (Figure8B). These trends suggest that being a great fighter is not
sufficienttobecomeachampion;fightersmustalsoembraceastylethatpotentiatestheirsuccess.
Figure8:UFCchampions
favorthemosteffective
prototypes.A)
Distributionoffittedwin
probabilitiesforall
fightersbasedontheir
estimatedstyle,correcting
forexperience.Defending
UFCchampions(alllisted
names)heavilyfavor
successfulprototypes.B)
Prototypemixturesofthe
fivechampionswiththe
greatestnumberoftitle
defenses.
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5.3.SheddinglightonsuccessinMMA
Applyingunsupervisedapproaches,wedefinedstyleinMMAandfoundthatdifferentelementsof
MMAfighters'gameshaverealizedvaryingsuccessinthecage.Lessonsemergefromthisanalysis.
Successfulstrategies:settingthepaceandemployingdynamicstriking
• Beingabletowinbydecisionisacrucialcomponentofeveryfighter'sgame.Winningbydecision
entailsnotonlyhavingsufficientcardiotofightfor15+minutesbutalsodoingsowithenoughof
an advantage to be declared the victor. Thus, winning by decision may be an indicator of the
overallfitnessandcaliberofafighter.Wealsonotethatfighterswhooverspecializeinsecuring
decisionvictoriesbecomelesseffective,thussecuringdecisionvictoriesshouldonlybeapartof
afighter'sstrategyratherthanhis/herprimarygoal.
• Winningbyusingkicks,elbowsandkneesisagreaterindicatorofsuccessthanvictorybyusing
punches,eventhoughpunchesarethemostfrequentwayoffinishingafight.ThisMuayThaistyle
couldbeadvantageousforseveralreasons.Kicks,elbowsandkneesareimportantforcontrolling
distance;theyaregenerallyacknowledgedasbeingmorepowerfulthanpunches,andtheymay
alsobeanindicatorofatop-levelstrikerwhowouldbedangerouswithpunches.
• The most successful submissions are the kimura, rear-naked choke, and other chokes applied
withthearms(excludingtheguillotine).Acommonfeatureofthesesuccessfulsubmissionsis
thattheyareappliedfromapositionofverystrongcontrol(back,mountorsidecontrol).Because
ofthis,ifanopponentescapesfromoneofthesesubmissions,thesubmittingfighterwillstillbe
leftinastrongposition.
Unsuccessfulstrategies:sacrificingpositionforsubmission
• Fighterswhospecializeinguillotinechokesorleglocksarelesssuccessfuloverall.Acommon
featureoftheseclassesofsubmissionsisthatshouldtheapplicationoftheseattacksfail,they
generallyleavetheattackerinaninferiorposition.
• Guillotine chokes are generally applied by pulling guard (or being forced down during a
takedown).Assuch,shouldtheattackfail,anopponentisinagenerallyadvantageouspositionto
passguardandsecureastrongcontrolposition.
• Leglocksarealsonotappliedfromthetraditionallydominantgrapplingpositions.Accordingly,
whentheyfail,anopponentoftenhasenoughspacetoeithersecureagoodpositionortorevert
toastandingposition.
• ThemostpoisonousstylewasanensembleoffinishesthatwerecommoninMMAaroundthe
year 2000, but have since decreased greatly in prevalence. These finishes could be
disadvantageous, but more likely, they simply date a fighter. If fighters have won by these
finishes,itislikelythattheyarepastthemostcompetitiveperiodoftheircareersandthatthisis
reflectedintheirrecords.
WhilemostmodernUFCchampionsarestrikersratherthangrapplersandwepointtosubsetsofthe
BJJ game that are ill-suited for MMA (the guillotine choke, leg locks, and to a lesser extent guard
attacks),wefindthatfighterswhospecializeinsubmissionsthatareappliedwithstrongpositional
controlcanbesuccessfulinMMA.Indeed,ratherthancommittingtoattacksthatsacrificeposition,
the most successful BJJ practitioners in MMA (like Ronaldo “Jacare” Souza, Rafael Dos Anjos and
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Demian Maia) blend BJJ with striking by challenging (or outright defeating) their opponents with
strikingbeforemovingtotheground.Then,whenontheground,thesefightersapplychokesfrom
strongpositions.
Whilethemannerbywhichafighterwinsisthepurestreflectionofhis/herindividualizedstyle,and
thissummarycanbeobtainedforamassivenumberoffighters,afighter'sstylecontributesnotonly
to how he/she wins a bout but also to how he/she performs throughout the fight. Such data is
increasinglycaptured,atleastfortopcompetitors,throughanalysisofvideofootagebyorganizations
suchasFightMetric[22].Thesedatacouldimprovetheaccuracywithwhichstyleisdefinedfortop
fightersaswellascaptureotherelementsofafighter'sstyle,suchaswrestlingability,thatdonot
directlyleadtocharacteristicfinishes.
6. Broaderapplicationsinsportsanalytics
Acommonelementoflatentvariablebasedapproachesthathavesoughttodefineathletes’styleshas
beentofindtrendsacrossathletesthatsummarizeeachathlete'sperformance.Onelimitationofsuch
methodsisthattheyaredirectlyfittothedata,andthusdonotaccountforstylisticcomponentsthat
have yet to manifest (or have, by chance, only been weakly demonstrated) in an athlete's
performance.WhenapplyingprinciplesinstylisticinferencetoMMA,asportthatisanextremecase
oflittleathlete-specificdata,itiscrucialtosharepoweracrossfightersusinginferencegroundedin
Bayesianmodeling.Whileothersportsarelikelymoreinsulatedfromsuchpathologiesbyvirtueof
possessingmoreper-athletedata,evenwhenalargeamountofdataisavailable,directlyfittingto
per-athleteobserveddatamightnotmostaccuratelyestimatethetruepropertiesoftheathlete(as
hasbeendemonstratedforbaseballbattingaverages).
By harnessing latent variable based methods, we characterized MMA fighters' styles based on
prototypesthatnaturallygroupfinishes.Ouranalysiscanbeappliedtoanycategoricalmeasureof
performance, including to other martial arts where fighters score points or win by using an
expressivesetofpossiblemoves(e.g.,BJJ,TaeKwonDo,Sumo,JudoandFencing).Ineachofthese
sports, the 3MA approach could help to systematize the styles of competitors and reveal factors
governingsuccess.
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