tab program description

Thispaperisintendedasatechnicalexplanation,includingpossibleformulas,fordesigninga Thispaperisintendedasatechnicalexplanation,includingpossibleformulas,fordesigninga newdebatetabulationprogrambasedonamatrixoptimizationalgorithm.Ihavespentalotof newdebatetabulationprogrambasedonamatrixoptimizationalgorithm.Ihavespentalotof timeexperimentingw

timeexperimentingwithdifferentformulasaithdifferentformulasandhavearrivedattheondhavearrivedattheonesIthinkworkbnesIthinkworkbest.Iam,est.Iam, howe

however,notacomputerver,notacomputerprogrammprogrammer.Ifanyonewanter.Ifanyonewantstostodeveldevelopaopaprogprogrambasedrambasedonon thesethese ideas,pleasedosoand

ideas,pleasedosoandsharewiththecomsharewiththecommunity.munity.

Section1:HowaMatrixOptimizationWorks Section1:HowaMatrixOptimizationWorks

Thekeyideathatmustbeundersto

Thekeyideathatmustbeunderstoodisodisthatcurrethatcurrentalgoritntalgorithmsforpairinghmsforpairing,whetherdoneby,whetherdoneby handonnotecardsorbyacom

handonnotecardsorbyacomputerprogram,aresequentiaputerprogram,aresequentialnotsimultaneous.Forexamlnotsimultaneous.Forexample,inple,in onecurren

onecurrentversionoftversionof apower-maapower-matchintchingalgorithmgalgorithm,debaters,debaters arerankedareranked fromfrom toptobottom,toptobottom, thenthedebatersarematchedtogethertwo-by-two:firstplacedebatessecond,thirddebates thenthedebatersarematchedtogethertwo-by-two:firstplacedebatessecond,thirddebates fou

fourthrth,, andand soon.Ifsoon.If amatcamatchh isprohisprohibiibitedted bytourbytournamnamentent rulrules—es— forfor exaexampmple,le, thethe firfirstandstand secondplacedebatersarefromthesameschool—thenthenextdebaterispicked,andfirst secondplacedebatersarefromthesameschool—thenthenextdebaterispicked,andfirst placedebatesthird. placedebatesthird. Everycurrentalgorithmusessomekindofsequentialprocess.Thelimitationisthatonlyone Everycurrentalgorithmusessomekindofsequentialprocess.Thelimitationisthatonlyone var

variabiablele cancan bebe conconsidsidereered,d, eveevenn ifif thathatt oneone varvariabiablele isis oftoftenen comcomposposeded ofof sevseveraerall difdifferferentent mea

measursures.es. ForFor exaexampmple,le, aa debdebateater’sr’s plaplacece inin thethe toutournarnamenmentt isis aa varvariabiable,le, whwhichich migmightht bebe composedofthemeasureswin-lossrecord,speakerpoints,opponentwins,etc.Itisimpossible composedofthemeasureswin-lossrecord,speakerpoints,opponentwins,etc.Itisimpossible forasequentialalgorithmtobalancetwovariablesatonce,suchasgeographyandskilllevel. forasequentialalgorithmtobalancetwovariablesatonce,suchasgeographyandskilllevel. Al

Alll aa seseququenentitialal alalgogoririththmm cacann dodo isis ococcacasisiononalallyly ststririkeke aa popotetentntiaiall mamatctchh foforr viviololatatiningg ononee cri

criterterionion oror anoanothether,r, e.ge.g.. elimeliminainatintingg aa mamatchtch becbecauausese thethe twotwo debdebateatersrs areare frofromm thethe samsamee state.

state.

Asimultaneousalgorithmcanoptimizeseveralvariablesatonce.Thebasicoutlineofsuchan Asimultaneousalgorithmcanoptimizeseveralvariablesatonce.Thebasicoutlineofsuchan alg

algoriorithmthm issimpissimpletoleto desdescricribe:be: stestep1p1 istoisto assassignign apointapoint valvalue,ue, orscororscore,basee,basedondon asmanasmanyy varia

variablesasdesiredblesasdesired,to,to everyeverypossiblematcpossiblematch;step2h;step2istopicktheistopickthe pairipairingthathasthehighestngthathasthehighest averagepointvaluepermatch.Step2isawell-knownandsolvedproblemincomputerscience averagepointvaluepermatch.Step2isawell-knownandsolvedproblemincomputerscience (se

(seee HunHungargarianian algalgoriorithmthm).There).Thereforfore,e, thethe onlonlyy proprobleblemm toto solsolveve isis howhow toto assassignign aa scoscorere toto everymatch.

everymatch.

Step1generatesamatrix,listingdebatersbyrowandbycolumn.Inaneven,side-constrained Step1generatesamatrix,listingdebatersbyrowandbycolumn.Inaneven,side-constrained roun

round,theteamsd,theteamsdueAffirmadueAffirmativecoultivecouldbelistedineachrow,anddbelistedineachrow,andtheteamsdueNegatheteamsdueNegativearetiveare listedbycolumn.Inanoddround,theteamscanberandomlyassignedtoeitherhalf,orthe listedbycolumn.Inanoddround,theteamscanberandomlyassignedtoeitherhalf,orthe programcouldassignteamsfromthesameschooltothesameside,sincetheyareblockedand programcouldassignteamsfromthesameschooltothesameside,sincetheyareblockedand cannotdebateeachother.Usingvariablesstoredforeachteam,theprogramcanpopulatethe cannotdebateeachother.Usingvariablesstoredforeachteam,theprogramcanpopulatethe

matrixwithdifferentscoresforeachpossiblematch.Inthecasewheretheteamsareblocked, matrixwithdifferentscoresforeachpossiblematch.Inthecasewheretheteamsareblocked, thescoreshouldbeset

thescoreshouldbesetto0.Instep2,thepto0.Instep2,theprogrampicksourogrampicksouttheoptimalsetofmttheoptimalsetofmatches.atches.

Section2:ParticularFormulasforParticularUses Section2:ParticularFormulasforParticularUses Formula1:StrengthofSchedule Formula1:StrengthofSchedule Theproblemwithahigh-highorhigh-lowpairingmethodisthatthestrengthofschedulecan Theproblemwithahigh-highorhigh-lowpairingmethodisthatthestrengthofschedulecan stillvarywidely,evenwithinabracket.Asevidence,lookattheopponentwinscolumninthe stillvarywidely,evenwithinabracket.Asevidence,lookattheopponentwinscolumninthe final

finalresultsofaresultsofa tourtournamenament;withint;withinthesamebrackenthesamebracket,say4-2s,oppont,say4-2s,opponentwinscanvaryentwinscanvaryfromfrom thelowteenstothehightwenties.Theproblemisthatthemethodmatchesteamsbyonlyone thelowteenstothehightwenties.Theproblemisthatthemethodmatchesteamsbyonlyone variable–strength(record,points)–andcannotconsiderasecondvariable–schedulestrength variable–strength(record,points)–andcannotconsiderasecondvariable–schedulestrength  –atthesamet  –atthesametime.ime. Thisisnoproblemforamatrixoptimizationmethod.ForapotentialmatchbetweenteamsA Thisisnoproblemforamatrixoptimizationmethod.ForapotentialmatchbetweenteamsA andB,theprogr

andB,theprogramneedsamneedstouseonlythestrengthtouseonlythestrengthsofteamsAandB,softeamsAandB, s s

 A

 Aandand s s B B,andalsothe,andalsothe strengthsoftheprioropponentsofteamA, strengthsoftheprioropponentsofteamA, s s C  C ,, s s _D D,, s s _E  E ,...,...

{

{

}}

,andofteamB,,andofteamB,  s s  N   N ,, s s_OO,, s s _P  P ,...,...

{

{

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.Then.Then theprogramcalculates theprogramcalculates! !   A

 Aofthesetoftheset

{

{

 s s B B,, s s_C C ,, s s _D D,, s s _E  E ,...,...

}}

andand! ! 

 B

 Bofthesetoftheset

{

{

 s s A A,, s s _N  N ,, s s_OO,, s s _P  P ,...,...

}}

.Each.Each

potentialmatchisscoredby potentialmatchisscoredby (1.1) (1.1)  scor scoree  sos  sos == 2 2 1 1 !  !   A  A + + 11 !  !   B  B withahigherscorebeingbetter.

withahigherscorebeingbetter.Therefore,thismethodlooksforthTherefore,thismethodlooksfortheopponentforteamAthaeopponentforteamAthatt mo

mostst incincreareasesses thethe stastandandardrd devdeviatiationion ofof itsits oppopponeonentnt setset –– i.ei.e.,., thathatt momostst balbalanancesces outout itsits sche

schedule–andlikewiseforteamB.Usingtheharmodule–andlikewiseforteamB.Usingtheharmonicmeantocombinnicmeantocombinethesetwostandarethesetwostandardd deviationsensuresthatonlymatchesthatbalanceoutbothteams’schedulescanreceivehigh deviationsensuresthatonlymatchesthatbalanceoutbothteams’schedulescanreceivehigh scores. scores. Strengthcouldbemeasuredinanyway:winsalone;winsplusafractionforspeakerpoints;or Strengthcouldbemeasuredinanyway:winsalone;winsplusafractionforspeakerpoints;or pointsfortheteam’srankorderoutofthenteamsatthetournament,wherefirstplace=n pointsfortheteam’srankorderoutofthenteamsatthetournament,wherefirstplace=n points,second=n–1,…,andlast=n–n=0.(Mypreferredwaytomeasureateam’sstrength points,second=n–1,…,andlast=n–n=0.(Mypreferredwaytomeasureateam’sstrength isdiscussedinalatersec

VariantA VariantA

Themethodforstrength-of-schedulepairingsstatedaboveignoresbrackets.Indeed,becauseit Themethodforstrength-of-schedulepairingsstatedaboveignoresbrackets.Indeed,becauseit istryingtopusheveryteam

istryingtopusheveryteamtoanequallybalancedschedtoanequallybalancedschedule,themethodstatedaboule,themethodstatedabovewillturnvewillturn atournamentintoakindofpartialroundrobin.Eachteamwillfaceopponentsfromthetop, atournamentintoakindofpartialroundrobin.Eachteamwillfaceopponentsfromthetop, middle,andbottombrackets.

middle,andbottombrackets.

Tocorrectthis,anadditionalfactor, Tocorrectthis,anadditionalfactor,winswins

 A

 A

!

!

winswins B B ,needstobeconsidered.Toforcewithin- ,needstobeconsidered.Toforcewithin-bracketpairings,thescoringneedstobeadjustedsuchthatthebestone-bracketpull-upmatch bracketpairings,thescoringneedstobeadjustedsuchthatthebestone-bracketpull-upmatch hasalowerscorethantheworstwithin-bracketpairing.Inthisway,theprogramwillchoosea hasalowerscorethantheworstwithin-bracketpairing.Inthisway,theprogramwillchoosea pul

pull-ul-upp mamatchtch onlonlyy ifitifit isnecesisnecessarsarybecauybecauseasea brabrackeckett hashas anuneveanunevennumbnnumberoferof teateams.ms. TheThe programwillthenchoosethebestpossiblepull-upmatch.Itisalsoworthconsideringmultiple programwillthenchoosethebestpossiblepull-upmatch.Itisalsoworthconsideringmultiple br

bracackekett pupullll-u-upsps asas wewell.ll. AlAlththououghgh ththesesee araree rararere,, sosomemetitimemess ththeyey araree nenececessssararyy atat smsmalalll to

toururnanamementntss wiwithth ononlyly aa fefeww papartrticicipipatatiningg scschohoolols.s. ItIt isis nonott didifffficiculultt toto hahaveve ththee prprogograramm cons

considerthesemultiderthesemultiplebrackiplebrackets,too.Thebesttwo-bets,too.Thebesttwo-brackracketpull-upetpull-upshouldhavealowerscoreshouldhavealowerscore thantheworstone-bracketpull-up,andsoon. thantheworstone-bracketpull-up,andsoon. Ifstrengthisturnedintoa0.01to1scale(forexample,bydividingteamrankpointsbyn),then Ifstrengthisturnedintoa0.01to1scale(forexample,bydividingteamrankpointsbyn),then thehigheststandarddeviationpossibleis0.5,andthereforethehighestpossiblescorefora thehigheststandarddeviationpossibleis0.5,andthereforethehighestpossiblescorefora matchis0.5.T

matchis0.5.Thescoringformulahescoringformulacouldbeadjustedtocouldbeadjustedto

(1.2)

(1.2)  score score

 sos

 sos..bracketsbrackets ==

2 2 1 1 !  !   A  A + + 11 !  !   B  B +

+0.50.5

_!!

roundsrounds

_"

_"

winswins

 A

 A

"

"

winswins B B

( (

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Thiswouldshiftthescalingup.After5rounds,forexample,afive-bracketpull-upmatchwould Thiswouldshiftthescalingup.After5rounds,forexample,afive-bracketpull-upmatchwould bescored0.01to0.5,afour-bra

bescored0.01to0.5,afour-bracketpull-upwouldbescoredcketpull-upwouldbescored0.51to1,…,andawithin-brack0.51to1,…,andawithin-bracketet matchwouldbescored2.51to3.Withinbrackets,theprogramwouldstilltrytopushteamsto matchwouldbescored2.51to3.Withinbrackets,theprogramwouldstilltrytopushteamsto equallybalancedschedules.Inotherwords,a3-1teamwithaneasyschedulewouldbepaired equallybalancedschedules.Inotherwords,a3-1teamwithaneasyschedulewouldbepaired againstastrong3-1team,whilea3-1teamwithatoughschedulewouldbepairedagainsta againstastrong3-1team,whilea3-1teamwithatoughschedulewouldbepairedagainsta weak3-1team.Inthisway,astrength-of-schedulepairinghelpsspeedthesortingprocessby weak3-1team.Inthisway,astrength-of-schedulepairinghelpsspeedthesortingprocessby ensuringthatteamsneithercoasttogoodrecordsduetosignificantlyweakerschedulesthan ensuringthatteamsneithercoasttogoodrecordsduetosignificantlyweakerschedulesthan theirbracketnorha

Formula2:Geographyplusinitialrankings Formula2:Geographyplusinitialrankings

Oneconsiderationforsomesta

Oneconsiderationforsomestateandnationaltournamteandnationaltournamentsisgeographicmixing.entsisgeographicmixing.Byrule,someByrule,some tournamentsb

tournamentsblockteamsfromlockteamsfromthesamegeograpthesamegeographicdistrictfrommhicdistrictfrommeeting.(Theseblockseeting.(Theseblockscanbecanbe treatedjustaswithin-schoolblocksare:byassigningthesematchesazero.)Sometournaments treatedjustaswithin-schoolblocksare:byassigningthesematchesazero.)Sometournaments merel

merelywantywant toencouragtoencouragee geogrgeographiaphiccmixinmixing.Ing.In manmanyycasecases,s,statstateeandnationaandnationall tourntournamenamentsts migh

mightalsoliketotalsoliketo runapartialroundrunapartialround robinrobin,whereeveryteamdebat,whereeveryteamdebatessometop,middleessometop,middle,and,and bottombrack

bottombracketteams(perhapetteams(perhapsbasedonrankinsbasedonrankingsfromtheregularseagsfromtheregularseason).son).

Theformula(1.1)outlinedabovealreadyrunsapartialroundrobin.Themeasureofstrength Theformula(1.1)outlinedabovealreadyrunsapartialroundrobin.Themeasureofstrength could

could comecomefromregularseasonfromregularseason pointpointsandnotsandnot varyvaryduringthetournamduringthetournament.ent. AllthatneedstoAllthatneedsto beaddedis

beaddedisameasureofgeographameasureofgeographicdiversityicdiversity.The.Thekeyconsidkeyconsideratioerationisnis thedistanthedistancebetweencebetween teams.Ifgeographicdistrictsareused,teamswithinadistrictcouldbeassigneda“distance”of teams.Ifgeographicdistrictsareused,teamswithinadistrictcouldbeassigneda“distance”of zero;teamsinadjacentdistricts,adistanceof1;andteamsinnon-adjacentdistricts,adistance zero;teamsinadjacentdistricts,adistanceof1;andteamsinnon-adjacentdistricts,adistance of2.Alternatively,theprogramcoulduseeachschool’sZIPcode,findlongitudeandlatitude of2.Alternatively,theprogramcoulduseeachschool’sZIPcode,findlongitudeandlatitude data,andassigntheactualdistancebetweentwoteams.Nomatterhowdistanceismeasured, data,andassigntheactualdistancebetweentwoteams.Nomatterhowdistanceismeasured, themetho

themethodofdof assigassigningning ascoreforgeographascoreforgeographicdiversiticdiversityisyis thesame.thesame. ForapotentialForapotentialmatchofmatchof teamAandteamB,whenteamAhadtheprioropponentCandteamBhadtheprioropponent teamAandteamB,whenteamAhadtheprioropponentCandteamBhadtheprioropponent D,theprogramwouldlookup

D,theprogramwouldlookup d d 

 AB

 AB,,d d  _AC  AC ,,d d  _AB AB,,d d  _BC  BC ,,d d  _BD BD

{

{

}}

,thedistancebetweeneachcombination,thedistancebetweeneachcombination ofteamslisted.Th

ofteamslisted.Theaveragedistancecreaeaveragedistancecreatedbythismatedbythismatchwouldbetchwouldbe

(2.1) (2.1) d d _avg avg == 66 2 2 d  d  _AB AB ++ 1 1 d  d  _AC  AC ++ 1 1 d  d  _AD AD ++ 1 1 d  d  _BC  BC ++ 1 1 d  d  _BD BD whereahigherscoreindicatemoregeographicspread.Theuseoftheharmonicmeanensures whereahigherscoreindicatemoregeographicspread.Theuseoftheharmonicmeanensures thatonlymatcheswherealldistancesarelongreceivehighscores.Thisfactorcouldbeusedin thatonlymatcheswherealldistancesarelongreceivehighscores.Thisfactorcouldbeusedin atournamentonitsownasawaytopairrounds,especiallyforrounds1and2,oritcouldbe atournamentonitsownasawaytopairrounds,especiallyforrounds1and2,oritcouldbe combinedwithafactorforstrengthofschedule. combinedwithafactorforstrengthofschedule. Formula3:Judges Formula3:Judges Themostexcitingapplicationofmatrixoptimizationisjudgeassignment.Ifthetournamenthas Themostexcitingapplicationofmatrixoptimizationisjudgeassignment.Ifthetournamenthas collectedeachteam’sjudgingpreferences,thesecanbeusedtoassigneachdebateamutually collectedeachteam’sjudgingpreferences,thesecanbeusedtoassigneachdebateamutually preferredjudge.Afterthedebatematcheshavepaired,theprogramwouldbuildamatrixof preferredjudge.Afterthedebatematcheshavepaired,theprogramwouldbuildamatrixof eachdebateversuseachjudge.IfteamAgavejudgejapreferenceof

preferenceof

preferenceof j j _B B(usuallyona1=mostpreferredto6=strikescale),thenthescorefordebate(usuallyona1=mostpreferredto6=strikescale),thenthescorefordebate ABtobegivenjudgejis

ABtobegivenjudgejis

(3.1)

(3.1)  judge judge _score score== 11 2

2 j j A A ₊₊₂₂ j j B B

whereahigherscoreindic

whereahigherscoreindicatesamorepreferredjudatesamorepreferredjudge.Ifatournamge.Ifatournamenthassurplusjudgesenthassurplusjudges,then,then several“blank”roundswillneedtobeaddedtomakethematrixsquare.Theseblankrounds several“blank”roundswillneedtobeaddedtomakethematrixsquare.Theseblankrounds shouldbescoredby

shouldbescoredby

(3.2)

(3.2)  judge judge _score score== 11

3 3++_rdsrds remain remain   theinverseoftheroundsofcommitmentajudgehasremaining.Thiswillputthejudgeswith theinverseoftheroundsofcommitmentajudgehasremaining.Thiswillputthejudgeswith thefewest

thefewestroundsofcommitroundsofcommitmentmentremaininremainingintoagintoarounroundoff,anditdoff,andit willpushjudgwillpushjudgeswiththeeswiththe mostroundsofcommitmentintoactualdebates.Finally,judgeswhohavealreadyseenateam mostroundsofcommitmentintoactualdebates.Finally,judgeswhohavealreadyseenateam orareblockedfromseeingateamshouldreceivea0forthatround.Thisneedstobeupdated orareblockedfromseeingateamshouldreceivea0forthatround.Thisneedstobeupdated duringatournament. duringatournament. VariantA VariantA The

The aboaboveve metmethodhod conconsidsidereeredd thethe prepreferferencenceses ofof eveeveryry teateamm equequallally.y. IfIf thethe goagoall isis toto givgivee higherprioritytoteamsinbreakrounds,thentheprogramneedstoincludeamultiplierin(3.1) higherprioritytoteamsinbreakrounds,thentheprogramneedstoincludeamultiplierin(3.1) toinflate

toinflatethethe judgjudgescoresfortheseteamsescoresfortheseteams.Forexample.Forexample,thisschemecould,thisschemecouldwork:eitherteamwork:eitherteam hastwo

hastwolosses=4points,eithlosses=4points,eitherteamhaserteamhasonelossbutneithonelossbutneitherhastwo=3points,boterhastwo=3points,bothteamshteams areundefe

areundefeatedated== 2points,andneith2points,andneitherteamcanmakeerteamcanmakeeliminatieliminationroundsonrounds== 1point.The(3.1)1point.The(3.1) formulawouldbemodifiedto

formulawouldbemodifiedto

(3.3)

(3.3)  judge judge _score score== multiplier multiplier  2

2jj A A₊₊₂₂jj B B

VariantB VariantB

Ofcourse,itisalsopossibletoaccomp

Ofcourse,itisalsopossibletoaccomplishsimilargoalswithoutteamlishsimilargoalswithoutteam’sjudgepreferences.Ifthe’sjudgepreferences.Ifthe tournamentranksallthejudgesonanexperiencescale,say,10pointsforveryexperiencedto0 tournamentranksallthejudgesonanexperiencescale,say,10pointsforveryexperiencedto0 pointsforanovicejudg

(3.4)

(3.4)  judge judge _score score==multiplier multiplier 

_!!

expexperienceerience

Itwouldmakenosensetoassignthesescoreswithonlytheexperiencevariable,sinceevery Itwouldmakenosensetoassignthesescoreswithonlytheexperiencevariable,sinceevery roundajudgecouldseewouldhavethesamescore. roundajudgecouldseewouldhavethesamescore. VariantC VariantC Perhapsthemostexcitingoptionisthatalldebatesinalldivisionsofalldebateeventscouldbe Perhapsthemostexcitingoptionisthatalldebatesinalldivisionsofalldebateeventscouldbe consideredsimultaneously.Theprogramwouldcreateamatrixwithalldebatematchesandall consideredsimultaneously.Theprogramwouldcreateamatrixwithalldebatematchesandall  judges.The

 judges.Thenewadditionalnewadditionalvariablewouldvariablewould bethebethe judge’sappropriatenessforjudge’sappropriatenessfor eachdivisioneachdivision forfor eachevent.Perhapsaninexperiencedjudgewouldbegivenascoreof10(mostappropriate) eachevent.Perhapsaninexperiencedjudgewouldbegivenascoreof10(mostappropriate) fornoviceLincolnDouglasdebatebutascoreof0(inappropriate)forvarsitypolicydebate.This fornoviceLincolnDouglasdebatebutascoreof0(inappropriate)forvarsitypolicydebate.This is,inessence,thesamethingasanexperiencescore,butgivenforeverydivisionandevent. is,inessence,thesamethingasanexperiencescore,butgivenforeverydivisionandevent. Theprogramwou

Theprogramwouldusejudgesinthemostapldusejudgesinthemostappropriatedivisionsandeventswpropriatedivisionsandeventswheretheycanstillheretheycanstill  judgedebates,

 judgedebates, butibutifftheyaretheyare blockedagainstblockedagainst allallthetheteamsinteamsin thatdivisionthatdivision ororevent,event,thenthethenthe programwoulds

programwouldseamlesslyslipthejudgeamlesslyslipthejudgesintothenextbestdesintothenextbestdivisionorevent.ivisionorevent.

Tomaketheformulacomplete,theprogramwouldalsoneedtoconsiderwhichdivisionand Tomaketheformulacomplete,theprogramwouldalsoneedtoconsiderwhichdivisionand eventhadthehighestpriorityforappropriatejudges.Giventhespeedandtechnicaldemands, eventhadthehighestpriorityforappropriatejudges.Giventhespeedandtechnicaldemands, thehighestpriorityevent

thehighestpriorityeventwouldmostlikelybwouldmostlikelybepolicydebate.Thepolicydebate.Theformulawouldtheneformulawouldthenbebe

(3.5)

(3.5)  judge judge _score score== prioritypriority

div

div..evev

!!

appropriatenessappropriateness

Thiscouldbecombinedwiththemultiplierforbreakroundsorevenmutualpreferencescores, Thiscouldbecombinedwiththemultiplierforbreakroundsorevenmutualpreferencescores, aslongasthesewereusedinalldivisionsofallevents. aslongasthesewereusedinalldivisionsofallevents. VariantD VariantD Thesamemethodcanbeusedforassigningjudgingpanels.Thejudgeswouldbeassignedone Thesamemethodcanbeusedforassigningjudgingpanels.Thejudgeswouldbeassignedone byone.Ifthetournamentusesthree-judgepanels,theprogramwoulddividethejudgesinto byone.Ifthetournamentusesthree-judgepanels,theprogramwoulddividethejudgesinto thr

threepoolseepools anandd aa foufourthrth setset ofunusofunusedjudgeedjudges.Thepoolss.Thepools coucouldbeldbe divdivideidedindin ananyy numnumberber ofof ways

ways::geoggeographyraphy,experienc,experience,stylistice,stylistic prefepreferencerences,sex,s,sex,age,etc.Eachpoolwouldbeage,etc.Eachpoolwouldbe assiassignedgned usingthematrixoptimizationmethod.

usingthematrixoptimizationmethod.

Formula4:IEs Formula4:IEs

Itisalsopossibletousema

ItisalsopossibletousematrixoptimizationtosetutrixoptimizationtosetupIEpanelsusingmupIEpanelsusingmultiplevariables.TheIErsltiplevariables.TheIErs wouldneedtobeaddedtothepanelonebyone.

Section3:ABetterMeasureofTeamStrength Section3:ABetterMeasureofTeamStrength

Thebette

ThebettermeasurrmeasureofeofteamteamstrengthstrengthII havecomhavecomeupwithiseupwithis weighweightedwins.tedwins.IfIf teamteamAA defeadefeatsts teamsB,C,andD

teamsB,C,andD,teamA’sweightedw,teamA’sweightedwinswouldbe3+winsofB+inswouldbe3+winsofB+winsofC+winsofD.IwinsofC+winsofD.Inthisnthis way,teamAearnsadditionalwinsforthestrengthofitsdefeatedopponents.IfteamAlosesto way,teamAearnsadditionalwinsforthestrengthofitsdefeatedopponents.IfteamAlosesto teamsEandF,teamA’sweightedlosseswouldbe2+lossesofE+lossesofF.TeamAearnsits teamsEandF,teamA’sweightedlosseswouldbe2+lossesofE+lossesofF.TeamAearnsits additionallossesifitlosestoweakopponents.Theratio

additionallossesifitlosestoweakopponents.Theratio ww

.

.

winswins w

w

.

.

winswins++ww

.

.

losseslosses

measuresateam’s measuresateam’s strengthona0to1scale. strengthona0to1scale. TheweightedwinsratioisNOTthesamethingasopponentwins.Opponentwinsmeasuresthe TheweightedwinsratioisNOTthesamethingasopponentwins.Opponentwinsmeasuresthe strengthofateam’sschedule;weightedwinsconsidersthestrengthofopponentstoadjustthe strengthofateam’sschedule;weightedwinsconsidersthestrengthofopponentstoadjustthe measureofateam’sstrength. measureofateam’sstrength.