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Hack 59. Rank with the Best of Them

Hack 59. Rank with the Best of Them

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forcompetitiveorganizationssuchassportsleaguesand

associations.Theproblemishowtosummarizeperformance

acrossavarietyofcategories,venues,andoccasions.

Therearethreemethodscommonlyusedintheworldofsports

tomakedeterminationsaboutwhoisthe"best."Allofthe

approachesmakesomeintuitivesense,thougheachmethod

hasitsownspecificadvantagesanddisadvantages.

First,let'stakealookatthenatureofthedataIhadto

analyze.Yourdatawilllikelybesimilar,whetheryourunyour

weeklyhomeMonopolygameoryouruntheProfessionalGolf

Association.Thoughpokerisnotasport,anyorganized

competitiveendeavorprovidesdataforrankings.Table5-16

showstheresultsfromeighttournamentsinmyownsummer

pokerleague.





5/14

5/21

5/28

6/4

6/11

6/18

6/25

7/2



Paul

6

3















Lisa

5

6















TableSummerpokerleaguedata

Billy

BJ Mark

Bruce

Cathy

4

3

2

1



4

5

7

2

1

5

4

1

3

2

4

6

3

7

2

4

5

6

1

2

5

4

2

3

1

1

4

3

5

2

1

5

4

3

2



Tim







5

3









David







1











Youcanseethatnineplayerstookpartinatleastone

tournament,butnoeventhadparticipationfromallplayers.Ifa

personreceivednopointsonagivennight,itwasbecauseshe

didn'tplay.Thisiscommonlythecaseinsportssuchasgolfand

tennisaswell.

Ontwooccasions,sevenpeopleplayed,butonotheroccasions,

asfewasfivesatdowntogether.Fourpeoplehaveplayedinall



eighttournaments.(Thesearethehard-coreplayerswhohave

toadmitthattheyhaveabitofaproblemrecognizingwhatis

importantinlife.)Oneplayer,David,playedinonlyone

tournament.

Thepointsundereachplayer'snameindicatetheorderinwhich

theywentout.Iftherearesixplayersandyougooutfirst,you

getonepointfortakinglastplace.Ifyouarethewinneramong

sixplayers,yougetsixpointsfortakingfirst.



Noticeacoupleofthingsaboutthispointsystem.First,yougetatleast

apointjustforshowingup.Second,yougetmorepointsforwinninga

tournamentwithmoreplayers.



How,then,torankplayersinthepokerleague?Herearethree

commonsolutions,allofwhichworktosomeextent.



Totalpoints

Thefirstthoughtthatcametomindinmysituationwasto

simplyaddupthepointsacrosstournamentsandrankplayers

basedontheirtotalpoints.Thisistheapproachtakenwhen

celebritiesarerankedbyincomeorbankrobbersarerankedby

theirnumberofcrimes.Justparticipatingalotmovesyouupin

theserankings.Tobegolferoftheyear,youhavetohave

playedinmanyevents,inadditiontoperformingOKinthem.



Meanperformance

Asecondmethodistoaveragethepointsbydividingthetotal

pointsbythenumberoftournamentsinwhichaplayer



participated.Thebeautyofproducinganaverageisthatyouget

anumberthatrepresentsatypicallevelofperformance.Thisis

idealformeasuringsomethingelusive,suchastalent.Your

averageperformanceatpoker(oranythingelse)shouldbethe

bestsingleindicatorofability.



Totalwins

Athirdmethod,thesimplestandmostcommonlyusedinteam

sports,istocountvictories.Theplayerwhowinsmostoftenis

thebestplayer.Thismethodworkswellfortournament-style

poker(thekindweplay)andanyeventsinwhichthereisone

competitorwhoistheclearwinner.



ComparingtheThreeMethods

Thougheachrankingapproachhassomeclearadvantagesand

doesthejobadequately,Table5-17showsthevaluesforeach

playerunderallthreerankingsystems.





Points

Mean

Wins



Paul

9

4.5

1



TableSummarizingpokerperformance

Lisa Billy

BJ

Mark

Bruce

Cathy

11

28

36

28

25

12

5.5

3.5

4.5 3.5

3.13

1.71

1

2

1

2

2

0



Tim

8

4.0

0



David

1

1.0

0



Allthreescoringsystemsmakesense.Butthequestionabout

whoisthebesthasadifferentanswerundereachofthethree

systems!Thisiscertainlyafrustratingfindingforapoker

scientistlikeme.Becauseonecoulddefendanyofthethree

methodsasthe"best"waytorank,itisabitofaparadoxthat

eachmethodproducesadifferent"best"pokerplayer.Table5-



18showshowtherankingsdifferundereachscoringmethod.





Points

Mean

Wins



Paul

7

2.5

4



Lisa

6

1

4



TablePokerrankings

Billy

BJ

Mark

Bruce

2.5

1

2.5

4

5.5

2.5 5.5

7

2

4

2

2



Cathy

5

8

6



Tim

8

4

6



David

9

9

6



Noticehowthe"bestplayer"isdifferentundereachsystem.BJ

isthebestunderthePointssystem.Lisaisthebestunderthe

Meansystem.ThreepeopletieforfirstundertheWinssystem,

butBJandLisaarenotamongthem.Theonlyrealagreement

acrossthethreemethodsisthatDavidisrankedastheworst

player.(Sorry,David,butnumbersdon'tlie.Andsorryabout

thepublicridicule.MaybeIcanmakeituptoyouwithafree

copyofthisbook?)



Ibroketieswhenassigningrankingsbyaveragingtherankingamong

thosewhoweretied.Inotherwords,Billy,Mark,andmyselfwereall

tiedforthenumberonerankingundertheWinssystem,sotheranksof

1,2,and3averageto2,andthatwasourranking.



Ifthreedifferentscoringsystemsresultinthreedifferent

rankings,itiscleartheycannotallbeequallyvalid.Theycannot

allproducescoresthattrulyreflectthevariableofinterest,

whichispoker-playingabilitydefinedinthesameway.The

solutiondoesnotinvolvepickingthesinglebestapproach.It

wasnotmygoaltoidentifythebestsystemandgowithit;my

goalwastoprovidevalidinformationandletothersinterpret

thedatahowtheywant.

Mysolutionwastoprovideallthreerankingsbasedonthethree



scoringmethods.Thatway,playerscouldchoosetofocuson

therankingresultsfromthemethodthatmakesthemostsense

tothem.



TheEndoftheStory

Thesystemthatmadethemostsensetotheplayersinmy

pokerleagueturnedouttobetheonethatrankedthemthe

highest.Imaginethat.

Isleepatnightsecureintheknowledgethatanyofthe

methodsisprobablyacceptableand"accurate."Afterall,none

ofthethreemethodsmakesthemistakeofidentifyingmeas

theonebestplayer.That'sgottobesomesortofvalidity

evidenceinandofitself!

Real-lifeprofessionalsportsorganizationshavedealtwiththe

advantagesanddisadvantagesofeachsystembycreating

compositepointsystems.Someofthetinkeringtoimprove

rankingsystemsintennisandgolf(andtournamentpoker,too)

includes:

Combiningperformancedataoveralongperiodoftime

Awardingmorepointsforwinningmoredifficult

tournaments

Usingboththemeanperformanceandtotalpointstogether

torewardexcellenceandfrequentparticipation

Itisabitironicthatthesesystemsthatarelikelyfairerand

moreaccurateareoftenperceivedbythepressandfansas

overlycomplexandcrazy.Attemptstomaketheranking

systemsmorevalidhaveresulted,often,inarejectionofthe

systemsbythepublicasinvalid.



Hack60.EstimatePibyChance



Statisticiansliketothinkthatanythingimportantcanbe

discoveredusingstatistics.Thatmightactuallybetrue,

sinceitturnsoutthatyoucanusestatisticstoestimate

thevalueofoneofthemostimportantbasicvaluesin

science:pi.

Theabilitytocalculatepiisoneoftheroutineskillsforall

buddinggeniuses.Iremember,forexample,thatdividing22by

7comesprettyclose.Thereareavarietyofotherways,some

moreaccuratethanothers.Myfavoritemethod,though,

requirestheelementofchanceandalong,lonelyseavoyageor

otherperiodofenforcedsolitude.Intrigued?Readon,Gilligan.

Beforeshowinghowtoestimatethevalueofpi,I'llbeginour

discussionbypresentingacoupleofbasicfactsfromgeometry.

Don'tpanic;Idon'tknowmuchaboutgeometry,sowewon't

spendalotoftimeonthis.I'lljustcoverthebasicsweneedto

appreciatethemagicofthishack.



Pi

Ingeometry,keyrelationshipshavebeenfoundbetweenpi,a

numberthatisroughly3.14159(symbolizedbyp),andtheway

variouspartsofacirclefittogether,asshowninFigure5-6.



Figure5-6.Calculatingpi



Forexample,ifyoutakethediameterofacircleandmultiplyit

bypi,youwillgetthecircumferenceofthecircle.Ifyoutake

theradiusofacircle,squareit,andmultiplythatvaluebypi,

youwillgetthecircle'sarea.

Allprettycool,perhaps,butitisprimarilyofinteresttothose

wholiketoplaywithgeometry,notwithstatistics.Butjustwait.



PiandFallingNeedles

Inthe1700s,Georges-LouisLeclercpresentedahalfgeometry/half-statisticspuzzletotheworld.HewastheCount

ofBuffon,orsomething,sothisproblemisknownasBuffon's

NeedleProblem.Hepresenteditgenerally,withoutspecifics,

andIsummarizeithere:

Imagineaneedlelandsrandomlyonadrawingoftwoparallel

horizontallines.Thelinesarefurtherapartthanthelengthof

theneedle.Whatarethechancesthattheneedlewilllandin

suchawaythatittouchesoneofthelines?

Thisisoneofthoseproblemsthatseemimpossibletosolvethe

firsttimeyouhearit,butitissolvable.There'snoneedto

spendanytimecalculatingthesolutionhere,thoughIcertainly



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