Basketball from perspective of nonlinear complex systems

UNIVERSITY OF LAS PALMAS DE GRAN CANARIA DEPARTAMENT OF PHYSICAL EDUCACTION Doc o al P og am: Mo o P axiology, Physical Educa ion and Spo T aining DOCTORAL THESIS BASKETBALL FROM THE PERSPECTIVE OF NON-LINEAR COMPLEX SYSTEMS YVES DE SAÁ GUERRA LAS PALMAS DE GRAN CANARIA, 2013 BASKETBALL FROM THE PERSPECTIVE OF NON-LINEAR COMPLEX SYSTEMS DOCTORAL THESIS By YVES DE SAÁ GUERRA, B.S. SCIENCES OF PHYSICAL ACTIVITY AND SPORT UNIVERSITY OF LAS PALMAS DE GRAN CANARIA, GRAN CANARIA, SPAIN 2013 Anexo II UNIVERSIDAD DE LAS PALMAS DE GRAN CANARIA Depa amen o de Educación Física. Facul ad de Educación Física P og ama de doc o ado: P axiología mo iz, educación ísica y en enamien o depo i o. Tí ulo de la Tesis BASKETBALL FROM THE PERSPECTIVE OF NON-LINEAR COMPLEX SYSTEMS Tesis Doc o al p esen ada po D. Y es de Saá Gue a Di igida po el D . D. Manuel Na a o Valdi ielso Codi igida po el D . D. Juan Manuel Ga cía Manso Codi igida po el D . D. Juan Manuel Ma ín González El Di ec o , El Codi ec o , El Codi ec o , El Doc o ando, ( i ma) ( i ma) ( i ma) ( i ma) Las Palmas de G an Cana ia, a _____ de_________________ de 2013 The mo e I know, he less I can a i m, ca ego ically. Cuan o más sé, menos puedo a i ma las cosas ca egó icamen e. Y es ACKNOWLEDGEMENTS This disse a ion would no ha e been possible wi hou he help o so many people in so many ways. I ca ied ou hanks o he commi men and dedica ion o a esea ch g oup led by D . Manuel Na a o Valdi ielso, D . Juan Manuel Ga cía Manso and D . Juan Manuel Ma ín. I is an hono be pa o his g oup and I would like o exp ess my since e g a i ude o my hesis di ec o D . Manuel Na a o Valdi ielso o his uncondi ional suppo , his ad ices and his academic o ien a ion, as well as o his help in he de elop o his hesis. Also I would like o exp ess my deepes admi a ion o my co-di ec o s D . Juan Manuel Ga cía Manso, because he augh me e e y hing I know abou aining and ad ised me abou spo in an exquisi e way. He has con eyed me all his s eng h and passion o he spo . I will ne e be able o emo e his seed; and o D . Juan Manuel Ma ín, because he de eloped he necessa y p og ams in Ma lab ha allow us o deepen in spo s eali y and ad ise wi h ma hema ics. I ha e been ascina ed by all o ou con e sa ions abou he uni e se and he sec e s o li e. I is a eal pleasu e o lis en o and lea n om him. Wi hou his help his hesis would no been possible. I eel p i ileged o sha ing pa o my li e wi h hese h ee g ea people. I also would like o acknowledge he collabo a ion and suppo o P o esso Ad ian Bejan, om Duke Uni e si y, USA. by his passiona e discussions and by his ou s anding ele ance in he scien i ic communi y. I was a eal pleasu e lea n om him du ing my s ay a Duke Uni e si y. I canno conclude his chap e wi hou o ex end my since e hanks o my ellows o he Labo a o y o Analysis and Planning o Spo s T aining, o Depa men o Physical Educa ion o his uni e si y, especially o D . Samuel Sa mien o Mon esdeoca o his suppo and iendship. I mus also acknowledge he suppo o he Uni e si y o Las Palmas de G an Cana ia, o suppo ing my esea ch by awa ding a g an o he de elopmen o his hesis. Finally, and by no means leas impo an , I would like o gi e hanks o my en i e amily o hei uncondi ional suppo e e y ime I needed i . They a e my ole models. They always ha e been. And always will be. Y es In oduc ion In ou e o o y o unde s and he spo eali y, we a e o ced o ques ion e e y hing ha we conside ue o s a ic. This leads us inexo ably, o y o isola e a phenomenon in o de o s udy be e . Bu a om ou pu pose, when we mo e owa ds his poin , we ealize ha we all in o he con adic ion o hoping o be able o unde s and a phenomenon by isola ing i om i s en i onmen . Following he asse e a ion: ” I am I and my ci cums ance” (O ega y Gasse , 1933), i is no possible o unde s and he spo phenomenon by isola ing he cons i uen elemen s o hei ela ionships wi h hei own uni e se. The e is a duali y elemen –en i onmen . F om his ela ionship migh eme ge new beha io s, which is known om he pe spec i e o complexi y, as an eme gency phenomenon o an eme gen beha io . I.e. in o de o eason his phenomenon, i would be good o mo e away om he de e minis ic and educ ionis classical model, and mo e on o s udy he sys ems om a mo e global concep ion (holis ic), allowing us o iden i y and desc ibe he p ocesses o new o ms o o ganiza ion, which is also use ul in spo . O ganizing he spo aining om a sys ema ic concep ion and concei e he a hle e, o he eam in ou case, as a sys em ha unc ions as a whole and ha is a ec ed by he su ounding en i onmen (Gambe a, 1989; Ma ín Ace o & Lago Peñas, 2005; Ga cía Manso & Ma ín González, 2008). Chaos heo y has p o ided a new ligh o obse e all hese sys ems, seemingly incomp ehensible o andom, because i is usual ha na u al sys ems a e chao ic. Chaos hides an in e nal o de ha is possible o ind (P igogine & Hol e, 1993). In complex sys ems, he p ocesses occu ing simul aneously in di e en scales o le els a e impo an , he in ica e and complex beha io o he sys em as a whole, depends on he uni s, al hough no di ec ly, because in complex sys ems he s uc u es ha e s ong ela ionships, o en in a non-linea manne (Vicsek, 2002; Goodwin, 2002; Ama al & O ino, 2004; Solé, 2009). I would be useless o conside he conscious hough as a me e sum o neu ons, o educe he beha io o a eam o he sum o he indi idual abili ies o his playe s indi idually. This de ini ion also dis inguishes he complex om he simple and complica ed, as can be any o he mechanism such as ha de e mines he ope a ion o a ca , an ai plane o a compu e . 6 Complexi y is a measu e o he numbe o possibili ies: he ways in which we ac o eac o he en i onmen . I is an impo an poin o he s udy o spo om he poin o iew o complex sys ems. Howe e , he ue complexi y o beha io , as seen in baske ball, occu s in he in e ac ion be ween i s elemen s and esponse possibili ies o each s uc u e. Spo pe o mance is he esul o he combina ion o many a iables ha some imes we know and domina e i h ough di e en analy ical me hods, and we y o unde s and e en be e o imp o e i . Seldom e ec i eness in spo shows a linea beha io . I would be e y easy o unde s and and ge be e . Ac ually, he e a e many ac ions we can conside p ope , e en whe he hese ac ions would epea , need no be consecu i e. The beha io o playe s, ball, coaches and many mo e aspec s, may condi ion he ou come. So i wo ks as any complex o ganiza ion, hence, mus be unde s and as a complex sys em. Es ablishing eam Spo s as complex sys ems, ou in en ion is o mo e away om classic and de e minis ic models, in o de o pass on a new pe spec i e o uni ying c i e ia and analysis and hus o add ess and esol e issues aised abo e, and inc ease pe o mance in di e en spo s collec i e. This model is based on how i s componen s a e ela ed in a p ecise and de e mined, and how hey eac o o he complex sys ems. In ac , all we seek is o ecognize and iden i y pa e ns o collec i e beha io and ela ionships among i s componen s, which make i , esemble sel -o ganizing complex sys ems non-linea in c i ical (Sys em O ganized C i ically). Acco ding o Goodwin, he ideal is o in es iga e he condi ions ha p omo e sel - o ganiza ion(Goodwin, 2002), in o de o ob ain he spo ing excellence. Complex sys ems a e he esul o an e olu iona y p ocess. Da win's ideas and he s udy o e olu ion ha e ocused on he compe i ion as a d i ing o ce o e olu iona y change. Playe s o example, when hey coope a e, compe e be e as a eam (Ba -Yam, 2003). Team pe o mance can be pos ula ed as win as many games as possible. I esul s om he synch onous in e ac ion o ce ain s a es o op imiza ion o sys ems ha make up, which also ha e a ecip ocal ela ionship wi h he eme ging and c i ical en i onmen : he compe i ion. The aim o he s udy is o obse e he beha io o he s uc u e (mac os uc u e) o baske ball as a sys em. We wan o ind ou how baske ball elemen s a e in e connec ed and how hey a ec each o he , analyzing he laws ha go e n hem. The e o e we ha e es ablished h ee le els o i ems o ca y ou he in es iga ion: league, games and baske ball eam as ne wo k. Baske ball Backg ound Baske ball om he pe spec i e o non-linea complex sys ems Y es de Saá Gue a 2013 Baske ball • Complexi y Baske ball Backg ound 9 3. Baske ball Backg ound 3.1 Baske ball His o y C ea ion o baske ball D . James Naismi h was a Canadian physical educa ion p o esso a he In e na ional Young Men's Ch is ian Associa ion T aining School (YMCA) in Sp ing ield, Massachuse s, USA. In Decembe o 1891, James Naismi h was asked by his di ec o o de ise an indoo game o he school´s 40 s uden s, o help keep hem physically ac i e be ween oo ball season and he sp ing ime ac i i ies o baseball and ack (Mu uzábal del Sola , 2012). D . Naismi h combined elemen s o ou doo games like oo ball and lac osse wi h he concep o a game he played in childhood, Duck on a Rock. To win Duck on Rock, playe s h ew s ones o hi a a ge placed on op o a la ge boulde (Naismi h, 1941). He se up peach baske s a ached o bo h ends o a gymnasium balcony on o a 3.05 me e s ele a ed ack, and used a ball in o de o sco e on i . The peach baske s e ained he ball a bo om and i had o be aken ou manually a e each poin sco ed un il he bo om o he baske was emo ed. The peach baske s we e used un il 1906 when hey we e inally eplaced by me al hoops wi h backboa ds. The backboa ds appea ed as p o ec ion, o p e en he an loca ed on he ailing o he galle y, whe e hung he baske s, could hinde he en y o he ball in he baske , which la e wen on o become a me al ing and a ne wo k wi hou holes, o lead in oday's ne wo ks (Tous Faja do, 1999). The ea ly playe s did no use d ibbling o he ball, excep o he “bounce pass” o eamma es. Passing he ball was he p ima y means o ball mo emen . D ibbling was e en ually in oduced bu limi ed by he asymme ic shape o ea ly balls. D ibbling only became a majo pa o he game a ound he 1950s, as manu ac u ing imp o ed he ball shape. The i s o icial game was played in he YMCA gymnasium in Albany, New Yo k on Janua y 20, 1892 wi h nine playe s. The game ended a 1-0. The sho was made om 7.6 me e s, on a 10 cou jus hal he size o a p esen day Na ional Baske ball Associa ion (NBA) cou . By 1897- 1898 eams o i e became s anda d (Naismi h, 1941). A ball and an ele a ed goal, hose a e he simple ing edien s o he spo ha now ha e playe s and abid ans in nea ly e e y pa o he wo ld. Acco ding o Alexande Wol , in his book 100 Yea s o Hoops (Wol , 1991), Naismi h d ew up he ules o he new game in “abou an hou ”. Nowadays baske ball is one o he wo ld's mos popula and widely iewed spo s (G i i hs, 2010). The Fi s 13 Rules o Baske ball Naismi h and Wheele w o e he i s 13 ules o he game (In e na ional Baske ball Fede a ion; FIBA, 2012). They we e published in he school newspape , The T iangle, o i s ime in 1892: 1. The ball may be h own in any di ec ion wi h one o bo h hands. 2. The ball may be ba ed in any di ec ion wi h one o bo h hands (ne e wi h he is ). 3. A playe canno un wi h he ball. The playe mus h ow i om he spo on which he ca ches i , allowance o be made o a man who ca ches he ball when unning a a good speed i he ies o s op. 4. The ball mus be held in o be ween he hands; he a ms o body mus no be used o holding i . 5. No shoulde ing, holding, pushing, ipping, o s iking in any way he pe son o an opponen shall be allowed; he i s in ingemen o his ule by any playe shall coun as a oul, he second shall disquali y him un il he nex goal is made, o , i he e was e iden in en o inju e he pe son, o he whole o he game, no subs i u e allowed. 6. A oul is s iking a he ball wi h he is , iola ion o ules 3, 4, and such as desc ibed in ule 5. Baske ball • Complexi y Baske ball Backg ound 11 7. I ei he side makes h ee consecu i e ouls, i shall coun a goal o he opponen s (consecu i e means wi hou he opponen s in he mean ime making a oul). 8. A goal shall be made when he ball is h own o ba ed om he g ounds in o he baske and s ays he e, p o iding hose de ending he goal do no ouch o dis u b he goal. I he ball es s on he edges, and he opponen mo es he baske , i shall coun as a goal. 9. When he ball goes ou o bounds, i shall be h own in o he ield o play by he pe son i s ouching i . In case o a dispu e, he umpi e shall h ow i s aigh in o he ield. The h owe -in is allowed i e seconds; i he holds i longe , i shall go o he opponen . I any side pe sis s in delaying he game, he umpi e shall call a oul on ha side. 10. The umpi e shall be judge o he men and shall no e he ouls and no i y he e e ee when h ee consecu i e ouls ha e been made. He shall ha e powe o disquali y men acco ding o ule 5. 11. The e e ee shall be judge o he ball and shall decide when he ball is in play, in bounds, o which side i belongs, and shall keep he ime. He shall decide when a goal has been made, and keep accoun o he goals wi h any o he du ies ha a e usually pe o med by a e e ee. 12. The ime shall be wo 15-minu e hal es, wi h i e minu es es be ween. 13. The side making he mos goals in ha ime shall be decla ed he winne . In case o a d aw, he game may, by ag eemen o he cap ains, be con inued un il ano he goal is made. 12 3.2 Baske ball Gene al Desc ip ion Some au ho s such as Knapp (1981), He nández (1994) and Ruiz (1999) analyze baske ball ega ding o o mal s uc u e and unc ionali y o he game i sel . I also can be classi ied as a eam spo o collec i e spo , o coope a ion-opposi ion, wi h unc ional s uc u e which de elops in a common space o bo h eams and simul aneous in e en ion on ball (He nández Mo eno, 1994) Baske ball is de ined as a eam spo played by wo eams o i e playe s each on he cou , and can be subs i u ed by bench playe s (each league allow a di e en numbe o bench playe s), p e iously selec ed by he coach. The aim o each eam is o sco e in he opponen s' baske and o p e en he o he eam om sco ing while ollowing a se o ules. The eam ha has sco ed he g ea e numbe o poin s a he end o playing ime shall be he winne (Rule 1. A . 1. FIBA, 2012). In he leagues we s udied (Na ional Baske ball Associa ion, NBA; Asociación de Clubs de Balonces o, ACB; and Na ional College A hle ic Associa ion, NCAA); he game is con olled by o icials and sco e o icials h oughou he O icial Baske ball Rules (NBA, 2012; NCAA, 2012a; FIBA, 2012b) (ACB uses FIBA Rules). Ne e heless he e a e some di e ences be ween he NBA, FIBA and NCAA ules, ega ding cou dimensions, ime ou s, ouls, o icials, e c. (see able 1). We ca ied ou his compa ison because hese h ee leagues a e he aim o s udy. In gene al, ules o he male gende and o he emale gende a e di e en . Only he FIBA ules es ablish he same ules o bo h gende s. Usually, eams play on a ma ked ec angula cou wi h a baske a each wid h end. The dimensions o he cou a e desc ibed by he o icial ules (NBA, 2012; FIBA, 2012b; NCAA, 2012b) (See Figu e 1, Figu e 2, Figu e 3 and Figu e 4). Baske ball • Complexi y Baske ball Backg ound 13 Figu e 1. Hoop dimensions. Sou ce: FIBA. Figu e 2: FIBA Baske ball Cou . Sou ce: FIBA. 20 2012). I also called ound- obin ou namen o all-play-all ou namen . In u n can be played in one single g oup (depend o he numbe s o eams) o can be di ided in se e al g oups, con e ences and/o di isions, acco ding o a p ea anged schedule. Play-o /Cup. The playo s o inals (we can also include cup ou namen s) in a spo a e a game o se ies o games played a e he egula season by he op eams classi ied, in o de o de e mine he league champion. Teams use o play in a b acke (ACB, 2012; NBA, 2012; NCAA, 2012a). A b acke is a ee diag am ha ep esen s he se ies o games played du ing a ou namen , named as such because i appea s o be a la ge numbe o in e connec ed (punc ua ional) b acke s. The e a e se e al o ma s o he play-o b acke s, meaning he numbe o games pe ound (1-1-1-1, 3-3-3, 7-7-7-7, e c.). In college baske ball is e y amous he ac o illing in b acke s, especially in NCAA baske ball, is e e ed o as b acke ology. Fede a ion. Fede a ions a e p i a e non-p o i o ganiza ions composed by adminis a i e sec ion, spo s clubs, a hle es, coaches, judges and e e ees and p o essional leagues, in o de o p omo e, p ac ice o con ibu e o he de elopmen o spo (Ley 10/1990, de 15 de oc ub e, del Depo e, A . 30. Minis e io de Educación, Cul u a y Depo e). The unc ions o he baske ball ede a ions a e he go e nmen , adminis a ion, managemen , o ganiza ion and egula ion o he spo o baske ball h oughou he e i o y co e ed, whe he in e na ional o domes ic, ega ding compe i ions and championships o ganized. As well as d awing up o he co esponding licenses ha a e equi ed o pa icipa e as playe , coach o e e ee, in compe i ions and championships o ganized. Baske ball • Complexi y Baske ball Backg ound 21 3.3.2. In e na ional s uc u es In baske ball he e a e se e al in e na ional s uc u es which y o egula e and p omo e wo ldwide baske ball h ough he na ional eams. FIBA (In e na ional Baske ball Fede a ion) The In e na ional Baske ball Fede a ion (FIBA) is he o ganiza ion ha is dedica ed o egula e he ules o baske ball wo ldwide, as well as holding egula compe i ions and e en s in he disciplines o baske ball (men and women). The associa ion was ounded in Gene a in 1932, wo yea s a e he spo was o icially ecognized by he IOC (In e na ional Olympic Commi ee). The name FIBA came om i s F ench name Fédé a ion In e na ionale de Baske ball, is an associa ion o na ional o ganiza ions which go e ns in e na ional compe i ion in baske ball. O iginally known as he Fédé a ion In e na ionale de Baske ball Ama eu (hence FIBA), he wo d “Ama eu ” was d opped in 1986 a e he dis inc ion be ween Ama eu s and P o essionals. The "BA" now ep esen s he i s wo le e s o baske ball. The main aim o he FIBA was o coo dina e ou namen s and eams. A gen ina, Czechoslo akia, G eece, I aly, La ia, Po ugal, Romania and Swi ze land we e he ounde membe s (FIBA, 2012). The FIBA Cen al Boa d is cu en ly composed o 23 membe s (22 ha e he igh o o e) and mee s wice yea ly. The FIBA Cen al Boa d has, among o he compe ences, he powe o es ablish he FIBA In e nal Regula ions. I also assigns he o ganiza ion o all FIBA Baske ball Wo ld Cup. FIBA coun s wi h 213 membe ede a ions. FIBA has o ganized a FIBA Wo ld Championship o men since 1950 and a Wo ld Championship o Women since 1953. Bo h e en s a e now held e e y ou yea s, al e na ing wi h he Olympics. 22 IOC (In e na ional Olympic Commi ee) The IOC coo dina es he ac i i ies o he Olympic Mo emen . I is also esponsible o supe ising and manages e e y hing abou he Olympics. Owns all he igh s associa ed wi h he Olympic symbols, lag, an hem, lemma, oa h and games. I con ols he igh s o b oadcas he games, ad e ising and o he ac i i ies acco ding o he Olympic Cha e . I is also he in e na ional body esponsible o o ganizing and selec ing he ci ies ha will hos he Olympic Games e e y 4 yea s (IOC, 2012). In de ail he ole o he IOC, acco ding o he Olympic Cha e (Sep embe 2004), is:  To encou age and suppo he p omo ion o e hics in spo as well as educa ion o you h h ough spo and o dedica e i s e o s o ensu ing ha , in spo , he spi i o ai play p e ails and iolence is banned.  To encou age and suppo he o ganiza ion, de elopmen and coo dina ion o spo and spo s compe i ions.  To ensu e he egula celeb a ion o he Olympic Games.  To coope a e wi h he compe en public o p i a e o ganiza ions and au ho i ies in he endea o o place spo a he se ice o humani y and he eby o p omo e peace.  To ake ac ion in o de o s eng hen he uni y and o p o ec he independence o he Olympic Mo emen .  To ac agains any o m o disc imina ion a ec ing he Olympic Mo emen .  To encou age and suppo he p omo ion o women in spo a all le els and in all s uc u es wi h a iew o implemen ing he p inciple o equali y o men and women.  To lead he igh agains doping in spo .  To encou age and suppo measu es p o ec ing he heal h o a hle es.  To oppose any poli ical o comme cial abuse o spo and a hle es.  To encou age and suppo he e o s o spo s o ganiza ions and public au ho i ies o p o ide o he social and p o essional u u e o a hle es.  To encou age and suppo he de elopmen o spo o all.  To encou age and suppo a esponsible conce n o en i onmen al issues, o p omo e sus ainable de elopmen in spo and o equi e ha he Olympic Games a e held acco dingly. Baske ball • Complexi y Baske ball Backg ound 23  To p omo e a posi i e legacy om he Olympic Games o he hos ci ies and hos coun ies.  To encou age and suppo ini ia i es blending spo wi h cul u e and educa ion.  To encou age and suppo he ac i i ies o he In e na ional Olympic Academy (IOA) and o he ins i u ions which dedica e hemsel es o Olympic educa ion. Baske ball appea ed o i s ime in Olympic Games in San Louis in 1904 as an exhibi ion game. Baske ball was included in Olympic Games in Be lin 1936 as Olympic Spo . Women´s baske ball is p esen in Olympic Games in Mon eal 1976. 3.3.3. Na ional S uc u es In e e y coun y, o ganiza ional s uc u es o baske ball a e designed in di e en ways. Acco ding ou in es iga ion we s udied he baske ball in USA and Spain, hence we desc ibe he o ganiza ional s uc u es o baske ball in USA and Spain as a ollows. 3.3.3.1. O ganiza ional s uc u e o baske ball in USA: Baske ball Associa ion o Ame ica The Baske ball Associa ion o Ame ica (BAA) was a p o essional baske ball league in No h Ame ica, ounded in 1946. The league me ged wi h se e al leagues such as he Na ional Baske ball League (NBL) in 1949, o ming he Na ional Baske ball Associa ion (NBA). Ele en ci ies a e o una e o welcome a eam o ep esen hem, a e essen ially ci ies loca ed on he coas : Nue a Yo k, Chicago, Bos on, P o idence, To on o, Cle eland, San Luis, Washing on, De oi , Pi sbu gh y Philadelphia. The i s game is played in he ci y o New Yo k and con on s New Yo k Knicks s. To on o Huskies. A ican Ame ican playe s do no s a o play un il 1950. The e we e se e al a emp s o c ea e o he p o essional leagues o o e h ow he NBA, highligh ing he ABA League. 24 Na ional Baske ball League The Na ional Baske ball League was ounded in 1898 in he USA and was he i s p o essional league in he wo ld. Six eams ook pa in i and he i s champions we e he T en on Na ionals, ollowed by he New Yo k Wande e s, he B is ol Pile D i e s and he Camden Elec ics. The Na ional League las ed i e seasons (1904), bu new leagues quickly we e o med h oughou New England and he Mid-A lan ic S a es, p ominen among hem we e he Philadelphia Baske ball League, Eas e n Baske Ball League, Hudson Ri e League, New Yo k S a e League and he In e s a e Baske Ball League. ABA (Ame ican Baske ball Associa ion) The Ame ican Baske ball Associa ion (ABA) was ounded as an al e na i e o he NBA in 1967. Teams we e c ea ed in di e en ci ies han NBA eams. The ABA was cha ac e ized by he colo o he ball ( ed, whi e and blue) and he manne o play. The ABA also in oduced se e al ules ha di e ed om he NBA. Among hem was he h ee-poin line. The NBA adop ed he h ee poin line in 1979-80. The ABA compe ed wi h he Na ional Baske ball Associa ion ( he NBA) o playe s, ans, and media a en ion. In June 1976, ou o he s onges ABA eams ( he New Yo k Ne s, Den e Nugge s, Indiana Pace s, and San An onio Spu s) joined he NBA (Sil e man, 2012). NCAA (Na ional Collegia e A hle ic Associa ion) The Na ional Collegia e A hle ic Associa ion (NCAA) is an associa ion o se e al ins i u ions, con e ences, o ganiza ions and indi iduals ha o ganizes he a hle ic p og ams o many colleges and uni e si ies in he Uni ed S a es. I is headqua e ed in Indianapolis, Indiana. The NCAA was ounded in 1906 o p o ec young people om he dange ous and exploi i e a hle ics p ac ices o he ime. In ha pe iod, he oo ball was used in o de o o ma ion and gang ackling, bu he e we e nume ous inju ies and dea hs and p omp ed many college and uni e si ies o discon inue he spo . The mos pa o he ans and people ela ed wi h oo ball hough ha college oo ball should be e o med o abolished. P esiden Theodo e Roose el summoned college a hle ics leade s o wo Whi e House con e ences o encou age e o ms. In Decembe 1905, in New Yo k Ci y, 62 colleges and Baske ball • Complexi y Baske ball Backg ound 25 uni e si ies became cha e membe s o he In e collegia e A hle ic Associa ion o he Uni ed S a es (IAAUS). I was o icially was cons i u ed Ma ch 31, 1906, bu in 1910 was enamed as Na ional Collegia e A hle ic Associa ion (NCAA). G adually, mo e ules commi ees we e o med and mo e championships we e c ea ed, including a baske ball championship in 1939 (NCAA, 2012c). One o he keys o success in college baske ball was ha he BAA had money and good game cou s, bu lacked o alen and expe ience playe s. This lack o playe s o ces he owne o c ea e a new policy o end he p oblem. This new policy is o ec ui he bes college playe s o shape and a championship ha combines he speed, alen and expe ience. The baske ball college championship in USA is di ided in h ee di isions (Di ision I, Di ision II and Di ision III). In u n, e e y di ision is di ided in con e ences o se e al eams each. Cu en ly, in he Di ision I a e in ol ed a o al o 344 eams ( he numbe a ies wi hin he season analyzed), di ided in 31 con e ences h ough all USA ( he numbe o eams pe con e ence is no homogenous). The compe i ion o ma o he NCAA desc ibed in his hesis makes e e ence only o he Di ision I o he men´s baske ball. The e is a egula phase ( egula league), and a playo . In he egula phase eams play agains he eams o he same con e ence. In addi ion, hey play ex a games ( ou namen s) in o de o ge mo e poin s o he playo classi ica ion. Mos o hese ou namen s a e he same e e y season. Some a e e y impo an in college baske ball communi y. Some o he mos popula a e: 2K Spo s Classic, Coaches s. Cance , Pue o Rico Tip-O , Pa adise Jam, CBE Classic, Maui In i a ional, NIT Season Tip-O , Old Spice Classic, 76 Classic, Legends Classic, ACC/Big Ten Challenge, Big12/Pac10 Ha dwood Se ies o he Jimmy V Baske ball Classic. A e he egula phase, he bes eams classi ied play a playo (only one game pe ound) o he na ional championship. The e a e wo ways o quali ying o he play-o ou namen . One is di ec ly and he o he is by in i a ion g an ed by he NCAA. 26 The di ec classi ica ion is ob ained by a anking made wi h he RPI index. The RPI index is a ma hema ical equa ion ha akes in o accoun he games won, games los , a se ies o nume ical cons an s and s eng h o he schedule. Once ob ained his anking he 31 bes will quali y di ec ly o he ou namen . The playo ou namen include 68 uni e si ies (31 champions + 37 in i ed) and is held in Ma ch (also called Ma ch Madness). The 68 eams a e di ided in o ou egions (Sou h, Eas , Wes and Midwes ) and o ganized in o a single elimina ion b acke . Each eam is anked wi hin i s egion. F om he 68 eams, 60 o hem go di ec ly o he second ound emaining he 8 lowe -seeded eams, which ha e ecei ed ewe o es om he NCAA; dispu e he ou emaining places in ou small games o he i s ound (called Fi s Fou ). A e an ini ial ou games be ween 8 lowe - anked eams, he ou namen akes place o e he cou se o h ee weekends, a p e-selec ed neu al si es a ound he Uni ed S a es. Lowe - anked eams a e placed in he b acke agains highe anked eams. Each weekend cu s h ee- ou hs o he eams, om a Round o 64, o a ound o 16 also called Swee Six een, o a Final wi h ou eams, called Final Fou . The Final ou is usually played on he i s weekend in Ap il. NBA (Na ional Baske ball Associa ion) The NBA is he men's p o essional baske ball league in No h Ame ica (Uni ed S a es and Canada). The league was ounded as he Baske ball Associa ion o Ame ica (BAA) in New Yo k Ci y on June 6, 1946 (NBA, s. .). The league adop ed he name Na ional Baske ball Associa ion (NBA) in 1949 a e me ging wi h he i al Na ional Baske ball League (NBL). The NBA is cu en ly he mos signi ican p o essional baske ball league in he Uni ed S a es o Ame ica, in e ms o popula i y, sala ies, alen , and le el o compe i ion (Pa e son, 1993; Hausman & Leona d, 1994). The NBA is a league o closed s uc u e (no p omo ions and demo ions), composed by 30 anchised membe s, which 29 a e loca ed in he Uni ed S a es and one in Canada. The cu en league o ganiza ion di ides he 30 eams in o wo con e ences o h ee di isions, wi h i e eams each. The cu en di isional alignmen was in oduced in he 2004–2005 season. Baske ball • Complexi y Baske ball Backg ound 27 Du ing he egula season, each eam plays 82 games, 41 a home and 41 away. A eam plays agains i s opponen s in i s own di ision o ou imes pe season (16 games), h ee o ou imes agains eams om he o he wo di isions in i s own con e ence (36 games) and wice agains eams in he o he con e ence, espec i ely (30 games). This asymme ic s uc u e means ha he s eng h o he schedule a ies signi ican ly among eams. The NBA o ganiza ion cha is cons i u ed by he CEO and di e en depa men s, which di ec ly depend on he CEO. The mos ema kable, ega ding ano he p o essional baske ball leagues, is ha in he NBA e e ees a e p o essionals and depend di ec ly on he NBA (do no depend on he Fede a ion). The NBA depa men s a e:  Baske ball Ope a ions  B oadcas Ope a ions  Communica ions  Communi y and Playe P og ams  C ea i e Se ices  E en s and A ac ions  Facili ies and Adminis a ion  Finance and Bene i s  Global Ma ke ing Pa ne ships  Global Me chandising G oup  Human Resou ces  In o ma ion Technology  In e ac i e Se ices  In e na ional  In e na ional Media Dis ibu ion  Legal  Legal and Business A ai s  Ma ke ing  D- League (NBA De elopmen League)  WNBA 28  NBA En e ainmen P oduc ion, P og amming and Pho os  Re e ee Ope a ions  Secu i y  S a egic De elopmen  Team Ma ke ing and Business Ope a ions Sou ce: NBA Sala y cap and D a The NBA has se e al mechanisms es ablished in o de o p e en eams, wi h la ge su pluses o p o i , can sign he bes playe s a ailable, he eby acili a ing he main enance o equali y in he league. The mo e ep esen a i e mechanisms a e he sala y cap and he d a . The lo e y d a p ocess has changed along he yea s, bu exis s since NBA ounda ion; unlike he sala y cup, which s a ed in he mid-1940s, (i was abolished a e only one season); and eins a ed in he 1984–85 season. Sala y cap The No h Ame ican p o essional spo s leagues (Majo League Baseball (MLB), Na ional Baske ball Associa ion (NBA), Na ional Foo ball League (NFL), and Na ional Hockey League (NHL) ha e an ag eemen o ule ha se s a limi on he amoun o money ha a eam can spend on playe pay olls, called sala y cap (Sco , Long, & Somppi, 1985). The sala y cap, in he NBA, s a ed in 1983 (Hill & G oo huis, 2001). A simple model shows ha a sala y cap can imp o e he compe i i e balance among clubs as well as he sala y dis ibu ion among playe s (Késenne, 2000). Fo ha eason, e e y anchise has o s udy ca e ully wha ma ke playe s could be in e es ing o his p ojec (depending on he eam´s p ojec ). The sala y cup is de ined by he league's collec i e ba gaining ag eemen (CBA). The sala y cap ensu es ha each anchise can only “shield” economically one o wo key playe s, who a e o en called anchise playe s. The e a e h ee kinds o egula ions: ha d sala y cap, so sala y cap (wi h luxu y ax), and luxu y ax (The NBA u ilizes a so sala y cap) (Scully, 1989; Késenne, 2000; Fo & Maxcy, 2003): Baske ball • Complexi y Baske ball Backg ound 29  Ha d sala y cap. A ha d sala y cap is whe e he league se s a maximum amoun o money allowed o playe sala ies, and no eam can exceed ha limi . A he beginning he sala y cap was no a ha d cap (Hill & G oo huis, 2001).  So sala y cap. A so sala y cap has a se limi o playe sala ies, bu he e a e se e al majo excep ions ha allow eams o exceed he sala y cap. Fo example, in he case o he NBA, eams can exceed he sala y cap when keeping playe s ha a e al eady on he eam (Die l, F anck, Lang, & Ra hke, 2010).  Luxu y ax: A luxu y ax sys em does no ha e a limi o how much money can be spen on playe sala ies. Howe e , he e is a ax le ied on money spen abo e a h eshold se by he Collec i e Ba gaining Ag eemen (CBA) be ween he playe s union and he owne s. Fo e e y dolla a eam spends abo e he ax h eshold, hey mus also pay some ac ion o he league. This sys em is used o discou age eams om g ea ly exceeding he ax h eshold, wi h he goal o ensu ing pa i y be ween la ge and small ma ke eams (Die l, Lang, & We ne , 2008). NBA lo e y d a The NBA D a is o he p ocedu e by which, in la e June each yea , he NBA anchises join o hei eams, playe s om U.S. uni e si ies o leagues in o he coun ies. These playe s a e usually ama eu U.S. college baske ball playe s, bu in e na ional playe s a e also eligible o be d a ed. College playe s who ha e inished hei ou -yea college eligibili y a e au oma ically eligible o selec ion, while he unde classmen ha e o decla e hei eligibili y and gi e up hei emaining college eligibili y. The i s d a ook place in 1950 wi h he aim o p o ide good playe s o he league (see abo e, NCAA). Teams could o ei hei i s - ound pick and selec a playe om hei immedia e geog aphical a ea, commonly known as a “ e i o ial pick” (NBA, 2007). In 1985 hey changed o a lo e y sys em, his NBA Lo e y sys em se he o de o selec ion o he non-playo eams (o he eams holding hei picks h ough ades) o he i s ound only. Teams picked in in e se o de o hei eco ds in he second ound in all succeeding ounds. In 1990, he NBA changed he o ma o he lo e y o gi e he eam wi h he wo s eco d he bes chance o landing he i s pick. Cu en ly NBA D a consis s o wo ounds, he i s and 36 FEB (Spanish Baske ball Fede a ion) I is a non-p o i o ganiza ion which in cha ge o p omo ion, managemen , and coo dina ion h oughou he na ional e i o y o baske ball, in all i s mani es a ions and a ia ions. The a m leagues a e managed by FEB (semi-p o essionals and a m leagues), and also manage he Au onomous Regions Fede a ions ( o ama eu compe i ions), bu he p o essional league o Spain is managed by ACB (associa ion o se e al spo clubs). All he pa icipan s in compe i ions o ganized by FEB a e in eg a ed in o he ede a ion, such as co po a ions, spo s, spo s clubs, a hle es, coaches and e e ees. And also i is esponsible o issuing all licenses necessa y o pa icipa ing in i s ac i i ies. The FEB is a ilia ed o FIBA as a membe , being obliged, he e o e o ollow i s s a u es and egula ions in all ma e s a ec ing he echnical o de and in e na ional ela ions. In e na ionally, he FEB is he ep esen a ion o Spanish baske ball in baske ball in e na ional o icial ac i i ies and compe i ions celeb a ed wi hin and ou side he Spanish e i o y. The FEB elabo a es he os e s o na ional eams (senio s and a m eams). The echnique s uc u es o he Baske ball Fede a ion a e: 1. Adminis a ion and ep esen a ion:  Gene al Assembly and Execu i e Commi ee  The p esiden 2. Managemen :  Execu i e Commi ee  Commi ee on Regional Fede a ions 3. Consul ing:  A ea execu i e commi ee 4. In e nal Managemen Sys em:  Gene al Sec e a y  Managemen  Those ha could be c ea ed o be e ul ill he ede a i e pu poses Baske ball • Complexi y Baske ball Backg ound 37 5. Technical-Spo ing  Gene al Sec e a y  Compe i ion  Re e ees  Coaches  Commi ee on Heal h and P e en ion o Doping 6. Discipline  Na ional Compe i ion Commi ee  Na ional Appeals Commi ee ACB (Baske ball Club Associa ion [Asociación de Clubs de Balonces o]) The ACB League is nowadays he main men's p o essional baske ball league in Spain. I began in 1957, wi h he name o Na ional League, and was o iginally o ganized by he Spanish Baske ball Fede a ion (FEB). In 1983-84 season he ACB was es ablished wi h i s own compe i ion o ma and eplaced he Na ional League. The league is a ed as one o he h ee "A" le el Eu opean na ional domes ic leagues in he ULEB League Rankings sys em (ACB, 2012). In Spain, he p o essional leagues a e p i a e s uc u es ha ca y ou unc ions o public in e es and a e supe ised by he Na ional Spo Council (Millán Ga ido, 2010). The ACB model p esen s an open s uc u e which means he e a e p omo ions and demo ions. The op eams classi ied in he egula season play a championship in a play-o o ma . The las anked eams a e elega ed o a lowe di ision and in u n a e eplaced o he wo op anked eams o he bo om ca ego y. The season 2011-12 ha e pa icipa ed a o al o 18 eams, bu his numbe has a ied in p e ious seasons ( om 13 un il 24 eams). The ACB spo model consis s in a egula season o double con on a ions (only wo games agains he same eam). The o de o each eam's i s -hal ix u es is epea ed in he second hal o he season. 38 Conce ning o he adminis a i e s uc u e, he ACB is he associa ion o se e al clubs (which ha e hei own eams ( a m eams) and e en o he di e en spo eams) wi h i s own o ganiza ional s uc u e. I means ha some eams can pa icipa e in o he Eu opean leagues simul aneously, such as Eu oleague, ULEB, e c. unlike he NBA and NCAA. Bu he pa icipa ion in hese leagues is ela ed wi h he s anding o he p e ious season. O he di e ence wi h he NBA and NCAA is ha he e e ees belong o he Spanish Baske ball Fede a ion (FEB). The cen al adminis a ion e ol es a ound a S ee ing Commi ee composed o i e membe s on di ec dependency on he Di ec o a e Gene al. The maximum esponsible has he Gene al Assembly as he ul ima e au ho i y o he inal decision. The i e membe s shall be di ec o s o he six s a egic a eas plus he Di ec o o e e ees. The cu en o ganiza ional s uc u e o he ACB comp ises six a eas: Compe i ion, Media, E en s, Ins i u ional, Comme cial and Adminis a ion. The las h ee sec ions also co e he es o he o ganiza ion ans e sally. The Compe i ion A ea includes he depa men s o Compe i ion, Scou ing and Ex e nal Rela ions, including wo k and ela ionship wi h he a ious in e na ional compe i ions: Eu oleague, ULEB and o he s. In on o his a ea will be he CEO o he ACB. The TV and Communica ion a ea con ains he depa men s o Communica ion, ACB.COM, Audio isuals and Tele ision. This new a ea was c ea ed wi h he pu pose o managing, in a comp ehensi e manne , he ela ionship wi h ele ision ope a o s o he Associa ion. The a eas o E en s Managemen and y o inc ease b and alue and cos o majo e en s in he ACB such as he Copa del Rey and he Supe copa, and also deal wi h Finance, In o ma ion Technology and S a is ics issues. The Ins i u ional A ea (Gene al Sec e a y) is esponsible o all legal a ai s and documen a ies o he Associa ion and manages he Human Resou ces Depa men and Gene al Se ices. Baske ball • Complexi y Baske ball Backg ound 39 Business A ea (Business and Ma ke ing) includes he depa men s o Ma ke ing and Business De elopmen . These a e key in gene a ing income o he o ganiza ion h ough na ional and in e na ional ag eemen s, sponso ship plan and i ness cen e s p omo ed by he Associa ion. Figu e 6. Cu en o ganiza ional s uc u e o he ACB. Sou ce: ACB. 3.3.4. S uc u al compa ison be ween beginning and he p esen The a chi ec u e o he baske ball ne wo k can p o ide us a new and good pe spec i e o how is he o ganiza ion o baske ball. Ce ain o ganiza ional and unc ional p inciples in complex sys ems a e uni e sal because some ne wo ks ha e simila i ies o o he biological and echnological ne wo ks (Solé & Goodwin, 2002). Hence we ep esen ed he baske ball ne wo k ( ocused on ACB and NBA) in he season 1983-1984 (Figu e 7), when he ACB was ounded, and in he season 2008-2009 (Figu e 8), in o de o unde s and hei e olu ion du ing all hese yea s. We simpli ied some s uc u es (such eu opean leagues) o be e unde s anding. 40 Figu e 7 Rep esen a ion o he ne wo k o ins i u ions and baske ball compe i ions. Some s uc u es (such Fa m Leagues and Eu opean leagues) ha e been simpli ied o be e unde s anding. The ci cles ep esen he compe i ions (leagues/ ou namen s/championships) and he squa es ep esen he ins i u ions. We can see how some s uc u es can be conside a e hyb ids, because hey a e compe i ions and o ganiza ions simul aneously. No all he ela ionships a e he same (some ela ionships a e unidi ec ional and o he s bidi ec ional). The e a e dissipa i e s uc u es (such he NCAA) and s uc u es whose goals a e o condense o concen a e he esou ces. Bu he mos ac i e s uc u es o he ne wo k a e hose ha consume and p oduce esou ces a he same ime (such he ACB). We no e ha he ne wo k is based on hese kinds o s uc u es. The iangle based on he ACB-NBA - Spo Clubs is he engine o he sys em. 1983-1984 Figu e 8. Illus a ion o baske ball ne wo k in he season 2008-2009. The ep esen a ions o he elemen s a e he same as be o e: ci cles symbolize he compe i ions (leagues/ ou namen s/championships), squa es ep esen he ins i u ions, and inally squa e plus ci cles a e hyb id s uc u es. The ne wo k has e ol ed: some s uc u es ha e disappea ed and new eme ged. E en some s uc u es ha e adap ed o he e olu ion in ime and he g own o he ne wo k by a bi u ca ion o one o i s s uc u es (such FIBA in FIBA Eu ope), in o de o p ese e he e ec i eness o he low wi hin he ne wo k. We a e s ill seeing how he iangle o med by ACB-NBA-Spo Clubs is he key o he ne wo k, bu whi he eme gency o mo e p o essional leagues, he ela ionships ha e been modi ied. The ne wo k has become mo e p o essional. 2008-2009 Baske ball • Complexi y Baske ball Backg ound 41 42 The Figu e 7 and he Figu e 8 ep esen ou p oposal o he baske ball ne wo k g aph using he s uc u es exis ing in Eu ope and USA and he ins i u ional ela ionships among hem. We p oposed playe s as aw ma e ial and he a ows as he channels wha hey ollow. Hence we can see, o example, he pa h ha a playe can ake om a domes ic league o he Wo ld Championship o he Olympic Games. We simpli ied some s uc u es such he Eu opean leagues (G eece League, I alia League, C oa ia League, e c.) and a m leagues in Spain (Gold LEB, Sil e LEB, EBA, e c.) inasmuch as we conside ed hem as simila s uc u es wi h he same p ope ies. A he g aph he ci cles ep esen he compe i ions (leagues/ ou namen s/championships). The squa es ep esen he ins i u ions, such as he FIBA o he IOC. The e some s uc u es which a e compe i ions wi h i s own o ganiza ional s uc u e ( hey can be independen o semi-independen om o he s uc u e such as ede a ion), o ins ance ACB o NBA, hence can be conside a e hyb ids, because hey a e compe i ions and ins i u ions simul aneously. The e is a s uc u e which we highligh ed as special, because has a composi ion e y di e en om he o he s, we e e o he spo clubs in Spain. This elemen possesses some p ope ies di e en ha a eam: a e managed by he p esiden , he boa d and gene al assembly. The e a e clubs de o ed exclusi ely o baske ball ac i i ies and o he clubs whe e he same spo club has se e al sec ions: oo ball, baske ball, olleyball, handball, hockey, e c. In addi ion, clubs ha e playe a ms, whe e hey ain young playe s and club eams compe ing in a ilia ed mino leagues, in o de o c ea e a young spo s a , o as a showcase o o he clubs. This s uc u e is mo e wide and de eloped han a eam, and enables he club maximize i s esou ces, sha ing common managemen s uc u es, acili ies, human esou ces, ma e ial esou ces, economic, e c. which enable o p epa e os e s much mo e compe i i e. The e a e some p oduc i e/dissipa e s uc u es (such he NCAA o FIBA) whose aim is o p oduce o sp ead he ma e ial in he ne wo k. On he o he hand he e a e s uc u es whose goals a e o condense o concen a e he esou ces, o ins ance he na ional ede a ions. Bu he mos ac i e s uc u es o he ne wo k a e hose ha consume and p oduce esou ces a he same ime (such he ACB). We no e ha he ne wo k is based on hese kinds o s uc u es, and he igu e o he spo clubs a e he mos ep esen a i e. The union o hese s uc u es is he engine o he sys em, as we can obse e a he iangle based on ACB-NBA-Clubs. Baske ball • Complexi y Baske ball Backg ound 43 Rega ding o he connec ions no all he ela ionships a e he same (some ela ionships a e unidi ec ional and o he s bidi ec ional). They indica e de low wi hin he ne wo k and he s uc u es which a e connec ed somehow. We can obse e how a he i s g aph (Figu e 7), in he season 1983-1984 he mos pa o he leagues a e ama eu such as he Spanish Fa m Leagues, he ACB (be o e become p o essional) and he NCAA, which p o ide playe s o he USA na ional eam (Olympic Games). These s uc u es a e which suppo he mos pa o he in e na ional compe i ions in addi ion o he NBA (p o essional league). The Figu e 8 ep esen s ou p oposal o he baske ball ne wo k in he season 2008-2009. We can see how some s uc u es ha e disappea ed and ha e appea ed new ones. The mos ema kable ac is ha he p o essionaliza ion o he baske ball has in luenced no ably in he a chi ec u e o he ne wo k. We can app ecia e how he playe s o he USA na ional eam, now a e p o ided by he NBA (p o essional league), and he same happens in Spain, he ACB (now p o essional) supplies playe s o he na ional Spanish eam (e en he NBA p o ides playe s o he Spanish na ional eam nowadays). An impo an ac is ha he FIBA has c ea ed a Eu opean delega ion (FIBA Eu ope was ounded in 2001). This p ocess o eme gency indica es a high g ow h o he low o in e na ional compe i ions be ween na ional eams in Eu ope, in ou case. Mo eo e , he Eu opean compe i ions ha e changed hei o ma o ha e been eplaced wi h new ones. As we men ioned, in he season 1983-1984 hese compe i ions we e played by spo clubs, in Eu ope. Bu in he season 2008-2009 a e played by na ional eams (compe i ions which a e dependen on he FIBA Eu ope). The mos ema kable ac ela ed wi h he design o he ne wo k, is he appea ance o wo hyb id s uc u es (as we called), he ULEB ( ounded in 1991) and he Eu oleague ( ounded in 2000). This eme gency is, p obably, one o he sou ces o he ne wo k es uc u a ion, because we ha e o ake in o accoun ha bo h o hese compe i ions a e played by spo clubs, ins ead o na ional eams. In ac , in 2000, majo p o essional spo clubs on Eu ope, led by he Spanish, I alians and G eeks, g ouped in he Union o Eu opean Baske ball Leagues (ULEB), spli o om he FIBA in 44 o de o o ganize a new Eu oleague wi h mode n managemen c i e ia. These clubs wan ed o ecei e mo e e enue om ele ision b oadcas ing igh s and me chandising han he o e ed by he FIBA. In addi ion, he NBA c ea ed i s own a m league: he NBA De elopmen League (NBA D- League); in which pa icipa ing eams plays hei own league ( he e is no p omo ions o he NBA). Some NBA eams sha e he esou ces bu playe s. This league depends on he NBA. We pas om 8 compe i ions, 6 ins i u ions and 4 hyb id s uc u es, in season 1983-1984, o 7 compe i ions, 8 ins i u ions and 6 hyb id s uc u es, in season 2008-2009. No e ha he enla gemen is ela ed wi h he hyb id s uc u es, which indica e us ha he p o essionaliza ion p o ides he appa i ion o hese kinds o elemen s. These a e good examples o he how a p ocess such as he p o essionaliza ion o a spo , can change he composi ion o he i s s uc u e. We also can app ecia e how he clus e ing is no homogeneous in he wo ne wo ks ( able 2). Table 2. Numbe o connec ions o he ne wo k nodes in he season 1983-1984 (le ) and in he season 2008-2009 ( igh ). 1983-1984 2008-2009 Agen Connec ions Agen Connec ions FIBA 8 ACB 8 ACB 7 NBA 8 Clubs 5 Spo Agencies 7 NBA 5 Clubs 6 NCAA 5 FIBA 6 Eu opean Leagues 4 FIBA Eu ope 6 FEB 4 Eu opean Leagues 5 IOC 4 NCAA 5 Fa m Leagues 3 FEB 4 USA Baske ball 3 IOC 4 COE 2 NBA D-League 4 Ko ac Cup 2 Fa m Leagues 3 Spo a Cup 2 Mino Leagues 3 CSD 2 USA Baske ball 3 Mino Leagues 2 COE 2 Eu oCup 1 CSD 2 Eu opean Championship 1 Eu oleague 2 Olympic Games 1 ULEB 2 Wo ld Championship 1 Di ision C 1 Eu oCup 1 FIBA Challenge 1 Wo ld Championship 1 Baske ball • Complexi y Baske ball Backg ound 45 The e a e o he s uc u es ha enhance he mo emen o playe s, which no pa icipa e di ec ly in he ne wo k, bu in luence s ongly in i . We e e o he Playe Agencies (o Spo Agen s). Thei aim is o localize playe s and mo e o ano he s uc u e o he ne wo k whe e is needed; in e u n o pa o he bene i c ea ed. This con i ms ha he na u e o he baske ball ne wo k has changed and ha he lows a e no cons an s. 3.4. Compe i ion Analysis Se e al disciplines such as economy, applied s a is ic, physics, e olu i e biology, social sciences, spo sciences, e c. ha e been acking he e olu ion o a sys em wi hin a con olled en i onmen , o en h ough analysing he in e ac ions o he agen s in ol ed. Ou aim was o in es iga e om an o e iew, he in e n dynamic o p o essional baske ball leagues s udying i s compe i i eness deg ee. The deg ee o equali y o he playing s eng hs o eams, compe i i e deg ee o compe i i e balance, is a cen al concep in he analysis o p o essional spo s leagues. The e is conside able in e es in cla i y he skewness and luc ua ions in compe i i e balance h oughou seasons; and analysing he e ec s o egula o y, ins i u ional and o he changes, as indica ed by he ex ensi e li e a u e on he subjec (Schmid & Be i, 2001; Fo & Maxcy, 2003; T. A. Rhoads, 2004; Goossens, 2006) and applied in di e en spo s such as baseball (Scully, 1989; Owen, Ryan, & Wea he s on, 2007), Ame ican oo ball (Humph eys, 2002), baske ball (Noll, 1988; Be i, B ook, F ick, Fenn, & Vicen e-Mayo al, 2005), ice hockey (Richa dson, 2000), Eu opean oo ball (socce ) (Halicioglu, 2006) o gol (T. Rhoads, 2005). The mos pa o wo ks deal wi h his phenomenon wi h ega d o he mechanics o he game i sel , meaning he game in isola ion, wi hou implica ions o he compe i ion (league)(Cha e jee & Yilmaz, 1999; McGa y, Ande son, Wallace, Hughes, & F anks, 2002; Lebed, 2006; McGa y & F anks, 2007; Passos e al., 2008; Passos, A aújo, Da ids, Milho, & Gou eia, 2009), ne e heless, ew wo ks do om he pe spec i e o compe i ion be ween eams in di e en spo s (Yilmaz & Cha e jee, 2000; Malaca ne & Mendes, 2000; Onody & de Baske ball • Complexi y Complex Sys ems Backg ound 53 4. Complex Sys ems Backg ound A complex sys em is a se o se e al elemen s (also called agen s) which a e ela ed among hem and whose links con ain in o ma ion hidden o he obse e . The es ablished ela ionships among hem a e mainly ype non-linea . These in e ac ions a e local in e ac ions. Tha is, a ec only he ela ionship be ween an agen and o he elemen s which su ound him, bu none o hem is awa e o he collec i e beha io (Goodwin, 2002; Vicsek, 2002; Ama al & O ino, 2004; Solé, 2009). These p ocesses, ha ake place simul aneously on di e en le els o scales, a e impo an . In ac , he way in wha i s uni s a e ela ed, g ea ly in luences in he ou pu o he en i e sys em. Tha is why he laws ha desc ibe he beha iou o a complex sys em a e quali a i ely di e en om hose ha go e n i s uni s (Ama al & O ino, 2004; Vicsek, 2002). As a esul o hese in e ac ions, new p ope ies eme ge ha canno be unde s ood om he indi idual ea u es o each elemen . These p ope ies a e called eme gen p ope ies. Tha is why a complex sys em mus be ea ed as a whole, om a holis ic concep ion, no jus he elemen s ha cons i u e i because in a complex sys em he whole is g ea e han he sum o he pa s. Complexi y is he esul o incessan adap i e p ocesses (Holland, 1995). 4.1. No-linea When he sys em is linea , he same s imulus always p oduces he same ou come. E e y ime he p ocess is epea ed, he same ou come will be ob ained. On he o he hand, i he sys em is non-linea , a s imulus can yield se e al esul s. Al hough he condi ions a e he same, he ou come o ou comes canno be known in ad ance (P igogine & Hol e, 1993; Solé & Goodwin, 2002; Ama al & O ino, 2004). The in e ela ionships o he componen s o he complex sys em a e go e ned by non-linea equa ions. As men ioned abo e, no always e ec i eness in spo shows a linea beha io , bu he e a e se e al ac ions ha can be conside ed e ec i e, and also do no ha e o be consecu i e. Complexi y, in i sel , is a measu e o he numbe o possibili ies. Such equa ions o en ha e a s ong dependence on ini ial condi ions o he sys em, which makes i e en mo e di icul o assess hei beha io 54 4.2. Sel -o ganiza ion The idea o sel -o ganiza ion can be exp essed as he gene al endency o a gi en sys em o gene a e beha io pa e ns om local in e ac ions o i s cons i uen elemen s and om he ela ionships wi h he en i onmen . I is he essen ial pa o any complex sys em and allows he sys em o eco e he balance, modi ied and adap ed o he su ounding en i onmen . Usually, he di e en sys em elemen s a e sel - egula ed by hemsel always seeking o op imize he o e all ope a ion o he assembly. The complex ne wo k o in e dependen sys ems in which human beings can o ganize, o example, is changing and eadjus ing o eali y ha co esponds o li e in each momen (Ga cía Manso & Ma ín González, 2008). Sel -o ganiza ion is a p ocess in which he in e nal o ganiza ion o a sys em inc eases in complexi y wi hou being guided o managed by an ou side sou ce. Sel -o ganizing sys ems usually display eme gen p ope ies. The o de and diso de need each o he , mu ually occu . They a e an agonis ic concep s bu complemen a y a he same ime. In some cases, some o diso de allows a di e en o de and some imes, iche . Fo example, an o ganism can pe sis as a esul o he dea h o i s cells, o an o ganiza ion is pe pe ua ed by he dismissal o i s membe s. The a ia ion and change a e ine i able and una oidable s ages h ough which e e y complex sys em mus a el o g ow and de elop. When his ans o ma ion is achie ed wi hou he in ol emen o ex e nal ac o s o he sys em, e e ed o a p ocess o sel -o ganiza ion (Nicolis & P igogine, 1977). Sel -o ganiza ion s ands ou as an essen ial pa o any complex sys em. I is he o m h ough which he sys em eco e s he balance, changing and adap ing o he su ounding en i onmen (i esponds o ex e nal agg essions ha seek o modi y i s s uc u e). In his kind o phenomena is essen ial he idea o le els. The in e ela ionships among he elemen s o a le el o igina e new ypes o elemen s on ano he le el which beha e qui e di e en ly, o example, om molecules o mac omolecules, mac omolecules in o cells and om he cells o issues. Thus, he sel -o ganizing sys em is buil as a esul o inc easing o de space- ime which is c ea ed in on di e en le els o laye s, one abo e he o he . To a la ge ex en , complex sys ems can be unde s ood like a machine ha gene a e o de , which equi es cons an ene gy in ake gene a ed by he chaos ha eeds (i is an open sys em Baske ball • Complexi y Complex Sys ems Backg ound 55 and dissipa i e). The sel -o ganizing complex sys ems a e conside ed adap i e because i can eac o ex e nal s imuli and esponding o any si ua ion ha h ea ens i s s abili y as a sys em. Thus, i expe iences luc ua ions. This has a limi , o cou se. I is said ha he sys em se les in o a s a e and when i is away om him ends o make e e y e o o e u n o he p e ious si ua ion. This happens o example wi h he human body cons an ly s i es o main ain he same body empe a u e. 4.3. C i ical S a e The gene al idea o sys em in a c i ical s a e can be unde s ood as a s a e close o he bounda y o ano he s a e (c i ical poin ). Meaning ha any sligh pe u ba ion, can lead o a new s a e (phase ansi ion). One o he mos amous example is he sandpile model by Bak–Tang–Wiesen eld (Bak, Tang, & Wiesen eld, 1987). The model desc ibes how a sandpile is builds up as g ains o sand andomly placed on o a pile. A he beginning small pe u ba ions only cause small esponses. Small a alanches ake place un il he pile eaches a c i ical s a e in which i s slope luc ua es abou a cons an angle o epose ( h eshold o c i ical poin ). I we add one sand g ain mo e, his can causes he slope exceeds he c i ical alue and o igina es a big a alanche. The a ia ion o he local slopes makes i impossible o p edic when his phenomenon will ake place. C i ical sys ems a e ea u ed by be in a delica ely balanced s a e which, in u n, is linked o he en i onmen , showing a g ea sensibili y (Jos , 2005). This si ua ion gi es hem a highly unp edic able beha io (chao ic, no andom). The mos pa o he complex sys ems a e uns able ( hey a e ou o he equilib ium s a e). This implies ha he sys ems canno sus ain hemsel es unless hey ecei e a cons an supply o ene gy (o de needs chaos and chaos needs o de . They canno exis wi hou each o he , as men ioned ea lie ). They demand adjus men s ollowing speci ic pa e ns. Any minimum a ia ion among composing elemen s can modi y unp edic ably, he in e ela ions and he e o e, he beha io o en i e sys em. Thus, he e olu ion o such sys ems is cha ac e ized by in e mi ency o luc ua ion (si ua ion in which he o de and diso de cons an ly al e na e). Thei e olu iona y s a es do no pass h ough con inuous and g adual p ocess, bu occu h ough eo ganiza ions and jumps. Each new s a e is only a ansi ion, a pe iod o en opic 56 es in he wo ds o Russian-Belgian Nobel P ize Ilya P igogine (P igogine & S enge s, 1984; P igogine & Hol e, 1993). These sys ems ne e each a global op imum, he minimum ene gy s a e. In gene al, g ow g adually un il hey each he limi o i s po en ial de elopmen . A ha momen , hey su e a diso de , a kind o up u e ha induces a agmen a ion o p e-exis ing o de . Bu hen, begin o eme ge egula i ies ha o ganize he sys em in acco dance wi h new laws, p oducing ano he kind o de elopmen . This beha io is ypical in na u al sys ems: o example, he ansi o he insec s, om egg o la a and om he e o he ch ysalis. Consequen ly, he o ganiza ion o complex sys ems is gi en a di e en le els. The laws go e ning he causali y o a gi en le el can be o ally di e en om a highe le el (Kau man, 1995; Bak, 1999). 4.4. Sel -O ganized Sys ems and Spo Unde hese kinds o limi si ua ions, in spo , a hle es and hei en i onmen ha e o make a big e o in o de o o e come he ci cums ances. In ha momen is when hey can eally lea n. I is a his ime when spo s sys ems c ea e new s a egies, aining plans and, he e o e, is when hey e ol e, change o beha e acco ding o he new eali y. Tha is, he i al y and compe i i eness a e he elemen s ha gene a e he c i ical beha io . When he sys em is sel -o ganized c i ically, in o ma ion lows be e among all pa s o he sys em (Solé, 2009). Mo eo e , hese kinds o sys ems ha e memo y and egula o y mechanisms ha adjus s he esponse o demand. These sys ems e ol e ying o op imize hei esou ces and end na u ally o be in hese s a es, he e o e, se e as a ac o s o he sys em (I ance ic & I ance ic, 2006). I.e., he ope a ion o sys em is he key and no he indi idual ea u es o i s elemen s. All we know wha is eally in e es ing in spo is he compe i ion. Compe i ion a ac s la ge masses o public, media and, equen ly, la ge amoun s o inancial esou ces. Usually, his leads spo s (especially in eli e) o play in a c i ical a ea (Ga cía Manso & Ma ín González, 2008), in he edge o he e o , isking, compe ing nex o he limi . This phenomenon p omo es ha spo e ol es. Playe s change hei game s yle, eams change ac ics, game dynamic changes as well, new aining me hodologies eme ge in o de o Baske ball • Complexi y Complex Sys ems Backg ound 57 suppo compe i ion equi emen s, e c. And e en we can see how some spo in oduce new ules (o modi y old ules) in o de o main ain compe i ion a ac i eness. Some ules such as o side in ugby, 24 seconds sho clock in baske ball, h ee ouches in olleyball, a s olen base in baseball, e c. a e a emp s o lead spo s o c i ical a eas. Because spo adap and he na u al endency leads o a hie a chical s uc u ing mo e o less de ined. Hence, he e o s o some spo s in o de o a oid hese kind o si ua ions. As men ioned agen s, who pa icipa e in hese spo sys ems, compe e among hem; and he na u al endency leads o hie a chical s uc u es, whe e some eams a e clea ly supe io o o he s. Theo e ically, his si ua ion could be ex ended in ime and ha dly be b oken by na u al means, because bes eams would con inue hogging he bes esou ces. This phenomenon is known as P e e en ial A achmen (Ba abási & Albe , 1999), o Snowball E ec o Sain Ma hew E ec . I is he popula he ich ge iche and he poo ge poo e . So, heo e ically, we can poin ou ha his si ua ion will con inue as long as no ex e nal sou ce modi ies he en i onmen in which he spo is de eloping ( ules, compe i ion spo model). Tha is why so impo an o igu e ou he ope a ion o he spo sys em, meaning league, game eam, e c. and how modi ica ions ( ules, new elemen s, e c.) a ec he en i e sys em. The c ea ion o modi ica ions o hese sys ems usually ollow ce ain laws, meaning ha some o hese phenomena p esen he same ea u es. One o he mos impo an examples is he appea ance o Powe Laws o hea y- ailed dis ibu ions. This dis ibu ion is ollowed by many na u al phenomena, o en ac al, a e also e iden in many no na u al sys ems. A lo o elemen s in e ac o p oduce a s uc u e o highe le el. These sys ems e ol e a om equilib ium and a e o en highly dissipa i e (sys ems a om equilib ium). The Powe Laws a e desc ibed by ma hema ical exp essions such as: Y=cXb Whe e X and Y a e wo a iables, o obse able quan i ies, c is a cons an and b is he scaling exponen . This kind o exp ession has wo p ope ies: 58 1) The loga i hmic ans o ma ion becomes a line (see Figu e 9): log(Y) = log(c) + b log(X) 2) I is in a ian o scale changes (scale- ee). Figu e 9. Example o a dis ibu ion (uppe panel) and i s log-log plo ans o ma ion (lowe panel). Phenomena wi h his ype o beha io (Powe Laws) a e also called scale- ee. By scale we mean he spa ial and empo al dimension o a phenomenon. The hypo hesis o scale ha ises in he con ex o he s udy o c i ical phenomena led o wo ca ego ies o p edic ions, bo h o which ha e been well e i ied by a la ge amoun o expe imen al da a on a ious sys ems. One 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0 50 100 150 200 250 100100.1 100.2 101 102 Baske ball • Complexi y Complex Sys ems Backg ound 59 o he mos impo an is he scaling law we ha e men ioned; i s use ulness lies in linking he a ious c i ical exponen s ha cha ac e ize he singula beha io o he o de pa ame e and esponse unc ions (Ama al & O ino, 2004). Mo eo e , his kind o dis ibu ion can poin ou phenomena such as ac ali y (Ba abási & Albe , 1999), sel -o ganized c i icali y (Dha , 1990; Bak, 1999), clus e ing (Newman, 2001a; Albe & Ba abási, 2002), alome ic laws (Wes , B own, & Enquis , 1997a), e c. In sho , hey indica e he possible p esence o complex sys ems. He e we p esen se e al examples o di e se elemen s which ollow Powe Laws: Figu e 10. Examples o Powe Laws: (a) Wo d equency, (b) Ci a ions o scien i ic pape s, (c) Web hi s, (d) Copies o books sold, (e) Telephone calls, ( ) Magni ude o ea hquakes, (g) Diame e o moon c a e s, (h) In ensi y o sola la es, (i) In ensi y o wa s, (j) Weal h o iches Ame icans, (k) F equencies o amily names and (l) Popula ions o US ci ies. Sou ce: (Newman, 2005). 60 Also in spo he e a e a lo o examples o his kind o dis ibu ions: a hle ics eco ds (Ka z & Ka z, 1999; Sa aglio & Ca bone, 2000), powe li ing (Ga cía Manso e al., 2008), goals dis ibu ions (Malaca ne & Mendes, 2000; Mendes e al., 2007), enu e leng hs o spo s manage s (Aid , Leong, Saslaw, & Sg oi, 2006), sco ing in baske ball (de Saá Gue a e al., 2013), e c. As we can see, spo in gene al is a good example o complexi y, he e o e we belie e accu a e o use his me hodology in o de o analyze baske ball. Gene al Resea ch Design Baske ball om he pe spec i e o non-linea complex sys ems Y es de Saá Gue a 2013 68 Finally we wan emphasize he applicabili y o he s udy in se e al ields. One o hem could be de eloping o in e en ion s a egies o c ea ing o compe i ion pa e ns, managemen and eaching o playe s, coaches and spo s o ganiza ions s a . And e en o he s a o en i ies ou side he spo , bu wi h links o he spo s ne wo k. In he academic ield, i is impo an because we can lea n and apply new echniques o o he ields o knowledge. To open new esea ch lines ha can be ollow by o he esea ches, and pass on hose disco e ies wi h he me hodologies o he uni e si y and scien i ic communi y. Following, we p esen (Figu e 11) he gene al esea ch design, in o de o a be e unde s anding o he documen : Baske ball • Complexi y Gene al Resea ch Design 69 Figu e 11: Gene al Design o Resea ch. Subjec o s udy Theo e ical Backg ound Hypo hesis Gene al Resea ch Design S udy 1 Objec i es Me hodology Resul s Discussion Conclusions S udy 2 Objec i es Me hodology Resul s Discussion Conclusions Gene al Conclusions Fu u e Resea ch Lines S udy 3 Objec i es Me hodology Resul s Discussion Conclusions Gene al Conclusions S udy 1. Baske ball league Baske ball om he pe spec i e o non-linea complex sys ems Y es de Saá Gue a 2013 Baske ball • Complexi y S udy 1. Baske ball League 73 6. S udy 1. Baske ball league 6.1 In o A league is a compe i i e model. In a spo , he design o he compe i i e model is as impo an as he p epa a ion o he subjec s in ol ed (playe s, coaches, o icials, e c.). Team pe o mance can be pos ula ed as winning as many games as possible. Final s anding is he esul o he way in which all eams in e ac o in a p e-schedule calenda . Tha is he compe i ion model. The e o e, he e ec i eness o a eam is closely condi ioned by is he esul o ce ain s a es o op imiza ion o sys ems ha compose i , which also ha e a ecip ocal ela ionship wi h eme ging and c i ical en i onmen : he compe i ion. The s udy o he equilib ium be ween sys ems which in e ac wi hin a same en i onmen , is a equen issue by disciplines such as economy, applied s a is ic, physics, e olu i e biology, social sciences, spo sciences, e c. Gi en he di icul y o p edic ing esul s o he games, and he e o e he inal s anding, we canno use a linea me hodology o analysis, i is necessa y o use a me hodology ha allows us o explo e he na u e o he compe i ion wi h as much de ail as possible, as is he heo y o complexi y. The poin is o unde s and he spo om a sys emic concep ion and concei e he a hle e, o eam in ou case, as a sys em ha wo ks as a whole ha is a ec ed by he su ounding en i onmen (Gambe a, 1989; Ga cía Manso & Ma ín González, 2008). The compe i i e model has a di ec in luence on he compe i ion ( ype o con on a ion), de elopmen and e alua ion so ha small changes can d ama ically al e he inal esul , gi en he close ela ionship be ween he compe i i e model and compe i ion (de Saá Gue a e al., 2012; Lebed, 2006). The eam abili y o compe e and how he championship wo ks (how compe i ion o ma is designed: con e ences, di isions, game schedule, league, playo , e c.) de e mine he le el o compe i i eness. Compe i i eness is a compa a i e concep o he abili y o s i e o a goal. The mo e balanced compe i ion, he g ea e he deg ee o compe i i eness, and ice e sa. This is an in e es ing because i e lec s he eali y o he compe i i e sys em, e.g. highe 74 budge s allow signing playe s o be e quali y. Whe eas igh e budge s do no allow hi e bes playe s, gi en hei high cos . One o he mos widesp ead ideas o explain he phenomenon o equali y among he compe i o s o he same championship is he concep o compe i i e balance. Compe i i e balance ep esen s he deg ee o equali y wi hin a championship. A cen al concep used in he economic analysis o p o essional spo s leagues, as indica ed by he ex ensi e li e a u e on he subjec (Schmid & Be i, 2001; Fo & Maxcy, 2003; T. A. Rhoads, 2004; Goossens, 2006). The idea o compe i i e balance is o y o measu e he deg ee o global compe i i eness in a gi en league. And speci ically has been applied in disciplines such as baseball (Owen e al., 2007; Scully, 1989), Ame ican oo ball (Benne & Fizel, 1995), baske ball (Noll, 1988; Be i e al., 2005), ice hockey (Richa dson, 2000), oo ball (socce ) (Halicioglu, 2006) o gol (T. Rhoads, 2005). Thus, g ea e compe i i e balance should lead o g ea e demand (Qui k & Fo , 1997; Goossens, 2006). Indeed, he mos compe i i e leagues end o be mo e a ac i e and gene a e mo e e enue ( icke s, sponso s, TV, e c.)(Szymanski, 2003) and his is closely ela ed wi h he spo model (Ribei o e al., 2010). The key elemen o he economic success in he p o essional spo is he inc ease in compe i i e balance. Each ime a compe i o eaches a e y high domain le el, he compe i ion equilib ium ( he way in which eams compe e) is b oken down, in he sense ha he unce ain y declines signi ican ly. In hese si ua ions, when he unce ain y o he ou come diminishes, he in e es o he compe i ion may educe conside ably. When his happens, he a endance o spec a o s o he games can dec ease and, consequen ly, access o inancial esou ces may be comp omised (Be i e al., 2005). Fo his eason, spo o ganiza ions which design spo compe i ions models (leagues), y o design s uc u es and ules which enable cope wi h a dec ease o compe i i eness in a championship. A ce ain le el o compe i i e balance seems easonable o hold he in e es o spec a o s and sponso s o all eams bu he de e mina ion o he op imal le el is e y complex. Some au ho s (Knowles e al., 1992; Rasche , 1999)no ed ha an a endance in Majo League Baseball is maximized when he p obabili y o he home eam winning is app oxima ely 0,6. I he home eam has a highe p obabili y o inding success, we can expec an a endance o decline. Consequen ly, gi en he impo ance o an a endance o a league’s inancial success, Baske ball • Complexi y S udy 1. Baske ball League 75 leagues a e expec ed o implemen ules and ins i u ions designed o add ess he ela i e s eng h o eams on he games. We s udied he esul s om di e en seasons o wo o he main p o essional baske ball leagues, he NBA (Na ional Baske ball Associa ion, USA) and he ACB (Baske ball Clubs Associa ion, Spain)and he esul s o one high le el ama eu league, he Di ision I o NCAA Men´s baske ball (Na ional Collegia e A hle ic Associa ion, USA). Da a ha e been ob ained om he o icial NBA, ACB and NCAA webpages (www.nba.com; www.acb.com and www.ncaa.com). The Spanish P o essional Baske ball League (ACB) is an open model league, whe e e e y season pa icipa ing eams a e eadjus ed aken in o accoun s p omo ions and demo ions o lowe ca ego ies. The eigh op anked eams play he play-o in o de o be p oclaimed champion o he league. The p o essional No h Ame ican League (NBA) is a anchise model compe i ion. Pa icipa ing eams a e di ided in wo con e ences (Eas e n and Wes e n). In u n, hese a e di ided in h ee di isions pe con e ence. When he egula season inishes, op anked eams will mee in he play-o o he i le. The NBA is a closed model whe e he e a e nei he p omo ions no demo ions. The NCAA (baske ball college championship in USA) is di ided in h ee di isions (Di ision I, Di ision II and Di ision III). In u n, e e y di ision is di ided in con e ences o se e al eams each. We only used he da a om he Di ision I o he men´s baske ball. We mus emembe ha he Di ision I o NCAA men´s baske ball is composed by a o al o 344 eams ( he numbe a ies in he season analyzed), di ided in 31 con e ences h ough all USA ( he numbe o eams pe con e ence is no homogenous). The aim o his s udy was o analyze, om an o e iew, he spo model and he in e n dynamic o se e al baske ball leagues (p o essionals and ama eu ) by s udying i s compe i i eness deg ee. Also we ied o de elop a model o he compe i i eness le el analysis in eam spo compe i ions, which would be use ul o assess hei compe i i eness le el based on he unce ain y le el ha migh exis o each con on a ion. 76 6.2. Me hodology Con on a ion ma ices Ou in e es is o ocus on s udying spo leagues, whe e each eam usually plays wice agains each o he eam (once a home, once away) in games acco ding o a p ea anged schedule. A se ies o games be ween a numbe N o eams, can be de ined by i s ma ix o con on a ion A = [Ai,j]N×N, wi h he same numbe o ows and columns. This is a double en ance ma ix whe e each ow and each column co espond o he esul s o each game be ween any wo eams. We shall use he subsc ip i o j o eams, i ≠ j, so we use Aij = 1 i eam i bea s eam j, and Aij = 0 o he wise. O he op ions such as ies o di e en alues o 0 o 1 a e no conside ed in his in oduc ion a he momen wi hou loss o gene ali y. F om his ma ix, a he end o he compe i ion, we ob ain he inal sco e R. See Table1 o an example o he ma ix N = 4. a b c d HW R a x 1 0 1 2 4 b 0 x 0 0 0 1 c 1 1 x 1 3 5 d 0 0 1 x 1 2 AL 1 2 1 2 AW=(N-1) - AL 2 1 2 1 Table 3. Example o a con on a ion ma ix wi h N=4 eams (a,b,c and d). The ows ep esen he games played (won o los ) by a eam a home. The columns ep esen he won o los games played by a eam away. HW (Home Wins) ep esen s he o al numbe o games won by he eams a home. AL (Away Los ) is he los games away. AW ep esen s he o al numbe o games won away. The inal sco e R is he sum o he home and away wins, R=HW+AW. The ow i o ma ix A ep esen s he poin s o games won o los by he eam i a home, while column j ep esen s he away games won o los by he same. The e o e he ho izon al sum:      N ji njiA 1 , Baske ball • Complexi y S udy 1. Baske ball League 77 ep esen s he numbe o games won by he eam i a home (ni), whe e N is he o al numbe o eams. No e ha A(i,j)=0, i i = j. Likewise, he e ical sum      N ji mijA 1 , ep esen s he numbe o away games los by i (nj). The e o e, he o al numbe o games won by he eam i will be   iii mNnR  1 The ec o R (sco e ec o ) ep esen s he esul s ob ained by each eam in each season. The esul ec o R beha es andomly, in he sense ha we do no know he inal esul , bu he esul s o p e ious seasons (his o ical pe o mance), may p o ide some clues. The alues o R his o ical o p e ious seasons di ided by he sum o all games can be conside ed o be a disc e e p obabili y dis ibu ion   N jj i iR R p 1 whe e pi indica es he p obabili y ha he i eam ge s a ce ain esul and he e o e can be conside ed as a pe o mance indica o . I he dis ibu ion is uni o m, all pi alues a e equal o simila o each o he , and all he eams ha e app oxima ely he same playing le el. This ep esen s a case whe e i is di icul o p edic he inal ou come. This may be conside ed o be highes possible pa i y among he eams (compe i i e balance). Howe e , i he e a e ce ain alues o pi g ea e han he es , i means ha he e a e some eams in he compe i ion wi h supe io pe o mance o o he eams. In he case o a uni o m dis ibu ion, any eam has an equal chance o winning. In e ms o s a is ical mechanics, such dis ibu ions a e ela ed o equilib ium si ua ions whe e all s uc u es and g adien s ha e been elimina ed. The diso de is maximum; he e o e he alues o en opy (S) a e also maximum. Following his analogy, i he sys em is isola ed, canno exchange ma e , ene gy o in o ma ion wi h i s en i onmen , all andom luc ua ions ha may occu and hus, all g adien s ha can be o med, end o be neglec ed. 84 u he om season o season. Thei alues oscilla e be ween wo league p o iles (seasons ha we e e y compe i i e and seasons ha we e less compe i i e). Figu e 14. En opy alues o he 10 season o NCAA men´s baske ball Di ision I. Da a display a no uni o m endency. Sn alues a y h oughou he seasons analyzed, showing an upwa d end in he las 4 seasons. E en hey each highe alues han he p e ious maximum. 01-02 02-03 03-04 04-05 05-06 06-07 07-08 08-09 09-10 10-11 0.956 0.958 0.96 0.962 0.964 0.966 0.968 0.97 Seasons Sn Values Baske ball • Complexi y S udy 1. Baske ball League 85 Figu e 15. Compa ison o he h ee leagues analyzed. We can no e ha he p o essional leagues ( he ACB and he NBA) ha e a highe le el o compe i i eness han he ama eu league (NCAA). Bu he in he las season, he alues a e close han e e . This ac is e y ele an and will be in e es ing o ind ou he eason o his beha io . In he analysis o he en opy alues o NCAA men´s baske ball Di ision I (Figu e 14) we can app ecia e ha he endency is no homogeneous. The e was a pe iod ( om 2004-2005 o 2005-2006) whe e he compe i i eness was maximum (0,967) wi h slopes e y simila o bo h sides. Bu he mos no able ac is ha he las 4 seasons he Sn alues inc ease up he highes alue o en opy (0,968). When we compa e he h ee leagues (Figu e 15), we can see ha he p o essional leagues p esen highe alues o en opy, wha means ha , in gene al, he compe i i eness is g ea e han he ama eu league. Nei he o hem display a homogeneous compo men bu all o hem oscilla e du ing he yea s analyzed. Only he NBA p esen s a mos s able beha io . As we men ioned abo e, hese asymme ies may be o igina ed by eadjus men s in he compe i i e sys em, enla gemen s, p omo ions and demo ions, e c. Bu possibly, he only ac ha seems o coincide in all he leagues is he economic c isis. Ne e heless, i gi es he imp ession o ha e di e en e ec s in he leagues analyzed. In he p o essional leagues ( he ACB and he NBA) he economic c isis lead o a dec ease (and o a s abiliza ion a he las seasons) on he 91-92 93-94 95-96 97-98 99-00 02-01 03-04 05-06 07-08 09-10 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 Sn Values Seasons NBA ACB NCAA 86 gene al compe i i eness le el, while on he o he hand, he NCAA expe imen an inc ease in he en opy alues du ing he same pe iod. Indeed, we can obse e ha in he season 2006-2007, he en opy o he NBA s a s o decline while he Sn alues o he NCAA g ow h. And he ACB su e s a d op d as ically du ing hese same yea s. I would be in e es ing o ind ou he eason o hese a ia ions so ma ked, bu i is necessa y u he deepen in o he causes ha o igina e his class o phenomena. ACB analysis The empo al e olu ion o hese alues can be seen in Figu e 13 and 15, and, as he da a show, he e a e some seasons in which he con on a ions ha e a high deg ee o unce ain y (1999– 2000, 2003–2004 and 2005–2006 seasons) and o he seasons wi h a comple ely di e en p o ile (seasons 1996–1997; 2000–2001 and 2008–2009). Issues such as he possibili y o elega ion, he majo budge a y di e ences o eams, high economic dependence on public ins i u ions, o he high ola ili y o he os e s a e some o he ac o s ha can in luence his beha io . This ma ke has become inc easingly ac i e, as baske ball has been p o essionalized, so much so ha a ew playe s emain mo e han i e seasons wi h he same eam in he ACB (A jonilla López, 2011) A eam’s pe o mance is mainly de e mined by wo ac o s: economics and he playe s hemsel es. Bo h a e closely linked. A la ge budge p o ides he abili y o sign supe io playe s and o make os e s balanced o a g ea e ex en . Teams wi h smalle budge s selec playe s wi h a supposedly lowe le el; consequen ly, hei os e s will be less balanced and less compe i i e. The eams make hei os e s based on budge and spo ing objec i es. These objec i es a e closely linked o he compe i i e ACB model (open model). Mo eo e , he absence o a sala y cap and a conspicuous dispa i y be ween budge s o eams can lead o la ge di e ences in he quali y o he os e s (spo ing po en ial g adien ). This makes di e ences in pe o mance insu moun able o some eams in he ACB, especially o p omo ed eams, whose budge s and os e s a e igh . The uns able p i a e and s a e economic suppo o some eams and no o he s, which ha e a solid economic suppo and he ma ke ing ( ading) o playe s, may in pa explain hese oscilla ions in he en opy, which a e cha ac e is ic o he ACB. Cu iously, Baske ball • Complexi y S udy 1. Baske ball League 87 he yea s wi h minimum alues o Sn coincide wi h Olympic yea s. This may be a opic o u u e s udy. NBA analysis The NBA has a mo e s able shape han he ACB. The e a e seasons wi h lowe compe i i eness (≤0,980) han he o e all a e age (0,983), and pe iods whe e he compe i i eness emains a highe le els ( ange: 0,985–0,990) (Figu e 13 and 15). In Figu e 13 and 15, we can clea ly see ha he e a e seasons ha do no co espond o he gene al end, showing a beha io ha is pe haps anomalous. These seasons a e 1993–1994 and 1996–1998, wi h he lowe alues o Sn, and he pe iod om 2001–2002 o 2006–2007 seasons, wi h highe le els. In he 1995–1998 seasons, Michael Jo dan’s Chicago Bulls achie ed he bes win eco d in he NBA egula season o da e (won–los : 72–10, 69–13, and 62–20 espec i ely). This pe o mance is p obably esponsible o he decline in compe i i eness in he league du ing his ime. Indeed, some au ho s men ion ha he 1990s we e he leas compe i i e decade in he his o y o he NBA(Be i e al., 2005). This seems a om i olous, as his so o phenomenon can be accompanied by a heigh ened a ac ion o a hle es and he gene al public (Ro ell, 2003), as well as p i a e companies and media, esul ing in a s ong economic impac (Fo bes, 2008; Ma hu , Ma hu , & Rangan, 1997). Ano he in e es ing a ea co e s he pe iod om he 2001–2002 o 2006–2007 seasons. These co espond wi h he enego ia ion o he sala y cap (1999–2005) when he Collec i e Ba gaining Ag eemen (CBA) was signed. This p obably had an impac on he o e all pe o mance o he league, because he objec i e o he sala y cap was o p e en eams wi h a la ge p o i su plus om signing he bes playe s a ailable, he eby acili a ing he equali y o e en ion in he league. This mechanism, oge he wi h he d a , is necessa y o each anchise o ca e ully selec which playe s may be in e es ed in he ma ke o i s pa icula anchise goal (depending on he backg ound o he eam). Consequen ly, each anchise is only able o “shield” economically one o wo playe s, commonly e e ed o as “ anchise playe s”. The decline o compe i i eness in he 2004–2005 season (see Figu e 13 and 15) may be a ibu ed o seasonal a iabili y, al hough i could also be caused by enla gemen o he pa icipa ing eams om 29 o 30, which u he es uc u ed he di isions in each con e ence. 88 This changed he o ma o he di isions: ins ead o ha ing wo di isions pe con e ence, he e would be h ee pe con e ence wi h i e eams each. The 2005–2006 season inc eased le els o compe i ion again, as shown in Figu e 13 and 15, possibly because all he Cen al Di ision eams quali ied o he playo s and his was he i s ime ha a di ision managed o place all o i s eams in he pos season since he Midwes Di ision did so 20 yea s ago. NCAA analysis Rega ding o he Figu es 14 and 15, we can see ha Da a display a no uni o m endency. Sn alues a y h oughou he seasons analyzed, showing an upwa d end in he las 4 seasons. E en hey each highe alues han he p e ious maximum. The Sn alue ange o all ime pe iod analyzed. We can see a pe iod whe e hey eached alues signi ican ly ele a ed and a inal pe iod whe e alues each hei maximum. In he NCAA pa icipa es a g ea numbe o eams wi h di e en pe o mance le els, hence when we analyze he en opy da a we ha e o ake in o accoun he he e ogenei y o he sample o he analysis o he esul s. The alues o Sn ange om 0.9679 o 0.9583. These alues a e qui e dis an om he alues o he p o essional leagues bu e en ha , he NCAA is he mos s able o he h ee (Sn NCAA mean=0.9631 ± 0.0033). The main poin we mus bea in mind when we analyze he NCAA is ha he eams ha quali y o he playo s using an index called Ra ing Pe cen age Index (RPI). The RPI is a quan i y used o ank spo s eams based upon a eam's wins and losses and i s s eng h o schedule. The cu en used o mula o de e mining he RPI o a college baske ball eam is as ollows. RPI = (WP * 0,25) + (OWP * 0,50) + (OOWP * 0,25) whe e WP is Winning Pe cen age, OWP is Opponen s´ Winning Pe cen age and OOWP is Opponen s´ Opponen s´ Winning Pe cen age. The WP is calcula ed by aking a eam´s wins di ided by he numbe o games i has played (i.e. wins plus losses). So NCAA s andings a e elabo a ed using he RPI ins ead o games won, as he es o leagues analyzed. This index ends o equilib a e he di e ences among di e en eams, so ha heo e ically a o s he weake eams and hampe s he eams wi h be e his o ical ajec o y. Baske ball • Complexi y S udy 1. Baske ball League 89 This cause ha a lo o he eams play ex a ou namen s in o de o imp o e i s index, hence he eams o NCAA do no play he same numbe o games du ing he egula phase. Also, he NCAA has he ea u e ha a single playe only can emain in he same uni e si y o ou yea s as maximum. The e is no a playe s ma ke as a he p o essional compe i ions. This ac condi ions he possible ad an age ha he “bes ” eams can ob ain o e he es o he pa icipan s. Theo e ically, he bes eams ec ui he bes playe s in o de o win mo e games and quali y o he playo . I his we e ue, he di e ences would be insu moun able o he es o he eams. Bu in eali y, he opposi e happens. The cu en endency o he playe s ha come om he high schools is o choose eams whe e hey a e going o play a lo o minu es and be he “s a ” o he eam, ins ead o choose a good baske ball p og am in a good uni e si y o college in o de o ecei e a good spo and academic o ma ion. Coaches men ion ha a e y la ge numbe o playe s concei e he college eam as a s ep in hei ca ee o he NBA, wha is a big mis ake, in he wo ds o he coaches. As an addi ional conside a ion, we obse e ha in bo h p o essional leagues he e is a signi ican decay in Sn in he las h ee seasons o a lowe limi , which appa en ly emains a ached o ha alue. I is possible ha his is ela ed o he cu en economic c isis in which some eams a e less a ec ed han o he s. This is less impo an in he NBA, as compa ed o he ACB, indica ing he sensi i i y o he wo di e en spo s models o ex e nal economic e ec s. We mus ake in o accoun ha he NBA is a p i a e league, and ha i wo ks as a company, while he ACB depends la gely on egional go e nmen subsidies, which pa icula ly a ec s some eams. Compa ison among eams o con e ences We used his p o ocol (no malized Shannon en opy) in o de o de e mina e he compe i i e balance in di e en ACB, NBA and NCAA seasons. This p oposal makes a ude analysis o compe i i e balance wi hou ega d o he anking in which eams comple e his phase o he league ( egula season), and disc imina es well be ween leagues. Fo his eason, we compa e 90 he alues o he win a io o (R) wi h wo ex eme heo e ical models ( andom o maximum compe i i eness and hie a chical o minimal compe i i eness) in o de o analyze he ac ual beha io o compe i i e balance in he wo p o essional baske ball leagues. We ha e disca ded he esul s o he NCAA because he esul s a e meaningless due o he league has many eams wi h di e en pe o mance le els. Figu e 16. Examples o (a) wo ACB seasons (2003–2004, 2005–2006) and (c) wo NBA seasons (2003–2004, 2005– 2006) ha we e mo e compe i i e compa ed o he heo e ical andom ex eme. Also shown a e examples o hie a chical (b) ACB seasons (2000–2001, 2008–2009) and (d) NBA (1996–1997, 1997–1998), compa ed wi h heo e ical models o each league. Each poin ep esen s he p obabili y alue p ob ained om he esul s a ay R o each season. The solid lines ep esen he wo ex eme cases. The s aigh line ep esen s he hie a chical case, while he o he (oblique line) esul s we e used o calcula e he p obabili y o he a e age esul ob ained in he andom p ocess. We ha e disca ded he esul s o NCAA in his analysis because he esul s can be e y con using due o he la ge numbe o pa icipa ing eams. Figu e 16 (a) shows he alues o he p obabili ies p o he 2003-04 season (o) and 2005-06 season (+) o he ACB, as an example o a e y compe i i e season. Figu e 16 (b) shows he seasons 2000-01 (o) and 2008-09 (+) o he ACB as an example o seasons wi h lowe Sn alues. The same esul s a e shown in Figu e 16 (c) o he NBA, o he seasons o 2003-04 (o) and 2005-06 (+), and 3 (d) o he seasons 1996-97 (o) and 1997-98 (+).We ha e disca ded he esul s o he NCAA because he esul s a e meaningless due o he league has many eams. Baske ball • Complexi y S udy 1. Baske ball League 91 The ACB shows a deg ee o compe i i eness away om he heo e ical hie a chical ex eme, in which compe i i eness is lowe . In he mos compe ed seasons, he en opy alues a e close o he heo e ical andom dis ibu ion, almos a he ail o he dis ibu ion. The es o he alues a e loca ed be ween he wo dis ibu ions, al hough, in he case o lowe en opy, he alues a e e y close o he ail o he hie a chical andom dis ibu ion. Simila beha io s can be obse ed in he NBA. 6.3.2. S a is ical and clus e analysis The Figu e 17 ep esen s he boxplo o he esul s R o all pa icipa ing eams, h ough he seasons analyzed in a io alues (wins/games played). 92 Figu e 17. Boxplo o en i ely esul s ob ained, exp essed in a io (wins/games played), o he all pa icipa ing eams in seasons analyzed. Uppe and middle plo s show up he p o essional leagues ACB and NBA. The bo om plo ep esen s he same a ion o he con e ences o he NCAA (no eams) which pa icipa e in he seasons analyzed. In he esul s o ACB we can dis inguish h ee clus e s o h ee zones. And e en wi hin some o hem, o he delimi a ion on a smalle scale, which a e occupied by eams o which can be called ansi ion eams. In he NBA ( he middle plo ) eams seem o ha e alues mo e simila o each o he . This sugges s a highe compe i i eness 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Values Column Numbe NCAA NBA 0.9 1 1 _ 1 0.8 1 , ¡ 1 ' 1 1 1 ." ~~ ( 1 111 1, ~:: 11 1 1 ~ ~~ + - 1 1 1 1 ¡ j ¡ 1 1 1 1 ] ] 1 0.4 1 1 1 1 1 1 + 1 + 1 1 11+11 11 + 0.3 1 1 1 1: 1 1 1 1 1 1 1 1 1 + 0.2 1 1 1 + 1 1 1 11 ] 1 1 1 1 1 1 1 1 ] 0.1 bL...L---"--..J'---:'-"----":-.L..L~~L..c'c_~~~~~~~~~ 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 Column Numbe + + + Baske ball • Complexi y S udy 1. Baske ball League 93 deg ee, because i ually any eam can each high le els o pe o mance. On he o he side, he NCAA plo (lowe plo ) shows a high homogeneous pe o mance le el and he da a cloud co e s a e y wide spec um. Bu despi e o ha da a show his endency, we ha e o ake in o accoun ha eams do no play he same numbe o games. Hence he boxplo seems poin ou he same alues. In all p o essional cases we obse e ha eams end is o clus e a ound hei pe o mance le el, al hough hei beha io is di e en . Seemingly in he ama eu league (NCAA), he alues a e much mo e s able. In he ACB (Figu e 17) we can dis inguish h ee zones. The i s one, wi h he bes pe o mance le el, he h ee bes eams a e loca ed wi h hei da a (da a cloud, mean, in e qua ile anges and con idence in e als) which a e clea ly abo e he es . These eams a e ollowing a model o compe i i e beha io ha we p e iously de ined as andom. They clus e and compe e o achie e i s place (a e y high deg ee o compe i i eness be ween hem). Also gi en he same si ua ion o he second g oup, whe e he ou eams ha e a simila pe o mance le el (a e y high compe i i eness among hese ou eams). The eams si ua ed in he middle sec o ( hi d g oup), s uggle among hem e y e en. Mo eo e , we can alk abou ansi ion eams, whe e he le el o pe o mance places hem in a bo de ing posi ion wi h he o he wo a eas o pe o mance. In he las pa o his g oup, he da a ha e li le s a is ical alue, because his is he zone which su e s mo e changes due o he p omo ions and demo ions. In he NBA he da a seems o poin ou a much mo e homogeneous beha io . The mos pa o he da a cloud, and he medians, a e si ua ed a ound o mean alues o a io, which indica es a high compe i i e balance. Occasionally he eams each unusually high alues (high consolida ed eams) o low alues (low consolida ed eams), some wi h a e y s ong sca e ing da a, sugges ing ha hey a e eams wi h good esul s and now hey ha e dec eased hei pe o mance, o ice e sa. In he NCAA, on he o he hand, he e is a low a iabili y in he boxplo alues. The mos pa o he con e ences each ange om 0 o 1 in he a io alues. Bu he poin is ha he eams do no play he same numbe o games. In he same con e ence we can ind eams wi h 16 games played agains eams wi h 4 games played. This is because he s andings a e elabo a ed by he RPI index, as we men ioned abo e. This o ce o se e al eams o play in ou namen s in o de o ge a be e punc ua ion. Tha is why he inal balance is so he e ogeneous. Ac ually his means ha in he NCAA he spo g adien s a e much mo e accen ua ed han he p o essional leagues. 100 We ha e o bea in mind ha NCAA pa icipa ing eams a e signi ican ly mo e (344) han ACB (16) and NBA (30). And he o ganiza ion o compe i ion is qui e di e en om p o essional leagues. Tha is why i seems he e a e wo compe i ions du ing egula phase, as indica e he log-log plo o a io dis ibu ion (Figu e 20) Figu e 96. Log-log plo o he NCAA a io dis ibu ion. The e a e a leas wo Powe Laws, which indica e di e en compe i ion dynamics. In o de o ind ou whe he eams a e ga he ed by hei pe o mance, we ca ied ou a non- hie a chical clus e ing analysis o pa i ional ealloca ion ype (k-means Ma lab unc ion) (Figu e 21), which places he poin s in space o be g ouped. These poin s a e assigned o he g oup ha is closes o hei cen oid. I is a me hod o clus e analysis which aims o pa i ion n obse a ions in o k clus e s in which each obse a ion belongs o he clus e wi h he nea es mean. This esul s in a pa i ioning o he da a space in o egions called Vo onoi cells. 10-1 100 101 102 Baske ball • Complexi y S udy 1. Baske ball League 101 Figu e 21. The uppe panel ep esen s he ACB clus e ing. We can obse e ha show up i e egions which a e clea ly ela ed wi h he eam pe o mance. The e a e some eams which a e clea ly loca ed in one egion (blue, black and ed zones), and occasionally each a di e en esul . Tha is, hey belong undoub edly o a egion. Teams 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 ACB Clus e Wins Team Numbe 0 5 10 15 20 25 30 0 10 20 30 40 50 60 70 80 Team Numbe Wins NBA Clus e • • • • • • • • • • ••• I ~~; • I • • • • • • • • • • • • ••• • • • . ~ : • '9" • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • l' • • · i . ~ ' q • • • I • • • • • • • • • • I .,. • • • • • • • • I • • • • • • • • • I • • • • • • • • • • • • • • • • • • I • • • I • • • • • • • • • • • • • • • . 10 . • -'Ol e. • • • •• • •• • • • • • • • • • • • • I ;1. • • • • • • • • I • • *11 • • • • • I I I i ~ 11 • • • • • • I I • .' . • • • • • • • • • • • • • 102 loca ed in g een and b own a eas can be conside ed ansi ion eams because some imes each be e o wo s esul s han o he seasons. E en we could conside hese wo egions as a single egion, ega ding he beha io o eams loca ed on i . The lowe panel ep esen s he NBA clus e ing. I is comple ely di e en o he ACB. The esul s poin ou ou egions ( ed, magen a, b own and g een) wi h a simila pe o mance. I means ha a eam can be on op o one o se e al seasons and subsequen seasons a he lowe s anding, o ice e sa. Mo eo e , exis s an eli e loca ed in an own egion by hei own esul s (black) bu seldom o he eams manage o achie e hese posi ions (blue). A ACB clus e ing (Figu e 21) comes up i e egions clea ly es ablished by pe o mance. We can see e e how he cen oids a e posi ioned hie a chically, which indica es s a i ica ion. The wo lowe egions ( ed and black) a e clea ly in eg a ed by ce ain eams which some ime eached be e esul s bu isibly belong o hose egions. The o he egion which is clea ly sepa a ed om he es is he blue egion. Teams loca ed in his a ea a e ma kedly supe io o he es . Teams loca ed in g een and b own a eas can be conside ed ansi ion eams because a imes each be e o wo s esul s han he co esponding esul s o hei zone. Thus we could conside hese wo egions as a single egion, ega ding he beha io o eams loca ed on i . The NBA clus e ing (Figu e 21) poin s ou six egions. The ed, magen a, b own and g een egions p esen a simila pe o mance le el, bu i is no clea wha eams a e in each one. I could means ha due o he in e n mechanisms o NBA eams a e a bad season can be compe i i e o he nex one. And eams wi h good esul s a e obliga ed o es uc u e hei os e season by season, in o de o keep good uns. In ac , we can no e ha some eams p esen e y good esul s (black and blue) and seasons no so good (some o hem wi h a e y ma ked da a sp ead). This can poin ou dynas ies. Fo ins ance, Chicago Bulls as long as Michael Jo dan emained in he os e he eam succeeded. Bu a e his e i emen , he Chicago Bulls ell in o a un o bad esul s. The NCAA uses he RPI index in o de o elabo a e s andings, so his kind o analysis o he NCAA does no e lec he eali y o he compe i ion sys em. Hence we conside ed ha he boxplo and clus e ing analysis o s andings do no p o ide ele an in o ma ion. ACB The ACB p esen s a s uc u e almos hie a chical. Pa icipa ing eams a e clus e ed by hemsel ega ding pe o mance le el (Figu e 21) and his phenomenon c ea es equency ba ie s (wins equency) o eams less powe ul. The ACB peak is a ound 0.40 a io (Figu es 17 and 19). Teams loca ed below his poin a e e y i egula and a e no able o o e come he le el o pe o mance needed o s i e in he middle o he league anking. This poin seems o wo k as Baske ball • Complexi y S udy 1. Baske ball League 103 a ba ie , unde s ood as a alue signi ican ly g ea e equency. I is ema kable ha mos o he eams a e placed in in e media e egions (Figu es 17 and 19), and only a ew eams a e posi ioned beyond he second ba ie (0.80 a io), which can be conside ed he mos compe ed a ea (Figu es 17 and 19). Teams loca ed abo e he ba ie s a e always he same (excep occasionally). So we can poin ou ha he highly compe i i e a ea is always occupied by he same eams and so on (Figu e 17, Figu e18, Figu e 19 and Table 4). Tha is, eams a e clus e ed a ound hei le el o pe o mance. The e o e, eams ha e o o e come ce ain ba ie s o pe o mance i hey wan o achie e highe le els o pe o mance The bes esul s in he ACB coincide indeed wi h highly consolida ed eams in his compe i ion and wi h a high pe o mance in se e al Eu opean Leagues. Resul s lowe o ACB da a co espond o eams which, in he seasons analyzed, we e poo ly consolida ed (Table 4). These di e en pe o mance egions (Figu e 21) could be o igina ed because ACB compe i ion model. The ACB is an open league model in which pa icipa ing eams a e adjus ed based on p omo ion and demo ions ( om o o lowe ca ego y), and whe e eigh op anked eams play he play-o . We mus ake in o accoun ha eams elabo a e hei os e depending on hei budge , and ha his is, almos always, ela ed o he esul s ob ained. The highe he budge , he be e playe s, coaches and s a can be hi e o ice e sa. A p io i, p omo ed eams ha e less compe i i e os e s, and also igh e budge s. Gi en i s open s uc u e, some eams (and i s unde lying s uc u es such as economic ne wo k, execu i e commi ee, playe ne wo k, a m o playe s, e c.) become mo e expe ienced h oughou all seasons. The exis ence o hese eams has an impac on he es , and abo e all on he less expe imen ed eams. Thus, he eams posi ioned on he ex emes a e closely ela ed: i he di e ences among he low anked eams and he op anked eams a e e y high, i is possible ha he up u n o he head is mo e e iden , because exis a high p obabili y ha he op anked eams de ea he bo om eams. Thus, he bes eams can imp o e hei winning a io. I can be deduce ha he e is a di e en c i icali y le el o each zone. This spo po en ial g adien is main ained by ene gy (playe s, coaches, money, e c.). This p o ides ha he 104 pe o mance di e ences o some ACB eams a e insu moun able especially o he newly p omo ed whose budge and empla es a e igh . The spo model g ea ly in luences he ma ke . The ac ha he eams end o clus e in zones i is no andom, bu ollows a phenomenon known as p e e en ial a achmen (Ba abási & Albe , 1999), also denomina ed Sain Ma hew e ec (Bunge, 2001; Ga cía Manso & Ma ín González, 2008), whe e he s ong eams eap mo e successes and less s ong eams will ha e less weal h. Ano he mechanism ha causes his beha io is memo y e ec , wha he sys ems p esen . The eams a e a ached o an a ac o as in some a eas o he anking (Figu e 17, Figu e18, Figu e 19 and Table 4). O he s easons o hese di e ences could be eams´ spo planning, spo aims es ablished o each season, os e , budge , ex e nal compe i ion (Eu opean Leagues, King´s Cup, ou namen s o pa icipa ions o playe s in na ional eams), e c. These aspec s may in luence subs an ially. NBA Gene ally, he NBA has an unce ain y deg ee bigge han he ACB (de Saá Gue a e al., 2012) and i s dynamic and s uc u e a e comple ely di e en . In he Figu es 17, 18 and 19 we obse e ha he mos pa o he da a a e loca ed nea by he 0.5 a io and how eams a e dispe sed o se e al egions (Figu e 21). The e a e no demo ions nei he p omo ions. In ac , hey wo s esul s ( a io <0.15) (Figu es 17 and 19) ha e some ad an age o he nex season. These eams akeo e op places o NBA d a , which means in a ein o cemen in hei os e . We also can see how o each he bes esul s is e y unlikely (Figu es 17and 19). We mus emembe ha he NBA seasons a e e y ex ensi e (82 games) and he play-o classi ica ion is e y ha d- ough . E en o ge a io esul s highe han 0.70 is in equen . The mos likely is ha he majo i y o he eams a e loca ed in medium zones (Figu es 17, 18 and 19). The same eam can be igh ing o ge play-o posi ions and he nex yea can be loca ed in a a io <0.5 o ice e sa (Figu es 17, 18 and 19). I.e. eaching highe alues han 0.70 o lowe han 0.25 is unlikely o he mos pa o he eams. No e ha he mos pa o he eams p esen simila Baske ball • Complexi y S udy 1. Baske ball League 105 pe o mance le els; ha is why clus e ing p esen egions simila whe e eams change hei loca ions du ing seasons analyzed (Figu e 21). The exis ence o his pe o mance dynamic could be also due o he spo model employed by he NBA. We mus ake in o accoun ha in he NBA pa icipa e much mo e eams han in he ACB (30 s. 18), and hey play much mo e games (82 s. 34). Mo eo e , he compe i i e s uc u e is diame ically opposi e. The e a e also mechanisms imposed by he NBA in o de o a oid eam monopoly (d a , sala y cap, ese e clause, e c.). The pu pose o hese measu es is o sa egua d always he compe i i e balance. The e o e i is possible ha he mos c i ical pa s o he compe i ion a e loca ed in he wo limi s, because hey a e a eas whe e eams a e posi ioned in hem will ge ewa ds (play-o and d a ). Pe haps due o i s compe i i e dynamic, he NBA is a good example o Red Queen hypo hesis p oposed by Van Valen (Van Valen, 1973): Fo an e olu iona y sys em, con inuing de elopmen is needed jus in o de o main ain i s i ness ela i e o he sys ems i is co-e ol ing wi h. I is an endless ace. All compe i o s need o imp o e o keep compe ing. ACB and NBA Compa ison The ACB and he NBA seem o p esen an in e se beha io . In he ACB case, mos compe ed egion is he medium a io a ea (lowes di e ences) and g een and b own egions (Figu e 21). In he NBA, he mos compe ed a ea is he op and he end o he s andings. Tha is why eams a e so dissemina ed in he clus e analysis (Figu e 21). We no e ha bo h cases a e cases o a highly compe i i e, bu opposi e easons: he ACB is an open model whe e he las classi ied is elega ed o ca ego y, hence he high deg ee o compe i i eness, while in he NBA, he poin is quali ying o he play-o o he i le o ying o ge a good place o he lo e y d a . In he ACB, we obse e ha eams a e clus e ed h oughou hei pe o mance le el as well (Figu e 21). The e a e eams clea ly placed in a pa icula a ea o he compe i ion, which could indica e he compe i i eness le el o he eam. The i s ou posi ions a e occupied almos en i ely by he same h ee eams and, occasionally, some eam was able o slip in o his eli e g oup (Figu e 17, 18 and 19). The e is a simila pa e n wi h he play-o posi ions ( he i s eigh posi ions), whe e we can obse e clea ly how he da a cloud and he us in e als o se e al eams a e encompassed in his a ea. The las posi ions a e mo e a ypical, because he las wo eams pass o a mino league and a e eplaced by o he wo di e en eams. The new 106 p omo ed eams, a p io i, ha e no he same pe o mance le el ha he eams o he middle zone. In he NBA, almos all eams ha e e e eached he play-o posi ions, al hough he e a e mo e eams ha belong o his a ea han o he s (Figu e 17). Thei da a a e less sca e ed and mo e i mly es ablished in his a ea. No e ha p ac ically all he eams ha e eached he op i e. Gi en he high andomness deg ee p esen in he ACB and he NBA, we can suppose ha he mos pa o he eams a e be ween he o de and chaos, known as c i ical s a e. A s a e o semi equilib ium whe e he mos insigni ican a ia ion can p oduce a change o s a e o a phase ansi ion, bu his is e y ha d o p edic (McGa y e al., 2002; Sche e e al., 2009). The deg ee o c i icali y a ies based on he zone o he s anding we a e s udying. Bu no e ha eams, e en hough he chao ic beha io o he compe i ion, always end o an a ac o ( eam clus e ing). The e o e, we can conside compe i i eness as an a ac o i sel . We mus emembe ha complex sys ems a e usually a om he equilib ium. E.g. he li ing o ganisms a e in con inuous igh wi h he en i onmen in o de o emain in a s a e a om equilib ium, i.e. ali e (Ama al & O ino, 2004). T ansla ed o ou con ex , i means ha o p ese e he high le el o compe i i eness, i is necessa y o keep igh ing agains he i als, and o in es huge amoun s o ene gy in o de o su i e in he compe i ion. 6.4. Conclusions The aim o ou s udy was o analyze spo compe i ions om a gene al poin o iew. This s udy show ha he analysis model (ma ix esul s using he Shannon en opy) o he s udy o compe i i eness le els in he sys em o league compe i ion is a use ul and highly sensi i e ool o de e mine he deg ee o o e all compe i i eness in he league, and o de ec small oscilla ions in i . This po en ially iden i ies minimum luc ua ions in he le el o compe i ion, which allows one o ocus a en ion on localized empo al changes and o in es iga e he mechanisms which cause i . This model shows ha bo h he ACB and he NBA p esen a high deg ee o compe i i eness. In bo h leagues he en opy le els a e high ( ange: 0.985–0.990), al hough hese pe iods a e Baske ball • Complexi y S udy 1. Baske ball League 107 mo e s able in he NBA. We can say ha bo h he ACB and he NBA a e e y compe i i e leagues whose eams a e well balanced wi hin each league. On he con a y, he ama eu league, he NCAA, p esen s a lowe compe i i eness deg ee (compa ed wi h he p o essional leagues). Bu i s endency is o inc ease du ing he las seasons. We can say bo h he ACB and he NBA a e e y compe i i e leagues wi h a high compe i i e balance and hey a e highly condi ioned by he spo model. The ac ha he ACB is an open league causes ha less powe ul eams sub ac compe i i eness o he en i e y. We should hink abou s a egies in o de o main ain o e en inc ease he deg ee o global compe i i eness o he league, like in he NBA. Despi e hese issues, he Spanish baske ball league (ACB) can be conside ed e y compe i i e. The NBA has speci ic mechanisms o ensu e high compe i i eness, such as he d a , he sala y cap, ese e clause, e c. Thei aim is o p ese e he compe i i e balance wi hin he sys em. I is a league wi h a high unce ain y on he inal esul ; hence, all eams ha e eal possibili ies o quali ying o he playo s. The case o he NCAA is p e y in e es ing as well. Despi e o he low en opy (compa ed o he p o essional leagues), NCAA is an a ac i e league ha gene a e expec a ion and a ac s housands o ans and media. Due o i s complica ed s uc u e and size (numbe o pa icipa ing eams), i is complica ed ca y ou an analysis o he en i e league. Bu we can obse e ha indeed, changes in i s spo model, such as enla gemen o di isions, ule changes, new punc ua ion sys em, e c., al e signi ican ly he compe i i eness le el h oughou yea s analyzed. 6.5. P ac ical P oposals As a p ac ical p oposal we sugges o use his me hodology in o de o s udy and compa e he compe i i e le el o di e en baske ball leagues and hei e olu ion in ime. I.e. di e en baske ball leagues such as Eu oleague, Eu ocup, NBAD-League, ABA, LEB, e c. This p ocess also can poin ou some e en s which can be he sou ce o such a ia ions. As ano he p ac ical p oposal, we pu o wa d o use his me hodology in o de o igu e ou how ule changes, enla gemen o pa icipa ing eams, compe i ion o ma es uc u ing (open o close, di isions, con e ences, e c.), o o he league mechanisms such as budge ules, 108 building up eams (hi ing), o e en p ac ices egula ions (i.e. NCAA) can modi y he compe i i e le el. 6.6. Limi a ions o he echniques used Shannon en opy, as me hodology o s udy he compe i i eness in baske ball, ca ies ou a coa se analysis o he compe i i e balance, in he sense ha he esul s do no ake in o accoun he eam ankings. On he o he hand, i we only use he a ios, boxplo s o e en clus e s in o de o analyze he compe i i eness deg ee, we will no know he league o e all. S udy 2. Baske ball Game Baske ball om he pe spec i e o non-linea complex sys ems Y es de Saá Gue a 2013 116 Scaling analysis indica es ha he p obabili y o ex eme e en s migh be es ima ed by ex apola ing o Powe Law dis ibu ions. Recen ly some au ho s (Pisa enko & So ne e, 2012; Sachs, Yode , Tu co e, Rundle, & Malamud, 2012; So ne e & Ouillon, 2012; Yukalo & So ne e, 2012) ha e ied o cha ac e ize he so-named D agon-King, ex eme e en s wi h impo an social implica ions, which exceed hese ex apola ions. One impo an limi a ion o his ool is he occu ence o he Powe Law beha io a he ail o he dis ibu ion whe e is desi able o ha e he bes accu acy (S ump & Po e , 2012). Howe e , he ails a e he zones o he empi ic dis ibu ions whe e he g ea es luc ua ions a e ound. On he o he hand, he scaling laws do no always p o ide he bes da a i . Al e na i e dis ibu ions o cha ac e ized hese en i onmen al phenomena a e he logno mal (Mi zenmache , 2004), he s e ched exponen ial (Lahe è e & So ne e, 1998) and o he unca ed Powe Law (Redne , 1998; Tsallis & Albuque que, 2000; Bu oughs & Tebbens, 2001). Despi e o hese di icul ies, he e a e some ad an ages o he applica ion o Powe Law and ac als ools o diagnos ic, cha ac e iza ion and e en p edic ion pu poses, a e he simplici y o he dis ibu ion and uni e sali y o i s sel -simila i y. This is qui e consis en wi h much o he li e a u e on ac als and scaling in ecologic, geophysics o economics sys ems. Fu he mo e, in he la es yea s; imp o ed s a is ical es s ha e p o ided s ong e idence o scaling laws o e a subs an ial (al hough limi ed) ange o scales (Clause , Shalizi, & Newman, 2009; Vi ka & Clause , 2012). One o he goals o his s udy is o p o ide a quali a i e backg ound on he lineal i o he powe law dis ibu ions and pa icula ly he log-log plo analysis. This allows he compa ison o analogous phenomena and he cha ac e iza ion o egions o e a simila en i onmen . Fo his pu pose we don' need o ha e absolu e ce ain y ha an empi ical da a se ollow a Powe Law. We used hese wo ools (Poisson dis ibu ion and Powe Law dis ibu ion) in ou analysis. They poin ed ou he beha io o he sys em (Baske ball). Baske ball • Complexi y S udy 2. Baske ball Game 117 P oblems wi h Poisson dis ibu ion 1. When he index o dispe sion is less han 1, he nega i e binomial dis ibu ion can be modeled o disc e e da a, because p esen s a longe ail han Poisson dis ibu ion. The ail o he dis ibu ion could dec ease slowe han in his case, so we should use he Powe Law o unca ed Powe Law dis ibu ion, which heo e ically, p esen a special meaning ega ding ex eme e en s dis ibu ions. 2. Ano he case is when  is no cons an du ing he game o in each minu e o he game. I also known ha i lambda ollows a gamma dis ibu ion, he p ocess is modeled by a nega i e binomial. We assume he games as easonably homogenous e en s. Bu , da a indica e ha no all he games a e compe ed (homogenous), meaning low compe i i eness, low eam quali y, pe iod o he season, os e di e ences, e c. The sys em beha io is nei he he same h oughou he game ime, no in he i s qua e , no he ou h qua e no e en las minu es o he game. No e ha depending on he game ime, can be conside ed as di e en games (subs i u ions, sco e di e ences, aul s, e c.). And also is i in luenced by ac ic decisions ( aul s, ime ou , de ense, e c.) 7.3 Resul s and discussion Applica ion o baske ball We s udied a o al o 5 seasons (1230 games pe season, wi h a o al o 6150 games) o he NBA egula season. In e e y game we analyzed he game ansc ip ion published by he NBA in which a e desc ibed in de ail, all e en s play by play (NBA). All he s a is ics e lec he incidences o game o de ed by he ime in which hey occu ed (ch onological o de ): wo and h ee poin sho s, ee h ows (made and missed), de ensi e and o ensi e ebounds, u no e s and s eals; iola ions (ou o bounds, ouls, echnical, e c.) subs i u ions, e c. F om all his in o ma ion we ocus on he analysis o poin ime in e als and sco ing. As we desc ibed in he me hodology, we p opose he use o he Index o Dispe sion and he  alue in o de o analyze baske ball games. We based ou s udy o analyze wha happens 118 e e y minu e o he game independen ly. Tha is, analyze he p obabili y p1(k) ha in he i s minu e k poin s a e sco ed, p2 (k) in he minu e 2, and so on, un il he minu e 48 o e e y game. Then we saw how a each o hese 48 cases p esen ed Poissonian beha io . To do his we calcula ed he mean numbe o poin s sco ed in e e y minu e o all games i : , 1 N ik k i n N    Being N he numbe o games (N=6150), and i=1, 2 , … 48. In addi ion, we also calcula ed he Index o Dispe sion, which as we saw is an indica o o he ex en o which andom a iable beha es like a Poisson p ocess. As we men ioned abo e, he Index o Dispe sion is he a io o he a iance o he poin s sco ed in e e y game e e y minu e o he mean alue o hese. The nex Figu e ep esen s bo h alues applied o he games s udied. Figu e 22. Index o Dispe sion o he poin sco ed by minu e. We can obse e ha he end o he alues is o ise o e ime. Only a he end o each qua e he e a e a signi ican ly inc ease, close o 1, bu only a he minu e 47 each he alue 1 (pu e Poisson). The minu e 48 is comple ely ou he ange o he es o he game, eaching alues highe han 1. The beha io o his minu e is e y complex. The uppe panel (a) ep esen s he numbe o poin s in e e y minu e (). This alue is low a he beginning o each qua e bu no e ha he alue inc eases along wi h ime, abo e all a he las qua e . 0 5 10 15 20 25 30 35 40 45 50 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Time in minu es 510 15 20 25 30 35 40 45 2 3 4 Time in minu es (a) Baske ball • Complexi y S udy 2. Baske ball Game 119 The mos in e es ing esul s a e ob ained when we analyze he game ime using he Index o Dispe sion and he  alue because poin s ou some e y in e es ing beha io s. The ime in e al used was 1 minu e because we conside ha lowe equencies, such as one second, o highe equencies such as 2 minu es we e no clea enough. The p o ile o hese g aphs may a y i he selec ed ime in e al is di e en han a minu e. Bu he cla i y o in e p e a ion o e ed by his choice, which includes all he ea u es ha we wan o emphasize, was wha led us o choose he ime in e al o 1 minu e, as he key o playing a baske ball game. The Index o Dispe sion (Figu e 22) displays ha he mos pa o he qua e s emain lowe han alue 1 (unde -dispe sed). Only a he end o each qua e he alues a e highe han in he es o he qua e . This means ha he beginning o e e y qua e is mo e p edic able han he end. No e ha he endency o all qua e s is o ise, o app oach o alue 1, o become mo e unp edic able. Bu only he minu e 47 p esen alue 1 in he Index o Dispe sion. To be p ecise, is a pu e Poisson p ocess. The game a his s age is comple ely andom. The minu e 48 equi es special a en ion. As we can obse e (Figu e 22), he minu e 48 exceed he alue 1 signi ican ly (o e -dispe sed). This sugges s ha he las minu e in a baske ball game is a comple ely di e en p ocess han he es o he game, meaning he game has changed i s dynamic. The uppe panel o Figu e 22 shows  (numbe o e en s pe ime) o e e y qua e . We can obse e han he numbe o e en s a he beginning o each qua e is low compa ed o he es o he qua e , bu he endency is o inc ease anyway. I is likely ha his ou come is because playe s, as agen s o he sys em, s a o in e ac a he beginning o he game. The e a e no p e ious si ua ions (no memo y om p e ious ac ions, because is he i s qua e ). We can e e o a ze o poin o base poin om which eme ge he cha ac e is ic ac ions o a baske ball game. Tha is, a sel -o ganiza ion p oblem. Also i is ema kable he di e ences a e hal ime, maybe caused by he adjus men s ca ied ou by coaches and echnical s a o by he game dynamic i sel ; as he las minu e, whe e he numbe o e en s conside ably highe han he es . 120 The playe s o bo h eams ha e p ede ined oles (by playe posi ion) and ins uc ions gi en by echnical s a , based on he in o ma ion abou he i al. Bu is he in e ac ion among hem h ough he game ime and he adap a ion o he eal game wha make eme ges game pa e ns: swi ch oles, mo o ask esolu ions, e c. This is known as a ibu ed ole. This ac mul iplies he possibili ies and he ac ions ca ied ou by playe s because playe s adap o he en i onmen . We ha e o bea in mind ha a he beginning o he i s qua e he sco e is 0 o bo h eams. This does no happen a he es o he qua e s, whe e he poin di e ences (i exis ) may es ablish u u e eam dynamics. I is possible ha hese cases a e a ec ed by memo y p ocesses depending on how big hese di e ences a e. Bu anyway, he e a e p e ious si ua ions on which o base s a egies. Mo eo e , he e is a a igue e ec o he sys em (e o s such as ouls, u no e s, e c.) and o he playe s (physical a igue, men al a igue, cogni i e a igue, e c.) which has an accumula i e e ec along he game ime and in luences in e nal p ocesses and eme gen beha io s. Hence he anomalies obse ed a he end o e e y qua e a e de i ed by hese kinds o p ocesses o mechanisms p obably. And abo e a e accen ua ed in he inal s ages o he ou h qua e . This may be because accumula ed eam ouls, bu ha e no di ec signi icance on he sco e (Figu e 22) un il he ouling eam is in he eam bonus (o oul penal y) si ua ion and ee h ows a e awa ded. In o de o a be e unde s anding, we based he analysis on he Index o Dispe sion. We ha e selec ed some cases wi h ep esen a i e Index o Dispe sion. As he Figu e 22 sugges s he gene al endency is o inc ease he le el o unce ain y du ing each qua e , abo e all a he end o he qua e , whe e he alues a e close o 1. Mo eo e , he minu e 47 each alue 1. We analyze wo minu es in e media e, minu es 6 and 32; one end o qua e , minu e 36 and he alues o minu e 47, wi h Index o Dispe sion alues 0.63, 0.75, 0.90 and 1.02 espec i ely. Baske ball • Complexi y S udy 2. Baske ball Game 121 Figu e 23. His og ams o he poin sco ed in he minu es 6, 32, 36 and 47, co esponding o Index o Dispe sion alues 0.63, 0.75, 0.90 and 1.02. The solid line ep esen s he Poisson heo e ical dis ibu ion. The wo uppe cases show unde -dispe sion, whe eas he lowe cases a e cases close o Index o Dispe sion=1, wi h a Poissonian beha io . We can obse e wo cases. The uppe Figu es co espond o he minu es 6 and 32, wi h Index o Dispe sion alues 0.63 and 0.75 espec i ely. We obse e ha does no i well o he Poisson dis ibu ion. The a iance is lowe han ha co esponding o he Poisson dis ibu ion (Index o Dispe sion<1), and da a a e clus e ed a ound mean alue, wi h less ze os and wi h a ail which d ops quicke han Poisson, cha ac e izing an unde -dispe sed Poisson dis ibu ion. In gene al his can means ha he momen s wi h an Index o Dispe sion lowe han 1 a e mo e p edic able han he es o he game. On he o he hand, he wo cases below, he end o he qua e (minu e 36), i s be e han he es o he qua e (Index o Dispe sion 0.90), bu wha eally ma ch wi h he heo e ical Poisson dis ibu ion is he minu e 47, wi h an Index o Dispe sion 1.02. No e ha he numbe o ze os ma ches be e and decays as Poisson. This ep esen s he mos unp edic able momen o he game, excep he las minu e, which will be ea ed sepa a ely because he na u e o he dis ibu ion is di e en . 0 1 2 3 4 5 6 7 500 1000 1500 2000 0 1 2 3 4 5 6 7 500 1000 1500 0 1 2 3 4 5 6 7 8 9 200 400 600 800 1000 1200 1400 1600 012345678910 200 400 600 800 1000 1200 1400 minu e 6 mean = 2.28 a = 1.43 ID = 0.63 minu e 32 mean = 2.32 a = 1.73 ID = 0.75 minu e 36 mean = 2.73 a = 2.46 ID = 0.90 minu e 47 mean = 2.60 a = 2.66 ID = 1.02 122 The esul s poin ou ha in he wo uppe cases he numbe o ze os is lowe han he heo e ical Poisson dis ibu ion, which is he heo e ical model we use as a base. Also he ail is educed, whe eas in he wo cases below i s be e . This seems an indica o o he isk assumed by eams. As we see la e in he Figu e 34, he numbe o 3 poin s and 1 poin s ( ouls) is highe in he end o each qua e , while 2 poin s a e dec easing h oughou he game. Teams end o isk mo e a hese imes, and de ensi e in ensi y inc eases (mo e ouls) which indica es g ea e likelihood o ailu es, mo e ze os han in p e ious imes o g ea e numbe o poin s (longe ail), meaning g ea andomness and explain i s p oximi y o he Poisson dis ibu ion. In he cases wi h less isk ( wo uppe subplo s), he game seems mo e p edic able and he numbe o ailu es is lowe ; which would jus i y he leas numbe o ze os and he sho e ail. And also explain i s p oximi y o he Poisson dis ibu ion. In his hesis, one o he objec i es is o compa e he esul s agains he Poisson model. This allows us o be e unde s and he concep o isk and o sepa a e he las minu e o each qua e on a baske ball game om he es o he game; and he ole o he las minu e, as discussed below. The nex igu e ep esen s he poin sco ed o he las minu e o he game, minu e 48, whose Index o Dispe sion alue is la ge han 1, o e -dispe sed: Figu e 24. His og am o poin sco ed in he las minu e o he game. The solid line ep esen s nega i e binomial dis ibu ion i ( i ing pa ame e s 3.81, 0.48; STD = 0.039). No e ha he alues a e be e i ed a he ail o he dis ibu ion. Fo u he analysis, we ca ied ou a log-log plo , in he uppe panel, which displays wo Powe Laws wi h a c osso e (s aigh s lines). The dashed line in he uppe panel ep esen s he nega i e binomial i . -2 0246810 12 14 16 18 0 200 400 600 800 1000 1200 F equency 101 102 103 Baske ball • Complexi y S udy 2. Baske ball Game 123 The analysis o he las minu e o he game in baske ball e eals some game ac s e y ema kable. The las minu e is o e -dispe sed (Index o Dispe sion la ge han 1; Figu e 22), which is associa ed wi h a nega i e binomial dis ibu ion. The p esence o a nega i e binomial dis ibu ion poin s ou he exis ence o clus e s o occu ences. Appa en ly, he poin equency dis ibu ion seems o ma ch wi h he heo e ical nega i e binomial dis ibu ion (solid line), pa icula ly a he ail o he his og am ( i ing pa ame e s 3.81; 0.48; STD = 0.03. STD is he quad a ic di e ence be ween he dis ibu ion and he da a ob ained). The appa en long ail beha io gi es he imp ession o indica ing he p esence o da a a emo ed om he mean, which migh indica e he p esence o a unca ed Powe Law. We mus ake in o accoun ha he e is only 1 minu e o eal game ime. To check his, we pe o med a log-log plo (uppe panel) and we obse e he alues a e i ed by wo Powe Laws. This migh means ha he e a e scaling phenomena, ega ding sco ing ime. The e is a zone 0 - 4 poin s, wi h a peak loca ed a ound 2 - 3 poin s. Bu beyond his egion, he i s Powe Law appea s; om 4 o 9 poin s app oxima ely. And a second one om 9 o 16 poin s wi h a highe slope ( unca ed). I.e. as he numbe o poin s sco ed inc eases he playing ime is educed d ama ically. The p esence o a c osso e poin s ou he p esence o se e al sco ing dynamics (mul i scale beha io ). We ha e o ake in o accoun ha he e is a ule in baske ball designed o p o ide c i icali y o he game. We a e alking abou he 24seconds ule. The aim o his ule is o o ce eams o sho , wha , ans e ed o he game, p o ides ideal condi ions o a c i ical si ua ion. In spo he e a e a lo o examples o ules whose aim is o p o ide c i icali y o he game, such as o side in oo ball o ugby, h ee ouches in olley-ball, he D-zone in handball e c. Bu as he Figu e 22 shows, in he las minu e in baske ball, he own na u e o he game u ns c i ical by i sel so signi ican ly, ha his ule ha gi es c i icali y o he game, makes no sense anymo e. A e he examina ion o game ime, we pe o med an analysis o  aluesnumbe o poin s pe minu e) and he Index o Dispe sion alue o each game (6150 games). 124 Figu e 25. QQ plo o poin s pe minu e mean alues s. No mal Dis ibu ion (uppe le ) and QQ plo o mean alues s. Gamma Dis ibu ion (uppe igh ). Rega ding hese plo s, no e ha he da a p esen a be e i by he gamma dis ibu ion. Wi h No mal Dis ibu ion he e a e some i egula i ies in he ails. Down le is ep esen ed he numbe de e en s pe minu e wi h a Gamma i ; and down igh he semi-log plo o he p e ious Figu e. The Figu e 25 ep esen s he  alues. The his og am (down le ) seems o be almos a No mal Dis ibu ion, bu i s be e wi h a Gamma Dis ibu ion (uppe plo s). We pe o med a isual es (QQ-plo ) wi h se e al dis ibu ions: No mal Dis ibu ion (QQ-plo up le ), Exponen ial Dis ibu ion, Weibull Dis ibu ion, e c. bu he da a i s be e by a Gamma Dis ibu ion (up igh ). The s a is ical alues o he dis ibu ion a e: mean = 2.280; STD= 0.251; a iance = 0.063; skewness = 0.284 and Ku osis = 3.174; Fo he Gamma dis ibu ion: Shape pa ame e k = 82.73 and Scale Pa ame e  = 0.027. The Gamma Dis ibu ion is an accu a e dis ibu ion o modeling he beha io o con inuous andom a iables wi h posi i ely skewed; i.e. a iables ha p esen a g ea e densi y o e en s o he le o he mean han o he igh . In ou case we can obse e ha he numbe o poin s pe game do no ollow a No mal Dis ibu ion bu is skewed o he igh , meaning ha he e a e mo e p obabili ies o sco e 2 2.5 3 2 2.5 3 QQ Plo o mean alues e sus Gamma 1.5 2 2.5 3 1.6 1.8 2 2.2 2.4 2.6 2.8 3 QQ Plo o mean alues e sus No mal 2 2.5 3 0 0.5 1 1.5 Poin s pe minu es F equency 2 2.5 3 10-3 10-2 10-1 100 Poin s pe minu es log( equency) Baske ball • Complexi y S udy 2. Baske ball Game 125 mo e poin s han he mean (mo e han 2.28 poin s; up 3.3 poin s pe minu e), al hough his p obabili y is low compa ed o he es , i can ake place. Figu e 26. His og am o Index o Dispe sion pe game. The dash line ep esen s he Gene alized Ex eme Value (GEV) Dis ibu ion. I seems ha i s well, bu when we pe o m he log-log plo (uppe panel) we no e ha he ail in no well i ed ((o) ep esen s he eal alues, () ep esen s he GEV alues. We can obse e ha he mos pa o he games a e loca ed a ound 1, which means hey a e e y unp edic able. Bu e en we can ind some games wi h alues la ge han 1 (o e -dispe sed). The Figu e 26 shows he His og am o Index o Dispe sion pe game, he mos pa o he alues a e loca ed close o 1, which poin s ou a Poisson p ocess. This means ha he numbe o poin s sco ed in he mos pa o he games ollows a Poisson p ocess. Bu mo eo e he e a e some cases whe e he dis ibu ion is o e -dispe sed. This indica es he p esence o ex eme e en s. In o de o check his we i ed by a Gene alized Ex eme Value dis ibu ion (GEV). The dash line ep esen s he Gene alized Ex eme Value (GEV) Dis ibu ion wi h shape pa ame e 0.0680; scale pa ame e 0.1786 and loca ion pa ame e 0.7150; and seems o i well excep a he i s alues. To ind ou whe he he da a ollows a GEV in he ail, we ca ied ou a log-log plo (Figu e 26 uppe panel) wi h he heo e ical GEV dis ibu ion (dash line agains eal alues (o)). No e ha he da a do no i well. On he o he hand, seems o i 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 100 10-2 100 132 ime. This is impo an o ake in o accoun when we wan o model o o p edic in o de o a be e unde s anding. Ou esul s poin ou ha we can conside h ee zones a leas . The i s one ex ends om he beginning (0 seconds) up app oxima ely 24 seconds. This a ea has a bell-shaped dis ibu ion, wi h a maximum a ound 20 seconds, wi h an i egula beha iou be o e 6 seconds and unca ed beyond 24 seconds. The Figu es 30 and 31 show us ha in a baske ball game, he mos likely ime be ween goals is a ound 20 seconds ( his seems logical conside ing he 24 seconds o possession). The beha iou o he ime in e als in his a ea deals wi h a kind o hy hm o play, ball possession, and sco ing ela ed wi h 24 seconds o possession, bu below 6 seconds, p esen s some pa icula i ies (see Figu es 30 and 31) as we men ioned abo e: The mos pa o ime di e ences o one o wo seconds a e p oduced a minu e 48. Poin s wi h one second di e ence only in he minu e 48. The sou ce o his pa icula i y can be he ee h ows sco ed due o he high numbe o ouls made a he end o he game ( o u he de ail see he oul sec ion); bu also because ime ou s called and s a egies o sco e quickly. This can be he ounda ion o he shape o he dis ibu ion a he beginning as we can see in Figu e 30 (b). The same Figu e displays ha he slope a ies in second 6, app oxima ely. Hence we can deduce ha he i s a ay co esponds wi h as b eaks and he second slope wi h plays mo e elabo a ed ( om 6 o 20 seconds). In ac , he mos common is o sco e a ound 20 seconds. This may be ela ed wi h ebounds: de ensi e ebounds, because allow building as b eaks quickly, and o ensi e ebounds because allow sco ing quickly wi h a high success a e, and u he elabo a ing successi e a acks. Hence, he s a egies o many eams a e o make ouls o a oid hese si ua ions, which c ea e se ious disad an ages be ween a eam and he o he . The second a ea, om 24 seconds up 100 seconds app oxima ely, shows a dec ease in a s aigh line (Figu es 30 (c) and 31), which co esponds o an exponen ial dis ibu ion, as we saw in he case III), sugges s ha he dis ibu ion ollows a Poisson p ocess, i.e., comple ely andom, wi hou memo y, o ime in e als la ge han 24 seconds. This is an in e es ing esul because i i is a Poisson phenomenon, i could ha e a ea u e called memo ylessness (also called e olu ion wi hou a e -e ec s): he numbe o goals Baske ball • Complexi y S udy 2. Baske ball Game 133 occu ing in any bounded in e al o ime a e ime is independen o he numbe o goals occu ing be o e ime . I means ha he ime in which each poin is sco ed is independen o he p e ious. The sco e becomes mo e andom. Beyond 100 seconds app oxima ely, as hi d egion, we can see how he da a a e sca e ed and he inal biased. I can be conside ed as a e phenomena (low p obabili y). Fo alues o e 100 seconds is possible ha i beha es as a Powe law. To e i y he ac ual beha io o he da a, we pe o med a log-log plo (uppe panel Figu e 31). The esul s poin ou ha he da a i s be e by a log-log plo , a e less sca e ed, which means ha can be conside ed a Powe Law. F om an o e iew we can obse e ha when he poin ime in e als each high alues, beha es as a Powe Law; he e o e we can say ha he e en s da a has memo y, whe eas when i beha es as Poisson dis ibu ion does no p esen such ea u e. This is impo an because can be ela ed wi h he sco ing di e ences be ween he winning eam and losing eam, and how he es ablish hei s a egies o each si ua ion. Winning eam ends o was e ime, o was e possessions (longe ime in e als). While he losing eam, ends o he opposi e. The s a egy o long ime in e als is ypically o eams ha wan o play o low sco es, whe e beha es as a Powe Law, hus hey a e mo e likely o win he game because he sys ems is mo e c i ical (SOC). We a e acing wo diame ically opposed endencies. This, as we ha e seen, can p esen a di ec ly in luence in he game. Mo eo e , i we ex apola e his o a highe le el, o a league o ins ance, i can p o ide us some clue abou how is he in e nal dynamic o he league: he ac ha exis s di e en dynamics well de ined ega ding poin ime in e als e lec s s ages o he game, p o iles games o ac ics employed by eams, which shows he spo eali y in baske ball. A league whe e he e a e la ge di e ences be ween hei eams, he sco e di e ences in hei games will be mo e p onounced, because heo e ically weake eams end o employ de ensi e s a egies h ough lowe sco es in o de o inc ease hei chances o ic o y. While on he o he hand, a mo e balanced league, whe e he di e ences be ween i s componen s a e no so ma ked, his end will no be so p onounced. 134 All eam spo s a e based on an a endance, pa icula ly baske ball, which ies o p o ide exci ing sco es. Hence he numbe o poin s pe minu e is as de e minan ac o when we analyze baske ball (Figu e 999). Figu e 32. Numbe o poin sco ed (Y-axis) wi h a ime di e ence o 1 seconds, 2 seconds, 3 seconds,…, up 30 seconds (X-axis). The dash line ep esen s poin s sco ed in he las minu e o he game, whe e i is clea ha mos poin s wi h one second di e ence is gi en in his pe iod. The o he alues co espond o he las minu e o he emaining qua e s (solid line ()) and o he y al minu e 47 (o), whose beha io was simila . When we s udy he numbe o poin s co esponding o each ime in e als in he las minu e o each qua e (Figu e 32), meaning numbe o poin s sco ed wi h one second di e ence be ween hem, wo second di e ence and so on un il 30 seconds, we a e we no e ha he e a e some signi ican di e ences. I seems ha he las minu e in he h ee i s qua e s solid line (), beha es simila . The alues inc ease up 14 seconds; hen emain s able un il 23 seconds. I can ha e sense by i sel , in he sense o his he mos p obably ime be ween poin s in hese pe iods, and p obably is ela ed wi h he 24 seconds o possession. Beyond 23 seconds d ops up 27 seconds. No e ha om 27 o 30 he equency p esen s simila alues. This s abili y poin s ou a sligh endency o ex end he ime be ween poin s a he end o e e y qua e . The minu e 47, solid line (o), ollow a simila dynamic as he p e ious cases. The equency inc eases up 14 seconds and s abilizes un il 18 seconds. A e ha , he equency d ops up 26 seconds whe e p esen s he same equency alues un il 30 seconds. 0 5 10 15 20 25 30 0 200 400 600 800 1000 1200 Baske ball • Complexi y S udy 2. Baske ball Game 135 The las minu e o he game, dash line, minu e 48 is comple ely di e en o he es o he las minu es o p e ious qua e s. The highes equency alue is o poin s wi h 1 second di e ences wi h a signi ican di e ence om o he qua e s. The equency alls up 3 seconds, bu inc ease again un il 5 seconds and is s ill high compa ed o he es o sample. The a ay o m 5 o 8 seconds is he mo e s able egion o his minu e. Bu beyond his a ea, he equency declines un il he end. E en he ange om 14 o 23 seconds is lowe han he es o qua e s as we can see in Figu e 32. I seems ha he ule o 24 seconds make no sense he e. The sho in e als a e nume ous han he la ge. Fo he en i e game ime he sho s wi h 1 second di e ence a e: 1 poin sho 1525 (89%); 2 poin sho s 137 (8%) and 3 poin sho s 48 (3%). The sou ce o his endency is he ouls and ee h ows, p obably. In ac , i we analyze he minu e 48 we obse e ha he sho s wi h 1 second di e ence a e: 1 poin sho 1128 (94.55%); 2 poin sho s 48 (4.02%) and 3 poin sho s 16 (1.34%). we ealize ha he issue o ouls is accen ua ed. The e o e i is in e es ing o check ou wha we e he inal sco e when he e we e sho s wi h one second di e ence. His og am o inal sco e di e ences when he e is some 1 second sco ing ime in e al in he las qua e : Figu e 33. Final sco e di e ences o games wi h 1 second di e ence in he las qua e . The da a a e clus e ed om 0 o 10 poin s app oxima ely. Mo e han 10 poin s is a a e e en . The peak alue is 5 poin s. -5 0 5 10 15 20 25 30 0 20 40 60 80 100 120 136 We no e ha he mos likely sco e di e ence is be ween 0 (we did no coun o e ime) and 10 poin s. Bu we can ind di e ences up 29 poin s. The 5.88% inish in a ie (0 poin s; wi h o e ime). A o al o 526 games o 918 (58%) inish wi h a di e ence be ween 3 and 7 poin s. And he 93% inish wi h less han 11 poin s F om an o e iew, he e we e 690 games wi h a leas a case o poin s wi h 1 second di e ence. In 184 games he e we e wo cases. In 41 he e we e h ee cases and in ou games he e we e only h ee cases. Sco ing Rega ding o sco e, he absolu e alue o sco e ( esul ) always g ows along wi h game, bu do no e ol e uni o mly. This is a eali y which is main ained on all baske ball games. Sco e uns and maximum alues achie ed by he eams may a y, bu always does inc emen ally. Bu wha ha eally se s he dynamics o he game is he poin di e ences be ween a eam and ano he du ing he game ime and, abo e all, a he end o he game. Fo ha eason, we analyzed he di e ences on he inal sco e o he whole sample analysis (6150 NBA games). The esul (Figu e 34) poin ou ha mos o he games (65%) ended wi h a di e ence be ween 1 and 11 poin s, 33% had a di e ence be ween 11 and 28 poin s, and only 2% did so wi h a di e ence o 28 o mo e poin s. To e i y whe he he da a ollowed a Powe Law ype dis ibu ion, we pe o med a log-log plo whose esul can be seen in he uppe panel o Figu e 34. Baske ball • Complexi y S udy 2. Baske ball Game 137 Figu e 34. Poin di e ence his og am exis ing in he inal sco e o each game s udied. The dis ibu ion is app oxima ely uni o m o alues less han 10-12. Fu he han his alue he dis ibu ion shows a possible beha io o long ail. Log-log plo o da a poin di e ence and equency. We can see ha he i s a ay p esen a homogeneous endency. A ound he alue o 10 poin s, an in e up ion in his end akes place; and a second one a a alue a ound 25-28 poin s. This sugges s he p esence o mo e han one Powe Law. 0-10 sco e di e ence F om 1 poin o 10 poin s app oxima ely, he dis ibu ion is almos uni o m, which co esponds wi h si ua ions o high unce ain y. I we exceed his sco e, om 10 o 28 poin s, he beha io appea s o ollow a Powe Law. This indica es ha he na u e o he game has changed. Finally, o e 28 poin s (a second Powe Law), he essence o he game changes adically, and he inal ou come is mo e p edic able. In b ie , esul s be ween 0 and 10 poin s a e simila : he game is ha d- ough . This poin s ou ha as long as he game emains be ween hese alues, he inal esul is unp edic able. This dynamic sugges s ha his a ea o poin di e ence (0-11 poin s) wo ks as an a ac o , because he sys em ( he game) ies o emain wi hin his na ow a ea h oughou he game ime. In ac , in a ound 20% (abou 1174) o he games assessed, eams did no exceed he maximum sco e di e ence o 11 poin s o he en i e y o he game. 138 Rega ding he inal esul , he numbe o games ha inished wi h a poin di e ence lowe o equal o 11 poin s was 3846 games (62% o he games analyzed), which co esponds wi h he i s cu in he log-log (Figu e 34). 2324 o hese games eached a maximum poin di e ence be ween 11 and 20 poin s, and 43% did i du ing he las qua e o he game. In 578 games (almos 25% o he 2324 games), a eam was able o o e come he di e ence (be ween 11-20 poin s) and win he game. This means ha he e we e eams able o o e come a signi ican di e ence (be ween 11-20 poin s) and e en win he game. I is possible ha by achie ing good sco e uns, he game is able o each he c i ical a ea, and, joined wi h s a egy a he end o he game ( inal qua e , ouls, ee h ows, ime-ou s, e c.), he combina ion needed o ha eam o win he game can be achie ed. 1831 games eached a maximum poin di e ence o mo e han 20 poin s. In 348 games (20%), his si ua ion was o e come and he game was loca ed in he a ea o a 0-11 poin di e ence. 34 games (9.7%) o hese cases won he game. In hese games, he maximum di e ence was eached be ween he 9-minu e and 44-minu e ma k. The esul s poin ou ha i is indeed e y di icul o o e come a 20-poin di e ence in he las 4 minu es o he game. 11-28 sco e di e ence Ou o he 2285 games wi h a sco e di e ence highe han 11 poin s, only 27 we e able o win and o e come he di e ences be ween 12 and 22 poin s. No e ha in his case, hese di e ences we e eached be o e he 24 h minu e (mos o hem in he second hal ), excep a case in which i was done in he 33 d minu e, bu in ha case he eam was losing by only 15 poin s. Hence, beyond 10 poin s he dynamic is comple ely di e en and is mo e p edic able. La ge han 28 sco e di e ence Nei he eam was able o ei he o e come he di e ence o achie e he egion o an 11-poin di e ence. This means ha i he e is mo e han a 28-poin di e ence, hen he e is a clea supe io i y o one eam o e ano he , so much so ha he game is qui e p edic able. We mus emembe ha he e is no a ixed c i e ion o iden i y non-linea complex sys ems o sel -o ganized c i icali y beha io s in spo s. Bu whe he a Powe Law appea s, i is possible we a e dealing wi h a non-linea complex sys em (Sa aglio & Ca bone, 2000; Ga cía Manso Baske ball • Complexi y S udy 2. Baske ball Game 139 e al., 2008). The log-log plo o he dis ibu ion o poin di e ence is b oken in o se e al Powe Laws; o ce ain cha ac e is ic alues ha can be conside ed h esholds o c i ical poin s, which means ha game dynamic, can be cha ac e ized by se e al c i ical phenomena, o wi h se e al scales (mul i-scale). The p esence o c osso e s in Powe Laws is an indica o o changes in he unde lying dynamic and sugges s ha pe haps we a e dealing wi h a phase ansi ion and c i ical exponen s (McGa y e al., 2002; Sche e e al., 2009). Re u ning o he gene al dis ibu ion o poin di e ences, i also can be modeled by a Nega i e Binomial (1.94; 0.15), as we can see in he nex Figu e: Figu e 35. His og am o he poin di e ences wi h a dash line as heo e ical nega i e binomial i . The main igu e ep esen s he log-log plo o he same da a, also wi h a nega i e binomial dis ibu ion (dash line). No e ha he his og am does no i well by he nega i e binomial dis ibu ion a he beginning bu i does a he ail. The log-log plo displays wo c oss o e . The i s one a ound 10 poin s and he second one a ound 28 poin s. We can obse e ha he his og am o he poin di e ences is ela i ely well i ed by a nega i e binomial dis ibu ion (Figu e 35) especially a he end. Nex i is shown he end o games in de ail. I he poin di e ence is less o equal o 10 poin s he dis ibu ion o he poin s sco ed a he las minu e o he game ollows a nega i e binomial dis ibu ion (Figu e 36). 101 100 101 102 010 20 30 40 50 0 0.01 0.02 0.03 0.04 0.05 0.06 140 Figu e 36. His og am o he poin s sco ed a he las minu e o games ended by 10 o less poin s di e ence. The dash line ep esen s he heo e ical nega i e binomial dis ibu ion. The his og am poin s ou ha he mos likely, o games ended by 10 o less poin s di e ence, is o sco e 3 o 4 poin s in he las minu e. Bu he dis ibu ion seems o i well o a nega i e binomial dis ibu ion wi h pa ame e s 5.9301 and 0.5306; mean = 5.2471; median = 5.0000; a iance = 9.9922 and STD = 3.1610. No e ha he e is a ail, bu he mos pa o he poin s sco ed a he las minu e a e loca ed bellow 10 poin s which poin owa d a g ea compe i i eness. The dis ibu ion o sho s sco ed was 1 poin sho s = 12008; 2 poin s sho s = 4954 and 3 poin s sho s = 1772. Fo he case o games ended wi h a di e ence be ween 11 and 28 poin s, he his og am is: -5 0 5 10 15 20 25 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Baske ball • Complexi y S udy 2. Baske ball Game 141 Figu e 37. His og am o poin s sco ed in he las minu e o games wi h a di e ence be ween 10 and 28 poin s. The dash line ep esen s he heo e ical nega i e binomial dis ibu ion. The Figu e 37 ep esen s he his og am o games ended wi h a poin di e ence om 10 o 28 poin s. We can obse e ha he highes p obabili y is o 1, 2 and 3 poin s and p esen s a ail as well. The s a is ical alues a e: mean = 2.5049; median = 2.0000; a iance = 3.0034 and STD = 1.7330 The Index o Dispe sion o hese da a was 1.19 which poin s ou an o e -dispe sed Poisson dis ibu ion. The dash line ep esen s he nega i e binomial dis ibu ion wi h pa ame e s 14.9755; 0.8567. The dis ibu ion o sho s sco ed was 1 poin sho s = 1732; 2 poin s sho s = 1693and 3 poin s sho s = 517. No e ha in his case he numbe o poin s is mo e clus e ed han he p e ious case. And he numbe o sho s sco ed is lowe as well. Fo he las case, he case o games ended wi h a poin di e ence highe han 28 poin s (Figu e 38), he s a is ic was: mean = 2.3212; median = 2.0000; a iance = 2.1241 and STD = 1.4574. The Index o Dispe sion is 0.91 (lowe han 1. Unde -dispe sed), e go i is no nega i e binomial, is mo e Poissonian. 012345678910 11 12 13 14 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35