Basketball from perspective of nonlinear complex systems

UNIVERSITY OF LAS PALMAS DE GRAN CANARIA DEPARTAMENT OF PHYSICAL EDUCACTION Doctoral Program: Motor Praxiology, Physical Education and Sport Training 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 Departamento de Educación Física. Facultad de Educación Física Programa de doctorado: Praxiología motriz, educación física y entrenamiento deportivo. Título de la Tesis BASKETBALL FROM THE PERSPECTIVE OF NON-LINEAR COMPLEX SYSTEMS Tesis Doctoral presentada por D. Yves de Saá Guerra Dirigida por el Dr. D. Manuel Navarro Valdivielso Codirigida por el Dr. D. Juan Manuel García Manso Codirigida por el Dr. D. Juan Manuel Martín González El Director, El Codirector , El Codirector, El Doctorando, (firma) (firma) (firma) (firma) Las Palmas de Gran Canaria, a _____ de_________________ de 2013 The more I know, the less I can affirm, categorically. Cuanto más sé, menos puedo afirmar las cosas categóricamente. Yves ACKNOWLEDGEMENTS This dissertation would not have been possible without the help of so many people in so many ways. It carried out thanks to the commitment and dedication of a research group led by Dr. Manuel Navarro Valdivielso, Dr. Juan Manuel García Manso and Dr. Juan Manuel Martín. It is an honor be part of this group and I would like to express my sincere gratitude to my thesis director Dr. Manuel Navarro Valdivielso for his unconditional support, his advices and his academic orientation, as well as for his help in the develop of this thesis. Also I would like to express my deepest admiration to my co-directors Dr. Juan Manuel García Manso, because he taught me everything I know about training and advised me about sport in an exquisite way. He has conveyed me all his strength and passion for the sport. I will never be able to remove his seed; and to Dr. Juan Manuel Martín, because he developed the necessary programs in Matlab that allow us to deepen in sports reality and advise with mathematics. I have been fascinated by all of our conversations about the universe and the secrets of life. It is a real pleasure to listen to and learn from him. Without his help this thesis would not been possible. I feel privileged for sharing part of my life with these three great people. I also would like to acknowledge the collaboration and support of Professor Adrian Bejan, from Duke University, USA. by his passionate discussions and by his outstanding relevance in the scientific community. I was a real pleasure learn from him during my stay at Duke University. I cannot conclude this chapter without to extend my sincere thanks to my fellows of the Laboratory for Analysis and Planning of Sports Training, of Department of Physical Education of this university, especially to Dr. Samuel Sarmiento Montesdeoca for his support and friendship. I must also acknowledge the support of the University of Las Palmas de Gran Canaria, for supporting my research by awarding a grant for the development of this thesis. Finally, and by no means least important, I would like to give thanks to my entire family for their unconditional support every time I needed it. They are my role models. They always have been. And always will be. Yves Introduction In our effort to try to understand the sport reality, we are forced to question everything that we consider true or static. This leads us inexorably, to try to isolate a phenomenon in order to study better. But far from our purpose, when we move towards this point, we realize that we fall into the contradiction of hoping of be able to understand a phenomenon by isolating it from its environment. Following the asseveration: ” I am I and my circumstance” (Ortega y Gasset, 1933), it is not possible to understand the sport phenomenon by isolating the constituent elements of their relationships with their own universe. There is a duality element–environment. From this relationship might emerge new behaviors, which is known from the perspective of complexity, as an emergency phenomenon or an emergent behavior. I.e. in order to reason this phenomenon, it would be good to move away from the deterministic and reductionist classical model, and move on to study the systems from a more global conception (holistic), allowing us to identify and describe the processes of new forms of organization, which is also useful in sport. Organizing the sport training from a systematic conception and conceive the athlete, or the team in our case, as a system that functions as a whole and that is affected by the surrounding environment (Gambetta, 1989; Martín Acero & Lago Peñas, 2005; García Manso & Martín González, 2008). Chaos theory has provided a new light to observe all these systems, seemingly incomprehensible or random, because it is usual that natural systems are chaotic. Chaos hides an internal order that is possible to find (Prigogine & Holte, 1993). In complex systems, the processes occurring simultaneously in different scales or levels are important, the intricate and complex behavior of the system as a whole, depends on the units, although not directly, because in complex systems the structures have strong relationships, often in a non-linear manner (Vicsek, 2002; Goodwin, 2002; Amaral & Ottino, 2004; Solé, 2009). It would be useless to consider the conscious thought as a mere sum of neurons, or reduce the behavior of a team to the sum of the individual abilities of his players individually. This definition also distinguishes the complex from the simple and complicated, as can be any other mechanism such as that determines the operation of a car, an airplane or a computer. 6 Complexity is a measure of the number of possibilities: the ways in which we act or react to the environment. It is an important point for the study of sport from the point of view of complex systems. However, the true complexity of behavior, as seen in basketball, occurs in the interaction between its elements and response possibilities of each structure. Sport performance is the result of the combination of many variables that sometimes we know and dominate it through different analytical methods, and we try to understand even better to improve it. Seldom effectiveness in sport shows a linear behavior. It would be very easy to understand and get better. Actually, there are many actions we can consider proper, even whether these actions would repeat, need not be consecutive. The behavior of players, ball, coaches and many more aspects, may condition the outcome. So it works as any complex organization, hence, must be understand as a complex system. Establishing team Sports as complex systems, our intention is to move away from classic and deterministic models, in order to pass on a new perspective for unifying criteria and analysis and thus to address and resolve issues raised above, and increase performance in different sports collective. This model is based on how its components are related in a precise and determined, and how they react to other complex systems. In fact, all we seek is to recognize and identify patterns of collective behavior and relationships among its components, which make it, resemble self-organizing complex systems non-linear in critical (System Organized Critically). According to Goodwin, the ideal is to investigate the conditions that promote selforganization(Goodwin, 2002), in order to obtain the sporting excellence. Complex systems are the result of an evolutionary process. Darwin's ideas and the study of evolution have focused on the competition as a driving force of evolutionary change. Players for example, when they cooperate, compete better as a team (Bar-Yam, 2003). Team performance can be postulated as win as many games as possible. It results from the synchronous interaction of certain states of optimization of systems that make up, which also have a reciprocal relationship with the emerging and critical environment: the competition. The aim of the study is to observe the behavior of the structure (macrostructure) of basketball as a system. We want to find out how basketball elements are interconnected and how they affect each other, analyzing the laws that govern them. Therefore we have established three levels or items to carry out the investigation: league, games and basketball team as network. Basketball Background Basketball from the perspective of non-linear complex systems Yves de Saá Guerra 2013 Basketball • Complexity Basketball Background 9 3. Basketball Background 3.1 Basketball History Creation of basketball Dr. James Naismith was a Canadian physical education professor at the International Young Men's Christian Association Training School (YMCA) in Springfield, Massachusetts, USA. In December of 1891, James Naismith was asked by his director to devise an indoor game for the school´s 40 students, to help keep them physically active between football season and the springtime activities of baseball and track (Muruzábal del Solar, 2012). Dr. Naismith combined elements of outdoor games like football and lacrosse with the concept of a game he played in childhood, Duck on a Rock. To win Duck on Rock, players threw stones to hit a target placed on top of a large boulder (Naismith, 1941). He set up peach baskets attached to both ends of a gymnasium balcony onto a 3.05 meters elevated track, and used a ball in order to score on it. The peach baskets retained the ball at bottom and it had to be taken out manually after each point scored until the bottom of the basket was removed. The peach baskets were used until 1906 when they were finally replaced by metal hoops with backboards. The backboards appeared as protection, to prevent the fan located on the railing of the gallery, where hung the baskets, could hinder the entry of the ball in the basket, which later went on to become a metal ring and a network without holes, for lead in today's networks (Tous Fajardo, 1999). The early players did not use dribbling of the ball, except for the “bounce pass” to teammates. Passing the ball was the primary means of ball movement. Dribbling was eventually introduced but limited by the asymmetric shape of early balls. Dribbling only became a major part of the game around the 1950s, as manufacturing improved the ball shape. The first official game was played in the YMCA gymnasium in Albany, New York on January 20, 1892 with nine players. The game ended at 1-0. The shot was made from 7.6 meters, on a 10 court just half the size of a present day National Basketball Association (NBA) court. By 18971898 teams of five became standard (Naismith, 1941). A ball and an elevated goal, those are the simple ingredients of the sport that now have players and rabid fans in nearly every part of the world. According to Alexander Wolff, in his book 100 Years of Hoops (Wolff, 1991), Naismith drew up the rules for the new game in “about an hour”. Nowadays basketball is one of the world's most popular and widely viewed sports (Griffiths, 2010). The First 13 Rules of Basketball Naismith and Wheeler wrote the first 13 rules of the game (International Basketball Federation; FIBA, 2012). They were published in the school newspaper, The Triangle, for first time in 1892: 1. The ball may be thrown in any direction with one or both hands. 2. The ball may be batted in any direction with one or both hands (never with the fist). 3. A player cannot run with the ball. The player must throw it from the spot on which he catches it, allowance to be made for a man who catches the ball when running at a good speed if he tries to stop. 4. The ball must be held in or between the hands; the arms or body must not be used for holding it. 5. No shouldering, holding, pushing, tripping, or striking in any way the person of an opponent shall be allowed; the first infringement of this rule by any player shall count as a foul, the second shall disqualify him until the next goal is made, or, if there was evident intent to injure the person, for the whole of the game, no substitute allowed. 6. A foul is striking at the ball with the fist, violation of rules 3, 4, and such as described in rule 5. Basketball • Complexity Basketball Background 11 7. If either side makes three consecutive fouls, it shall count a goal for the opponents (consecutive means without the opponents in the mean time making a foul). 8. A goal shall be made when the ball is thrown or batted from the grounds into the basket and stays there, providing those defending the goal do not touch or disturb the goal. If the ball rests on the edges, and the opponent moves the basket, it shall count as a goal. 9. When the ball goes out of bounds, it shall be thrown into the field of play by the person first touching it. In case of a dispute, the umpire shall throw it straight into the field. The thrower-in is allowed five seconds; if he holds it longer, it shall go to the opponent. If any side persists in delaying the game, the umpire shall call a foul on that side. 10. The umpire shall be judge of the men and shall note the fouls and notify the referee when three consecutive fouls have been made. He shall have power to disqualify men according to rule 5. 11. The referee shall be judge of the ball and shall decide when the ball is in play, in bounds, to which side it belongs, and shall keep the time. He shall decide when a goal has been made, and keep account of the goals with any other duties that are usually performed by a referee. 12. The time shall be two 15-minute halves, with five minutes rest between. 13. The side making the most goals in that time shall be declared the winner. In case of a draw, the game may, by agreement of the captains, be continued until another goal is made. 12 3.2 Basketball General Description Some authors such as Knapp (1981), Hernández (1994) and Ruiz (1999) analyze basketball regarding to formal structure and functionality of the game itself. It also can be classified as a team sport or collective sport, of cooperation-opposition, with functional structure which develops in a common space for both teams and simultaneous intervention on ball (Hernández Moreno, 1994) Basketball is defined as a team sport played by two teams of five players each on the court, and can be substituted by bench players (each league allow a different number of bench players), previously selected by the coach. The aim of each team is to score in the opponents' basket and to prevent the other team from scoring while following a set of rules. The team that has scored the greater number of points at the end of playing time shall be the winner (Rule 1. Art. 1. FIBA, 2012). In the leagues we studied (National Basketball Association, NBA; Asociación de Clubs de Baloncesto, ACB; and National College Athletic Association, NCAA); the game is controlled by officials and score officials throughout the Official Basketball Rules (NBA, 2012; NCAA, 2012a; FIBA, 2012b) (ACB uses FIBA Rules). Nevertheless there are some differences between the NBA, FIBA and NCAA rules, regarding court dimensions, time outs, fouls, officials, etc. (see table 1). We carried out this comparison because these three leagues are the aim of study. In general, rules for the male gender and for the female gender are different. Only the FIBA rules establish the same rules for both genders. Usually, teams play on a marked rectangular court with a basket at each width end. The dimensions of the court are described by the official rules (NBA, 2012; FIBA, 2012b; NCAA, 2012b) (See Figure 1, Figure 2, Figure 3 and Figure 4). Basketball • Complexity Basketball Background 13 Figure 1. Hoop dimensions. Source: FIBA. Figure 2: FIBA Basketball Court. Source: FIBA. 20 2012). It also called round-robin tournament or all-play-all tournament. In turn can be played in one single group (depend of the numbers of teams) or can be divided in several groups, conferences and/or divisions, according to a prearranged schedule. Play-off/Cup. The playoffs or finals (we can also include cup tournaments) in a sport are a game or series of games played after the regular season by the top teams classified, in order to determine the league champion. Teams use to play in a bracket (ACB, 2012; NBA, 2012; NCAA, 2012a). A bracket is a tree diagram that represents the series of games played during a tournament, named as such because it appears to be a large number of interconnected (punctuational) brackets. There are several formats for the play-off brackets, meaning the number of games per round (1-1-1-1, 3-3-3, 7-7-7-7, etc.). In college basketball is very famous the fact of filling in brackets, especially in NCAA basketball, is referred to as bracketology. Federation. Federations are private non-profit organizations composed by administrative section, sports clubs, athletes, coaches, judges and referees and professional leagues, in order to promote, practice or contribute to the development of sport (Ley 10/1990, de 15 de octubre, del Deporte, Art. 30. Ministerio de Educación, Cultura y Deporte). The functions of the basketball federations are the government, administration, management, organization and regulation of the sport of basketball throughout the territory covered, whether international or domestic, regarding competitions and championships organized. As well as drawing up of the corresponding licenses that are required to participate as player, coach or referee, in competitions and championships organized. Basketball • Complexity Basketball Background 21 3.3.2. International structures In basketball there are several international structures which try to regulate and promote worldwide basketball through the national teams. FIBA (International Basketball Federation) The International Basketball Federation (FIBA) is the organization that is dedicated to regulate the rules of basketball worldwide, as well as holding regular competitions and events in the disciplines of basketball (men and women). The association was founded in Geneva in 1932, two years after the sport was officially recognized by the IOC (International Olympic Committee). The name FIBA came from its French name Fédération Internationale de Basketball, is an association of national organizations which governs international competition in basketball. Originally known as the Fédération Internationale de Basketball Amateur (hence FIBA), the word “Amateur” was dropped in 1986 after the distinction between Amateurs and Professionals. The "BA" now represents the first two letters of basketball. The main aim of the FIBA was to coordinate tournaments and teams. Argentina, Czechoslovakia, Greece, Italy, Latvia, Portugal, Romania and Switzerland were the founder members (FIBA, 2012). The FIBA Central Board is currently composed of 23 members (22 have the right to vote) and meets twice yearly. The FIBA Central Board has, among other competences, the power to establish the FIBA Internal Regulations. It also assigns the organization of all FIBA Basketball World Cup. FIBA counts with 213 member federations. FIBA has organized a FIBA World Championship for men since 1950 and a World Championship for Women since 1953. Both events are now held every four years, alternating with the Olympics. 22 IOC (International Olympic Committee) The IOC coordinates the activities of the Olympic Movement. It is also responsible for supervising and manages everything about the Olympics. Owns all the rights associated with the Olympic symbols, flag, anthem, lemma, oath and games. It controls the rights to broadcast the games, advertising and other activities according to the Olympic Charter. It is also the international body responsible for organizing and selecting the cities that will host the Olympic Games every 4 years (IOC, 2012). In detail the role of the IOC, according to the Olympic Charter (September 2004), is:  To encourage and support the promotion of ethics in sport as well as education of youth through sport and to dedicate its efforts to ensuring that, in sport, the spirit of fair play prevails and violence is banned.  To encourage and support the organization, development and coordination of sport and sports competitions.  To ensure the regular celebration of the Olympic Games.  To cooperate with the competent public or private organizations and authorities in the endeavor to place sport at the service of humanity and thereby to promote peace.  To take action in order to strengthen the unity and to protect the independence of the Olympic Movement.  To act against any form of discrimination affecting the Olympic Movement.  To encourage and support the promotion of women in sport at all levels and in all structures with a view to implementing the principle of equality of men and women.  To lead the fight against doping in sport.  To encourage and support measures protecting the health of athletes.  To oppose any political or commercial abuse of sport and athletes.  To encourage and support the efforts of sports organizations and public authorities to provide for the social and professional future of athletes.  To encourage and support the development of sport for all.  To encourage and support a responsible concern for environmental issues, to promote sustainable development in sport and to require that the Olympic Games are held accordingly. Basketball • Complexity Basketball Background 23  To promote a positive legacy from the Olympic Games to the host cities and host countries.  To encourage and support initiatives blending sport with culture and education.  To encourage and support the activities of the International Olympic Academy (IOA) and other institutions which dedicate themselves to Olympic education. Basketball appeared for first time in Olympic Games in San Louis in 1904 as an exhibition game. Basketball was included in Olympic Games in Berlin 1936 as Olympic Sport. Women´s basketball is present in Olympic Games in Montreal 1976. 3.3.3. National Structures In every country, organizational structures of basketball are designed in different ways. According our investigation we studied the basketball in USA and Spain, hence we describe the organizational structures of basketball in USA and Spain as a follows. 3.3.3.1. Organizational structure of basketball in USA: Basketball Association of America The Basketball Association of America (BAA) was a professional basketball league in North America, founded in 1946. The league merged with several leagues such as the National Basketball League (NBL) in 1949, forming the National Basketball Association (NBA). Eleven cities are fortunate to welcome a team to represent them, are essentially cities located on the coast: Nueva York, Chicago, Boston, Providence, Toronto, Cleveland, San Luis, Washington, Detroit, Pittsburgh y Philadelphia. The first game is played in the city of New York and confronts New York Knicks vs. Toronto Huskies. African American players do not start to play until 1950. There were several attempts to create other professional leagues to overthrow the NBA, highlighting the ABA League. 24 National Basketball League The National Basketball League was founded in 1898 in the USA and was the first professional league in the world. Six teams took part in it and the first champions were the Trenton Nationals, followed by the New York Wanderers, the Bristol Pile Drivers and the Camden Electrics. The National League lasted five seasons (1904), but new leagues quickly were formed throughout New England and the Mid-Atlantic States, prominent among them were the Philadelphia Basketball League, Eastern Basket Ball League, Hudson River League, New York State League and the Interstate Basket Ball League. ABA (American Basketball Association) The American Basketball Association (ABA) was founded as an alternative to the NBA in 1967. Teams were created in different cities than NBA teams. The ABA was characterized by the color of the ball (red, white and blue) and the manner of play. The ABA also introduced several rules that differed from the NBA. Among them was the three-point line. The NBA adopted the three point line in 1979-80. The ABA competed with the National Basketball Association (the NBA) for players, fans, and media attention. In June 1976, four of the strongest ABA teams (the New York Nets, Denver Nuggets, Indiana Pacers, and San Antonio Spurs) joined the NBA (Silverman, 2012). NCAA (National Collegiate Athletic Association) The National Collegiate Athletic Association (NCAA) is an association of several institutions, conferences, organizations and individuals that organizes the athletic programs of many colleges and universities in the United States. It is headquartered in Indianapolis, Indiana. The NCAA was founded in 1906 to protect young people from the dangerous and exploitive athletics practices of the time. In that period, the football was used in order to formation and gang tackling, but there were numerous injuries and deaths and prompted many college and universities to discontinue the sport. The most part of the fans and people related with football thought that college football should be reformed or abolished. President Theodore Roosevelt summoned college athletics leaders to two White House conferences to encourage reforms. In December 1905, in New York City, 62 colleges and Basketball • Complexity Basketball Background 25 universities became charter members of the Intercollegiate Athletic Association of the United States (IAAUS). It was officially was constituted March 31, 1906, but in 1910 was renamed as National Collegiate Athletic Association (NCAA). Gradually, more rules committees were formed and more championships were created, including a basketball championship in 1939 (NCAA, 2012c). One of the keys to success in college basketball was that the BAA had money and good game courts, but lacked of talent and experience players. This lack of players forces the owner to create a new policy to end the problem. This new policy is to recruit the best college players to shape and a championship that combines the speed, talent and experience. The basketball college championship in USA is divided in three divisions (Division I, Division II and Division III). In turn, every division is divided in conferences of several teams each. Currently, in the Division I are involved a total of 344 teams (the number varies within the season analyzed), divided in 31 conferences through all USA (the number of teams per conference is not homogenous). The competition format of the NCAA described in this thesis makes reference only to the Division I of the men´s basketball. There is a regular phase (regular league), and a playoff. In the regular phase teams play against the teams of the same conference. In addition, they play extra games (tournaments) in order to get more points for the playoff classification. Most of these tournaments are the same every season. Some are very important in college basketball community. Some of the most popular are: 2K Sports Classic, Coaches vs. Cancer, Puerto Rico Tip-Off, Paradise Jam, CBE Classic, Maui Invitational, NIT Season Tip-Off, Old Spice Classic, 76 Classic, Legends Classic, ACC/Big Ten Challenge, Big12/Pac10 Hardwood Series or the Jimmy V Basketball Classic. After the regular phase, the best teams classified play a playoff (only one game per round) for the national championship. There are two ways of qualifying for the play-off tournament. One is directly and the other is by invitation granted by the NCAA. 26 The direct classification is obtained by a ranking made with the RPI index. The RPI index is a mathematical equation that takes into account the games won, games lost, a series of numerical constants and strength of the schedule. Once obtained this ranking the 31 best will qualify directly for the tournament. The playoff tournament include 68 universities (31 champions + 37 invited) and is held in March (also called March Madness). The 68 teams are divided into four regions (South, East, West and Midwest) and organized into a single elimination bracket. Each team is ranked within its region. From the 68 teams, 60 of them go directly to the second round remaining the 8 lower-seeded teams, which have received fewer votes from the NCAA; dispute the four remaining places in four small games of the first round (called First Four). After an initial four games between 8 lower-ranked teams, the tournament takes place over the course of three weekends, at pre-selected neutral sites around the United States. Lowerranked teams are placed in the bracket against higher ranked teams. Each weekend cuts threefourths of the teams, from a Round of 64, to a round of 16 also called Sweet Sixteen, to a Final with four teams, called Final Four. The Final four is usually played on the first weekend in April. NBA (National Basketball Association) The NBA is the men's professional basketball league in North America (United States and Canada). The league was founded as the Basketball Association of America (BAA) in New York City on June 6, 1946 (NBA, s. f.). The league adopted the name National Basketball Association (NBA) in 1949 after merging with the rival National Basketball League (NBL). The NBA is currently the most significant professional basketball league in the United States of America, in terms of popularity, salaries, talent, and level of competition (Patterson, 1993; Hausman & Leonard, 1994). The NBA is a league of closed structure (no promotions and demotions), composed by 30 franchised members, which 29 are located in the United States and one in Canada. The current league organization divides the 30 teams into two conferences of three divisions, with five teams each. The current divisional alignment was introduced in the 2004–2005 season. Basketball • Complexity Basketball Background 27 During the regular season, each team plays 82 games, 41 at home and 41 away. A team plays against its opponents in its own division for four times per season (16 games), three or four times against teams from the other two divisions in its own conference (36 games) and twice against teams in the other conference, respectively (30 games). This asymmetric structure means that the strength of the schedule varies significantly among teams. The NBA organization chart is constituted by the CEO and different departments, which directly depend on the CEO. The most remarkable, regarding another professional basketball leagues, is that in the NBA referees are professionals and depend directly on the NBA (do not depend on the Federation). The NBA departments are:  Basketball Operations  Broadcast Operations  Communications  Community and Player Programs  Creative Services  Events and Attractions  Facilities and Administration  Finance and Benefits  Global Marketing Partnerships  Global Merchandising Group  Human Resources  Information Technology  Interactive Services  International  International Media Distribution  Legal  Legal and Business Affairs  Marketing  DLeague (NBA Development League)  WNBA 28  NBA Entertainment Production, Programming and Photos  Referee Operations  Security  Strategic Development  Team Marketing and Business Operations Source: NBA Salary cap and Draft The NBA has several mechanisms established in order to prevent teams, with large surpluses of profit, can sign the best players available, thereby facilitating the maintenance of equality in the league. The more representative mechanisms are the salary cap and the draft. The lottery draft process has changed along the years, but exists since NBA foundation; unlike the salary cup, which started in the mid-1940s, (it was abolished after only one season); and reinstated in the 1984–85 season. Salary cap The North American professional sports leagues (Major League Baseball (MLB), National Basketball Association (NBA), National Football League (NFL), and National Hockey League (NHL) have an agreement or rule that sets a limit on the amount of money that a team can spend on player payrolls, called salary cap (Scott, Long, & Somppi, 1985). The salary cap, in the NBA, started in 1983 (Hill & Groothuis, 2001). A simple model shows that a salary cap can improve the competitive balance among clubs as well as the salary distribution among players (Késenne, 2000). For that reason, every franchise has to study carefully what market players could be interesting for his project (depending on the team´s project). The salary cup is defined by the league's collective bargaining agreement (CBA). The salary cap ensures that each franchise can only “shield” economically one or two key players, who are often called franchise players. There are three kinds of regulations: hard salary cap, soft salary cap (with luxury tax), and luxury tax (The NBA utilizes a soft salary cap) (Scully, 1989; Késenne, 2000; Fort & Maxcy, 2003): Basketball • Complexity Basketball Background 29  Hard salary cap. A hard salary cap is where the league sets a maximum amount of money allowed for player salaries, and no team can exceed that limit. At the beginning the salary cap was not a hard cap (Hill & Groothuis, 2001).  Soft salary cap. A soft salary cap has a set limit to player salaries, but there are several major exceptions that allow teams to exceed the salary cap. For example, in the case of the NBA, teams can exceed the salary cap when keeping players that are already on the team (Dietl, Franck, Lang, & Rathke, 2010).  Luxury tax: A luxury tax system does not have a limit to how much money can be spent on player salaries. However, there is a tax levied on money spent above a threshold set by the Collective Bargaining Agreement (CBA) between the players union and the owners. For every dollar a team spends above the tax threshold, they must also pay some fraction to the league. This system is used to discourage teams from greatly exceeding the tax threshold, with the goal of ensuring parity between large and small market teams (Dietl, Lang, & Werner, 2008). NBA lottery draft The NBA Draft is for the procedure by which, in late June each year, the NBA franchises join to their teams, players from U.S. universities or leagues in other countries. These players are usually amateur U.S. college basketball players, but international players are also eligible to be drafted. College players who have finished their four-year college eligibility are automatically eligible for selection, while the underclassmen have to declare their eligibility and give up their remaining college eligibility. The first draft took place in 1950 with the aim to provide good players to the league (see above, NCAA). Teams could forfeit their first-round pick and select a player from their immediate geographical area, commonly known as a “territorial pick” (NBA, 2007). In 1985 they changed to a lottery system, this NBA Lottery system set the order of selection for the non-playoff teams (or the teams holding their picks through trades) for the first round only. Teams picked in inverse order of their records in the second round in all succeeding rounds. In 1990, the NBA changed the format of the lottery to give the team with the worst record the best chance of landing the first pick. Currently NBA Draft consists of two rounds, the first and 36 FEB (Spanish Basketball Federation) It is a non-profit organization which in charge of promotion, management, and coordination throughout the national territory of basketball, in all its manifestations and variations. The farm leagues are managed by FEB (semi-professionals and farm leagues), and also manage the Autonomous Regions Federations (for amateur competitions), but the professional league of Spain is managed by ACB (association of several sport clubs). All the participants in competitions organized by FEB are integrated into the federation, such as corporations, sports, sports clubs, athletes, coaches and referees. And also it is responsible for issuing all licenses necessary for participating in its activities. The FEB is affiliated to FIBA as a member, being obliged, therefore to follow its statutes and regulations in all matters affecting the technical order and international relations. Internationally, the FEB is the representation of Spanish basketball in basketball international official activities and competitions celebrated within and outside the Spanish territory. The FEB elaborates the rosters of national teams (seniors and farm teams). The technique structures of the Basketball Federation are: 1. Administration and representation:  General Assembly and Executive Committee  The president 2. Management:  Executive Committee  Committee on Regional Federations 3. Consulting:  Area executive committee 4. Internal Management System:  General Secretary  Management  Those that could be created to better fulfill the federative purposes Basketball • Complexity Basketball Background 37 5. Technical-Sporting  General Secretary  Competition  Referees  Coaches  Committee on Health and Prevention of Doping 6. Discipline  National Competition Committee  National Appeals Committee ACB (Basketball Club Association [Asociación de Clubs de Baloncesto]) The ACB League is nowadays the main men's professional basketball league in Spain. It began in 1957, with the name of National League, and was originally organized by the Spanish Basketball Federation (FEB). In 1983-84 season the ACB was established with its own competition format and replaced the National League. The league is rated as one of the three "A" level European national domestic leagues in the ULEB League Rankings system (ACB, 2012). In Spain, the professional leagues are private structures that carry out functions of public interest and are supervised by the National Sport Council (Millán Garrido, 2010). The ACB model presents an open structure which means there are promotions and demotions. The top teams classified in the regular season play a championship in a play-off format. The last ranked teams are relegated to a lower division and in turn are replaced for the two top ranked teams of the bottom category. The season 2011-12 have participated a total of 18 teams, but this number has varied in previous seasons (from 13 until 24 teams). The ACB sport model consists in a regular season of double confrontations (only two games against the same team). The order of each team's first-half fixtures is repeated in the second half of the season. 38 Concerning to the administrative structure, the ACB is the association of several clubs (which have their own teams (farm teams) and even other different sport teams) with its own organizational structure. It means that some teams can participate in other European leagues simultaneously, such as Euroleague, ULEB, etc. unlike the NBA and NCAA. But the participation in these leagues is related with the standing of the previous season. Other difference with the NBA and NCAA is that the referees belong to the Spanish Basketball Federation (FEB). The central administration revolves around a Steering Committee composed of five members on direct dependency on the Directorate General. The maximum responsible has the General Assembly as the ultimate authority for the final decision. The five members shall be directors of the six strategic areas plus the Director of referees. The current organizational structure of the ACB comprises six areas: Competition, Media, Events, Institutional, Commercial and Administration. The last three sections also cover the rest of the organization transversally. The Competition Area includes the departments of Competition, Scouting and External Relations, including work and relationship with the various international competitions: Euroleague, ULEB and others. In front of this area will be the CEO of the ACB. The TV and Communication area contains the departments of Communication, ACB.COM, Audiovisuals and Television. This new area was created with the purpose of managing, in a comprehensive manner, the relationship with television operators of the Association. The areas of Events Management and try to increase brand value and cost of major events in the ACB such as the Copa del Rey and the Super copa, and also deal with Finance, Information Technology and Statistics issues. The Institutional Area (General Secretary) is responsible for all legal affairs and documentaries of the Association and manages the Human Resources Department and General Services. Basketball • Complexity Basketball Background 39 Business Area (Business and Marketing) includes the departments of Marketing and Business Development. These are key in generating income for the organization through national and international agreements, sponsorship plan and fitness centers promoted by the Association. Figure 6. Current organizational structure of the ACB. Source: ACB. 3.3.4. Structural comparison between beginning and the present The architecture of the basketball network can provide us a new and good perspective of how is the organization of basketball. Certain organizational and functional principles in complex systems are universal because some networks have similarities to other biological and technological networks (Solé & Goodwin, 2002). Hence we represented the basketball network (focused on ACB and NBA) in the season 1983-1984 (Figure 7), when the ACB was founded, and in the season 2008-2009 (Figure 8), in order to understand their evolution during all these years. We simplified some structures (such european leagues) for better understanding. 40 Figure 7 Representation of the network of institutions and basketball competitions. Some structures (such Farm Leagues and European leagues) have been simplified for better understanding. The circles represent the competitions (leagues/tournaments/championships) and the squares represent the institutions. We can see how some structures can be considerate hybrids, because they are competitions and organizations simultaneously. Not all the relationships are the same (some relationships are unidirectional and others bidirectional). There are dissipative structures (such the NCAA) and structures whose goals are to condense or concentrate the resources. But the most active structures of the network are those that consume and produce resources at the same time (such the ACB). We note that the network is based on these kinds of structures. The triangle based on the ACB-NBA - Sport Clubs is the engine of the system. 1983-1984 Figure 8. Illustration of basketball network in the season 2008-2009. The representations of the elements are the same as before: circles symbolize the competitions (leagues/tournaments/championships), squares represent the institutions, and finally square plus circles are hybrid structures. The network has evolved: some structures have disappeared and new emerged. Even some structures have adapted to the evolution in time and the grown of the network by a bifurcation of one of its structures (such FIBA in FIBA Europe), in order to preserve the effectiveness of the flow within the network. We are still seeing how the triangle formed by ACB-NBA-Sport Clubs is the key of the network, but whit the emergency of more professional leagues, the relationships have been modified. The network has become more professional. 2008-2009 Basketball • Complexity Basketball Background 41 42 The Figure 7 and the Figure 8 represent our proposal of the basketball network graph using the structures existing in Europe and USA and the institutional relationships among them. We proposed players as raw material and the arrows as the channels what they follow. Hence we can see, for example, the path that a player can take from a domestic league to the World Championship or the Olympic Games. We simplified some structures such the European leagues (Greece League, Italia League, Croatia League, etc.) and farm leagues in Spain (Gold LEB, Silver LEB, EBA, etc.) inasmuch as we considered them as similar structures with the same properties. At the graph the circles represent the competitions (leagues/tournaments/championships). The squares represent the institutions, such as the FIBA or the IOC. There some structures which are competitions with its own organizational structure (they can be independent or semi-independent from other structure such as federation), for instance ACB or NBA, hence can be considerate hybrids, because they are competitions and institutions simultaneously. There is a structure which we highlighted as special, because has a composition very different from the others, we refer to the sport clubs in Spain. This element possesses some properties different that a team: are managed by the president, the board and general assembly. There are clubs devoted exclusively to basketball activities and other clubs where the same sport club has several sections: football, basketball, volleyball, handball, hockey, etc. In addition, clubs have player farms, where they train young players and club teams competing in affiliated minor leagues, in order to create a young sport star, or as a showcase for other clubs. This structure is more wide and developed than a team, and enables the club maximize its resources, sharing common management structures, facilities, human resources, material resources, economic, etc. which enable to prepare rosters much more competitive. There are some productive/dissipate structures (such the NCAA or FIBA) whose aim is to produce or spread the material in the network. On the other hand there are structures whose goals are to condense or concentrate the resources, for instance the national federations. But the most active structures of the network are those that consume and produce resources at the same time (such the ACB). We note that the network is based on these kinds of structures, and the figure of the sport clubs are the most representative. The union of these structures is the engine of the system, as we can observe at the triangle based on ACB-NBA-Clubs. Basketball • Complexity Basketball Background 43 Regarding to the connections not all the relationships are the same (some relationships are unidirectional and others bidirectional). They indicate de flow within the network and the structures which are connected somehow. We can observe how at the first graph (Figure 7), in the season 1983-1984 the most part of the leagues are amateur such as the Spanish Farm Leagues, the ACB (before become professional) and the NCAA, which provide players to the USA national team (Olympic Games). These structures are which support the most part of the international competitions in addition to the NBA (professional league). The Figure 8 represents our proposal for the basketball network in the season 2008-2009. We can see how some structures have disappeared and have appeared new ones. The most remarkable fact is that the professionalization of the basketball has influenced notably in the architecture of the network. We can appreciate how the players for the USA national team, now are provided by the NBA (professional league), and the same happens in Spain, the ACB (now professional) supplies players for the national Spanish team (even the NBA provides players for the Spanish national team nowadays). An important fact is that the FIBA has created a European delegation (FIBA Europe was founded in 2001). This process of emergency indicates a high growth of the flow to international competitions between national teams in Europe, in our case. Moreover, the European competitions have changed their format or have been replaced with new ones. As we mentioned, in the season 1983-1984 these competitions were played by sport clubs, in Europe. But in the season 2008-2009 are played by national teams (competitions which are dependent on the FIBA Europe). The most remarkable fact related with the design of the network, is the appearance of two hybrid structures (as we called), the ULEB (founded in 1991) and the Euroleague (founded in 2000). This emergency is, probably, one of the sources of the network restructuration, because we have to take into account that both of these competitions are played by sport clubs, instead of national teams. In fact, in 2000, major professional sport clubs on Europe, led by the Spanish, Italians and Greeks, grouped in the Union of European Basketball Leagues (ULEB), split off from the FIBA in 44 order to organize a new Euroleague with modern management criteria. These clubs wanted to receive more revenue from television broadcasting rights and merchandising than the offered by the FIBA. In addition, the NBA created its own farm league: the NBA Development League (NBA DLeague); in which participating teams plays their own league (there is not promotions to the NBA). Some NBA teams share the resources but players. This league depends on the NBA. We past from 8 competitions, 6 institutions and 4 hybrid structures, in season 1983-1984, to 7 competitions, 8 institutions and 6 hybrid structures, in season 2008-2009. Note that the enlargement is related with the hybrid structures, which indicate us that the professionalization provides the apparition of these kinds of elements. These are good examples of the how a process such as the professionalization of a sport, can change the composition of the its structure. We also can appreciate how the clustering is not homogeneous in the two networks (table 2). Table 2. Number of connections of the network nodes in the season 1983-1984 (left) and in the season 2008-2009 (right). 1983-1984 2008-2009 Agent Connections Agent Connections FIBA 8 ACB 8 ACB 7 NBA 8 Clubs 5 Sport Agencies 7 NBA 5 Clubs 6 NCAA 5 FIBA 6 European Leagues 4 FIBA Europe 6 FEB 4 European Leagues 5 IOC 4 NCAA 5 Farm Leagues 3 FEB 4 USA Basketball 3 IOC 4 COE 2 NBA D-League 4 Korac Cup 2 Farm Leagues 3 Sporta Cup 2 Minor Leagues 3 CSD 2 USA Basketball 3 Minor Leagues 2 COE 2 EuroCup 1 CSD 2 European Championship 1 Euroleague 2 Olympic Games 1 ULEB 2 World Championship 1 Division C 1 EuroCup 1 FIBA Challenge 1 World Championship 1 Basketball • Complexity Basketball Background 45 There are other structures that enhance the movement of players, which no participate directly in the network, but influence strongly in it. We refer to the Player Agencies (or Sport Agents). Their aim is to localize players and move to another structure of the network where is needed; in return for part of the benefit created. This confirms that the nature of the basketball network has changed and that the flows are not constants. 3.4. Competition Analysis Several disciplines such as economy, applied statistic, physics, evolutive biology, social sciences, sport sciences, etc. have been tracking the evolution of a system within a controlled environment, often through analysing the interactions of the agents involved. Our aim was to investigate from an overview, the intern dynamic of professional basketball leagues studying its competitiveness degree. The degree of equality of the playing strengths of teams, competitive degree or competitive balance, is a central concept in the analysis of professional sports leagues. There is considerable interest in clarify the skewness and fluctuations in competitive balance throughout seasons; and analysing the effects of regulatory, institutional and other changes, as indicated by the extensive literature on the subject (Schmidt & Berri, 2001; Fort & Maxcy, 2003; T. A. Rhoads, 2004; Goossens, 2006) and applied in different sports such as baseball (Scully, 1989; Owen, Ryan, & Weatherston, 2007), American football (Humphreys, 2002), basketball (Noll, 1988; Berri, Brook, Frick, Fenn, & Vicente-Mayoral, 2005), ice hockey (Richardson, 2000), European football (soccer) (Halicioglu, 2006) or golf (T. Rhoads, 2005). The most part of works deal with this phenomenon with regard to the mechanics of the game itself, meaning the game in isolation, without implications to the competition (league)(Chatterjee & Yilmaz, 1999; McGarry, Anderson, Wallace, Hughes, & Franks, 2002; Lebed, 2006; McGarry & Franks, 2007; Passos et al., 2008; Passos, Araújo, Davids, Milho, & Gouveia, 2009), nevertheless, few works do from the perspective of competition between teams in different sports (Yilmaz & Chatterjee, 2000; Malacarne & Mendes, 2000; Onody & de Basketball • Complexity Complex Systems Background 53 4. Complex Systems Background A complex system is a set of several elements (also called agents) which are related among them and whose links contain information hidden to the observer. The established relationships among them are mainly type non-linear. These interactions are local interactions. That is, affect only the relationship between an agent and to the elements which surround him, but none of them is aware of the collective behavior (Goodwin, 2002; Vicsek, 2002; Amaral & Ottino, 2004; Solé, 2009). These processes, that take place simultaneously on different levels or scales, are important. In fact, the way in what its units are related, greatly influences in the output of the entire system. That is why the laws that describe the behaviour of a complex system are qualitatively different from those that govern its units (Amaral & Ottino, 2004; Vicsek, 2002). As a result of these interactions, new properties emerge that cannot be understood from the individual features of each element. These properties are called emergent properties. That is why a complex system must be treated as a whole, from a holistic conception, not just the elements that constitute it because in a complex system the whole is greater than the sum of the parts. Complexity is the result of incessant adaptive processes (Holland, 1995). 4.1. No-linear When the system is linear, the same stimulus always produces the same outcome. Every time the process is repeated, the same outcome will be obtained. On the other hand, if the system is non-linear, a stimulus can yield several results. Although the conditions are the same, the outcome or outcomes cannot be known in advance (Prigogine & Holte, 1993; Solé & Goodwin, 2002; Amaral & Ottino, 2004). The interrelationships of the components of the complex system are governed by non-linear equations. As mentioned above, not always effectiveness in sport shows a linear behavior, but there are several actions that can be considered effective, and also do not have to be consecutive. Complexity, in itself, is a measure of the number of possibilities. Such equations often have a strong dependence on initial conditions of the system, which makes it even more difficult to assess their behavior 54 4.2. Self-organization The idea of self-organization can be expressed as the general tendency of a given system to generate behavior patterns from local interactions of its constituent elements and from the relationships with the environment. It is the essential part of any complex system and allows the system to recover the balance, modified and adapted to the surrounding environment. Usually, the different system elements are self-regulated by themself always seeking to optimize the overall operation of the assembly. The complex network of interdependent systems in which human beings can organize, for example, is changing and readjusting to reality that corresponds to live in each moment (García Manso & Martín González, 2008). Self-organization is a process in which the internal organization of a system increases in complexity without being guided or managed by an outside source. Self-organizing systems usually display emergent properties. The order and disorder need each other, mutually occur. They are antagonistic concepts but complementary at the same time. In some cases, some of disorder allows a different order and sometimes, richer. For example, an organism can persist as a result of the death of its cells, or an organization is perpetuated by the dismissal of its members. The variation and change are inevitable and unavoidable stages through which every complex system must travel to grow and develop. When this transformation is achieved without the involvement of external factors to the system, referred to a process of self-organization (Nicolis & Prigogine, 1977). Self-organization stands out as an essential part of any complex system. It is the form through which the system recovers the balance, changing and adapting to the surrounding environment (it responds to external aggressions that seek to modify its structure). In this kind of phenomena is essential the idea of levels. The interrelationships among the elements of a level originate new types of elements on another level which behave quite differently, for example, from molecules to macromolecules, macromolecules into cells and from the cells to tissues. Thus, the self-organizing system is built as a result of increasing order space-time which is created in on different levels or layers, one above the other. To a large extent, complex systems can be understood like a machine that generate order, which requires constant energy intake generated by the chaos that feeds (it is an open system Basketball • Complexity Complex Systems Background 55 and dissipative). The self-organizing complex systems are considered adaptive because it can react to external stimuli and responding to any situation that threatens its stability as a system. Thus, it experiences fluctuations. This has a limit, of course. It is said that the system settles into a state and when it is away from him tends to make every effort to return to the previous situation. This happens for example with the human body constantly strives to maintain the same body temperature. 4.3. Critical State The general idea of system in a critical state can be understood as a state close to the boundary of another state (critical point). Meaning that any slight perturbation, can lead to a new state (phase transition). One of the most famous example is the sandpile model by Bak–Tang–Wiesenfeld (Bak, Tang, & Wiesenfeld, 1987). The model describes how a sandpile is builds up as grains of sand randomly placed onto a pile. At the beginning small perturbations only cause small responses. Small avalanches take place until the pile reaches a critical state in which its slope fluctuates about a constant angle of repose (threshold or critical point). If we add one sand grain more, this can causes the slope exceeds the critical value and originates a big avalanche. The variation of the local slopes makes it impossible to predict when this phenomenon will take place. Critical systems are featured by be in a delicately balanced state which, in turn, is linked to the environment, showing a great sensibility (Jost, 2005). This situation gives them a highly unpredictable behavior (chaotic, not random). The most part of the complex systems are unstable (they are out of the equilibrium state). This implies that the systems cannot sustain themselves unless they receive a constant supply of energy (order needs chaos and chaos needs order. They cannot exist without each other, as mentioned earlier). They demand adjustments following specific patterns. Any minimum variation among composing elements can modify unpredictably, the interrelations and therefore, the behavior of entire system. Thus, the evolution of such systems is characterized by intermittency or fluctuation (situation in which the order and disorder constantly alternate). Their evolutionary states do not pass through continuous and gradual process, but occur through reorganizations and jumps. Each new state is only a transition, a period of entropic 56 rest in the words of Russian-Belgian Nobel Prize Ilya Prigogine (Prigogine & Stengers, 1984; Prigogine & Holte, 1993). These systems never reach a global optimum, the minimum energy state. In general, grow gradually until they reach the limit of its potential development. At that moment, they suffer a disorder, a kind of rupture that induces a fragmentation of pre-existing order. But then, begin to emerge regularities that organize the system in accordance with new laws, producing another kind of development. This behavior is typical in natural systems: for example, the transit of the insects, from egg to larva and from there to the chrysalis. Consequently, the organization of complex systems is given at different levels. The laws governing the causality of a given level can be totally different from a higher level (Kauffman, 1995; Bak, 1999). 4.4. Self-Organized Systems and Sport Under these kinds of limit situations, in sport, athletes and their environment have to make a big effort in order to overcome the circumstances. In that moment is when they can really learn. It is at this time when sports systems create new strategies, training plans and, therefore, is when they evolve, change or behave according to the new reality. That is, the rivalry and competitiveness are the elements that generate the critical behavior. When the system is self-organized critically, information flows better among all parts of the system (Solé, 2009). Moreover, these kinds of systems have memory and regulatory mechanisms that adjusts the response to demand. These systems evolve trying to optimize their resources and tend naturally to be in these states, therefore, serve as attractors of the system (Ivancevic & Ivancevic, 2006). I.e., the operation of system is the key and not the individual features of its elements. All we know what is really interesting in sport is the competition. Competition attracts large masses of public, media and, frequently, large amounts of financial resources. Usually, this leads sports (especially in elite) to play in a critical area (García Manso & Martín González, 2008), in the edge of the error, risking, competing next to the limit. This phenomenon promotes that sport evolves. Players change their game style, teams change tactics, game dynamic changes as well, new training methodologies emerge in order to Basketball • Complexity Complex Systems Background 57 support competition requirements, etc. And even we can see how some sport introduce new rules (or modify old rules) in order to maintain competition attractiveness. Some rules such as offside in rugby, 24 seconds shot clock in basketball, three touches in volleyball, a stolen base in baseball, etc. are attempts to lead sports to critical areas. Because sport adapt and the natural tendency leads to a hierarchical structuring more or less defined. Hence, the efforts of some sports in order to avoid these kind of situations. As mentioned agents, who participate in these sport systems, compete among them; and the natural tendency leads to hierarchical structures, where some teams are clearly superior to others. Theoretically, this situation could be extended in time and hardly be broken by natural means, because best teams would continue hogging the best resources. This phenomenon is known as Preferential Attachment (Barabási & Albert, 1999), or Snowball Effect or Saint Matthew Effect. It is the popular the rich get richer and the poor get poorer. So, theoretically, we can point out that this situation will continue as long as no external source modifies the environment in which the sport is developing (rules, competition sport model). That is why so important to figure out the operation of the sport system, meaning league, game team, etc. and how modifications (rules, new elements, etc.) affect the entire system. The creation or modifications of these systems usually follow certain laws, meaning that some of these phenomena present the same features. One of the most important examples is the appearance of Power Laws or heavy-tailed distributions. This distribution is followed by many natural phenomena, often fractal, are also evident in many not natural systems. A lot of elements interact to produce a structure of higher level. These systems evolve far from equilibrium and are often highly dissipative (systems far from equilibrium). The Power Laws are described by mathematical expressions such as: Y=cXb Where X and Y are two variables, or observable quantities, c is a constant and b is the scaling exponent. This kind of expression has two properties: 58 1) The logarithmic transformation becomes a line (see Figure 9): log(Y) = log(c) + b log(X) 2) It is invariant to scale changes (scale-free). Figure 9. Example of a distribution (upper panel) and its log-log plot transformation (lower panel). Phenomena with this type of behavior (Power Laws) are also called scale-free. By scale we mean the spatial and temporal dimension of a phenomenon. The hypothesis of scale that rises in the context of the study of critical phenomena led to two categories of predictions, both of which have been well verified by a large amount of experimental data on various systems. 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 Basketball • Complexity Complex Systems Background 59 of the most important is the scaling law we have mentioned; its usefulness lies in linking the various critical exponents that characterize the singular behavior of the order parameter and response functions (Amaral & Ottino, 2004). Moreover, this kind of distribution can point out phenomena such as fractality (Barabási & Albert, 1999), self-organized criticality (Dhar, 1990; Bak, 1999), clustering (Newman, 2001a; Albert & Barabási, 2002), alometric laws (West, Brown, & Enquist, 1997a), etc. In short, they indicate the possible presence of complex systems. Here we present several examples of diverse elements which follow Power Laws: Figure 10. Examples of Power Laws: (a) Word frequency, (b) Citations of scientific papers, (c) Web hits, (d) Copies of books sold, (e) Telephone calls, (f) Magnitude of earthquakes, (g) Diameter of moon craters, (h) Intensity of solar flares, (i) Intensity of wars, (j) Wealth of richest Americans, (k) Frequencies of family names and (l) Populations of US cities. Source: (Newman, 2005). 60 Also in sport there are a lot of examples of this kind of distributions: athletics records (Katz & Katz, 1999; Savaglio & Carbone, 2000), power lifting (García Manso et al., 2008), goals distributions (Malacarne & Mendes, 2000; Mendes et al., 2007), tenure lengths of sports managers (Aidt, Leong, Saslaw, & Sgroi, 2006), scoring in basketball (de Saá Guerra et al., 2013), etc. As we can see, sport in general is a good example of complexity, therefore we believe accurate to use this methodology in order to analyze basketball. General Research Design Basketball from the perspective of non-linear complex systems Yves de Saá Guerra 2013 68 Finally we want emphasize the applicability of the study in several fields. One of them could be developing of intervention strategies for creating of competition patterns, management and teaching for players, coaches and sports organizations staff. And even for the staff of entities outside the sport, but with links to the sports network. In the academic field, it is important because we can learn and apply new techniques to other fields of knowledge. To open new research lines that can be follow by other researches, and pass on those discoveries with the methodologies to the university and scientific community. Following, we present (Figure 11) the general research design, in order to a better understanding of the document: Basketball • Complexity General Research Design 69 Figure 11: General Design of Research. Subject of study Theoretical Background Hypothesis General Research Design Study 1 Objectives Methodology Results Discussion Conclusions Study 2 Objectives Methodology Results Discussion Conclusions General Conclusions Future Research Lines Study 3 Objectives Methodology Results Discussion Conclusions General Conclusions Study 1. Basketball league Basketball from the perspective of non-linear complex systems Yves de Saá Guerra 2013 Basketball • Complexity Study 1. Basketball League 73 6. Study 1. Basketball league 6.1 Intro A league is a competitive model. In a sport, the design of the competitive model is as important as the preparation of the subjects involved (players, coaches, officials, etc.). Team performance can be postulated as winning as many games as possible. Final standing is the result of the way in which all teams interact of in a pre-schedule calendar. That is the competition model. Therefore, the effectiveness of a team is closely conditioned by is the result of certain states of optimization of systems that compose it, which also have a reciprocal relationship with emerging and critical environment: the competition. The study of the equilibrium between systems which interact within a same environment, is a frequent issue by disciplines such as economy, applied statistic, physics, evolutive biology, social sciences, sport sciences, etc. Given the difficulty of predicting results of the games, and therefore the final standing, we cannot use a linear methodology for analysis, it is necessary to use a methodology that allows us to explore the nature of the competition with as much detail as possible, as is the theory of complexity. The point is to understand the sport from a systemic conception and conceive the athlete, or team in our case, as a system that works as a whole that is affected by the surrounding environment (Gambetta, 1989; García Manso & Martín González, 2008). The competitive model has a direct influence on the competition (type of confrontation), development and evaluation so that small changes can dramatically alter the final result, given the close relationship between the competitive model and competition (de Saá Guerra et al., 2012; Lebed, 2006). The team ability to compete and how the championship works (how competition format is designed: conferences, divisions, game schedule, league, playoff, etc.) determine the level of competitiveness. Competitiveness is a comparative concept of the ability to strive for a goal. The more balanced competition, the greater the degree of competitiveness, and vice versa. This is an interesting because it reflects the reality of the competitive system, e.g. higher 74 budgets allow signing players of better quality. Whereas tighter budgets do not allow hire best players, given their high cost. One of the most widespread ideas to explain the phenomenon of equality among the competitors of the same championship is the concept of competitive balance. Competitive balance represents the degree of equality within a championship. A central concept used in the economic analysis of professional sports leagues, as indicated by the extensive literature on the subject (Schmidt & Berri, 2001; Fort & Maxcy, 2003; T. A. Rhoads, 2004; Goossens, 2006). The idea of competitive balance is to try to measure the degree of global competitiveness in a given league. And specifically has been applied in disciplines such as baseball (Owen et al., 2007; Scully, 1989), American football (Bennett & Fizel, 1995), basketball (Noll, 1988; Berri et al., 2005), ice hockey (Richardson, 2000), football (soccer) (Halicioglu, 2006) or golf (T. Rhoads, 2005). Thus, greater competitive balance should lead to greater demand (Quirk & Fort, 1997; Goossens, 2006). Indeed, the most competitive leagues tend to be more attractive and generate more revenue (tickets, sponsors, TV, etc.)(Szymanski, 2003) and this is closely related with the sport model (Ribeiro et al., 2010). The key element of the economic success in the professional sport is the increase in competitive balance. Each time a competitor reaches a very high domain level, the competition equilibrium (the way in which teams compete) is broken down, in the sense that the uncertainty declines significantly. In these situations, when the uncertainty of the outcome diminishes, the interest of the competition may reduce considerably. When this happens, the attendance of spectators to the games can decrease and, consequently, access to financial resources may be compromised (Berri et al., 2005). For this reason, sport organizations which design sport competitions models (leagues), try to design structures and rules which enable cope with a decrease of competitiveness in a championship. A certain level of competitive balance seems reasonable to hold the interest of spectators and sponsors for all teams but the determination of the optimal level is very complex. Some authors (Knowles et al., 1992; Rascher, 1999)noted that fan attendance in Major League Baseball is maximized when the probability of the home team winning is approximately 0,6. If the home team has a higher probability of finding success, we can expect fan attendance to decline. Consequently, given the importance of fan attendance to a league’s financial success, Basketball • Complexity Study 1. Basketball League 75 leagues are expected to implement rules and institutions designed to address the relative strength of teams on the games. We studied the results from different seasons of two of the main professional basketball leagues, the NBA (National Basketball Association, USA) and the ACB (Basketball Clubs Association, Spain)and the results of one high level amateur league, the Division I of NCAA Men´s basketball (National Collegiate Athletic Association, USA). Data have been obtained from the official NBA, ACB and NCAA webpages (www.nba.com; www.acb.com and www.ncaa.com). The Spanish Professional Basketball League (ACB) is an open model league, where every season participating teams are readjusted taken into accounts promotions and demotions to lower categories. The eight top ranked teams play the play-off in order to be proclaimed champion of the league. The professional North American League (NBA) is a franchise model competition. Participating teams are divided in two conferences (Eastern and Western). In turn, these are divided in three divisions per conference. When the regular season finishes, top ranked teams will meet in the play-off for the title. The NBA is a closed model where there are neither promotions nor demotions. The NCAA (basketball college championship in USA) is divided in three divisions (Division I, Division II and Division III). In turn, every division is divided in conferences of several teams each. We only used the data from the Division I of the men´s basketball. We must remember that the Division I of NCAA men´s basketball is composed by a total of 344 teams (the number varies in the season analyzed), divided in 31 conferences through all USA (the number of teams per conference is not homogenous). The aim of this study was to analyze, from an overview, the sport model and the intern dynamic of several basketball leagues (professionals and amateur) by studying its competitiveness degree. Also we tried to develop a model for the competitiveness level analysis in team sport competitions, which would be useful to assess their competitiveness level based on the uncertainty level that might exist for each confrontation. 76 6.2. Methodology Confrontation matrices Our interest is to focus on studying sport leagues, where each team usually plays twice against each other team (once at home, once away) in games according to a prearranged schedule. A series of games between a number N of teams, can be defined by its matrix of confrontation A = [Ai,j]N×N, with the same number of rows and columns. This is a double entrance matrix where each row and each column correspond to the results of each game between any two teams. We shall use the subscript i or j for teams, i ≠ j, so we use Aij = 1 if team i beats team j, and Aij = 0 otherwise. Other options such as ties or different values of 0 or 1 are not considered in this introduction at the moment without loss of generality. From this matrix, at the end of the competition, we obtain the final score R. See Table1 for an example of the matrix 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 of a confrontation matrix with N=4 teams (a,b,c and d). The rows represent the games played (won or lost) by a team at home. The columns represent the won or lost games played by a team away. HW (Home Wins) represents the total number of games won by the teams at home. AL (Away Lost) is the lost games away. AW represents the total number of games won away. The final score R is the sum of the home and away wins, R=HW+AW. The row i of matrix A represents the points for games won or lost by the team i at home, while column j represents the away games won or lost by the same. Therefore the horizontal sum:      N ji njiA 1 , Basketball • Complexity Study 1. Basketball League 77 represents the number of games won by the team i at home (ni), where N is the total number of teams. Note that A(i,j)=0, if i = j. Likewise, the vertical sum      N ji mijA 1 , represents the number of away games lost by i (nj). Therefore, the total number of games won by the team i will be   iii mNnR  1 The vector R (score vector) represents the results obtained by each team in each season. The result vector R behaves randomly, in the sense that we do not know the final result, but the results of previous seasons (historical performance), may provide some clues. The values of R historical or previous seasons divided by the sum of all games can be considered to be a discrete probability distribution   N jj i iR R p 1 where pi indicates the probability that the i team gets a certain result and therefore can be considered as a performance indicator. If the distribution is uniform, all pi values are equal or similar to each other, and all the teams have approximately the same playing level. This represents a case where it is difficult to predict the final outcome. This may be considered to be highest possible parity among the teams (competitive balance). However, if there are certain values of pi greater than the rest, it means that there are some teams in the competition with superior performance to other teams. In the case of a uniform distribution, any team has an equal chance of winning. In terms of statistical mechanics, such distributions are related to equilibrium situations where all structures and gradients have been eliminated. The disorder is maximum; therefore the values of entropy (S) are also maximum. Following this analogy, if the system is isolated, cannot exchange matter, energy or information with its environment, all random fluctuations that may occur and thus, all gradients that can be formed, tend to be neglected. 84 further from season to season. Their values oscillate between two league profiles (seasons that were very competitive and seasons that were less competitive). Figure 14. Entropy values of the 10 season of NCAA men´s basketball Division I. Data display a no uniform tendency. Sn values vary throughout the seasons analyzed, showing an upward trend in the last 4 seasons. Even they reach higher values than the previous 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 Basketball • Complexity Study 1. Basketball League 85 Figure 15. Comparison of the three leagues analyzed. We can note that the professional leagues (the ACB and the NBA) have a higher level of competitiveness than the amateur league (NCAA). But the in the last season, the values are closer than ever. This fact is very relevant and will be interesting to find out the reason for this behavior. In the analysis of the entropy values of NCAA men´s basketball Division I (Figure 14) we can appreciate that the tendency is not homogeneous. There was a period (from 2004-2005 to 2005-2006) where the competitiveness was maximum (0,967) with slopes very similar to both sides. But the most notable fact is that the last 4 seasons the Sn values increase up the highest value of entropy (0,968). When we compare the three leagues (Figure 15), we can see that the professional leagues present higher values of entropy, what means that, in general, the competitiveness is greater than the amateur league. Neither of them display a homogeneous comportment but all of them oscillate during the years analyzed. Only the NBA presents a most stable behavior. As we mentioned above, these asymmetries may be originated by readjustments in the competitive system, enlargements, promotions and demotions, etc. But possibly, the only fact that seems to coincide in all the leagues is the economic crisis. Nevertheless, it gives the impression to have different effects in the leagues analyzed. In the professional leagues (the ACB and the NBA) the economic crisis lead to a decrease (and to a stabilization at the last seasons) on the 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 general competitiveness level, while on the other hand, the NCAA experiment an increase in the entropy values during the same period. Indeed, we can observe that in the season 2006-2007, the entropy of the NBA starts to decline while the Sn values of the NCAA growth. And the ACB suffers a drop drastically during these same years. It would be interesting to find out the reason for these variations so marked, but it is necessary further deepen into the causes that originate this class of phenomena. ACB analysis The temporal evolution of these values can be seen in Figure 13 and 15, and, as the data show, there are some seasons in which the confrontations have a high degree of uncertainty (1999– 2000, 2003–2004 and 2005–2006 seasons) and other seasons with a completely different profile (seasons 1996–1997; 2000–2001 and 2008–2009). Issues such as the possibility of relegation, the major budgetary differences of teams, high economic dependence on public institutions, or the high volatility of the rosters are some of the factors that can influence this behavior. This market has become increasingly active, as basketball has been professionalized, so much so that a few players remain more than five seasons with the same team in the ACB (Arjonilla López, 2011) A team’s performance is mainly determined by two factors: economics and the players themselves. Both are closely linked. A large budget provides the ability to sign superior players and to make rosters balanced to a greater extent. Teams with smaller budgets select players with a supposedly lower level; consequently, their rosters will be less balanced and less competitive. The teams make their rosters based on budget and sporting objectives. These objectives are closely linked to the competitive ACB model (open model). Moreover, the absence of a salary cap and a conspicuous disparity between budgets of teams can lead to large differences in the quality of the rosters (sporting potential gradient). This makes differences in performance insurmountable for some teams in the ACB, especially for promoted teams, whose budgets and rosters are tight. The unstable private and state economic support of some teams and not others, which have a solid economic support and the marketing (trading) of players, may in part explain these oscillations in the entropy, which are characteristic of the ACB. Curiously, Basketball • Complexity Study 1. Basketball League 87 the years with minimum values of Sn coincide with Olympic years. This may be a topic for future study. NBA analysis The NBA has a more stable shape than the ACB. There are seasons with lower competitiveness (≤0,980) than the overall average (0,983), and periods where the competitiveness remains at higher levels (range: 0,985–0,990) (Figure 13 and 15). In Figure 13 and 15, we can clearly see that there are seasons that do not correspond to the general trend, showing a behavior that is perhaps anomalous. These seasons are 1993–1994 and 1996–1998, with the lower values of Sn, and the period from 2001–2002 to 2006–2007 seasons, with higher levels. In the 1995–1998 seasons, Michael Jordan’s Chicago Bulls achieved the best win record in the NBA regular season to date (won–lost: 72–10, 69–13, and 62–20 respectively). This performance is probably responsible for the decline in competitiveness in the league during this time. Indeed, some authors mention that the 1990s were the least competitive decade in the history of the NBA(Berri et al., 2005). This seems far from frivolous, as this sort of phenomenon can be accompanied by a heightened attraction to athletes and the general public (Rovell, 2003), as well as private companies and media, resulting in a strong economic impact (Forbes, 2008; Mathur, Mathur, & Rangan, 1997). Another interesting area covers the period from the 2001–2002 to 2006–2007 seasons. These correspond with the renegotiation of the salary cap (1999–2005) when the Collective Bargaining Agreement (CBA) was signed. This probably had an impact on the overall performance of the league, because the objective of the salary cap was to prevent teams with a large profit surplus from signing the best players available, thereby facilitating the equality of retention in the league. This mechanism, together with the draft, is necessary for each franchise to carefully select which players may be interested in the market for its particular franchise goal (depending on the background of the team). Consequently, each franchise is only able to “shield” economically one or two players, commonly referred to as “franchise players”. The decline of competitiveness in the 2004–2005 season (see Figure 13 and 15) may be attributed to seasonal variability, although it could also be caused by enlargement of the participating teams from 29 to 30, which further restructured the divisions in each conference. 88 This changed the format of the divisions: instead of having two divisions per conference, there would be three per conference with five teams each. The 2005–2006 season increased levels of competition again, as shown in Figure 13 and 15, possibly because all the Central Division teams qualified for the playoff s and this was the first time that a division managed to place all of its teams in the postseason since the Midwest Division did so 20 years ago. NCAA analysis Regarding to the Figures 14 and 15, we can see that Data display a no uniform tendency. Sn values vary throughout the seasons analyzed, showing an upward trend in the last 4 seasons. Even they reach higher values than the previous maximum. The Sn value range for all time period analyzed. We can see a period where they reached values significantly elevated and a final period where values reach their maximum. In the NCAA participates a great number of teams with different performance levels, hence when we analyze the entropy data we have to take into account the heterogeneity of the sample for the analysis of the results. The values of Sn range from 0.9679 to 0.9583. These values are quite distant from the values of the professional leagues but even that, the NCAA is the most stable of the three (Sn NCAA mean=0.9631 ± 0.0033). The main point we must bear in mind when we analyze the NCAA is that the teams that qualify for the playoffs using an index called Rating Percentage Index (RPI). The RPI is a quantity used to rank sports teams based upon a team's wins and losses and its strength of schedule. The current used formula for determining the RPI of a college basketball team is as follows. RPI = (WP * 0,25) + (OWP * 0,50) + (OOWP * 0,25) where WP is Winning Percentage, OWP is Opponents´ Winning Percentage and OOWP is Opponents´ Opponents´ Winning Percentage. The WP is calculated by taking a team´s wins divided by the number of games it has played (i.e. wins plus losses). So NCAA standings are elaborated using the RPI instead of games won, as the rest of leagues analyzed. This index tends to equilibrate the differences among different teams, so that theoretically favors the weaker teams and hampers the teams with better historical trajectory. Basketball • Complexity Study 1. Basketball League 89 This cause that a lot of the teams play extra tournaments in order to improve its index, hence the teams of NCAA do not play the same number of games during the regular phase. Also, the NCAA has the feature that a single player only can remain in the same university for four years as maximum. There is not a players market as at the professional competitions. This fact conditions the possible advantage that the “best” teams can obtain over the rest of the participants. Theoretically, the best teams recruit the best players in order to win more games and qualify for the playoff. If this were true, the differences would be insurmountable for the rest of the teams. But in reality, the opposite happens. The current tendency of the players that come from the high schools is to choose teams where they are going to play a lot of minutes and be the “star” of the team, instead of choose a good basketball program in a good university or college in order to receive a good sport and academic formation. Coaches mention that a very large number of players conceive the college team as a step in their career to the NBA, what is a big mistake, in the words of the coaches. As an additional consideration, we observe that in both professional leagues there is a significant decay in Sn in the last three seasons to a lower limit, which apparently remains attached to that value. It is possible that this is related to the current economic crisis in which some teams are less affected than others. This is less important in the NBA, as compared to the ACB, indicating the sensitivity of the two different sports models to external economic effects. We must take into account that the NBA is a private league, and that it works as a company, while the ACB depends largely on regional government subsidies, which particularly affects some teams. Comparison among teams or conferences We used this protocol (normalized Shannon entropy) in order to determinate the competitive balance in different ACB, NBA and NCAA seasons. This proposal makes a rude analysis of competitive balance without regard to the ranking in which teams complete this phase of the league (regular season), and discriminates well between leagues. For this reason, we compare 90 the values of the win ratio of (R) with two extreme theoretical models (random of maximum competitiveness and hierarchical of minimal competitiveness) in order to analyze the actual behavior of competitive balance in the two professional basketball leagues. We have discarded the results of the NCAA because the results are meaningless due to the league has many teams with different performance levels. Figure 16. Examples of (a) two ACB seasons (2003–2004, 2005–2006) and (c) two NBA seasons (2003–2004, 2005– 2006) that were more competitive compared to the theoretical random extreme. Also shown are examples of hierarchical (b) ACB seasons (2000–2001, 2008–2009) and (d) NBA (1996–1997, 1997–1998), compared with theoretical models of each league. Each point represents the probability value p obtained from the results array R of each season. The solid lines represent the two extreme cases. The straight line represents the hierarchical case, while the other (oblique line) results were used to calculate the probability of the average result obtained in the random process. We have discarded the results of NCAA in this analysis because the results can be very confusing due to the large number of participating teams. Figure 16 (a) shows the values of the probabilities p of the 2003-04 season (o) and 2005-06 season (+) of the ACB, as an example of a very competitive season. Figure 16 (b) shows the seasons 2000-01 (o) and 2008-09 (+) of the ACB as an example of seasons with lower Sn values. The same results are shown in Figure 16 (c) for the NBA, for the seasons of 2003-04 (o) and 2005-06 (+), and 3 (d) for the seasons 1996-97 (o) and 1997-98 (+).We have discarded the results of the NCAA because the results are meaningless due to the league has many teams. Basketball • Complexity Study 1. Basketball League 91 The ACB shows a degree of competitiveness away from the theoretical hierarchical extreme, in which competitiveness is lower. In the most competed seasons, the entropy values are close to the theoretical random distribution, almost at the tail of the distribution. The rest of the values are located between the two distributions, although, in the case of lower entropy, the values are very close to the tail of the hierarchical random distribution. Similar behaviors can be observed in the NBA. 6.3.2. Statistical and cluster analysis The Figure 17 represents the boxplot of the results R of all participating teams, through the seasons analyzed in ratio values (wins/games played). 92 Figure 17. Boxplot of entirely results obtained, expressed in ratio (wins/games played), of the all participating teams in seasons analyzed. Upper and middle plots show up the professional leagues ACB and NBA. The bottom plot represents the same ration of the conferences of the NCAA (not teams) which participate in the seasons analyzed. In the results of ACB we can distinguish three clusters or three zones. And even within some of them, other delimitation on a smaller scale, which are occupied by teams to which can be called transition teams. In the NBA (the middle plot) teams seem to have values more similar to each other. This suggests a higher competitiveness 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 Number NCAA NBA 0.9 1 1 _ 1 0.8 1 , ¡ 1 ' 1 1 1 ." ~~ ( 1 111 1, ~:: 11 1 1 ~ ~~f + - 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 Number + + + Basketball • Complexity Study 1. Basketball League 93 degree, because virtually any team can reach high levels of performance. On the other side, the NCAA plot (lower plot) shows a high homogeneous performance level and the data cloud covers a very wide spectrum. But despite of that data show this tendency, we have to take into account that teams do not play the same number of games. Hence the boxplot seems point out the same values. In all professional cases we observe that teams tend is to cluster around their performance level, although their behavior is different. Seemingly in the amateur league (NCAA), the values are much more stable. In the ACB (Figure 17) we can distinguish three zones. The first one, with the best performance level, the three best teams are located with their data (data cloud, mean, interquartile ranges and confidence intervals) which are clearly above the rest. These teams are following a model of competitive behavior that we previously defined as random. They cluster and compete to achieve first place (a very high degree of competitiveness between them). Also given the same situation for the second group, where the four teams have a similar performance level (a very high competitiveness among these four teams). The teams situated in the middle sector (third group), struggle among them very even. Moreover, we can talk about transition teams, where the level of performance places them in a bordering position with the other two areas of performance. In the last part of this group, the data have little statistical value, because this is the zone which suffers more changes due to the promotions and demotions. In the NBA the data seems to point out a much more homogeneous behavior. The most part of the data cloud, and the medians, are situated around of mean values of ratio, which indicates a high competitive balance. Occasionally the teams reach unusually high values (high consolidated teams) o low values (low consolidated teams), some with a very strong scattering data, suggesting that they are teams with good results and now they have decreased their performance, or vice versa. In the NCAA, on the other hand, there is a low variability in the boxplot values. The most part of the conferences reach range from 0 to 1 in the ratio values. But the point is that the teams do not play the same number of games. In the same conference we can find teams with 16 games played against teams with 4 games played. This is because the standings are elaborated by the RPI index, as we mentioned above. This force to several teams to play in tournaments in order to get a better punctuation. That is why the final balance is so heterogeneous. Actually this means that in the NCAA the sport gradients are much more accentuated than the professional leagues. 100 We have to bear in mind that NCAA participating teams are significantly more (344) than ACB (16) and NBA (30). And the organization of competition is quite different from professional leagues. That is why it seems there are two competitions during regular phase, as indicate the log-log plot of ratio distribution (Figure 20) Figure 96. Log-log plot of the NCAA ratio distribution. There are at least two Power Laws, which indicate different competition dynamics. In order to find out whether teams are gathered by their performance, we carried out a nonhierarchical clustering analysis of partitional reallocation type (k-means Matlab function) (Figure 21), which places the points in space to be grouped. These points are assigned to the group that is closest to their centroid. It is a method of cluster analysis which aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean. This results in a partitioning of the data space into regions called Voronoi cells. 10-1 100 101 102 Basketball • Complexity Study 1. Basketball League 101 Figure 21. The upper panel represents the ACB clustering. We can observe that show up five regions which are clearly related with the team performance. There are some teams which are clearly located in one region (blue, black and red zones), and occasionally reach a different result. That is, they belong undoubtedly to a region. Teams 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 ACB Cluster Wins Team Number 0 5 10 15 20 25 30 0 10 20 30 40 50 60 70 80 Team Number Wins NBA Cluster • • • • • • • • • • ••• 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 located in green and brown areas can be considered transition teams because sometimes reach better or worst results than other seasons. Even we could consider these two regions as a single region, regarding the behavior of teams located on it. The lower panel represents the NBA clustering. It is completely different to the ACB. The results point out four regions (red, magenta, brown and green) with a similar performance. It means that a team can be on top for one or several seasons and subsequent seasons at the lower standing, or vice versa. Moreover, exists an elite located in an own region by their own results (black) but seldom other teams manage to achieve these positions (blue). At ACB clustering (Figure 21) comes up five regions clearly established by performance. We can see eve how the centroids are positioned hierarchically, which indicates stratification. The two lower regions (red and black) are clearly integrated by certain teams which sometime reached better results but visibly belong to those regions. The other region which is clearly separated from the rest is the blue region. Teams located in this area are markedly superior to the rest. Teams located in green and brown areas can be considered transition teams because at times reach better or worst results than the corresponding results to their zone. Thus we could consider these two regions as a single region, regarding the behavior of teams located on it. The NBA clustering (Figure 21) points out six regions. The red, magenta, brown and green regions present a similar performance level, but it is not clear what teams are in each one. It could means that due to the intern mechanisms of NBA teams after a bad season can be competitive for the next one. And teams with good results are obligated to restructure their roster season by season, in order to keep good runs. In fact, we can note that some teams present very good results (black and blue) and seasons not so good (some of them with a very marked data spread). This can point out dynasties. For instance, Chicago Bulls as long as Michael Jordan remained in the roster the team succeeded. But after his retirement, the Chicago Bulls fell into a run of bad results. The NCAA uses the RPI index in order to elaborate standings, so this kind of analysis for the NCAA does not reflect the reality of the competition system. Hence we considered that the boxplot and clustering analysis of standings do not provide relevant information. ACB The ACB presents a structure almost hierarchical. Participating teams are clustered by themself regarding performance level (Figure 21) and this phenomenon creates frequency barriers (wins frequency) for teams less powerful. The ACB peak is around 0.40 ratio (Figures 17 and 19). Teams located below this point are very irregular and are not able to overcome the level of performance needed to strive in the middle of the league ranking. This point seems to work as Basketball • Complexity Study 1. Basketball League 103 a barrier, understood as a value significantly greater frequency. It is remarkable that most of the teams are placed in intermediate regions (Figures 17 and 19), and only a few teams are positioned beyond the second barrier (0.80 ratio), which can be considered the most competed area (Figures 17 and 19). Teams located above the barriers are always the same (except occasionally). So we can point out that the highly competitive area is always occupied by the same teams and so on (Figure 17, Figure18, Figure 19 and Table 4). That is, teams are clustered around their level of performance. Therefore, teams have to overcome certain barriers of performance if they want to achieve higher levels of performance The best results in the ACB coincide indeed with highly consolidated teams in this competition and with a high performance in several European Leagues. Results lower of ACB data correspond to teams which, in the seasons analyzed, were poorly consolidated (Table 4). These different performance regions (Figure 21) could be originated because ACB competition model. The ACB is an open league model in which participating teams are adjusted based on promotion and demotions (from or to lower category), and where eight top ranked teams play the play-off. We must take into account that teams elaborate their roster depending on their budget, and that this is, almost always, related to the results obtained. The higher the budget, the better players, coaches and staff can be hire or vice versa. A priori, promoted teams have less competitive rosters, and also tighter budgets. Given its open structure, some teams (and its underlying structures such as economic network, executive committee, player network, farm of players, etc.) become more experienced throughout all seasons. The existence of these teams has an impact on the rest, and above all on the less experimented teams. Thus, the teams positioned on the extremes are closely related: if the differences among the low ranked teams and the top ranked teams are very high, it is possible that the upturn of the head is more evident, because exist a high probability that the top ranked teams defeat the bottom teams. Thus, the best teams can improve their winning ratio. It can be deduce that there is a different criticality level for each zone. This sport potential gradient is maintained by energy (players, coaches, money, etc.). This provides that the 104 performance differences for some ACB teams are insurmountable especially to the newly promoted whose budget and templates are tight. The sport model greatly influences the market. The fact that the teams tend to cluster in zones it is not random, but follows a phenomenon known as preferential attachment (Barabási & Albert, 1999), also denominated Saint Matthew effect (Bunge, 2001; García Manso & Martín González, 2008), where the strong teams reap more successes and less strong teams will have less wealth. Another mechanism that causes this behavior is memory effect, what the systems present. The teams are attached to an attractor as in some areas of the ranking (Figure 17, Figure18, Figure 19 and Table 4). Others reasons for these differences could be teams´ sport planning, sport aims established for each season, roster, budget, external competition (European Leagues, King´s Cup, tournaments or participations of players in national teams), etc. These aspects may influence substantially. NBA Generally, the NBA has an uncertainty degree bigger than the ACB (de Saá Guerra et al., 2012) and its dynamic and structure are completely different. In the Figures 17, 18 and 19 we observe that the most part of the data are located nearby the 0.5 ratio and how teams are dispersed for several regions (Figure 21). There are not demotions neither promotions. In fact, they worst results (ratio <0.15) (Figures 17 and 19) have some advantage for the next season. These teams takeover top places of NBA draft, which means in a reinforcement in their roster. We also can see how to reach the best results is very unlikely (Figures 17and 19). We must remember that the NBA seasons are very extensive (82 games) and the play-off classification is very hard-fought. Even to get ratio results higher than 0.70 is infrequent. The most likely is that the majority of the teams are located in medium zones (Figures 17, 18 and 19). The same team can be fighting to get play-off positions and the next year can be located in a ratio <0.5 or vice versa (Figures 17, 18 and 19). I.e. reaching higher values than 0.70 or lower than 0.25 is unlikely for the most part of the teams. Note that the most part of the teams present similar Basketball • Complexity Study 1. Basketball League 105 performance levels; that is why clustering present regions similar where teams change their locations during seasons analyzed (Figure 21). The existence of this performance dynamic could be also due to the sport model employed by the NBA. We must take into account that in the NBA participate much more teams than in the ACB (30 vs. 18), and they play much more games (82 vs. 34). Moreover, the competitive structure is diametrically opposite. There are also mechanisms imposed by the NBA in order to avoid team monopoly (draft, salary cap, reserve clause, etc.). The purpose of these measures is to safeguard always the competitive balance. Therefore it is possible that the most critical parts of the competition are located in the two limits, because they are areas where teams are positioned in them will get rewards (play-off and draft). Perhaps due to its competitive dynamic, the NBA is a good example of Red Queen hypothesis proposed by Van Valen (Van Valen, 1973): For an evolutionary system, continuing development is needed just in order to maintain its fitness relative to the systems it is co-evolving with. It is an endless race. All competitors need to improve to keep competing. ACB and NBA Comparison The ACB and the NBA seem to present an inverse behavior. In the ACB case, most competed region is the medium ratio area (lowest differences) and green and brown regions (Figure 21). In the NBA, the most competed area is the top and the end of the standings. That is why teams are so disseminated in the cluster analysis (Figure 21). We note that both cases are cases of a highly competitive, but opposite reasons: the ACB is an open model where the last classified is relegated of category, hence the high degree of competitiveness, while in the NBA, the point is qualifying for the play-off for the title or trying to get a good place for the lottery draft. In the ACB, we observe that teams are clustered throughout their performance level as well (Figure 21). There are teams clearly placed in a particular area of the competition, which could indicate the competitiveness level of the team. The first four positions are occupied almost entirely by the same three teams and, occasionally, some team was able to slip into this elite group (Figure 17, 18 and 19). There is a similar pattern with the play-off positions (the first eight positions), where we can observe clearly how the data cloud and the trust intervals of several teams are encompassed in this area. The last positions are more atypical, because the last two teams pass to a minor league and are replaced by other two different teams. The new 106 promoted teams, a priori, have not the same performance level that the teams of the middle zone. In the NBA, almost all teams have ever reached the play-off positions, although there are more teams that belong to this area than others (Figure 17). Their data are less scattered and more firmly established in this area. Note that practically all the teams have reached the top five. Given the high randomness degree present in the ACB and the NBA, we can suppose that the most part of the teams are between the order and chaos, known as critical state. A state of semi equilibrium where the most insignificant variation can produce a change of state or a phase transition, but this is very hard to predict (McGarry et al., 2002; Scheffer et al., 2009). The degree of criticality varies based on the zone of the standing we are studying. But note that teams, even though the chaotic behavior of the competition, always tend to an attractor (team clustering). Therefore, we can consider competitiveness as an atractor itself. We must remember that complex systems are usually far from the equilibrium. E.g. the living organisms are in continuous fight with the environment in order to remain in a state far from equilibrium, i.e. alive (Amaral & Ottino, 2004). Translated to our context, it means that to preserve the high level of competitiveness, it is necessary to keep fighting against the rivals, and to invest huge amounts of energy in order to survive in the competition. 6.4. Conclusions The aim of our study was to analyze sport competitions from a general point of view. This study show that the analysis model (matrix results using the Shannon entropy) for the study of competitiveness levels in the system of league competition is a useful and highly sensitive tool to determine the degree of overall competitiveness in the league, and to detect small oscillations in it. This potentially identifies minimum fluctuations in the level of competition, which allows one to focus attention on localized temporal changes and to investigate the mechanisms which cause it. This model shows that both the ACB and the NBA present a high degree of competitiveness. In both leagues the entropy levels are high (range: 0.985–0.990), although these periods are Basketball • Complexity Study 1. Basketball League 107 more stable in the NBA. We can say that both the ACB and the NBA are very competitive leagues whose teams are well balanced within each league. On the contrary, the amateur league, the NCAA, presents a lower competitiveness degree (compared with the professional leagues). But its tendency is to increase during the last seasons. We can say both the ACB and the NBA are very competitive leagues with a high competitive balance and they are highly conditioned by the sport model. The fact that the ACB is an open league causes that less powerful teams subtract competitiveness to the entirety. We should think about strategies in order to maintain or even increase the degree of global competitiveness of the league, like in the NBA. Despite these issues, the Spanish basketball league (ACB) can be considered very competitive. The NBA has specific mechanisms to ensure high competitiveness, such as the draft, the salary cap, reserve clause, etc. Their aim is to preserve the competitive balance within the system. It is a league with a high uncertainty on the final result; hence, all teams have real possibilities for qualifying for the playoffs. The case of the NCAA is pretty interesting as well. Despite of the low entropy (compared to the professional leagues), NCAA is an attractive league that generate expectation and attracts thousands of fans and media. Due to its complicated structure and size (number of participating teams), it is complicated carry out an analysis of the entire league. But we can observe that indeed, changes in its sport model, such as enlargement of divisions, rule changes, new punctuation system, etc., alter significantly the competitiveness level throughout years analyzed. 6.5. Practical Proposals As a practical proposal we suggest to use this methodology in order to study and compare the competitive level of different basketball leagues and their evolution in time. I.e. different basketball leagues such as Euroleague, Eurocup, NBAD-League, ABA, LEB, etc. This process also can point out some events which can be the source of such variations. As another practical proposal, we put forward to use this methodology in order to figure out how rule changes, enlargement of participating teams, competition format restructuring (open or close, divisions, conferences, etc.), or other league mechanisms such as budget rules, 108 building up teams (hiring), or even practices regulations (i.e. NCAA) can modify the competitive level. 6.6. Limitations of the techniques used Shannon entropy, as methodology to study the competitiveness in basketball, carries out a coarse analysis of the competitive balance, in the sense that the results do not take into account the team rankings. On the other hand, if we only use the ratios, boxplots or even clusters in order to analyze the competitiveness degree, we will not know the league overall. Study 2. Basketball Game Basketball from the perspective of non-linear complex systems Yves de Saá Guerra 2013 116 Scaling analysis indicates that the probability of extreme events might be estimated by extrapolating of Power Law distributions. Recently some authors (Pisarenko & Sornette, 2012; Sachs, Yoder, Turcotte, Rundle, & Malamud, 2012; Sornette & Ouillon, 2012; Yukalov & Sornette, 2012) have tried to characterize the so-named Dragon-King, extreme events with important social implications, which exceed these extrapolations. One important limitation of this tool is the occurrence of the Power Law behavior at the tail of the distribution where is desirable to have the best accuracy (Stumpf & Porter, 2012). However, the tails are the zones of the empiric distributions where the greatest fluctuations are found. On the other hand, the scaling laws do not always provide the best data fit. Alternative distributions to characterized these environmental phenomena are the lognormal (Mitzenmacher, 2004), the stretched exponential (Laherrère & Sornette, 1998) and other truncated Power Law (Redner, 1998; Tsallis & Albuquerque, 2000; Burroughs & Tebbens, 2001). Despite of these difficulties, there are some advantages of the application of Power Law and fractals tools for diagnostic, characterization and even prediction purposes, are the simplicity of the distribution and universality of its self-similarity. This is quite consistent with much of the literature on fractals and scaling in ecologic, geophysics or economics systems. Furthermore, in the latest years; improved statistical tests have provided strong evidence of scaling laws over a substantial (although limited) range of scales (Clauset, Shalizi, & Newman, 2009; Virkar & Clauset, 2012). One of the goals of this study is to provide a qualitative background on the lineal fit of the power law distributions and particularly the log-log plot analysis. This allows the comparison of analogous phenomena and the characterization of regions over a similar environment. For this purpose we don't need to have absolute certainty that an empirical data set follow a Power Law. We used these two tools (Poisson distribution and Power Law distribution) in our analysis. They pointed out the behavior of the system (Basketball). Basketball • Complexity Study 2. Basketball Game 117 Problems with Poisson distribution 1. When the index of dispersion is less than 1, the negative binomial distribution can be modeled for discrete data, because presents a longer tail than Poisson distribution. The tail of the distribution could decrease slower than in this case, so we should use the Power Law or truncated Power Law distribution, which theoretically, present a special meaning regarding extreme events distributions. 2. Another case is when  is not constant during the game or in each minute of the game. It also known that if lambda follows a gamma distribution, the process is modeled by a negative binomial. We assume the games as reasonably homogenous events. But, data indicate that not all the games are competed (homogenous), meaning low competitiveness, low team quality, period of the season, roster differences, etc. The system behavior is neither the same throughout the game time, nor in the first quarter, nor the fourth quarter not even last minutes of the game. Note that depending on the game time, can be considered as different games (substitutions, score differences, faults, etc.). And also is it influenced by tactic decisions (faults, time out, defense, etc.) 7.3 Results and discussion Application to basketball We studied a total of 5 seasons (1230 games per season, with a total of 6150 games) of the NBA regular season. In every game we analyzed the game transcription published by the NBA in which are described in detail, all events play by play (NBA). All the statistics reflect the incidences of game ordered by the time in which they occurred (chronological order): two and three point shots, free throws (made and missed), defensive and offensive rebounds, turnovers and steals; violations (out of bounds, fouls, technical, etc.) substitutions, etc. From all this information we focus on the analysis of point time intervals and scoring. As we described in the methodology, we propose the use of the Index of Dispersion and the  value in order to analyze basketball games. We based our study to analyze what happens 118 every minute of the game independently. That is, analyze the probability p1(k) that in the first minute k points are scored, p2 (k) in the minute 2, and so on, until the minute 48 for every game. Then we saw how far each of these 48 cases presented Poissonian behavior. To do this we calculated the mean number of points scored in every minute for all games i : , 1 N ik k i n N    Being N the number of games (N=6150), and i=1, 2 , … 48. In addition, we also calculated the Index of Dispersion, which as we saw is an indicator of the extent to which random variable behaves like a Poisson process. As we mentioned above, the Index of Dispersion is the ratio of the variance of the points scored in every game every minute to the mean value of these. The next Figure represents both values applied to the games studied. Figure 22. Index of Dispersion of the point scored by minute. We can observe that the trend of the values is to rise over time. Only at the end of each quarter there are a significantly increase, closer to 1, but only at the minute 47 reach the value 1 (pure Poisson). The minute 48 is completely out the range of the rest of the game, reaching values higher than 1. The behavior of this minute is very complex. The upper panel (a) represents the number of points in every minute (). This value is low at the beginning of each quarter but note that the value increases along with time, above all at the last quarter. 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 minutes 510 15 20 25 30 35 40 45 2 3 4 Time in minutes (a) Basketball • Complexity Study 2. Basketball Game 119 The most interesting results are obtained when we analyze the game time using the Index of Dispersion and the  value because points out some very interesting behaviors. The time interval used was 1 minute because we consider that lower frequencies, such as one second, or higher frequencies such as 2 minutes were not clear enough. The profile of these graphs may vary if the selected time interval is different than a minute. But the clarity of interpretation offered by this choice, which includes all the features that we want to emphasize, was what led us to choose the time interval of 1 minute, as the key to playing a basketball game. The Index of Dispersion (Figure 22) displays that the most part of the quarters remain lower than value 1 (under-dispersed). Only at the end of each quarter the values are higher than in the rest of the quarter. This means that the beginning of every quarter is more predictable than the end. Note that the tendency of all quarters is to rise, to approach to value 1, to become more unpredictable. But only the minute 47 present value 1 in the Index of Dispersion. To be precise, is a pure Poisson process. The game at this stage is completely random. The minute 48 requires special attention. As we can observe (Figure 22), the minute 48 exceed the value 1 significantly (over-dispersed). This suggests that the last minute in a basketball game is a completely different process than the rest of the game, meaning the game has changed its dynamic. The upper panel of Figure 22 shows  (number of events per time) of every quarter. We can observe than the number of events at the beginning of each quarter is low compared to the rest of the quarter, but the tendency is to increase anyway. It is likely that this outcome is because players, as agents of the system, start to interact at the beginning of the game. There are not previous situations (no memory from previous actions, because is the first quarter). We can refer to a zero point o base point from which emerge the characteristic actions of a basketball game. That is, a self-organization problem. Also it is remarkable the differences after halftime, maybe caused by the adjustments carried out by coaches and technical staff or by the game dynamic itself; as the last minute, where the number of events considerably higher than the rest. 120 The players of both teams have predefined roles (by player position) and instructions given by technical staff, based on the information about the rival. But is the interaction among them through the game time and the adaptation to the real game what make emerges game patterns: switch roles, motor task resolutions, etc. This is known as attributed role. This fact multiplies the possibilities and the actions carried out by players because players adapt to the environment. We have to bear in mind that at the beginning of the first quarter the score is 0 for both teams. This does not happen at the rest of the quarters, where the point differences (if exist) may establish future team dynamics. It is possible that these cases are affected by memory processes depending on how big these differences are. But anyway, there are previous situations on which to base strategies. Moreover, there is a fatigue effect of the system (errors such as fouls, turnovers, etc.) and of the players (physical fatigue, mental fatigue, cognitive fatigue, etc.) which has an accumulative effect along the game time and influences internal processes and emergent behaviors. Hence the anomalies observed at the end of every quarter are derived by these kinds of processes or mechanisms probably. And above are accentuated in the final stages of the fourth quarter. This may be because accumulated team fouls, but have no direct significance on the score (Figure 22) until the fouling team is in the team bonus (or foul penalty) situation and free throws are awarded. In order to a better understanding, we based the analysis on the Index of Dispersion. We have selected some cases with representative Index of Dispersion. As the Figure 22 suggests the general tendency is to increase the level of uncertainty during each quarter, above all at the end of the quarter, where the values are closer to 1. Moreover, the minute 47 reach value 1. We analyze two minutes intermediate, minutes 6 and 32; one end of quarter, minute 36 and the values of minute 47, with Index of Dispersion values 0.63, 0.75, 0.90 and 1.02 respectively. Basketball • Complexity Study 2. Basketball Game 121 Figure 23. Histograms of the point scored in the minutes 6, 32, 36 and 47, corresponding to Index of Dispersion values 0.63, 0.75, 0.90 and 1.02. The solid line represents the Poisson theoretical distribution. The two upper cases show under-dispersion, whereas the lower cases are cases close to Index of Dispersion=1, with a Poissonian behavior. We can observe two cases. The upper Figures correspond to the minutes 6 and 32, with Index of Dispersion values 0.63 and 0.75 respectively. We observe that does not fit well to the Poisson distribution. The variance is lower than that corresponding to the Poisson distribution (Index of Dispersion<1), and data are clustered around mean value, with less zeros and with a tail which drops quicker than Poisson, characterizing an under-dispersed Poisson distribution. In general this can means that the moments with an Index of Dispersion lower than 1 are more predictable than the rest of the game. On the other hand, the two cases below, the end of the quarter (minute 36), fits better than the rest of the quarter (Index of Dispersion 0.90), but what really match with the theoretical Poisson distribution is the minute 47, with an Index of Dispersion 1.02. Note that the number of zeros matches better and decays as Poisson. This represents the most unpredictable moment of the game, except the last minute, which will be treated separately because the nature of the distribution is different. 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 minute 6 mean = 2.28 var = 1.43 ID = 0.63 minute 32 mean = 2.32 var = 1.73 ID = 0.75 minute 36 mean = 2.73 var = 2.46 ID = 0.90 minute 47 mean = 2.60 var = 2.66 ID = 1.02 122 The results point out that in the two upper cases the number of zeros is lower than the theoretical Poisson distribution, which is the theoretical model we use as a base. Also the tail is reduced, whereas in the two cases below fits better. This seems an indicator of the risk assumed by teams. As we see later in the Figure 34, the number of 3 points and 1 points (fouls) is higher in the end of each quarter, while 2 points are decreasing throughout the game. Teams tend to risk more at these times, and defensive intensity increases (more fouls) which indicates greater likelihood of failures, more zeros than in previous times or greater number of points (longer tail), meaning great randomness and explain its proximity to the Poisson distribution. In the cases with less risk (two upper subplots), the game seems more predictable and the number of failures is lower; which would justify the least number of zeros and the shorter tail. And also explain its proximity to the Poisson distribution. In this thesis, one of the objectives is to compare the results against the Poisson model. This allows us to better understand the concept of risk and to separate the last minute of each quarter on a basketball game from the rest of the game; and the role of the last minute, as discussed below. The next figure represents the point scored of the last minute of the game, minute 48, whose Index of Dispersion value is larger than 1, over-dispersed: Figure 24. Histogram of point scored in the last minute of the game. The solid line represents negative binomial distribution fit (fitting parameters 3.81, 0.48; STD = 0.039). Note that the values are better fitted at the tail of the distribution. For further analysis, we carried out a log-log plot, in the upper panel, which displays two Power Laws with a crossover (straights lines). The dashed line in the upper panel represents the negative binomial fit. -2 0246810 12 14 16 18 0 200 400 600 800 1000 1200 Frequency 101 102 103 Basketball • Complexity Study 2. Basketball Game 123 The analysis of the last minute of the game in basketball reveals some game facts very remarkable. The last minute is over-dispersed (Index of Dispersion larger than 1; Figure 22), which is associated with a negative binomial distribution. The presence of a negative binomial distribution points out the existence of clusters of occurrences. Apparently, the point frequency distribution seems to match with the theoretical negative binomial distribution (solid line), particularly at the tail of the histogram (fitting parameters 3.81; 0.48; STD = 0.03. STD is the quadratic difference between the distribution and the data obtained). The apparent long tail behavior gives the impression of indicating the presence of data far removed from the mean, which might indicate the presence of a truncated Power Law. We must take into account that there is only 1 minute of real game time. To check this, we performed a log-log plot (upper panel) and we observe the values are fitted by two Power Laws. This might means that there are scaling phenomena, regarding scoring time. There is a zone 0 - 4 points, with a peak located around 2 - 3 points. But beyond this region, the first Power Law appears; from 4 to 9 points approximately. And a second one from 9 to 16 points with a higher slope (truncated). I.e. as the number of points scored increases the playing time is reduced dramatically. The presence of a crossover points out the presence of several scoring dynamics (multi scale behavior). We have to take into account that there is a rule in basketball designed to provide criticality to the game. We are talking about the 24seconds rule. The aim of this rule is to force teams to shot, what, transferred to the game, provides ideal conditions for a critical situation. In sport there are a lot of examples of rules whose aim is to provide criticality to the game, such as offside in football or rugby, three touches in volley-ball, the D-zone in handball etc. But as the Figure 22 shows, in the last minute in basketball, the own nature of the game turns critical by itself so significantly, that this rule that gives criticality to the game, makes no sense anymore. After the examination of game time, we performed an analysis of valuesnumber of points per minute) and the Index of Dispersion value of each game (6150 games). 124 Figure 25. QQ plot of points per minute mean values vs. Normal Distribution (upper left) and QQ plot of mean values vs. Gamma Distribution (upper right). Regarding these plots, note that the data present a better fit by the gamma distribution. With Normal Distribution there are some irregularities in the tails. Down left is represented the number de events per minute with a Gamma fit; and down right the semi-log plot of the previous Figure. The Figure 25 represents the values. The histogram (down left) seems to be almost a Normal Distribution, but fits better with a Gamma Distribution (upper plots). We performed a visual test (QQ-plot) with several distributions: Normal Distribution (QQ-plot up left), Exponential Distribution, Weibull Distribution, etc. but the data fits better by a Gamma Distribution (up right). The statistical values of the distribution are: mean = 2.280; STD= 0.251; variance = 0.063; skewness = 0.284 and Kurtosis = 3.174; For the Gamma distribution: Shape parameter k = 82.73 and Scale Parameter  = 0.027. The Gamma Distribution is an accurate distribution for modeling the behavior of continuous random variables with positively skewed; i.e. variables that present a greater density of events to the left of the mean than to the right. In our case we can observe that the number of points per game do not follow a Normal Distribution but is skewed to the right, meaning that there are more probabilities to score 2 2.5 3 2 2.5 3 QQ Plot of mean values versus Gamma 1.5 2 2.5 3 1.6 1.8 2 2.2 2.4 2.6 2.8 3 QQ Plot of mean values versus Normal 2 2.5 3 0 0.5 1 1.5 Points per minutes Frequency 2 2.5 3 10-3 10-2 10-1 100 Points per minutes log(rrequency) Basketball • Complexity Study 2. Basketball Game 125 more points than the mean (more than 2.28 points; up 3.3 points per minute), although this probability is low compared to the rest, it can take place. Figure 26. Histogram of Index of Dispersion per game. The dash line represents the Generalized Extreme Value (GEV) Distribution. It seems that fits well, but when we perform the log-log plot (upper panel) we note that the tail in not well fitted ((o) represents the real values, () represents the GEV values. We can observe that the most part of the games are located around 1, which means they are very unpredictable. But even we can find some games with values larger than 1 (over-dispersed). The Figure 26 shows the Histogram of Index of Dispersion per game, the most part of the values are located close to 1, which points out a Poisson process. This means that the number of points scored in the most part of the games follows a Poisson process. But moreover there are some cases where the distribution is over-dispersed. This indicates the presence of extreme events. In order to check this we fitted by a Generalized Extreme Value distribution (GEV). The dash line represents the Generalized Extreme Value (GEV) Distribution with shape parameter 0.0680; scale parameter 0.1786 and location parameter 0.7150; and seems to fit well except at the first values. To find out whether the data follows a GEV in the tail, we carried out a log-log plot (Figure 26 upper panel) with the theoretical GEV distribution (dash line against real values (o)). Note that the data do not fit well. On the other hand, seems to fit 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 time. This is important to take into account when we want to model or to predict in order to a better understanding. Our results point out that we can consider three zones at least. The first one extends from the beginning (0 seconds) up approximately 24 seconds. This area has a bell-shaped distribution, with a maximum around 20 seconds, with an irregular behaviour before 6 seconds and truncated beyond 24 seconds. The Figures 30 and 31 show us that in a basketball game, the most likely time between goals is around 20 seconds (this seems logical considering the 24 seconds of possession). The behaviour of the time intervals in this area deals with a kind of rhythm of play, ball possession, and scoring related with 24 seconds of possession, but below 6 seconds, presents some particularities (see Figures 30 and 31) as we mentioned above: The most part of time differences of one or two seconds are produced at minute 48. Points with one second difference only in the minute 48. The source of this particularity can be the free throws scored due to the high number of fouls made at the end of the game (for further detail see the foul section); but also because time outs called and strategies to score quickly. This can be the foundation of the shape of the distribution at the beginning as we can see in Figure 30 (b). The same Figure displays that the slope varies in second 6, approximately. Hence we can deduce that the first array corresponds with fast breaks and the second slope with plays more elaborated (from 6 to 20 seconds). In fact, the most common is to score around 20 seconds. This may be related with rebounds: defensive rebounds, because allow building fast breaks quickly, and offensive rebounds because allow scoring quickly with a high success rate, and further elaborating successive attacks. Hence, the strategies of many teams are to make fouls to avoid these situations, which create serious disadvantages between a team and the other. The second area, from 24 seconds up 100 seconds approximately, shows a decrease in a straight line (Figures 30 (c) and 31), which corresponds to an exponential distribution, as we saw in the case III), suggests that the distribution follows a Poisson process, i.e., completely random, without memory, for time intervals larger than 24 seconds. This is an interesting result because if it is a Poisson phenomenon, it could have a feature called memorylessness (also called evolution without after-effects): the number of goals Basketball • Complexity Study 2. Basketball Game 133 occurring in any bounded interval of time after time t is independent of the number of goals occurring before time t. It means that the time in which each point is scored is independent of the previous. The score becomes more random. Beyond 100 seconds approximately, as third region, we can see how the data are scattered and the final biased. It can be considered as rare phenomena (low probability). For values over 100 seconds is possible that it behaves as a Power law. To verify the actual behavior of the data, we performed a log-log plot (upper panel Figure 31). The results point out that the data fits better by a log-log plot, are less scattered, which means that can be considered a Power Law. From an overview we can observe that when the point time intervals reach high values, behaves as a Power Law; therefore we can say that the events data has memory, whereas when it behaves as Poisson distribution does not present such feature. This is important because can be related with the scoring differences between the winning team and losing team, and how the establish their strategies for each situation. Winning team tends to waste time, to waste possessions (longer time intervals). While the losing team, tends to the opposite. The strategy of long time intervals is typically of teams that want to play to low scores, where behaves as a Power Law, thus they are more likely to win the game because the systems is more critical (SOC). We are facing two diametrically opposed tendencies. This, as we have seen, can present a directly influence in the game. Moreover, if we extrapolate this to a higher level, to a league for instance, it can provide us some clue about how is the internal dynamic of the league: the fact that exists different dynamics well defined regarding point time intervals reflects stages of the game, profiles games or tactics employed by teams, which shows the sport reality in basketball. A league where there are large differences between their teams, the score differences in their games will be more pronounced, because theoretically weaker teams tend to employ defensive strategies through lower scores in order to increase their chances of victory. While on the other hand, a more balanced league, where the differences between its components are not so marked, this trend will not be so pronounced. 134 All team sports are based on fan attendance, particularly basketball, which tries to provide exciting scores. Hence the number of points per minute is as determinant factor when we analyze basketball (Figure 999). Figure 32. Number of point scored (Y-axis) with a time difference of 1 seconds, 2 seconds, 3 seconds,…, up 30 seconds (X-axis). The dash line represents points scored in the last minute of the game, where it is clear that most points with one second difference is given in this period. The other values correspond to the last minute of the remaining quarters (solid line ()) and to the y al minute 47 (o), whose behavior was similar. When we study the number of points corresponding to each time intervals in the last minute for each quarter (Figure 32), meaning number of points scored with one second difference between them, two second difference and so on until 30 seconds, we are we note that there are some significant differences. It seems that the last minute in the three first quarters solid line (), behaves similar. The values increase up 14 seconds; then remain stable until 23 seconds. It can have sense by itself, in the sense of this the most probably time between points in these periods, and probably is related with the 24 seconds of possession. Beyond 23 seconds drops up 27 seconds. Note that from 27 to 30 the frequency presents similar values. This stability points out a slight tendency to extend the time between points at the end of every quarter. The minute 47, solid line (o), follow a similar dynamic as the previous cases. The frequency increases up 14 seconds and stabilizes until 18 seconds. After that, the frequency drops up 26 seconds where presents the same frequency values until 30 seconds. 0 5 10 15 20 25 30 0 200 400 600 800 1000 1200 Basketball • Complexity Study 2. Basketball Game 135 The last minute of the game, dash line, minute 48 is completely different to the rest of the last minutes of previous quarters. The highest frequency value is for points with 1 second differences with a significant difference from other quarters. The frequency falls up 3 seconds, but increase again until 5 seconds and is still high compared to the rest of sample. The array form 5 to 8 seconds is the more stable region of this minute. But beyond this area, the frequency declines until the end. Even the range from 14 to 23 seconds is lower than the rest of quarters as we can see in Figure 32. It seems that the rule of 24 seconds make no sense here. The short intervals are numerous than the large. For the entire game time the shots with 1 second difference are: 1 point shot 1525 (89%); 2 point shots 137 (8%) and 3 point shots 48 (3%). The source of this tendency is the fouls and free throws, probably. In fact, if we analyze the minute 48 we observe that the shots with 1 second difference are: 1 point shot 1128 (94.55%); 2 point shots 48 (4.02%) and 3 point shots 16 (1.34%). we realize that the issue of fouls is accentuated. Therefore it is interesting to check out what were the final score when there were shots with one second difference. Histogram of final score differences when there is some 1 second scoring time interval in the last quarter: Figure 33. Final score differences of games with 1 second difference in the last quarter. The data are clustered from 0 to 10 points approximately. More than 10 points is a rare event. The peak value is 5 points. -5 0 5 10 15 20 25 30 0 20 40 60 80 100 120 136 We note that the most likely score difference is between 0 (we did not count overtime) and 10 points. But we can find differences up 29 points. The 5.88% finish in a tie (0 points; with over time). A total of 526 games of 918 (58%) finish with a difference between 3 and 7 points. And the 93% finish with less than 11 points From an overview, there were 690 games with at least a case of points with 1 second difference. In 184 games there were two cases. In 41 there were three cases and in four games there were only three cases. Scoring Regarding to score, the absolute value of score (result) always grows along with game, but do not evolve uniformly. This is a reality which is maintained on all basketball games. Score runs and maximum values achieved by the teams may vary, but always does incrementally. But what that really sets the dynamics of the game is the point differences between a team and another during the game time and, above all, at the end of the game. For that reason, we analyzed the differences on the final score of the whole sample analysis (6150 NBA games). The result (Figure 34) point out that most of the games (65%) ended with a difference between 1 and 11 points, 33% had a difference between 11 and 28 points, and only 2% did so with a difference of 28 or more points. To verify whether the data followed a Power Law type distribution, we performed a log-log plot whose result can be seen in the upper panel of Figure 34. Basketball • Complexity Study 2. Basketball Game 137 Figure 34. Point difference histogram existing in the final score of each game studied. The distribution is approximately uniform for values less than 10-12. Further than this value the distribution shows a possible behavior of long tail. Log-log plot of data point difference and frequency. We can see that the first array present a homogeneous tendency. Around the value of 10 points, an interruption in this trend takes place; and a second one at a value around 25-28 points. This suggests the presence of more than one Power Law. 0-10 score difference From 1 point to 10 points approximately, the distribution is almost uniform, which corresponds with situations of high uncertainty. If we exceed this score, from 10 to 28 points, the behavior appears to follow a Power Law. This indicates that the nature of the game has changed. Finally, over 28 points (a second Power Law), the essence of the game changes radically, and the final outcome is more predictable. In brief, results between 0 and 10 points are similar: the game is hard-fought. This points out that as long as the game remains between these values, the final result is unpredictable. This dynamic suggests that this area of point difference (0-11 points) works as an attractor, because the system (the game) tries to remain within this narrow area throughout the game time. In fact, in around 20% (about 1174) of the games assessed, teams did not exceed the maximum score difference of 11 points for the entirety of the game. 138 Regarding the final result, the number of games that finished with a point difference lower or equal to 11 points was 3846 games (62% of the games analyzed), which corresponds with the first cut in the log-log (Figure 34). 2324 of these games reached a maximum point difference between 11 and 20 points, and 43% did it during the last quarter of the game. In 578 games (almost 25% of the 2324 games), a team was able to overcome the difference (between 11-20 points) and win the game. This means that there were teams able to overcome a significant difference (between 11-20 points) and even win the game. It is possible that by achieving good score runs, the game is able to reach the critical area, and, joined with strategy at the end of the game (final quarter, fouls, free throws, time-outs, etc.), the combination needed for that team to win the game can be achieved. 1831 games reached a maximum point difference of more than 20 points. In 348 games (20%), this situation was overcome and the game was located in the area of a 0-11 point difference. 34 games (9.7%) of these cases won the game. In these games, the maximum difference was reached between the 9-minute and 44-minute mark. The results point out that it is indeed very difficult to overcome a 20-point difference in the last 4 minutes of the game. 11-28 score difference Out of the 2285 games with a score difference higher than 11 points, only 27 were able to win and overcome the differences between 12 and 22 points. Note that in this case, these differences were reached before the 24th minute (most of them in the second half), except a case in which it was done in the 33rd minute, but in that case the team was losing by only 15 points. Hence, beyond 10 points the dynamic is completely different and is more predictable. Larger than 28 score difference Neither team was able to either overcome the difference or achieve the region of an 11-point difference. This means that if there is more than a 28-point difference, then there is a clear superiority of one team over another, so much so that the game is quite predictable. We must remember that there is not a fixed criterion to identify non-linear complex systems or self-organized criticality behaviors in sports. But whether a Power Law appears, it is possible we are dealing with a non-linear complex system (Savaglio & Carbone, 2000; García Manso Basketball • Complexity Study 2. Basketball Game 139 et al., 2008). The log-log plot of the distribution of point difference is broken into several Power Laws; for certain characteristic values that can be considered thresholds or critical points, which means that game dynamic, can be characterized by several critical phenomena, or with several scales (multi-scale). The presence of crossovers in Power Laws is an indicator of changes in the underlying dynamic and suggests that perhaps we are dealing with a phase transition and critical exponents (McGarry et al., 2002; Scheffer et al., 2009). Returning to the general distribution of point differences, it also can be modeled by a Negative Binomial (1.94; 0.15), as we can see in the next Figure: Figure 35. Histogram of the point differences with a dash line as theoretical negative binomial fit. The main figure represents the log-log plot of the same data, also with a negative binomial distribution (dash line). Note that the histogram does not fit well by the negative binomial distribution at the beginning but it does at the tail. The log-log plot displays two cross over. The first one around 10 points and the second one around 28 points. We can observe that the histogram of the point differences is relatively well fitted by a negative binomial distribution (Figure 35) especially at the end. Next it is shown the end of games in detail. If the point difference is less or equal to 10 points the distribution of the points scored at the last minute of the game follows a negative binomial distribution (Figure 36). 101 100 101 102 010 20 30 40 50 0 0.01 0.02 0.03 0.04 0.05 0.06 140 Figure 36. Histogram for the points scored at the last minute for games ended by 10 or less points difference. The dash line represents the theoretical negative binomial distribution. The histogram points out that the most likely, for games ended by 10 or less points difference, is to score 3 or 4 points in the last minute. But the distribution seems to fit well to a negative binomial distribution with parameters 5.9301 and 0.5306; mean = 5.2471; median = 5.0000; variance = 9.9922 and STD = 3.1610. Note that there is a tail, but the most part of the points scored at the last minute are located bellow 10 points which point toward a great competitiveness. The distribution for shots scored was 1 point shots = 12008; 2 points shots = 4954 and 3 points shots = 1772. For the case of games ended with a difference between 11 and 28 points, the histogram 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 Basketball • Complexity Study 2. Basketball Game 141 Figure 37. Histogram for points scored in the last minute for games with a difference between 10 and 28 points. The dash line represents the theoretical negative binomial distribution. The Figure 37 represents the histogram of games ended with a point difference from 10 to 28 points. We can observe that the highest probability is for 1, 2 and 3 points and presents a tail as well. The statistical values are: mean = 2.5049; median = 2.0000; variance = 3.0034 and STD = 1.7330 The Index of Dispersion for these data was 1.19 which points out an over-dispersed Poisson distribution. The dash line represents the negative binomial distribution with parameters 14.9755; 0.8567. The distribution for shots scored was 1 point shots = 1732; 2 points shots = 1693and 3 points shots = 517. Note that in this case the number of points is more clustered than the previous case. And the number of shots scored is lower as well. For the last case, the case for games ended with a point difference higher than 28 points (Figure 38), the statistic was: mean = 2.3212; median = 2.0000; variance = 2.1241 and STD = 1.4574. The Index of Dispersion is 0.91 (lower than 1. Under-dispersed), ergo it is not negative binomial, is more Poissonian. 012345678910 11 12 13 14 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35