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Parity and Predictability of Competitions Parity and Predictability of Competitions E. Ben-Naim,1, ¤ F. Vazquez,1, 2, y and S. Redner1, 2, z 1Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 2Department of Physics, Boston University, Boston, Massachusetts 02215 We present an extensive statistical analysis of the results of all sports competitions in ¯ve major sports leagues in England and the United States. We characterize the parity among teams by the variance in the winning fraction from season-end standings data and quantify the predictability of games by the frequency of upsets from game results data. We introduce a novel mathematical model in which the underdog team wins with a ¯xed upset probability. This model quantitatively relates the parity among teams with the predictability of the games, and it can be used to estimate the upset frequency from standings data. What is the most competitive sports league? We 1 answer this question via an extensive statistical survey (a) of game results in ¯ve major sports. Previous stud- 0.8 ies have separately characterized parity (Fort 1995) and predictability (Stern 1997, Wesson 2002, Lundh 2006) of 0.6 sports competitions. In this investigation, we relate par- F(x) 0.4 NFL ity with predictability using a novel theoretical model in NBA NHL which the underdog wins with a ¯xed upset probability. MLB Our results provide further evidence that the likelihood 0.2 of upsets is a useful measure of competitiveness in a given 0 sport (Wesson 2002, Lundh 2006). This characterization 0 0.2 0.4 0.6 0.8 1 complements the myriad of available statistics on the out- x comes of sports events (Albert 2005, Stern 1991, Gembris FIG. 1: Winning fraction distribution (curves) and the best- 2002). ¯t distributions from simulations of our model (circles). For We studied the results of nearly all regular sea- clarity, FA, that lies between MLB and NHL, is not displayed. son competitions in 5 major professional sports leagues in England and the United States (table I): the pre- mier soccer league of the English Football Associa- winning fraction. Here denotes the average over all tion (FA), Major League Baseball (MLB), the National teams and all years usingh¢iseason-end standings. In our Hockey League (NHL), the National Basketball Associa- de¯nition, gives a quantitative measure for parity in tion (NBA), and the National Football League (NFL). a league (Fort 1995). For example, in baseball, where NFL data includes the short-lived AFL. Incomplete the winning fraction x typically falls between 0:400 and seasons, such as the quickly abandoned 1939 FA sea- 0:600, the variance is = 0:084. As shown in ¯gures 1 son, and nineteenth-century results for the National and 2a, the winning fraction distribution clearly distin- League in baseball were not included. In total, we ana- guishes the ¯ve leagues. It is narrowest for baseball and lyzed more than 300,000 games in over a century (data widest for football. source: http://www.shrpsports.com/, http://www.the- Do these results imply that MLB games are the most english-football-archive.com/). competitive and NFL games the least? Not necessarily! The length of the season is a signi¯cant factor in the vari- ability in the winning fraction. In a scenario where the I. QUANTIFYING PARITY outcome of a game is random, i.e., either team can win with equal probability, the total number of wins performs The winning fraction, the ratio of wins to total games, a simple random walk, and the standard deviation is quanti¯es team strength. Thus, the distribution of win- inversely proportional to the square root of the number ning fraction measures the parity between teams in a of games played. Generally, the shorter the season, the league. We computed F (x), the fraction of teams with a larger . Thus, the small number of games is partially winning fraction of x or lower at the end of the season, responsible for the large variability observed in the NFL. as well as = x2 x 2, the standard deviation in ph i ¡ h i II. QUANTIFYING PREDICTABILITY ¤Electronic address: [email protected] yElectronic address: [email protected] To account for the varying season length and reveal the zElectronic address: [email protected] true nature of the sport, we set up arti¯cial sports leagues 2 league years games hgamesi q qmodel 0.28 0.26 (a) FA 1888-2005 43350 39.7 0.102 0.452 0.459 0.24 NFL MLB 1901-2005 163720 155.5 0.084 0.441 0.413 NBA 0.22 NHL MLB NHL 1917-2004 39563 70.8 0.120 0.414 0.383 0.20 FA NBA 1946-2005 43254 79.1 0.150 0.365 0.316 σ 0.18 NFL 1922-2004 11770 14.0 0.210 0.364 0.309 0.16 0.14 TABLE I: Summary of the sports statistics data. Listed are 0.12 the time periods, total number of games, average number of 0.10 games played by a team in a season (hgamesi), variance in 0.08 1900 1920 1940 1960 1980 2000 the win-percentage distribution (), measured frequency of year upsets (q), and upset probability obtained using the theoreti- 0.48 cal model (qmodel). The fraction of ties in soccer, hockey, and (b) football is 0:246, 0:144, and 0:016, respectively. 0.46 0.44 0.42 where teams, paired at random, play a ¯xed number of q 0.40 games. In this simulation model, the team with the bet- 0.38 ter record is considered as the favorite and the team with 0.36 FA the worse record is considered as the underdog. The out- MLB 0.34 NHL come of a game depends on the relative team strengths: NBA with \upset probability" q < 1=2, the underdog wins, 0.32 NFL 0.30 but otherwise, the favorite wins. If the two teams have 1900 1920 1940 1960 1980 2000 the same fraction of wins, one is randomly selected as the year winner. We note that a similar methodology was utilized by FIG. 2: (a) The cumulative variance in the winning fraction Wesson who focused on the upset likelihood as a func- distribution (for all seasons up to a given year) versus time. tion of the ¯nal point spread in soccer (Wesson 2002). (b) The cumulative frequency of upsets q, measured directly Also, an equivalent de¯nition of the upset frequency was from game results, versus time. very recently employed by Lundh to characterize how competitive tournaments are in a variety of team sports (Lundh 2006). wins, the greater the disparity between teams. Perfect Our analysis of the nonlinear master equations that de- parity is achieved when q = 1=2, where the outcome scribe the evolution of the distribution of team win/loss of a game is completely random. However, for a ¯nite records shows that decreases both as the season length and realistic number of games per season, such as those increases and as games become more competitive, i.e., that occur in sports leagues, we ¯nd that the variance as q increases. This theory is described in the appendix is larger than the in¯nite game limit given in Eq. (2). and more generally in Ben-Naim et al. 2006. The basic As a function of the number of games, the variance de- quantity to characterize team win/loss records is F (x), creases monotonically, and it ultimately reaches the lim- the fraction of teams that have a winning fraction that iting value (2). is less than or equal to x. In a hypothetical season with We run numerical simulations of these arti¯cial sports an in¯nite number of games, the winning fraction distri- leagues by simply following the rules of our theoretical bution is uniform model. In a simulated game, the records of each team are updated according to the following rule: if the two 0 0 < x < q teams have a di®erent fraction of wins, the favorite wins 8 x q with probability 1 q and the underdog wins with prob- F (x) = > ¡ q < x < 1 q (1) <1 2q ¡ ability q. If the t¡wo teams are equal in strength, the ¡ 1 1 q < x: winner is chosen at random. Using the simulations, we > ¡ : determined the value of qmodel that gives the best match From the de¯nition of the upset probability, the lowest between the distribution F (x) from the simulations to winning fraction must equal q, while the largest winning the actual sports statistics (¯gure 1). Generally, there is fraction must be 1 q. good agreement between the simulations results and the By straightforward¡ calculation from F (x), the stan- data, as quanti¯ed by qmodel (table I). dard deviation is a linear function of the upset proba- To characterize the predictability of games di- bility rectly from the game results data, we followed the 1=2 q chronologically-ordered results of all games and recon- = ¡ : (2) structed the league standings at any given day. We then p3 measured the upset frequency q by counting the fraction Thus, the larger the probability that the stronger team of times that the team with the worse record on the game 3 date actually won (table I). Games between teams with 1 no record (start of a season) or teams with equal records NFL were disregarded. Game location was ignored and so was 0.8 NHL the margin of victory. In soccer, hockey, and football, MLB ties were counted as 1/2 of a victory for the underdog 0.6 and 1/2 of a victory for the favorite.
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