Volume 10, Issue 2

In this paper we consider the volleyball under the assumption that the probability of winning a single rally is independent of the other rallies and constant during the game. Fixing two parameters which indicate the probabilities of winning a rally for the serving team, we derive the exact expression of the probability of winning a set and a match, even in the case of sets ending with a tie break, in the present rally point- and in the former side-out scoring systems. Furthermore we are able to evaluate by a simple direct computation the mean duration of the sets in both the scoring systems, obtaining, as well known in the practice, that the change in the scoring system reduced the (expected) length of the matches. The analysis of the matches between the 4 top teams of the Italian Volley League in the period 2001–2012 shows that the model is adequate and accurate in describing the game.

In 2012, 3 out of 10 singles players in the top 100 on the Association of Tennis Professionals (ATP) World Tour were 30 years old or older – nearly a four-fold increase over 20 years ago, suggesting that the “old at 30” view in men’s tennis may be an old reality. In this paper, I investigate aging patterns among top ATP singles players between 1991 and 2012 and consider how surface effects, career length, and age at peak performance have influenced aging trends. Following a decade and a half of little change, the average age of top singles players has increased at a pace of 0.34 years per season since the mid-2000s, reaching an all-time high of 27.9 years in 2012. Underlying this age shift was a coincident rise in the proportion of 30-and-overs (29% in 2012) and the virtual elimination of teenagers from the top 100 (0% in 2012). Because the typical age players begin competing professionally has varied little from 18 years in the past two decades, career length has increased in step with player age. Demographics among top players on each of today’s major surfaces indicate that parallel aging trends have occurred on clay, grass, and hard court from the late 2000s forward. As a result of the changing age demographic over the past decade, the age of tennis’s highest-ranked singles players is now comparable to the age of elite long-distance runners. This evolution likely reflects changes in tennis play that have made endurance and fitness increasingly essential for winning success.

Sports ranking systems are often viewed as inadequate for judging the quality of the teams or players involved. Meanwhile, statistical models have been shown to produce more accurate ratings for those competitors, based on their ability to forecast future results. However, whilst predictive power is a desirable property of any official ranking system, these systems must also be fair, transparent and insensitive to bias. Additional requirements may also be required, such as promoting major tournaments and deciding seedings. By considering rankings for ATP tennis players, we propose that statistical models can be used to improve the existing ranking system, in such a way that the resulting rankings are fair and usable by the governing body. In many cases, there is a trade-off between predictive power and other desirable properties, and so compromise is required to produce a system that can be implemented successfully.

We analyze a sports league that wishes to augment its traditional double round-robin tournament into a longer season. The method for doing so, chosen by the top Swedish handball league Elitserien, is to form two divisions that hold an additional single round-robin tournament to start the season. This format introduces new constraints since pairs of teams in the same division meet three times during the season, while others only meet twice. Though motivated by the concerns of a specific league, the requirements addressed are general enough to be useful for other leagues. We enumerate the number of minimum break home-away pattern sets that satisfy the league’s requirements, not all of which are schedulable. We propose a sequence of increasingly restrictive necessary conditions that remove most of the unschedulable home-away pattern sets from consideration. We lastly discuss the final steps of assigning teams to a schedulable home-away pattern set; such an approach was used to construct the 2013–2014 Elitserien schedule.

Standings in sports are obtained by applying a system of rules to evaluate the performance of the participants in a competition. We consider standings that result from assigning an ordinal rank to each competitor according to their performance. We develop an integer programming model for standings that allows us to calculate the number of points needed to guarantee a team the i th position, as well as the minimum number of points that could yield the i th place. The model is very general and can thus be adapted to many types of sports. We discuss examples coming from football (soccer), ice hockey, and Formula 1. We answer various questions and debunk a few myths along the way. Are 40 points enough to avoid relegation in the German Bundesliga? Do 95 points guarantee the participation of a team in the NHL playoffs? Moreover, in the season restructuration that was under consideration in November 2012, would it be easier or harder to access the playoffs? Is it possible to win the Formula 1 World Championship without winning at least one race or without even climbing once on the podium? Finally, we observe that the optimal solutions of the aforementioned model are associated to extreme situations which are unlikely to happen. Thus, to get closer to realistic scenarios, we enhance the model by adding some constraints inferred from the results of the previous years.

Competitive balance is a key issue for any professional sport league substantiated by its effect on demand for league games or other associated products. This work focuses on the measurement of between-seasons competitive balance, the longest time-wise dimension, which captures the relative quality of teams across seasons. The review of the existing indices examines their applicability in the context of European football. Given that domestic championships are multi-prized tournaments, as opposed to the more common North American unitary structure with a single prize, a set of specially designed indices that capture the complex structure of European football are introduced. An empirical investigation, using data from the English Premier League, further elucidates the main features of all appropriate indices by exploring their value and trend. It may be inferred that between-seasons competitive balance in England worsens through seasons mainly due to the very low ranking mobility of the top teams through seasons.

The Referee Assignment Problem (RAP) is a novel arising problem in sports management, in which a limited number of referees with different qualifications and availabilities should be assigned to a set of games already scheduled, in order to respect a list of constraints. Number and nature of these constraints may significantly vary for sports, nation and type of league. Almost each tournament has its own particular set of constraints to be satisfied, therefore it is very difficult to generalize this problem. The goal of the problem is to find a feasible assignment, i.e., a configuration which allows to respect all the constraints given. An extension of the RAP is the Fair Referee Assignment Problem (FRAP), in which the objective is to minimize the violation of a set of soft (optional) constraints, while satisfying all the hard (mandatory) ones. In this work, the Italian Major Soccer League, the so-called SERIE A, is addressed, and an integer programming model for the related FRAP is proposed. Soft and hard constraints have been formulated according to the rules suggested by the AIA (Italian Referee Association) which is in charge of referee assignment for the SerieA. The model has been tested on a real instance taken from the season 2011/2012. Results obtained show the efficacy and the effectiveness of the model.

This paper uses a statistical analysis of match sprint outcomes to guide tactical decisions in this highly tactical contest and to provide competitors and coaches with a potential, marginal gain. Logistic regression models are developed to predict the probability of the leading rider winning at different points of the race, based on how the race proceeds up to each point. Key tactics are successfully identified from the models, including how the leading rider might hold the lead and how the following rider might optimize overtaking.

The ratio of uphill pace to downhill pace in a foot race up and down a single mountain is used as a measure of a competitor’s descending skills – those qualities which are needed for fast descent but not for ascent. For the set of competitors in each of 44 races on seven courses of differing gradients and terrain roughnesses, we calculate the variance of this pace ratio and do linear regressions of pace ratio on finish time and on competitors’ age. The variances tend to be greater for races on steeper and rougher terrain, indicating a greater influence of descending skills on actual descent speeds in these races. The regression analysis shows a clear negative correlation with finish time, indicating that faster finishers tend to be those with better descending skills, but there is little evidence of correlation with age. Significant differences between the sexes are only found in races on the most difficult terrain, where men display better descending skills than women.

Multiple models are discussed for ranking teams in a league and introduce a new model called the Oracle method. This is a Markovovian method that can be customized to incorporate multiple team traits into its ranking. Using a foresight prediction of NFL game outcomes for the 2002–2013 seasons, it is shown that the Oracle method correctly picked 64.1% of the games under consideration, which is higher than any of the methods compared, including ESPN Power Rankings, Massey, Colley, and PageRank.

Before any play of a National Football League (NFL) game, the probability that a given team will win depends on many situational variables (such as time remaining, yards to go for a first down, field position and current score) as well as the relative quality of the two teams as quantified by the Las Vegas point spread. We use a random forest method to combine pre-play variables to estimate Win Probability (WP) before any play of an NFL game. When a subset of NFL play-by-play data for the 12 seasons from 2001 to 2012 is used as a training dataset, our method provides WP estimates that resemble true win probability and accurately predict game outcomes, especially in the later stages of games. In addition to being intrinsically interesting in real time to observers of an NFL football game, our WP estimates can provide useful evaluations of plays and, in some cases, coaching decisions.

This paper analyzes a unique data set consisting of all penalty calls in the National Hockey League between the 1995–1996 and 2001–2002 seasons. The primary finding is the prevalence of “reverse calls:” if previous penalties have been on one team, then the next penalty is more likely to be on the other. This pattern is consistent with a simple behavioral rationale based on the fundamental difficulty of refereeing a National Hockey League game. Statistical modeling reveals that the identity of the next team to be penalized also depends on a variety of other factors, including the score, the time in the game, the time since last penalty, which team is at home, and whether one or two referees are calling the game. There is also evidence of differences among referees in their tendency to reverse calls.

Sports leagues consist of conferences subdivided into divisions. Teams play a number of games within their divisions and fewer games against teams in different divisions and conferences. Usually, a league structure remains stable from one season to the next. However, structures change when growth or contraction occurs, and realignment of the four major professional sports leagues in North America has occurred more than 25 times since 1967. In this paper, we describe a method for realigning sports leagues that is flexible, adaptive, and that enables construction of schedules that minimize travel while satisfying other criteria. We do not build schedules; we develop league structures which support the subsequent construction of efficient schedules. Our initial focus is the NHL, which had a need for realignment following the recent move of the Atlanta Thrashers to Winnipeg, but our methods can be adapted to virtually any situation. We examine a variety of scenarios for the NHL, and apply our methods to the NBA, MLB, and NFL. We find the biggest improvements for MLB and the NFL, where adopting the best solutions would reduce league travel by about 20%.

Using NHL data for the 1997–1998 through 2011–2012 seasons, we examine the effect of age on scoring performance and plus-minus for NHL skaters (non-goalies) and on save percentage for goaltenders. We emphasize fixed-effects regression methods that estimate a representative age-performance trajectory. We also use a method based on the best performances over time, a method based on the age distribution of NHL players, and a “naïve” specification that does not correct for selection bias. In addition we estimate individual age-performance relationships to obtain a distribution of peak ages. All methods provide similar results (with small but understandable differences) except the naïve specification, which yields implausible results, indicating that correcting for selection bias is very important. Our best estimate of the scoring peak age is between 27 and 28 for forwards and between 28 and 29 for defencemen. Both forwards and defencemen exhibit near-peak performance over a wide range, going from about 24 to 32 and 24 to 34, respectively. Goaltenders display little systematic performance variation over most of the age range from the early 20s to late 30s.

Subjective ratings of the best drivers in the history of Formula One are common, but objective analyses are hampered by the difficulties involved in comparing drivers who raced for different teams and in different eras. Here, we present a new method for comparing performances within and between eras. Using a statistical model, we estimate driver and team contributions to performance, as well as the effects of competition with other drivers. By adjusting for team and competition effects, underlying driver performances are revealed. Using this method, we compute adjusted scoring rates for 1950–2013. Driver performances are then compared using: (i) peak performances for 1-year, 3-year, and 5-year intervals; and (ii) number of championships. Overall, these comparisons rank Clark, Stewart, Fangio, Alonso, and Schumacher as the five greatest drivers. We confirm the model’s accuracy by comparing its performance predictions to 2010–2013 lap-time data. The results of the analysis are generally in good agreement with expert opinions regarding driver performances. However, the model also identifies several undervalued and overvalued driver performances, which are discussed. This is the first objective method for comparing Formula One drivers that has yielded sensible results. The model adds a valuable perspective to previous subjective analyses.

This paper examines the performance of goalkeepers in the English Premier League. A commonly used metric to assess goalkeeper ability is the saves-to-shots ratio. It is shown that goalkeepers playing in weak teams have to defend against on-target shots that have a relatively high probability of scoring as compared to their counterparts in higher-performing teams. This tends to produce a downwards bias in the save-to-shots ratio of goalkeepers in weak teams, an effect which may lead to their ability being underestimated. A logistic regression model is used to adjust goalkeepers’ saves-to-shots ratios for differences in save difficulty. The adjusted ratio is more stable across seasons, suggesting that it is a more reliable estimate of true goalkeeper ability than the unadjusted ratio.

Soccer analytics often follow one of two approaches: 1) regression models on number of shots taken or goals scored to predict match winners, or 2) spatial and/or temporal analysis of plays for evaluation of strategy. We propose a new model to evaluate skill importance in soccer. Play by play data were collected on 22 NCAA Division I Women’s Soccer matches with a new skill notation system. Using a Bayesian approach, we model play sequences as discrete absorbing Markov chains. Using posterior distributions, we estimate the probability of 35 distinct offensive skills leading to a shot during a single possession.