Mathematics and Sports: Extending the Colley Method to Generate Predictive Football Rankings

Among the many mathematical ranking systems publish ed in college football, the method used by Wes Colley is notable for its elegance. It invo lves setting up a matrix system in a relatively sim ple way, then solving it to determine a ranking. However, t he Colley rankings are not particularly strong at predicting the outcomes of future games. We discus s the reasons why ranking college football teams is difficult, namely weak connections (as 120 teams ea ch play 11-14 games) and divergent strengths-ofschedule. Then, we attempt to extend this method t o improve the predictive quality, partially by appl ying margin-of-victory and home-field advantage in a log ical manner. Each team's games are weighted unequally, to emphasize the outcome of the most inf ormative games. This extension of the Colley metho d is developed in detail, and its predictive accuracy during a recent season is assessed. Many algorithmic ranking systems in collegiate Amer ican football publish their results online each season. Kenneth Massey compares the results of over one hundred such systems (see [9]), and David Wilson’s site [14] lists many rankings by category. A variety of methods are used, and some are dependent on complex tools from statistics or mathematics. For example, Massey’s ratings [10] use maximum likelihood estimation. A few methods, including those of Richard Billingsley [3], are computed recursively, so that each week’s ratings are a function of the prev ious week’s ratings and new results. Some high-profile rankings, such as those of USA To day oddsmaker Jeff Sagarin [12], use methods that are not fully disclosed, for propr ieta y reasons. Despite the different approaches, nearly all ranking methods use the same imple data set, the scores of games played during the current season. A few also use o th r statistics, such as yardage gained. College football’s Bowl Championship Series (BCS), which matches top teams in financially-lucrative postseason games, including a unofficial national championship game, chooses its teams using a hybrid ranking, cur rently including two human polls and six computer rankings [5]. The use of victory margins in BCS-affiliated computer rankings was prohibited following the 2001 season [ 4], out of concern that coaches might run up huge margins, violating good sportsmanship. There are two opposing philosophies in ranking meth ods, leading to retrodictive and predictive rankings. Retrodictive rankings aim to reflect mo st accurately the results of the current season in hindsight (minimizing violations, cases of a lower-ranked team defeating a higher-ranked one). Predictive ranking s attempt to identify the strongest teams at the present, so as to forecast the winners of upcoming contests. Most predictive rankings use margin-of-victory and also consider ho me-field advantage, both in previous and future games. To achieve reasonable early-seas on re ults, predictive ranking methods typically carry over data from the previous season, perhaps with adjustments made for returning or departing players and coachin g changes. Retrodictive rankings start from scratch each season, so they are not pub lished until each team has played several games. Jay Coleman’s MinV ranking [6, 7], designed to achieve optimal retrodictive results, is superior to any other rank ing in that category [9]. No predictive ranking algorithm has consistently outperformed the Las Vegas oddsmakers [1], who have a strong financial interest in accurately asse ssing upcoming games. Many ranking systems seek a balance of predictive and retrodicti ve quality, attempting to give insight into future contests while remaining faithful to pa st results. Among the six Bowl Championship Series computer ran kings, the Colley Matrix, designed by astrophysicist Wes Colley, is unique. Colley’s algorithm [8], rooted in linear algebra, is elegant in its simplicity. His ranking s are relatively easy to reproduce, and Colley’s method involves neither margin-of-victory nor home-field advantage. This method is not a strong predictor [2], nor is it designed to be, raising the questio n of whether it could be extended to create a predictive algorithm based on linear algebra. In such an attempt, we must accept losing the elegance of Colley’s original method, and will need to use additional data, such as victory margin s and home-field advantage. To extend Colley’s method in this way, we must firs t understand how it works. The method begins by rating each team using a modif ied winning percentage, then adjusts the ratings according to the quality of opp osition a team has faced. Each team effectively starts the season with a win and a loss , to avoid having undefeated teams automatically at the top (and winless teams at the bottom) regardless of schedule strength. All teams are placed in a fixed, arbitrary order, s o that each row of a square matrix system Ax = b relates to a particular team. Before any games ar pl yed, A is a diagonal matrix with 2’s for all diagonal entries, and b is a vector consisting of all ones, as shown in (1). If we solve the system, we find that xj = 0.5 for each j, agreeing with a winning percentage of 0.5, from an implied 1-1 initial reco rd. Thus, all teams are considered equal prior to the start of the season.