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The Bowl Championship Series: a Mathematical Review Thomas Callaghan, Peter J

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The Bowl Championship Series: a Mathematical Review Thomas Callaghan, Peter J The Bowl Championship Series: A Mathematical Review Thomas Callaghan, Peter J. Mucha, and Mason A. Porter Introduction sensus. Meanwhile, the top non-BCS squads have On February 29, 2004, the college football Bowl never been invited to a BCS bowl. Predictably, some Championship Series (BCS) announced a proposal have placed blame for such predicaments squarely to add a fifth game to the “BCS bowls” to improve on the “computer nerds” whose ranking algorithms access for midmajor teams ordinarily denied invi- form part of the BCS formula [7], [14]. Although we tations to these lucrative postseason games. Al- have no part in the BCS system and the moniker though still subject to final approval, this agree- may be accurate in our personal cases, we provide ment is expected to be instituted with the new BCS here a mathematically inclined review of the BCS. contract just prior to the 2006 season. We briefly discuss its individual components, com- There aren’t too many ways that things could pare it with a simple algorithm defined by ran- have gone worse this past college football season dom walks on a biased graph, attempt to predict with the BCS Standings governing which teams whether the proposed changes will truly lead to in- play in the coveted BCS bowls. The controversy over creased BCS bowl access for non-BCS schools, and USC’s absence from the BCS National Champi- conclude by arguing that the true problem with the onship game, despite being #1 in both polls, gar- BCS Standings lies not in the computer algorithms nered most of the media attention [12], but it is the but rather in misguided addition. yearly treatment received by the “non-BCS” mid- Motivation for the BCS major schools that appears to have finally gener- ated changes in the BCS system [15]. The National Collegiate Athletic Association (NCAA) Created from an abstruse combination of polls, neither conducts a national championship in Di- computer rankings, schedule strength, and qual- vision I-A football nor is directly involved in the cur- ity wins, the BCS Standings befuddle most fans rent selection process. For decades, teams were se- and sportswriters, as we repeatedly get “national lected for major bowl games according to championship” games between purported “#1” and traditional conference pairings. For example, the “#2” teams in disagreement with the polls’ con- Rose Bowl featured the conference champions from the Big Ten and Pac-10. Consequently, a match be- Thomas Callaghan is an undergraduate majoring in ap- tween the #1 and #2 teams in the nation rarely oc- plied mathematics, Peter Mucha is assistant professor of curred. This frequently left multiple undefeated mathematics, and Mason Porter is a VIGRE visiting assis- teams and cochampions—most recently Michigan tant professor, all at Georgia Institute of Technology. Peter and Nebraska in 1997. It was also possible for a Mucha’s email address is [email protected] team with an easier schedule to go undefeated work was partially supported by NSF VIGRE grant DMS- without having played a truly “major” opponent and 0135290 as a Research Experiences for Undergraduate pro- ject and by a Georgia Tech Presidential Undergraduate be declared champion by the polls, though the last Research Award. The simulated monkeys described herein two schools outside the current BCS agreement to do not know that they live on Georgia Tech computers. No do so were BYU in 1984 and Army in 1945. actual monkeys were harmed in the course of this inves- The BCS agreement, forged between the six tigation. major “BCS” conferences (the Pac-10, Big 12, Big SEPTEMBER 2004 NOTICES OF THE AMS 887 Ten, ACC, SEC, and Big East, plus Notre Dame as outcomes to seed a 12-team playoff. Indeed, an independent), was instituted in 1998 in an at- Division I-A college football is one of the only lev- tempt to fix such problems by matching the top two els of any sport that does not currently determine NCAA Division I-A teams in an end-of-season BCS its champion via a multigame playoff format.1 National Championship game. The BCS Standings, Ranking teams is further complicated by the Divi- tabulated by The National Football Foundation [18], sion I-A conference structure, as teams play most selects the champions of the BCS conferences plus of their games within their own conferences, which two at-large teams to play in four end-of-season vary significantly in their level of play. To make mat- “BCS bowl games”, with the top two teams playing ters worse, even the notion of “top 2” teams is in a National Championship game that rotates woefully nebulous: Should these be the two teams among those bowls. Those four bowl games—Fiesta, who had the best aggregate season or those play- Orange, Rose, and Sugar—generate more than $100 ing best at the end of the season? million annually for the six BCS conferences, but less than 10 percent of this windfall trickles down The BCS Formula and Its Components to the other five (non-BCS) Division I-A conferences In the past, national champions were selected by [13]. With the current system guaranteeing a BCS polls, which have been absorbed as one component bowl bid to a non-BCS school only if that school fin- of the BCS formula. However, they have been ac- ishes in the top 6 in the Standings, those confer- cused of bias towards the traditional football pow- ences have complained that their barrier to ap- ers and of making only conservative changes among pearing in a BCS bowl is unfairly high [20]. teams that repeatedly win. In attempts to provide Moreover, the money directly generated by the BCS unbiased rankings, many different systems have bowls is only one piece of the proverbial pie, as the been promoted by mathematically and statistically schools that appear in such high-profile games re- inclined fans. A subset of these algorithms com- ceive marked increases in both donations and ap- prise the second component of the official BCS plications. Standings. Many of these schemes are sufficiently Born from a desire to avoid controversy, the complicated mathematically that it is virtually im- short history of the BCS has been anything but un- possible for lay sports enthusiasts to understand controversial. In 2002 precisely two major teams them. Worse still, the essential ingredients of some (Miami and Ohio State) went undefeated during of the algorithms currently used by the BCS are not the regular season, so it was natural for them to publicly declared. This state of affairs has inspired play each other for the championship. In 2000, the creation of software to develop one’s own rank- 2001, and 2003, however, three or four teams each ings using a collection of polls and algorithms [21] year were arguably worthy of claiming one of the and comical commentary on “faking” one’s own two invites to the championship game. Meanwhile, mathematical algorithm [11]. none of the non-BCS schools have ever been invited Let’s break down the cause of all this confusion. to play in a BCS bowl. Tulane went undefeated in The BCS Standings are created from a sum of four 1998 but finished 10th in the BCS Standings. Sim- numbers: polls, computer rankings, a strength of ilarly, Marshall went undefeated in 1999 but fin- schedule multiplier, and the number of losses by ished 12th in the BCS. In 2003, with no undefeated each team. Bonus points for “quality wins” are also teams and six one-loss teams, the three BCS one- awarded for victories against highly ranked teams. loss teams (Oklahoma, LSU, and USC) finished 1st The smaller the resulting sum for a given team, the through 3rd (respectively) in the BCS Standings, higher that team will be ranked in the BCS Stand- whereas the three non-BCS one-loss teams finished ings. 11th (Miami of Ohio), 17th (Boise State), and 18th The first number in the sum is the mean rank- (TCU). ing earned by a team in the AP Sportswriters Poll The fundamental difficulty in accurately rank- and the USA Today/ESPN Coaches Poll. ing or even agreeing on a system of ranking the Di- The second factor is an average of computer vision I-A college football teams lies in two factors: rankings. Seven sources currently provide the the paucity of games played by each team and the algorithms selected by the BCS. The lowest large disparities in the strength of individual sched- computer ranking of each team is removed, and the ules. With 117 Division I-A football teams, the remaining six are averaged. The sources of the 10–13 regular season games (including conference participating ranking systems have changed over the short history of the system, most recently when the tournaments) played by each team severely limits BCS mandated that the official computer ranking the quantity of information relative to, for exam- ple, college and professional basketball and base- 1The absence of a Division I-A playoff is itself quite con- ball schedules. While the 32 teams in the profes- troversial, but we do not intend to address this issue here. sional National Football League (NFL) each play 16 Rather, we are more immediately interested in possible so- regular season games against 13 distinct oppo- lutions under the constraint of the NCAA mandate against nents, the NFL subsequently uses regular season playoffs. 888 NOTICES OF THE AMS VOLUME 51, NUMBER 8 Simple Random Walker Rankings Consider independent random walkers who each cast a single vote for the team they believe is the best. Each walker occasionally considers changing its vote by examining the outcome of a single game selected randomly from those played by their favorite team, recasting its vote for the winner of that game with probability p (and for the loser with probability 1 − p).
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