Tanley Brown History of Math Final Paper Math & Football 10/20/2014 American Football Playoffs: Does the Math Matter? History of Football While many people consider baseball the great American pastime, it is undeniable that one of the most prominent American sports is football. Originally an Ivy League sport, football was played by boys from Harvard and Yale as early as 1830. Back in the day, football had many participants and was much more similar to modern day rugby than to modern day football. Overtime, rules were added, number of players was diminished, and safety became more important. However, it wasn t until 1878 that Walter Camp suggested several changes to football making it much more similar to the game we know and love today (The History of Football in America). As with any sport, there are inherently obvious ways that math is involved. For instance, distance is highly important in the game of football. Were distance not established mathematically, football fields would all be different sizes and teams could easily obtain or prevent first downs for themselves and the other team, respectively. Mathematics is also used when odds makers determine the odds for various matchups and create the line of the game. However, that is just the tip of the iceberg when it comes to mathematics in football. History of Ranking Systems The first known person to rank college football was Frank Dickinson from the University of Illinois. He did this for the first time in the year of 1924. He determined a ranking system
Brown 2 which awarded points based on the number of games that each team won and the quality of the opponent which was played in those victories. Ultimately, Dickinson began this largely because of a passion for college football, so after some time, as his age became an issue, and his overall interest in college football declined, Dickinson ended his career of ranking the teams of college football. He ranked college football teams from 1924 to 1940, and in this time period other methods for ranking college football teams were created. This indicates the growing interest in ranking systems and ultimately would lead to the establishment of a nationally recognized ranking system (Stern). The Associated Press (AP) poll was used to rank college football teams beginning in 1936. This system compiled the opinions of national sportswriters to create a listing of the top teams in the nation. And, shortly after this was instituted, a similar poll was started by United Press International (UPI), a competitor of the Associate Press (Stern). This poll ranked teams based on the opinions of the nation s coaches and is today known as the Coaches poll. Both of these polls are still used to rank teams, although little importance is currently given to the AP poll s rankings of teams. Also, during this time bowl games began to occur. This was a way for teams to compete against each other after the regular season. Teams were invited to play and the event would serve as a money-making venture for both the host city and the bowl s hosting organization. This is still true today, which came into play when college football was considering the move to a playoff system. The first bowl game was the Rose Bowl in the early 1990s and other games would soon begin around 1935. The AP poll and the UPI poll began to release their national champion results after the bowl games in 1968 and 1974,
Brown 3 respectively. This led to an increased importance of bowl games for both the teams and the organizing parties (Stern). Bowl Championship Series Rankings For instance, in college football, the Bowl Championship Series (BCS) rankings are extremely important. These rankings list the top 25 teams in America each and every Sunday of football season. People anxiously await the rankings each week, which surely makes the BCS rankings one of the most highly anticipated mathematical results of the 21 st century. However, no one truly knows the math behind the BCS rankings because a portion of the formula is a secret. While being the National Champion of college football is not the most important thing for a university, it can be considered important because of the subsequent benefits for the champion team and champion team s university. It was estimated that the top team of the nation received $17.5 million in 2010 (Teasley). In addition to the large monetary gain for the winner s school and conference, the winning school can expect to obtain more notoriety which will in turn increase donations from alumni and increase applications to the school. This will also increase athletic notability of the football program, which will likely lead to more NLIs (National Letter of Intent) and more high-quality athletes. In turn, this could help the team become national champion again, helping the university in the same ways again.
Brown 4 Figure 1: The BCS rankings from December 3 rd, 2013. This serves as an example of how the BCS rankings are released each week, and what information is included in these releases. In 2010, BCS scores were calculated with three components. Two human polls: the Harris poll and the Coaches poll. These values are calculated by taking a percent value with a denominator of the number of voters in the poll multiplied by the number of spaces available (25). Then, each numerator is determined by multiplying the number of votes at each value. For instance, if Virginia Tech were to be scored on the coaches poll which has 61 voting coaches, and allows points for 25 spots, they would be ranked out of (61)(25) = 1,525 available points in the denominator. If they received 48 votes for the number one spot (at 25 points per vote) and 13 votes at the number two spot (at 24 points per vote), they would have a numerator valued at (48)(25) + (13)(24) = 1,512. This would lead to an overall Coaches Poll score of 1,512/1,525 =.9915 (Teasley). This is done for both the
Brown 5 Coaches Poll and the Harris poll. The Harris poll has 113 voting members of various backgrounds, and it replaced the AP poll in BCS calculations. And, the third element of the BCS rankings is the computer rankings. These are found in a similar fashion to the two human polls. There are six programs which are employed to rank teams from highest to lowest, with the first 25 teams receiving points. The top and bottom score for each team is then removed to account for error in the system. This is where many consider errors to be found in the system because it is based on a small number of ballots (4) and BCS does not release the algorithms it uses to determine this final component of the rankings. Prior to this, there were several other renditions to the formula. Past renditions have included information on strength of schedule and on number of losses. Overtime these renditions were dropped in order to better the system. Additionally, the formulas which are used for the computer polls and consistently undergoing improvements through minute changes by the programmers. As such, the ranking system is never truly the same from year to year, and is theoretically improving due to these changes throughout the years. This system has several errors involved. For instance, in 2008, it could be argued that Texas was disadvantaged by the system in place. There were small percentage differences that became large as the rankings were converted to placements of 1-25. As such, Texas was not ranked ahead of Oklahoma at the end of the season, even when it is easily argued that Texas did in fact belong in front of Oklahoma. As a result of this, Texas did not play in the conference championship, and thus did not have the chance to re-play
Brown 6 Missouri. Because Texas beat Missouri earlier that season by a large margin, they could be predicted to do this again, which would have placed them in the national championship. Therefore, it can be concluded that the BCS algorithms could have led to the crowning of a false National Champion in 2008. Random Walker Rankings One theory about the Bowl Championship Series rankings is that random walkers could select team rankings better than the BCS algorithm. In historical analyses, it is often found that random walkers select rankings on par with, or even better than, the BCS algorithms. This method also has the added benefit of being extremely simplified when compared with the series of computer rankings and poll rankings that are considered for the BCS. The Simple Rules for Each Monkey (1) Pick a game played by your favorite team; that is, the team you are currently casting your single vote for. (2) Flip a weighted coin that is more likely to come up heads. [How much more likely? What percentage of the time does it come up heads? That is the one number we can modify.] (3) Completely forgetting which team you voted for before, go with the winner of the game if heads, the loser if tails, changing your vote if necessary. (4) Return to step (1). Figure 2: An explanation of the random walking rules from the Random Walker Rankings blog written by Peter Mucha. In further explanation from the above figure, a monkey, or random walker, would select a team. Each monkey gets one singular vote, and they must always cast it for their favorite team. The team they select originally is their first favorite team. Then, based on the percentages set by the simulation executer, they flip a coin which is weighted. For instance,
Brown 7 if the simulation executer selected a 75% chance of the coin indicating the winning team, then the coin would have a 75% chance of landing heads up when the records indicate that a certain team has won. The monkey flips the coin, and then, based on which side the coin lands on, either maintains their allegiance to their original team, or switches to the victorious team. Note, that the monkey could switch allegiance to a losing team because the coin could still land tails up, which would indicate the monkey s allegiance now being for the losing team. This process is completed throughout the season and the votes of various monkeys are compiled to indicate the overall rankings at various points throughout the season and at the conclusion of the season. While this concept is fairly simple, mathematical analyses of 33 football seasons has indicated that random walkers are equitably accurate in their rankings. However, because it is impossible to conclusively state that a certain team should have been number one or number two in a certain season, it is not possible to conclude that certain systems are better than their counterparts. This leads to inconclusive comparisons, but still intriguing information. This system is unique because it already incorporates strength of schedule into its considerations. This is accounted for because games which are played against highly ranked opponents are more likely to have many monkeys following them. This means, that the win/loss coin would be largely in favor of the team which pulled the upset, thus representing strength of schedule.
Brown 8 Figure 3: Random Walker Ranking results from November 14 th, 2014 on Peter Mucha s blog. This image shows the way random walker data is commonly presented. The author does indicate that other variables could certainly be accounted for in the weight of the coin. For instance, the system could have been considered to fail when it looked into the champions of the 2001 football season. The monkeys backed Tennessee as a contender, but the national consensus was that Tennessee did not belong in the contentions. This was because of their loss in the SEC Championship game against LSU. Because this loss came late in the season, it was deemed more significant by society. Margin of victory could also be accounted for in a similar manner (Mucha). Despite the possibility of increased accuracy, Mucha explains that the simpler formula is more ideal because it negates the need for him to assign random values to strength of schedule and margin of victory, thus only leaving one variable- win/loss weight. Ultimately, Peter Mucha ascertains the following question and response: Can a bunch of monkeys rank football teams as well as the systems in use now? Perhaps they can. And, even with its strengths and weakness, this system appears to be comparable to
Brown 9 the Bowl Championship Series rankings. As such, it is important when considering the mathematics behind football rankings. Football Today Recently, college football underwent a drastic change in end of the season protocol. Instead of all teams going to bowl games, some teams would be attending playoff games for the National Championship. The idea behind the concept is to avoid ranking systems selecting teams because of certain aspects and in spite of certain aspects. There is a committee that considers a wealth of information including review of video, statistics, and their own expertise to guide them in their deliberations. They will emphasize obvious factors like win-loss records, strength of schedule, conference championships won, headto-head results, and results against common opponents (College Football Playoff). Thus, instead of relying entirely on the math, the selection committee is merely guided by the math. This is technically a move away from mathematical thought, and ultimately, historically important for football. Because it is a departure from the Bowl Championship Series rankings, it is a departure from portions of mathematics that have been included in football for years upon years. Despite this, it can be argued the mathematics has a continuing effect on the selection of contenders for the national championship. This is because the selection committee does consider mathematical concepts and formulas; for example, the algorithm they use to determine strength of schedule. As such, though the face of mathematics in football is changing, it is still highly relevant.
Brown 10 Random Walkers and College Football Playoff Rankings According to the College Football Playoff (CFP) rankings for week 13 (the week of November 14 th ) are as follows: (1) Alabama, (2) Oregon, (3) Florida State, (4) Mississippi State, (5) TCU, (6) Ohio State, (7) Baylor, (8) Ole Miss, (9) UCLA, (10) Georgia, (11) Michigan State, (12) Kansas State, (13) Arizona State, and (14) Auburn. For comparison purposes with random walker data, the first 14 of the top 25 are listed. However, according to Peter Mucha s ranking system, the top 14 teams as follows: (1) Alabama, (2) Oregon, (3) Florida State, (4) Mississippi State, (5) Mississippi, (6) TCU, (7) UCLA, (8) Georgia, (9) Auburn, (10) Ohio State, (11) Baylor, (12) Arizona, (13) Missouri, and (14) Arizona. If we compare these two lists, we have the following teams in the same spot: Alabama (1), Oregon (2), Florida State (3), and Mississippi State (4). Note, that these are the first four teams, which indicates the accuracy of random walking in determining the current national champion rankings. When comparing the distance between random walker data points and CFP data points there is a maximum difference of 7 for Missouri which is ranked 20 th in the BCS rankings and 13 th in the random walker rankings. However, the average distance between data points in the two polls is 2.29. With respect to these recent changes, it would seem that random walker rankings are no longer as significant. This is because they are the most accurate across the first few spaces of the rankings. After this, with the maximum distance of seven points in the most recent week of rankings, the rankings are no longer as accurate. Because playoffs would include more than a couple of teams, the rankings would need to be more accurate, even as the distance from the first few spots increased. For this reason, it is mathematically
Brown 11 important that the playoff committee not consider random walker rankings, even though these rankings could have been substitutes for the BCS rankings.
Brown 12 Works Cited Callaghan, Thomas, Peter J. Mucha, and Mason A. Porter. "The Bowl Championship Series: A Mathematical Review." Notices of the AMS 51.8 (2004): 887-93. Web. 19 Oct. 2014. "College Football Playoff." College Football Playoff. N.p., n.d. Web. 18 Oct. 2014. "The History of Football in America." Understanding American Football.com. N.p., n.d. Web. 17 Oct. 2014. <http://www.understanding-american-football.com/history_of_football.html>. Mucha, Peter J. "How Well Can Monkeys Rank Football Teams?" Web log post. Random Walker Rankings. N.p., Dec. 2003. Web. 18 Oct. 2014. Stern, Hal S. "Statistics and the College Football Championship." The American Statistician 58.3 (2004): 179-85. Web. 19 Sept. 2014. Teasley, C.E. Wynn, and Martin Hornyak. "BCS or Just BS: How College Football Could Crown The Wrong National Champion? Just Do The Math -- Correctly!" American Journal of Business Education 3.3 (2010): 1-7. Web. 17 Oct. 2014. Palumbo, Lou. "2013 Current BCS Rankings at of Week 15." Bankroll Sports Picks Blog RSS. Bankroll Sports, n.d. Web. 19 Nov. 2014.