This assignment concerns matrix algebra, linear systems, and least-squares technique application. One problem that has perplexed folks for quite some time is how to rank things, especially sports teams. In American sports, this is a big issue when it comes to college football.
At the highest level, there are 130 teams that are competing for a National Championship. There is no reasonable way for all of the teams to play each other. So, ranking the teams is particularly tricky.
In 1997, Kenneth Massey proposed a method of ranking in an undergraduate honors project. His idea was very simple: when team A wins against team B by a number of points p, this determines an equation rA – rB = p, where rA and rB are the rating values of the two teams. Any set of games and teams will develop a system of equations. However, the system will not have a solution – when team A and B play again, there will probably be a different result. Massey used the Least-Squares Method to develop a set of solutions that were the closest match to the original system. Since there was no unique solution, he imposed the condition that all of the ratings must add up to zero. In this solution, the team with the most positive rating was ranked first, and the team with the most negative rating was ranked last. In 1999, when Massey was a graduate student at Virginia Tech, his method was used as part of the formula for declaring a National Champion in college football. The method was included until 2013 when it was replaced by the current playoff system.
One criticism of this method is that it uses the point difference p to help rank the teams. One could argue that there is no real difference in a team winning by 50 points and a team winning by 40 points, but there is a real difference between a team winning by 13 points and a team winning by 3 points. In 2002, Wesley Colley proposed a method that did not depend on point differences at all. The mathematics behind the creation of the method is a bit more involved than Massey’s Method, but it still involves solving a system of equations. We would not say that Colley’s Method is better or worse than Massey’s Method, but that it is different.
One thing to note about Massey’s Method is that it was designed to be predictive, i.e. if Team A were to play Team F, then the outcome would be a certain way. Colley’s Method was not designed to be predictive. In fact, Massey’s Method can weight certain games more than others to take into account special circumstances – star player out, early-season game, win on the road. Mathematics majors at Davidson College use weighted Massey’s Method to make very good predictions for the NCAA Basketball Tournament.
Use the games below to rank the teams.
1) In the Atlantic Coast Conference, there are four colleges in North Carolina and they are usually the ones that are included when referring to a “mythical State Championship”. However, I have listed all of the Football Bowl Subdivision teams in North Carolina and their games against each other from a hypothesized set.
a. Write the system of equations – one equation for each game.
b. Compute the Massey matrix, M, and the vector of point differences, d.
c. Impose the condition that the sum of all the ratings is zero and solve the system.
d. Interpret the solution from part c.
e. Write the Colley matrix C, and vector b.
f. Solve the system.
g. Interpret the solution from part f.
h. Since it is harder for a team to win on the road, increase the weight on games in which the away team won by 30% (i.e. weight = 1.3) and use the Massey Method to rank the teams. Make sure to interpret the numerical solution.
At the highest level, there are 130 teams that are competing for a National Championship. There is no reasonable way for all of the teams to play each other. So, ranking the teams is particularly tricky.
In 1997, Kenneth Massey proposed a method of ranking in an undergraduate honors project. His idea was very simple: when team A wins against team B by a number of points p, this determines an equation rA – rB = p, where rA and rB are the rating values of the two teams. Any set of games and teams will develop a system of equations. However, the system will not have a solution – when team A and B play again, there will probably be a different result. Massey used the Least-Squares Method to develop a set of solutions that were the closest match to the original system. Since there was no unique solution, he imposed the condition that all of the ratings must add up to zero. In this solution, the team with the most positive rating was ranked first, and the team with the most negative rating was ranked last. In 1999, when Massey was a graduate student at Virginia Tech, his method was used as part of the formula for declaring a National Champion in college football. The method was included until 2013 when it was replaced by the current playoff system.
One criticism of this method is that it uses the point difference p to help rank the teams. One could argue that there is no real difference in a team winning by 50 points and a team winning by 40 points, but there is a real difference between a team winning by 13 points and a team winning by 3 points. In 2002, Wesley Colley proposed a method that did not depend on point differences at all. The mathematics behind the creation of the method is a bit more involved than Massey’s Method, but it still involves solving a system of equations. We would not say that Colley’s Method is better or worse than Massey’s Method, but that it is different.
One thing to note about Massey’s Method is that it was designed to be predictive, i.e. if Team A were to play Team F, then the outcome would be a certain way. Colley’s Method was not designed to be predictive. In fact, Massey’s Method can weight certain games more than others to take into account special circumstances – star player out, early-season game, win on the road. Mathematics majors at Davidson College use weighted Massey’s Method to make very good predictions for the NCAA Basketball Tournament.
Use the games below to rank the teams.
1) In the Atlantic Coast Conference, there are four colleges in North Carolina and they are usually the ones that are included when referring to a “mythical State Championship”. However, I have listed all of the Football Bowl Subdivision teams in North Carolina and their games against each other from a hypothesized set.
| Date | Home team | Away Team | ||
| Aug. 31 | NC State | 42 | East Carolina | 14 |
| Sept. 7 | App. State | 33 | Charlotte | 21 |
| Sept. 13 | Wake Forest | 17 | UNC | 14 |
| Sept. 21 | UNC | 28 | App. State | 30 |
| Oct. 26 | UNC | 20 | Duke | 14 |
| Nov. 2 | Wake Forest | 13 | NC State | 7 |
| Nov. 23 | Wake Forest | 28 | Duke | 27 |
| Nov. 30 | NC State | 17 | UNC | 15 |
b. Compute the Massey matrix, M, and the vector of point differences, d.
c. Impose the condition that the sum of all the ratings is zero and solve the system.
d. Interpret the solution from part c.
e. Write the Colley matrix C, and vector b.
f. Solve the system.
g. Interpret the solution from part f.
h. Since it is harder for a team to win on the road, increase the weight on games in which the away team won by 30% (i.e. weight = 1.3) and use the Massey Method to rank the teams. Make sure to interpret the numerical solution.