Massey Method. Introduction. The Process

Transcription

1 Massey Method Introduction Massey s Method, also referred to as the Point Spread Method, is a rating method created by mathematics professor Kenneth Massey. It is currently used to determine which teams in the NCAA college football league qualify to play in each of the Bowl Championship Series games. This method is mainly based off the theory of least squares and it incorporates not only how many total games played by each team, but also takes into consideration how many games a certain team plays another. For teams that have played against each other it then also incorporates the difference in points scored between the two. The Process The math that the Massey Method is driven by is that of the least squares solution to a system of equations. The least squares method can be described in summary by the following equation: r i r j = y k where y k is the margin of victory for game k and r i and r j are the rating of the teams i and j respectively. The difference in the ratings is used to predict the margin of victory between the two. For example, if team i had a rating of 20 and team j had a rating of 5, one would expect team i to win by a margin of 5. However, since we do not know the ranking of each team before hand Massey tweaks this method so that one can employ it to discover the rating of a team. The ratings of the teams are unknown, but we do know who played who and what the margin of victory was. Thus, Massey employed an equation of the same form of the least squares method, that creates a system of m linear equation and n unknowns. This equation is written as: Xr = y where the matrix X is a m x n matrix, m being the number of games played and n being the number of teams. This matrix consists of nearly all zeros except for a in location i and a - in location j, meaning that team i beat team j that game. Vector y contains the margin for victory, and vector r contains the unknown ratings. However, Massey noticed

2 that this system is highly overdetermined and in order to solve the system of equations he multiplied each side by the transpose of X, therefore normalizing the system. He then rewrote the equations as: Mr = p where M=X(X T ) and p=y(x T ) Another problem that Massey encountered, is that after a few games have been played, the matrix M has a rank n?, which means that there are infinite solutions to the system. We need a single solution so we can obtain the rating of each team. To fix this problem, Massey employed the method of replacing the last row of the matrix to a row of all ones and the corresponding entry in the point differential vector to a zero, so that the rank is no longer less than n, allowing the system to now be solved. An Example We are now going to apply the Massey ranking method in an example. The following table lists the scores of games from an imaginary season involving teams A, B, C, D, and E. The Win or Loss in each box of the table refers to the outcome Teams Team A Team B Team C Team D Team E Team A - Win 6-2 Loss 4-57 Win 3-2 Win 27-2 Team B Loss Loss 30-3 Win 36-2 Loss 35-0 Team C Win 57-4 Win Loss -29 Loss 4-37 Team D Loss 2-3 Loss 2-36 Win Loss 0-39 Team E Loss 2-27 Win 35-0 Win 37-4 Win of the game relative to the team listed in the corresponding row, where the team listed in the corresponding column is their opponent. We can see that each team played every other team once, for a total of four games played by every team. We 2

3 first want to create the matrix X. For our example X looks like so: X = Team A Team B Team C Team D Team E The first row, for example, corresponds to the game played between Team A and Team B. Team A defeated Team B, thus the column for Team A has a and the column for Team B has a -. Next we must create our matrix y, which is as follows: y = Making Xr = y 3

4 r A r B * r C r D = r E Here, again, the game represented by the first row is that between Team A and Team B, where A defeated B. The number in the first row of the vector y is the point difference of this game. A defeated B with a score of 6 to 2, as shown in the table above, thus the point differential is 4. Now the next step would be multiplying each side by the transpose of X so we get our final equation of Mr = p where M=X(X T ) and p=y(x T ) The matrix M becomes an n x n matrix with the total number of games played by each team along the diagonal; that is, for team i, entry M i i is the total number of games played by team i. The entry M i j for i?= j is equal to the negation of the number of games that teams i and j played against each other. We see that the rows and columns of this matrix sum to zero, so they are linearly dependent. The matrix p becomes a n x matrix of simply the total point differential for each team. We can calculate the point differential between two teams that played a game against each other by subtracting one team?s score from their opponent?s score. For Massey?s Method, we will want to calculate the total point differential for each team; that is, for each team, we want to sum the point differences of every game that team played. Let us start by calculating the total point differential for each team. For example, we begin with Team A: they beat Team B, Team D, and Team E, only losing to Team C. We express this in the following system of equations: Team A - Team B = 6-2 = 4 Team A - Team C = 4-57 = -6 Team A - Team D = 3-2 = Team A - Team E = 27-2 = 6 If we add these individual point differences, we get the total point differential for Gettysburg for this series of games, and find it is -5. Doing these calculations for 4

5 the other teams, we find that Team B has a total point differential of -29, Team C of -0, Team D of -37, and Team E of. Therefore, we see matrix M become: M = Team A Team B Team C Team D Team E Team A 4 Team B 4 Team C 4. Team D 4 Team E 4 and matrix p becomes: Now employing the last step that Massey calls for in order to allow this system of equations to be solved, we must replace the last row of the matrix M to a row of all ones and the corresponding entry in the point differential vector to a zero, so that the rank is no longer less than n. Doing so we obtain a final equation of 4 r A 5 4 r B * r C r D r E = 37 0 Solving this you get the final ranking to be that Team E is first, then Team A, Team C, Team B, and Team D is last. Advantages and Disadvantages One advantage of this method is that by multiplying matrix X by the transpose we are able to obtain M, which is a square matrix. This allows for analysis to be much simpler. It is a straightforward method that only requires the use of the number of games played and the scores of each game. Also, this method does not require that every team plays the same number of games, as other methods often do. Another advantage is that because it only utilizes the point differential, and 5

6 not the fact if a team won or lose, ties are automatically accounted for. A disadvantage of this is method is that the matrix has to be altered in order for one to see a solution. Although, this is still a fairly simple fix. Potential Improvements One potential improvement that can be made is to create a cap for the point differential. If one team beats another by a significant amount, the winning team?s victory margin is very large, therefore perhaps causing their rating to increase drastically, and be less accurate. One way to prevent this increase if they score much higher than their opponent is to cap the point differential. If the cap is set to a certain number, any game in which the winning team beats the opponent by a margin greater than the cap, it will only be counted as a win by whatever has been chosen for the cap. Conclusion Massey s method of ranking is based upon the mathematical idea of the least squares method. It takes into consideration only the point differential between two teams, and the number of games they have played. It is a simple method that is easy to compute, and only calls for one simple alteration to be made. Additional Readings For further information on this method feel free to read the book Who s #? by Amy N. Langville, along with Kevin Massey s dissertation. Written by: Sabrina Marell Edited by: Alex Grun 6