The Massey Ranking Algorithm

Today we will see an application of least squares that is not finding a best fitting curve!

Ranking is a simple idea: put a list of things in order of importance. A good deal of money and effort is put into ranking:

Sports teams
Internet pages (how does a serch engine know which of a million pages about linear algebra it should show us first?)
Business (what things should we prioritize in our marketing?)
Universities (US News and World Report rankings)
Video games (I'm Bronze in Overwatch)

In this lesson, we'll learn about a widely used algorithm to rank sports teams.

Ranking Sports Teams: Inconsistent

Ranking sports teams is a difficult problem because it usally involves inconsistent data. Consider this small example from the 2025 college football season:

Wake beat NC State 34-24
NC State beat VaTech 32-21
VaTech beat Wake 30-23

Clearly, Wake is better than NC State, and NS State is better than VaTech, and VaTech is better than Wake...this is inconsistent! Now imagine a season with over 800 games! It's very overdetermined.

Massey Ranking Algorithm

In the 1990's, Ken Massey, then a graduate student in mathematics and Virginia Tech, developed a ranking system based on least squares solutions.

The issues described above is that the data are inconsistent. Remember, this is precisely what a least squares approach is for! Since there is no solution when there are so many equations (hundreds of games) and so few unknowns (number of teams), we will attempt to fit the sports data with a "best fitting" solution.

If we want to take a least squares approach to a football season, we need a way to turn games into equations. Here's what Massey proposed:

If Wake beats NCState 34-24, we write $\large r_W-r_{NCState}=10$.
Here, the unknowns are the rankings: $\large r_W$ for Wake, and $\large r_{NCState}$ for NC State.
The 10 in the equation is the margin of victory for the game.
Essentially, we can read this equation as "If Wake beat NC State 34-24, then Wake is 10 points better than NC State".

We will have one equation like the one above for every game played in the season!

Running Scenario: 2024 SEC football

As we develop this method, it will be helpful to keep a running example. We'll use a set of games from the 2024 SEC college football season. There were 6 teams that all finished with a 5-3 record (that's 5 wins and 3 losses) in the conference that season. How do we rank them if they had the same record? Here's the information we'll use:

Team	Bama	LSU	USC	A&M	Ole Miss	Mizzou
Bama	-	42-13	27-25	-	-	34-0
LSU	13-42	-	36-33	23-38	29-26	-
USC	25-27	33-36	-	44-20	3-27	34-30
A&M	-	38-23	20-44	-	-	41-10
Ole Miss	-	26-29	27-3	-	-	-
Mizzou	0-34	-	30-34	10-41	-	-

In this small case, we have $\large n=6$ teams and $\large m=10$ games. Our overdetermined system of equations is:

$$\large \begin{array}{cccccccccccr} r_1 & - & r_2 & & & & & & & & & = & 29 \\ r_1 & & & - & r_3 & & & & & & & = & 2 \\ r_1 & & & & & & & & & - & r_6 & = & 34 \\ & & r_2 & - & r_3 & & & & & & & = & 3 \\ & & r_2 & & & - & r_4 & & & & & = & -15\\ & & r_2 & & & & & - & r_5 & & & = & 3 \\ & & & & r_3 & - & r_4 & & & & & = & 24 \\ & & & & r_3 & & & - & r_5 & & & = & -24 \\ & & & & r_3 & & & & & - & r_6 & = & 4 \\ & & & & & & r_4 & & & - & r_6 & = & 11 \\ \end{array} $$

If we write this as $\large A\vec{x}=\vec{b}$, we have:

$$\large \begin{bmatrix} 1 & -1 & 0 & 0 & 0 & 0\\ 1 & 0 & -1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & -1\\ 0 & 1 & -1 & 0 & 0 & 0\\ 0 & 1 & 0 & -1 & 0 & 0\\ 0 & 1 & 0 & 0 & -1 & 0\\ 0 & 0 & 1 & -1 & 0 & 0\\ 0 & 0 & 1 & 0 & -1 & 0\\ 0 & 0 & 1 & 0 & 0 & -1\\ 0 & 0 & 0 & 1 & 0 & -1\\ \end{bmatrix}\begin{bmatrix} r_1\\ r_2\\ r_3\\ r_4\\ r_5\\ r_6\\ \end{bmatrix}=\begin{bmatrix} 29\\ 2\\ 34\\ 3\\ -15\\ 3\\ 24\\ -24\\ 4\\ 11\\ \end{bmatrix} $$