## Giving the NFL a rating

Algebra covers solution techniques to solve linear systems like 2x – 3y = 5 and 3x + 2y = 1. Removing y, for instance, solving for x and then substituting the value for x to find y yields the solution x = 1 and y = -1.

Another way to represent the equations is as a matrix system, with its two unknowns x and y:

$left(begin{array}{rr} 2 & -3 \ 3 & 2 \ end{array} right) left( begin{array}{r} x \ y end{array} right) = left( begin{array}{r} 5 \ 1 end{array} right).$

The matrix notation serves as useful bookkeeping, which will help us keep track as we form a larger linear system below.

Time for some linear thinking as we rank sports teams with the Colley method, which is a math technique used by the Bowl Championship Series, the organization that determines which college football teams are invited to which bowl games. The Colley method was created by astrophysicist Wesley Colley who saw problems with ranking by winning percentage and developed this method.

This blog entry will bypass the derivation of the method, although interesting, in order to focus on the mechanics of forming the associated linear system. This will enable us to focus on the ease and accessibility of the method. Interested in the derivation? Then, see the essay, Bracketology: How can math help?, from Math Awareness Month 2010. The article uses the Colley method to create personalized brackets for March Madness. The article also derives another technique used by the BCS called the Massey method and gives techniques that adapt each method to model momentum in a season.

In this article, we will focus on a fictional series of games within the South Division of the NFL. We’ll represent the records of the teams by a graph. An arrow from team A to team B indicates that team A beat team B in a game. So, we see below that the Panthers beat both the Buccaneers and the Falcons but did not even play the Saints. If you are unfamiliar with the teams then note that in clockwise order starting in the upper lefthand corner, the teams are the Saints, the Buccaneers, the Panthers and the Falcons.

First, we will form a $4 times 4$ matrix necessary to rank the teams. We’ll represent this matrix as a table. Each row of the table and each column corresponds to one of the teams. We have written the team’s name that is associated with each row and column in the table below.

Saints

Buccaneers

Panthers

Falcons
Saints

Buccaneers

Panthers

Falcons

Now the question is what numbers to enter into the table.

For the entry in row 1 (Saints) and column 2 (Buccaneers), we put the number of times the two teams played multiplied by -1. Since the Saints and Buccaneers played once, we put -1. This is done in a similar way for all the entries except those on the diagonal. The entry in row 1 (Saints) and column 1 (Saints) equals the sum of 2 and the total number of games played by the Saints, which equals 4. Performing this task for all the teams creates the following table:

Saints

Buccaneers

Panthers

Falcons
Saints

4

-1

0

-1
Buccaneers

-1

5

-1

-1
Panthers

0

-1

4

-1
Falcons

-1

-1

-1

5

Now, we have only to form the vector for the righthand side of the linear equation. Each row in that vector corresponds to the same team that it did in the matrix. So, row 1 in the vector associates with the Saints. Let W equal the number of wins by the Saints and L equal the number of losses. Then in the first row of the vector, we place the number 1 + 1/2(W – L). The Saints had 1 win and 1 loss so we place a 1 in that entry in the vector. Continuing in this way forms the vector:

$left(begin{array}{r} 1 \ 1/2 \ 2 \ 1/2 end{array} right)$

In this way, we can express the linear system in matrix form as:

$left(begin{array}{rrrr} 4 & -1 & 0 & -1 \ -1 & 5 & -1 & -1 \ 0 & -1 & 4 & -1 \ -1 & -1 & -1 & 5 \ end{array}right) left(begin{array}{r} S \ B \ P \ F end{array} right) = left(begin{array}{r} 1 \ 1/2 \ 2 \ 1/2 end{array} right)$

While a rather modest system compared to what we would use if we rated all the NFL teams, this system is still rather big for solving by hand. Given you are reading this blog, I’ll assume you have online access and can use WolframAlpha. WolframAlpha enables one to solve such systems. Here is the WolframAlpha entry that ranks the South Division of the NFL!

We that the solution is:

$left(begin{array}{r} 0.458 \ 0.417 \ 0.708 \ 0.417 \ end{array}right)$

These are the ratings for the teams. The higher the rating, the better ranked the team. Therefore, the teams are ranked (from best to worst) Panthers, Saints and for a tie for third between the Buccaneers and Falcons.

This model would predict that the Panthers would easily beat any of the other three teams. Would that be the case? Should you book a flight to Vegas to play the odds? Keep in mind we have linear equations which means we are assuming life is linear. Just like the contour of the linemen, football is often not linear. A lot of variables beyond a win-loss record can play into a game. Perhaps one team plays better at home or a team struggles in the cold or….What then? Underneath this method is a model which can be altered but even so, it will be a model, which is intended to reflect but not exactly replicate life. So, give the NFL your best rating and still sit back and enjoy the games as you never know on any given Sunday what might happen!