How the Massey Method Ranks College Football Teams Using Linear Algebra

The Bowl Championship Series (BCS) is a scoring system for NCAA college football that combines human polls and computer models to decide which teams play in which bowl games, often sparking controversy.

Original Massey Rating Method

In 1997, Kenneth Massey, then an undergraduate at Bluefield College, introduced a ranking method based on the least‑squares principle. He later became a mathematics professor and refined his sports‑ranking models, one of which is used by the BCS.

Massey's Main Idea

The core concept is an idealized equation where the difference between two teams' ratings predicts the winning margin in a game. For each game, this yields a linear equation, forming a large, sparse system with many unknown team ratings.

Because the system is highly over‑determined and inconsistent, a least‑squares solution is obtained via the normal equations, providing the best linear unbiased estimate of the rating vector.

Applying the Massey Method to an Example

Consider five teams that each play every other team once. The resulting matrix has off‑diagonal entries of –1 and diagonal entries equal to the number of games played (4). To ensure a unique solution, Massey adds the constraint that the sum of all ratings equals zero.

The solved system yields a ranking and rating list for the five teams.

Advanced Features of the Massey Method

Beyond the basic ratings, Massey derives two additional vectors: the total points scored (offensive rating) and total points allowed (defensive rating). By decomposing the Massey matrix into a diagonal matrix (games played) and an off‑diagonal matrix (head‑to‑head matchups), these vectors are extracted through algebraic manipulation.

The final equations separate a team's total season points into contributions from its offensive rating multiplied by games played, minus the sum of opponents' defensive ratings.

Summary of Symbols Used

n: number of teams in the league.

m: total number of games played.

A: game‑outcome matrix (A_ij = 1 if team i beats j, –1 if i loses to j, 0 otherwise).

D: diagonal matrix containing each team’s total games played.

G: off‑diagonal matrix recording head‑to‑head match counts.

M: the symmetric M‑matrix (Massey matrix) used in the least‑squares system.

r: rating vector (unknown team ratings).

p: vector of cumulative point differentials for each team.

s: total points scored vector (offensive scores).

a: total points allowed vector (defensive scores).

d: adjusted point‑differential vector (with one element set to zero for full rank).