# Sports ranking methods, 2

This is the second of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses Keener’s method (see J.P. Keener, The Perron-Frobenius theorem and the ranking of football, SIAM Review 35 (1993)80-93 for details).

See the first post in the series for a discussion of the data we’re using to explain this method. We recall the table of results:

 X\Y Army Bucknell Holy Cross Lafayette Lehigh Navy Army x 14-16 14-13 14-24 10-12 8-19 Bucknell 16-14 x 27-30 18-16 23-20 10-22 Holy Cross 13-14 30-27 x 19-15 17-13 9-16 Lafayette 24-14 16-18 15-19 x 12-23 17-39 Lehigh 12-10 20-23 13-17 23-12 x 12-18 Navy 19-8 22-10 16-9 39-17 18-12 x Win-loss digraph of the Patriot league mens baseball from 2015

Suppose T teams play each other. Let $A=(a_{ij})_{1\leq i,j\leq T}$ be a non-negative square matrix determined by the results of their games, called the preference matrix. In his 1993 paper, Keener defined the score of the $i$th team to be given by $s_i = \frac{1}{n_i}\sum_{j=1}^T a_{ij}r_j,$

where $n_i$ denotes the total number of games played by team $i$ and ${\bf r}=(r_1,r_2,\dots ,r_T)$ is the rating vector (where $r_i\geq 0$ denotes the rating of team $i$).

One possible preference matrix the matrix $A$ of total scores obtained from the pre-tournament table below: $A = \left(\begin{array}{rrrrrr} 0 & 14 & 14 & 14 & 10 & 8 \\ 16 & 0 & 27 & 18 & 23 & 28 \\ 13 & 30 & 0 & 19 & 27 & 43 \\ 24 & 16 & 15 & 0 & 12 & 17 \\ 12 & 20 & 43 & 23 & 0 & 12 \\ 19 & 42 & 30 & 39 & 18 & 0 \end{array}\right),$

(In this case, $n_i=4$ so we ignore the $1/n_i$ factor.)

In his paper, Keener proposed a ranking method where the ranking vector ${\bf r}$ is proportional to its score. The score is expressed as a matrix product $A{\bf r}$, where $A$ is a square preference matrix. In other words, there is a constant $\rho>0$ such that $s_i=\rho r_i$, for each $i$. This is the same as saying $A {\bf r} = \rho {\bf r}$.

The Frobenius-Perron theorem implies that $S$ has an eigenvector ${\bf r}=(r_1,r_2,r_3,r_4,r_5,r_6)$ having positive entries associated to the largest eigenvalue $\lambda_{max}$ of $A$, which has (geometric) multiplicity $1$. Indeed, $A$ has maximum eigenvalue $\lambda_{max}= 110.0385...$, of multiplicity $1$, with eigenvector ${\bf r}=(1, 1.8313\dots , 2.1548\dots , 1.3177\dots , 1.8015\dots , 2.2208\dots ).$

Therefore the teams, according to Kenner’s method, are ranked,

Army $<$ Lafayette $<$ Lehigh $<$ Bucknell $<$ Holy Cross $<$ Navy.

This gives a prediction failure rate of just $6.7\%$.