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 |

Suppose T teams play each other. Let 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 th team to be given by

where denotes the total number of games played by team and is the rating vector (where denotes the rating of team ).

One possible preference matrix the matrix of total scores obtained from the pre-tournament table below:

(In this case, so we ignore the factor.)

In his paper, Keener proposed a ranking method where the ranking vector is proportional to its score. The score is expressed as a matrix product , where is a square preference matrix. In other words, there is a constant such that , for each . This is the same as saying .

The Frobenius-Perron theorem implies that has an eigenvector having positive entries associated to the largest eigenvalue $\lambda_{max}$ of , which has (geometric) multiplicity . Indeed, has maximum eigenvalue , of multiplicity , with eigenvector

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 .