How the Keener Method Quantifies Team Strength Using Eigenvectors

Keener Method

James P. Keener introduced his rating method in a 1993 SIAM Review paper, using non‑negative statistics from competitions to generate numerical scores for each team, which can then be ordered into rankings.

Rules for Strength and Rating

A team’s strength should be judged based on its interactions with opponents and the opponents’ strengths.

In a given league, a team’s rating should be directly proportional to its strength; that is, there exists a constant \(k\) such that for every team \(i\), \(r_i = k s_i\).

Choosing Attributes that Reflect Strength

To turn the two rules into a computable mechanism, we select measurable attributes (e.g., number of wins, draws, points scored, cumulative rushing yards, passing yards) that allow relative comparisons of team strength.

These attributes must be updated throughout the season, and a decision must be made whether early‑season values should be weighted equally with later‑season values.

Laplace’s Continuation Rule

Raw statistics often need smoothing; for example, using total scores instead of raw points prevents teams with high offensive output but weak defense from being over‑rated. A Laplace‑type smoothing adjusts extreme values to avoid unrealistic probabilities.

Preference Functions

To mitigate excessive gaps between strong and weak teams, a non‑linear preference function can be applied, reshaping the relationship between raw statistics and ratings. The function’s shape can be customized to increase or decrease its effect on different parts of the rating scale.

Normalization

If teams play different numbers of matches, a normalization step adjusts scores to account for unequal game counts, preventing teams with more games from gaining an unfair advantage.

Matrix Formulation

After defining attributes, applying preference and normalization, the data can be arranged in a matrix \(A\) where each entry reflects the interaction between two teams. The rating vector \(r\) becomes an eigenvector of \(A\), and the proportionality constant \(k\) is the corresponding eigenvalue.

Perron‑Frobenius Theorem

For a non‑negative, irreducible matrix \(A\), the Perron‑Frobenius theorem guarantees a unique positive eigenvalue (the Perron value) and a unique positive eigenvector (the Perron vector), which serve as the constant \(k\) and the rating vector respectively.

Practical Issues

General matrices may yield many eigenvalues, requiring a choice of the appropriate one.

Eigenvalues can be complex or negative, which are unsuitable for rating purposes.

Even real positive eigenvalues may correspond to eigenvectors containing negative components.

Computing eigenvalues/eigenvectors can be computationally intensive and may require expensive software packages.

These challenges highlight the need for careful implementation when applying the Keener method to real‑world ranking problems.

Source : "Who Ranks First? The Science of Evaluation and Ranking" by R. Lanville and M. Meyer, Mechanical Industry Press, 2014.