Note on Computing Ratings from Eigenvectors

TL;DR

This paper presents a method to compute player ratings from game results by reducing the problem to eigenvector calculations, linking it to Markov chains, and proposing an efficient power method algorithm.

Contribution

It introduces a novel eigenvector-based approach for rating computation and develops an efficient power method algorithm for large sparse matrices.

Findings

The eigenvector approach accurately reflects player ratings.

The power method efficiently computes ratings for large datasets.

The method connects rating computation with Markov chain stationary distributions.

Abstract

We consider the problem of computing ratings using the results of games played between a set of n players, and show how this problem can be reduced to computing the positive eigenvectors corresponding to the dominant eigenvalues of certain n by n matrices. There is a close connection with the stationary probability distributions of certain Markov chains. In practice, if n is large, then the matrices involved will be sparse, and the power method may be used to solve the eigenvalue problems efficiently. We give an algorithm based on the power method, and also derive the same algorithm by an independent method.