Note on Computing Ratings from Eigenvectors

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.