The algebraic approach to some ranking problems

The problem of ranking a set of elements, namely giving a “rank” to the elements of the set, may arise in very different contexts and may be handled in some possible different ways, depending on the ways these elements are set in competition the ones against the others. For example there are contexts in which we deal with an even paired competition, in the sense the pairings are evenly matched: if we think for example of a national soccer championship, each team is paired with every other team the same number of times. Sometimes we may deal with an uneven paired competition: think for example of the UEFA Champions League, in which the pairings are not fully covered, but just some pairings are set, by means of a random selection process for example. Mathematically based ranking schemes can be used and may show interesting connections between the ranking problems and classical theoretical results. In this working paper we first show how a linear scheme in the ranking process directly takes to some fundamental Linear Algebra concepts and results, mainly the eigenvalues and eigenvectors of linear transformations and Perron–Frobenius theorem. We apply also the linear ranking model to a numerical simulation taking the data from the Italian soccer championship 2015-2016. We finally point out some interesting differences in the final ranking by comparing the actual placements of the teams at the end of the contest with the mathematical scores provided to teams by the theoretical model.