A linear model for a ranking problem

We continue with the application of the linear model described in [2016WP22] to the Italian soccer championship. We consider the simulation taking the data from the final results of the 2016-2017 championship. We intend first to see if the model gives rise again to some discrepancies with the official final results and we want to study the reasons for that. 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. In this working paper we deal with a so called even paired competition, where the pairings are evenly matched: in a national soccer championship actually each team is paired with every other team the same number of times. A mathematically based ranking scheme can be easily defined in order to get the scores for all the teams. The underlined structure of the model depends on the existence and uniqueness of a particular eigenvalue of the preference matrix. At this point the Perron – Frobenius theorem is involved. In the previous working paper [2016WP22] we showed how in the ranking process some fundamental Linear Algebra concepts and results are important, mainly the dominant eigenvalue and a corresponding eigenvector. The linear ranking model was also applied to a first numerical simulation. This gave evidence of some discrepancies between the actual final placements of teams and the ones provided by the model. We want to go here into a more detailed study about this aspect.