Ranking Theory with Application to Popular Sports.

Abstract

The rank of an object is its relative importance to the other objects in a finite set of size n. Often a rank is an integer assigned from the set { 1,2,...,n}. Ideally an assignment of available ranks ({1,2,...,n}) to n objects is one-to-one. However in certain circumstances it is possible that more than one object is assigned the same rank. A ranking model is a method of determining a way in which the ranks are assigned. Typically a ranking model uses information available to determine a rating for each object. The ratings carry more information than the ranks; they provide us with the degree of relative importance of each object. Once we have the ratings the assignment of ranks can be as simple as sorting the objects in descending order of the corresponding ratings. Ranking models can be used for a number of applications such as sports, web search, literature search, etc. The type of ranking investigated in this work has close ties with the Method of Paired Comparison. Oftentimes the information that is the easiest to obtain or naturally available is the relative preference of the objects taken two at a time. The information is then summarized in a weighted directed graph and hence as the corresponding matrix. A number of ranking models makes use of the matrix representation of paired comparisons to compute ratings of the individual objects. Two ranking models proposed and investigated in this work start with forming nonegative matrices that do represent certain pairwise type comparisons. The models have different approaches to computing the rating scores. The Offense-Defense Model makes use of the Sinkhorn-Knopp Theorem on equivalence matrix balancing, whereas the Generalized Markov model is based on Markov Chain theory. Both models are then used to compute the ratings of the National Football League teams, National Collegiate Athletic Association football and basketball teams. The ratings are used to perform game predictions. The proposed models are not specific to sports and can be applied to any situation consisting of a set of objects and a set of pairwise information. However, picking team sports as a ranking application allows for unrestricted access of free and abundant data. The game predictions experiments consisted of both foresight and hindsight predictions. All the experiments included the proposed models as well as several current sports ranking methods.