Ranking Teams in American Football through Total Enumeration

Abstract

The American National Football League (NFL) stands as a significant sporting event with profound economic and social implications. Due to the physically demanding nature of the sport and the associated risks to player health, a full round-robin play format is not feasible within the league. Therefore, achieving fair and merit-based rankings of teams is imperative. This paper proposes a total enumeration approach as a precise solution to the ranking formulation put forth by Cassady et al., despite the NP-hard nature of the quadratic assignment problem inherent in the mathematical model. Total enumeration not only furnishes an exact solution for leagues with a small number of teams but also serves as a benchmark for evaluating the efficacy of heuristic methods for larger-scale problems. The implementation of the total enumeration algorithm in VBA facilitates performance analysis, with its efficiency assessed for a league comprising nine teams through statistical modeling using the JMP software package. Future research avenues include the incorporation of pruning strategies inspired by the quadratic assignment problem literature and the exploration of heuristic methods for practical league applications. Additionally, the development of a hypothetical, probabilistic game offers an experimental framework to assess team rankings, enhancing our understanding of competitive dynamics within the NFL.