Standings in sports competitions using integer programming

Standings in sports are obtained by applying a system of rules to evaluate the performance of the participants in a competition. We consider standings that result from assigning an ordinal rank to each competitor according to their performance. We develop an integer programming model for standings that allows us to calculate the number of points needed to guarantee a team the ith position, as well as the minimum number of points that could yield the ith place. The model is very general and can thus be adapted to many types of sports. We discuss examples coming from football (soccer), ice hockey, and Formula 1. We answer various questions and debunk a few myths along the way. Are 40 points enough to avoid relegation in the German Bundesliga? Do 95 points guarantee the participation of a team in the NHL playoffs? Moreover, in the season restructuration that was under consideration in November 2012, would it be easier or harder to access the playoffs? Is it possible to win the Formula 1 World Championship without winning at least one race or without even climbing once on the podium? Finally, we observe that the optimal solutions of the aforementioned model are associated to extreme situations which are unlikely to happen. Thus, to get closer to realistic scenarios, we enhance the model by adding some constraints inferred from the results of the previous years.