Probabilistic transitivity in sports

Abstract

We seek to find the statistical model that most accurately describes empirically observed results in sports. The idea of a transitive relation concerning the team strengths is implemented by imposing a set of constraints on the outcome probabilities. We theoretically investigate the resulting optimization problem and draw comparisons to similar problems from the existing literature including the linear ordering problem and the isotonic regression problem. Our optimization problem turns out to be very complicated to solve. We propose a branch and bound algorithm for an exact solution and for larger sets of teams a heuristic method for quickly finding a "good" solution. Finally we apply the described methods to panel data from soccer, American football and tennis and also use our framework to compare the performance of empirically applied ranking schemes.1 kinds of data sets can arise. An experiment in a league where each team plays each other team a fixed number of times will be called complete. That this property is not naturally given can be seen for instance in American college football, where each team of the 119 teams (in division 1) plays only a small fraction of their competitors.An important attribute of a ranking is that it expresses a transitive relation between all of its objects. This means that if object or team A precedes B and B precedes C, it automatically implies that A precedes C. In contrast to this, paired comparison data can include circular relations, which seem to be inconsistent with this property. In a tournament it is possible that A beats B, B beats C, but C beats A. It is easy to imagine that as the number of teams rises, the probability of the occurrence of such inconsistencies rapidly increases. Especially in the not very recent literature many suggestions have been made to overcome these inconsistencies and find a ranking with a good fit according to different concepts. A good overview of the classical models for obtaining rankings from data sets gives Brunk [1960b]. One approach that deserves attention is the one proposed by Slater [1961]. Here the observed number of inconsistencies (in the sense mentioned above) is minimized. This nontrivial problem later became known as a particular form of the so called linear ordering problem. For a good survey on the linear ordering problem see for example Charon and Hudry [2010].The major issue concerning the mentioned approaches is that despite all of them having some intuitive appeal, they seem to be rather arbitrary in finding the "right" ranking. The difference of our approach is that we assume that there actually exists a correct ranking. Of course we cannot directly observe it, but we can try to find the ranking which is most likely identical to it. To be more precise, we first of all make the assumption that the outcome of each match follows a trinomial distribution, with a fixed probability for a loss, a tie, and a win. These unobservable probabilities fulfill a certain form of transitivity. Applying the respective conditions we can th...