Probabilistic Transitivity in Sports

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.