If the Warriors beat the Rockets and the Rockets beat the Spurs, does that mean that the Warriors are better than the Spurs? Sophisticated fans would argue that the Warriors are better by the transitive property, but could diehard Spurs fans make a legitimate argument that their team is better despite this chain of evidence? We attempt to answer whether the assumption of transitivity in pairwise comparisons should hold by modeling without the assumption.
Applications of pairwise comparisons reach far beyond sports. Examples include ranking political candidates, selecting a Pope, experimentally predicting animal behavior, and exploring dominance relations within or among species. We focus on the setting where all pairs of items, teams, players, or objects have been compared to one another twice.
We propose a novel linear model (CRSP) whose latent bilinear fixed effect allows us to estimate deviations from our transitive model (C). We discuss the mathematical details of the model, including an eigendecomposition that enables an estimate of the latent bilinear fixed effect. We also propose a generalized version of the bilinear fixed effect and demonstrate an application thereof.