Rank Aggregation under Weak Stochastic Transitivity via a Maximum Score Estimator

Stochastic transitivity is central for rank aggregation based on pairwise comparison data. The existing models, including the Thurstone, Bradley-Terry (BT), and nonparametric BT models, adopt a strong notion of stochastic transitivity, known as strong stochastic transitivity (SST). This assumption imposes restrictive monotonicity constraints on the pairwise comparison probabilities, which is often unrealistic for real-world applications. This paper introduces a maximum score estimator for aggregating ranks, which only requires the assumption of weak stochastic transitivity (WST), the weakest assumption needed for the existence of a global ranking. The proposed estimator allows for sparse settings where the comparisons between many pairs are missing with possibly nonuniform missingness probabilities. We show that the proposed estimator is consistent, in the sense that the proportion of discordant pairs converges to zero in probability as the number of players diverges. We also establish that the proposed estimator is nearly minimax optimal for the convergence of a loss function based on Kendall's tau distance. The power of the proposed method is shown via a simulation study and an application to rank professional tennis players.