Rank Aggregation under Weak Stochastic Transitivity via a Maximum Score Estimator

2025-10-08

AI Summary

Existing pairwise comparison ranking models—such as Thurstone’s and Bradley–Terry—rely on the strong stochastic transitivity (SST) assumption, whose stringent monotonicity constraint often fails under real-world sparse and non-uniformly missing comparison data. To address this limitation, we propose the first maximum-score estimation method that requires only weak stochastic transitivity (WST), the minimal assumption sufficient for global rankability. Our method minimizes Kendall’s tau distance as the loss function and is theoretically shown to be consistent with near-minimax optimal convergence rate. Extensive experiments demonstrate that our approach significantly outperforms SST-based models on both synthetic benchmarks and a real-world professional tennis player ranking task—particularly in settings characterized by sparse and non-uniform pairwise comparisons. This work bridges a critical gap between theoretical assumptions and practical data conditions in preference learning and ranking inference.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Preferences
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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.

Problem

• Aggregating ranks under weak stochastic transitivity assumption
• Handling sparse pairwise comparisons with nonuniform missingness
• Establishing consistent and minimax optimal rank estimators

Innovation

• Maximum score estimator for rank aggregation
• Uses weak stochastic transitivity assumption
• Handles sparse pairwise comparison data