Pairwise Comparisons without Stochastic Transitivity: Model, Theory and Applications

2025-01-13

AI Summary

Conventional paired-comparison models (e.g., Bradley–Terry, Thurstone) rely on global linear ordering assumptions, limiting their predictive accuracy in real-world settings with multi-dimensional skills and strategic intransitivities—such as competitive gaming. Method: We propose the first general modeling framework based on (nearly) low-rank skew-symmetric matrices, explicitly abandoning stochastic transitivity and global ranking assumptions to capture rich, non-transitive preference structures. Our approach integrates spectral theory of skew-symmetric matrices, maximum likelihood estimation, and sparse low-dimensional modeling. Contribution/Results: Theoretically, our estimator achieves the minimax optimal convergence rate. Empirically, it significantly outperforms the classical Bradley–Terry model on both synthetic benchmarks and diverse real-world datasets—including esports and sports—while demonstrating superior statistical efficiency and robustness. This work establishes a new paradigm for modeling complex, strategic decision-making under intransitive preferences.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Preferences
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Most statistical models for pairwise comparisons, including the Bradley-Terry (BT) and Thurstone models and many extensions, make a relatively strong assumption of stochastic transitivity. This assumption imposes the existence of an unobserved global ranking among all the players/teams/items and monotone constraints on the comparison probabilities implied by the global ranking. However, the stochastic transitivity assumption does not hold in many real-world scenarios of pairwise comparisons, especially games involving multiple skills or strategies. As a result, models relying on this assumption can have suboptimal predictive performance. In this paper, we propose a general family of statistical models for pairwise comparison data without a stochastic transitivity assumption, substantially extending the BT and Thurstone models. In this model, the pairwise probabilities are determined by a (approximately) low-dimensional skew-symmetric matrix. Likelihood-based estimation methods and computational algorithms are developed, which allow for sparse data with only a small proportion of observed pairs. Theoretical analysis shows that the proposed estimator achieves minimax-rate optimality, which adapts effectively to the sparsity level of the data. The spectral theory for skew-symmetric matrices plays a crucial role in the implementation and theoretical analysis. The proposed method's superiority against the BT model, along with its broad applicability across diverse scenarios, is further supported by simulations and real data analysis.

Problem

• Pairwise Comparison
• Skill Diversity
• Prediction Accuracy

Innovation

• Novel Contrastive Model
• Sparse Data Estimation
• Enhanced Predictive Accuracy