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Pairwise Comparisons without Stochastic Transitivity: Model, Theory and Applications

arXiv 2501.07437 ↗First posted 13 Jan 2025stat.ML cs.LGcited by 0

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Abstract (arXiv)

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.

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References · 22

  1. 2022Training language models to follow instructions with human feedbackcited by 15,712
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10 more
  1. 2013Matrix Completion via Max-Norm Constrained Optimizationcited by 18
  2. 2022Uncertainty Quantification of MLE for Entity Ranking with Covariatescited by 16
  3. 2017Structured Latent Factor Analysis for Large-scale Data: Identifiability, Estimability, and Their Implicationscited by 8
  4. 2020A General Pairwise Comparison Model for Extremely Sparse Networkscited by 7
  5. 2017Asymptotically Optimal Sequential Design for Rank Aggregationcited by 5
  6. 2020Determining the Number of Factors in High-dimensional Generalized Latent Factor Modelscited by 5
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  8. 2016Bayesian Decision Process for Cost-Efficient Dynamic Ranking via Crowdsourcingcited by 3
  9. 2024A Latent Variable Approach to Learning High-dimensional Multivariate longitudinal Datacited by 3
  10. 2021Modelling intransitivity in pairwise comparisons with application to baseball datacited by 2

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