# Pairwise Comparisons without Stochastic Transitivity: Model, Theory and Applications

arXiv: https://arxiv.org/abs/2501.07437 · HTML: https://mlsanity.com/paper/2501.07437/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 53 references that are in the map.
- First posted on arXiv 13 Jan 2025.

## Builds on (most cited first, 22 of 22)

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- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Principled Reinforcement Learning with Human Feedback from Pairwise or K-wise Comparisons](https://mlsanity.com/paper/2301.11270/) (2023)
- [Simple, Robust and Optimal Ranking from Pairwise Comparisons](https://mlsanity.com/paper/1512.08949/) (2015)
- [Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence](https://mlsanity.com/paper/1505.01462/) (2015)
- [Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization](https://mlsanity.com/paper/1902.07698/) (2019)
- [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017)
- [Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons](https://mlsanity.com/paper/1504.07218/) (2015)
- [Stochastically Transitive Models for Pairwise Comparisons: Statistical and Computational Issues](https://mlsanity.com/paper/1510.05610/) (2015)
- [A Max-Norm Constrained Minimization Approach to 1-Bit Matrix Completion](https://mlsanity.com/paper/1309.6013/) (2013)
- [Lagrangian Inference for Ranking Problems](https://mlsanity.com/paper/2110.00151/) (2021)
- [Matrix Completion via Max-Norm Constrained Optimization](https://mlsanity.com/paper/1303.0341/) (2013)
- [Uncertainty Quantification of MLE for Entity Ranking with Covariates](https://mlsanity.com/paper/2212.09961/) (2022)
- [Structured Latent Factor Analysis for Large-scale Data: Identifiability, Estimability, and Their Implications](https://mlsanity.com/paper/1712.08966/) (2017)
- [A General Pairwise Comparison Model for Extremely Sparse Networks](https://mlsanity.com/paper/2002.08853/) (2020)
- [Asymptotically Optimal Sequential Design for Rank Aggregation](https://mlsanity.com/paper/1710.06056/) (2017)
- [Determining the Number of Factors in High-dimensional Generalized Latent Factor Models](https://mlsanity.com/paper/2010.02326/) (2020)
- [Modeling outcomes of soccer matches](https://mlsanity.com/paper/1807.01623/) (2018)
- [Bayesian Decision Process for Cost-Efficient Dynamic Ranking via Crowdsourcing](https://mlsanity.com/paper/1612.07222/) (2016)
- [A Latent Variable Approach to Learning High-dimensional Multivariate longitudinal Data](https://mlsanity.com/paper/2405.15053/) (2024)
- [Modelling intransitivity in pairwise comparisons with application to baseball data](https://mlsanity.com/paper/2103.12094/) (2021)

## 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.