Pairwise Comparisons without Stochastic Transitivity: Model, Theory and Applications

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.

💡 Research Summary

The paper addresses a fundamental limitation of most pairwise‑comparison models, such as the Bradley‑Terry (BT) and Thurstone models, which assume stochastic transitivity. Stochastic transitivity forces the existence of a hidden global ranking and imposes monotone constraints on the pairwise win probabilities. In many real‑world settings—e‑sports, multi‑skill sports, or crowdsourced preference tasks—players or items may excel in different dimensions, leading to intransitive cycles (A beats B, B beats C, yet C beats A). When this assumption is violated, traditional models can suffer from poor predictive performance.

To overcome this, the authors propose a general family of models that do not require stochastic transitivity. They represent the matrix of latent “strengths” as a skew‑symmetric matrix (M\in\mathbb{R}^{n\times n}) (so (M=-M^\top)). The win probability for a pair ((i,j)) is linked through a logistic function: (\pi_{ij}=g(m_{ij})) with (g(x)=1/(1+e^{-x})). Skew‑symmetry automatically guarantees (\pi_{ij}=1-\pi_{ji}). The key structural assumption is that (M) is approximately low‑rank. Rather than fixing an exact rank, the authors enforce a nuclear‑norm constraint (|M|_* \le C_n n), where (C_n) may grow slowly with (n). This convex constraint captures the idea that only a few latent factors drive most of the variability while allowing many weak factors to be present.

Given observed counts (y_{ij}) of wins for player (i) against (j) (with total comparisons (n_{ij})), the likelihood is binomial: \