Pairwise Comparisons without Stochastic Transitivity: Model, Theory and Applications (2501.07437v1)

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.

• The paper relaxes the assumption of stochastic transitivity by using a low-dimensional skew-symmetric matrix to model pairwise probabilities.
• It develops computational algorithms with likelihood-based estimation for sparse data, achieving minimax-rate optimality in various applications.
• The model leverages spectral theory and nuclear norm regularization, demonstrating superior prediction in both sports and e-sport datasets compared to traditional approaches.

Overview of "Pairwise Comparisons without Stochastic Transitivity: Model, Theory and Applications"

The paper "Pairwise Comparisons without Stochastic Transitivity: Model, Theory and Applications" by Sze Ming Lee and Yunxiao Chen presents a novel approach to modeling pairwise comparison data by relaxing the assumption of stochastic transitivity. This research introduces a statistical framework that extends the Bradley-Terry (BT) and Thurstone models, acknowledging that the assumption of an underlying global ranking among items may not always hold, especially in contexts involving multiple skills or strategies.

Key Contributions and Methodological Advancements

1. Relaxation of Stochastic Transitivity: The authors propose a framework that does not impose the constraint of stochastic transitivity. Instead of assuming a global ranking order, this model employs an approximately low-dimensional skew-symmetric matrix to determine pairwise probabilities. This approach significantly expands the applicability of traditional pairwise comparison models to scenarios where intransitivity is a factor.
2. Efficiency in Sparse Data Contexts: A notable advancement is the development of computational algorithms and likelihood-based estimation methods designed for sparse data, where only a small fraction of pairs are observed. The estimator introduced achieves minimax-rate optimality, effectively adapting to the sparsity levels. This is crucial in applications such as predicting outcomes in sports tournaments, where data sparsity is common.
3. Use of Skew-Symmetric Matrix: The approach utilizes the spectral theory of skew-symmetric matrices, which is central to both the theoretical analysis and the implementation of this model. The leveraging of this mathematical foundation distinguishes it from other models that rely heavily on rank-based assumptions.

Theoretical Implications

The theoretical contributions include demonstrating that the proposed estimator is capable of achieving optimal convergence rates under various sparsity scenarios. The model operates under a nuclear norm constraint which serves as a form of regularization, controlling the complexity of the parameter matrix. This formulation enhances the flexibility of the model, allowing it to handle diverse real-world data sets without the restrictive transitivity assumption.

Empirical Analysis and Applications

Empirical evaluation using datasets from the e-sport StarCraft II and professional tennis underscores the practical efficiency of the proposed model. The findings suggest that the absence of stochastic transitivity does not significantly impair the model's accuracy, indicating its robustness across different contexts. The model achieved superior performance in predicting pairwise outcomes compared to traditional models, especially where data exhibit intransitive properties.

Future Directions and Research Opportunities

The paper opens several avenues for future research, including the integration of covariate information and time dynamics into the model, enhancing predictive capabilities in evolving scenarios. Additionally, further exploration could involve extending the model to accommodate more complex parametric structures, potentially incorporating machine learning techniques to handle even higher-dimensional data contexts.

Conclusion

Overall, this paper provides a significant step forward in the methodology of pairwise comparisons, especially in its application to scenarios where the assumption of global rankings is impractical. By leveraging skew-symmetric matrices and addressing the challenges of sparse data, this framework sets a groundwork for future advancements in the analysis of pairwise comparison situations in various domains.