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.

翻译：大多数成对比较的统计模型，包括Bradley-Terry（BT）模型、Thurstone模型及其众多扩展，均建立在相对较强的随机传递性假设之上。该假设要求所有参与者/团队/项目之间存在一个未观测的全局排序，且比较概率需满足该全局排序所隐含的单调约束。然而，随机传递性假设在许多现实世界的成对比较场景中并不成立，尤其是在涉及多重技能或策略的博弈中。因此，依赖该假设的模型可能具有次优的预测性能。本文提出了一种无需随机传递性假设的成对比较数据统计模型通用族，显著拓展了BT模型与Thurstone模型。在该模型中，成对概率由一个（近似）低维斜对称矩阵决定。我们开发了基于似然的估计方法与计算算法，能够处理仅含少量已观测对比例的稀疏数据。理论分析表明，所提估计量达到极小化极大速率最优性，并能有效适应数据的稀疏程度。斜对称矩阵的谱理论在方法实现与理论分析中起着关键作用。通过模拟实验与真实数据分析，进一步验证了所提方法相对于BT模型的优越性及其在多类场景中的广泛适用性。