Intransitivity is a critical issue in pairwise preference modeling. It refers to the intransitive pairwise preferences between a group of players or objects that potentially form a cyclic preference chain and has been long discussed in social choice theory in the context of the dominance relationship. However, such multifaceted intransitivity between players and the corresponding player representations in high dimensions is difficult to capture. In this paper, we propose a probabilistic model that jointly learns each player's d-dimensional representation (d>1) and a dataset-specific metric space that systematically captures the distance metric in Rd over the embedding space. Interestingly, by imposing additional constraints in the metric space, our proposed model degenerates to former models used in intransitive representation learning. Moreover, we present an extensive quantitative investigation of the vast existence of intransitive relationships between objects in various real-world benchmark datasets. To our knowledge, this investigation is the first of this type. The predictive performance of our proposed method on different real-world datasets, including social choice, election, and online game datasets, shows that our proposed method outperforms several competing methods in terms of prediction accuracy.

翻译：不可传递性是成对偏好建模中的一个关键问题。它指的是一组玩家或对象之间可能形成循环偏好链的不可传递成对偏好，长期以来在社会选择理论中关于支配关系的背景下被广泛讨论。然而，玩家之间这种多方面的不可传递性及其在高维空间中的相应表示难以捕捉。本文提出了一种概率模型，该模型联合学习每个玩家的d维表示（d>1）以及一个数据集特定的度量空间，该度量空间系统性地捕捉了嵌入空间在Rd上的距离度量。有趣的是，通过在度量空间中施加额外的约束，我们提出的模型可以退化为先前用于不可传递表示学习的模型。此外，我们对各种现实世界基准数据集中对象之间广泛存在的不可传递关系进行了广泛的定量研究。据我们所知，此类研究尚属首次。我们提出的方法在不同现实世界数据集（包括社会选择、选举和在线游戏数据集）上的预测性能表明，该方法在预测准确性方面优于多种竞争方法。