Abstract Reciprocal relations are binary relations Q with entries Q ( i , j ) ∈ [ 0 , 1 ] , and such that Q ( i , j ) + Q ( j , i ) = 1 . Relations of this kind occur quite naturally in various domains, such as preference modeling and preference learning. For example, Q ( i , j ) could be the fraction of voters in a population who prefer candidate i to candidate j. In the literature, various attempts have been made at generalizing the notion of transitivity to reciprocal relations. In this paper, we compare three important frameworks of generalized transitivity: g-stochastic transitivity, T-transitivity, and cycle-transitivity. To this end, we introduce E-transitivity as an even more general notion. We also use this framework to extend an existing hierarchy of different types of transitivity. As an illustration, we study transitivity properties of probabilities of pairwise preferences, which are induced as marginals of an underlying probability distribution on rankings (strict total orders) of a set of alternatives. In particular, we analyze the interesting case of the so-called Babington Smith model, a parametric family of distributions of that kind.

中文翻译：

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广义传递性：概念的系统比较与巴宾顿史密斯模型中偏好的应用

摘要 互易关系是二元关系Q，其项Q ( i , j ) ∈ [ 0, 1 ] ，且Q ( i , j ) + Q ( j , i ) = 1 。这种关系在各个领域都很自然地发生，例如偏好建模和偏好学习。例如，Q ( i , j ) 可以是人口中更喜欢候选人 i 而非候选人 j 的选民比例。在文献中，已经进行了各种尝试将传递性的概念推广到互惠关系。在本文中，我们比较了广义传递性的三个重要框架：g-随机传递性、T-传递性和循环传递性。为此，我们将电子传递性作为一个更一般的概念引入。我们还使用这个框架来扩展不同类型传递性的现有层次结构。作为例证，我们研究成对偏好概率的传递特性，这些特性是作为一组备选方案的排名（严格的总阶数）的潜在概率分布的边际诱导的。特别是，我们分析了所谓的 Babington Smith 模型的有趣案例，该模型是此类分布的参数族。