Many applications, e.g. in content recommendation, sports, or recruitment, leverage the comparisons of alternatives to score those alternatives. The classical Bradley-Terry model and its variants have been widely used to do so. The historical model considers binary comparisons (victory or defeat) between alternatives, while more recent developments allow finer comparisons to be taken into account. In this article, we introduce a probabilistic model encompassing a broad variety of paired comparisons that can take discrete or continuous values. We do so by considering a well-behaved subset of the exponential family, which we call the family of generalized Bradley-Terry (GBT) models, as it includes the classical Bradley-Terry model and many of its variants. Remarkably, we prove that all GBT models are guaranteed to yield a strictly convex negative log-likelihood. Moreover, assuming a Gaussian prior on alternatives' scores, we prove that the maximum a posteriori (MAP) of GBT models, whose existence, uniqueness and fast computation are thus guaranteed, varies monotonically with respect to comparisons (the more A beats B, the better the score of A) and is Lipschitz-resilient with respect to each new comparison (a single new comparison can only have a bounded effect on all the estimated scores). These desirable properties make GBT models appealing for practical use. We illustrate some features of GBT models on simulations.

翻译：许多应用领域（如内容推荐、体育竞技或人才招聘）通过比较不同选项来为其评分。经典的布拉德利-特里模型及其变体已被广泛用于此目的。传统模型仅考虑二元比较（胜或负），而近年来的发展允许纳入更精细的比较。本文提出了一种概率模型，涵盖可离散或连续取值的广泛成对比较类型。为此，我们选取指数族中的一个良好子集，将其命名为广义布拉德利-特里模型族（简称GBT模型），因其包含经典布拉德利-特里模型及其众多变体。值得关注的是，我们证明了所有GBT模型的负对数似然函数严格凸。此外，在选项得分服从高斯先验的假设下，我们证明了GBT模型的最大后验估计（MAP）具有以下特性：存在性、唯一性及快速计算性均得到保证；该估计随比较结果单调变化（A击败B越频繁，A的得分越高）；且对每次新增比较具有利普希茨韧性（单次新增比较仅对所有估计得分产生有界影响）。这些优良性质使GBT模型极具实用价值。我们通过模拟实验展示了GBT模型的若干特性。