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（例如内容推荐、体育竞技或招聘）通过比较备选项来对其评分。经典Bradley-Terry模型及其变体已被广泛用于此目的。传统模型仅考虑备选项之间的二元比较（胜负），而近期发展允许纳入更精细的比较。本文提出了一种概率模型，可涵盖取值离散或连续的多种成对比较类型。通过考虑指数族的一个良性子集（称为广义Bradley-Terry（GBT）模型族，因其包含经典Bradley-Terry模型及其众多变体），我们证明所有GBT模型均能保证负对数似然函数严格凸。此外，假设备选项得分服从高斯先验，我们证明GBT模型的最大后验估计（MAP）——其存在性、唯一性及快速计算均得以保证——随比较结果单调变化（A越胜B，A得分越高），且对每次新比较具有Lipschitz弹性（单个新比较对全部估计得分的影响有界）。这些优良性质使GBT模型具有实用价值。我们通过仿真实验展示了GBT模型的若干特性。