将非互反比较中的偏差分为尺度变化与噪声,实现更合理的排名推断。
Noisy Nonreciprocal Pairwise Comparisons: Scale Variation, Noise Calibration, and Admissible Ranking Regions
- 区分非互反性中的尺度变化与随机噪声,构建可解释的加性模型。
- 通过高斯扰动建模,估计噪声水平并评估尺度变化是否可控。
- 给出严格排序的可接受区域概率,适合需稳健决策的场景。
成对比较广泛应用于决策分析、偏好建模与评估问题。许多实际场景中,观测到的比较矩阵并非互反。传统做法常将非互反性视为缺陷并立即修正。本文提出不同视角:部分非互反性反映评价尺度的真实变化,另一部分源于随机扰动。我们引入一个加性模型,其中未知的基础比较矩阵具有一致性但未必互反。互反分量承载全局排名信息,对称分量描述可能的尺度变化。在该结构化矩阵周围添加随机扰动,展示如何估计噪声水平、评估尺度变化是否适度,并为基于成对比较的严格排名分配可接受区域的概率。我们还将此方法与强制投影至互反矩阵的粗暴方法进行对比,后者一次性抑制所有对称信息。此处采用高斯扰动模型并非因人类决策严格服从正态分布,而是因为观察到的判断误差往往由多个微小因素累积而成。在此背景下,中心极限定理为高斯噪声提供了自然启发式依据。这使得能够推导出显式估计器和概率评估,同时保持模型在决策问题中的可解释性。
原文摘要 · Abstract (English)
Pairwise comparisons are widely used in decision analysis, preference modeling, and evaluation problems. In many practical situations, the observed comparison matrix is not reciprocal. This lack of reciprocity is often treated as a defect to be corrected immediately. In this article, we adopt a different point of view: part of the nonreciprocity may reflect a genuine variation in the evaluation scale, while another part is due to random perturbations. We introduce an additive model in which the unknown underlying comparison matrix is consistent but not necessarily reciprocal. The reciprocal component carries the global ranking information, whereas the symmetric component describes possible scale variation. Around this structured matrix, we add a random perturbation and show how to estimate the noise level, assess whether the scale variation remains moderate, and assign probabilities to admissible ranking regions in the sense of strict ranking by pairwise comparisons. We also compare this approach with the brutal projection onto reciprocal matrices, which suppresses all symmetric information at once. The Gaussian perturbation model is used here not because human decisions are exactly Gaussian, but because observed judgment errors often result from the accumulation of many small effects. In such a context, the central limit principle provides a natural heuristic justification for Gaussian noise. This makes it possible to derive explicit estimators and probability assessments while keeping the model interpretable for decision problems.
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