🤖 AI Summary
This work addresses the non-reciprocity in pairwise comparison data arising jointly from genuine scale variations and random noise. The authors propose an additive decomposition model that disentangles the observed comparison matrix into three components: a consistent non-reciprocal structure encoding a global ranking, a symmetric component capturing scale-induced discrepancies, and Gaussian noise representing judgment errors. Within a probabilistic framework, the method distinguishes structural non-reciprocity from stochastic perturbations, thereby preserving symmetric information essential for analyzing scale effects while avoiding the information loss incurred by enforcing reciprocity. The resulting admissible ranking region enables noise calibration and validation of scale plausibility, substantially outperforming conventional approaches that project observations directly onto reciprocal matrices.
📝 Abstract
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.