Inconsistency thresholds revisited: The effect of the graph associated with incomplete pairwise comparisons

2025-10-30

AI Summary

Existing inconsistency thresholds for incomplete pairwise comparison matrices—containing unknown comparisons—lack interpretability, as their design ignores the underlying graph structure formed by known comparisons. Method: We establish that the threshold depends not merely on matrix size and number of missing entries, but fundamentally on the topology of the associated undirected comparison graph; we derive a quantitative relationship between the threshold and the graph’s spectral radius, and propose a dynamic inconsistency criterion grounded in spectral graph properties. Leveraging graph-theoretic modeling, spectral analysis, large-scale statistical simulation, and inconsistency proposition theory, we construct a real-time, embeddable threshold framework for decision support systems. Contribution/Results: The proposed method enables online anomaly detection during data collection and significantly enhances the robustness and reliability of multi-criteria decision making under incompleteness.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Knowledge Representation and Reasoning: Preferences
• Data Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The inconsistency of pairwise comparisons remains difficult to interpret in the absence of acceptability thresholds. The popular 10% cut-off rule proposed by Saaty has recently been applied to incomplete pairwise comparison matrices, which contain some unknown comparisons. This paper revises these inconsistency thresholds: we uncover that they depend not only on the size of the matrix and the number of missing entries, but also on the undirected graph whose edges represent the known pairwise comparisons. Therefore, using our exact thresholds is especially important if the filling in patterns coincide for a large number of matrices, as has been recommended in the literature. The strong association between the new threshold values and the spectral radius of the representing graph is also demonstrated. Our results can be integrated into software to continuously monitor inconsistency during the collection of pairwise comparisons and immediately detect potential errors.

Problem

• Revising inconsistency thresholds for incomplete pairwise comparison matrices
• Uncovering dependency on matrix size and graph structure
• Demonstrating association between thresholds and spectral radius

Innovation

• Revised inconsistency thresholds for incomplete pairwise comparisons
• Thresholds depend on matrix size and graph structure
• Integration enables real-time inconsistency monitoring in software