Recursive Multilevel Decomposition of Pairwise Comparison Matrices: Algorithms and Inconsistency Diagnostics
Abstract
Positive reciprocal pairwise comparison matrices are studied in entrywise logarithmic coordinates. Starting from the established consistent-residual split, we show that, for any fixed ordering, the residual matrix admits a unique top-down decomposition into mutually Frobenius-orthogonal layers. The construction applies to every positive reciprocal PC matrix of size n ≥ 3, without additional spectral, sparsity, or genericity assumptions. An explicit active-matrix recursion and an equivalent cumulative parameter-extraction procedure have computational complexities O(n3) and O(n2), respectively. The squared Frobenius norms of the layers define an ordering-conditioned residual-energy profile that provides structural information complementary to the geometric-mean priorities and the Koczkodaj inconsistency indicator (Kii). Entrywise exponentiation yields an exact Hadamard-product reconstruction of the original reciprocal matrix. Numerical experiments confirm agreement between the two computational procedures, exact reconstruction, mutual orthogonality, and numerical accuracy to machine precision.
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BibTeX
@misc{carboga2026recursive,
title = {{Recursive Multilevel Decomposition of Pairwise Comparison Matrices: Algorithms and Inconsistency Diagnostics}},
author = {Mert Çarboğa and Waldemar W. Koczkodaj and Yusuf Yaylı},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03184},
eprint = {2610.03184},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03184},
}
RIS
TY - UNPB
TI - Recursive Multilevel Decomposition of Pairwise Comparison Matrices: Algorithms and Inconsistency Diagnostics
AU - Mert Çarboğa
AU - Waldemar W. Koczkodaj
AU - Yusuf Yaylı
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03184
UR - https://arxiv.org/abs/2610.03184
AB - Positive reciprocal pairwise comparison matrices are studied in entrywise logarithmic coordinates. Starting from the established consistent-residual split, we show that, for any fixed ordering, the residual matrix admits a unique top-down decomposition into mutually Frobenius-orthogonal layers. The construction applies to every positive reciprocal PC matrix of size n ≥ 3, without additional spectral, sparsity, or genericity assumptions. An explicit active-matrix recursion and an equivalent cumulative parameter-extraction procedure have computational complexities O(n^3) and O(n^2), respectively. The squared Frobenius norms of the layers define an ordering-conditioned residual-energy profile that provides structural information complementary to the geometric-mean priorities and the Koczkodaj inconsistency indicator (Kii). Entrywise exponentiation yields an exact Hadamard-product reconstruction of the original reciprocal matrix. Numerical experiments confirm agreement between the two computational procedures, exact reconstruction, mutual orthogonality, and numerical accuracy to machine precision.
ER -