Incomplete pairwise comparison matrices based on graphs with average degree approximately 3

Abstract

A crucial, both from theoretical and practical points of view, problem in preference modelling is the number of questions to ask from the decision maker. We focus on incomplete pairwise comparison matrices based on graphs whose average degree is approximately 3 (or a bit more), i.e., each item is compared to three others in average. In the range of matrix sizes we considered, $$n=5,6,7,8,9,10$$ n = 5 , 6 , 7 , 8 , 9 , 10 , this requires from 1.4n to 1.8n edges, resulting in completion ratios between 33% ($$n=10$$ n = 10 ) and 80% ($$n=5$$ n = 5 ). We analyze several types of union of two spanning trees (three of them building on additional ordinal information on the ranking), 2-edge-connected random graphs and 3-(quasi-)regular graphs with minimal diameter (the length of the maximal shortest path between any two vertices). The weight vectors are calculated from the natural extensions, to the incomplete case, of the two most popular weighting methods, the eigenvector method and the logarithmic least squares. These weight vectors are compared to the ones calculated from the complete matrix, and their distances (Euclidean, Chebyshev and Manhattan), rank correlations (Kendall and Spearman) and similarity (Garuti, cosine and dice indices) are computed in order to have cardinal, ordinal and proximity views during the comparisons. Surprisingly enough, only the union of two star graphs centered at the best and the second best items perform well among the graphs using additional ordinal information on the ranking. The union of two edge-disjoint spanning trees is almost always the best among the analyzed graphs.

Comparative Analysis of the Existence and Uniqueness Conditions of Parameter Estimation in Paired Comparison Models

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Evaluating the capacity of paired comparison methods to aggregate rankings of separate groups

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Proposals for the Sequences of Pairwise Comparisons

“…This way we can estimate the preferences of the decision maker in a problem the best possible way. Szádoczki et al (2022b) have compared and proposed some special patterns of comparison sets, while other studies also included additional prior information to the analysis (Szádoczki et al, 2022a). However, they only focused on special cases, and compared patterns with different number of comparisons, while we are determining the optimal sets of comparisons among all sets with the same cardinality.…”