Abstract
Pairwise comparison matrices often exhibit inconsistency, therefore many indices have been suggested to measure their deviation from a consistent matrix. A set of axioms has been proposed recently that is required to be satisfied by any reasonable inconsistency index. This set seems to be not exhaustive as illustrated by an example, hence it is expanded by adding two new properties. All axioms are considered on the set of triads, pairwise comparison matrices with three alternatives, which is the simplest case of inconsistency. We choose the logically independent properties and prove that they characterize, that is, uniquely determine the inconsistency ranking induced by most inconsistency indices that coincide on this restricted domain. Since triads play a prominent role in a number of inconsistency indices, our results can also contribute to the measurement of inconsistency for pairwise comparison matrices with more than three alternatives.
Highlights
All axioms are considered on the set of triads, pairwise comparison matrices with three alternatives, which is the simplest case of inconsistency
Since triads play a prominent role in a number of inconsistency indices, our results can contribute to the measurement of inconsistency for pairwise comparison matrices with more than three alternatives
Pairwise comparisons play an important role in a number of decision analysis methods such as the Analytic Hierarchy Process (AHP) (Saaty 1977, 1980)
Summary
Pairwise comparisons play an important role in a number of decision analysis methods such as the Analytic Hierarchy Process (AHP) (Saaty 1977, 1980) They naturally emerge in country (Petróczy 2019) and higher education (Csató and Tóth 2019) rankings, in voting systems (Caklovicand Kurdija 2017), as well as in sport tournaments This work aims to connect these two research directions by placing the axioms of Brunelli (2017)—which is itself an extended set of the properties proposed by Brunelli and Fedrizzi (2015)—and Csató (2018a) into a single framework They will be considered on the domain of triads, that is, pairwise comparison matrices with only three alternatives.
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