Triple perturbed consistent matrix and the efficiency of its principal right eigenvector

Abstract

Let A be a pairwise comparison matrix obtained from a consistent one by perturbing three entries above the main diagonal, $x,y,z$, and the corresponding reciprocal entries, in a way that there is a submatrix of size 2 containing the three perturbed entries and not containing a diagonal entry. In this paper we describe the relations among $x,y,z$ with which A always has its principal right eigenvector efficient. Previously, and only for a few cases of this problem, R. Fernandes and S. Furtado (2022) proved the efficiency of the principal right eigenvector of A. In this paper, we continue to use the strong connectivity of a certain digraph associated with A and its principal right eigenvector to characterize the vector efficiency. For completeness, we show that the existence of a sink in this digraph is equivalent to the inefficiency of the principal right eigenvector of A.