Efficiency analysis of natural cardinal ranking vectors for pairwise comparisons and the universal efficiency of the Perron geometric mean

In models using pairwise (ratio) comparisons among alternatives, a cardinal ranking vector should be deduced from a reciprocal matrix. The right Perron eigenvector (RP) was traditionally used, though several other options have emerged. We consider some alternatives, namely the entry-wise reciprocal of the left Perron vector (LP), the left singular vector (LS), the entry-wise reciprocal of the right singular vector (RS), the arithmetic and geometric means of RP and LP (AP and GP), and of LS and RS (AS and GS). All eight of these vectors produce the natural vector in the consistent case. We compare them empirically, in terms of frequency of efficiency, for random reciprocal matrices, as a function of the number of alternatives. The vector GP is proved to be universally efficient. This provides a new proposal for the cardinal ranking vector. The vector GS performs better than the remaining six vectors, though all of them have a high frequency of efficiency. We show that, for reciprocal matrices obtained from consistent matrices by modifying one column and the corresponding row, all eight vectors are efficient. Moreover, the cone generated by the columns is efficient. This class of matrices includes a type of double perturbed consistent matrices, that is, reciprocal matrices obtained from consistent ones by modifying two pairs of symmetrically located entries. We also show the efficiency of the studied vectors for the double perturbed consistent matrices that are not column perturbed.