Efficiency of the singular vector of a reciprocal matrix and comparison to the Perron vector

In models using pair-wise (ratio) comparisons among alternatives (reciprocal matrices A) to deduce a cardinal ranking vector, the right Perron eigenvector was traditionally used, though several other options have emerged. We propose and motivate another alternative, the left singular vector (Perron eigenvector of AAT). Theory is developed. We show that for reciprocal matrices obtained from consistent matrices by modifying one column, and the corresponding row, the cone generated by the columns is efficient, implying that the Perron vector and the left singular vector of such matrices are always efficient. In addition, the two vectors are compared empirically for random matrices. With regard to efficiency, the left singular vector consistently outperforms the right Perron eigenvector, though both perform well for large numbers of alternatives.