An \( n \)-by-\( n \) positive matrix \( A = (a_{ij}) \) is called "reciprocal" if \( a_{ji}=1/a_{ij} \). The entries represent pair-wise ratio comparisons among \( n \) alternatives, and appear in many models in which alternatives are to be ranked, either ordinally or cardinally. A positive vector \( w \) is called efficient for such a matrix if the consistent matrix formed from it is a Pareto optimal approximation. We survey recent advances in the subject, including how to determine the efficient vectors. This work is joint with Susana Furtado.
Several interesting mathematical questions are mentioned.