  In prioritization schemes, based on pairwise comparisons, such as the
Analytical Hierarchy Process, it is necessary to extract a cardinal ranking
vector from a reciprocal matrix that is unlikely to be consistent. It is
natural to choose such a vector only from efficient ones. One of the most used
ranking methods employs the (right) Perron eigenvector of the reciprocal matrix
as the vector of weights. It is known that the Perron vector may not be
efficient. Here, we focus on extending arbitrary reciprocal matrices and show,
constructively, that two different extensions of any fixed size always exist
for which the Perron vector is inefficient and for which it is efficient, with
the following exception. If B is consistent, any reciprocal matrix obtained
from B by adding one row and one column has efficient Perron vector. As a
consequence of our results, we obtain families of reciprocal matrices for which
the Perron vector is inefficient. These include known classes of such matrices
and many more. We also characterize the 4-by-4 reciprocal matrices with
inefficient Perron vector. Some prior results are generalized or completed.
