  Pairwise comparisons are used in a wide variety of decision situations where
the importance of alternatives should be measured on a numerical scale. One
popular method to derive the priorities is based on the right eigenvector of a
multiplicative pairwise comparison matrix. We consider two monotonicity axioms
in this setting. First, increasing an arbitrary entry of a pairwise comparison
matrix is not allowed to result in a counter-intuitive rank reversal, that is,
the favoured alternative in the corresponding row cannot be ranked lower than
any other alternative if this was not the case before the change (rank
monotonicity). Second, the same modification should not decrease the normalised
weight of the favoured alternative (weight monotonicity). Both properties are
satisfied by the geometric mean method but violated by the eigenvector method.
The axioms do not uniquely determine the geometric mean. The relationship
between the two monotonicity properties and the Saaty inconsistency index are
investigated for the eigenvector method via simulations. Even though their
violation turns out not to be a usual problem even for heavily inconsistent
matrices, all decision-makers should be informed about the possible occurrence
of such unexpected consequences of increasing a matrix entry.
