Description
Pairwise comparisons form a basis of preference modeling and decision
analysis with a wide range of applications in multiple criteria
decision making, group decision making, ranking and voting.
A specific model of pairwise comparisons, proposed by Saaty in 1977,
is discussed in the talk. A pairwise comparison matrix is built up
from numerical answers on questions like ‘How many times criterion i
is more important than criterion j?’ or ‘How many times action i is
better than action j?’. The problem of weighting is then to find a
vector such that the pairwise ratios of its coordinates are as close
as possible to the corresponding matrix elements. The eigenvector, the
least squares and the logarithmic least squares methods are classical
ways of weighting, besides dozens of other proposals.
Incomplete pairwise comparison matrices offer the possibility of
ignoring some (in certain applications more than 90%) of all the
possible pairs in the questionnaire. Due to this reduction,
essentially larger weighting and ranking problems can be considered
and solved. For example, we recently developed a ranking method of top
tennis players.
The Pareto optimality of a weight vector, derived from a pairwise
comparison matrix, is a natural and desirable property. A weight
vector is called Pareto optimal if no other weight vector is at least
as good in approximating the elements of the pairwise comparison
matrix, and strictly better in at least one position. The least
squares method and the logarithmic least squares method always yield
efficient weight vectors, while the principal right eigenvector can be
inefficient. It is still open to find a necessary and sufficient
condition for the Pareto optimality of the eigenvector.
Main references
Bozóki, S., Fülöp, J. (2017): Efficient weight vectors from pairwise
comparison matrices, European Journal of Operational Research, online
first, DOI 10.1016/j.ejor.2017.06.033
Bozóki, S., Csató, L., Temesi, J. (2016): An application of incomplete
pairwise comparison matrices for ranking top tennis players, European
Journal of Operational Research, 248(1), 211–218, with an online
appendix at http://www.sztaki.hu/%7Ebozoki/tennis/appendix.pdf
Bozóki, S., Fülöp, J., Rónyai, L. (2010): On optimal completions of
incomplete pairwise comparison matrices, Mathematical and Computer
Modelling, 52(1-2), 318–333