3,727 research outputs found
Notes on the existence of solutions in the pairwise comparisons method using the Heuristic Rating Estimation approach
Pairwise comparisons are a well-known method for modelling of the subjective
preferences of a decision maker. A popular implementation of the method is
based on solving an eigenvalue problem for M - the matrix of pairwise
comparisons. This does not take into account the actual values of preference.
The Heuristic Rating Estimation (HRE) approach is a modification of this method
in which allows modelling of the reference values. To determine the relative
order of preferences is to solve a certain linear equation system defined by
the matrix A and the constant term vector b (both derived from M). The article
explores the properties of these equation systems. In particular, it is proven
that for some small data inconsistency the A matrix is an M-matrix, hence the
equation proposed by the HRE approach has a unique strictly positive solution.Comment: 8 page
Clustering and Inference From Pairwise Comparisons
Given a set of pairwise comparisons, the classical ranking problem computes a
single ranking that best represents the preferences of all users. In this
paper, we study the problem of inferring individual preferences, arising in the
context of making personalized recommendations. In particular, we assume that
there are users of types; users of the same type provide similar
pairwise comparisons for items according to the Bradley-Terry model. We
propose an efficient algorithm that accurately estimates the individual
preferences for almost all users, if there are
pairwise comparisons per type, which is near optimal in sample complexity when
only grows logarithmically with or . Our algorithm has three steps:
first, for each user, compute the \emph{net-win} vector which is a projection
of its -dimensional vector of pairwise comparisons onto an
-dimensional linear subspace; second, cluster the users based on the net-win
vectors; third, estimate a single preference for each cluster separately. The
net-win vectors are much less noisy than the high dimensional vectors of
pairwise comparisons and clustering is more accurate after the projection as
confirmed by numerical experiments. Moreover, we show that, when a cluster is
only approximately correct, the maximum likelihood estimation for the
Bradley-Terry model is still close to the true preference.Comment: Corrected typos in the abstrac
On Axiomatization of Inconsistency Indicators for Pairwise Comparisons
We examine the notion of inconsistency in pairwise comparisons and propose an
axiomatization which is independent of any method of approximation or the
inconsistency indicator definition (e.g., Analytic Hierarchy Process, AHP). It
has been proven that the eigenvalue-based inconsistency (proposed as a part of
AHP) is incorrect.Comment: Enhanced text, with 21 pages and 3 figures, proves that arbitrarily
inaccurate pairwise matrices are considered acceptable by theories with a
inconsistency based on the principal eigenvalue (e.g., AHP). CPC (corner
pairwise comparisons) matrix is the crucial part of this study as it
invalidates any eigenvalue-based inconsistency. All comments are highly
appreciate
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