Characterization of the Row Geometric Mean Ranking with a Group Consensus Axiom

Abstract

An axiomatic approach is applied to the problem of extracting a ranking of the alternatives from a pairwise comparison ratio matrix. The ordering induced by row geometric mean method is proved to be uniquely determined by three independent axioms, anonymity (independence of the labelling of alternatives), responsiveness (a kind of monotonicity property) and aggregation invariance, which requires the preservation of group consensus, that is, the pairwise ranking between two alternatives should remain unchanged if unanimous individual preferences are combined by geometric mean.

New preference violation indices for the condition of order preservation

“…The examination of properties of inconsistency indices attracted many recent studies, see e.g. [1], [15], [16], [17], [23], [25] [43], [48], [52]. Their authors call the desirable properties, which inconsistency indices should satisfy, "axioms", and propose their own axiomatic systems.…”

“…The following proposition states that when a pairwise comparison matrix A is the COP consistent and all preferences are abided, but intensied by a factor b > 1, then the matrix A remains the COP consistent (the feature also satised by numerically consistent matrices), if the priority deriving method is the geometric mean method, which is due to the scale invariance of the GMM, see [21] and [23]. Proposition 6.…”

“…In recent years, a number of studies focused on the problem of inconsistency of pairwise comparisons, properties of inconsistency indices and their relationships, as well as on proposition of axiomatic systems for inconsistency indices, see e.g. [1], [10], [11], [13], [15], [16], [17], [23], [24], [25], [33] [35], [38], [39], [40], [41], [42], [46], [47], [48], [49], [52], [56], [57], [58], or [59].…”

Coherent weights for pairwise comparison matrices and a mixed-integer linear programming problem

“…F is not restricted max-max M-transitive (e.g. f 14 = 0.667 < max{ f 12 , f 24 } = 0.9, see (20)). Moreover, the F -mean vector w m F (F) = (0.627, 0.634, 0.414, 0.327) is not a coherent vector for the actual ranking because…”

“…[3,4,12,14,30,31,36]), axiomatic approaches are applied for the derivation of the weights (see e.g. [2,18,20,21,[23][24][25]). Cavallo and D'Apuzzo [12] provide a suitable weighting vector w m G (G) for a PCM defined over an Alo-group; it has an intuitive meaning because its i-th component is the mean of the preference intensities of x i over all others elements x j , and satisfies further properties [12], such as the independence of scale inversion condition.…”