Inconsistency evaluation in pairwise comparison using norm-based distances
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Inconsistency evaluation in pairwise comparison using norm-based distances Michele Fedrizzi1
· Nino Civolani2 · Andrew Critch3
Received: 14 December 2019 / Accepted: 11 August 2020 © The Author(s) 2020
Abstract This paper studies the properties of an inconsistency index of a pairwise comparison matrix under the assumption that the index is defined as a norm-induced distance from the nearest consistent matrix. Under additive representation of preferences, it is proved that an inconsistency index defined in this way is a seminorm in the linear space of skew-symmetric matrices and several relevant properties hold. In particular, this linear space can be partitioned into equivalence classes, where each class is an affine subspace and all the matrices in the same class share a common value of the inconsistency index. The paper extends in a more general framework some results due, respectively, to Crawford and to Barzilai. It is also proved that norm-based inconsistency indices satisfy a set of six characterizing properties previously introduced, as well as an upper bound property for group preference aggregation. Keywords Inconsistency index · Pairwise comparison matrix · Norm · Distance JEL Classification C44 · D7
1 Introduction Pairwise comparison over a set of alternatives X = {x1 , . . . , xn } is a well known and powerful method for preference elicitation in a decision problem. An important characteristic of this method is the capability of dealing with the imprecision of the collected data due to the unavoidable inconsistency of human judgements. Each entry ai j of a pairwise comparison matrix, PCM in the following, A = (ai j )n×n quantifies
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Michele Fedrizzi [email protected]
1
Department of Industrial Engineering, University of Trento, Via Sommarive 77, 38123 Trento, Italy
2
University of Trento, Trento, Italy
3
Center for Human-Compatible AI, University of California, Berkeley, California, USA
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the degree of preference of alternative xi over alternative x j . Two widely used representations of preferences are the so-called multiplicative and additive ones. In the multiplicative approach (Saaty 1977), ai j is the relative preference of alternative xi over alternative x j , and therefore, it estimates the ratio between the weight wi of xi and wi Conversely, in the additive approach, ai j estimates the the weight w j of x j , ai j ≈ w j difference between the weights of xi and x j , respectively, ai j ≈ wi − w j . Therefore, different assumptions in preference quantification correspond to different meaning of the entries ai j . It has been proved that the multiplicative and the additive representations are isomorphic and, therefore, equivalent. In fact, a multiplicative PC M (ai j )n×n can be easily transformed into a corresponding additive PC M by componentwise applying the logarithmic function, thus obtaining (ln(ai j ))n×n . Details on this isomorphism can be found in (Barzilai 1998; Cavallo and D’Apuzzo 2009). In the multiplicative approach, a pairwise comparison
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