A cardinal dissensus measure based on the Mahalanobis distance

Highlights

• We address the problem of measuring the degree of consensus/dissensus.

• Experts express their opinions on alternatives or issues by means of cardinal evaluations.

• We propose a new class of distance-based consensus model, the family of the Mahalanobis dissensus measures.

• An application with discussion over a real empirical example is presented.

Abstract

In this paper we address the problem of measuring the degree of consensus/dissensus in a context where experts or agents express their opinions on alternatives or issues by means of cardinal evaluations. To this end we propose a new class of distance-based consensus model, the family of the Mahalanobis dissensus measures for profiles of cardinal values. We set forth some meaningful properties of the Mahalanobis dissensus measures. Finally, an application over a real empirical example is presented and discussed.

Introduction

In Decision Making Theory and its applications, consensus measurement and its reaching in a society (i.e., a group of agents or experts) are relevant research issues. Many studies investigating the aforementioned subjects have been carried out under several frameworks (see Cabrerizo, Moreno, Pérez, Herrera-Viedma, 2010, Dong, Xu, Li, 2008, Dong, Xu, Li, Feng, 2010, Dong, Zhang, 2014, Fedrizzi, Fedrizzi, Marques Pereira, 2007, Fu, Yang, 2012, Herrera-Viedma, Herrera, Chiclana, 2002, Liu, Liao, Yang, 2015, Palomares, Estrella-Liébana, Martínez, Herrera, 2014, Wu, Chiclana, 2014a, Wu, Chiclana, 2014b, Wu, Chiclana, Herrera-Viedma, 2015 among others) and based on different methodologies (Chiclana, Tapia García, del Moral, Herrera-Viedma, 2013, Cook, 2006, Eklund, Rusinowska, de Swart, 2008, Eklund, Rusinowska, Swart, 2007, Fedrizzi, Fedrizzi, Marques Pereira, 2007, Fu, Yang, 2011, Fu, Yang, 2010, Gong, Zhang, Forrest, Li, Xu, 2015, González-Pachón, Romero, 1999, Liu, Liao, Yang, 2015, Palomares, Martínez, 2014 among others).

Since the seminal contribution by Bosch (2005) several authors have addressed the consensus measurement topic from an axiomatic perspective. Earlier analyses can be mentioned, e.g., Hays (1960) or Day and McMorris (1985). This issue is also seen as the problem of combining a set of ordinal rankings to obtain an indicator of their ‘consensus’, a term with multiple possible meanings (Martínez-Panero, 2011).

Generally speaking, the usual axiomatic approaches assume that each individual expresses his or her opinions through ordinal preferences over the alternatives. A group of agents is characterized by the set of their preferences – their preference profile. Then a consensus measure is a mapping which assigns to each preference profile a number between 0 and 1. The assumption is made that the higher the values, the more consensus in the profile.

Technical restrictions on the preferences provide various approaches in the literature. In most cases the agents are presumed to linearly order the alternatives (see Bosch, 2005 or Alcalde-Unzu & Vorsatz, 2013). Since this assumption seems rather demanding (especially as the number of alternatives grows), an obvious extension is to allow for ties. This is the case where the agents have complete preorders on the alternatives (e.g., García-Lapresta & Pérez-Román, 2011). Alcantud, de Andrés Calle, Cascón, 2015, Alcantud, de Andrés Calle, Cascón, 2013a take a different position. They study the case where agents have dichotomous opinions on the alternatives, a model that does not necessarily require pairwise comparisons.

Notwithstanding the use of different ordinal preference frameworks, the problem of how to measure consensus is an open-ended question in several research areas. This fact is due to that methodology used in each case is a relevant element in the problem addressed. To date various methods have been developed to measure consensus under ordinal preference structures based on distances and association measures like Kemeny’s distance, Kendall’s coefficient, Goodman-Kruskal’s index and Spearman’s coefficient among others (see e.g., Cook, Seiford, 1982, Goodman, Kruskal, 1979, Kemeny, 1959, Kendall, Gibbons, 1990, Spearman, 1904).

In this paper we first tackle the analysis of coherence that derives from profiles of cardinal rather than ordinal evaluations. Modern convention applies the term cardinal to measurements that assign significance to differences (cf., Basu, 1982, Chiclana, Herrera-Viedma, Alonso, Herrera, 2009, High, Bloch, 1989). In contrast ordinal preferences only permit to order the alternatives from best to worst without any additional information. To see how this affects the analysis of our problem, let us consider a naive example of a society with two agents. They evaluate two public goods with monetary amounts. One agent gives a value of 1€ for the first good and 2€ for the second good. The other agent values these goods at 10€ and 90€ respectively. If we only use the ordinal information in this case, we should conclude that there is unanimity in the society: all members agree that ‘good 2 is more valuable than good 1’. However the agents disagree largely. Therefore, the subtleties of cardinality clearly have an impact when we aim at measuring the cohesiveness of cardinal evaluations.

Unlike previous references, we adopt the notion of dissensus measure as the fundamental concept. This seems only natural because it resembles more the notion of a “measure of statistical dispersion”, in the sense that 0 captures the natural notion of unanimity as total lack of variability among agents, and then increasingly higher numbers mean more disparity among evaluations in the profile.¹

In order to build a particular dissensus measure we adopt a distance-based approach. Firstly, one computes the distances between each pair of individuals. Then all these distances are aggregated. In our present proposal the distances (or similarities) are computed through the Mahalanobis distance (Mahalanobis, 1936). We thus define the class of Mahalanobis dissensus measures.

The Mahalanobis distance plays an important role in Statistics and Data Analysis. It arises as a natural generalization of the Euclidean distance. A Mahalanobis distance accounts for the effects of differences in scales and associations among magnitudes. Consequently, building on the well-known performance of the Mahalanobis distance, our novel proposal seems especially fit for the cases when the measurement units of the issues are different, e.g., performance appraisal processes when employees are evaluated attending to their productivity and their leadership capacity; or where the issues are correlated. For example, evaluation of related public projects. An antecedent for the weaker case of profiles of preferences has been provided elsewhere, cf. Alcantud, de Andrés Calle, and González-Arteaga (2013b), and an application to comparisons of real rankings on universities worldwide is developed. Here we apply our new indicator to a real situation, namely, economic forecasts made by several agencies. Since the forecasts concern economic quantities, they have an intrinsic value which is naturally cardinal and also there are relations among them.

The paper is structured as follows. In Section 2, we introduce basic notation and definitions. In Section 3, we set forth the class of the Mahalanobis dissensus measures and their main properties. Section 4 provides a comparison of several Mahalanobis dissensus measures. Next, a practical application with discussion is given in Section 5. Finally, we present some concluding remarks. Appendices contain proofs of some properties and a short review in matrix algebra.