# A cardinal dissensus measure based on the Mahalanobis distance

TL;DR: A new class of distance-based consensus model, the family of the Mahalanobis dissensus measures for profiles of cardinal values, is proposed.

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.

## Summary (3 min read)

### 1. 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.
- Notwithstanding the use of different ordinal preference frameworks, the problem of how to measure consensus is an open-ended question in several research areas.
- They evaluate two public goods with monetary amounts.
- The Mahalanobis distance plays an important role in Statistics and Data Analysis.
- In Section 3, the authors set forth the class of the Mahalanobis dissensus measures and their main properties.

### 2. Notation and definitions

- This section is devoted to introduce some notation and a new concept in order to compare group cohesiveness: namely, dissensus measures.
- The authors partially borrow notation and definitions from Alcantud et al. (2013b).
- The authors consider that each expert evaluates each alternative by means of a quantitative value.
- N×k A profile M ∈ MN×k is unanimous if the evaluations for all the alternatives are the same across experts.
- The terms consensus and dissensus should not be taken as formal antonyms, especially because a universally accepted definition of consensus is not available and the authors do not intend to give an absolute concept of dissensus.

### 3. The class of Mahalanobis dissensus measures and its properties

- The authors interest is to cover the specific characteristics in cardinal profiles, like possible differences in scales, and correlations among the issues.
- Before providing their main definition, the authors recover the definition of the Mahalanobis distance on which their measure is based.
- The off-diagonal elements of Σ permit to account for cross relations among the issues or alternatives.
- The authors have only used the fact that the permutation matrix Pπ is orthogonal.

### 3.1. Some particular specifications

- Some special instances of Mahalanobis dissensus measures have specific interpretations.
- This ex- pression uses the square of the Euclidean distance between real-valued vectors, thus it recovers a version of the consensus measure for ordinal preferences based on this distance (Cook and Seiford (1982)).
- Henceforth δI is called the Euclidean dissensus measure.
- This particular specification of the dissensus measure allows to incorporate different weights to the alternatives.
- This fact increases the richness of the analysis in comparison with the (square of the) Euclidean distance.

### 3.2. Some properties of the class of Mahalanobis dissensus measures

- Measuring cohesiveness by means of the Mahalanobis dissensus measure ensures some interesting operational features.
- The following properties hold true: 1. Neutrality .
- If for a particular size N of a society the Mahalanobis dissensus measures associated with two matrices coincide for all possible profiles, then the corresponding dissensus measures are equal.
- So an important question arises about if the scale choice disturbs the cohesiveness measures.
- If the assessments of the new agent coincide with the average of the original agents’ evaluations for each alternative, then the minimal increment of the dissensus measure is obtained.

### 4. Comparison of Mahalanobis dissensus measures

- In practical situations the authors could potentially use various Mahalanobis dissensus measures for profiles of cardinal information.
- Theorems 1 and 2 below identify conditions on matrices that ensure consistent comparisons between Mahalanobis dissensus measures, whatever the number of agents.
- Nevertheless, Theorem 2 below proves that a partial converse of Theorem 1 holds true under a technical restriction on the definite matrices.
- Therefore it is not true that δΣ1(M) ≥ δΣ2(M) holds throughout.
- Moreover, distance dΣ is always between the values of the corresponding distances dλ1I and dλkI .

### 5. Discussion on practical application using a real example

- The authors are interested in assessing the cohesiveness of the forecasts of various magnitudes for the Spanish Economy in 2014: GDP (Gross Domestic Product), Unemployment Rate, Public Deficit, Public Debt and Inflation.
- These forecasts have been published by different institutions and organizations, and each one was made at around the same time.
- Next, the authors select a suitable reference matrix and finally they make the computations of the Mahalanobis dissensus measures.

### 5.1. Reference matrix

- Once the profiles have been fixed, the following step to compute their Mahalanobis dissensus measures is to avail oneself of a suitable reference matrix Σ.
- This matrix contains the variances and covariances among the statistical variables, therefore, those characteristics are brought into play in this distance.
- One exception is the unlikely case when the data are generated by a known multivariate probability distribution.
- It seems natural to produce such a matrix from historical macroeconomic data corresponding to the issues under inspection.
- The ellipses slant upward (resp., downward) show a positive (resp., negative) correlation.

### 5.2. Computation of the dissensus

- Now the authors calculate the Mahalanobis dissensus measures associated with Σ for the profiles of the forecasts for the Spanish Economy, namely, M (S), M (A) and M (lS).
- Table 6 provides these items for comparison.

### 5.3. Other simpler approaches: Drawbacks or limitations

- The choice of the reference matrix is a key point in the application of the Mahalanobis dissensus measure.
- However the choice of the identity matrix as the reference matrix discards much relevant information.
- The authors remove the effects of the interdependence among the economic magnitudes on the dissensus measure.
- Table 7 shows the dissensus measures derived from the three matrices mentioned above, Σ, I and Σσ.

### 6. Concluding remarks

- The authors use the general concept of dissensus measure and introduce one particular formulation based on the Mahalanobis distance for numerical vectors, namely the Mahalanobis dissensus measure.
- The authors provide some properties which make their proposal appealing.
- T. González-Arteaga acknowledges financial support by the Spanish Ministerio de Economı́a y Competitividad (Project ECO2012-32178).
- The authors define the permutation matrix Pπ whose rows are eπ(i).

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