José Luis García-Lapresta

A general class of simple majority decision rules based on linguistic opinions

In this paper we have introduced a class of decision rules related to simple majority, by considering individual intensities of preference. These intensities will be shown by means of linguistic labels. In order to compare the amount of opinion obtained by each alternative, we have considered the total ordered monoid generated by the sums of the original labels, according to an addition and an ordering. In this general framework different sets of linguistic labels can be employed and these sets can be represented by means of diverse mathematical objects. Moreover, on these mathematical representations of linguistic labels several orderings can be considered. Thus, flexibility is an important feature of this new class of group decision making procedures. Some examples of putting in practice the simple majority decision rules based on linguistic labels are provided, and the main properties of these voting systems are analyzed. It is worth emphasizing that these properties are satisfied for any total ordered monoid, regardless of the mathematical representation of linguistic labels or the ordering used to compare collective opinions.

Majority decisions based on difference of votes

Abstract In this paper a class of voting procedures, located between simple and unanimous majorities, is introduced and characterized. Given two alternatives, the winning alternative is the one with a number of votes exceeding that obtained by the other in a previously fixed quantity. Moreover, a subclass of these voting procedures has been considered, by demanding additionally a number of votes greater than a previously fixed threshold. The main results of this paper are characterizations of these voting procedures through aggregation functions of fuzzy preferences associated with quasiarithmetic means and ordered weighted averaging (OWA) operators.

A multi-granular linguistic model for management decision-making in performance appraisal

The performance appraisal is a relevant process to keep and improve the competitiveness of companies in nowadays. In spite of this relevance, the current performance appraisal models are not sufficiently well-defined either designed for the evaluation framework in which they are defined. This paper proposes a performance appraisal model where the assessments are modelled by means of linguistic information provided by different sets of reviewers in order to manage the uncertainty and subjectivity of such assessments. Therefore, the reviewers could express their assessments in different linguistic scales according to their knowledge about the evaluated employees, defining a multi-granular linguistic evaluation framework. Additionally, the proposed model will manage the multi-granular linguistic labels provided by appraisers in order to compute collective assessments about the employees that will be used by the management team to make the final decision about them.

Measuring Consensus in Weak Orders

In this chapter we focus our attention in how to measure consensus in groups of agents when they show their preferences over a fixed set of alternatives or candidates by means of weak orders (complete preorders). We have introduced a new class of consensus measures on weak orders based on distances, and we have analyzed some of their properties paying special attention to seven well-known distances.

The self-dual core and the anti-self-dual remainder of an aggregation operator

In most decisional models based on pairwise comparison between alternatives, the reciprocity of the individual preference representations expresses a natural assumption of rationality. In those models self-dual aggregation operators play a central role, in so far as they preserve the reciprocity of the preference representations in the aggregation mechanism from individual to collective preferences. In this paper we propose a simple method by which one can associate a self-dual aggregation operator to any aggregation operator on the unit interval. The resulting aggregation operator is said to be the self-dual core of the original one, and inherits most of its properties. Our method constitutes thus a new characterization of self-duality, with some technical advantages relatively to the traditional symmetric sums method due to Silvert. In our framework, moreover, every aggregation operator can be written as a sum of a self-dual core and an anti-self-dual remainder which, in some cases, seems to give some indication on the dispersion of the variables. In order to illustrate the method proposed, we apply it to two important classes of continuous aggregation operators with the properties of idempotency, symmetry, and stability for translations: the OWA operators and the exponential quasiarithmetic means.

Consensus-based clustering under hesitant qualitative assessments

In this paper, we consider that agents judge the feasible alternatives through linguistic terms - when they are confident in their opinions - or linguistic expressions formed by several consecutive linguistic terms - when they hesitate. In this context, we propose an agglomerative hierarchical clustering process where the clusters of agents are generated by using a distance-based consensus measure.

Favoring Consensus and Penalizing Disagreement in Group Decision Making

In this paper we introduce a multi-stage decision making procedure where decision makers’ opinions are weighted by their contribution to the agreement after they sort alternatives into a fixed finite scale given by linguistic categories, each one having an associated numerical score. We add scores obtained for each alternative using an aggregation operator. Based on distances among vectors of individual and collective scores, we assign an index to decision makers showing their contributions to the agreement. Opinions of negative contributors are excluded and the process is reinitiated until all decision makers contribute positively to the agreement. To obtain the final collective weak order on the set of alternatives, we weigh the scores that decision makers assign to alternatives by indices corresponding to their contribution to the agreement.

Measuring consensus in a preference-approval context

We consider measuring the degree of homogeneity for preference-approval profiles which include the approval information for the alternatives as well as the rankings of them. A distance-based approach is followed to measure the disagreement for any given two preference-approvals. Under the condition that a proper metric is used, we propose a measure of consensus which is robust to some extensions of the ordinal framework. This paper also shows that there exists a limit for increasing the homogeneity level in a group of individuals by simply replicating their preference-approvals.

Classical inequality indices, welfare and illfare functions, and the dual decomposition

Abstract In the traditional framework, social welfare functions depend on the mean income and on the income inequality. An alternative illfare framework has been developed to take into account the disutility of unfavorable variables. The illfare level is assumed to increase with the inequality of the distribution. In some social and economic fields, such as those related to employment, health, education, or deprivation, the characteristics of the individuals in the population are represented by bounded variables, which encode either achievements or shortfalls. Accordingly, both the social welfare and the social illfare levels may be assessed depending on the framework we focus on. In this paper we propose a unified dual framework in which welfare and illfare levels can both be investigated and analyzed in a natural way. The dual framework leads to the consistent measurement of achievements and shortfalls, thereby overcoming one important difficulty of the traditional approach, in which the focus on achievements or shortfalls often leads to different inequality rankings. A number of welfare functions associated with inequality indices are OWA operators. Specifically this paper considers the welfare functions associated with the classical inequality measures due to Gini, Bonferroni, and De Vergottini. These three indices incorporate different value judgments in the measurement of inequality, leading to different behavior under income transfers between individuals in the population. In the bounded variables representation, we examine the dual decomposition and the orness degree of the three classical welfare/illfare functions in the standard framework of aggregation functions on the [ 0 , 1 ] n domain. The dual decomposition of each welfare/illfare function into a self-dual central index and an anti-self-dual inequality index leads to the consistent measurement of achievements and shortfalls.

Linguistic-based voting through centered OWA operators

Two linguistic-based voting systems have been introduced in recent years, namely: Majority Judgement (Balinski and Laraki in http://ceco.polytechnique.fr/jugement-majoritaire.html, 2007a) and Range Voting (Smith in http://www.math.temple.edu/~wds/homepage/rangevote.pdf, 2000). The keys for them are aggregation procedures based on the median and the arithmetic mean of the grades assessed to the alternatives, respectively. In this paper a comprehensive framework based on centered OWA operators (Yager in Soft Comput 11:631–639, 2007a) and the 2-tuple model (Herrera and Martínez in IEEE Trans Fuzzy Syst 8:746–752, 2000) is provided to enclose such distinct approaches. In addition, we show how to avoid some drawbacks of Majority Judgement and Range Voting by means of the use of suitable aggregation functions.