Modelling influence in group decision making

Abstract

Group decision making has been widely studied since group decision making processes are very common in many fields. Formal representation of the experts’ opinions, aggregation of assessments or selection of the best alternatives has been some of main areas addressed by scientists and researchers. In this paper, we focus on another promising area, the study of group decision making processes from the concept of influence and social networks. In order to do so, we present a novel model that gathers the experts’ initial opinions and provides a framework to represent the influence of a given expert over the other(s). With this proposal it is feasible to estimate both the evolution of the group decision making process and the final solution before carrying out the group discussion process and consequently foreseeing possible actions.

Appendices

Appendix 1: Fuzzy quantifiers and their use to model fuzzy majority

Fuzzy majority is a soft majority concept expressed by a fuzzy quantifier. This fuzzy quantifier is manipulated by means of a fuzzy logic-based calculus of linguistically quantified propositions. Hence, the use of fuzzy-majority-guided aggregation operators allows us to incorporate the concept of majority into the computation of the solution.

Quantifiers are used to represent the amount of items satisfying a given predicate. Classic logic defines two quantifiers, there exists and for all, however, this can be seen as an important drawback because human discourse is much richer and more diverse. In order to provide a more flexible knowledge representation tool, Zadeh introduced the concept of fuzzy quantifiers (Zadeh 1983).

Zadeh suggested that the semantics of a fuzzy quantifier can be captured using fuzzy subsets for its representation. Moreover, he differentiated between two types of fuzzy quantifiers: absolute and relative ones. In this model, we have focused on relative quantifiers, such as most, at least half, etc., since they can symbolise any quantifier of natural language. These quantifiers can be represented by fuzzy subsets of the unit interval $\left[ 0,1\right]$. For any $r\in \left[ 0,1\right]$, $Q\left( r\right)$ indicates the degree in which the proportion r is compatible with the meaning of the quantifier it represents.

A relative quantifier $Q:\left[ 0,1\right] \rightarrow \left[ 0,1\right]$ satisfies

$$\begin{aligned} Q\left( 0\right) =0\,\exists r\in \left[ 0,1\right] \,such\,that\,Q\left( r\right) =1 \end{aligned}$$

Yager (1996) identified two categories of relative quantifiers: regular increasing monotone (RIM) quantifiers and regular decreasing monotone (RDM) quantifiers. The first category is characterised by quantifiers such as all, most, many, at least $\alpha$; and the second one by at most one, few, at most $\alpha$, being the former one, the category used in this proposal.

A RIM quantifier satisfies

$$\begin{aligned} \forall a,b\,if\,a>b\,then\,Q\left( a\right) \ge Q\left( b\right) . \end{aligned}$$

Yager (1996) considers the parameterised family of RIM quantifiers

$$\begin{aligned} Q\left( r\right) =r^{\alpha }, \, \alpha \ge 0 \end{aligned}$$

When this family of RIM quantifiers is used with OWA and IOWA operators, it is important to realise that $\alpha < 1$ to associate high weighting values with high consistent ones. In particular, in this paper, we use the RIM function $Q\left( r\right) =r^{1/2}$.

Appendix 2: Example of group decision making

Let $E=\left\{ e_{1},e_{2},e_{3},e_{4} \right\}$ be the group of four experts and $X=\left\{ x_{1},x_{2},x_{3},x_{4}\right\}$ be the set of four alternatives. This group of experts, E, express their preferences about the set of alternatives, X, by means of fuzzy preference relations, $\left\{ P_{1},P_{2},P_{3},P_{4}\right\}$, $P_{k}=\left[ p_{ij}^{k}\right]$, $p_{ij}^{k} \in \left[ 0,1\right]$, which are additive reciprocal.

Consider the following preferences over the set of alternatives X:

$$\begin{aligned}&P_{1}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0.5 &{} 0.3 &{} 0.7 &{} 0.1 \\ 0.7 &{} 0.5 &{} 0.6 &{} 0.6 \\ 0.3 &{} 0.4 &{} 0.5 &{} 0.2 \\ 0.9 &{} 0.4 &{} 0.8 &{} 0.5 \\ \end{array}\right) P_{2}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0.5 &{} 0.4 &{} 0.6 &{} 0.2 \\ 0.6 &{} 0.5 &{} 0.7 &{} 0.4 \\ 0.4 &{} 0.3 &{} 0.5 &{} 0.1 \\ 0.8 &{} 0.6 &{} 0.9 &{} 0.5 \\ \end{array}\right) \\&P_{3}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0.5 &{} 0.5 &{} 0.7 &{} 0 \\ 0.5 &{} 0.5 &{} 0.8 &{} 0.4 \\ 0.3 &{} 0.2 &{} 0.5 &{} 0.2\\ 1 &{} 0.6 &{} 0.8 &{} 0.5 \\ \end{array}\right) P_{4}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0.5 &{} 0.4 &{} 0.7 &{} 0.8 \\ 0.6 &{} 0.5 &{} 0.4 &{} 0.3 \\ 0.3 &{} 0.6 &{} 0.5 &{} 0.1 \\ 0.2 &{} 0.7 &{} 0.9 &{} 0.5 \\ \end{array}\right) , \end{aligned}$$

and their respective expert’s importance $I=\lbrace 0.75,1,0.5,0.25\rbrace$.

We use the fuzzy linguistic quantifier ’most of’ defined by $Q\left( r\right) =r^{1/2}$ (Yager 1996), with its corresponding weighting vector $W=\left( 0.5, 0.21, 0.16, 0.13 \right)$. By using the I-IOWA operator, the following collective preference relation is computed:

$$\begin{aligned} P_{c}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0.5 &{} 0.4 &{} 0.65 &{} 0.23 \\ 0.6 &{} 0.5 &{} 0.65 &{} 0.43 \\ 0.35 &{} 0.35 &{} 0.5 &{} 0.14 \\ 0.77 &{} 0.57 &{} 0.86 &{} 0.5 \\ \end{array}\right) .\end{aligned}$$

Now, if we use the quantifier-guided dominance degree, $QGDD_{i}$ with the weighting vector $W_{QGDD}=(0.5,0.21,0.16,0.13)$, the following utility vector is obtained:

$$\begin{aligned} U^{c}=\left( \begin{array}{c} 0.52 \\ 0.59 \\ 0.4 \\ 0.75 \\ \end{array}\right) , \end{aligned}$$

and, therefore, the final solution is

$$\begin{aligned} \left( x_{4},x_{2},x_{1},x_{3} \right) , \end{aligned}$$

i.e. $x_4$ is the preferred alternative, $x_2$ is the second one, etc.