Keywords

1 Introduction

The notion of consensus is central to decision making models involving the aggregation of individual preferences. We can distinguish essentially two complementary readings of the consensus concept. In general terms, it refers to the consensual preference resulting from the aggregation scheme, whether or not preference aggregation is formulated as an iterative consensus reaching process. More specifically, the notion of consensus refers to the construction of consensus (dissensus) measures, which express the level of agreement (disagreement) present in the collective profile of individual preferences.

In general, consensual aggregation models involve some form of explicit or implicit averaging of the individual preferences. In the context of aggregation theory, comprehensive reviews of averaging functions can be found in [2, 7, 21, 33, 34].

In our approach we are primarily interested in the class of aggregation schemes which are based on consensus (dissensus) measures, often constructed on the basis of some binary distance acting pairwise on the individual preferences.

In this respect, the recent literature on the use of penalty functions in aggregation [3,4,5,6, 17, 18, 20] provides a suggestive framework in which to describe the interrelation between aggregation functions and consensus (dissensus) measures.

Further interesting investigation on the construction and applications of consensus (dissensus) measures can be found in [1, 8,9,10,11, 13,14,15,16, 19, 23,24,26, 46, 47, 49,50,53].

In the tradition of the fuzzy approach to consensus in the aggregation of individual preferences [31, 32, 35, 36, 40, 41], the soft consensus model was originally proposed in [37,38,39] and later reformulated in [27,28,29,30]. The soft consensus model is based on a dissensus measure constructed from pairwise square differences, composed with a subadditive scaling function (substituting the linguistic quantifiers in the original version of the model), whose role is that of emphasizing small (attenuating large) preference differences by means of a smooth thresholding effect.

In this paper we wish to revisit the soft consensus model and investigate the formulation of the soft dissensus measure in relation with the notion of multidistance, recently introduced in [22, 42,43,44,45, 48]. The concept of multidistance is as an extension of the classical concept of binary distance, obtained by means of a generalization of the triangular inequality.

With respect to the traditional soft consensus model, here the idea is to construct a new multidistance dissensus measure directly from the pairwise absolute value differences and the subadditive scaling function, keeping the traditional character of the soft dissensus measure but avoiding the square differences in the functional form.

The paper is organized as follows. In Sect. 2 we briefly review the soft consensus model and the construction of the traditional soft dissensus measure. In Sect. 3 we review the basic notions regarding multidistances and in Sect. 4 we introduce the new multidistance dissensus measure, with a detailed study of the subadditive scaling function. Finally, in Sect. 5 we present some concluding remarks and notes on future research.

2 The Soft Consensus Model

In this section we present a brief review of the traditional soft consensus model in the formulation introduced in [27]. Our point of departure is a set of individual fuzzy preference relations. If \(A = \{a_1,\dots ,a_m\}\) is a set of decisional alternatives and \(I =\{1,\dots , n\}\) is a set of individuals, the fuzzy preference relation \(R_i\) of individual i is given by its membership function \(R_i : A \times A \rightarrow [0,1]\) with

$$\begin{aligned} \begin{array}{ll} R_i (a_k, a_l) = \quad 1 \quad &{} \text{ if }\, a_k\,\text{ is } \text{ definitely } \text{ preferred } \text{ over }\, a_l \\ R_i (a_k, a_l) \in (0.5, 1) \quad &{} \text{ if }\, a_k\,\text{ is } \text{ preferred } \text{ over }\, a_l \\ R_i (a_k, a_l) = \quad 0.5 \quad &{} \text{ if } \, a_k \, \text{ is } \text{ considered } \text{ indifferent } \text{ to }\, a_l \\ R_i (a_k, a_l) \in (0, 0.5) \quad &{} \text{ if } \, a_l \, \text{ is } \text{ preferred } \text{ over }\, a_k \\ R_i (a_k, a_l) = \quad 0 \quad &{} \text{ if } \, a_l \, \text{ is } \text{ definitely } \text{ preferred } \text{ over }\, a_k, \end{array} \end{aligned}$$

where \(i=1,\dots ,n\) and \(k,l=1,\dots ,m\). Each individual fuzzy preference relation \(R_i\) can be represented by a matrix \([r^i_{kl}]\), \(r^i_{kl} = R_i (a_k, a_l)\) which is commonly assumed to be reciprocal, that is \(r^i_{kl} + r^i_{lk} = 1\). Clearly, this implies \(r^i_{kk} = 0.5\) for all \(i=1,\dots ,n\) and \(k=1,\dots ,m\).

The general case \(A = \{a_1,\dots ,a_m\}\) for the set of decisional alternatives is discussed in [27, 28]. Here, for the sake of simplicity, we assume that the alternatives available are only two (\(m=2\)), which means that each individual preference relation \(R_i\) has only one degree of freedom, denoted by \( x_i = r^i_{12} \).

In the framework of the soft consensus model, assuming \(m=2\), the degree of dissensus between individuals i and j as to their preferences between the two alternatives is measured by

$$\begin{aligned} V_{ij} = g((x_i - x_j)^2) \qquad i,j =1,\ldots ,n \end{aligned}$$
(1)

where \(g : [0,1] \rightarrow \mathbb {R}\) is a scaling function defined as

$$\begin{aligned} g (u) = \frac{1}{\alpha }\ln \bigg (\frac{1}{1 + e^{- \alpha (u - \beta )}}\bigg ) \qquad u\in [0,1] \, . \end{aligned}$$
(2)

In the scaling function formula above, \(\beta \in (0,1)\) is a threshold parameter and \(\alpha \in (0,\infty )\) is a free parameter which controls the polarization of the sigmoid function \(g' : [0,1] \rightarrow (0,1)\) given by

$$\begin{aligned} g' (u) = \frac{1}{1 + e^{\alpha (u - \beta )}} \qquad u\in [0,1]. \end{aligned}$$
(3)

In the network representation of the soft consensus model [27], each decision maker \(i=1,\dots ,n\) is represented by a pair of connected nodes, a primary node (dynamic) and a secondary node (static). The n primary nodes form a fully connected subnetwork and each of them encodes the individual opinion of a single decision maker. The n secondary nodes, on the other hand, encode the individual opinions originally declared by the decision makers, denoted \(s_{i} \in [0,1]\), and each of them is connected only with the associated primary node.

The iterative process of preference change corresponds to the gradient descent optimization of a cost function W, depending on both the present and the original network configurations. The value of W combines a measure V of the overall dissensus in the present network configuration with a measure U of the overall change from the original network configuration.

The various interactions involving node i are modulated by interaction coefficients whose role is to quantify the strength of the interaction. The consensual interaction between primary nodes i and j is modulated by the interaction coefficient \(v_{ij} \in (0,1)\), whereas the inertial interaction between primary node i and the associated secondary node is modulated by the interaction coefficient \(u_{i} \in (0,1)\). In the soft consensus model the values of these interaction coefficients are given by the derivative \(g'\) of the scaling function according to

$$\begin{aligned} v_{ij} = g' ((x_i - x_j)^2) \qquad i,j =1,\ldots ,n \end{aligned}$$
(4)
$$\begin{aligned} v_{i} = \sum _{j(\ne i)=1}^n v_{ij} / (n-1), \quad u_{i} = g' ((x_i - s_i)^2) \qquad i =1,\ldots ,n. \end{aligned}$$
(5)

The average preference \(\bar{x}_{i}\) of the context of individual i is given by

$$\begin{aligned} \bar{x}_{i} = \frac{ \sum _{j(\ne i)=1}^n v_{ij}\, x_{j}}{\sum _{j(\ne i)=1}^n v_{ij}} \qquad i =1,\ldots ,n \end{aligned}$$
(6)

and represents the average preference of the remaining decision makers as seen by decision maker \(i=1,...,n\).

The construction of the cost function W that drives the dynamics of the soft consensus model is as follows. The individual dissensus cost is given by

$$\begin{aligned} V_i (\mathbf x ) = \sum _{j(\ne i)=1}^n V_{ij} / (n-1) \qquad i =1,\ldots ,n \end{aligned}$$
(7)

and the individual opinion changing cost is

$$\begin{aligned} U_i (\mathbf x ) = g ( ( x_{i} - s_i )^2 ) \qquad i =1,\ldots ,n. \end{aligned}$$
(8)

Summing over the various decision makers we obtain the collective dissensus cost V and inertial cost U,

$$\begin{aligned} V(\mathbf x ) = \frac{1}{4} \; \sum _{i=1}^n V_i (\mathbf x ), \quad U(\mathbf x ) = \frac{1}{2} \; \sum _{i=1}^n U_i (\mathbf x ) \end{aligned}$$
(9)

with conventional multiplicative factors 1 / 4 and 1 / 2. The full cost function is then

$$\begin{aligned} W(\mathbf x ) = V(\mathbf x ) + U(\mathbf x ). \end{aligned}$$
(10)

The consensual network dynamics acts on the individual opinion variables \(x_{i}\) through the iterative process

$$\begin{aligned} x_i \leadsto x_{i}' = x_{i} - \gamma \frac{\partial W}{\partial x_{i}} \qquad i=1, \ldots , n. \end{aligned}$$
(11)

Analyzing the effect of the two dynamical components V and U separately we obtain

$$\begin{aligned} \frac{\partial V}{\partial x_{i}} = v_{i} (x_{i} - {\bar{x}}_{i})\qquad i=1, \ldots , n \end{aligned}$$
(12)

where the coefficients \(v_{i}\) were defined in (5) and the average preference \({\bar{x}}_{i}\) was defined in (6), and therefore

$$\begin{aligned} x_{i}' = (1 - \gamma \, v_{i}) x_{i} + \gamma \, v_{i} {\bar{x}}_{i} \qquad i=1, \ldots , n. \end{aligned}$$
(13)

On the other hand, we obtain

$$\begin{aligned} \frac{\partial U}{\partial x_{i}} = u_{i} (x_{i} - s_{i}) \qquad i=1, \ldots , n \end{aligned}$$
(14)

where the coefficients \(u_{i}\) were defined in (5), and therefore

$$\begin{aligned} x_{i}' = (1 - \gamma \, u_{i}) x_{i} + \gamma \, u_{i} s_{i } \qquad i=1, \ldots , n. \end{aligned}$$
(15)

The full dynamics associated with the cost function \(W = V + U\) acts iteratively according to

$$\begin{aligned} x_{i}' = (1 - \gamma \, (v_{i} + u_{i})) x_{i} + + \gamma \, v_{i} {\bar{x}}_{i} + \gamma \, u_{i} s_{i} \qquad i=1, \ldots , n \end{aligned}$$
(16)

and the decision maker i is in dynamical equilibrium, in the sense that \(x_{i}' = x_{i}\), if the following stability equation holds,

$$\begin{aligned} x_{i} = (v_{i} {\bar{x}}_{i} + u_{i} s_{i}) / (v_{i} + u_{i}) \qquad i=1, \ldots , n \end{aligned}$$
(17)

that is, if the present opinion \(x_{i}\) coincides with an appropriate weighted average of the original opinion \(s_{i}\) and the average opinion value \({\bar{x}}_{i}\) for \(i=1, \ldots , n\).

3 The Multidistance Framework

The definition of multidistance has been introduced by Martín and Mayor in [43, 44] as an extension of the classical notion of binary distance to the case of more than two points.

Consider a domain \(X\subseteq \mathbb {R}\), with points in \(X^n\) being denoted as \(\mathbf x =(x_1, \ldots , x_n)\). The multidistance definition given in [44] is as follows.

Definition 1

Given a domain \(X\subseteq \mathbb {R}\), a multidistance is a function

$$ D:\bigcup _{n\ge 2} X^{n} \rightarrow \mathbb {R} $$

with the following properties

  1. (P1)

    \(D(x_{1},\ldots ,x_{n})=0\) if and only if \(x_{i}=x_{j}\) for all \(i,j=1,\ldots ,n\)

  2. (P2)

    \(D(x_{1},\ldots ,x_{n})=D(x_{\pi (1)},\ldots ,x_{\pi (n)})\) for any permutation \(\pi \) of \(1,\ldots ,n\)

  3. (P2)

    \(D(x_{1},\ldots ,x_{n}) \le D(x_{1},y)+ \cdots + D(x_{n},y)\) for all \(y \in X\)

for all \(x_{1},\ldots ,x_{n} \in X\) and \(n \ge 2\). Note that (P1), (P2) and (P3) extend the usual distance axioms. In particular, (P3) generalizes the triangle inequality.

An important class of multidistances, the functionally expressible multidistances, are studied in [45, 48]. Applications of multidistances to the problem of consensus measuring can be found in [16, 22].

Starting from the results obtained in [13, 14] in [16] some connections between m-ary adjacency relations and multidistances were highlighted. It has been shown how m-ary adjacency relations can be modeled on the basis of OWA-based multidistances, and some consensus related optimization problems on m-ary adjacency relations are equivalent to corresponding multidistance minimization problems.

In this paper, a multidistance dissensus measure is introduced as an extension of the relationship between the dissensus measure in the traditional soft consensus model proposed in [27] and the multidistance approach to consensus introduced in [16]. This measure is based on a binary distance defined by means of a subadditive function whose effect is that of emphasizing small distances and attenuating large distances.

There are several methods to construct multidistances. As suggested in [44] given a binary distance \(d(x_i , x_j)\), a multidistance may be defined on the basis of the pairwise binary distances, multiplying their sum by a sufficiently small value \( \lambda (n)\) depending on n. This type of multidistance is called sum–based multidistance.

Proposition 1

(Martín and Mayor [44]) A function \(D :\bigcup _{n\ge 2} X^{n} \rightarrow \mathbb {R}\) defined as

$$ D(x_1,...,x_n) = \lambda (n)\displaystyle {\sum _{i,j=1}^n} d(x_i , x_j) \qquad n\ge 2 $$

is a multidistance if and only if the coefficient \(\lambda (n)\) satisfies \(\lambda (2)=1/2\) and

$$\begin{aligned} 0<\lambda (n)\le \frac{1}{2(n-1)} \qquad n\ge 3 \end{aligned}$$
(18)

where \(x_1, \ldots , x_n\in X\).

In this paper we use the domain \(X=[0,1]\) equipped with the classical distance \(d(x,y)=|x-y|\in [0,1]\), for \(x,y \in [0,1]\), with the usual triangular inequalities \(|x + y|\le |x| +|y|\) and \(d(x,y)\le d(x,z)+d(y,z)\), for all \(x,y,z \in [0,1]\).

Moreover, we consider the particular coefficient choice

$$\begin{aligned} \lambda (n)=\frac{1}{n(n-1)} \end{aligned}$$
(19)

which corresponds to constructing the sum-based multidistance by averaging pairwise binary distances.

Consider now an increasing and subadditive function \( f: [0,1]\rightarrow \mathbb {R}\). Due to the subadditivity of the function f, the composition of the distance d with the function f yields a new distance denoted \(d_f(x,y)= f(d(x,y))\), which satisfies the triangle inequality \(d_f(x,y)\le d_f(x,z)+d_f(y,z)\). This is obtained as follows,

$$\begin{aligned} d(x,y)\le d(x,z)+d(y,z) \end{aligned}$$
(20)
$$\begin{aligned} f(d(x,y))\le f(d(x,z)+d(y,z))\le f(d(x,z))+f(d(y,z)) \end{aligned}$$
(21)

where the first inequality is due to the increasingness of f and the second inequality is due to the subadditivity of f. Finally, we obtain

$$\begin{aligned} d_f(x,y)\le d_f(x,z)+d_f(y,z). \end{aligned}$$
(22)

We consider the construction of multidistances based on the binary distance \(d_f\), in particular by averaging pairwise binary distances. In this way we define a multidistance \(D_f:\bigcup _{n\ge 2}[0,1]^n\rightarrow \mathbb {R}\) as

$$\begin{aligned} D_f(x_1,...,x_n) = \frac{1}{n(n-1)} \, \sum _{i,j=1}^n d_f(x_i,x_j) \, . \end{aligned}$$
(23)

Consider for instance the case \(n=3\). The multidistance \(D_f\) is given by

$$\begin{aligned} D_f(\mathbf x ) = \frac{1}{6}\, \sum _{i,j=1}^3 d_f(x_i,x_j)= \end{aligned}$$
(24)
$$ =\frac{1}{3} \Big (d_f(x_1,x_2)+d_f(x_1,x_3)+d_f(x_2,x_3)\Big )= $$
$$ =\frac{1}{3} \Big (f(d(x_1,x_2))+f(d(x_1,x_3))+f(d(x_2,x_3)) \Big )\, . $$

Notice that each binary distance term is of the form

$$\begin{aligned} d_f(x_i,x_j) = c_f(x_i,x_j) \cdot d(x_i,x_j) \end{aligned}$$
(25)

where each classical binary distance \(d(x_i,x_j)\) is multiplied by a coefficient

$$\begin{aligned} c_f(x_i,x_j) = f(d(x_i,x_j) / d(x_i,x_j) \end{aligned}$$
(26)

depending on the choice of the function f.

The multidistance \(D_f\) corresponds to a linear combination of the classical binary distances d, with non negative coefficients \(c_f\). However, notice that these coefficients do not have unit sum and therefore the multidistance \(D_f\) does not correspond to a weighted mean of of the classical binary distances d.

4 The Soft Dissensus Measure in the Multidistance Framework

The traditional soft consensus model in group decision making [27, 28] is based on a non linear dissensus measure whose role is that of emphasizing small distances and attenuating large distances in the preference domain.

In this section we reformulate the soft dissensus measure as a sum-based multidistance, in the approach introduced in [12]. In the new multidistance framework, the usual binary distance is composed with a non linear subadditive function \( f: [0,1]\rightarrow \mathbb {R}\) defined as

$$\begin{aligned} f(u)= \frac{2}{\alpha }\ln \bigg (\frac{1+e^{\alpha \beta }}{1+e^{-\alpha (u-\beta )}}\bigg ) \qquad \alpha \in (0, \infty ) \quad \beta \in [0, 1] \end{aligned}$$
(27)

for all \(u \in [0,1]\). The two parameters are \(\alpha \in (0, \infty )\) and \(\beta \in [0, 1]\), but we can extend the domain of the former by defining \(f(u)=u\) for \(\alpha =0\), which in fact corresponds to the asymptotic form of definition (27) at \(\alpha =0^+\).

In Fig. 1 we plot the function f(u) with \(u \in [0,1]\) for various choices of the parameters \(\alpha \), \(\beta \). In each plot the diagonal line \(f(u)=u \in [0,1]\) is associated with the case \(\alpha =0\).

Fig. 1
figure 1

The function f(u) for \(u\!\in \![0,1]\). Each plot is associated with a choice of the parameter \(\beta \) and shows the graph of f(u) for various choices of the parameter \(\alpha \)

Full size image

In the following result we determine the values of the function f at the boundaries of its domain [0, 1], for any choice of the parameters \(\alpha \), \(\beta \).

Proposition 2

In relation with the function f defined in (27), we obtain \(f(0)=0\) for any choice of the parameters \(\alpha \in (0, \infty )\) and \(\beta \in [0, 1]\), and

  • \(f(1) < 1\) for all \(\beta \in [0,1/2)\)

  • \(f(1)=1\) for \(\beta =1/2\)

  • \(f(1) > 1\) for all \(\beta \in (1/2, 1]\)

for any choice of the parameter \(\alpha \in (0, \infty )\). In particular, the values of f(1) for \(\beta =0,1\) are

$$ f(1)=\frac{2}{\alpha }\ln \bigg (\frac{2}{1+e^{-\alpha }}\bigg ), \; \beta =0 \qquad f(1)= \frac{2}{\alpha }\ln \bigg (\frac{1+e^{\alpha }}{2}\bigg ), \; \beta =1\, . $$

The limit at \(\alpha =0^+\) is \(f(1)=1\) in both cases \(\beta =0,1\), whereas the limit at \(\alpha =\infty \) is \(f(1)=0\) for \(\beta =0\) and \(f(1)=2\) for \(\beta =1\).

Proof

From definition (27) we obtain immediately that \(f(0)=0\) for any choice of the parameters \(\alpha \), \(\beta \), plus also

$$\begin{aligned} f(1)= 1+\frac{2}{\alpha }\ln \bigg (\frac{1+e^{\alpha \beta }}{e^{\alpha /2}+e^{\alpha \beta }e^{-\alpha /2}}\bigg ) \end{aligned}$$
(28)

which leads immediately to \(f(1)=1\) for \(\beta =1/2\). Otherwise, writing \(N = 1+e^{\alpha \beta }\) for the numerator and \(D = e^{\alpha /2}+e^{\alpha \beta }e^{-\alpha /2}\) for the denominator of the logarithm, it follows that

$$\begin{aligned} N-D = (1-e^{-\alpha /2})(e^{\alpha \beta }-e^{\alpha /2})\, . \end{aligned}$$
(29)

Considering the second factor in the product, we conclude that the logarithmic term in (29) is negative (\(N < D\)) for \(\beta \in [0,1/2)\) and is positive (\(N > D\)) for \(\beta \in (1/2, 1]\). The asymptotic limits of f(1) with respect to the parameter \(\alpha \) for \(\beta =0,1\) can be obtained straightforwardly by means of l’Hospital’s rule. \(\square \)

The function f is continuous, strictly increasing and strictly concave in \(u\in [0,1]\) for any choice of the parameters \(\alpha , \beta \). Continuity is clear from definition (27) and the other properties follow directly from the first and second derivatives of f,

$$\begin{aligned} f\,'(u)= \frac{2}{1+e^{\alpha (u-\beta ) }} \qquad \quad f\,'(u)\in (0,2) \end{aligned}$$
(30)
$$\begin{aligned} f\,''(u)= -\frac{2\alpha e^{\alpha (u + \beta )}}{(e^{\alpha \beta } + e^{\alpha u } )^2} \qquad \quad f\,''(u)\in (-\alpha /2 ,0) \end{aligned}$$
(31)

Notice that \(f\,'(u=\beta )=1\) and \(f\,''(u=\beta )=-\alpha /2 \). Moreover, we can show that \(f\,''(u)= -\alpha f\,'(u) (2-f\,'(u))/2\) for any choice of the parameters \(\alpha \), \(\beta \). In the case \(\alpha = 0\), for any choice of \(\beta \), we have the linear form \(f(u)=u\) for all \(u\in [0,1]\).

Given that f is (strictly) increasing and \(f(0)=0\) for any choice of the parameters \(\alpha ,\beta \), we have that \(f(u)\ge 0\) for all \(u\in [0,1]\). Moreover, we can write

$$\begin{aligned} f(u) = 2u + \frac{2}{\alpha } \ln \bigg (\frac{1+e^{\alpha \beta }}{e^{\alpha u }+e^{\alpha \beta }}\bigg ) \end{aligned}$$
(32)

for all \(u\in [0,1]\), where the logarithmic term is always non positive. Therefore, we obtain \(0\le f(u)\le 2u\) for all \(u\in [0,1]\) and any choice of the parameters \(\alpha ,\beta \).

The function f is subadditive, in the sense that \(f(u+v)\le f(u)+f(v)\). The proof is as follows: assuming \(u, \, v \in [0, 1]\) and \(u+v\ne 0\), concavity of f implies

$$\begin{aligned} f(u)\ge \frac{v}{u+v}\, f(0) + \frac{u}{u+v}\, f(u+v) = \frac{u}{u+v}\, f(u+v) \end{aligned}$$
(33)
$$\begin{aligned} f(v)\ge \frac{u}{u+v}\, f(0) + \frac{v}{u+v}\, f(u+v) = \frac{v}{u+v}\, f(u+v) \end{aligned}$$
(34)

and therefore we obtain \(f(u)+f(v)\ge f(u+v)\) for \(u, \, v \in [0,1]\).

The composition of the distance d with the subadditive function f yields a new distance denoted

$$\begin{aligned} d_f(x,y)= f(d(x,y)) \end{aligned}$$
(35)

satisfying the triangle inequality \(d_f(x,y)\le d_f(x,z)+d_f(y,z)\) as in (20)–(22).

We define the multidistance \(D_f\) by averaging pairwise binary distances \(d_f\),

$$\begin{aligned} D_f(\mathbf x ) = \frac{1}{n(n-1)}\, \sum _{i,j=1}^n d_f(x_i,x_j) = \frac{1}{n(n-1)}\, \sum _{i,j=1}^n f(d(x_i,x_j))\, . \end{aligned}$$
(36)

This sum–based multidistance is a natural nonlinear measure of dissensus, analogous but not equivalent to the traditional soft dissensus measure V in (9). The new soft dissensus measure, however, has a more appealing geometrical interpretation as a multidistance.

Finally, recall that each term in (36) is of the form \(d_f(x_i,x_j) = c_f(x_i,x_j) \cdot d(x_i,x_j)\), where each binary distance \(d(x_i,x_j)\) is multiplied by a coefficient \(c_f(x_i,x_j) = f(d(x_i,x_j) / d(x_i,x_j)\) depending on the choice of the function f.

In other words, each term is of the form \( f(u) = (f(u) / u) \cdot u \), where each single binary distance u is multiplied by a coefficient f(u) / u which is decreasing with respect to the distance u.

In Fig. 2 we plot the function f(u) / u with various choices of the parameters \(\alpha \) and \(\beta \). In each plot the horizontal line is associated with \(\alpha =0\) and the remaining lines are associated with \(\alpha =1, 2, 3, 4\). In the case \(\alpha =0\) all pairwise distances have the same weight 1 / 6 and thus the multidistance corresponds to the weighted average. In the case \(\alpha =1, 2, 3, 4\) the function f(u) / u is monotonically decreasing with respect to pairwise distances, in which a larger weight is assigned to a small distance and a smaller weight is given to a large distance.

The multidistance is defined as the weighted sum of pairwise distances. In this approach, the sum-based multidistance is closely related with the disensus measure in the soft consensus model. There is essentially a single difference: the basic pairwise distance \(d_f(x,y)\) involves \(|x-y|\) and not \((x-y)^2\), which is not a binary distance.

Fig. 2
figure 2

The function \(f(u)\!/\!u\) for \(u\!\in \! [0,1]\). Each plot is associated with a choice of the parameter \(\beta \) and shows the graph of \(f(u)\!/\!u\) for various choices of the parameter \(\alpha \)

Full size image

5 Conclusions

We introduce a multidistance measure of dissensus within the framework of the soft consensus model of group decision making. The multidistance dissensus measure is based on a fundamental binary distance \(d_f\) associated with a subadditive function f over the domain \(X=[0,1]\), with \(d_f(x,y)=f(|x-y|)\). This subadditive function has the effect of emphasizing small distances and attenuating large distances, in analogy with the subadditive scaling function g which plays a central role in the traditional soft consensus model [27, 28, 30].