Ordinal proximity measures in the context of
unbalanced qualitative scales and some applications to
consensus and clustering
Jos´e Luis Garc´ıa-Lapresta
PRESAD Research Group, IMUVA, Dept. de Econom´ıa Aplicada, Universidad de
Valladolid, Spain
David P´erez-Rom´an
PRESAD Research Group, Dept. de Organizaci´on de Empresas y Comercializaci´on e
Investigaci´on de Mercados, Universidad de Valladolid, Spain
Abstract
In this paper, we introduce ordinal proximity measures in the setting of unbal-
anced qualitative scales by comparing the proximities between linguistic terms
without numbers, in a purely ordinal approach. With this new tool, we pro-
pose how to measure the consensus in a set of agents when they assess a set
of alternatives through an unbalanced qualitative scale. We also introduce an
agglomerative hierarchical clustering procedure based on these consensus mea-
sures.
Keywords: decision making; qualitative scales; proximity; difference measure-
ment; consensus; clustering.
1. Introduction
In different decision-making problems, agents have to show their opinions
on a set of alternatives and then an aggregation procedure is used for generat-
ing a collective outcome: a winning alternative, several winning alternatives, a
ranking on the set of alternatives, etc.
The agents opinions can be provided in very different ways: the favorite
alternative, a subset of acceptable alternatives, a ranking on the set of alterna-
tives, an assessment for each alternative, etc.
When agents assess independently each alternative, the corresponding as-
sessments can be of different nature depending on the context: numerical values,
intervals of real numbers, fuzzy numbers, linguistic terms, etc.
Email addresses: lapresta@eco.uva.es (Jos´e Luis Garc´ıa-Lapresta), david@emp.uva.es
(David P´erez-Rom´an)
Preprint submitted to Applied Soft Computing January 23, 2015
Qualitative scales are formed by linguistic terms. Usually, these scales are
balanced and uniform: there are the same number of positive and negative
terms, and adjacent terms are equidistant (for instance, ‘very bad’, ‘bad’, ‘ac-
ceptable’, ‘good’ and ’very good’). However, sometimes the qualitative scales
are unbalanced: there are different number of positive terms compared to nega-
tive ones
1
, and it is not clear how to measure the nearness between the linguistic
terms of an unbalanced qualitative scale
2
.
In this paper, we do not assign numerical distances between linguistic terms,
but we propose to make pairwise comparisons of psychological proximities be-
tween them. This approach has some similarities with difference measurement
within the classical measurement theory (see Krantz et al. [32, chapter 4] and
Roberts [40, section 3.3]), and also with non-metric multidimensional scaling,
where only the ranks of the psychological distances or proximities are known
(see Bennett and Hays [5], Shepard [41], Coombs [12], Kruskal and Wish [33],
Cox and Cox [13] and Borg and Groenen [7, chapter 9], among others). We have
also to mention Bossert et al. [9] that consider ordinal measures of distances in
the analysis of diversity.
In order to explain how the mentioned comparisons can be made, we con-
sider, as an example, that some journals use the linguistic terms ‘reject’, ‘major
revision’, ‘minor revision’ and ‘accept’ in the evaluation of papers. It has no
sense to assign numerical values neither to these terms nor to distances be-
tween terms. However, an author may feel that the psychological proximity
between ‘minor revision’ and ‘accept’ is bigger than the psychological proxim-
ity between ‘minor revision’ and ‘major revision’. Obviously, this author could
compare psychological proximities between the rest of pairs of linguistic terms.
Initially, this task may seem hard, because there are 16
2
= 256 possible pair-
wise comparisons. Fortunately, it is not necessary to compare all the pairs
3
: the
psychological proximity between two terms does not depend on the order these
terms are presented; the psychological proximity between a term and itself is
always the same and it is bigger than the psychological proximity between two
different terms; etc.
Taking into account the previous ideas, we propose the notion of ordinal
proximity measure as a mapping that assigns an element of a chain to each pair
of psychological proximities between linguistic terms, satisfying four indepen-
dent properties: all the elements in the chain correspond to some psychological
proximity, i.e., no element in the chain is superfluous; psychological proximities
are symmetric, i.e., the order of the pairs is irrelevant in the comparison; the
maximum psychological proximity is reached when comparing a linguistic term
with itself; and given three different linguistic terms, the degree of proximity
1
For instance, Herrera et al. [24] consider the following nine linguistic terms: ‘none’, ‘low’,
‘medium’, ‘almost high’, ‘high’, ‘quite high’, ‘very high’ and ‘total’.
2
Nevertheless, within a fuzzy approach, some cardinal proposals on unbalanced qualitative
scales can be found in Herrera et al. [24] and Cabrerizo et al. [11], among others.
3
In Remark 1 we show that, with four linguistic terms, only between three and six com-
parisons are needed.
2
between the lowest and the highest terms should be smaller than the degrees
of proximity between the lowest and the intermediate terms and between the
intermediate and the highest terms.
Once the ordinal proximity measuring model has been introduced, we pro-
pose consensus measures and agglomerative hierarchical clustering procedures
when a group of agents evaluate the alternatives through a qualitative scale,
taking into account the ordinal proximities between individual assessments.
Given a subset of agents and a subset of alternatives, we define the degree
of consensus as the upper median of the proximities between all the pairs of in-
dividual assessments. We propose a sequential tie-breaking process and provide
some properties of the degrees of consensus.
We have also devised an agglomerative hierarchical clustering procedure
where agents are grouped into clusters by defining the similarity between two
groups of agents with respect to a subset of alternatives as the degree of consen-
sus in the merged group. We have illustrated our proposal from the qualitative
marks obtained by a group of students in several subjects.
The rest of the paper is organized as follows. Section 2 is devoted to intro-
duce and analyze ordinal proximity measures. In Section 3 we propose some
applications to consensus and clustering. And Section 4 includes some conclud-
ing remarks.
2. Ordinal proximity measures
Let A = {1, . . . , m}, with m ≥ 2, be a set of agents and let X = {x
1
, . . . , x
n
},
with n ≥ 2, be the set of alternatives which have to be evaluated. Each agent
assigns a linguistic term to every alternative within a finite linguistic ordered
scale L = {l
1
, . . . , l
g
}, arranged from the lowest to the highest terms
4
, where
the granularity of L is at least 3 (g ≥ 3).
2.1. The model
Consider that the psychological proximity between l
r
∈ L and l
s
∈ L is
represented by π
rs
and let ∆ = {π
rs
| r, s ∈ {1, . . . , g}} be the set of all
possible psychological proximities between linguistic terms
5
.
Although we do not associate numbers to psychological proximities, we as-
sume that it is possible to compare psychological proximities between linguistic
terms through an asymmetric and transitive binary relation on ∆, where
π
rs
π
tu
means that the psychological proximity between l
r
and l
s
is bigger
than the psychological proximity between l
t
and l
u
.
We consider that the following properties should be satisfied for all r, s, t, u ∈
{1, . . . , g}:
4
For instance, Balinski and Laraki [4] consider the following six linguistic terms: ‘to reject’
(l
1
), ‘poor’ (l
2
), ‘acceptable’ (l
3
), ‘good’ (l
4
), ‘very good’ (l
5
) and ‘excellent’ (l
6
).
5
At this stage we do not specify what kind of mathematical objects represent psychological
proximities.
3
1. If neither π
rs
π
tu
nor π
tu
π
rs
, then π
rs
= π
tu
.
2. π
sr
= π
rs
.
3. π
rr
= π
ss
.
4. If s 6= t, then π
rr
π
st
.
5. If r < s < t, then π
rs
π
rt
and π
st
π
rt
.
6. If r < s and (r, s) 6= (1, g), then π
rs
π
1g
.
We now introduce a formal notion of proximity between linguistic terms with
values on a finite chain (linear order) ∆ = {δ
1
, . . . , δ
h
}, with δ
1
· · · δ
h
,
that captures the properties introduced above. The elements of ∆ have no
meaning and they only represent different degrees or proximity, being δ
1
and δ
h
the maximum and minimum degrees of proximity, respectively.
As usual in the setting of linear orders, δ
r
≺ δ
s
means δ
s
δ
r
; δ
r
δ
s
means δ
r
≺ δ
s
or δ
r
= δ
s
; and δ
r
δ
s
means δ
r
δ
s
or δ
r
= δ
s
.
First we assume that all the elements of ∆ are relevant because they are
reached as the degree of proximity between at least a pair of linguistic terms
(exhaustiveness). We also assume that the proximity between a pair of linguistic
terms does not depend on the order these terms are presented (symmetry),
and the maximum proximity between linguistic terms is only reached when
comparing a term with itself. Additionally, we assume that, given three different
linguistic terms, the degree of proximity between the lowest and the highest
terms should be smaller than the degrees of proximity between the lowest and
the intermediate terms and also between the intermediate and the highest terms
(monotonicity).
Definition 1. An ordinal proximity measure on L with values in ∆ is a map-
ping π : L
2
−→ ∆, where π(l
r
, l
s
) = π
rs
means the degree of proximity between
l
r
and l
s
, satisfying the following conditions:
1. Exhaustiveness: For every δ ∈ ∆, there exist l
r
, l
s
∈ L such that δ = π
rs
.
2. Symmetry: π
sr
= π
rs
, for all r, s ∈ {1, . . . , g}.
3. Maximum proximity: π
rs
= δ
1
⇔ r = s, for all r, s ∈ {1, . . . , g}.
4. Monotonicity: min{π
rs
, π
st
} π
rt
, for all r, s, t ∈ {1, . . . , g} such that
r < s < t.
Every ordinal proximity measure can be represented by a g × g symmetric
matrix with coefficients in ∆, where the elements in the main diagonal are
π
rr
= δ
1
, r = 1, . . . , g:
π
11
· · · π
1s
· · · π
1g
· · · · · · · · · · · · · · ·
π
r1
· · · π
rs
· · · π
rg
· · · · · · · · · · · · · · ·
π
g 1
· · · π
g s
· · · π
g g
= (π
rs
) .
This matrix is called proximity matrix.
4
Taking into account the conditions of Definition 1, it is only necessary to
show the upper half proximity matrix
δ
1
π
12
π
13
· · · π
1(g−1)
π
1g
δ
1
π
23
· · · π
2(g−1)
π
2g
· · · · · · · · ·
δ
1
π
(g −1)g
δ
1
.
Proposition 1. The four conditions appearing in Definition 1 are independent.
Proof:
1. The matrix
δ
1
δ
2
δ
3
δ
2
δ
1
δ
3
δ
3
δ
3
δ
1
satisfies conditions 1, 2 and 3, but not condition 4: min{π
12
, π
23
} = δ
3
=
π
13
.
2. The matrix
δ
1
δ
3
δ
4
δ
3
δ
1
δ
3
δ
4
δ
3
δ
2
satisfies conditions 1, 2 and 4, but not condition 3: π
33
= δ
2
6= δ
1
.
3. The matrix
δ
1
δ
2
δ
4
δ
3
δ
1
δ
2
δ
4
δ
2
δ
1
satisfies conditions 1, 3 and 4, but not condition 2: π
12
= δ
2
6= δ
3
= π
21
.
4. The matrix
δ
1
δ
2
δ
4
δ
2
δ
1
δ
2
δ
4
δ
2
δ
1
satisfies conditions 2, 3 and 4, but not condition 1: δ
3
6= π
rs
for all
r, s ∈ {1, . . . , g}.
2.2. Some results
In the following proposition we establish that the minimum proximity be-
tween linguistic terms is only reached when comparing the extreme linguistic
terms l
1
and l
g
.
Proposition 2. For all r, s ∈ {1, . . . , g}, π
rs
= δ
h
⇔ (r, s) ∈ {(1, g), (g, 1)}.
5
