A LINGUISTIC GROUP DECISION MAKING METHOD
BASED ON DISTANCES
Edurne Falc´o and Jos´e Luis Garc´ıa-Lapresta
PRESAD Research Group, Dept. of Applied Economics, University of Valladolid, Valladolid, Spain
Keywords:
Group decision making, Linguistic assessments, Distances.
Abstract:
It is common knowledge that the political voting systems suffer inconsistencies and paradoxes such that Arrow
has shown in his well-known Impossibility Theorem. Recently Balinski and Laraki have introduced a new
voting system called Majority Judgement (MJ) which tries to solve some of these limitations. In MJ voters
have to asses the candidates through linguistic terms belonging to a common language. From this information,
MJ assigns as the collective assessment the lower median of the individual assessments and it considers a
sequential tie-breaking method for ranking the candidates. The present paper provides an extension of MJ
focused to reduce some of the drawbacks that have been detected in MJ by several authors. The model assigns
as the collective assessment a label that minimizes the distance to the individual assessments. In addition, we
propose a new tie-breaking method also based on distances.
1 INTRODUCTION
Social Choice Theory shows that there does not ex-
ist a completely acceptable voting system for electing
and ranking alternatives. The well-known Arrow Im-
possibility Theorem (Arrow, 1963) proves with math-
ematic certainty that no voting system simultaneously
fulfills certain desirable properties
1
.
Recently (Balinski and Laraki, 2007a; Balinski
and Laraki, 2007c) have proposed a voting system
called Majority Judgement (MJ) which tries to avoid
these unsatisfactory results and allows the voters to
assess the alternatives through linguistic labels, as
Excellent, Very good, Good, . .., instead of rank or-
der the alternatives. Among all the individual assess-
ments given by the voters, MJ chooses the median as
the collective assessment. Balinski and Laraki also
describe a tie-breaking process which compares the
number of labels above the collective assessment and
those below of it. These authors also have an exper-
imental analysis of MJ (Balinski and Laraki, 2007b)
carried out in Orsay during the 2007 French presiden-
tial election. In that paper the authors show some in-
teresting properties of MJ and they advocate that this
1
Any voting rule that generates a collective weak order
from every profile of weak orders, and satisfies indepen-
dence of irrelevant alternatives and unanimity is necessarily
dictatorial, insofar as there are at least three alternatives and
three voters.
voting system is easily implemented and that it avoids
the necessity for a second round of voting.
Desirable properties and advantages have been
attributed to MJ against the classical Arrow frame-
work of preferences’ aggregation. Among them are
the possibility that voters show more faithfully and
properly their opinions than in the conventional vot-
ing systems, anonymity, neutrality, independence of
irrelevant alternatives, etc. However, some authors
(Felsenthal and Machover, 2008), (Garc´ıa-Lapresta
and Mart´ınez-Panero, 2009) and (Smith, 2007) have
shown several paradoxes and inconsistencies of MJ.
In this paper we propose an extension of MJ which
diminishes some of the MJ inconveniences. The ap-
proach of the paper is distance-based, both for gener-
ating a collective assessment of each alternative and
in the tie-breaking process that provides a weak order
on the set of alternatives. As in MJ we consider that
individuals assess the alternatives through linguistic
labels and we propose as the collective assessment
a label that minimizes the distance to the individual
assessments. These distances between linguistic la-
bels are induced by a metric of the parameterized
Minkowski family. Depending on the specific met-
ric we use, the discrepancies between the collective
and the individual assessments are weighted in a dif-
ferent manner, and the corresponding outcome can be
different.
The paper is organized as follows. In Section 2,
458
Falcó E. and Luis García-Lapresta J..
A LINGUISTIC GROUP DECISION MAKING METHOD BASED ON DISTANCES.
DOI: 10.5220/0003273104580463
In Proceedings of the 3rd International Conference on Agents and Artificial Intelligence (ICAART-2011), pages 458-463
ISBN: 978-989-8425-40-9
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

the MJ voting system is formally explained. Section
3 introduces our proposal, within a distance-based ap-
proach (the election of the collective assessment for
each alternative and the tie-breaking method). In Sec-
tion 4 we include an illustrative example showing the
influence of the metric used in the proposed method
and its differences with respect to MJ and Range Vot-
ing (Smith, 2007). Finally, in Section 5 we collect
some conclusions.
2 MAJORITY JUDGEMENT
We consider
2
a finite set of voters V = {1, . . . , m},
with m ≥ 2, who evaluate a finite set of alternatives
X = {x
1
, . . . , x
n
}, with n ≥ 2. Each alternative is as-
sessed by each voter through a linguistic term be-
longing to an ordered finite scale L = {l
1
, . . . , l
g
},
with l
1
< · ·· < l
g
and granularity g ≥ 2. Each voter
assesses the alternatives in an independent way and
these assessments are collected by a matrix

v
i
j

,
where v
i
j
∈ L is the assessment that the voter i gives
to the alternative x
j
.
MJ chooses for each alternative the median of the
individual assessment as the collective assessment.
To be precise, the single median when the number
of voters is odd and the lower median in the case
that the number of voters is even. We denote with
l(x
j
) the collective assessment of the alternative x
j
.
Given that several alternatives might share the same
collective assessment, Balinski and Laraki (Balinski
and Laraki, 2007a) propose a sequential tie-breaking
process. This can be described through the following
terms (Garc´ıa-Lapresta and Mart´ınez-Panero, 2009):
N
+
(x
j
) = #{i ∈ V | v
i
j
> l(x
j
)},
N
−
(x
j
) = #{i ∈ V | v
i
j
< l(x
j
)}
and
t(x
j
) =





−1, if N
+
(x
j
) < N
−
(x
j
),
0, if N
+
(x
j
) = N
−
(x
j
),
1, if N
+
(x
j
) > N
−
(x
j
).
Taking into account the collective assessments and
the previous indices, we define a weak order  on X
in the following way: x
j
 x
k
if and only if one of the
following conditions hold:
1. l(x
j
) > l(x
k
).
2
The current notation is similar to the one introduced by
(Garc´ıa-Lapresta and Mart´ınez-Panero, 2009). This allows
us to describe the MJ process, presented by (Balinski and
Laraki, 2007a), in a more precise way.
2. l(x
j
) = l(x
k
) and t(x
j
) > t(x
k
).
3. l(x
j
) = l(x
k
), t(x
j
) = t(x
k
) = 1 and
N
+
(x
j
) > N
+
(x
k
).
4. l(x
j
) = l(x
k
), t(x
j
) = t(x
k
) = 1, N
+
(x
j
) = N
+
(x
k
)
and N
−
(x
j
) ≤ N
−
(x
k
).
5. l(x
j
) = l(x
k
), t(x
j
) = t(x
k
) = 0 and
m− N
+
(x
j
) − N
−
(x
j
) ≥ m− N
+
(x
k
) − N
−
(x
k
).
6. l(x
j
) = l(x
k
), t(x
j
) = t(x
k
) = −1 and
N
−
(x
j
) < N
−
(x
k
).
7. l(x
j
) = l(x
k
), t(x
j
) = t(x
k
) = −1,
N
−
(x
j
) = N
−
(x
k
) and N
+
(x
j
) ≥ N
+
(x
k
).
The asymmetric and symmetric parts of  are de-
fined in the usual way:
x
j
≻ x
k
⇔ not x
k
 x
j
x
j
∼ x
k
⇔ (x
j
 x
k
and x
k
 x
j
).
Next an example of how MJ works is shown.
Example 1. Consider three alternatives x
1
, x
2
and x
3
that are evaluated by seven voters through a set of six
linguistic terms L = {l
1
, . . . , l
6
}, the same set used in
MJ (Balinski and Laraki, 2007b), whose meaning is
shown in Table 1.
Table 1: Meaning of the linguistic terms.
l
1
To reject
l
2
Poor
l
3
Acceptable
l
4
Good
l
5
Very good
l
6
Excellent
The assessments obtained for each alternative are
collected and ranked from the lowest to the highest in
Table 2.
Table 2: Assessments of Example 1.
x
1
l
1
l
1
l
3
l
5
l
5
l
5
l
6
x
2
l
1
l
4
l
4
l
4
l
4
l
5
l
6
x
3
l
1
l
3
l
4
l
4
l
5
l
5
l
5
For ranking the three alternatives, first we take the
median of the individual assessments, that will be the
collective assessment for each one of the mentioned
alternatives: l(x
1
) = l
5
, l(x
2
) = l
4
and l(x
3
) = l
4
.
Given that x
1
has the best collective assessment, it
will be the one ranked in first place. However, the
alternatives x
2
and x
3
share the same collective as-
sessment, we need to turn to the tie-breaking pro-
cess, where we obtain N
+
(x
2
) = 2, N
−
(x
2
) = 1 and
A LINGUISTIC GROUP DECISION MAKING METHOD BASED ON DISTANCES
459

t(x
2
) = 1; N
+
(x
3
) = 3, N
−
(x
3
) = 2 and t(x
3
) =
1. Since both alternatives have the same t (t(x
2
) =
t(x
3
) = 1), we should compare their N
+
: N
+
(x
2
) =
2 < 3 = N
+
(x
3
). Therefore, the alternative x
3
defeats
the alternative x
2
, and the final order is x
1
≻ x
3
≻ x
2
.
3 DISTANCE-BASED METHOD
In this section the alternative method to MJ that we
propose through a distance-based approach is intro-
duced. The first step for ranking the alternatives is
to assign a collective assessment l(x
j
) ∈ L to each
alternative x
j
∈ X. For its calculation, the vectors
(v
1
j
, . . . , v
m
j
) that collect all the individual assessments
for each alternative x
j
∈ X are taken into account.
The proposal, that is detailed below, involves
how to choose a l(x
j
) ∈ L that minimizes the dis-
tance between the vector of individual assessments
(v
1
j
, . . . , v
m
j
) and the vector (l(x
j
), . . . , l(x
j
)) ∈ L
m
.
The election of that term is performed in an indepen-
dent way for each alternative. This guarantees the ful-
fillment of the independence of irrelevant alternatives
principle
3
.
Once a collective assessment l(x
j
) has been as-
sociated with each alternative x
j
∈ X , we rank the
alternatives according to the ordering of L. Given the
possible existence of ties, we also propose a sequen-
tial tie-breaking process based on the difference be-
tween the distance of l(x
j
) to the assessments higher
than l(x
j
) and the distance of l(x
j
) to the assess-
ments lower than l(x
j
).
3.1 Distances
A distance or metric on R
m
is a mapping
d : R
m
× R
m
−→ R
that fulfills the following conditions for all
(a
1
, . . . , a
m
), (b
1
, . . . , b
m
), (c
1
, . . . , c
m
) ∈ R
m
:
1. d((a
1
, . . . , a
m
), (b
1
, . . . , b
m
)) ≥ 0.
2. d((a
1
, . . . , a
m
), (b
1
, . . . , b
m
)) = 0 ⇔
(a
1
, . . . , a
m
) = (b
1
, . . . , b
m
).
3. d((a
1
, . . . , a
m
), (b
1
, . . . , b
m
)) =
d((b
1
, . . . , b
m
), (a
1
, . . . , a
m
)).
3
This principle says that the relative ranking between
two alternatives would only depend on the preference or as-
sessments on these alternatives and must not be affected by
other alternatives, that must be irrelevant on that compari-
son.
4. d((a
1
, . . . , a
m
), (b
1
, . . . , b
m
)) ≤
d((a
1
, . . . , a
m
), (c
1
, . . . , c
m
)) +
d((c
1
, . . . , c
m
), (b
1
, . . . , b
m
)).
Given a distance d : R
m
×R
m
−→ R, the distance
on L
m
induced by d is the mapping
¯
d :L
m
×L
m
−→ R
defined by
¯
d((l
a
1
, . . . , l
a
m
), ((l
b
1
, . . . , l
b
m
)) =
d((a
1
, . . . , a
m
), (b
1
, . . . , b
m
)).
A relevant class of distances in R
m
is constituted
by the family of Minkowski distances {d
p
| p ≥ 1},
which are defined by
d
p
((a
1
, . . . , a
m
), (b
1
, . . . , b
m
)) =
m
∑
i=1
|a
i
− b
i
|
p
!
1
p
,
for all (a
1
, . . . , a
m
), (b
1
, . . . , b
m
) ∈ R
m
.
We choose this family due to the fact that it is
parameterized and it includes from the well-known
Manhattan (p = 1) and Euclidean (p = 2) distances,
to the limit case, the Chebyshev distance (p = ∞).
The possibility of choosing among different values of
p ∈ (1,∞) gives us a very flexible method, and we
can choose the most appropriate p according to the
objectives we want to achieve with the election.
Given a Minkowski distance on R
m
, we consider
the induced distance on L
m
which works with the as-
sessments vector through the subindexes of the corre-
sponding labels:
¯
d
p
((l
a
1
, . . . , l
a
m
), (l
b
1
, . . . , l
b
m
)) =
d
p
((a
1
, . . . , a
m
), (b
1
, . . . , b
m
)).
Clearly, this approach means that the labels that
form L are equidistant. In this sense, the distance be-
tween two labels’ vectors is based on the number of
positions that we need to cover to go from one to an-
other, in each of its components. To move from l
a
i
to l
b
i
we need to cover |a
i
− b
i
| positions. For in-
stance between l
5
and l
2
we need to cover |5−2| = 3
positions.
3.2 Election of a Collective Assessment
for each Alternative
Our proposal is divided into several stages. First
we assign a collective assessment l(x
j
) ∈ L to each
alternative x
j
∈ X which minimizes the distance
between the vector of the individual assessments,
(v
1
j
, . . . , v
m
j
) ∈ L
m
, and the vector of m replicas of the
desired collective assessment, (l(x
j
), . . . , l(x
j
)) ∈ L
m
.
For this, first we establish the set L(x
j
) of all the
labels l
k
∈ L satisfying
¯
d
p
((v
1
j
, . . . , v
m
j
), (l
k
, . . . , l
k
)) ≤
¯
d
p
((v
1
j
, . . . , v
m
j
), (l
h
, . . . , l
h
)),
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
460

for each l
h
∈ L, where (l
h
, . . . , l
h
) and (l
k
, . . . , l
k
) are
the vectors of m replicas of l
h
and l
k
, respectively.
Thus, L(x
j
) consists of those labels that minimize
the distance to the vector of individual assessments.
Notice that L(x
j
) = {l
r
, . . . , l
r+s
} is always an inter-
val, because it contains all the terms from l
r
to l
r+s
,
where r ∈ {1, . . . , g} and 0 ≤ s ≤ g−r. Two different
cases are possible:
1. If s = 0, then L(x
j
) contains a single label, which
will automatically be the collective assessment
l(x
j
) of the alternative x
j
.
2. If s > 0, then L(x
j
) has more than one label. In
order to select the most suitable label of L(x
j
),
we now introduce L
∗
(x
j
), the set of all the labels
l
k
∈ L(x
j
) that fulfill
¯
d
p
((l
k
, . . . , l
k
), (l
r
, . . . , l
r+s
))≤
¯
d
p
((l
h
, . . . , l
h
), (l
r
, . . . , l
r+s
)),
for all l
h
∈ L(x
j
), where (l
k
, . . . , l
k
) and
(l
h
, . . . , l
h
) are the vectors of s + 1 replicas of l
k
and l
h
, respectively.
(a) If the cardinality of L(x
j
) is odd, then L
∗
(x
j
)
has a unique label, the median term, that will
be the collective assessment l(x
j
).
(b) If the cardinality of L(x
j
) is even, then L
∗
(x
j
)
has two different labels, the two median terms.
In this case, similarly to the proposal of (Balin-
ski and Laraki, 2007a), we consider the low-
est label in L
∗
(x
j
) as the collective assessment
l(x
j
).
It is worth pointing out two different cases when
we are using induced Minkowski distances.
1. If p = 1, we obtain the same collective assess-
ments that those given by MJ, the median
4
of the
individual assessments. However, the final results
are not necessarily the same that in MJ because we
use a different tie-breaking process, as is shown
later.
2. If p = 2, each collective assessment is the clos-
est label to the “mean” of the individual assess-
ments
5
, which is the one chosen in the Range Vot-
ing (RV) method
6
(see (Smith, 2007)).
4
It is more precise to speak about the interval of medi-
ans, because if the assessments’ vector has an even number
of components, then there are more than one median. See
(Monjardet, 2008).
5
The chosen label is not exactly the arithmetic mean of
the individual assessments, because we are working with a
discrete spectrum of linguistic terms and not in the continu-
ous one of the set of real numbers.
6
RV works with a finite scale given by equidistant real
numbers, and it ranks the alternatives according to the arith-
metic mean of the individual assessments.
Notice that when we choose p ∈ (1, 2), we find
situations where the collective assessment is located
between the median and the “mean”. This allows us
to avoid some of the problems associated with MJ and
RV.
3.3 Tie-breaking Method
Usually there exist more alternatives than linguistic
terms, so it is very common to find several alterna-
tives sharing the same collective assessment. But irre-
spectively of the number of alternatives, it is clear that
some of them may share the same collective assess-
ment, even when the individual assessments are very
different. For these reasons it is necessary to intro-
duce a tie-breaking method that takes into account not
only the number of individual assessments above or
below the obtained collective assessment (as in MJ),
but the positions of these individual assessments in
the ordered scale associated with L.
As mentioned above, we will calculate the differ-
ence between two distances: one between l(x
j
) and
the assessments higher than l(x
j
) and another one be-
tween l(x
j
) and the assessments lower than the l(x
j
).
Let v
+
j
and v
−
j
the vectors composed by the assess-
ments v
i
j
from

v
1
j
, . . . , v
m
j

higher and lower than
the term l(x
j
), respectively. First we calculate the two
following distances:
D
+
(x
j
) =
¯
d
p

v
+
j
, (l(x
j
), . . . , l(x
j
))

,
D
−
(x
j
) =
¯
d
p

v
−
j
, (l(x
j
), . . . , l(x
j
))

,
where the number of components of (l(x
j
), . . . ,l(x
j
))
is the same that in v
+
j
and in v
−
j
, respectively (obvi-
ously, the number of components of v
+
j
and v
−
j
can
be different).
Once these distances have been determined, a new
index D(x
j
) ∈ R is calculated for each alternative
x
j
∈ X: the difference between the two previous dis-
tances:
D(x
j
) = D
+
(x
j
) − D
−
(x
j
).
By means of this index, we provide a kind of com-
pensation between the individual assessments that are
bigger and smaller than the collectiveassessment, tak-
ing into account the position of each assessment in the
ordered scale associated with L.
For introducing our tie-breaking process, we fi-
nally need the distance between the individual assess-
ments and the collective one:
E(x
j
) =
¯
d
p

(v
1
j
, . . . , v
m
j
), (l(x
j
), . . . , l(x
j
))

.
A LINGUISTIC GROUP DECISION MAKING METHOD BASED ON DISTANCES
461

Notice that for each alternative x
j
∈ X, E(x
j
)
minimizes the distance between the vector of individ-
ual assessments and the linguistic labels in L, such as
has been considered above in the definition of L(x
j
).
The use of the index E(·) is important in the
tie-breaking process because if two alternatives share
the same couple (l(·), D(·)), the alternative with the
lower E(·) is the alternative whose individual assess-
ments are more concentrated around the collective as-
sessment, i.e., the consensus is higher.
Summarizing, for ranking the alternatives we will
consider the following triplet
T(x
j
) = (l(x
j
), D(x
j
), E(x
j
)) ∈ L× R × [0, ∞)
for each alternative x
j
∈ X. The sequential process
works in the following way:
1. We rank the alternatives through the collective as-
sessments l(·). The alternatives with higher col-
lective assessments will be preferred to those with
lower collective assessments.
2. If several alternativesshare the same collective as-
sessment, then we break the ties through the D(·)
index. The alternatives with a higher D(·) will be
preferred.
3. If there are still ties, we break them through the
E(·) index, in such a way such that the alterna-
tives with a lower E(·) will be preferred.
Formally, the sequential process can be introduced
by means of the lexicographic weak order  on X
defined by x
j
 x
k
if and only if
1. l(x
j
) ≥ l(x
k
) or
2. l(x
j
) = l(x
k
) and D(x
j
) > D(x
k
) or
3. l(x
j
) = l(x
k
), D(x
j
) = D(x
k
) and E(x
j
) ≤ E(x
k
).
Remark. Although it is possible that ties still exist,
whenever two or more alternatives share T(·), these
cases are very unusual when considering metrics with
p > 1.
7
For instance, consider seven voters that assess
two alternatives x
1
and x
2
by means of the set of lin-
guistic terms given in Table 1. Table 3 includes these
assessments arranged from the lowest to the highest
labels.
Table 3: Individual assessments.
x
1
l
2
l
2
l
2
l
2
l
4
l
4
l
6
x
2
l
2
l
2
l
2
l
2
l
3
l
5
l
6
It is easy to see that for p = 1 we have T(x
1
) =
T(x
2
) = (l
2
, 8, 8), then x
1
∼ x
2
(notice that MJ and
7
The Manhattan metric (p = 1) produces more ties than
the other metrics in the Minkowski family because of the
simplicity of its calculations.
RV also provide a tie). However, if p > 1, the tie
disappears. So, we have x
2
≻ x
1
, excepting wherever
p ∈ (1.179, 1.203), where x
1
≻ x
2
.
4 AN ILLUSTRATIVE EXAMPLE
This section focus on how the election of the parame-
ter p is relevant in the final ranking of the alternatives.
We show this fact through an example. We consider a
case where the median of the individual assessments
is the same for all the alternatives. In this example
we use the set of six linguistic terms L = {l
1
, . . . , l
6
}
whose meaning is shown in Table 1.
As mentioned above, the sequential process for
ranking the alternativesis based on the triplet T(x
j
) =
(l(x
j
), D(x
j
), E(x
j
)) for each alternative x
j
∈ X.
However, by simplicity, in the following example we
only show the first two components, (l(x
j
), D(x
j
)). In
this example we also obtain the outcomes provided by
MJ and RV.
Example 2. Table 4 includes the assessments given
by six voters to four alternatives x
1
, x
2
, x
3
and x
4
ar-
ranged from the lowest to the highest labels.
Table 4: Assessments in Example 2.
x
1
l
1
l
2
l
4
l
4
l
4
l
6
x
2
l
1
l
1
l
3
l
4
l
6
l
6
x
3
l
2
l
2
l
2
l
4
l
5
l
6
x
4
l
1
l
1
l
4
l
5
l
5
l
5
Notice that the mean of the individual assess-
ments’ subindexes is the same for the four alterna-
tives, 3.5. Since RV ranks the alternatives accord-
ing to this mean, it produces a tie x
1
∼ x
2
∼ x
3
∼ x
4
.
However, it is clear that this outcome might not seem
reasonable, and that other rankings could be justified.
Using MJ, where l(x
1
) = l(x
4
) = l
4
> l
3
= l(x
2
) >
l
2
= l(x
3
) and, according to the MJ tie-breaking pro-
cess, we have t(x
1
) = −1 < 1 = t(x
4
). Thus, MJ pro-
duces the outcome x
4
≻ x
1
≻ x
2
≻ x
3
.
Table 5: (l(x
j
), D(x
j
)) in Example 2.
p = 1 p = 1.25 p = 1.5
x
1
(l
4
, −3) (l
4
, −2.375) (l
4
, −2.008)
x
2
(l
3
, 10) (l
3
, 2.264) (l
3
, 1.888)
x
3
(l
2
, 9) (l
3
, 2.511) (l
3
, 2.254)
x
4
(l
4
, −3) (l
4
, −2.815) (l
4
, −2.682)
We now consider the distance-based procedure for
six values of p.In Table 6 we can see the influence of
these values on (l(x
j
), D(x
j
)), for j = 1, 2, 3, 4.
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
462

Table 6: (l(x
j
), D(x
j
)) in Example 2.
p = 1.75 p = 2 p = 5
x
1
(l
4
, −1.770) (l
3
, 1.228) (l
3
, 0.995)
x
2
(l
3
, 1.669) (l
3
, 1.530) (l
3
, 1.150)
x
3
(l
3
, 2.104) (l
3
, 2.010) (l
4
, −0.479)
x
4
(l
4
, −2.585) (l
3
, 0.777) (l
3
, 0.199)
For p = 1 we have T(x
1
) = (l
4
, −3, 7), T(x
2
) =
(l
3
, 10, 11), T(x
3
) = (l
2
, 9, 9) and T(x
4
) = (l
4
, −3, 9).
Then, we obtain x
1
≻ x
4
≻ x
2
≻ x
3
, a different out-
come than obtained using MJ. For p = 1.25, p = 1.5
and p = 1.75 we obtain x
1
≻ x
4
≻ x
3
≻ x
2
; and for
p = 2 and p = 5 we have x
3
≻ x
2
≻ x
1
≻ x
4
.
5 CONCLUDING REMARKS
In this paper we have presented an extension of
the Majority Judgement voting system developed by
(Balinski and Laraki, 2007a; Balinski and Laraki,
2007b; Balinski and Laraki, 2007c). This extension is
based on a distance approach but it also uses linguis-
tic labels to evaluate the alternatives. We choose as
the collective assessment for each alternative a label
that minimizes the distance to the individual assess-
ments. It is important to note that our proposal coin-
cides in this aspect with Majority Judgement when-
ever the Manhattan metric is used.
We also provide a tie-breaking process through the
distances between the individual assessments higher
and lower than the collective one. This process is
richer than the one provided by Majority Judgement,
that only counts the number of alternatives above
or below the collective assessment, irrespectively of
what they are. We also note that our tie-breaking
process is essentially different to Majority Judgement
even when the Manhattan metric is considered.
It is important to note that using the distance-
based approach we pay attention to all the individual
assessments that have not been chosen as the collec-
tive assessment. With the election of a specific met-
ric of the Minkowski family we are deciding how to
evaluate these other assessments. This aspect pro-
vides flexibility to our extension and it allows to de-
vise a wide class of voting systems that may avoid
some of the drawbacks related to Majority Judgement
and Range Voting without losing their good features.
This becomes specially interesting when the value of
the parameter p in the Minkowski family belongs to
the open interval (1, 2), since p = 1 and p = 2 cor-
respond to the Manhattan and the Euclidean metrics,
respectively, just the metrics used in Majority Judge-
ment and Range Voting. For instance, the election of
p = 1.5 allows us to have a kind of compromise be-
tween both methods.
As shown in the previous examples, when the
value of parameter p increases, the distance-based
procedure focuses more and more on the extreme as-
sessments. However, if the individual assessments are
well balanced on both sides, the outcome is not very
affected by the parameter p.
In further research we will analyze the proper-
ties of the presented extension of Majority Judgement
within the Social Choice framework.
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