Fuzzy Sets and Systems 151 (2005) 269 – 284
www.elsevier.com/locate/fss
Individual-valued preferences and their aggregation:
consistency analysis in a real case
José Luis García-Lapresta∗ , Luis Carlos Meneses
Departamento de Economía Aplicada (Matemáticas), Facultad de CC EE y EE, Universidad de Valladolid, Avda Valle de
Esgueva 6, 47011 Valladolid, Spain
Received 13 June 2003; received in revised form 9 June 2004; accepted 29 September 2004
Communicated by Fodor
Available online 30 October 2004
Abstract
In this paper, we have analyzed the accomplishment of several consistency conditions in a real decision case.
A group of students showed their intensities of preference among the alternatives by means of linguistic labels
represented by real numbers. The absolute and relative fulfillments of some kinds of fuzzy transitivity properties
have been studied for individual and collective preferences. Collective preferences have been obtained by means of
a wide class of neutral and stable for translations aggregation rules, which transports reciprocity from individual
preferences to the collective preference. We notice that, in the real case studied, the aggregate preferences reach
higher consistency properties than individual preferences.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Individual decision-making; Graded preferences; Linguistic labels; Fuzzy transitivity; Rational behavior; Group
decision-making; Aggregation operators; Quasiarithmetic means
1. Introduction
Given two alternatives, we can ask an individual if she prefers one alternative to another or if she is
indifferent. If she does not declare indifference, and that person prefers an alternative, then she could
show, with more detail, her preference: slight, high, absolute, etc. But, as it happens in conventional
∗ Corresponding author. Tel.: +34 983 184 391; fax: +34 983 423 299.
E-mail address: [email protected] (J.L. García-Lapresta).
0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2004.09.016
270
J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
preferences, individuals can also have inconsistent opinions when they show intensities of preference
over more than two alternatives. See for instance [24], where an empirical analysis of individual rational
behavior based on several fuzzy transitivity properties can be found.
The main purpose of this paper is to analyze, in a real case, the fulfillment of 6 consistency conditions
related to fuzzy transitivity, both in individual decisions and in the collective ones based on some aggregation rules. The real case is based on the graded preferences of 85 first year students over 6 degrees, at
the time of their registration in the Faculty of Economics and Business Administration of the University
of Valladolid (Spain). Students compared the degrees by pairs and they showed intensities of preference
among the alternatives by means of linguistic labels represented by real numbers. Due to this representation, our analysis has been based on the fuzzy set theory (see for instance [9,30]), especially on fuzzy
preference relations. We have to emphasize that students showed sincerely their preferences about a very
interesting issue for them in the crucial moment of the entry into the University.
According to [1], it is impossible to find aggregation rules that provide social consistent decisions
satisfying some reasonable properties. In spite of the good properties of the arithmetic mean (see [13]),
this aggregation rule does not assure consistent decisions in the framework of fuzzy preferences (cycles and
intransitivities can appear in the aggregate preference). This is why we have analyzed the accomplishment
of several fuzzy transitivity properties, not only in the individual decisions, but also in the group opinion
provided by the arithmetic mean aggregation rule. Moreover, we have considered a class of aggregation
rules related to exponential quasiarithmetic means, introduced in [15], all of them reciprocal and stable
for translations. Reciprocity ensures that if all the individuals reverse their preferences, then the group
preference is also reversed. Stability for translations guarantees that if each individual increases the
intensity of preference between two alternatives in a fixed quantity, then the group intensity of preference
is also increased in the same quantity. We have also analyzed classical transitivity in some ordinary
preference relations (-cuts) associated with the aggregate fuzzy preference relation.
Several analyses and references about quasiarithmetic means and other aggregation operators can be
found in [5, 12 Chapter 5]. We note that the problem of consistency in the fuzzy group decision-making
has been considered in [6,17], among others.
The paper is organized as follows. In Section 2, we introduce notation and some concepts related to
fuzzy preferences and aggregation rules. In Section 3, we set up some consistency properties of ordinary
and fuzzy preferences. Section 4 is devoted to explain the main characteristics of the real case decision
problem. In Section 5 we present the results, and in Section 6 we present some conclusions.
2. Fuzzy preferences and aggregation rules
Let X = {x1 , . . . , xn } be a set of alternatives and assume that m individuals show their preferences over
the pairs of X, with n 3 and m 3. Suppose that each individual k ∈ {1, . . . , m} compares all the pairs of
alternatives of X and declares her intensities of preference by means of a fuzzy binary relation on X, R k ,
defined by its membership function R k (xi , xj ) = rijk ∈ [0, 1] for every xi , xj ∈ X. This index rijk means
the intensity of preference with which individual k prefers xi over xj , being 1, 0.5 or 0 depending on
whether this individual prefers absolutely xi to xj , is indifferent between xi and xj , or prefers absolutely
xj to xi , respectively (see [3]). Other numbers different to 0, 0.5 and 1 are allowed for neither extreme
preferences nor indifference, in the sense that the closer is the number to 1, the more xi is preferred to
xj , and the closer is the number to 0, the more xj is preferred to xi .
J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
271
Moreover, we suppose that R k is reciprocal, i.e., rijk +rjki = 1 for every xi , xj ∈ X. By R(X) we denote
the set of all the reciprocal fuzzy binary relations on X. If R ∈ R(X), we say that R is a fuzzy preference
relation on X. Justifications of the use of the reciprocity axiom can be found in [3,13,19,20,26], among
others.
Given R ∈ R(X), it is easy to see that, for every ∈ [0.5, 1), the ordinary binary relation on X, P ,
defined by xi P xj ⇒ rij > , is asymmetric, i.e., if xi P xj , then not xj P xi . Thus, P is an ordinary
preference relation on X, the -cut of R. The indifference relation associated with P reflects absence
of preference, and it is defined by xi I xj ⇔ neither xi P xj nor xj P xi , i.e., rij and rj i . By
reciprocity, these conditions are equivalent to 1 − rij . Consequently, for each pair of alternatives
xi , xj ∈ X one and only one of the following situations occurs: xi P xj (rij > ), xi I xj (1 − rij ),
xj P xi (rj i > , i.e., rij < 1 − ).
We note that if R is not reciprocal, then P is not necessarily asymmetric, which means that at least
a pair of alternatives would be mutually preferred. This is why we will require that both individual and
group preferences be reciprocal.
An aggregation rule is a function F : R(X)m → R(X) which assigns the collective fuzzy preference
relation, R̄ = F (R 1 , . . . , R m ) ∈ R(X), to each profile (R 1 , . . . , R m ) ∈ R(X)m of individual fuzzy
preferences. With r̄ij we denote the collective preference between xi and xj according to R̄.
In this paper we only consider neutral aggregation rules, those providing an egalitarian treatment
to alternatives: for every pair of profiles (R 1 , . . . , R m ), (S 1 , . . . , S m ) ∈ R(X)m and every alternak is satisfied for all k ∈ {1, . . . , m}, then r̄ = s̄ . Obviously,
tives xi , xj , xp , xq ∈ X, if rijk = spq
ij
pq
F : R(X)m → R(X) is neutral if and only if there exists a function f : [0, 1]m → [0, 1] such that
r̄ij = f (rij1 , . . . , rijm ) for all alternatives xi , xj ∈ X.
Since R̄ is reciprocal, we have that for all (a1 , . . . , am ) ∈ [0, 1]m :
f (1 − a1 , . . . , 1 − am ) = 1 − f (a1 , . . . , am ).
Every function f : [0, 1]m → [0, 1] verifying the previous condition will be considered reciprocal and
it will naturally define a neutral aggregation rule F : R(X)m → R(X) such that r̄ij = f (rij1 , . . . , rijm ).
In this paper, we will consider neutral aggregation rules stable for translations: for every (a1 , . . . , am ) ∈
[0, 1]m and t ∈ [−1, 1]:
f (a1 + t, . . . , am + t) = f (a1 , . . . , am ) + t,
whenever (a1 + t, . . . , am + t) ∈ [0, 1]m and f (a1 + t, . . . , am + t) ∈ [0, 1].
The class of neutral and stable for translations aggregation rules provide an adequate tool to assign a
collective fuzzy preference to each profile of individual fuzzy preferences, preserving reciprocity. First of
all, neutrality ensures that the collective intensity of preference between a pair of alternatives is given by
means of a reciprocal function f : [0, 1]m → [0, 1], taking into account only the individual intensities
of preference between that pair of alternatives. This fact guarantees not only an egalitarian treatment to
alternatives, but also the fulfillment of the axiom of independence of irrelevant alternatives (see [1,13]).
On the other hand, stability for translations transfers to the collective preference the same unanimous
positive or negative increase of the individual intensities of preference.
Among the neutral aggregation rules that are stable for translations, we have considered the arithmetic
mean, because of its good properties (see [13]), and a class of aggregation rules related to exponential
quasiarithmetic means, introduced in [15].
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J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
Given an increasing and bijective function : [0, 1] → [0, 1], the quasiarithmetic mean associated
with is the function f : [0, 1]m → [0, 1] defined by
m
k=1 (ak )
−1
.
f (a1 , . . . , am ) = m
According to [19] (see [12, pp. 117–118]), the quasiarithmetic means generated by the exponential
functions
(a) =
e a − 1
,
e − 1
>0
and the arithmetic mean, generated by the identity function (a) = a:
m
m
a k
ak
1
k=1 e
and f0 (a1 , . . . , am ) = k=1
f (a1 , . . . , am ) = ln
m
m
are the only quasiarithmetic means satisfying stability for translations.
In [14], it is proven that the quasiarithmetic mean associated with is reciprocal if and only if (1−a) =
1 − (a) for all a ∈ [0, 1]. Then, the only reciprocal function of the family {f | 0} is the arithmetic
mean, f0 . However, according to [15], the symmetric part of f , the function fˆ : [0, 1]m → [0, 1]
defined by
m
e a k
1
ˆ
f (a1 , . . . , am ) =
ln mk=1 −a ,
k
2
k=1 e
is reciprocal and stable for translations; moreover, in that paper is established that for m = 2, fˆ coincides
with the arithmetic mean, and that for m > 2, fˆ is not a quasiarithmetic mean.
We are now going to justify that the function fˆ tends to the average of the minimum and maximum
values of the components of each vector when tends to infinity.
Proposition. Given a vector (a1 , . . . , am ) ∈ [0, 1]m , let a∗ = min{a1 , . . . , am } and a ∗ = max{a1 , . . . ,
am }. Then:
a∗ + a ∗
.
fˆ∞ (a1 , . . . , am ) = lim fˆ (a1 , . . . , am ) =
→∞
2
Proof.
m
− a k
e a k
ln m
eak − ln m
1
k=1
k=1
k=1 e
lim fˆ (a1 , . . . , am ) = lim
ln m −a = lim
.
k
→∞
→∞ 2
→∞
2
k=1 e
Applying the L’Hospital rule, we have
lim fˆ (a1 , . . . , am ) = lim
→∞
→∞
m
ak
k=1 ak e
m
ak
k=1 e
+
2
m
ak e−ak
k=1
m
−ak
k=1 e
J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
273
m
m
a k
− a k
1
1
k=1 ak e
k=1 ak e
lim m
lim
=
+
m
a k
− a k
2 →∞
2 →∞
k=1 e
k=1 e
∗
m
−
a
a
a
− a k
k
e
e ∗ m
1
1
k=1 ak e
k=1 ak e
=
+
lim
lim
∗
m
a k
− a k
2 →∞ e−a
2 →∞ ea∗ m
k=1 e
k=1 e
m
m
∗)
(a
−a
−
(a
k −a∗ )
ak e k
1
1
k=1 ak e
lim k=1
lim
=
+
m
m
(ak −a ∗ )
−(ak −a∗ )
2 →∞
2 →∞
k=1 e
k=1 e
∗
1
1
a∗ + a
.
= a ∗ + a∗ =
2
2
2
We immediately see that the function fˆ∞ : [0, 1]m → [0, 1] is reciprocal and stable for translations.
Then, fˆ defines a neutral aggregation rule stable for translations.
Because of the aforementioned reasons, in the aggregation of individual preferences we will consider
the stable for translations neutral aggregation rules associated with the functions f0 , fˆ ( > 0) and fˆ∞ .
Given two alternatives xi , xj ∈ X, the above mentioned aggregation rules assign the collective intensity
of preference between xi and xj in the following manner:
m
k
m
k
erij
1
k=1 rij
k=1
0
1
m
1
m
, r̄ij = fˆ (rij , . . . , rij ) =
ln ,
r̄ij = f0 (rij , . . . , rij ) =
−rijk
m
m
2
e
k=1
r̄ij∞ = fˆ∞ (rij1 , . . . , rijm ) =
min{rij1 , . . . , rijm } + max{rij1 , . . . , rijm }
2
.
3. Consistency conditions
In the classical preference modeling, transitivity is the starting point to tackle the analysis of rationality.
An ordinary binary relation P on X is transitive if xi P xj and xj P xk implies xi P xk , for all xi , xj , xk ∈
X. The main consistency assumption in the probabilistic and fuzzy approaches to decision theory is still
transitivity. However, in both frameworks a wide class of transitivity conditions generalizes the classical
property. On this, see [2,7,9–12,16,18,20–30], among others. Now we introduce the fuzzy transitivity
properties considered in the real case studied. Let ∗ be a binary operation on [0.5, 1], i.e., a ∗ b ∈ [0.5, 1]
for all a, b ∈ [0.5, 1], with the following properties:
• Commutativity: a ∗ b = b ∗ a for all a, b ∈ [0.5, 1].
• Monotonicity: (a a and b b ) ⇒ a ∗ b a ∗ b , for all a, a , b, b ∈ [0.5, 1].
• Continuity: small changes in variables a, b produce small changes in the result a ∗ b.
We say that R ∈ R(X) is weak max-∗ transitive if the following holds:
(rij > 0.5 and rj k > 0.5) ⇒ (rik > 0.5 and rik rij ∗ rj k )
for all xi , xj , xk ∈ X.
Obviously, the ordinary preference relation P0.5 associated with every weak max-∗ transitive R ∈ R(X)
is transitive.
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J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
We note that max-∗ transitivity for a fuzzy binary relation R was initially defined by demanding
rik rij ∗ rj k . “Weak” (or “restricted”) conditions are considered in [7,26], among others, when certain
additional hypotheses are required. In this paper we consider preference intensities greater than 0.5.
In order to introduce concrete weak max-∗ transitive properties, we consider 6 commutative, monotonous and continuous binary operations on [0.5, 1]:
a ∗1 b = 0.5,
a ∗2 b = max{a + b − 1, 0.5}, a ∗3 b = max{ab, 0.5},
a+b
a ∗4 b = min{a, b}, a ∗5 b =
, a ∗6 b = max{a, b}.
2
We say that R ∈ R(X) verifies property Ti if R is weak max-∗i transitive.
It is easily seen that a ∗1 b a ∗2 b a ∗3 b a ∗4 b a ∗5 b a ∗6 b, for all a, b ∈ [0.5, 1], i.e.,
T6 ⇒ T5 ⇒ T4 ⇒ T3 ⇒ T2 ⇒ T1 . Moreover, T1 is equivalent to P0.5 being transitive, and P is
transitive for all ∈ [0.5, 1) whenever R satisfies T4 , T5 or T6 . This is due to the fact that T4 is equivalent
to P being transitive for all ∈ [0.5, 1).
4. A real case
In order to check the consistency properties in a real case, we have made a survey to 85 students. These
students were questioned about their preferences over the following degrees:
(A)
(B)
(C)
(D)
(E)
(F)
Business Administration and Management (5 years).
Business Administration (3 years).
Law (5 years).
Business Administration, Management and Law (6 years).
Labor Relations (3 years).
Economics (5 years).
The survey was conducted at the same time that the students were registering for the first year of the
Faculty of Economics and Business Administration of the University of Valladolid (Spain), in any of the
degrees A, D or F (the other 3 degrees, B, C and E, are in other Faculties, but they have some similarities
with A, D and F).
Students had to compare each pair of alternatives through four modalities of preference: “totally”,
“highly”, “rather” and “slightly”, when they preferred one alternative to another; in absence of preference
between alternatives they could declare “indifference”.
Then, we assigned a number from 0 to 1 to each of the 9 modalities of preference or indifference: the
intensity of preference between xi and xj , rij , can be of the 9 terms; taking into account reciprocity, the
intensity of preference between xj and xi is defined by rj i = 1 − rij .
In order to know the possible influence of the real numbers associated with the linguistic labels over
the accomplishment of the Ti properties, we have considered two different assignments (see Table 1).
By simplicity, in semantics 1 the real numbers associated with consecutive terms have a constant step,
0.125. However, individuals can feel different distances between consecutive linguistic labels. For this
reason, the steps of the numerical representation appearing in semantics 2 are variable (0.022, 0.130,
0.131, 0.217). These numbers are related to the semantics provided by Bonissone and Decker [4], where
vagueness is greater around indifference than in the proximities of extreme preferences. Our assignments
J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
275
Table 1
Two semantics for 9 terms
Term
Semantics 1
Semantics 2
xi is totally preferred to xj
xi is highly preferred to xj
xi is rather preferred to xj
xi is slightly preferred to xj
xi is indifferent to xj
xj is rather preferred to xi
xj is slightly preferred to xi
xj is highly preferred to xi
xj is totally preferred to xi
rij
rij
rij
rij
rij
rij
rij
rij
rij
rij
rij
rij
rij
rij
rij
rij
rij
rij
= 1 (rj i = 0)
= 0.875 (rj i = 0.125)
= 0.750 (rj i = 0.250)
= 0.625 (rij = 0.375)
= 0.500 (rj i = 0.500)
= 0.375 (rj i = 0.625)
= 0.250 (rj i = 0.750)
= 0.125 (rj i = 0.875)
= 0 (rj i = 1)
= 1 (rj i = 0)
= 0.978 (rj i = 0.022)
= 0.848 (rj i = 0.152)
= 0.717 (rj i = 0.283)
= 0.500 (rj i = 0.500)
= 0.283 (rj i = 0.717)
= 0.152 (rj i = 0.848)
= 0.022 (rj i = 0.978)
= 0 (rj i = 1)
are similar to the associated real numbers provided by Delgado et al. [8] to the trapezoidal fuzzy numbers
given by Bonissone and Decker [4]. Consequently, steps decrease when terms are moving towards extreme
preferences.
Since the set of alternatives has 6 elements, each student had to compare 15 pairs of alternatives.
Then, the total number of compared pairs was 1275. This information was processed by means of several
computer programs in order to know the consistency level, related to the 6 fuzzy transitivity properties,
reached by students. Thus, 1700 triplets of alternatives were involved in these analyses.
5. The results
Our empirical analysis is divided in two different parts. First, we obtain the collective intensities
of preference among the different pairs of alternatives by means of several aggregation rules, and the
orderings associated with the corresponding 0.5-cuts. On the other hand, we check each one of the 6 kinds
of fuzzy transitivity on the individual and collective preferences. Moreover, we analyze the fulfillment of
the ordinary transitivity for several -cuts associated with the fuzzy preferences.
5.1. Aggregation of the individual preferences
In order to obtain the collective opinion among the 6 alternatives, we have considered the aggregation
rules associated with the arithmetic mean, f0 , the symmetric part of the exponential quasiarithmetic
means fˆ , for several values of , and the limit case fˆ∞ . The exponential quasiarithmetic means, f ,
have not been considered because they are not reciprocal, and consequently they do not define properly
aggregation rules.
In Tables 2 and 3 we show the collective intensities of preference between all the pairs of alternatives,
by considering the aggregation rules associated with f0 , fˆ for = 1, 2, 3, 4, 5, 10, 20, 30, 40, 50, 100,
200, 300, 400, and fˆ∞ , according to the two semantics. Figs. 1a and b show graphically these outcomes
for semantics 1.
We can note that the collective intensities of preference tend to the limit value provided by fˆ∞ . These
tendencies are monotonous, increasingly or decreasingly, except for the pairs of alternatives (A,D) and
(C,E) (see Fig. 2). The pair (A,D) is the only one where the sense of the collective preference changes
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J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
Table 2
Collective intensities of preference with semantics 1: values
(A,B) (A,C) (A,D) (A,E)
(A,F)
(B,C) (B,D)
(B,E)
(B,F)
(C,D)
(C,E)
(C,F)
(D,E)
(D,F)
(E,F)
f0
fˆ1
fˆ2
fˆ3
fˆ4
fˆ5
0.800
0.797
0.788
0.772
0.752
0.729
0.822
0.821
0.817
0.810
0.800
0.788
0.529
0.528
0.525
0.520
0.516
0.512
0.866
0.866
0.865
0.864
0.863
0.861
0.649
0.645
0.637
0.626
0.613
0.601
0.609
0.607
0.600
0.591
0.580
0.569
0.293
0.295
0.302
0.312
0.324
0.336
0.703
0.702
0.698
0.692
0.683
0.673
0.388
0.390
0.394
0.399
0.405
0.410
0.206
0.207
0.209
0.213
0.218
0.225
0.538
0.539
0.540
0.542
0.545
0.548
0.310
0.313
0.322
0.335
0.349
0.363
0.766
0.765
0.760
0.753
0.743
0.731
0.600
0.597
0.590
0.581
0.571
0.562
0.294
0.296
0.302
0.311
0.321
0.332
fˆ10
fˆ20
fˆ30
fˆ40
fˆ50
0.638
0.574
0.551
0.538
0.531
0.718
0.644
0.617
0.604
0.595
0.502
0.498
0.498
0.499
0.499
0.846
0.814
0.795
0.784
0.777
0.562
0.534
0.523
0.517
0.514
0.531
0.506
0.501
0.500
0.500
0.375
0.402
0.413
0.418
0.422
0.623
0.581
0.571
0.568
0.567
0.430
0.450
0.462
0.470
0.476
0.267
0.319
0.338
0.347
0.353
0.555
0.542
0.530
0.522
0.518
0.414
0.452
0.467
0.475
0.480
0.673
0.619
0.600
0.591
0.585
0.533
0.517
0.511
0.509
0.507
0.370
0.399
0.411
0.417
0.421
fˆ100
fˆ200
fˆ300
fˆ400
0.515
0.508
0.505
0.504
0.579
0.571
0.568
0.567
0.499
0.500
0.500
0.500
0.764
0.757
0.755
0.753
0.507
0.503
0.502
0.502
0.500
0.500
0.500
0.500
0.430
0.434
0.435
0.436
0.565
0.564
0.563
0.563
0.488
0.494
0.496
0.497
0.364
0.370
0.371
0.372
0.509
0.504
0.503
0.502
0.490
0.495
0.497
0.497
0.574
0.568
0.566
0.565
0.503
0.502
0.501
0.501
0.429
0.433
0.435
0.435
fˆ∞
0.500
0.563
0.500
0.750 0.500 0.500
0.438
0.563 0.500
0.375
0.500 0.500 0.563 0.500 0.438
Table 3
Collective intensities of preference with semantics 2: values
(A,B) (A,C) (A,D) (A,E)
(A,F)
(B,C) (B,D)
(B,E)
(B,F)
(C,D)
(C,E)
(C,F)
(D,E)
(D,F)
(E,F)
f0
fˆ1
fˆ2
fˆ3
fˆ4
fˆ5
0.859
0.853
0.836
0.808
0.773
0.738
0.876
0.873
0.864
0.849
0.828
0.802
0.543
0.541
0.535
0.528
0.523
0.518
0.929
0.928
0.927
0.925
0.922
0.919
0.681
0.675
0.659
0.639
0.619
0.603
0.648
0.644
0.631
0.615
0.598
0.584
0.252
0.259
0.275
0.298
0.320
0.341
0.770
0.767
0.756
0.740
0.720
0.698
0.372
0.375
0.385
0.397
0.409
0.419
0.140
0.143
0.149
0.160
0.175
0.193
0.538
0.538
0.538
0.539
0.540
0.540
0.273
0.279
0.296
0.318
0.341
0.362
0.819
0.815
0.804
0.786
0.763
0.739
0.626
0.621
0.608
0.592
0.577
0.565
0.258
0.264
0.278
0.298
0.318
0.337
fˆ10
fˆ20
fˆ30
fˆ40
fˆ50
0.629
0.565
0.544
0.534
0.527
0.690
0.603
0.572
0.556
0.546
0.507
0.502
0.501
0.500
0.500
0.888
0.829
0.802
0.788
0.780
0.559
0.532
0.522
0.517
0.513
0.542
0.519
0.511
0.508
0.506
0.399
0.437
0.454
0.463
0.469
0.618
0.561
0.541
0.532
0.526
0.447
0.467
0.476
0.481
0.484
0.282
0.354
0.379
0.391
0.399
0.541
0.535
0.528
0.522
0.517
0.422
0.459
0.472
0.478
0.482
0.646
0.578
0.555
0.543
0.536
0.533
0.515
0.510
0.508
0.506
0.394
0.436
0.453
0.463
0.469
fˆ100
fˆ200
fˆ300
fˆ400
0.515
0.508
0.505
0.504
0.528
0.519
0.516
0.515
0.499
0.499
0.499
0.499
0.764
0.757
0.755
0.753
0.507
0.503
0.502
0.502
0.501
0.500
0.500
0.500
0.480
0.485
0.486
0.487
0.516
0.512
0.512
0.512
0.490
0.494
0.496
0.497
0.412
0.418
0.420
0.421
0.509
0.504
0.503
0.502
0.490
0.495
0.497
0.497
0.523
0.517
0.515
0.514
0.503
0.502
0.501
0.501
0.480
0.485
0.486
0.487
fˆ∞
0.500
0.511
0.500
0.750 0.500 0.500
0.489
0.511 0.500
0.424
0.500 0.500 0.511 0.500 0.489
J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
277
0.90
0.90
(A,B)
0.80
0.70
(C,D)
0.70
(A,D)
0.60
(A,E)
0.50
(A,F)
(C,E)
(C,F)
0.60
(D,E)
0.50
(D,F)
(B,C)
0.40
(B,F)
0.80
(A,C)
0.40
(E,F)
(B,D)
0.30
0.30
(B,E)
0.20
(a)
0.20
f0
fˆ10
fˆ20
fˆ30
fˆ100
fˆ300
f0
(b)
fˆ10
fˆ20
fˆ30
fˆ100
fˆ300
Fig. 1. (a) and (b) Collective intensities of preference with semantics 1: graphical representation.
0.56
0.55
0.54
0.53
(A,D)
0.52
(C,E)
0.51
0.50
0.49
f0
fˆ10
fˆ20
fˆ30
fˆ100
fˆ300
Fig. 2. Collective intensities of preference with semantics 1 in the pairs (A,D) and (C,E): graphical representation.
depending on the aggregation rules we use. Initially,A is preferred to D, but when increases, the collective
intensity of preference decreases until the collective preference is reversed, and finally D is preferred to A;
subsequently, the collective intensity of preference increases slightly, tending to the limit value provided
by fˆ∞ . In the other pathological pair, (C,E), the collective intensity of preference momentarily increases
with , but from a certain value the intensity decreases and tends to the limit value assigned by fˆ∞ .
According to the Proposition, the collective intensity of preference between two alternatives provided
by fˆ tends to the average of the maximum and minimum individual intensities when increases. This
fact shows us that the collective intensity assigned by fˆ∞ (or by fˆ for high values of ) could not be
representative of the majority opinion. For instance, if 84 individuals rather prefer an alternative to another
and 1 individual totally prefers the second alternative to the first one, taking a high value of , the second
alternative would be declared better than the first one, according to fˆ .
Figs. 3 and 4 show us that the 0.5-cut associated with the collective preference given by the arithmetic
mean provides the same ranking of alternatives with the two semantics: A, D, F, B, C, E. On the other
hand, if we consider collective intensities of preference greater than 0.5, those given by semantics 2 are
0 , but only for a difference of 0.001.
bigger than those given by semantics 1, except r̄CE
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J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
A
0.529
0.649
D
0.600
0.800
0.822
F
0.866
0.707
0.794
0.612
0.766
0.690
0.703
B
0.609
0.706
C
0.538
E
Fig. 3. Collective intensities of preference with semantics 1 for the arithmetic mean aggregation rule: graphical representation.
The aforementioned ranking remains with the aggregation rules associated with fˆ for low values of
: smaller than 10 and 50 for semantics 1 and 2, respectively. But, for higher values of , the collective
preferences between A and D are reversed and the new ranking for the 0.5-cut is D, A, F, B, C, E.
5.2. Consistency analysis
Now we check the individual fulfillment of each fuzzy transitivity property Ti , with i = 1, . . . , 6. We
have considered two different approaches: on the one hand, the absolute fulfillment of the properties, taking
into account the percentages of students who satisfy each property (in all the triplets of alternatives); on
the other hand, we have considered a relative measure of the accomplishment of each property, regarding
the percentage of triplets xi , xj , xk verifying
(rij > 0.5 and rj k > 0.5) ⇒ (rik > 0.5 and rik rij ∗ rj k ).
Table 4 contains percentages of absolute and relative fulfillment of each fuzzy transitivity property Ti for
individual fuzzy preferences, according to the two semantics. Obviously, the absolute accomplishment
of each property Ti is smaller than the relative one. Notice that differences between these percentages
increase with i. We also note that the results coincide in the two semantics for T1 , T4 , T5 and T6 . The
results are the same for T1 , T4 and T6 , because the fulfillment of these properties does not depend on the
semantics we use. However, the accomplishment of the other properties could depend on the semantics.
In our empirical case, T5 has obtained the same outcomes in both semantics, but T2 and T3 have achieved
J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
279
A
0.543
D
0.681
0.626
0.859
0.876
F
0.929
0.747
0.860
0.628
0.819
B
0.727
0.648
0.742
0.770
C
0.537
E
Fig. 4. Collective intensities of preference with semantics 2 for the arithmetic mean aggregation rule: graphical representation.
Table 4
Percentages of individual fulfillment of Ti with the two semantics: values
T1
T2
T3
T4
T5
T6
Semantics
1
2
1
2
1
2
1
2
1
2
1
2
Absolute
Relative
78.82
98.59
78.82
98.59
75.29
98.12
56.47
96.47
56.47
96.47
50.59
96.06
50.59
96.06
50.59
96.06
25.88
87.82
25.88
87.82
18.82
84.12
18.82
84.12
different fulfillment levels, being T2 the more sensitive to the semantics. So, in the absolute case there
is a difference near 20% between the percentages of students satisfying this property, depending on the
semantics we use. This is due to the fact that the real numbers associated with the linguistic labels of the
semantics 2 are greater than or equal to those used in the semantics 1 (see Table 1). Consequently, it is
more difficult to satisfy this property with semantics 2.
This part of our empirical study is related to another one appearing in [24], where 44 students compared
all the possible pairs of alternatives that could be arranged in a set of 5 alternatives; then 440 pairs and 440
triplets were involved. As commented before, in our study 85 students show their preferences over the
pairs of a set of 6 alternatives; hence, 1275 pairs and 1700 triplets are implicated. We have to note that the
fuzzy transitivity properties analyzed in [24] are stronger than those included in our analysis, because in
that paper our requirement of individual intensities of preference being greater than 0.5 becomes greater
than or equal to 0.5. The properties (S), (0.5), (M) and (W) of [24] are similar (but stronger) to our T6 ,
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J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
Table 5
Percentages of collective fulfillment of Ti with the two semantics: values
T1 –T4
T5
T6
Semantics
1
2
1
2
1
2
f0
fˆ1
fˆ2
fˆ3
fˆ4
fˆ5
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
95
95
90
90
90
90
90
90
90
90
90
85
fˆ10
fˆ20
fˆ30
fˆ40
fˆ50
100
100
100
100
100
100
100
100
100
100
100
90
90
90
90
100
95
95
95
90
90
80
80
80
80
85
90
85
85
85
fˆ100
fˆ200
fˆ300
fˆ400
100
100
100
100
100
100
100
100
90
90
90
90
85
85
85
85
80
80
85
85
80
80
80
80
fˆ∞
100
100
100
100
100
100
100%
75%
Absolute
Relative
50%
A. Mean
25%
0%
T1
T2
T3
T4
T5
T6
Fig. 5. Percentages of fulfillment of Ti with semantics 1: graphical representation.
T5 , T4 and T1 , respectively. It is worth to emphasize that the relative fulfillment of these properties has
been very similar in both empirical analyses: 70.2% in (S) versus 84.12% in T6 ; 86.6% in (0.5) versus
87.72% in T5 ; 93.6% in (M) versus 96.06% in T4 ; and 97.7% in (S) versus 98.59% in T1 .
Table 5 shows percentages of relative fulfillment of each property Ti for collective preferences, according to the considered aggregation rules. Figs. 5 and 6 show the accomplishment of each property Ti
graphically, both for individuals and for the aggregation rule associated with the arithmetic mean.
J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
281
100%
75%
Absolute
Relative
50%
A. Mean
25%
0%
T1
T2
T3
T4
T5
T6
Fig. 6. Percentages of fulfillment of Ti with semantics 2: graphical representation.
Table 6
Percentages of fulfillment of ordinary transitivity in the -cuts with the two semantics: values
0.5
Semantics
1
2
1
2
1
2
1
2
1
2
Absolute
Relative
78.82
98.59
78.82
98.59
78.82
98.59
78.82
98.59
82.35
98.94
78.82
98.59
72.94
98.06
82.35
98.94
98.82
99.94
72.94
98.06
0.6
0.7
0.8
0.9
In spite of individual inconsistencies, we have to point out the extraordinary fulfillment of the consistency properties by the collective preferences: in both semantics, for each aggregation rule T1 , T2 , T3 and
T4 are totally satisfied; T5 is verified in all the triplets for the aggregation rules associated with f0 and fˆ
for 10; for > 10, T5 is not satisfied for no more than 3 triplets. We note that T6 is the only property
which is not verified for any aggregation rule, but for no more than 4 triplets.
In most cases, percentages of relative fulfillment of the properties Ti decrease or remain constant for
the aggregation rules associated with fˆ whenever increases. However, this behavior is not general: for
example, with semantics 2 there are more triplets satisfying T6 for = 20 than for < 20.
With regard to the aggregation rule associated with fˆ∞ , it is worth to emphasize that it verifies all the
properties. This behavior could seem surprising, because the fulfillment of the Ti properties decreases
when increases. We note that, in the limit case fˆ∞ , the collective intensity of preference is usually 0.5
(indifference), so the properties are satisfied automatically.
Table 6 and Figs. 7 and 8 show the accomplishment of the ordinary transitivity in some -cuts associated
with the individual fuzzy preferences. According to Table 5, all the collective preferences satisfy T4 ; then,
all the -cuts associated with collective fuzzy preferences are transitive.
We note that the two semantics provide the same results for the 0.5 and 0.6 cuts. Although percentages
of relative fulfillment are similar in both semantics, the absolute accomplishment of the properties is more
sensible to the use of different semantics: in the 0.8-cut there is a difference of 9.41%, and in the 0.9-cut
the difference is 25.88%. We emphasize that, for all the -cuts considered and for the two semantics, the
fulfillment of the transitivity is total in the aggregate preferences. It is worth to attract the attention on the
fact that for the two semantics and for each aggregation rule, all the -cuts are transitive.
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J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
100%
Absolute
Relative
A. Mean
90%
80%
70%
0.5
0.6
0.7
0.8
0.9
Fig. 7. Percentages of fulfillment of ordinary transitivity in the -cuts with semantics 1: graphical representation.
100%
Absolute
90%
Relative
A. Mean
80%
70%
0.5
0.6
0.7
0.8
0.9
Fig. 8. Percentages of fulfillment of ordinary transitivity in the -cuts with semantics 2: graphical representation.
Finally, notice that there is not a monotonic behavior in the fulfillment of transitivity in -cuts. In fact,
this accomplishment is independent of the values of . For instance, if an individual strongly prefers xi
to xj and xj to xk and, simultaneously, slightly prefers xi to xk , then all the considered -cuts, except for
= 0.7, are transitive for the semantics 1; however, all the considered -cuts, except for = 0.8, are
transitive for the semantics 2.
6. Concluding remarks
When a group opinion has to be constructed taking into account individual preferences among alternatives, it is essential to choose an appropriate aggregation rule in order to avoid undesirable outcomes.
With this purpose, in this paper we have considered neutral and stable for translations aggregation rules,
which transmit reciprocity from individual fuzzy preferences to the collective one. Within this class of aggregation rules, we have taken into account those associated with the arithmetic mean, f0 , the symmetric
part of the exponential quasiarithmetic means, fˆ , and the limit case fˆ∞ .
J.L. García-Lapresta, L.C. Meneses / Fuzzy Sets and Systems 151 (2005) 269 – 284
283
In order to allow the individuals to show their graded preferences among the alternatives, we have
considered linguistic labels represented by real numbers, with two different semantics related to two
different approaches. It is worth to emphasize that both the individual and collective outcomes have been
very similar in the two semantics. Among the results obtained in our empirical study, we note that the
ranking provided by the 0.5-cut associated with the collective preference generated by f0 and fˆ , for low
values of (smaller than 10 and 50 for semantics 1 and 2, respectively), is the same; for higher values
of the ranking is very similar to the first one. But using fˆ∞ and extreme values of could produce
non-representative outcomes, because collective indifferences appear.
A purpose of the paper has been to compare individual and collective rational behavior according
to 6 fuzzy transitivity properties. While individuals do not satisfy any property for some triplets, all the
collective fuzzy preferences provided by the considered aggregation rules fully verify the 4 first properties,
included the more usual assumption of consistency, the weak max–min transitivity. And the other two
properties have been accomplished by a high percentage of individuals.
Acknowledgements
The financial support of the Junta de Castilla y León (Consejería de Educación y Cultura, Proyecto
VA057/02) and the Spanish Ministerio de Ciencia y Tecnología, Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica (I+D+I) (Proyecto BEC2001-2253) and ERDF are gratefully
acknowledged.
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