INDIVIDUAL VALUED PREFERENCES AND THEIR AGGREGATION:
CONSISTENCY ANALYSIS IN A REAL CASE ∗
José Luis GARCÍA-LAPRESTA§ & Luis Carlos MENESES
Departamento de Economía Aplicada (Matemáticas)
Universidad de Valladolid, Spain
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.
Keywords: individual decision-making, graded preferences, linguistic labels, fuzzy transitivity, rational
behavior, group decision-making, aggregation operators, quasiarithmetic means.
Short title: Consistency analysis in a real case.
∗
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.
§
Corresponding author. Avda. Valle de Esgueva 6, 47011 Valladolid, Spain. Phone: +34 983 184 391,
Fax: +34 983 423 299. Email: [email protected]
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 preferences, individuals can also have
inconsistent opinions when they show intensities of preference over more than two
alternatives. See for instance [25], 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, 31]), 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
2
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 ( α -cuts) associated with the
aggregate fuzzy preference relation.
Several analyses and references about quasiarithmetic means and other aggregation
operators can be found in [5] and [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
3
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 , x j ) = rijk ∈ [ 0,1] for every xi , x j ∈ X . This index rijk means the intensity
of preference with which individual k prefers xi over x j , being 1, 0.5 or 0 depending on
whether this individual prefers absolutely xi to x j , is indifferent between xi and x j , or
prefers absolutely x j 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 x j , and the closer is the number to
0, the more x j is preferred to xi .
Moreover, we suppose that R k is reciprocal, i.e., rijk + rjik = 1 for every xi , x j ∈ 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, 21, 27], among others.
Given R ∈ R(X ) , it is easy to see that, for every α ∈ [0.5, 1) , the ordinary binary relation
on X, Pα , defined by xi Pα x j ⇔ rij > α , is asymmetric, i.e., if xi Pα x j , then not x j Pα xi .
Thus, Pα is an ordinary preference relation on X, the α -cut of R. The indifference
relation associated with Pα reflects absence of preference, and it is defined by
xi I α x j ⇔ neither xi Pα x j nor x j Pα xi , i.e., rij ≤ α and rji ≤ α . By reciprocity, these
conditions are equivalent to 1 − α ≤ rij ≤ α . Consequently, for each pair of alternatives
x i , x j ∈ X one and only one of the following situations occurs: xi Pα x j ( rij > α ),
xi I α x j ( 1 − α ≤ rij ≤ α ), x j Pα xi ( rji > α , i.e., rij < 1 − α ).
4
We note that if R is not reciprocal, then Pα 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
( R ,… , R ) ∈ R ( X )
1
m
relation,
m
R = F ( R1 ,… , R m ) ∈ R ( X ) ,
to
each
profile
of individual fuzzy preferences. With rij we denote the collective
preference between xi and x j according to R .
In this paper we only consider neutral aggregation rules, those providing an egalitarian
treatment to alternatives: for every pair of profiles
( R ,… , R ) , ( S ,… , S ) ∈ R ( X )
1
m
1
m
m
and every alternatives xi , x j , x p , xq ∈ X , if rijk = s kpq is satisfied for all k ∈ {1,… , m} ,
then rij = s pq . Obviously, F : R( X ) m → R( X ) is neutral if and only if there exists a
function f : [ 0,1] → [ 0,1] such that rij = f ( rij1 ,… , rijm ) for all alternatives xi , x j ∈ X .
m
Since R 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] → [ 0,1] verifying the previous condition will be considered
m
reciprocal and it will naturally define a neutral aggregation rule F : R( X ) m → R( X )
such that rij = 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] :
5
f ( a1 + t ,… , am + t ) = f ( a1 ,… , am ) + t ,
whenever ( a1 + t ,… , am + t ) ∈ [ 0,1] and f ( a1 + t ,… , am + t ) ∈ [ 0,1] .
m
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] → [ 0,1] , taking into account only the individual intensities of
m
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].
Given an increasing and bijective function ϕ : [ 0,1] → [ 0,1] , the quasiarithmetic mean
associated to ϕ is the function f : [ 0,1] → [ 0,1] defined by
m
⎛ m
⎞
⎜ ∑ ϕ ( ak ) ⎟
⎟.
f ( a1 ,… , am ) = ϕ −1 ⎜ k =1
m
⎜
⎟
⎜
⎟
⎝
⎠
According to [19] (see [12, pp. 117-118]), the quasiarithmetic means generated by the
exponential functions,
6
ϕ (a) =
eβ a −1
, β > 0,
eβ −1
and the arithmetic mean, generated by the identity function ϕ ( a ) = a :
m
f β ( a1 ,… , am ) =
1
β
m
∑ e β ak
ln
k =1
f 0 ( a1 ,… , am ) =
and
m
∑a
k =1
k
m
are the only quasiarithmetic means satisfying stability for translations.
In [14] is proven that the quasiarithmetic mean associated to ϕ is reciprocal if and only
if ϕ (1 − a ) = 1 − ϕ ( a ) for all a ∈ [ 0,1] . Then, the only reciprocal function of the family
{f
β
β ≥ 0} is the arithmetic mean, f 0 . However, according to [15], the symmetric part
m
of f β , the function fˆβ : [ 0,1] → [ 0,1] defined by
m
1
ln
fˆβ ( a1 ,… , am ) =
2β
∑ eβ
k =1
m
∑e
ak
,
− β ak
k =1
is reciprocal and stable for translations; moreover, in that paper is established that for
m = 2 , fˆβ coincides with the arithmetic mean, and that for m > 2 , fˆβ is not a
quasiarithmetic mean.
We are now going to justify that the function fˆβ tends to the average of the minimum
and maximum values of the components of each vector when β tends to infinity.
Proposition. Given a vector
( a1 ,… , am ) ∈ [0,1]
m
a* = max {a1 ,… , am } . Then:
7
, let a* = min {a1 ,… , am } and
a + a*
.
fˆ∞ ( a1 ,… , am ) = lim fˆβ ( a1 ,… , am ) = *
β →∞
2
m
1
ln
Proof: lim fˆβ ( a1 ,… , am ) = lim
β →∞
β →∞ 2 β
m
∑ eβ ak
k =1
m
∑e
m
ln ∑ e β ak − ln ∑ e − β ak
= lim
k =1
− β ak
k =1
2β
β →∞
.
k =1
Applying the L’Hospital rule, we have:
m
∑ a eβ
k =1
m
lim fˆβ ( a1 ,… , am ) = lim
β →∞
m
ak
k
∑ eβ
+
∑a e β
k =1
m
∑e β
ak
k =1
− ak
=
k =1
β →∞
2
m
1
= lim
2 β →∞
− ak
k
∑ ak e β a k
k =1
m
∑e
β ak
m
∑
1
+ lim k =1m
2 β →∞
∑e
k =1
=
1
lim
2 β →∞
e− β a
e
*
k =1
m
*
∑e
β ak
m
+
1
lim
2 β →∞
e β a* ∑ ak e − β ak
k =1
m
e
β a*
k =1
m
1
= lim
2 β →∞
∑ ak e
k =1
m
∑e
=
− β ak
k =1
m
∑ ak eβ ak
−βa
ak e − β ak
(
k =1
)
m
∑
1
k =1
lim
+
m
β
→∞
β ( ak − a* )
2
β ak − a*
∑e
=
− β ak
ak e
− β ( ak − a* )
∑ e β(
k =1
−
ak − a* )
=
a + a*
1 * 1
.
a + a* = *
2
2
2
k =1
m
We immediately see that the function fˆ∞ : [ 0,1] → [ 0,1] is reciprocal and stable for
translations. Then, fˆβ 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
8
functions f 0 , fˆβ ( β > 0 ) and fˆ∞ . Given two alternatives xi , x j ∈ X , the above
mentioned aggregation rules assign the collective intensity of preference between xi
and x j in the following manner:
m
r ij0 = f 0 ( rij1 ,… , rijm ) =
m
∑ rijk
k =1
m
,
1
rij β = fˆβ ( rij1 ,… , rijm ) =
ln
2β
∑e
k =1
m
∑e
β rijk
− β rijk
,
k =1
rij
∞
= 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 x j and
x j P xk implies xi P xk , for all xi , x j , 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, 10, 11, 12, 16, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31], 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] .
9
–
( a ≤ a ' and b ≤ b' ) ⇒ a ∗ b ≤ a '∗b' ,
Monotonicity:
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 rjk > 0.5) ⇒ (rik > 0.5 and rik ≥ rij ∗ rjk ) ,
for all xi , x j , x k ∈ X .
Obviously, the ordinary preference relation P0.5 associated with every weak max- ∗
transitive R ∈ R(X ) is transitive.
We note that max- ∗ transitivity for a fuzzy binary relation R was initially defined by
demanding rik ≥ rij ∗ r jk . “Weak” (or “restricted”) conditions are considered in [7, 27],
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 ∗4 b = min{a, b} ,
a ∗5 b =
a+b
,
2
a ∗3 b = max{ab, 0.5} ,
a ∗6 b = max{a , b} .
We say that R ∈ R(X ) verifies property Ti if R is weak max- ∗ i transitive.
10
It is easily seen that
a, b ∈ [ 0.5, 1] , i.e., T6
a *1 b ≤ a *2 b ≤ a *3 b ≤ a *4 b ≤ a *5 b ≤ a *6 b , for
⇒ T5
⇒ T4
⇒ T3
⇒ T2
all
⇒ T1 . Moreover, T1 is
equivalent to P0.5 being transitive, and Pα is transitive for all α ∈ [0.5, 1) whenever R
satisfies T4 , T5 or T6 . This is due to the fact that T4 is equivalent to Pα being transitive
for all α ∈ [ 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) Business Administration and Management (5 years).
B) Business Administration (3 years).
C) Law (5 years).
D) Business Administration, Management and Law (6 years).
E) Labor Relations (3 years).
F) 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).
11
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 x j , rij , can be of the 9 terms;
taking into account reciprocity, the intensity of preference between x j and xi is defined
by r ji = 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).
******** INSERT TABLE 1 HERE ********
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 [4], where vagueness is greater around indifference
than in the proximities of extreme preferences. Our assignments are similar to the
associated real numbers provided by [8] to the trapezoidal fuzzy numbers given by [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
12
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 α-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, f 0 , the symmetric part of the
exponential quasiarithmetic means fˆβ , for several values of β , and the limit case fˆ∞ .
The exponential quasiarithmetic means, f β , 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 f 0 , fˆβ for
β = 1, 2, 3, 4, 5, 10, 20, 30, 40, 50, 100, 200, 300, 400 , and fˆ∞ , according to the two
semantics. Figures 1a and 1b show graphically these outcomes for semantics 1.
13
******** INSERT TABLE 2 HERE ********
******** INSERT TABLE 3 HERE ********
******** INSERT FIGURE 1a HERE ********
******** INSERT FIGURE 1b HERE ********
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 Figure 2). The pair (A,D) is the only one
where the sense of the collective preference changes depending on the aggregation rules
we use. Initially, A is preferred to D, but when β 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 β , 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 fˆβ tends to the average of the maximum and minimum
individual intensities when β increases. This fact shows us that the collective intensity
assigned by fˆ∞ (or by fˆβ for high values of β ) 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
14
value of β , the second alternative would be declared better than the first one, according
to fˆβ .
******** INSERT FIGURE 2 HERE ********
Figures 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 bigger than those given by
semantics 1, except rCE0 , but only for a difference of 0.001.
******** INSERT FIGURE 3 HERE ********
******** INSERT FIGURE 4 HERE ********
The aforementioned ranking remains with the aggregation rules associated with fˆβ for
low values of β : smaller than 10 and 50 for semantics 1 and 2, respectively. But, for
higher values of β , 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
15
relative measure of the accomplishment of each property, regarding the percentage of
triplets x i , x j , x k verifying
(rij > 0.5 and rjk > 0.5) ⇒ (rik > 0.5 and rik ≥ rij ∗ rjk ) .
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 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.
******** INSERT TABLE 4 HERE ********
This part of our empirical study is related to another one appearing in [25], 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
16
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 [25] 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 [25] are similar (but stronger) to our T6 , 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. Figures 5 and 6 show the
accomplishment of each property Ti graphically, both for individuals and for the
aggregation rule associated with the arithmetic mean.
******** INSERT TABLE 5 HERE ********
******** INSERT FIGURE 5 HERE ********
******** INSERT FIGURE 6 HERE ********
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 f 0 and fˆβ for β ≤ 10 ; for β > 10 , T5 is not
17
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 fˆβ whenever β increases. However,
this behavior is not general: for example, with semantics 2 there are more triplets
satisfying T6 for β = 20 than for β < 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 β 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 Figures 7 and 8 show the accomplishment of the ordinary transitivity in
some α -cuts associated with the individual fuzzy preferences. According to Table 5, all
the collective preferences satisfy T4 ; then, all the α -cuts associated with collective
fuzzy preferences are transitive.
******** INSERT TABLE 6 HERE ********
******** INSERT FIGURE 7 HERE ********
******** INSERT FIGURE 8 HERE ********
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
18
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 α -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
α -cuts are transitive.
Finally, notice that there is not a monotonic behavior in the fulfillment of transitivity in
α -cuts. In fact, this accomplishment is independent of the values of α . For instance, if
an individual strongly prefers xi to x j and x j to xk and, simultaneously, slightly
prefers xi to xk , then all the considered α -cuts, except for α = 0.7 , are transitive for
the semantics 1; however, all the considered α -cuts, except for α = 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, f 0 , the symmetric part of
the exponential quasiarithmetic means, fˆβ , and the limit case fˆ∞ .
19
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 f 0 and
fˆβ , for low values of β (smaller than 10 and 50 for semantics 1 and 2, respectively), is
the same; for higher values of β the ranking is very similar to the first one. But using
fˆ∞ and extreme values of β 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.
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24
Term
Semantics 1
Semantics 2
xi is totally preferred to x j
rij = 1 ( rji = 0 )
rij = 1 ( rji = 0 )
xi is highly preferred to x j
rij = 0.875 ( rji = 0. 125)
rij = 0.978 ( r ji = 0.022 )
xi is rather preferred to x j
rij = 0.750 ( rji = 0. 250)
rij = 0.848 ( r ji = 0.152 )
xi is slightly preferred to x j
rij = 0.625 ( rji = 0. 375)
rij = 0.717 ( r ji = 0.283 )
xi is indifferent to x j
rij = 0.500 ( rji = 0. 500)
rij = 0.500 ( rji = 0. 500)
x j is rather preferred to xi
rij = 0.375 ( rji = 0. 625)
rij = 0.283 ( r ji = 0.717 )
x j is slightly preferred to xi
rij = 0.250 ( rji = 0. 750)
rij = 0.152 ( r ji = 0.848 )
x j is highly preferred to xi
rij = 0.125 ( rji = 0. 875)
rij = 0.022 ( r ji = 0.978 )
x j is totally preferred to xi
rij = 0 ( rji = 1 )
rij = 0 ( rji = 1 )
Table 1. Two semantics for 9 terms.
25
(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
fˆ
1
fˆ2
fˆ
3
fˆ4
fˆ
5
fˆ10
fˆ
20
fˆ30
fˆ
40
fˆ50
fˆ
100
fˆ200
fˆ
300
fˆ400
fˆ
∞
0.800 0.822
0.529 0.866 0.649 0.609 0.293 0.703 0.388 0.206 0.538 0.310 0.766 0.600 0.294
0.797 0.821
0.528 0.866 0.645 0.607 0.295 0.702 0.390 0.207 0.539 0.313 0.765 0.597 0.296
0.788 0.817
0.525 0.865 0.637 0.600 0.302 0.698 0.394 0.209 0.540 0.322 0.760 0.590 0.302
0.772 0.810
0.520 0.864 0.626 0.591 0.312 0.692 0.399 0.213 0.542 0.335 0.753 0.581 0.311
0.752 0.800
0.516 0.863 0.613 0.580 0.324 0.683 0.405 0.218 0.545 0.349 0.743 0.571 0.321
0.729 0.788
0.512 0.861 0.601 0.569 0.336 0.673 0.410 0.225 0.548 0.363 0.731 0.562 0.332
0.638 0.718
0.502 0.846 0.562 0.531 0.375 0.623 0.430 0.267 0.555 0.414 0.673 0.533 0.370
0.574 0.644
0.498 0.814 0.534 0.506 0.402 0.581 0.450 0.319 0.542 0.452 0.619 0.517 0.399
0.551 0.617
0.498 0.795 0.523 0.501 0.413 0.571 0.462 0.338 0.530 0.467 0.600 0.511 0.411
0.538 0.604
0.499 0.784 0.517 0.500 0.418 0.568 0.470 0.347 0.522 0.475 0.591 0.509 0.417
0.531 0.595
0.499 0.777 0.514 0.500 0.422 0.567 0.476 0.353 0.518 0.480 0.585 0.507 0.421
0.515 0.579
0.499 0.764 0.507 0.500 0.430 0.565 0.488 0.364 0.509 0.490 0.574 0.503 0.429
0.508 0.571
0.500 0.757 0.503 0.500 0.434 0.564 0.494 0.370 0.504 0.495 0.568 0.502 0.433
0.505 0.568
0.500 0.755 0.502 0.500 0.435 0.563 0.496 0.371 0.503 0.497 0.566 0.501 0.435
0.504 0.567
0.500 0.753 0.502 0.500 0.436 0.563 0.497 0.372 0.502 0.497 0.565 0.501 0.435
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 2. Collective intensities of preference with semantics 1. Values.
26
(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
fˆ
1
fˆ2
fˆ
3
fˆ4
fˆ
5
fˆ10
fˆ
20
fˆ30
fˆ
40
fˆ50
fˆ
100
fˆ200
fˆ
300
fˆ400
fˆ
∞
0.859 0.876 0.543 0.929 0.681 0.648 0.252 0.770 0.372 0.140 0.538 0.273 0.819 0.626 0.258
0.853 0.873 0.541 0.928 0.675 0.644 0.259 0.767 0.375 0.143 0.538 0.279 0.815 0.621 0.264
0.836 0.864 0.535 0.927 0.659 0.631 0.275 0.756 0.385 0.149 0.538 0.296 0.804 0.608 0.278
0.808 0.849 0.528 0.925 0.639 0.615 0.298 0.740 0.397 0.160 0.539 0.318 0.786 0.592 0.298
0.773 0.828 0.523 0.922 0.619 0.598 0.320 0.720 0.409 0.175 0.540 0.341 0.763 0.577 0.318
0.738 0.802 0.518 0.919 0.603 0.584 0.341 0.698 0.419 0.193 0.540 0.362 0.739 0.565 0.337
0.629 0.690 0.507 0.888 0.559 0.542 0.399 0.618 0.447 0.282 0.541 0.422 0.646 0.533 0.394
0.565 0.603 0.502 0.829 0.532 0.519 0.437 0.561 0.467 0.354 0.535 0.459 0.578 0.515 0.436
0.544 0.572 0.501 0.802 0.522 0.511 0.454 0.541 0.476 0.379 0.528 0.472 0.555 0.510 0.453
0.534 0.556 0.500 0.788 0.517 0.508 0.463 0.532 0.481 0.391 0.522 0.478 0.543 0.508 0.463
0.527 0.546 0.500 0.780 0.513 0.506 0.469 0.526 0.484 0.399 0.517 0.482 0.536 0.506 0.469
0.515 0.528 0.499 0.764 0.507 0.501 0.480 0.516 0.490 0.412 0.509 0.490 0.523 0.503 0.480
0.508 0.519 0.499 0.757 0.503 0.500 0.485 0.512 0.494 0.418 0.504 0.495 0.517 0.502 0.485
0.505 0.516 0.499 0.755 0.502 0.500 0.486 0.512 0.496 0.420 0.503 0.497 0.515 0.501 0.486
0.504 0.515 0.499 0.753 0.502 0.500 0.487 0.512 0.497 0.421 0.502 0.497 0.514 0.501 0.487
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
Table 3. Collective intensities of preference with semantics 2. Values.
27
0.90
(A,B)
0.80
(A,C)
0.70
(A,D)
0.60
(A,E)
0.50
(A,F)
(B,C)
0.40
(B,D)
0.30
(B,E)
0.20
f0
fˆ10
fˆ 20
fˆ30
fˆ100
fˆ300
Figure 1a. Collective intensities of preference with semantics 1. Graphical representation.
28
0.90
0.80
(B,F)
0.70
(C,D)
(C,E)
0.60
(C,F)
0.50
(D,E)
0.40
(D,F)
0.30
(E,F)
0.20
f0
fˆ10
fˆ20
fˆ30
fˆ100
fˆ300
Figure 1b. Collective intensities of preference with semantics 1 (continuation). Graphical
representation.
29
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
Figure 2. Collective intensities of preference with semantics 1 in the pairs (A,D) and (C,E).
Graphical representation.
30
A
0.529
0.649
D
0.600
0.800
0.822
F
0.866
0.612
0.707
0.794
0.766
0.690
B
0.609
0.703
0.706
C
0.538
E
Figure 3. Collective intensities of preference with semantics 1 for the arithmetic mean aggregation
rule. Graphical representation.
31
A
0.543
0.681
D
0.626
0.859
0.876
F
0.929
0.628
0.747
0.860
0.819
0.727
B
0.648
0.742
0.770
C
0.537
E
Figure 4. Collective intensities of preference with semantics 2 for the arithmetic mean aggregation
rule. Graphical representation.
32
T1
Semantics
1
T2
2
1
T3
2
1
T4
2
1
T5
2
1
T6
2
1
2
Absolute 78.82 78.82 75.29 56.47 56.47 50.59 50.59 50.59 25.88 25.88 18.82 18.82
Relative 98.59 98.59 98.12 96.47 96.47 96.06 96.06 96.06 87.82 87.82 84.12 84.12
Table 4. Percentages of individual fulfillment of Ti with the two semantics. Values.
33
T1 - T4
T5
T6
Semantics
1
2
1
2
1
2
f0
fˆ
100
100
100
100
95
90
100
100
100
100
95
90
fˆ2
fˆ
100
100
100
100
90
90
100
100
100
100
90
90
fˆ4
fˆ
100
100
100
100
90
90
100
100
100
100
90
85
fˆ10
fˆ
100
100
100
100
90
85
100
100
90
95
80
90
fˆ30
fˆ
100
100
90
95
80
85
100
100
90
95
80
85
fˆ50
fˆ
100
100
90
90
80
85
100
100
90
85
80
80
fˆ200
fˆ
100
100
90
85
80
80
100
100
90
85
85
80
fˆ400
fˆ
100
100
90
85
85
80
100
100
100
100
100
100
1
3
5
20
40
100
300
∞
Table 5. Percentages of collective fulfillment of Ti with the two semantics. Values.
34
100%
75%
Absolute
Relative
A. Mean
50%
25%
0%
T1
T2
T3
T4
T5
T6
Figure 5. Percentages of fulfillment of Ti with semantics 1. Graphical representation.
35
100%
75%
Absolute
Relative
A. Mean
50%
25%
0%
T1
T2
T3
T4
T5
T6
Figure 6. Percentages of fulfillment of Ti with semantics 2. Graphical representation.
36
α
Semantics
0.5
1
0.6
0.7
0.8
0.9
2
1
2
1
2
1
2
1
2
Absolute 78.82
78.82
78.82
78.82
82.35
78.82
72.94
82.35
98.82
72.94
Relative
98.59
98.59
98.59
98.94
98.59
98.06
98.94
99.94
98.06
98.59
Table 6. Percentages of fulfillment of ordinary transitivity in the α -cuts with the two semantics.
Values.
37
100%
Absolute
Relative
A. Mean
90%
80%
70%
0.5
0.6
0.7
0.8
0.9
Figure 7. Percentages of fulfillment of ordinary transitivity in the α -cuts with semantics 1.
Graphical representation.
38
100%
Absolute
Relative
A. Mean
90%
80%
70%
0.5
0.6
0.7
0.8
0.9
Figure 8. Percentages of fulfillment of ordinary transitivity in the α -cuts with semantics 2.
Graphical representation.
39