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Fuzzy Sets and Systems 155 (2005) 372 – 389 www.elsevier.com/locate/fss Fuzzy strict preference and social choice Louis Aimé Fono a , , Nicolas Gabriel Andjiga b a Département de Mathématiques et Informatique, Université de Douala, Faculté des Sciences, BP 24157 Douala, Cameroun b Département de Mathématiques, Université deYaoundé I, Ecole Normale Supérieure, BP 47Yaoundé, Cameroun Received 26 November 2003; received in revised form 20 April 2005; accepted 2 May 2005 Available online 26 May 2005 Abstract In this paper we generalize some classical factorizations of a fuzzy relation into a symmetric component (indif- ference) and an asymmetric and regular component (regular fuzzy strict preference). From the above notions, we establish two properties of any regular fuzzy strict preference of a max--transitive fuzzy relation, which are then used to obtain new fuzzy versions of Gibbard’s oligarchy theorem and Arrow’s impossibility theorem. © 2005 Elsevier B.V. All rights reserved. Keywords: Arrow impossibility theorem; Fuzzy preference; Gibbard’s oligarchy theorem; Max--transitivity; Regular fuzzy strict preference 1. Introduction Dutta [9] and Richardson [17] establish fuzzy versions of Gibbard’s oligarchy theorem and Arrow’s impossibility theorem. We summarize their results in Table 1. Let us remark that: (i) Dutta uses the min-transitivity and the L-transitivity which are two classical examples of standard transitivity called max--transitivity, where is a t-norm (as indicated in Definitions 1 and 2, and Remark 1). He also uses P 1 which is a classical example of a regular fuzzy strict preference (as indicated in Section 3). Corresponding author. Tel.: +237 602 29 15. E-mail addresses: [email protected] (L.A. Fono), [email protected] (N.G. Andjiga). 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.05.001

Fuzzy strict preference and social choice

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Fuzzy Sets and Systems155 (2005) 372–389www.elsevier.com/locate/fss

Fuzzy strict preference and social choice

Louis Aimé Fonoa,∗, Nicolas Gabriel Andjigab

aDépartement de Mathématiques et Informatique, Université de Douala, Faculté des Sciences, BP 24157 Douala, CamerounbDépartement de Mathématiques, Université deYaoundé I, Ecole Normale Supérieure, BP 47Yaoundé, Cameroun

Received 26 November 2003; received in revised form 20 April 2005; accepted 2 May 2005Available online 26 May 2005

Abstract

In this paper we generalize some classical factorizations of a fuzzy relation into a symmetric component (indif-ference) and an asymmetric and regular component (regular fuzzy strict preference). From the above notions, weestablish two properties of any regular fuzzy strict preference of a max-∗-transitive fuzzy relation, which are thenused to obtain new fuzzy versions of Gibbard’s oligarchy theorem and Arrow’s impossibility theorem.© 2005 Elsevier B.V. All rights reserved.

Keywords:Arrow impossibility theorem; Fuzzy preference; Gibbard’s oligarchy theorem; Max-∗-transitivity; Regular fuzzystrict preference

1. Introduction

Dutta [9] and Richardson [17] establish fuzzy versions of Gibbard’s oligarchy theorem and Arrow’simpossibility theorem. We summarize their results in Table 1.

Let us remark that:

(i) Dutta uses the min-transitivity and the L-transitivity which are two classical examples of standardtransitivity called max-∗-transitivity, where∗ is a t-norm (as indicated in Definitions 1 and 2, andRemark 1). He also usesP1 which is a classical example of a regular fuzzy strict preference (asindicated in Section 3).

∗ Corresponding author. Tel.: +237 602 29 15.E-mail addresses:[email protected](L.A. Fono),[email protected](N.G. Andjiga).

0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2005.05.001

L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389 373

Table 1

References Hypotheses Hypotheses Properties Type of regular Conclusionon individual on social of FAR fuzzy strictpreferences preferences preference

Proposition 3.7 Reflexive, connected Reflexive, connected IIA and PCP1 Oligarchy(Dutta) and min-transitive and min-transitive

Proposition 3.9 Reflexive, connected Reflexive, connected IIA and PCP1 There exists a(Dutta) and min-transitive and min-transitive non-dictatorial FAR

Proposition 3.12 Reflexive, connected Reflexive, connected IIA, PC andP1 Dictator(Dutta) and min-transitive and min-transitive PR

Proposition 3.14 Reflexive, connected Reflexive, connected IIA, PC andP1 There exists a(Dutta) and L-transitive and L-transitive PR non-dictatorial FAR

Proposition 2.1 Reflexive, s-connected Reflexive, s-connected IIA and PCP Dictator(Richardson) and m-transitive and m-transitive

Proposition 2.2 Reflexive, connected Reflexive, connected IIA, PC andP2 There exists a(Richardson) and L-transitive and L-transitive PR non-dictatorial FAR

(ii) Richardson uses the strong connectedness which is a very restrictive hypothesis, and the m-transitivity(see Definition 2) which is not a standard transitivity. It seems unreasonable that ifR is m-transitive,[R(x, y) �= 1 orR(y, z) �= 1] should yield arbitrary values toR(x, z). Thus, although them-transitivity is a hypothesis weaker than any standard transitivity, it presents some intuitive dif-ficulties.

The aim of this paper is to generalize these two social choice results by using any max-∗-transitivity,any generic regular fuzzy strict preferenceP and the notion of connectedness instead of the notion ofstrong connectedness. These results are summarized in Table2.

To achieve our goal, we initially generalize some factorizations ofR into I andP introduced by [9] and[17], and characterize by means of max-∗-transitivity two properties ofP.

It is important to emphasize that many crisp properties can be generalized in different ways to thefuzzy framework. Authors such as Barrett et al. [3], Banerjee [1] and Tang [21] among others establishsome fuzzy Arrowian results with formalisms and/or assumptions distinct from those presented above.However, all these works in fuzzy social choice are based on properties (IIA, PC, PR) introduced byBarrett et al. [3]. In addition, Blin [5], De Baets et al. [7,8], Bufardi [6] and Llamazares [13,14] tacklethe problem of factorization with many assumptions different from those we use in this paper.

The paper is organized as follows: In Section 2, we define basic concepts and properties on fuzzyoperators and fuzzy relations. In Section 3, we give a general method of decomposition of a fuzzyrelation into a symmetric component and an asymmetric and regular component. We determine all themax-∗-transitive fuzzy relations whose regular fuzzy strict preferences satisfy pos-transitivity, and thennegative transitivity. In Section 4, we obtain new fuzzy versions of Gibbard’s oligarchy result and Arrow’sresult. Section 5 contains some concluding remarks.

374 L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389

Table 2

References Hypotheses Hypotheses Properties Type of regular Conclusionon individual on social of FAR fuzzy strictpreferences preferences preference

Theorem 9 Reflexive, Reflexive, IIA and PC P Oligarchyconnected, connected,-*-transitive max-∗-transitiveandQ∗

1 andQ∗1

Proposition 11 Reflexive, Reflexive, IIA and PC P1 There exists aconnected, connected, non-dictatorialmax-∗-transitive max-∗-transitive FARandQ∗

1 andQ∗1

Theorem 12 Reflexive, Reflexive, IIA, PC and P Dictatorconnected, connected, PRmax-*-transitive max-*-transitiveandQ∗

1 andQ∗1

Theorem 13 Reflexive, Reflexive, IIA and PC P Dictatorconnected, connected,max-*-transitive max-*-transitive,andQ∗

1 Q∗1 andQ∗

2

For the comparison of the two tables, see Remark 5.

2. Preliminaries

LetA be a given set of alternatives. We assume that| A | �3.

Definition 1 (Fodor and Roubens[10] , Gwét[12] , Llamazares[13,14]).

(1) A t-norm is a function∗: [0,1] × [0,1] → [0,1] satisfying for allx, y, z, u ∈ [0,1];(i) x ∗ 1 = x, (ii) x ∗ y�u ∗ z if x�u andy�z, (iii) x ∗ y = y ∗ x and (iv)(x ∗ y) ∗ z = x ∗ (y ∗ z).

(2) A t-conorm is a function⊕ : [0,1]× [0,1] → [0,1] satisfying for allx, y, z, u∈ [0,1], (i) x⊕0 = x,(ii) x ⊕ y�u⊕ z if x�u andy�z, (iii) x ⊕ y = y ⊕ x and (iv)(x ⊕ y)⊕ z = x ⊕ (y ⊕ z).

(3) A t-conorm⊕ is strict if ∀x, y ∈ [0,1], ∀z ∈ [0,1[ , x < y impliesx ⊕ z < y⊕ z.(4) Let ∗ be a continuous t-norm. The quasi-inverse of∗ is the internal composition law denoted by|

and defined over[0,1] by x | y = max{t ∈ [0,1] , x ∗ t�y} for all x, y ∈ [0,1].(5) Let⊕ be a continuous t-conorm. The quasi-subtraction of⊕ is the internal composition law denoted

by � and defined over[0,1] by x�y = min{t ∈ [0,1] , x ⊕ t�y} for all x, y ∈ [0,1].

Example 1. (1) The Zadeh’s min t-norm and the Zadeh’s max t-conorm are, respectively, denoted by∧and∨ (i.e., for alla, b ∈ [0,1] , a ∨ b = max(a, b) anda ∧ b = min(a, b)).

(2) The Lukasiewicz’s t-norm is defined by∀a, b ∈ [0,1] , a ∗ b = 0 ∨ (a + b − 1).

We assume throughout that t-norm∗ and t-conorm⊕ are continuous functions. We need thefollowing properties (for proofs, one may consult[10,12]). Let∗ be a t-norm and| its quasi-inverse.

L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389 375

For alla, b ∈ [0,1],

(i) a ∗ b � a ∧ b, (ii) b�a | b, (iii) a ∗ (a | b) = a ∧ b. (1)

Definition 2. A fuzzy binary relation is a functionR: A× A→ [0,1].

• R is reflexive if∀x ∈ A, R(x, x) = 1.• R is connected if∀x, y ∈ A, R(x, y)+ R(y, x)�1.• R is strongly connected (s-connected) if∀x, y ∈ A, R(x, y) ∨ R(y, x) = 1.• A fuzzy weak preference relation (FWPR) is a reflexive and connected fuzzy binary relation.• Let ∗ be a t-norm.R is max-∗-transitive if∀x, y, z ∈ A, R(x, z)�R(x, y) ∗ R(y, z).• R is m-transitive if∀x, y, z ∈ A, (R(x, y) = 1 andR(y, z) = 1) implyR(x, z) = 1.• Rsatisfies conditionT (see[11,20]) if

∀x, y, z ∈ A, ( R(x, y) = R(y, x) = R(y, z) = R(z, y) ) impliesR(x, z) = R(z, x).We assume throughout thatR is an FWPR.

Remark 1. LetRbe an FWPR and∗ be a t-norm:(i) For all (x, y) ∈ A× A, R(x, y) is the degree to whichx is at least as good asy.(ii) R is a crisp binary relation if∀x, y ∈ A, R(x, y) ∈ {0,1}. In this case,xRydenotesR(x, y) = 1.(iii) A strong connected fuzzy binary relation is a particular case of an FWPR. In other words, strong

connectedness implies reflexivity and connectedness.(iv) If ∗ is the Zadeh’s min t-norm, the max-∗-transitivity becomes the min-transitivity (i.e.,∀x, y, z ∈

A, R(x, z)�R(x, y) ∧ R(y, z)).(v) If ∗ is the Lukasiewicz’s t-norm, the max-∗-transitivity becomes the L-transitivity (i.e.,∀x, y, z ∈ A,

R(x, z)�R(x, y)+ R(y, z)− 1).

Let us end this section by establishing the following lemma which will be useful in the next sectionand which needs the following four reals of[0,1]:

�∗1(x, y, z) = R(z, y) ∗ R(y, x),

�∗2(x, y, z) = R(x, y) ∗ R(y, z),

�∗3(x, y, z) = (R(y, z) | R(y, x)) ∧ (R(x, y) | R(z, y)),

�∗4(x, y, z) = (R(y, x) | R(y, z)) ∧ (R(z, y) | R(x, y)),

where∗ is a t-norm,| its quasi-inverse,R is an FWPR andx, y, z ∈ A.

Lemma 1. Let R be an FWPR, ∗ be a t-norm and{x, y, z} ⊆ A.

R is max-∗-transitive on{x, y, z} ⇔{

(i) R(x, z) ∈ [�∗

2(x, y, z), �∗4(x, y, z)

],

(ii) R(z, x) ∈ [�∗

1(x, y, z), �∗3(x, y, z)

].

(2)

Proof. (⇒): The max-∗-transitivity ofRon {x, y, z} impliesR(x, z) ∗ R(z, y)�R(x, y) andR(y, x) ∗R(x, z)�R(y, z). Thus R(x, z)�R(z, y) | R(x, y) and R(x, z)�R(y, x) | R(y, z). Then, we

376 L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389

obtainR(x, z)��∗4(x, y, z). The inequalityR(x, y) ∗ R(y, z)�R(x, z) impliesR(x, z) ∈ [

�∗2(x, y, z) ,

�∗4(x, y, z)

].

Analogously, we can show thatR(z, x) ∈ [�∗

1(x, y, z) , �∗3(x, y, z)

].

(⇐): With (iii) of ( 1), the proof of the converse is analogous.�

3. Regular fuzzy strict preference

3.1. Review

Definition 3 (SeeBanerjee[1] ,Dutta[9] , Fodor andRoubens[10] , FonoandGwét[11] , Richardson[17] ).Let I andP be two fuzzy binary relations onA.

(i) I is symmetric if∀x, y ∈ A, I (x, y) = I (y, x).(ii) P is asymmetric if∀x, y ∈ A, P (x, y) ∧ P(y, x) = 0.

Although this definition of asymmetry is classic, it is a particular case of that used in[6,13,14] (∀x, y ∈A, P (x, y) ∗ P(y, x) = 0 where∗ is a t-norm).

LetRbe a crisp weak preference relation.Rcan be decomposed, in only one way, into crisp indifferenceI and crisp strict preferenceP iff (see [8,14,17,19])

I is symmetric,P is asymmetric,R = I ∪ P andI ∩ P = ∅. (3)

WhenR is an FWPR, there are many factorizations ofR into a symmetric componentI and an asym-metric componentP. Let us recall the following ones:

• Dutta[9] uses the Zadeh’s max t-conorm to model a fuzzy union and replaces the conditionI ∩P = ∅by the following one:

P is simple [ i.e.,∀x, y ∈ A, R(x, y) = R(y, x) impliesP(x, y) = P(y, x)]. (4)

He obtains the following result[9, Proposition 2.5]:If R, I and P are fuzzy relations satisfying(i) for x, y ∈ A, R(x, y) = I (x, y) ∨ P(x, y), (ii) P issimple, (iii) I is symmetric and(iv) P is asymmetric, then I and P can be uniquely derived from R asfollows: ∀x, y ∈ A, I (x, y) = R(x, y) ∧ R(y, x) andP(x, y) = P1(x, y), where

P1(x, y) ={R(x, y) if R(x, y) > R(y, x),0 otherwise.

(5)

• Richardson[17] uses the Lukasiewicz’s t-conorm and t-norm to model a fuzzy union and a fuzzyintersection. He obtains the following result [17, Proposition 1.1]:If R, I andP are fuzzy relations satisfying(i) for x, y ∈ A,R(x, y) = I (x, y)⊕P(x, y), (ii) ∀x, y ∈ A,I (x, y) ∗P(x, y) = 0, i.e.,I (x, y)+P(x, y)�1, (iii) I is symmetric and(iv) P is asymmetric, then Iand P can be uniquely derived fromR as follows: ∀x, y ∈ A, I (x, y)=R(x, y)∧R(y, x) andP(x, y)= P2(x, y), where

P2(x, y) = 0 ∨ (R(x, y)− R(y, x)). (6)

L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389 377

• Richardson[17] uses a generic t-conorm to model a fuzzy union and replaces the condition “I∩P = ∅”by the condition “P is simple”. He obtains the following result [17, Propositions 1.2 and 1.3]:If R, I and P are fuzzy relations satisfying

(i) ∀x, y ∈ A, R(x, y) = I (x, y)⊕ P(x, y),(ii)P is asymmetric and I is symmetric,(iii)P is simple,

(7)

then

∀x, y ∈ A,{

(i) I (x, y) = R(x, y) ∧ R(y, x),(ii) P is regular(i.e., R(x, y) � R(y, x) ⇔ P(x, y) = 0 ).

(8)

We can remark thatP1 andP2 are regular.Theregular fuzzy strict preferenceP is well defined when the fuzzy union is modelized by a particular

t-conorm (see (5), (6)). But, in general, the previous factorizations do not stipulate how one can obtain thevalue ofP(x, y)whenP(x, y) > 0. This leads us to establish a new factorization in the next subsection.

3.2. New factorization of an FWPR

In this subsection,⊕ is a t-conorm and� is its quasi-subtraction. To establish our factorization, wegive the following definition and we need the lemma below.

Definition 4. LetRbe an FWPR, andI andP are two fuzzy binary relations.I andP are, respectively, “ fuzzy indifference ofR” and “ fuzzy strict preference ofR” if R, I andP

satisfy (7).

Let us remark that when the fuzzy relationsR, I andP become crisp, (3) and (7) are equivalent.

Lemma 2. Let R be an FWPR andx, y ∈ A.

(i) If R(x, y) > R(y, x), then the equation

R(x, y) = R(y, x) ⊕ X whereX ∈ [0,1] (9)

has at least one solution. The realR(y, x)�R(x, y) is its lowest solution.(ii) Furthermore, if ⊕ is the Zadeh’s max t-conorm or⊕ is a strict t-conorm, then the realR(y, x)�R

(x, y) is the unique solution of Eq. (9).

Proof. Suppose thatR(x, y) > R(y, x) and let us show that Eq. (9) has at least one solution.Consider the functiongdefined over[0,1] byg(t) = R(y, x)⊕t .g is continuous,g(0) = R(y, x)⊕0 =

R(y, x) andg(1) = R(y, x)⊕ 1 = 1; thusg takes all values betweenR(y, x) and 1. In particular,g hasthe valueR(x, y) for R(x, y) > R(y, x). Eq. (9) has therefore at least one solution.

The proofs of the other results are obvious.�

Let us now give the new factorization.

378 L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389

Proposition 3. Let R be an FWPR, and I and P are two fuzzy binary relations.The two following statements are equivalent:

(1) I and P are, respectively, “fuzzy indifference of R” and“ fuzzy strict preference of R”.

(2) ∀x, y ∈ A,

(i) I (x, y) = R(x, y) ∧ R(y, x),

(ii) R(x, y) � R(y, x) ⇔ P(x, y) = 0,

(iii) R(x, y) > R(y, x) ⇔P(x, y) > 0andP(x, y) is a solution of Eq. (9)

. (10)

Proof. (1)⇒ (2): Richardson’s results (Propositions 1.2 and 1.3, see (7) and (8)) imply (i) and (ii) of(10). (i) of (7), (i) of (10), (ii) of (8) and the first result of Lemma 2 imply (iii) of (10).

(2)⇒ (1): (i) of (10) implies thatI is symmetric. (ii) and (iii) of (10) imply thatP is asymmetric.P issimple (see (4)) because, by (ii) of (10),R(x, y) = R(y, x) impliesP(x, y) = P(y, x) = 0. It remainsto show (i) of (7). We distinguish two cases:R(y, x)�R(y, x) andR(y, x) < R(x, y).

If R(y, x)�R(x, y), then (i) and (ii) of (10) imply thatI (x, y)⊕ P(x, y) = R(x, y)⊕ 0 = R(x, y).If R(y, x) < R(x, y), then (iii) of (10) implies thatP(x, y) is a solution of Eq. (9). Thus (i) of (10)

implies thatI (x, y)⊕ P(x, y) = R(x, y). �

Remark 2. (i) Proposition 3 generalizes Richardson’s results (see (7) and (8)).(ii) Since the solution of Eq. (9) is not unique, Proposition 3 gives for any t-conorm, a class of regular

fuzzy strict preferencesPof an FWPRR. It also stipulates that we obtain an expression ofP(x, y) (whenP(x, y) > 0) by taking one of the solutions of Eq. (9).

We can notice that:

• If a fuzzy union is modelized by⊕, then we can obtainP by choosing, for any (x, y) ∈ A2, the lowestvalue ofP(x, y) defined byP(x, y) = P3(x, y) = R(y, x)�R(x, y).Furthermore,

• If ⊕ is the Zadeh’s max t-conorm, thenP3 = P1.• If ⊕ is the Lukasiewicz’s t-conorm, thenP3 = P2.

The following result gives some additional conditions on t-conorm⊕ to obtain a unique expressionof P.

Corollary 4. Let R be an FWPR, and I and P are two fuzzy binary relations.If ⊕ is a strict t-conormor the Zadeh’smax t-conorm, then the two following statements are equivalent:

(1) I and P are, respectively, “fuzzy indifference of R” and“ fuzzy strict preference of R”.

(2) ∀x, y ∈ A,{

(i) I (x, y) = R(x, y) ∧ R(y, x),(ii) P(x, y) = P3(x, y) = R(y, x)�R(x, y).

Proof. The proof of the corollary is deduced from Proposition 3 and Lemma 2.�

L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389 379

Remark 3. If ⊕ is the Zadeh’s max t-conorm, then Corollary 4 becomes the factorization established byDutta (see (5)).

In the continuation of this paper, we will assume that a fuzzy strict preference of a given FWPR isdefined by any regularP.

Let us define two usual properties ofP (see [11,17,19]).

Definition 5. (i) P is pos-transitive if

∀x, y, z ∈ A, (P (x, y) > 0 andP(y, z) > 0) imply P(x, z) > 0.

(ii) P is negative transitive if

∀x, y, z ∈ A, (P (x, y) = 0 andP(y, z) = 0) imply P(x, z) = 0.

In the rest of this section, we determine all the max-∗-transitive FWPRs whose fuzzy strict preferencesP satisfy pos-transitivity, and then negative transitivity. We will proceed as follows:

(i) In Section 3.3, we introduce and analyze two new conditions on FWPR.(ii) In Section 3.4, we use these conditions to establish these two results.

(iii) In Section 3.5, we point out intuitive meanings, an example and remarks to illustrate and explainthese conditions and results.

3.3. New conditions on FWPR

Definition 6. LetRbe an FWPR and∗ be a t-norm.

(i) RsatisfiesQ∗1 if for all x, y, z ∈ A,

(R(x, y) > R(y, x) andR(y, z) > R(z, y)) imply

R(x, z) ∈ [

�∗2(x, y, z), �∗

3(x, y, z)]

andR(z, x) ∈ [

�∗2(x, y, z), �∗

3(x, y, z)]

⇒ R(x, z) > R(z, x)

.

(ii) RsatisfiesQ∗2 if for all x, y, z ∈ A,

(R(x, y) > R(y, x) andR(y, z) = R(z, y))or

(R(x, y) = R(y, x) andR(y, z) > R(z, y))

imply

R(x, z) ∈ [

�∗2(x, y, z) , �

∗3(x, y, z)

]and

R(z, x) ∈ [�∗

2(x, y, z) , �∗3(x, y, z)

] ⇒ R(x, z) > R(z, x)

.

Analysis of these conditions in particular cases gives the following results. The first result shows thatany strong connected FWPR satisfiesQ∗

1 andQ∗2; the second result shows that if∗ is the Zadeh’s min

380 L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389

t-norm, then any FWPR satisfiesQ∗1; and the third one gives conditions on∗ andR such thatQ∗

2 andTare equivalent.

Proposition 5. Let R be an FWPR and∗ be a t-norm.(i) If R is strongly connected, then R satisfies conditionsQ∗

1 andQ∗2.

(ii) If ∗ is the Zadeh’s min t-norm, then R satisfies conditionQ∗1.

(iii) If ∗ is the Zadeh’s min t-norm and R is min-transitive, then conditionsQ∗2 and T are equivalent.

Proof. Let x, y, z ∈ A. The proof of (i) is obvious since�∗3(x, y, z) < �∗

2(x, y, z).(ii) Since the Zadeh’s min t-norm and

R(x, y) > R(y, x) andR(y, z) > R(z, y) (11)

give�∗3(x, y, z) < �∗

2(x, y, z), we have the second result.(iii) Suppose that∗ is the Zadeh’s min t-norm andR is min-transitive. It is obvious to show that:(iii 1) If

(R(x, y) > R(y, x) andR(y, x) < R(y, z) = R(z, y))or

(R(z, y) < R(x, y) = R(y, x) andR(y, z) > R(z, y))

,

then�∗3(x, y, z) < �∗

2(x, y, z).(iii 2) If

R(x, y) > R(y, x) � R(y, z) = R(z, y)or

R(x, y) = R(y, x) � R(z, y) < R(y, z)

, (12)

then�∗3(x, y, z) = �∗

2(x, y, z) = R(x, y) ∧ R(y, z).(iii 1) and(iii 2) imply that conditionQ∗

2 is equivalent to the following one:

∀ x, y, z ∈ A, R(x, y) > R(y, x) � R(y, z) = R(z, y)

orR(x, y) = R(y, x) � R(z, y) < R(y, z)

imply

(R(x, z) = R(z, x) = R(x, y) ∧ R(y, z)) is not possible.

(13)

It remains to show that (13) and conditionT are equivalent.It is obvious to show that ifR does not satisfy (13), thenR does not satisfy conditionT. Also, the

min-transitivity ofR implies the converse.�

We now determine all the max-∗-transitive FWPRs whose regular fuzzy strict preferencesP satisfypos-transitivity, and then negative transitivity.

3.4. Properties of fuzzy strict preference

We now establish, usingQ∗1 andQ∗

2, the two key results of this paper.

L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389 381

The first key result determines all the max-∗-transitive FWPRs whose fuzzy strict preferences arepos-transitive.

Proposition 6. Let R be an FWPR and∗ be a t-norm.If R is max-∗-transitive, then

(R satisfies conditionQ∗1)⇔ (P is pos-transitive).

Proof. (⇒): AsP is regular, it suffices to show that∀x, y, z∈ A,(R(x, y) > R(y, x) andR(y, z) > R(z, y)) imply R(x, z) > R(z, x).

Let x, y, z ∈ A. Suppose (11) and show thatR(x, z) > R(z, x).With the max-∗-transitivity ofRon {x, y, z}, Lemma 1 implies

(i) R(x, z) ∈ [�∗

2(x, y, z), �∗4(x, y, z)

],

(ii ) R(z, x) ∈ [�∗

1(x, y, z), �∗3(x, y, z)

]. (14)

We distinguish two cases: (1)�∗3(x, y, z) < �∗

2(x, y, z) or R(z, x) < �∗2(x, y, z) � �∗

3(x, y, z) or�∗

2(x, y, z) � �∗3(x, y, z) < R(x, z) and (2)�∗

2(x, y, z) � �∗3(x, y, z) andR(x, z), R(z, x) ∈ [

�∗2(x, y, z) ,

�∗3(x, y, z)

].

In the first case, (14) impliesR(x, z) > R(z, x).In the second case, conditionQ∗

1 impliesR(x, z) > R(z, x).(⇐): The proof of the converse is obvious.�

The second key result determines all the max-∗-transitive FWPRs whose fuzzy strict preferences arenegative transitive.

Proposition 7. Let R be an FWPR and∗ be a t-norm.If R is max-∗-transitive, then

(R satisfies conditionsQ∗1 andQ

∗2)⇔ (P is negative transitive).

Proof. (⇒): As P is regular, it suffices to show that∀x, y, z ∈ A, (R(z, y)�R(y, z) andR(y, x)�R(x, y)) imply R(z, x)�R(x, z).

Let x, y, z ∈ A such thatR(z, y)�R(y, z) andR(y, x)�R(x, y).We distinguish three cases:

(c1) If R(x, y) > R(y, x) andR(y, z) > R(z, y), then Proposition 6 and conditionQ∗1 imply R(z, x) <

R(x, z).(c2) If

(R(x, y) > R(y, x) andR(y, z) = R(z, y))or

(R(x, y) = R(y, x) andR(y, z) > R(z, y))

, (15)

then we show thatR(z, x) < R(x, z).

382 L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389

As R is max-∗-transitive on{x, y, z}, Lemma 1 implies (14). We distinguish two cases: (c21)�∗

3(x, y, z) < �∗2(x, y, z) or R(z, x) < �∗

2(x, y, z) � �∗3(x, y, z) or �∗

2(x, y, z) � �∗3(x, y, z) <

R(x, z) and (c22) �∗2(x, y, z) � �∗

3(x, y, z) andR(x, z), R(z, x) ∈ [�∗

2(x, y, z), �∗3(x, y, z)

].

In the first case (c21), (14) impliesR(z, x) < R(x, z).In the second case (c22), conditionQ∗

2 impliesR(z, x) < R(x, z).(c3) Let us show that(R(x, y) = R(y, x) andR(y, z) = R(z, y)) imply R(x, z) = R(z, x). Assume to

the contrary thatR(x, y) = R(y, x),R(y, z) = R(z, y) andR(x, z) �= R(z, x). We distinguish twocases: (c31) R(x, z) > R(z, x) and (c32) R(x, z) < R(z, x).

In the first case (c31), the equalityR(y, z) = R(z, y) and the case (c2) of this proof implyR(x, y) >R(y, x). This contradicts the equalityR(x, y) = R(y, x).

Analogously, we can have a contradiction for the second case (c32).(⇐):Suppose thatP is negative transitive and show thatRsatisfies conditionsQ∗

1 andQ∗2. Let x, y, z ∈ A.

Suppose (11) or (15) and show thatR(x, z),R(z, x) ∈ [�∗

2(x, y, z), �∗3(x, y, z)

]implyR(x, z) > R(z, x).

It suffices to show thatR(x, z) > R(z, x).(11) or (15) and the negative transitivity ofP imply R(x, z)�R(z, x). Assume to the contrary that

R(x, z) = R(z, x). We distinguish two cases: (1)R(x, y) > R(y, x) andR(y, z)�R(z, y) and (2)R(x, y) = R(y, x) andR(y, z) > R(z, y).

In the first case, we haveR(y, z)�R(z, y) andR(z, x)�R(x, z). The negative transitivity ofP impliesR(y, x)�R(x, y). This contradictsR(x, y) > R(y, x).

In the second case, we haveR(z, x)�R(x, z) andR(x, y)�R(y, x). The negative transitivity ofPimpliesR(z, y)�R(y, z). This contradictsR(y, z) > R(z, y). �

Let us give the two previous results in the two particular cases in whichR is strongly connected and∗is the Zadeh’s min t-norm.

Corollary 8. Let R be an FWPR and∗ be a t-norm.(1)

If

R is strongly connected

andR is max-∗-transitive

, thenP is pos-transitive and negative transitive.

(2) Suppose that∗ is the Zadeh’s min t-norm.(i) If R is min-transitive, then P is pos-transitive.

(ii) If R is min-transitive, then

R satisfies conditionT ⇔ P is negative transitive.

Proof. The proof is deduced from Propositions 5, 6 and 7.�

3.5. Some explanations on the previous conditions and results

Let ∗ be a given t-norm,R be an FWPR,P be any regular fuzzy strict preference ofR, I be theindifference ofRandx, y, z be three alternatives ofA.

L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389 383

In this subsection, for a best understanding of our conditions and results, we assume that (see also[2,11,18–20]):

(i) R(x, y) > R(y, x) (or P(x, y) > 0) is interpreted as “x is strictly preferred to y with the degreeP(x, y)”;

(ii) R(x, y)�R(y, x) (i.e.,P(y, x) = 0) is interpreted as “x is preferred to y”; and(iii) R(x, y) = R(y, x) (i.e.,P(x, y) = P(y, x) = 0) is interpreted as “x is equivalent to y with the

degreeI (x, y)”.Thus, Definition 5 is seen as follows:

(i) P is pos-transitive means that:

If x is strictly preferred toy andy is strictly preferred toz,thenx is strictly preferred toz. (16)

(ii) P is negative transitive means that:

If x is preferred toy andy is preferred toz, thenx is preferred toz. (17)

It is well known that ifR is a crisp ordering (a reflexive, connected and transitive crisp binary relation)onA, then the two previous intuitive statements ((16) and (17)) given by Definition 5 are true.

Since a crisp ordering is a particular case of a∗-fuzzy ordering (i.e., max-∗-transitive FWPR), it isnecessary to seek if these statements are satisfied by any∗- fuzzy ordering.

The following example shows that these statements are not always preserved (true) for any∗-fuzzyordering.

Example 2. Let ∗ be a product t-norm defined by∀a, b ∈ [0,1] , a ∗ b = a × b, the quasi-inverse| of ∗is defined by

a

b=

{1 if a = 01 ∧ b

aif a > 0

(see[10]),A = {x, y, z} andR a∗-fuzzy ordering onAdefined by∀d ∈ A, R(d, d) = 1;R(x, y) = 0.9,R(y, x) = 0.6,R(y, z) = 0.8,R(z, y) = 0.65,R(x, z) = 0.7209 andR(z, x) = 0.721.

ForR, we have:

(i) “ x is strictly preferred toy andy is strictly preferred toz”, but “x is not strictly preferred toz”.(ii) “ x is preferred toy andy is preferred toz, but “x is not preferred toz”.

Therefore, what additional conditions can verify a given∗-fuzzy orderingR to preserve these twointuitive statements ((16) and (17))?

In Section 3.4, we show (Propositions 6 and 7) that these two intuitive statements are preserved iff agiven∗-fuzzy orderingRsatisfies, respectively,Q∗

1 andQ∗2. That is why those conditions are introduced

in Definition 6.Moreover, we show in Corollary 8 that: (i) any strong connected∗-fuzzy ordering preserves the two

intuitive statements; (ii) any min-transitive FWPR preserves the first intuitive statement (16); and (iii) ifwe consider min-transitive FWPRs, only those satisfying conditionTpreserve the second statement (17).

384 L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389

These results generalize those of crisp case since any crisp ordering satisfies conditionsQ∗1 andQ∗

2.So, these results and these conditions lead us to notice that, contrary to crisp case, we can have the

following three undesirable situations with the fuzzy case:

(s1) “If x is strictly preferred toy, y is strictly preferred tozand the two degreesR(x, z), R(z, x) belongto the interval

[�∗

2(x, y, z), �∗3(x, y, z)

], thenx is not always strictly preferred toz”,

(s2) “If x is strictly preferred toy, y is equivalent toz and the two degreesR(x, z), R(z, x) belong to[�∗

2(x, y, z), �∗3(x, y, z)

], thenx is not always strictly preferred toz”,

(s3) “If x is equivalent toy, y is strictly preferred toz and the two degreesR(x, z), R(z, x) belong to[�∗

2(x, y, z) , �∗3(x, y, z)

], thenx is not always strictly preferred toz”.

In particular, the given∗-fuzzy orderingR of Example 2 does not satisfy conditionQ∗1. In fact,

�∗2(x, y, z) = R(x, y) ∗ R(y, z) = R(x, y) × R(y, z) = 0.72; �∗

3(x, y, z) = (R(y, z) | R(y, x)) ∧(R(x, y) | R(z, y)) = R(y, x)/R(y, z) ∧ R(z, y)/R(x, y) � 0.722. Thus, for thisR, x is strictly pre-ferred toy, y is strictly preferred toz and the two degreesR(x, z) andR(z, x) belong to the interval[0.72, 0.722]; butx is not strictly preferred toz.

We determine in Proposition 5, some particular and classical∗-fuzzy orderings satisfying conditionsQ∗

1 andQ∗2.

Let us now end this subsection by the following remarks.

Remark 4. (i) Richardson[17, Proposition 1.4] shows that ifR is m-transitive and strongly connected,then any regularP is pos-transitive. Proposition 6 shows that, by weakening strong connectedness toobtain connectedness, conditionQ∗

1 is necessary and sufficient to obtain pos-transitivity of any regularfuzzy strict preference of a given max-∗-transitive FWPR.

(ii) Richardson [17, Proposition 1.5] shows that ifR is m-transitive and strongly connected, then anyregularP is negatively transitive. Thus, in comparison with Proposition 7, we show that, by weakeningstrong connectedness to obtain connectedness, conditionsQ∗

1 andQ∗2 are necessary and sufficient to

obtain negative transitivity of any regular fuzzy strict preference of a given max-∗-transitive FWPR.(iii) Fono and Gwét [11, Lemma 2] show that ifR is min-transitive, thenP2 is negative transitive if

and only ifRsatisfies conditionT. This result is a particular case of the third result of Corollary 8 whenP = P2.

In the last section, we establish new versions of fuzzy counterpart of Gibbard’s oligarchy theorem andArrow’s impossibility theorem which are summarized in Table 2.

4. New fuzzy social choice results

In the following subsection, we recall some useful definitions and notations of the Social Choice theory(see [1–3,9,15,17,18,21]).

4.1. Definitions and notations

N = {1, . . . , i, . . . , n} is the set of voters in a society. We assume that|N | �3.2N is the set of all non-empty subsets ofN and an element of 2N is called a coalition of voters.

L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389 385

Given two sets of FWPRsF andG, a fuzzy aggregation rule (FAR) is a functionf : FN → G, wherethe elements ofFN are indicated by(R1, R2, . . . , Ri, . . . , Rn) = (Ri)i∈N = RN and are called theprofiles. Whenf is an FAR, we denotef (RN) = R.PRi andPR are, respectively, regular fuzzy strict preference ofRi andR.For anyx, y ∈ A, RN ∈ FN andS ∈ 2N , we simply write “RS(x, y) > 0” instead of “∀i ∈ S,

Ri(x, y) > 0”.We also have the following definitions.

Definition 7. Let f : FN → G be an FAR andS ∈ 2N .

(i) f satisfies:(i1) Pareto criterion (PC) if∀RN ∈ FN , ∀x, y ∈ A, PR(x, y)� mini ∈N PRi (x, y).(i2) Independence of irrelevant alternatives (IIA) if∀RN,QN ∈ FN , ∀x, y ∈ A, [∀i ∈ N , PRi (x, y) =

PQi (x, y) andPRi (y, x) = PQi (y, x)] imply [PR(x, y) = PQ(x, y) andPR(y, x) = PQ(y, x)].(i3) Positive responsiveness (PR) if∀RN,QN ∈ FN , ∀x, y ∈ A,

R(x, y) = R(y, x)

and

∃ j ∈ N,

(∀i ∈ N, i �= j, Ri = Qi)

and (PRj (x, y) = 0 andPQj (x, y) > 0)

or(PRj (y, x) > 0 andPQj (y, x) = 0)

imply PQ(x, y) > 0.

(ii) f is dictatorial if∃i ∈ N , ∀RN ∈ FN , ∀x, y ∈ A, PRi (x, y) > 0 impliesPR(x, y) > 0.(iii) Let i ∈ N andx, y ∈ A. i is almost semi-decisive over(x, y) if ∀RN ∈ FN , (PRi (x, y) > 0 and

PRN−{i}(y, x) > 0) imply PR(y, x) = 0.(iv) S is almost decisive over(x, y) if ∀RN ∈ FN, (PRS (x, y) > 0 andPRN−S (y, x) > 0) imply

PR(x, y) > 0.(v) S is decisive if∀RN ∈ FN, ∀x, y ∈ A, PRS (x, y) > 0 impliesPR(x, y) > 0.

(vi) Let i ∈ N . i has a veto if∀RN ∈ FN , ∀x, y ∈ A, PRi (x, y) > 0 impliesPR(y, x) = 0.(vii) S is an oligarchy ifS is decisive and for alli ∈ S, i has a veto.

Let ∗ be a t-norm. We denote by:H ∗ the set of max-∗-transitive FWPR’s (∗-fuzzy orderings);H ∗

1 the set of max-∗-transitive FWPR’s satisfying conditionQ∗1;

H ∗1,2 the set of max-∗-transitive FWPR’s satisfying conditionsQ∗

1 andQ∗2;

SFO∗ the set of strong connected and max-∗-transitive FWPR’s (strong∗-fuzzy orderings); andCO the set of all crisp orderings.We haveCO⊂ SFO∗ ⊂ H ∗

1,2 ⊂ H ∗1 ⊂ H ∗.

We also note that the setsCO, SFO∗ andH ∗ are traditional and usual (see[1,3,4,8,9,11,17,18,20]).

386 L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389

4.2. Gibbard’s oligarchy theorem

Let us now give our fuzzy version of Gibbard’s oligarchy theorem which is the first main result of thispaper.

Theorem 9. Let∗ be a t-norm and P be any regular fuzzy strict preference.If f : H ∗N

1 → H ∗1 is an FAR satisfying IIA and PC, then there exists a unique oligarchy.

Theorem 9 is due to the following traditional lemma which is a fuzzy version of the crisp veto-fieldexpansion lemma.

Lemma 10(Veto-field expansion lemma). Let ∗ be a t-norm, P be any regular fuzzy strict preference,f : H ∗N

1 → H ∗1 be an FAR satisfying IIA and PC, i ∈ N andx, y ∈ A.

If i is almost semi-decisive over(x, y), then i has a veto.

Proof. With Proposition 6, the proof is exactly analogous to that of the crisp veto-field expansion lemma[15,16,19]. �

Analysis of the previous theorem suggests examining whether the oligarchy result can lead to that ofthe dictatorship.

4.3. From Gibbard to Arrow’s result

The following result shows that the replacement ofP byP1 on the hypotheses of the previous theoremgives, in a way similar to Dutta’s result [9, Proposition 3.9], nondictatorial FARs. This is the second mainresult of our paper.

Proposition 11. Let∗ be a t-norm and fuzzy strict preference be defined byP1.There exists a nondictatorial FARf : H ∗N

1 → H ∗1 satisfying IIA and PC.

Proof. Let � ∈ ]1/2,1[. Considerf : H ∗N1 → H ∗

1 defined by∀RN ∈ H ∗N1 , f (RN) = R, where

∀x, y ∈ A (x �= y),

R(x, x) = 1 andR(x, y) ={

1 if ∀i ∈ N, Ri(x, y) > Ri(y, x),� otherwise.

As Dutta shows thatf is IIA, PC and nondictatorial, it remains to show thatf is well defined, i.e.,R ∈ H ∗

1 . R is reflexive and connected.Let us show thatR is max-∗-transitive.Assume to the contrary thatR is not max-∗-transitive. Then∃x, y, z ∈ A, R(x, z) < R(x, y)∗R(y, z).

ThusR(x, z) = � andR(x, y) = R(y, z) = 1. By the definition ofR, we have

(a) ∃j ∈ N, Rj (x, z) < Rj (z, x),(b) ∀i ∈ N, Ri(x, y) > Ri(y, x), Ri(y, z) > Ri(z, y). (18)

L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389 387

Since∀i ∈ N, Ri ∈ H ∗1 , (b) of (18) and Proposition 6 imply∀i ∈ N, Ri(x, z) > Ri(z, x). This

contradicts (a) of (18).Let us show thatRsatisfies conditionQ∗

1. Letx, y, z ∈ A. It suffices to show that (11) implies�∗3(x, y, z)

< �∗2(x, y, z). Eq. (11) and the definition ofR imply R(x, y) = R(y, z) = 1. Thus�∗

3(x, y, z) =R(y, x) ∧ R(z, y) = � < 1 = �∗

2(x, y, z). �

Therefore, if we add the condition PR on the hypotheses of the previous theorem, we obtain a fuzzycounterpart of the crispquasi-transitive positive-responsive dictatorship theorem(see [15,19]) which isthe third main result of our paper.

Theorem 12. Let∗ be a t-norm and P be a regular fuzzy strict preference.If f : H ∗N

1 → H ∗1 satisfies IIA, PC and PR, then f is dictatorial.

Proof. With Theorem 9, the proof is exactly analogous to that of the corresponding result by Dutta[9,Proposition 3.12]. �

By removing the condition PR with FAR and by adding the conditionQ∗2 to social preferences, we

obtain another fuzzy counterpart of Arrow’s impossibility theorem which is the fourth and the last mainresult of our paper.

Theorem 13. Let∗ be a t-norm and P be any regular fuzzy strict preference.If f : H ∗N

1 → H ∗1,2 satisfies IIA and PC, then f is dictatorial.

Theorem 13 is due to the two following traditional lemmas.The first lemma is a fuzzy version of the crisp field expansion lemma.

Lemma 14(Field expansion lemma). Let ∗ be a t-norm, P be any regular fuzzy strict preference, f :H ∗N

1 → H ∗1 be an FAR satisfying IIA and PC, S ∈ 2N andx, y ∈ A.

If S is almost decisive over(x, y), then S is decisive.

Proof. With Proposition 6, the proof is exactly analogous to the first stage of the proof of the correspondingresult by Richardson[17, Proposition 2.1]. �

The second lemma is a fuzzy version of the crisp group contraction lemma.

Lemma 15(Group contraction lemma). Let ∗ be a t-norm, P be any regular fuzzy strict preference,f : H ∗N

1 → H ∗1,2 be an FAR satisfying IIA and PC, andS ∈ 2N (|S| �2).

If S is decisive, then∃K ∈ 2S, K �= S and K is decisive.

Proof. With Proposition 7, the proof is exactly analogous to the second stage of the proof of the corre-sponding result by Richardson[17, Proposition 2.1]. �

Let us end this section by the following remark which gives a comparison of the two tables.

388 L.A. Fono, N.G. Andjiga / Fuzzy Sets and Systems 155 (2005) 372–389

Remark 5. (i) Theorem 9 becomes Dutta’s result[9, Proposition 3.7], if the regular fuzzy strict preferenceP is defined byP1 and if∗ is the Zadeh’s min t-norm.

(ii) Proposition 11 becomes Dutta’s result [9, Proposition 3.9] if∗ is the Zadeh’s min t-norm.(iii) Theorem 12 becomes Dutta’s result [9, Proposition 3.12] if the regular fuzzy strict preferenceP is

defined byP1 and if∗ is the Zadeh’s min t-norm.(iv) Proposition 2.1 of Richardson generalizes Theorem 13 inSFO∗ which is a subset ofH ∗

1 ; but inH ∗

1 , it is not true as the theorem.(i), (ii), (iii) and (iv) mean that we generalize the results of Dutta and Richardson.

5. Concluding remarks

This paper establishes new properties on the fuzzy binary relations in Section 3, and proposes theirapplications in Social Sciences in Section 4.

In Section 3, we establish how one can obtain regular fuzzy strict preference of a fuzzy binary relation(Proposition 3). The originality of this factorization is that it generalizes some classical ones. Moreover,we show that any regular fuzzy strict preference of a given∗-fuzzy orderingR is pos-transitive (respec-tively, negative transitive) if and only ifR satisfiesQ∗

1 (respectively,Q∗1 andQ∗

2) (Propositions 6 and 7,respectively).

Using these properties, we obtain, in Section 4, a general solution to the following question: Let∗ bea t-norm. Can we obtain an oligarchy or a dictator if individual and social preferences are modelized byreflexive, connected and max-∗-transitive fuzzy binary relations, if fuzzy strict preference is defined byany regularP, and if an FAR satisfies IIA and PC?

We show that a unique oligarchy and a nondictatorial FAR appear when preferences satisfy conditionQ∗1

(Theorem 9 and Proposition 11, respectively). We also show that if conditions on FAR are supplementedby positive responsiveness or conditions on social preferences are supplemented byQ∗

2, the possibilityof oligarchy is translated to the dictator (Theorems 12 and 13, respectively).

The new conditionsQ∗1 andQ∗

2 used do not seem intuitive or natural. They become classic for theZadeh’s min t-norm or particular fuzzy binary relations (Proposition 5). We believe that, contrary toclassical and crisp transitivity on{0,1}, max-∗-transitivity on[0,1] does not handle certain problemsrelating to rationality on triplets (Example 2 and Section 3.5). It is therefore in view of covering thisshortcoming that we impose these conditions.

Acknowledgements

We sincerely thank Professors B. Mbih, H. Gwét, M. Armstrong, S.F. Kouam, L. Diffo, M. Salles andB. Llamazares for their help and advice. We are also indebted to the referees for their helpful comments.

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