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Fuzzy Sets and Systems 160 (2009) 2658 – 2672 www.elsevier.com/locate/fss Fuzzy arrow-type results without the Pareto principle based on fuzzy pre-orders Louis Aimé Fono a , , 1 , Véronique Donfack-Kommogne a , Nicolas Gabriel Andjiga b a Département de Mathématiques et Informatique, Faculté des Sciences, Université de Douala, B.P. 24157 Douala, Cameroun b Département de Mathématiques, Ecole Normale Supérieure, Université de Yaoundé I, B.P. 47 Yaoundé, Cameroun Received 21 February 2008; received in revised form 22 September 2008; accepted 21 December 2008 Available online 3 January 2009 Abstract In this paper, we introduce appropriate properties of fuzzy preferences and fuzzy aggregation rules. We use them to provide fuzzy counterparts of Malawski and Zhou’s [A note on social choice theory without the pareto principle, Social Choice and Welfare 16 (1994)] and Wilson’s [Social choice theory without the pareto principle, Journal of Economic Theory 5 (1972) 478–486] impossibility results concerning the aggregation of individual preferences, which do not assume the Pareto principle. By weakening conditions on fuzzy social preferences, we obtain a possibility result. © 2009 Elsevier B.V. All rights reserved. Keywords: Binary fuzzy relation; Wilson’s theorem; T-transitivity; Minimal and regular fuzzy strict preference; Pareto principle 1. Introduction Arrow’s Impossibility theorem and its variants are well studied when individual and social preferences are crisp total pre-orders (complete and transitive crisp binary relations) on the set of alternatives [15,16,19,22]. In the last 20 years, economists recognized that individuals and society could have vague preferences. Therefore, Barrett et al.’s [5] framework was the first in the chain of results which use fuzzy set theory to investigate if crisp Arrowian results (such as Arrow’s Impossibility theorem and Gibbard’s oligarchy theorem) are preserved when individual and social preferences are vague. The main papers of this recent literature are: Abdelaziz et al. [1], Banerjee [3], Barrett et al. [4], Dutta [8], Fono and Andjiga [10], Geslin et al. [13], Richardson [17] and Tang [21]. However, all these fuzzy Arrowian results require the Pareto principle as a basic assumption. This raises the question of whether or not the Pareto principle can be disregarded in the general development of the fuzzy social choice theory. In this paper, our objective is modest. Instead of attempting to completely answer this question, we investigate in what extent the results of Malawski and Zhou [15], and Wilson theorem [22] (results which are summarized in Table 1) are preserved when individual and social preferences are fuzzy and represented by weakly complete Corresponding author. Tel.: +237 952 17 83. E-mail address: [email protected] (L.A. Fono). 1 The author thanks AUF (Agence Universitaire de la Francophonie). This work was achieved when he was Visiting Researcher at Université de Caen-France under the Research grants “Bourse Post-doctorale de la Francophonie 2005–2006”. 0165-0114/$-see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2008.12.012

Fuzzy arrow-type results without the Pareto principle based on fuzzy pre-orders

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Page 1: Fuzzy arrow-type results without the Pareto principle based on fuzzy pre-orders

Fuzzy Sets and Systems 160 (2009) 2658–2672www.elsevier.com/locate/fss

Fuzzy arrow-type results without the Pareto principle based onfuzzy pre-orders

Louis Aimé Fonoa,∗,1, Véronique Donfack-Kommognea, Nicolas Gabriel Andjigab

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

Received 21 February 2008; received in revised form 22 September 2008; accepted 21 December 2008Available online 3 January 2009

Abstract

In this paper, we introduce appropriate properties of fuzzy preferences and fuzzy aggregation rules. We use them to provide fuzzycounterparts of Malawski and Zhou’s [A note on social choice theory without the pareto principle, Social Choice and Welfare 16(1994)] andWilson’s [Social choice theorywithout the pareto principle, Journal of EconomicTheory 5 (1972) 478–486] impossibilityresults concerning the aggregation of individual preferences, which do not assume the Pareto principle. By weakening conditionson fuzzy social preferences, we obtain a possibility result.© 2009 Elsevier B.V. All rights reserved.

Keywords: Binary fuzzy relation; Wilson’s theorem; T-transitivity; Minimal and regular fuzzy strict preference; Pareto principle

1. Introduction

Arrow’s Impossibility theorem and its variants are well studied when individual and social preferences are crisp totalpre-orders (complete and transitive crisp binary relations) on the set of alternatives [15,16,19,22].In the last 20 years, economists recognized that individuals and society could have vague preferences. Therefore,

Barrett et al.’s [5] frameworkwas thefirst in the chain of resultswhich use fuzzy set theory to investigate if crispArrowianresults (such as Arrow’s Impossibility theorem and Gibbard’s oligarchy theorem) are preserved when individual andsocial preferences are vague. The main papers of this recent literature are: Abdelaziz et al. [1], Banerjee [3], Barrettet al. [4], Dutta [8], Fono and Andjiga [10], Geslin et al. [13], Richardson [17] and Tang [21].However, all these fuzzy Arrowian results require the Pareto principle as a basic assumption. This raises the question

of whether or not the Pareto principle can be disregarded in the general development of the fuzzy social choice theory.In this paper, our objective is modest. Instead of attempting to completely answer this question, we investigate

in what extent the results of Malawski and Zhou [15], and Wilson theorem [22] (results which are summarized inTable 1) are preserved when individual and social preferences are fuzzy and represented by weakly complete

∗Corresponding author. Tel.: +2379521783.E-mail address: [email protected] (L.A. Fono).

1 The author thanks AUF (Agence Universitaire de la Francophonie). This work was achieved when he was Visiting Researcher at Université deCaen-France under the Research grants “Bourse Post-doctorale de la Francophonie 2005–2006”.

0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2008.12.012

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L.A. Fono et al. / Fuzzy Sets and Systems 160 (2009) 2658–2672 2659

Table 1

Ref. Hypotheses on individualpreferences

Properties of CAR Hypotheses on socialpreferences

Conclusion

Proposition 1(Malawski and Zhou)

Reflexive, complete,and transitive

IIA and NI Reflexive, complete,and transitive

f is either null,or PO, or APO

Wilson Reflexive, complete,and transitive

IIA and NI Reflexive, complete,and transitive

f is either null, or dicta-torial, or anti-dictatorial

Proposition 2(Malawski and Zhou)

Reflexive, complete,and transitive

IIA, SNI and SNN Reflexive, complete,and quasitransitive

∃K ⊆ N/ f is K − PO

Table 2

Ref. Hypotheses on individualpreferences

Properties of FAR Hypotheses on socialpreferences

Decision

Theorem 1 Reflexive, weakly complete,T-transitive and CT

1

IIA and NI Reflexive, weakly complete,T-transitive, CT

1 and CT2

f is either null, or PO, or APO

Theorem 2 Reflexive, weakly complete,T-transitive and CT

1

IIA and NI Reflexive, weakly complete,T-transitive, CT

1 and CT2

f is either null, or dictatorial,or anti-dictatorial

Proposition 2 Reflexive, weakly complete,T-transitive and CT

1

IIA and NI Reflexive, weakly complete,T-transitive, CT

1

f is non-null, non-dictatorial,non-anti-dictatorial

Theorem 3 Reflexive, weakly complete,T-transitive and CT

1

IIA, SNI and SNN Reflexive, weakly complete,T-transitive, and CT

1

∃K ⊆ N/ f is K − PO

T-pre-orders (reflexive, weakly complete and T-transitive fuzzy relations) where T is a t-norm. Our results are summa-rized in Table 2.It is important to notice that it is very difficult to examine whether crisp Arrowian’s results are preserved when

individual and social preferences are vague since many notions of crisp set theory can be generalized in differentways to the fuzzy framework. Depending the way that crisp properties are generalized, one could obtain positive ornegative results in the study of fuzzy Arrow-type problems. For instance see Dutta [8], Fono and Andjiga [10] andGeslin et al. [13].All the literature of fuzzy social choice theory (see [1,3,4,8,10,13,17,20,21]) has two branches. In one branch

of this literature, vague preferences are represented by fuzzy strict preferences (see [3,4,13,20,21]) and this linewill not be considered in this paper. In the other branch vague preferences are modelled by fuzzy weak preferencerelations (FWPRs) (see [1,8,10,17]), and fuzzy strict preferences are obtained by factorization of a given fuzzy weakpreference.In this second line of research we adopt, to establish Arrowian results, the first step is to consider a fuzzy strict

preference of a given FWPR. Thus, we use in our paper the factorization recently established by Fono and Andjiga[10]. We notice that their decomposition generalizes the ones of Dutta [8] and Richardson [17].To achieve our goal, that is, to study if results of Table 1 are preserved in fuzzy case, the paper is organized as follows:

In Section 2, we give some basic concepts and properties of fuzzy operators and fuzzy relations. We recall two usefulresults of a fuzzy strict preference established by Fono and Andjiga [10]. Section 3 has three parts. In Section 3.1, werecall some notations and definitions on fuzzy social choice. We introduce new fuzzy versions of some properties of acrisp aggregation rule (CAR) such as non-imposition (NI), strictly non-imposition (SNI), null, strictly non-null (SNN),Pareto and anti-dictatorial. In Section 3.2, we establish fuzzy counterparts of the first result of Malawski and Zhou, andWilson’s theorem when individual and social preferences are modelled by weakly complete T-pre-orders. Contrary tothe crisp case where individual and social preferences are crisp total pre-orders, in our results, social preferences satisfyone more condition than individual preferences. In Section 3.3, we show that if we cancel this condition (i.e., social

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and individual preferences have the same properties), the possibility of obtaining negative results can disappear andwe obtain fuzzy counterpart of the second result of Malawski and Zhou. Section 4 contains some concluding remarks.

2. Preliminaries

Throughout A is a given set of alternatives such that |A|�3.

2.1. Basic notions on fuzzy operators and fuzzy relations

Let us recall some useful operators and examples [2,9–11,14].A t-norm T (resp. a t-conorm S,) is an increasing, commutative, associative binary operation on [0, 1] with neutral

element 1 (resp. 0). The dual of a t-norm T is a t-conorm denoted T ∗.The two usual t-norms are: the minimum operator denoted TM = ∧ and the Łukasiewicz t-norm denoted TL

(∀a, b ∈ [0, 1], TL(a, b) = max(a + b − 1, 0)). Their duals are: the maximum operator denoted SM = T ∗M = ∨, and

the Łukasiewicz t-conorm denoted SL = T ∗L and defined by ∀a, b ∈ [0, 1], SL(a, b) = T ∗

L (a, b) = min(a + b, 1).Throughout the paper, we assume that T and S are continuous.Let T be a t-norm. The R-implicator of T is the binary operator IT on [0, 1] defined by IT (v, w) = max{t ∈

[0, 1], T (v, t)�w} for all v, w ∈ [0, 1].The R-implicator of TM is defined by

ITM (v, w) ={1 if v�w,

w if v > w.

The R-implicator of TL is defined by ITL (v, w) = min(1, 1 − v + w).Let S be a t-conorm. The residual coimplicator of S is the binary operator JS on [0, 1] defined by JS(v, w) = min{t ∈

[0, 1], S(v, t)�w} for all v, w ∈ [0, 1]. The residual coimplicator of SM is defined by

JSM (v, w) ={0 if v�w,

w if v < w.

The residual coimplicator of SL is defined by JSL (v, w) = max(0, w − v).Let us recall some useful notions and properties on fuzzy relations (see [6,8–11,17,18]).A (binary) fuzzy relation is a function R : A × A −→ [0, 1]. R is reflexive if ∀x ∈ A, R(x, x) = 1. R is weakly

complete if ∀x, y ∈ A, R(x, y)+ R(y, x)�1.A FWPR is a reflexive and weakly complete fuzzy relation. R is stronglycomplete if ∀x, y ∈ A,max(R(x, y), R(y, x)) = 1. R is asymmetric if ∀x, y ∈ A, R(x, y) > 0 ⇒ R(y, x) = 0. Theconverse of R is the binary fuzzy relation R−1 defined by ∀x, y ∈ A, R−1(x, y) = R(y, x). Let T be a t-norm. R isT-transitive if ∀x, y, z ∈ A, R(x, z)�T (R(x, y), R(y, z)).

Remark 1. Let R be an FWPR and T be a t-norm.

(1) For all (x, y) ∈ A × A, R(x, y) is the degree to which x is at least as good as y.(2) R is a crisp binary relation if ∀x, y ∈ A, R(x, y) ∈ {0, 1}. In this case, xRy denotes R(x, y) = 1, and weak

completeness becomes crisp completeness (i.e.,∀x, y ∈ A, xRy or yRx).(3) Strong completeness implies reflexivity and weak completeness.(4) If T = TM, the T-transitivity becomes the min-transitivity (i.e., ∀x, y, z ∈ A, R(x, z)� min(R(x, y), R(y, z))).(5) R−1 is an FWPR.(6) If R is T-transitive, then R−1 is T-transitive.

In addition to weakly complete T-pre-orders and crisp total pre-orders, we will use the following types of fuzzypre-orders:A strongly complete T-pre-order is a strongly complete and T-transitive fuzzy relation. In the particular case where

T = TM,we simply say (i) weakly complete fuzzy pre-order instead of weakly complete TM-pre-order, and (ii) stronglycomplete fuzzy pre-order instead of strongly complete TM-pre-order.

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L.A. Fono et al. / Fuzzy Sets and Systems 160 (2009) 2658–2672 2661

We recall that, for any T, weakly complete T-pre-orders and strongly complete T-pre-orders are standard fuzzyextensions of crisp total pre-orders. Furthermore, strongly complete T-pre-orders and weakly complete fuzzy pre-orders are particular cases of weakly complete T-pre-orders.We assume throughout that individual and social preferences are vague and modelled by fuzzy pre-orders.As we said in the Introduction, in the line of research we adopt, to establish our fuzzy versions of crisp Arrowian

results, we need to consider a strict component of an FWPR. For that, we end this section with the review on regularfuzzy strict components of a given FWPR (see [10,11,17]).Fono and Andjiga [10] determined for any t-conorm S, a class of regular fuzzy strict preferences of a given FWPR

R. Each class has a minimal element called the minimal regular fuzzy strict preference of R associated to S and definedas follows:Let S be a t-conorm and JS be its residual coimplicator. The minimal regular fuzzy strict component P of R associated

with S is defined by

∀x, y ∈ A, P(x, y) = JS(R(y, x), R(x, y)). (1)

It is important to recall that P is regular, that is,

∀x, y ∈ A, R(x, y)�R(y, x) ⇔ P(x, y) = 0. (2)

Interpretation 1. [Barrett et al. [4], Dutta [8]] Let R be an FWPR, P be any minimal regular fuzzy strict preference ofR and x, y ∈ A.

If R(x, y) > R(y, x) then we say “x is strictly preferred to y with the degree P(x, y)”.

Let us recall usual examples of minimal regular fuzzy strict preferences of a given FWPR defined by (1) in threeparticular cases.

Example 1. Let R be an FWPR and S be a t-conorm.

(1) Theminimal regular fuzzy strict preference ofR associatedwith themaximumoperator SM is defined by ∀x, y ∈ A,

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

(3)

PM is fuzzy strict preference proposed by Dutta [8].(2) The minimal regular fuzzy strict preference of R associated with the Łukasiewicz t-conorm SL is defined by

∀x, y ∈ A,

PL(x, y) = JSL (R(y, x), R(x, y)) = 0 ∨ (R(x, y) − R(y, x)). (4)

PL is fuzzy strict preference introduced by Richardson [17].(3) If R is a crisp weak preference relation, then for any t-conorm S, P becomes the unique and well-known crisp strict

preference of R defined by

∀x, y ∈ A, x Py ⇔ (x Ry and not(yRx)). (5)

In the next section, we explain how Fono and Andjiga obtained necessary and sufficient conditions on a weaklycomplete T-pre-order R such that any minimal regular strict component PR of R satisfies fuzzy version of crisp pos-transitivity and fuzzy version of crisp negative transitivity.

2.2. Review on two results of a minimal regular strict component of an FWPR

Let T a t-norm, R be a weakly complete T-pre-order and PR be any minimal regular strict component of R.

(1) When R becomes crisp, that is, R is a crisp total pre-order, then PR is defined by (5) and satisfies the two followingwell-known properties:

(i) PR is pos-transitive, i.e., for all x, y, z ∈ A, “if x is strictly preferred to y and y is strictly preferred to z, thenx is strictly preferred to z”. More formally, for all x, y, z ∈ A (x PR y and yPRz) ⇒ x PRz.

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2662 L.A. Fono et al. / Fuzzy Sets and Systems 160 (2009) 2658–2672

(ii) PR is negative transitive, i.e., for all x, y, z ∈ A, “if x is not strictly preferred to y and y is not strictly preferredto z, then x is not strictly preferred to z”. More formally, for all x, y, z ∈ A (not(x PR y) and not(yPRz)) ⇒not(x PRz), i.e., x PRz ⇒ (x PR y or yPRz).

It is important to notice that crisp Arrowian results are based on these two properties.A main question is to wonder if any minimal regular strict component PR of a weakly complete T-pre-order R

satisfies fuzzy versions of these two properties.Fono and Andjiga [10] introduced the following fuzzy versions of these properties.

Definition 1 (Fono and Andjiga [10, Definition 5, p. 379]). Let R be an FWPR and PR be any minimal regular com-ponent of R.

(1) PR is pos-transitive if

∀x, y, z ∈ A (PR(x, y) > 0 and PR(y, z) > 0) ⇒ PR(x, z) > 0.

(2) PR is negative transitive if

∀x, y, z ∈ A (PR(x, y) = 0 and PR(y, z) = 0) ⇒ PR(x, z) = 0.

Fono and Andjiga [10, Corollary 8, p. 382] showed that in the particular case where R is a strongly completeT-pre-order, PR satisfies these two properties.Furthermore, in the particular case where R is a weakly complete fuzzy pre-order, i.e., T = TM, Fono and Andjiga

[10, Corollary 8, p. 382] showed that (i) PR is pos-transitive, and (ii) PR is negative transitive iff R satisfies condition Cwhere condition C is defined by (see [10,11,20]) ∀x, y, z ∈ A, R(x, y) = R(y, x) = R(y, z) = R(z, y) ⇒ R(x, z) =R(z, x).In the general case, i.e., if R is a weakly complete T-pre-order, Fono and Andjiga [10, Section 3.5, Example 2,

p. 383] showed there exists some minimal regular strict components of R which violate each of these two properties.Therefore, in order to determine the set of all weakly complete T-pre-orders for which any minimal regular strict

component satisfies each of these two properties, they introduced two following conditions on a FWPR.To recall these conditions, we need the two following reals (see [10]):{

�R2 (x, y, z) = min(R(x, y), R(y, z)),

�T,R3 (x, y, z) = min[IT (R(y, z), R(y, x)), IT (R(x, y), R(z, y))].

(6)

And we have the two conditions.

Definition 2 (Fono and Andjiga [10, Definition 6, p. 379]). (1) R satisfies CT1 if ∀x, y, z ∈ A,

(R(x, y) > R(y, x) and R(y, z) > R(z, y))

⇓⎛⎜⎜⎝⎛⎜⎜⎝

R(x, z) ∈ [�R2 (x, y, z), �

T,R3 (x, y, z)]

and

R(z, x) ∈ [�R2 (x, y, z), �

T,R3 (x, y, z)]

⎞⎟⎟⎠ ⇒ R(x, z) > R(z, x)

⎞⎟⎟⎠ .

(7)

(2) R satisfies CT2 if ∀x, y, z ∈ A,⎛

⎝ (R(x, y) > R(y, x) and R(y, z) = R(z, y))or

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

⎞⎠

⇓⎛⎝⎛⎝ R(x, z) ∈ [�R

2 (x, y, z), �T,R3 (x, y, z)]

andR(z, x) ∈ [�R

2 (x, y, z), �T,R3 (x, y, z)]

⎞⎠ ⇒ R(x, z) > R(z, x)

⎞⎠ .

(8)

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L.A. Fono et al. / Fuzzy Sets and Systems 160 (2009) 2658–2672 2663

They showed (Proposition 5, p. 380) that (i) CT1 and CT

2 are satisfied by any strongly complete T-pre-order, (ii) any

weakly complete fuzzy pre-order satisfies CTM1 , and (iii) any weakly complete fuzzy pre-order R satisfies CTM

2 iff Rsatisfies C.

The following result, established by Fono and Andjiga, shows that (i) only PR of weakly complete T-pre-orderssatisfying CT

1 are pos-transitive, and (ii) only PR of weakly complete T-pre-orders satisfying CT1 and CT

2 are negativetransitive.

Proposition 1 (Fono and Andjiga [10, Proposition 6, p. 381]). Let R be a weakly complete T-pre-order and PR be anyminimal regular fuzzy strict preference of R.

(1) (R satisfies condition CT1 ) ⇔ (PR is pos-transitive).

(2) (R satisfies conditions CT1 and CT

2 ) ⇔ (PR is negative transitive).

With the previous results, they obtained the following sets of fuzzy pre-orders: for a t-norm T, HT the set of weaklycomplete T-pre-orders, HT

1 the set of weakly complete T-pre-orders satisfying condition CT1 , HT

1,2 the set of weakly

complete T-pre-orders satisfying conditions CT1 and CT

2 ,SFOT the set of strongly complete T-pre-orders and CO theset of all crisp total pre-orders.In the particular case where T = TM, HT andHT

1 become simplyH the sets of weakly complete fuzzy pre-orders,the setHT

1,2 becomes the setHC of weakly complete fuzzy pre-orders satisfying condition C, SFOT becomes simplySFO the sets of strongly complete fuzzy pre-orders.We have the following diagram of sets inclusions:

CO ⊂ SFO ⊂ HC ⊂ H = H∩ ∩ ∩ ∩

SFOT ⊂ HT1,2 ⊂ HT

1 ⊂ HT. (9)

Let us end this section by establishing the following useful lemma.

Lemma 1. Let R be an FWPR.

(1) If R satisfies CT1 , then R−1 satisfies CT

1 .(2) (PR)−1 = PR−1 .

Proof. Obvious (see [7]). �

In the next section, we introduce notations and definitions of fuzzy social choice theory. We establish our four mainresults which are summarized in Table 2.

3. New properties and results of fuzzy social choice

In the following subsection, we recall some useful notations and definitions of social choice theory[3–5,8,10,13,17,18]. We also introduce some new definitions that will be useful in this paper.

3.1. Notations, first definitions, and domain and range of a FAR

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 subsetsof N and an element of 2N is called a coalition of voters. Given two sets of FWPRs F and G, a fuzzy aggregation rule(FAR) is a function f : FN −→ G, where the elements of FN are indicated by (R1, R2, . . . , Rn) = (Ri )i∈N = RN

and are called profiles. When f is an FAR, we denote f (RN ) = R.

PRi and PR are, respectively, minimal regular fuzzy strict preference of Ri and R.

For any x, y ∈ A, RN ∈ FN and L ∈ 2N , we simply write “PRL (x, y) > 0” instead of “∀i ∈ L , PRi (x, y) > 0”.

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Let K be a subset of N , RN ∈ FN , and f be an FAR: we have

∀i ∈ N , RiK ={Ri if i ∈ K ,

(Ri )−1 if i ∈ N − K ,RNK = (RiK )i∈N

and f K the FAR defined by ∀QN ∈ FN , f K (QN ) = f (QNK ).We notice that, for K = N , QiK = Qi and f K = f , and for K = ∅, QiK = (Qi )−1.We recall some definitions.

Definition 3 (Dutta [8], Fono and Andjiga [10], Richardson [17]). Let f : FN −→ G be an FAR, L ∈ 2N anda, b ∈ A.

(1) f satisfies Pareto condition (PC) if

∀RN ∈ FN , ∀x, y ∈ A, PR(x, y)� mini∈N

PRi (x, y).

(2) f is dictatorial if

∃i ∈ N , ∀RN ∈ FN , ∀x, y ∈ A, PRi (x, y) > 0 ⇒ PR(x, y) > 0.

(3) f satisfies independence of irrelevant alternatives (IIA) if ∀RN , QN ∈ FN , ∀x, y ∈ A,(∀ j ∈ N ,

{PR j (x, y) = PQ j (x, y)

PR j (y, x) = PQ j (y, x)

)⇒{PR(x, y) = PQ(x, y),

PR(y, x) = PQ(y, x).

(4) L is almost decisive over (a, b) if

∀RN ∈ FN (PRL (a, b) > 0 and PRN−L (b, a) > 0) ⇒ PR(a, b) > 0. (10)

(5) L is decisive if

∀RN ∈ FN , ∀x, y ∈ A, PRL (x, y) > 0 ⇒ PR(x, y) > 0.

The intuitive meaning of the well-known crisp condition of Pareto criterion is “If everyone strictly prefers onealternative to a second, then do the society”. Up to now, all the frameworks (see [3,5,8,10,13,17,18,21]) in fuzzy socialchoice theory use condition PC as a fuzzy version of crisp Pareto criterion. However, in the fuzzy case, if everyonestrictly prefers one alternative to the other, the previous condition PC requires that the society do so with a thresholdwhich is the minimum of degrees of individual strict preferences of the two alternatives. This is more demanding andrestrictive for the society.In this paper, we introduce and use condition Pareto optimal (PO), as a new fuzzy version of the crisp Pareto criterion.

PO does not require a threshold to the society, and is weaker than the usual condition PC. We also introduce new fuzzyversions of some crisp properties of a CAR.

Definition 4. Let f : FN −→ G be an FAR, L ∈ 2N , a, b ∈ A and K be a subset of N .

(1) f is:

(i) PO if

∀RN ∈ FN , ∀x, y ∈ A, PRN (x, y) > 0 ⇒ PR(x, y) > 0.

(ii) Anti-Pareto optimal (APO) if

∀RN ∈ FN , ∀x, y ∈ A, PRN (x, y) > 0 ⇒ PR(y, x) > 0.

(iii) K-Pareto optimal (K-PO) if

∀x, y ∈ A, ∀RN ∈ FN , PRNK (x, y) > 0 ⇒ PR(x, y) > 0.

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L.A. Fono et al. / Fuzzy Sets and Systems 160 (2009) 2658–2672 2665

(iv) Locally Pareto optimal (locally PO) if

∃x, y ∈ A, ∃RN ∈ FN , PRN (x, y) > 0 and PR(x, y) > 0.

(v) Locally anti-Pareto optimal (locally APO) if

∃x, y ∈ A, ∃RN ∈ FN , PRN (x, y) > 0 and PR(y, x) > 0.

(2) f is:

(i) Null if ∀x, y ∈ A, ∀RN ∈ FN , PR(x, y) = PR(y, x) = 0.(ii) SNN if ∃x, y ∈ A, ∃RN ∈ FN ,

(∀i ∈ N , PRi (x, y) > 0 or PRi (y, x) > 0) and PR(x, y) > 0.

(3) f is:

(i) NI if

∀x, y ∈ A, ∃RN ∈ FN , PR(x, y) > 0 or PR(x, y) = PR(y, x) = 0.

(ii) SNI if ∀x, y ∈ A, ∃RN ∈ FN , PR(x, y) > 0.(iii) Anti-dictatorial if

∃i ∈ N , ∀RN ∈ FN , ∀x, y ∈ A, PRi (x, y) > 0 ⇒ PR(y, x) > 0.

(4) L is:

(i) Almost anti-decisive over (a, b) if

∀RN ∈ FN (PRL (a, b) > 0 and PRN−L (b, a) > 0) ⇒ PR(b, a) > 0. (11)

(ii) Anti-decisive if ∀RN ∈ FN , ∀x, y ∈ A, PRL (x, y) > 0 ⇒ PR(y, x) > 0.

Remark 2. (1) (i) If K = ∅, then condition K-PO becomes condition APO.(ii) If K = N , then condition K-PO becomes condition PO.(2) It is important to notice that all the frameworks [3,5,8,10,13,17,18,21] established with PC are also true with PO.

Let us end this section by recalling the domain and the range of an FAR which we need in this paper.To establish fuzzy versions of Arrow Impossibility theorem and Gibbard Oligarchy theorem, Fono and Andjiga [10]

introduced a general (defined by means of T-transitivity) domain and range of an FAR. In this paper, we use theirdomain and range.

More precisely, according to diagram (9), for a t-norm T, the set HT1Nof profiles of weakly complete T-pre-orders

is the general domain, and the range is HT1,2 orHT

1 .

In the particular case where T = TM, the domain HT1Nbecomes the set HN of profiles of weakly complete fuzzy

pre-orders and, the rangesHT1,2 andHT

1 , respectively, becomeHC andH. It is important to notice that these particulardomains and ranges are those used earlier by the two first scholars Dutta [8] and Richardson [17] to establish theirArrowian results.In the following subsection, we establish the two first results of Table 2.

3.2. First fuzzy Arrowian results without the Pareto principle

The following first main result establishes a fuzzy version of Malawski and Zhou’s [15] impossibility result.

Theorem 1. Let T be a t-norm and f : HT1N −→ HT

1,2 be an FAR.If f satisfies IIA and NI, then f is either null, or PO, or APO.

This result is due to the following lemma.

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2666 L.A. Fono et al. / Fuzzy Sets and Systems 160 (2009) 2658–2672

Lemma 2. Let T be a t-norm and f : HT1N −→ HT

1,2 be an FAR satisfying IIA.If f is non-null, then f is locally PO or locally APO.

Proof. Suppose that f is non-null and let us show that⎛⎜⎜⎝

(i) ∃{a, b} ⊂ A, ∃WN ∈ HT1N, PWN (a, b) > 0 and PW (a, b) > 0

or

(ii) ∃{a, b} ⊂ A, ∃WN ∈ HT1N, PWN (a, b) > 0 and PW (b, a) > 0

⎞⎟⎟⎠ . (12)

Since f is non-null, then ∃x, y ∈ A, ∃RN ∈ HT1Nsuch that PR(x, y) > 0 or PR(y, x) > 0. Let z ∈ A and the profile

R′N ∈ HT1Nsatisfying on {x, y, z} the following conditions:

⎧⎪⎨⎪⎩

(i) PR′N (x, z) > 0,

(ii) PR′N (y, z) > 0,

(iii) ∀k ∈ N , PR′k (x, y) = PRk (x, y) and PR′k (y, x) = PRk (y, x).

(13)

The definition of PR′ on {x, z} and {y, z} gives{(i) PR′ (x, z) > 0 or PR′ (z, x) > 0 or PR′ (x, z) = PR′ (z, x) = 0,

(ii) PR′ (y, z) > 0 or PR′ (z, y) > 0 or PR′ (y, z) = PR′ (z, y) = 0.(14)

Since PR(x, y) > 0 or PR(y, x) > 0, we distinguish two cases:First case: Suppose that PR(x, y) > 0. Since f is IIA, then (iii) of (13) and the previous inequality imply

PR′ (x, y) > 0. (15)

From (14), we distinguish five cases:

(1) If PR′ (x, z) > 0, then for a = x, b = z and WN = R′N , (i) of (13) and the previous inequality give (i) of (12).(2) If PR′ (z, x) > 0, then for a = x, b = z and WN = R′N , (i) of (13) and the previous inequality give (ii) of (12).(3) If PR′ (y, z) > 0, then for a = y, b = z and WN = R′N , (ii) of (13) and the previous inequality give (i) of (12).(4) If PR′ (z, y) > 0, then for a = y, b = z and WN = R′N , (ii) of (13) and the previous inequality give (ii) of (12).(5) If PR′ (x, z) = PR′ (z, x) = 0 and PR′ (y, z) = PR′ (z, y) = 0. Since f (R′N ) = R′ ∈ HT

1,2, then the secondresult of Proposition 1 implies that PR′ is negative transitive. Hence, the two previous inequalities imply thatPR′ (x, y) = PR′(y, x) = 0. This contradicts (15).

The proof of the second case where PR(y, x) > 0 is analogous to the previous one (see [7]). �

Proof of the Theorem. With Proposition 1 and Lemma 2, the proof is exactly analogous to that of the correspondingresult by Malawski and Zhou [15, Proposition 1, p. 104]. More precisely, one can show that if f is IIA, NI and locallyPO (respectively, locally APO), then f is PO (respectively, APO) [7]. �

With the previous result, we deduce a fuzzy version of Wilson theorem which is the second main result of thispaper.

Theorem 2. Let T be a t-norm, f : HT1N −→ HT

1,2 be an FAR.If f is IIA and NI, then f is either null, or dictatorial, or anti-dictatorial.

This theorem is due to the two following lemmas: the first one gives fuzzy versions of the Field Expansion Lemmafor paretian and anti-paretian CAR.

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Lemma 3. Let T be a t-norm, f : HT1N −→ HT

1 be an FAR satisfying IIA, x, y ∈ A and L ∈ 2N .

(1) If f is APO and L is almost anti-decisive over (x, y), then L is anti-decisive.(2) If f is PO and L is almost decisive over (x, y), then L is decisive.

Proof. (1) Suppose that f is APO and L is almost anti-decisive over (x, y), and let us show that L is anti-decisive,

i.e, ∀QN ∈ HT1N, ∀a, b ∈ A, PQL (a, b) > 0 ⇒ PQ(b, a) > 0. Let QN ∈ HT

1N

and {a, b} ⊂ A such thatPQL (a, b) > 0, and let us show that PQ(b, a) > 0.

Consider RN ∈ HT1Nsatisfying on {x, y}, PRL (x, y) > 0 and PRN−L (y, x) > 0. Thus, since L is almost anti-decisive

over (x, y), we have PR(y, x) > 0.Let us consider the profile WN ∈ HT

1N(withW = f (WN )) satisfying on {x, y, a, b} the following conditions:⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

PWL (x, y) = PRL (x, y), PWL (a, x) > 0, PWL (a, b) = PQL (a, b),

PWL (y, b) > 0, PWN−L (y, x) = PRN−L (y, x),

PWN−L (a, x) > 0, PWN−L (a, b) = PQN−L (a, b),

PWN−L (b, a) = PQN−L (b, a), PWN−L (y, b) > 0.

(16)

We distinguish two steps:First step: Let us show that

PW (b, a) > 0. (17)

Eq. (16) implies PWN (a, x) > 0 (since PWL (a, x) > 0 and PWN−L (a, x) > 0). Since f satisfies APO, then

PW (x, a) > 0. (18)

Eq. (16) implies PWN (y, b) > 0 (since PWL (y, b) > 0 and PWN−L (y, b) > 0). Since f satisfies APO, then

PW (b, y) > 0. (19)

Eq. (16) implies PWL (y, x) = PRL (y, x) = 0 and PWN−L (x, y) = PRN−L (x, y) = 0 (because PWL (x, y) =PRL (x, y) > 0, PWN−L (y, x) = PRN−L (y, x) > 0 and PR is asymmetric).Thus, we obtain PWN (y, x) = PRN (y, x) and PWN (x, y) = PRN (x, y). IIA and the two previous equalities give

PW (y, x) = PR(y, x). And because PR(y, x) > 0, we have

PW (y, x) > 0. (20)

Since W = f (WN ) ∈ HT1 , then the first result of Proposition 1 implies PW is pos-transitive on A. Thus, the

pos-transitivity of PW , (19), (20) and (18) imply (17).Second step: Let us show that PQ(b, a) > 0.Eq. (16) implies that PWL (b, a) = PQL (b, a) = 0 (because PWL (a, b) = PQL (a, b) > 0 and PW is asymmet-

ric), PWN−L (a, b) = PQN−L (a, b), PWL (a, b) = PQL (a, b) and PWN−L (b, a) = PQN−L (b, a). Thus, PWN (a, b) =PQN (a, b) and PWN (b, a) = PQN (b, a). Consequently, IIA and the two previous equalities imply PW (b, a) = PQ(b, a).Thus, (17) and the previous equality imply PQ(b, a) > 0.

(2) The proof of this result is similar to the first one. �

The second lemma gives fuzzy versions of the crisp Group Contraction Lemma for paretian and anti-paretian rules.

Lemma 4. Let T be a t-norm, f : HT1N −→ HT

1,2 be an FAR satisfying IIA and L ∈ 2N (with |L|�2).

(1) If f is APO and L is anti-decisive, then ∃K ∈ 2N , K�L and K is anti-decisive.(2) If f is PO and L is decisive, then ∃K ∈ 2N , K�L and K is decisive.

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2668 L.A. Fono et al. / Fuzzy Sets and Systems 160 (2009) 2658–2672

Proof. (1) Suppose that L is anti-decisive and let us find K ∈ 2N such that K ⊂ L and K anti-decisive. As |L|�2,

consider a partition L1 and L2 of L. Let {x, y, z} ⊂ A and the profile RN ∈ HT1Nsatisfying on {x, y, z} the following

conditions:⎧⎪⎪⎨⎪⎪⎩

PRL1 (x, y) > 0, PRL1 (y, z) > 0,

PRL2 (x, y) > 0, PRL2 (z, x) > 0,

PRN−L (y, z) > 0, PRN−L (z, x) > 0.

(21)

Eq. (21) implies that PRL1 (x, z) > 0 (because RN ∈ HT1N, the first result of Proposition 1 implies that ∀i ∈ N , PRi

is pos-transitive).Since PRL (x, y) > 0 (because PRL1 (x, y) > 0 and PRL2 (x, y) > 0) and L anti-decisive, then PR(y, x) > 0. Since

R = f (RN ) ∈ H�1,2, then the second result of Proposition 1 implies that PR is negative transitive on A. The negative

transitivity of PR and the previous inequality imply that PR(z, x) > 0 or PR(y, z) > 0. Then, we have two cases andlet us determine K in every case:First case: Suppose that PR(z, x) > 0, then with the profile RN , the alternatives {x, z} and the coalition L1, we have

PRL1 (x, z) > 0, PRN−L1 (z, x) > 0 and PR(z, x) > 0. Thus, L1 is almost anti-decisive on {x, z}. By the first result ofLemma 3, L1 is anti-decisive. Hence K = L1.

Second case: Suppose that PR(y, z) > 0, then with the profile RN , the alternatives {y, z} and the coalition L2, wehave PRL2 (z, y) > 0, PRN−L2 (y, z) > 0 and PR(y, z) > 0. Thus, L2 is almost anti-decisive on {y, z}. By the first resultof Lemma 3, L2 is anti-decisive. Hence K = L2.

(2) The proof of this result is similar to the first one. �

We now establish the theorem.

Proof of the Theorem. Since f is IIA andNI, then Theorem 1 implies that f is either null, or APO, or PO.We distinguishtwo cases:First case: If f is PO, let us show that f is dictatorial. Let us determine i ∈ N such that

∀x, y ∈ A, ∀QN ∈ HT1N, PRi (x, y) > 0 ⇒ PR(x, y) > 0.

We firstly show that N is decisive. Let QN ∈ HT1Nand a, b ∈ A. Suppose that PQN (a, b) > 0 and let us show that

PQ(a, b) > 0.Since PQN (a, b) > 0 and f satisfies PO, then PQ(a, b) > 0. Thus, N is decisive.We now determine the voter i.Because N is decisive and |N |�2, then by the second result of Lemma 4, N contains a decisive coalition N1. If

|N1 = 1, then its unique element i is the dictator. If |N1|�2, then we apply the second result of Lemma 4 on N1. SinceN is finite, we will obtain by this procedure the voter i.Second case: If f is APO, let us show that f is anti-dictatorial.

Determine a voter i in N such that ∀RN ∈ HT1N, ∀x, y ∈ A, PRi (x, y) > 0 ⇒ PR(y, x) > 0.

We firstly show that N is anti-decisive. Let QN ∈ HT1Nand a, b ∈ A such that PQN (a, b) > 0. Since PQN (a, b) > 0

and f is APO, then PQ(b, a) > 0. Hence, N is anti-decisive.Let us determine the anti-dictator i. N anti-decisive and |N |�2, by the first result of Lemma 4, N contains an anti-

decisive coalition N1. If |N1| = 1, then its unique element i is the anti-dictator. If |N1|�2, then we apply the firstresult of Lemma 4 to N1. Since N is finite, then we will obtain by this method the anti-dictator i. �

For the minimum operator TM, the two previous theorems become:

Corollary 1. If f : HN −→ HC is an FAR satisfying IIA and NI, then

(1) f is either null, or PO, or APO.(2) f is either null, or dictatorial, or anti-dictatorial.

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Proof. Since T = TM, then HT1 = H and HT

1,2 = HC and Theorems 1 and 2 become the first and second results,respectively. �

Let us end this section with this interesting remark.

Remark 3. (1)After a careful check of the results of the pioneers asBanerjee [3], Barrett et al. [4], Dutta [8], Richardson[17], Fono and Andjiga [10], one can think that the obtention of a dictatorial FAR need PO and PC the fuzzy versionsof Pareto principle. However, the two previous results show that as in crisp case, a fuzzy version of condition NI leadsus to obtain not only dictatorial rules, but anti-dictatorial and null ones also.(2) The corollary gives results obtained in Fono et al. [12]. In other words, our two main results generalize, for a

general t-norm, their results.

The two previous results establish fuzzy counterparts of Malawski and Zhou’s result [15, Proposition 1, p. 104], andWilson’s theorem.However, individual preferences are weakly complete T-pre-orders satisfying condition CT

1 and social preferencesare weakly complete T-pre-orders satisfying both conditions CT

1 and CT2 . It is important to seek what happens if we

cancel condition CT2 .

In the following section, the study of this question give us two results.

3.3. Fuzzy Arrowian results without the Pareto principle and condition CT2

The first result of this section shows that if one cancels condition CT2 in the previous theorem (i.e., individual and

social preferences satisfies the same condition), then the possibility of the obtention of nullness, anti-dictator anddictator could disappear. This is our third main result.

Proposition 2. Let T be a t-norm.

There exists an FAR f : HT1N −→ HT

1 satisfying IIA, NI and which is non-null, non-anti-dictatorial and non-dictatorial.

Proof. Let � ∈] 12 , 1[. Consider f : HT1N −→ HT

1 defined by ∀RN ∈ HT1N, f (RN ) = R, where ∀x, y ∈ A (x � y),

R(x, x) = 1 and R(x, y) ={1 if ∀i ∈ N , Ri (x, y) > Ri (y, x),

� otherwise.

Fono and Andjiga [10, Proposition 11, p. 386] showed that f is well defined, IIA and non-dictatorial. It remains to showthat f is NI, non-null, and non-anti-dictatorial.

(1) Let us show that f is NI, i.e, ∀x, y ∈ A, ∃ RN ∈ HT1Nsuch that PR(x, y) > 0 or PR(x, y) = PR(y, x) = 0.

Let x, y ∈ A.We consider RN ∈ HT1Nsatisfying on {x, y} the following conditions: ∀i ∈ N , Ri (x, y) > Ri (y, x).

We have R(x, y) = 1 and R(y, x) = �. Since R(x, y) > R(y, x), then PR(x, y) > 0.

(2) Let us show that f is non-null. Let us find a, b ∈ A and RN ∈ HT1Nsuch that PR(a, b) > 0 or PR(b, a) > 0.

Consider a, b ∈ A and RN ∈ HT1Nsatisfying on {a, b} the following conditions: ∀i ∈ N , Ri (a, b) > Ri (b, a).

The definition of f gives R(a, b) = 1 and R(b, a) = �. Thus R(a, b) > R(b, a), i.e, PR(a, b) > 0. Hence, f isnon-null.

(3) Let us show that f is non-anti-dictatorial, i.e., ∀i ∈ N , ∃x, y ∈ A, ∃RN ∈ HT1N, PRi (x, y) > 0 and PR(y, x) = 0.

Let i ∈ N , a, b ∈ A and a profile QN ∈ HT1N, satisfying on {a, b} the following conditions:{

Qi (a, b) > Qi (b, a),

∀ j ∈ N − {i}, Q j (a, b)�Q j (b, a).

Since Qi (a, b) > Qi (b, a), then PQi (a, b) > 0. We also have not(∀k ∈ N , Qk(b, a) > Qk(a, b)), thus Q(b, a) =Q(a, b) = �. Since PQ is regular, we have PQ(b, a) = 0. �

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The second result of this subsection shows that, if in Theorem 1, one cancels CT2 , strengthens condition NI to obtain

condition SNI and adds condition SNN to the FAR, then we obtain a fuzzy counterpart of the second result of Malawskiand Zhou [15, Proposition 2, p. 106]. More precisely, it stipulates that an FAR is K-Pareto optimal (K ⊆ N ) if the FARis IIA, SNI, SNN and its domain and range areHT

1 .

Theorem 3. If f : HT1N −→ HT

1 is an FAR satisfying IIA, SNI and SNN, then there exists a subset K of N such that fis K-Pareto optimal.

This fourth and last main result of our paper can give possibilities to obtain dictator (if K = N ) and anti-dictator (ifK = ∅).To establish this result, we need the two following lemmas. The first one give the link between three properties of

the FAR f K and those of f.

Lemma 5. Let f : HT1N −→ HT

1 be an FAR and K be a subset of N.

(1) If f is IIA, then is f K .(2) If f is SNI, then is f K .(3) If f is SNN, then there exists a subset K1 of N such that f K1 is locally PO.

Proof. (1) Let RN , QN ∈ HT1N, x, y ∈ A such that

∀ j ∈ N , PR j (x, y) = PQ j (x, y) and PR j (y, x) = PQ j (y, x). (22)

Let us show that Pf K (RN )(x, y) = Pf K (QN )(x, y) and Pf K (RN )(y, x) = Pf K (QN )(y, x). We have two steps:First step: Let us show that

PRNK (x, y) = PQNK (x, y) and PRNK (y, x) = PQNK (y, x), (23)

i.e., ∀i ∈ N , PRiK (x, y) = PQiK (x, y) and PRiK (y, x) = PQiK (y, x).By the definition of RNK , we have

∀i ∈ N , PRiK ={

PRi if i ∈ K

P(Ri )−1 if i ∈ N − Kand PQiK =

{PQi if i ∈ K ,

P(Qi )−1 if i ∈ N − K .

If i ∈ K , (22) implies that PRiK (x, y) = PQiK (x, y) and PRiK (y, x) = PQiK (y, x).If i ∈ N−K , (22) implies that (PRi )−1(x, y) = (PQi )−1(x, y) and (PRi )−1(y, x) = (PQi )−1(y, x).The second result

of Lemma 1 implies that P(Ri )−1 (x, y) = P(Qi )−1 (x, y) and P(Ri )−1 (y, x) = P(Qi )−1 (y, x), i.e., PRiK (x, y) = PQiK (x, y)and PRiK (y, x) = PQiK (y, x).

Second step: Since f is IIA, then (23) implies that Pf (RNK )(x, y) = Pf (QNK )(x, y) and Pf (RNK )(y, x) = Pf (QNK )(y, x). Hence Pf K (RN )(x, y) = Pf K (QN )(x, y) and Pf K (RN )(y, x) = Pf K (QN )(y, x).

(2) Let x, y ∈ A, we find QN ∈ HT1Nsuch that Pf K (QN )(x, y) > 0, i.e., Pf (QNK )(x, y) > 0. Since f is SNI, then

for x, y ∈ A, there exists RN ∈ HT1Nsuch that Pf (RN )(x, y) > 0.

Let us define QN by

∀i ∈ N , Qi ={

Ri if i ∈ K ,

(Ri )−1 if i ∈ N − K .

We have QNK = RN and Pf (RN )(x, y) = Pf (QNK )(x, y) > 0.(3) Let us show that if f is SNN, then there exists a subset K1 of N such that f K1 is locally PO. Since f is SNN, then

∃{a, b} ⊂ A, ∃RN ∈ HT1Nsuch that

(∀i ∈ N , PRi (a, b) > 0 or PRi (b, a) > 0) and PR(a, b) > 0. (24)

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Let K1 = {i ∈ N , PRi (a, b) > 0} and consider the profile QN defined by

∀i ∈ N , Qi ={

Ri if i ∈ K1,

(Ri )−1 if i ∈ N − K1.

We have QNK1 = RN , PQN (a, b) > 0 and PR(a, b) = Pf (RN )(a, b) = Pf (QNK1 )(a, b) = Pf K1 (QN )(a, b). Since RN ∈HT

1N, then the first result of Lemma 1 implies that QN ∈ HT

1N. Thus,HT

1NandHT

1 are, respectively, the domain and

the range of f K1 . Eq. (24) becomes ∃{a, b} ⊂ A, ∃QN ∈ HT1Nsuch that PQN (a, b) > 0 and Pf K1 (QN )(a, b) > 0, i.e.,

f K1 is locally PO. �

We now establish the second lemma.

Lemma 6. Let T be a t-norm and f : HT1N −→ HT

1 be an FAR satisfying IIA and SNI.If f is locally PO, then f is PO.

Proof. Suppose that f is locally PO, i.e.,

∃{a, b} ⊂ A, ∃WN ∈ HT1N, PWN (a, b) > 0 and PW (a, b) > 0 (25)

and let us show that f is PO. Let {s, t} ⊂ A, QN ∈ HT1Nsuch that PQN (s, t) > 0. Let us show that PQ(s, t) > 0. We

distinguish five cases: {s, t}∩ {a, b} = ∅, (s = a and t = b), (s = b and t = a), (s = a and t � b) and (s � a and t = b).First case: if {s, t} ∩ {a, b} = ∅.

Since f is SNI, for {s, a} ⊂ A and {b, t} ⊂ A, ∃K N , LN ∈ HT1Nsuch that{

(i) PK (s, a) > 0,

(ii) PL (b, t) > 0.(26)

Consider the profile Q′N ∈ HT1Nsatisfying on {a, b, s, t} the following conditions:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(i) ∀k ∈ N , PQ′k (a, b) = PWk (a, b) and PQ′k (b, a) = PWk (b, a),

(ii) ∀k ∈ N , PQ′k (s, t) = PQk (s, t) and PQ′k (t, s) = PQk (t, s),

(iii) ∀k ∈ N , PQ′k (s, a) = PKk (s, a) and PQ′k (a, s) = PKk (a, s),

(iv) ∀k ∈ N , PQ′k (b, t) = PLk (b, t) and PQ′k (t, b) = PLk (t, b).

(27)

Let us show that PQ′(s, t) > 0. Since f is IIA, (25) and (i) of (27) imply that PQ′ (a, b) > 0. Also by IIA, (iii) of (27),(iv) of (27) and (26) give{

(i) PQ′(s, a) > 0,

(ii) PQ′ (b, t) > 0.(28)

Since f (Q′N ) = Q′ ∈ HT1 , the first result of Proposition 1 implies that PQ′ is pos-transitive. Thus, the pos-transitivity

of PQ′ , (i) of (28), the inequality PQ′(a, b) > 0 and (ii) of (28) imply that PQ′(s, t) > 0. Since f is IIA, then (ii) of (27)and the previous inequality imply that PQ(s, t) > 0.The proof of the other four cases are analogous to the previous ones (see [7]). �

We now establish the Theorem.

Proof of the Theorem. If f is IIA, SNI and SNN, then Lemma 5 implies that f K is IIA, SNI and locally PO. AndLemma 6 implies that f K is PO. �

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2672 L.A. Fono et al. / Fuzzy Sets and Systems 160 (2009) 2658–2672

In the particular case where T = TM, our fourth main result becomes:

Corollary 2. If f : HN −→ H is an FAR satisfying IIA, SNI and SNN, then there exists a subset K of N such that f isK-PO.

Proof. Since T = TM, then HT1 = H and Theorem 3 gives the result. �

4. Concluding remarks

The results of Malawski and Zhou [15], and Wilson [22] are preserved when a FAR satisfies some appropriateproperties, individual preferences are weakly complete T-pre-orders satisfying conditionCT

1 and social preferences areweakly complete T-pre-orders satisfying conditions CT

1 and CT2 (Theorems 1 and 2, respectively). In addition, without

condition CT2 on social preferences, one can obtain non-dictatorial (Proposition 2) or dictatorial rules (Theorem 3).

Thereby, condition CT2 plays a central role in our results.

Previous fuzzy versions of Arrow’s theorem have relied heavily on the Pareto principle as a basic postulate. Ourpresent results seem to prove that, as in crisp case, the essential significance of Arrow’s theorem is not diminished ifone abandons the Pareto principle.

Acknowledgements

A previous version of this paper [12] was presented at the meeting of the society for Social Choice and Welfare inTurkey in July 2006 (Istanbul Bilgi University). We are very grateful to several participants for their comments.

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