1 Introduction FUZZY VERSIONS OF SOME ARROW 0 S TYPE RESULTS 1. Louis Aimé Fono a,2, Véronique Kommogne b and Nicolas Gabriel Andjiga c

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1 FUZZY VERSIONS OF SOME ARROW 0 S TYPE RESULTS 1 Louis Aimé Fono a,2, Véronique Kommogne b and Nicolas Gabriel Andjiga c a;b Département de Mathématiques et Informatique, Université de Douala, Faculté des Sciences, BP Douala, Cameroun. c Département de Mathématiques, Université de Yaoundé I, Ecole Normale Supérieure, BP 47 Yaoundé, Cameroun. Abstract Malawski et al. [8] and Wilson [12] establish some variants of Arrow s type results when individual and collective preferences are crisp. In this paper, we introduce new properties on fuzzy aggregation rules and generalize each of these results when individual and collective preferences are fuzzy. Keywords: Fuzzy preference; Wilson s theorem; max-min transitivity; minimal and regular fuzzy strict preference. 1 Introduction Malawski et al. [8] shows that an crisp agregation rule (CAR) is either null or Pareto or Antiweak pareto if its domain and range are crisp orderings (re exive, complete and transitive crisp binary relations) on the set of alternatives A (with jaj 3) and if the given CAR satis es Independence of Irrelevant Alternatives (IIA) and Non Imposition (NI). This result generalizes the Wilson s result [12] which stipulates that with the same conditions, a CAR is either null or dictatorial or antidictatorial. The purpose of this paper is to investigate in what extent these two results are preserved when individual and collective preferences are fuzzy. Contrary to the frameworks of Barrett et al. [2], and Tang [11] who modelled individual and collective vague preferences by fuzzy strict preferences, we assume these vague preferences are here represented by fuzzy weak preferences whose fuzzy strict components are obtained by factorization as in [3, 9, 6]. The paper is organized as follows: In section 2, we give some basic concepts and properties on fuzzy operators and fuzzy relations. In section 3, we recall some notations of fuzzy social choice and de nitions on fuzzy aggregation rule (FAR). We introduce fuzzy versions of some properties of a FAR such as NI, null and antidictatorial, and establish our two mains results. The paper ends in section 4 with some concluding remarks. 1 The rst author sincerely thanks the "Agence Universitaire de la Francophonie". This version was completed when he was Visiting Researcher at CREM, University of Caen under the research grant "Bourse Post-Doctorant de l AUF, ". 2 Corresponding author: lfono2000@yahoo.fr 1

2 2 Preliminaries In this paper, we assume as in [3, 9, 6] that individual and collective preferences are given by fuzzy weak preference relation on the set of alternatives A (with jaj 3) which is formally de ned by: De nition 1 A fuzzy binary relation is a function R : A A! [0; 1]. R re exive if 8x 2 A; R(x; x) = 1. R is connected if 8x; y 2 A, R(x; y) + R(y; x) 1. A Fuzzy Weak Preference Relation (FWPR ) is a re exive and connected fuzzy binary relation. Let us recall that: (i) For any x; y 2 A, R(x; y) is the degree to which "x is at least as good as y" ; (ii) R is a crisp binary relation if 8x; y 2 A, R(x; y) 2 f0; 1g, and this case, xry denotes R(x; y) = 1. We need the following notions and properties on fuzzy operators and fuzzy binary relations: De nition 2 (see [4, 7, 5, 6]) A mapping S: [0; 1][0; 1]! [0; 1] is a triangular conorm (or t-conorm) if it is a commutative, associative, non-decreasing in each argument and S(a; 1) = a for all a 2 [0; 1]. Let S be a continuous t- conorm. The quasi-substraction of S is the internal composition law denoted by and de ned over [0; 1] by a b = minft 2 [0; 1], S(a; t) bg for all a; b 2 [0; 1]. The most usual t-conorms and their quasi-subtractions are: Example 3 1) The Zadeh s max t-conorm S M is de ned by for all a; b 2 [0; 1], S M (a; b) = max(a; b). Its quasi-substraction is M de ned by: 8a; b 2 [0; 1], a if a > b a M b = 0 otherwise. 2) The Lukasiewicz t-conorm S L is de ned by for all a; b 2 [0; 1], S L (a; b) = max(0; a+b 1). Its quasi-substraction is L de ned by: 8a; b 2 [0; 1], a Lb = max(0; b a). De nition 4 (see [5, 6]) R is strongly connected if 8x; y 2 A, max(r(x; y); R(y; x)) = 1. The converse of R is the FWPR denoted R 1 and de ned by: 8x; y 2 A, R 1 (x; y) = R(y; x). R satis es condition T if 8x; y; z 2 A, R(x; y) = R(y; x) = R(y; z) = R(z; y) implies R(x; z) = R(z; x). R is max-min transitive if 8x; y; z 2 A, R(x; z) min(r(x; y), R(y; z)). 2

3 Strong connectedness implies connectedness (see [5, 6]). It is easy to show that for any two FWPRs R and Q, R = Q i R 1 = Q 1 (1) To establish fuzzy versions of Arrowian results, we need to de ne fuzzy strict preference P R of a given FWPR R and to determine two useful results of P R. Dutta [3] and Richardson [9] propose the two following fuzzy strict preferences: 8x; y 2 A, 8 >< >: i) P Z R (x; y) = R(x; y) if R(x; y) > R(y; x) 0 otherwise ii) PR L (x; y) = max(0, R(x; y) R(y; x)) Recently, Fono et al. [6] determine for a given t-conorm S, a class of regular fuzzy strict components of a given FWPR R. Each class has a minimal element called the miminal regular fuzzy strict preference P R associated with S de ned as follows: De nition 5 (Fono et Andjiga [6] ) Let S be a continuous t-conorm, its quasisubstraction and R be an FWPR. The minimal regular fuzzy strict component P R of R associated with S is de ned by : 8x; y 2 A, P R (x; y) = R(y; x) R(x; y) (2) Let us remark that the minimal regular fuzzy strict preference P R of R associated with the Zadeh s max t-conorm S M (resp. the Lukasiewicz s t-conorm S L ) is PR Z (resp. P R L ). Moreover, if R becomes crisp, then for any t-conorm S, the minimal regular fuzzy strict preference P R of R associated with S becomes the unique well-known crisp strict preference of R de ned by: For all x; y 2 A, xp R y,(xry and not(yrx)). Throughout this paper, we will use any minimal and regular fuzzy strict preference of R denoted P R, that is, the minimal and regular fuzzy strict preference of R associated with any continuous t-conorm S. Therefore, it obvious to show that: If R is an FWPR on A, P R 1 any minimal and regular fuzzy strict preference of R 1, then P 1 R = P R 1 (3) As raised Richardson [9], any P R satis es the following properties: P R is regular, that is, 8x; y 2 A; R(x; y) R(y; x), P R (x; y) = 0 P R is simple, that is, 8x; y 2 A; R(x; y) = R(y; x) ) P R (x; y) = P R (y; x). Let us recall these interpretations given by [9, 5, 6]: (i) for any x; y 2 A such that R(x; y) > R(y; x), i.e., P R (x; y) > 0, the real P R (x; y) is the degree to which 3

4 x is strictly preferred to y ; (ii) for any x; y 2 A such that R(x; y) = R(y; x), i.e., P R (x; y) = P R (y; x) = 0, then x is equivalent to y. Let us end this section by recalling two useful properties and results on P R introduced by Fono and Andjiga : De nition 6 ([6], page 379, De nition 5) P R is pos-transitive if 8x; y; z 2 A, (P R (x; y) > 0 and P R (y; z) > 0) imply P R (x; z) > 0. P R is negative transitive if 8x; y; z 2 A, (P R (x; y) = 0 and P R (y; z) = 0) imply P R (x; z) = 0. Proposition 7 (Fono and Andjiga [6], Corollary 8, page 382 ) Let R be an FWPR and P R be any minimal and regular fuzzy strict preference of R. 1) If R is max-min transitive, then P R is pos-transitive. 2) If R is max-min transitive, then R satis es condition T () P R is negative transitive 3 Fuzzy social choice results In the following subsection, we recall some de nitions and notations of social choice theory [12, 2, 3, 8, 1, 9, 6]. We also introduce some new de ntions that will be useful in this paper. 3.1 De nitions, notations, domain and range of a FAR N = f1; :::; i; :::; ng is the set of voters in a society. We assume that jnj 3: 2 N is the set of all non-empty subsets of N and an element of 2 N is called a coalition of voters. Given two sets of FWPRs F and G, an fuzzy aggregation rule (FAR) is a function f : F N! G; where the elements of F N are labelled by ( R 1 ; :::; R i ; :::; R n ) = (R i ) i2n = R N and are called the pro les. When f is an FAR, we denoted f(r N ) = R: P R i and P R are, respectively, any minimal and regular fuzzy strict preference of R i and R, that are, the minimal and regular fuzzy strict preferences of R i and R associated with any t-conorm S. For any x; y 2 A; R N 2 F N and S 2 2 N, we simply write " R S (x; y) 0" instead of " 8i 2 S; R i (x; y) 0": We also have the following de nitions: De nition 8 Let f : F N! G be an FAR and S 2 2 N. (i) f satis es Independence of Irrelevant Alternatives (IIA) if 8R N, Q N 2 F N, 8x; y 2 A, 0 1 P R j (x; y) = P Q j (x; 2 N; and A PR (x; y) = P ) Q (x; y) P P R j (y; x) = P Q j (y; x) R (y; x) = P Q (y; x) 4

5 (ii) f is dictatorial if 9i 2 N, 8R N 2 F N, 8x; y 2 A, P R i(x; y) > 0 ) P R (x; y) > 0. (iii) f satis es: (iii 1 ) Pareto Optimality (PO) if 8 R N 2 F N, 8x; y 2 A, P R N (x; y) > 0 =) P R (x; y) > 0. (iii 2 ) Anti-Pareto Optimality (APO) if 8 R N 2 F N ; 8x; y 2 A; P R N (x; y) > 0 =) P R (y; x) > 0. (iii 3 ) Nullness if 8x; y 2 A, 8R N 2 F N, P R (x; y) = P R (y; x) = 0. (iii 4 ) Non-Imposition (NI) if 8x; y 2 A, 9R N 2 F N, P R (x; y) > 0 or P R (x; y) = P R (y; x). (iv) f is anti-dictatorial if 9i 2 N=8R N 2 F N, 8x; y 2 A, P R i(x; y) > 0 ) P R (y; x) > 0. (v) S is almost anti-decisive over (x; y) if 8R N 2 F N, (P R S (x; y) > 0 and P R N S (y; x) > 0) =) P R (y; x) > 0. (vi) S is anti-decisive if 8R N 2 P N ; 8x; y 2 A, P R S (x; y) > 0 ) P R (y; x) > 0. Remark 9 Except the two rst de nitions, the others de nitions are new in the fuzzy social choice theory. However (PO) is a weak version of Pareto Creterion (PC), that is, 8R N 2 F N, P R (x; y) min P Ri(x; y) (see [3, 1, 9, 6]). i2n When preferences are modelled by FWPRs, the authors such as Dutta [6], Richardson [9] and Fono et al. [6] use condition PC to establish fuzzy versions of the Arrow s impossibility theorem and Gibbard s oligarchy theorem. However, it is important to note that: - their results remain valid with condition (PO). Thus, we obtain a slight generalization of their results by weakenning condition PC by condition PO. - the intuitive meaning of the well-known crisp condition of Pareto Criterion is " If everyone strictly prefers one alternative to a second, then do the society". We think that PO express it more than PC. Therefore, even if PC is usual, PO is a natural fuzzy version of the crisp Pareto Criterion. For these reasons, we will use in this paper PO as a fuzzy version of the crisp pareto criterion. Fono et al. [6] introduce a domain and a range of a FAR to establish fuzzy versions of Arrow impossibility theorem and Gibbard Oligarchy theorem. We will use these domain and range to establish our results. To recall them, we introduce the following notations: H the set of max-min transitive FWPR s (fuzzy orderings), H T the set of max-min transitive FWPR s satisfying condition T, SF O is the set of strong connected and max-min transitive FWPR s (strong fuzzy orderings) and CO the set of all crisp orderings. 5

6 We have the following relations between these sets: CO SF O H T H In this work, the domain of any FAR is H N and the range is H T, therefore our domain (resp. range) contains the well-known crisp Arrowian domain CO N (resp. range CO). 3.2 New fuzzy social choice results Let us now establish our rst main result which is a fuzzy version of the result due to Malawski et al. [8]. Theorem 10 If f H N! H T is an FAR satisfying IIA and NI, then f is either null or pareto optimal or anti-pareto optimal. This result is obtained through to the following lemma. Lemma 11 Let f H N! H T be an FAR satisfying IIA. If f is not null, then 9fa; bg A, 9W N 2 H N, i) P W N (a; b) > 0 and P W (a; b) > 0 or ii) P W N (a; b) > 0 and P W (b; a) > 0 1 A (4) Proof: Suppose that f is not null and let us determine fa; bg A and W N 2 H N satisfying i) or ii) of (4). Since f is not Null, then 9R N 2 H N and 9x; y 2 A such that P R (x; y) > 0 or P R (y; x) > 0 (5) Let z 2 A. Let us consider the pro le R 0N 2 H N satisfying on fx; y; zg the following conditions: 8 < By de nition of P R 0 : i) P R 0 N (x; z) > 0 ii) P R 0 N (y; z) > 0 iii) P R 0 N = P R N on fx; yg on fx; zg and fy; zg, we have: (6) i) PR 0 (x; z) > 0 or P R 0 (z; x) > 0 or P R 0 (x; z) = P R 0 (z; x) = 0 ii) P R 0 (y; z) > 0 or P R 0 (z; y) > 0 or P R 0 (y; z) = P R 0 (z; y) = 0 (7) With (5), we distinguish two cases: 1 st case: Suppose that P R (x; y) > 0. 6

7 Since f is IIA, then iii) of (6) and the inequality P R (x; y) > 0 imply: P R 0 (x; y) > 0 (8) From (7), we distinguish ve cases: 1) if P R 0 (x; z) > 0, then for a = x; b = z and W N = R 0N, i) of (6) and the previous inequality give i) of (4). 2) if P R 0 (z; x) > 0; then for a = x; b = z and W N = R 0N, i) of (6) and the previous inequality give ii) of (4). 3) if P R 0 (y; z) > 0, then for a = y; b = z and W N = R 0N, ii) of (6) and the previous inequality give i) of (4). 4) if P R 0 (z; y) > 0, then for a = y; b = z and W N = R 0N, ii) of (6) and the previous inequality give ii) of (4). 5) if P R 0 (x; z) = P R 0 (z; x) = 0 and P R 0 (y; z) = P R 0 (y; z) = 0. Since f(r 0N ) = R 0 2 H T, the second result of proposition 7 implies that P R 0 is negatively transitive. Thus the two previous equalities and the negative transitivity of P R 0 imply P R 0 (x; y) = P R 0 (y; x) = 0. This contradicts (8). This means that this case is impossible. 2 nd case: Suppose that P R (y; x) > 0. The proof of this case is analogous to the previous case. Proof of theorem 10: If f is not Null, then lemma 11 implies (4). Thus we have two cases: - If (i) of (4), then as f satis es IIA, (1), (3) and proposition 7 imply that f is pareto optimal. - If (ii) of (4), then as f satis es IIA, (1), (3) and proposition 7 imply that f is anti-pareto optimal. We note the proofs of these two previous cases are exactly analogous to those of the result of Malawski et al. [8]. With the previous result, we deduce our second main theorem which is a fuzzy version of Wilson s theorem [12]. Theorem 12 If f H N! H T is an FAR satisfying IIA and NI, then f is either null, or dictatorial, or anti-dictatorial. To prove this theorem, we will rst establish two lemmas which can be respectively seem as a eld expansion lemma and a group contraction lemma [9, 6]. Lemma 13 Let f H N! H T be an FAR satisfying IIA and APO, x; y 2 A, S 2 2 N and S almost antidecisive over (x; y). Then S is antidecisive. Proof: With proposition 7, the proof is exactly analogous to the proof the eld expansion lemma established by Fono et al. (see [6], page 387, Lemma 13). 7

8 Lemma 14 Let f H N! H T be an FAR satisfying IIA and APO, S 2 2 N ( jsj 2 ) with S antidecisive. Then, 9K 2 2 N, K 6= S and K antidecisive. Proof: With proposition 7, the proof is exactly analogous to the proof the group contraction lemma established by Fono et al. (see [6], page 387, Lemma 14). Proof of theorem 12: By theorem 10, f is Null or PO or APO. Then we have two cases: - If f is PO, then as f H N! H T is an FAR satisfying IIA and PO, theorem 13 of Fono et al. ([6], page 387) implies that f is dictatorial. - If f is APO, then as f H N! H T is an FAR satisfying IIA and APO, the two previous lemmas imply that f is anti-dictatorial. 4 Concluding remarks Our contribution can be seen as another fuzzy version of the Arrow s General possibility theorem. Speci cally, the results of Malaski et al. and Wilson are preserved given appropriate properties on vague preferences and on FAR. Previous fuzzy versions of Arrow s theorem have relied heavily on the Pareto Principle as a basic postulate. Our present results ( theorems 10 and 12), as Malaski et al. s result and Wilson s result on crisp case, seem to prove that the essential signi cance of Arrow s theorem is not disminished if one abandons the Pareto Principle. Our two theorems are, of course, weaker than Arrow s theorem, but the fact remains that Arrow s other conditions su ce to exclude all of the democratic fuzzy social choice processes of interest. References [1] Banerjee A. (1994), Fuzzy preferences and Arrow-type problems in social choice, Social Choice and Welfare 11 : [2] Barrett C. R., Pattanaik P. K. and Salles M. (1986), On the structure of fuzzy social welfare functions, Fuzzy Sets and Systems 19 : [3] Dutta B. (1987), Fuzzy preferences and social choice, Mathematical Social Sciences 13 : [4] Fodor J. and Roubens M. (1994), Fuzzy preference modelling and Multicriteria Decision Support, (Kluwer Academic Publishers, Dordrecht). [5] Fono L.A. and Gwét H. (2003), On strict lower and upper sections of fuzzy orderings, Fuzzy sets and Systems 139 :

9 [6] Fono L.A. and Andjiga N. G. (2005), Fuzzy strict preference and social choice, Fuzzy sets and Systems, 155: [7] Gwét H. (1997), Normalized conditional possibility distributions and informational connection between fuzzy variables, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 5, N o 2 : [8] Malawski M. and Zhou L. (1994), A note on social choice theory without the Pareto principle, Social Choice and Welfare 11: [9] Richardson G. (1998), The structure of fuzzy preferences, Social Choice and Welfare 15 : [10] Sen A. (1986), Social choice theory, in: Arrow K. J., Intriligator M. D. (Eds), Handbook of Mathematical Economics, Vol. 3, Elsevier, North- Holland, pp [11] Tang F. F. (1994), Fuzzy preferences and social choice, Bull. Econom. Res. 46: [12] Wilson R. (1972), Social choice theory without the Pareto principle, Journal of Econ Theory 5: