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On the manipulability of the fuzzy social choice functions
Fuzzy Sets and Systems 159 (2008) 177 – 184 www.elsevier.com/locate/fss
On the manipulability of the fuzzy social choice functions Fouad Ben Abdelaziza, c,∗ , José Rui Figueirab , Olfa Meddeba a LARODEC, Institut Supérieur de Gestion, University of Tunis, 41, Rue de la liberté, 2000 Le Bardo, Tunisia b CEG-IST, Center for Management Studies, Instituto Superior Técnico, Technical University of Lisbon, Tagus Park, Av. Cavaco Silva, 2780-990
Porto Salvo, Portugal c Visiting School of Engineering, American University of Sharjah, P.O. Box 26666, Sharjah-UAE
Received 23 March 2006; received in revised form 19 June 2007; accepted 19 June 2007 Available online 3 July 2007
Abstract In many social decision-making contexts, a manipulator has incentives to change the social choice in his favor by strategically misrepresenting his preference. Gibbard [Manipulation of voting schemes: a general result, Econometrica 41(4) (1973) 587–601] and Satterthwaite [Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J. Econom. Theory 10 (1975) 187–217] have shown that any non-dictatorial voting choice procedure is vulnerable to strategic manipulation. This paper extends their result to the case of fuzzy weak preference relations. For this purpose, the best alternative set is defined in three ways and consequently three generalizations of the Gibbard–Satterthwaite theorem to the fuzzy context are provided. © 2007 Elsevier B.V. All rights reserved. Keywords: Fuzzy preference relations; Fuzzy social choice functions; Fuzzy strategy-proofness
1. Introduction A voting choice procedure is known to be subject to strategic manipulation when an individual reveals a non-sincere preference relation in order to change the social choice in his favor. Gibbard [11] and Satterthwaite [14] (henceforth G–S) proved independently that any non-dictatorial voting choice procedure is manipulable whenever the set of alternatives contains at least three elements. Since the publication of this negative result, researchers have been attempting to avoid it by relaxing its original assumptions [1,17]. This is the case when individuals have some difficulties to express clearly their preferences over the set of alternatives but they can, however, specify a preference degree between 0 and 1 for each ordered pair of alternatives [2,9,10,18]. Therefore, the choice of a single alternative requires the use of a fuzzy social choice function (FSCF). This paper intends to confirm the negative G–S’s result even when individuals are not crisp in their preferences over the set of alternatives. We are considering in our contribution, in opposition to Tang [17], weak fuzzy preferences and three types of transitivity of the preference relations. To our best knowledge, this is the first result that deals with FSCFs on fuzzy weak preference domains. ∗ Corresponding author. LARODEC, Institut Supérieur de Gestion, University of Tunis, 41, Rue de la liberté, 2000 Le Bardo, Tunisia.
Tel.: +216 961 3 962766; fax: +216 916 1 750214. E-mail addresses: [email protected] (F. Ben Abdelaziz), fi[email protected] (J.R. Figueira), [email protected] (O. Meddeb). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.06.015
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One cannot introduce fuzzy manipulation without first defining how alternatives must be compared and how the social choice must be interpreted. The concept of best alternative set, as proposed by Dutta et al. [8], is reintroduced to provide three possible answers to our concerns. Consequently, three new definitions of fuzzy manipulability are proposed. In each case, the strategic manipulation of an FSCF by an individual is possible if two conditions are fulfilled. The first condition is that the sincere social choice does not belong to the best alternative set of the manipulator. The second one is that there exists a fuzzy relation securing the manipulator an outcome at least as good as the sincere social choice. Moreover, we define the concept of a dictator for an FSCF as the individual who secures the social choice in his best alternative set in all situations. Under these new definitions, we state our main result which is that G–S’s negative result remains true. The paper is organized as follows. Section 2 presents the main concepts, definitions, and notation. The fundamental concept of best alternative set is also introduced. Section 3 presents the different definitions of fuzzy manipulability and dictatorship. An illustrative example is provided in each of the above sections. Section 4 establishes the impossibility results regarding the strategy-proofness of FSCFs. The last section provides concluding remarks. 2. Concepts: definitions and notations We introduce, in this section, three fuzzy preference relations and their best alternative sets. Consider a finite set of alternatives, X = {x, y, z, . . .} and a finite set of individuals N = {1, 2, . . . , i, . . . , n} with |X|3. The social choice problem consists of finding the best alternative in X according to the preferences of all individuals in N. The best alternative is also called the social choice. It is assumed here that individuals have difficulties to express their preferences, i.e. fuzzy rather than crisp preferences should be considered. Fuzzy binary relations (FBRs) can thus, be introduced to model the vagueness or the fuzzy aspect of the preferences. These fuzzy relations can be defined as fuzzy sets on X 2 = X × X with a membership function, R. The formal definitions of crisp and FBRs and some fundamental properties are introduced next [9–11,13]. Definition 1 (Crisp binary relation). A crisp binary relation (CBR) on X is a function R : X 2 → {0, 1}. It is said to be, (i) connected, if for all x, y ∈ X, R(x, y) = 1 or R(y, x) = 1; (ii) transitive, i.e. for all x, y, z ∈ X, if R(x, y) = 1 and R(y, z) = 1, then R(x, z) = 1; (iii) anti-symmetric, i.e. for all x, y ∈ X, if R(x, y) = 1 and R(y, x) = 1, then x = y. Definition 2 (Fuzzy binary relation). A fuzzy binary relation (FBR) on X is a function R : X2 → [0, 1]. It is said to be, (i) (ii) (iii) (iv)
reflexive, if for all x ∈ X, R(x, x) = 1; connected, if for all x, y ∈ X, R(x, y) + R(y, x)1; max–min transitive, if for all x, y, z ∈ X, R(x, z) min{R(x, y), R(y, z)}; weak max–min transitive, if for all x, y, z ∈ X, [R(x, y) R(y, x) ∧ R(y, z) R(z, y)] ⇒ [R(x, z) min{R (x, y), R(y, z)}].
There are several versions of transitivity for fuzzy preference relations [3–6]. Definition 3 (Fuzzy weak preference relation). A fuzzy weak preference relation (FWPR) on X is a reflexive FBR. Let R be an FWPR. For all x, y ∈ X, R(x, y), is interpreted as the degree to which alternative x is “at least as good as” alternative y [12]. Definition 4 (Crisp linear order). Let R be a CBR. R is a crisp linear order (CLO) if it is connected, anti-symmetric, and transitive. Let Ri be a CLO with respect to individual i ∈ N . The best alternative set for individual i coincides with the most preferred element. It can be stated in the following manner: P0 (X, Ri ) = {x ∈ X | ∀ y ∈ X, Ri (x, y) = 1}.
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Unlike the case of a CLO, there are different ways to define “at least as good as” based on an FWPR [8]. Thus, “the alternative x is at least as good as alternative y”, if: (1) Simple degree rule (SDR) Ri (x, y)Ri (y, x). The best alternative set can be defined as follows: P1 (X, Ri ) = {x ∈ X | ∀ y ∈ X, Ri (x, y) Ri (y, x)}. (2) Confidence threshold rule (CTR) Ri (x, y)
where ∈]0, 1/2].
The best alternative set can be defined as follows: P2 (X, Ri ) = {x ∈ X | ∀ y ∈ X, Ri (x, y) }. (3) Dominance degree rule (DDR) d(Ri , X)(x)d(Ri , X)(y), where, d(Ri , X)(x) = minz∈X {Ri (x, z)}, is a dominance degree of the alternative x with respect to all the remaining alternatives in X. The best alternative set can be defined as follows: P3 (X, Ri ) = x ∈ X | d(Ri , X)(x) = max{d(Ri , X)(y)} . y∈X
Let us introduce the following notation: • • • • •
Dc is the set of all connected FWPRs. D0 is the set of all CLOs on X. D1 is the set of all FWPRs in Dc satisfying max–min transitivity when adopting SDR. D2 is the set of all FWPRs in Dc satisfying weak max–min transitivity when adopting CTR. D3 is the set of all FWPRs in D1 when adopting DDR.
Remark 1 (Dutta et al. [8], Peleg [13]). Since any CLO satisfies reflexivity and the max–min transitivity implies the weak max–min transitivity, we have D0 ⊂ D1 = D3 ⊂ D2 .
(1)
The following result on the composition of the best alternative sets can be stated [8,12,15,16]. Proposition 1. Let A be a non-empty subset of X and R be an FWPR. (i) If R ∈ D , then P (A, R) = ∅, for all ∈ {0, 1, 2, 3}. (ii) If R ∈ D0 , then P0 (A, R) = P1 (A, R) = P2 (A, R) = P3 (A, R). Throughout the paper, RN = (R1 , R2 , . . . , Ri , . . . , Rn ) is a profile of individuals’ preference relations. Example 1. Consider the following illustrative example with X = {a, b, c} and N = {1, 2, 3, 4, 5}. The relations, Ri , for i ∈ {1, 2, 3, 4, 5} belong to D1 and are presented in the following tables: R1
a
b
c
R2
a
b
c
R3
a
b
c
a b c
1 0.4 0.4
0.7 1 0.4
0.8 0.6 1
a b c
1 0.6 0.65
0.7 1 0.65
0.65 0.6 1
a b c
1 0.7 0.6
0.65 1 0.6
1 0.9 1
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R4
a
b
c
R5
a
b
c
a b c
1 0.65 1
0.7 1 0.9
0.6 0.6 1
a b c
1 0.7 0.65
0.6 1 0.6
0.64 0.65 1
For example, according to R5 , R5 (b, a) > R5 (a, b) and R5 (b, c) > R5 (c, b). Thus, P1 (X, R5 ) = {b}. When a confidence threshold = 0.45, P2 (X, R1 ) = {a}. For individual 4, the alternative c has the maximal dominance degree, d(R4 , X)(a) = d(R4 , X)(b) = 0.6 and d(R4 , X)(c) = 0.9. Thus, P3 (X, R4 ) = {c}. 3. Fuzzy social choice functions This section introduces fundamental definitions on FSCFs. Three types of FSCFs and dictatorship are presented along with the corresponding manipulation procedures. Definition 5 (Fuzzy social choice function). An FSCF is a function that associates a single alternative (in X) to a profile of individuals’ preference relations. Definition 6 (-Fuzzy social choice function). Let ∈ {0, 1, 2, 3}. An -FSCF is an FSCF such that its domain is Dn . In other words, when each individual expresses an FWPR in D , an -FSCF defines a social choice. It should be noticed that a 0-FSCF can be viewed as a voting procedure. Example 2. Reconsider the previous example. Each individual’s FWPR, Ri , i ∈ {1, 2, 3, 4, 5}, belongs to D , for ∈ {1, 2, 3}. Apply the arithmetic mean function as a fuzzy social welfare function to obtain the fuzzy social relation R s . Then, consider P1 (X, R s ) = {a} as the social choice. Such an FSCF can be viewed as an -FSCF, for ∈ {1, 2, 3}. Rs
a
b
c
a b c
1 0.61 0.66
0.67 1 0.63
0.73 0.67 1
Now, consider the following notation: • • • •
(RN | Ri ) is the profile of individuals’ preference relations (R1 , . . . , Ri−1 , Ri , Ri+1 , . . . , Rn ). ) is the profile of individuals’ preference relations (R , . . . , R (RN | Ri , Ri+1 1 i−1 , Ri , Ri+1 , . . . , Rn ). (RN | R1 , . . . , Rk ) for k ∈ {1, . . . , n}, is the profile of individuals’ preference relations (R1 , . . . , Rk , Rk+1 , . . . , Rn ). R N is the profile of individuals’ preference relations (R1 , . . . , Ri , . . . , Rn ).
Consider the context where a profile of individuals’ preference relations has to be expressed in Dn , and the social choice is obtained by the use of an -FSCF, , for ∈ {0, 1, 2, 3}. Recall that when each individual i ∈ N expresses a preference relation, Ri , that belongs to D , he considers P (X, Ri ) as his best alternative set. Suppose that an individual m ∈ N, with a sincere preference relation, Rm , comes to know the (n−1) preference relations declared by the remaining individuals and the -FSCF, . Therefore, he can anticipate the outcome (RN ). The question to be asked by the individual m is whether or not the outcome (RN ) can be considered as a good social choice on the basis of his sincere preference relation, Rm . The answer to this question is to check if the outcome (RN ) belongs to the m’s best alternative set P (X, Rm ). In the case where the outcome (RN ) does not belong to the set P (X, Rm ), the individual m seeks for a social choice at least good as the outcome (RN ) on the basis of his preference relation, Rm . If there exists , in D , such that the outcome (R | R ) is better than the outcome (R ), the individual m a binary relation, Rm N N m . Consequently, the -manipulability of an -FSCF can formally be can manipulate the -FSCF, , by revealing Rm introduced as follows.
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Definition 7 (-Manipulability). Let be an -FSCF for ∈ {0, 1, 2, 3}. ∈ D such that (R | (i) The function 0 is said to be 0-manipulable if there exists m ∈ N , RN ∈ D0n , and Rm 0 0 N Rm ) = 0 (RN ) and Rm (0 (RN | Rm ), 0 (RN )) = 1. ∈ D such that there (ii) The function 1 is said to be 1-manipulable if there exists m ∈ N , RN ∈ D1n , and Rm 1 exists x ∈ X, such that Rm (x, 1 (RN )) > Rm (1 (RN ), x), 1 (RN | Rm ) = 1 (RN ), and Rm (1 (RN | ), (R ))R ( (R ), (R | R )). Rm m 1 N 1 N 1 N m ∈ D such that for a (iii) The function 2 is said to be 2-manipulable if there exists m ∈ N , RN ∈ D2n , and Rm 2 fixed ∈]0, 1/2], there exists x ∈ X, such that Rm (2 (RN ), x) < , 2 (RN | Rm ) = 2 (RN ) and Rm (2 (RN | ), (R )). Rm 2 N ∈ D there exists (iv) The function 3 is said to be 3-manipulable if there exists m ∈ N , RN ∈ D3n , and Rm 3 x ∈ X, such that d(Rm , X)(x) > d(Rm , X) (3 (RN )), 3 (RN | Rm ) = 3 (RN ), and d(Rm , X)(3 (RN | ))d(R , X)( (R )). Rm N m 3
Remark 2. The 0-manipulability of a 0-FSCF coincides with the one of G–S. In addition, for ∈ {1, 2, 3}, the -manipulability of an -FSCF implies its 0-manipulability when its domain is restricted to the set D0n . Definition 8 (-Strategy-proofness). Let be an -FSCF for ∈ {0, 1, 2, 3}. The function is said to be -strategyproof, if is not -manipulable. Example 3. Reconsider the previous example. (1) Individual 5 can 1-manipulate the FSCF. Indeed, the alternative a does not belong to his best alternative set P1 (X, R5 ) = {b}. Therefore, he can reveal the non-sincere fuzzy relation R5 to obtain b as the social choice. R5
a
b
c
a b c
1 0.95 0.9
0 1 0
0 1 1
(2) Let = 0.45 be a confidence threshold for all individuals. Alternative a belongs to every best alternative set P2 (X, Ri ), for all i ∈ {1, 2, 3, 4, 5}. Thus, no individual can 2-manipulate the FSCF in this situation. (3) Individual 4 can 3-manipulate the FSCF. Indeed, alternative a does not belong to his best alternative set P3 (X, R4 ) = {c}. Therefore, he can reveal the non-sincere fuzzy relation R4 to obtain c as the social choice. R4
a
b
c
a b c
1 0.9 0.95
0 1 1
0 0 1
In the following definition we introduce and generalize the G–S’s dictatorship [11,14] to the fuzzy context. For ∈ {0, 1, 2, 3}, an individual d ∈ N , having a preference relation, Rd in D , is considered to be an -dictator for the FSCF, , if the outcome (RN ) belongs to his best alternative set P (X, Rd ) for any RN = (R1 , R2 , . . . , Rd , . . . , Rn ). If an -dictator exists for an -FSCF, , then is considered to be -dictatorial. Consequently, the -dictatorship of an -FSCF can formally be introduced as follows. Definition 9 (-Dictatorship). Let be an -FSCF for ∈ {0, 1, 2, 3}. (i) The function 0 is said to be 0-dictatorial if there exists d ∈ N such that for every RN ∈ D0n , ∀x ∈ X, Rd (0 (RN ), x) = 1. (ii) The function 1 is said to be 1-dictatorial if there exists d ∈ N such that for every RN ∈ D1n , ∀x ∈ X, Rd (1 (RN ), x)Rd (x, 1 (RN )).
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(iii) The function 2 is said to be 2-dictatorial if there exists d ∈ N such that for every RN ∈ D2n , for a fixed ∈]0, 1/2], ∀x ∈ X, Rd (2 (RN ), x) . (iv) The function 3 is said to be 3-dictatorial if there exists d ∈ N such that for every RN ∈ D3n , ∀x ∈ X, d(Rd , X)(3 (RN ))d(Rd , X)(x). Remark 3. The 0-dictatorship of a 0-FSCF coincides with the G–S’s dictatorship. In addition, for ∈ {1, 2, 3}, the -dictatorship of an -FSCF implies its 0-dictatorship when its domain is restricted to the set D0n . 4. Impossibility results This section presents the strategy-proofness theorem of G–S and generalizes it to the -FSCFs for ∈ {1, 2, 3}. Theorem 1 (Gibbard [11], Satterthwaith [14]). Let 0 : D0n → X be a 0-FSCF such that 0 (D0n ) = X. If 0 is 0-strategy-proof, then it is 0-dictatorial. The following lemma will be useful for the proof of our main theorem. Lemma 1. Let A be a non-empty subset of X and R be a CBR on X. If R satisfies the following conditions: (i) R is a CLO on A; (ii) R(x, y) = 1, and R(y, x) = 0, for all x ∈ A, y ∈ X\A; (iii) R is a CLO on X\A; then R is a CLO on X and P0 (X, R) ⊆ A. R is a CLO on X, extended from A in such a way that the elements of A are preferred to the elements in X\A. Proof. • R is a CLO if it fulfills: ◦ Reflexivity (i) ⇒ for all x ∈ A, R(x, x) = 1, (iii) ⇒ for all x ∈ X\A, R(x, x) = 1. Thus, for all x ∈ X, R(x, x) = 1. ◦ Connectedness (i) ⇒ for all x, y ∈ A, R(x, y) + R(y, x)1, (ii) ⇒ for all x ∈ A, y ∈ X\A, R(x, y) + R(y, x) 1, (iii) ⇒ for all x, y ∈ X\A, R(x, y) + R(y, x)1. Thus, for all x, y ∈ X, R(x, y) + R(y, x)1. ◦ Transitivity: Let x, y, z ∈ X, such that R(x, y) = 1 and R(y, z) = 1 and show that R(x, z) = 1. Assume that z ∈ A and x ∈ X\A. If y ∈ A, then R(x, y) = 0. This contradicts the equality R(x, y) = 1. If y ∈ X\A, then R(y, z) = 0. This contradicts the equality R(y, z) = 1. Therefore, we have either x, y ∈ A, or x, z ∈ X\A or (x ∈ A and z ∈ X\A). In each of three cases, we have R(x, z) = 1. • Let x ∈ P0 (X, R). Suppose that x does not belong to A, i.e. x ∈ X\A. Thus, there exists y ∈ A, such that R(x, y) = 0. This leads to a contradiction with the assumption that x ∈ P0 (X, R). Thus, P0 (X, R) is included in A. In the following we show that the G–S’s result can be extended to the fuzzy context. We establish that any -strategyproof -FSCF is -dictatorial. The proof follows the main steps as in [17]. Theorem 2. Let be an -FSCF such that (D0n ) = X and ∈ {1, 2, 3}. If is -strategy-proof, then it is -dictatorial. Proof. Here, we detail only the proof for = 1 since it can be performed in almost the same way for each ∈ {1, 2, 3}.
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Consider a 1-strategy-proof 1-FSCF, 1 : D1n → X. Let 0 : D0n → X be a 0-FSCF such that for all RN ∈ D0n , 0 (RN ) = 1 (RN ). The 0-FSCF is 0-strategy-proof because of the 1-strategy-proofness of 1 . Thus, according to Theorem 1, 0 is 0-dictatorial. Let individual 1 be the 0-dictator for 0 . In the remainder we show that individual 1 is also 1-dictator for 1 . Let RN be any profile of individuals’ preference relations in D1n . Let x0 be 1 (RN ). Consider P1 (X, R1 ) = {x ∈ X | ∀ y ∈ X, R1 (x, y)R1 (y, x)} be the best alternative set of individual 1. Next, it will be proved that x0 belongs to P1 (X, R1 ). Let R N ∈ D0n be a profile of individuals’ CLOs such that: • For i = 1 ⎧ ⎨ Ri is a CLO on P1 (X, R1 ), R (x, y) = 1 and Ri (y, x) = 0 ⎩ i Ri is a CLO on X\P1 (X, R1 ).
if x ∈ P1 (X, R1 ), y ∈ X\P1 (X, R1 ),
• For all i = 1 ⎧ ⎨ Ri is a CLO on P1 (X, R1 ), R (y, x) = 1 and Ri (x, y) = 0 ⎩ i Ri is a CLO on X\P1 (X, R1 ).
if x ∈ P1 (X, R1 ), y ∈ X\P1 (X, R1 ),
According to Lemma 1, Ri is a CLO for all i ∈ N and P0 (X, R1 ) ⊆ P1 (X, R1 ). Suppose that xk = 1 (RN | R1 , R2 , . . . , Ri , . . . , Rk ) is the social choice when the k first individuals change their preference relations Ri into Ri in order to contradict individual 1. Note that k ∈ {0, 1, . . . , n}. If k = n, then xn = 1 (R N ) = 0 (R N ). Thus, xn belongs to P0 (X, R1 ) because of the dictatorship of 1 by 0 . Therefore, xn belongs to P1 (X, R1 ). Now, suppose that j denotes the least k in {0, 1, . . . , i, . . . , n} such that xk ∈ P1 (X, R1 ). To have x0 in P1 (X, R1 ), it is needed to show that j = 0. The proof is made by contradiction. Suppose that j 1. • If j = 1, then x1 = 1 (RN | R1 ) ∈ P1 (X, R1 ),
(2)
/ P1 (X, R1 ). x0 = 1 (RN ) ∈
(3)
(2) ⇒ ∀ y ∈ X, R1 (x1 , y)R1 (y, x1 ). Thus, R1 (x1 , x0 ) R1 (x0 , x1 ). (3) ⇒ There exists x ∈ X such that R1 (x, x0 ) > R1 (x0 , x). Consequently, 1 is 1-manipulable by individual 1 at RN . • If j > 1, then xj = 1 (RN | R1 , R2 , . . . , Rj ) ∈ P1 (X, R1 ), xj −1 = 1 (RN | R1 , . . . , Ri , . . . , Rj −1 ) ∈ / P1 (X, R1 ). Therefore, Rj (xj −1 , xj ) = 1, and Rj (xj , xj −1 ) = 0. Thus, Rj (xj −1 , xj ) > Rj (xj , xj −1 ). Consider the situation where (RN | R1 , R2 , . . . , Rj ) is the profile of individuals’ preference relations. If individual j declares a fuzzy preference Rj instead of a crisp relation Rj , then 1 (RN | R1 , R2 , . . . , Rj ) is changed in his favor. Consequently, 1 is 1-manipulable by individual j at (RN | R1 , R2 , . . . , Rj ). We conclude that j must be equal to 0. Thus, x0 = (RN ) ∈ P1 (X, R1 ), for any RN ∈ D1n . It follows that individual 1 is also a 1-dictator for 1 . 5. Conclusion This paper generalizes to the fuzzy context the well-known result of G–S on the manipulability of the crisp social choice function. The paper shows how an individual could manipulate a social choice even if the preferences of the
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individuals are ambiguous. Depending on the type of transitivity of the fuzzy preference relations, three definitions of strategy-proofness have been proposed. It has been proved that for all the three cases, the G–S’s theorem remains valid. Further researches should be carried out to examine the validity of the G–S’s theorem under other domains of FSCFs and to determine the minimum requirements for fuzzy strategy-proofness. Acknowledgments This work has benefited from the Luso–Tunisian bilateral cooperation (2004–2007). The authors are very grateful to the anonymous referees for their valuable comments and suggestions that helped to improve the quality of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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On the manipulability of the fuzzy social choice functions Fouad Ben Abdelaziza, c,∗ , José Rui Figueirab , Olfa Meddeba a LARODEC, Institut Supérieur de Gestion, University of Tunis, 41, Rue de la liberté, 2000 Le Bardo, Tunisia b CEG-IST, Center for Management Studies, Instituto Superior Técnico, Technical University of Lisbon, Tagus Park, Av. Cavaco Silva, 2780-990
Porto Salvo, Portugal c Visiting School of Engineering, American University of Sharjah, P.O. Box 26666, Sharjah-UAE
Received 23 March 2006; received in revised form 19 June 2007; accepted 19 June 2007 Available online 3 July 2007
Abstract In many social decision-making contexts, a manipulator has incentives to change the social choice in his favor by strategically misrepresenting his preference. Gibbard [Manipulation of voting schemes: a general result, Econometrica 41(4) (1973) 587–601] and Satterthwaite [Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J. Econom. Theory 10 (1975) 187–217] have shown that any non-dictatorial voting choice procedure is vulnerable to strategic manipulation. This paper extends their result to the case of fuzzy weak preference relations. For this purpose, the best alternative set is defined in three ways and consequently three generalizations of the Gibbard–Satterthwaite theorem to the fuzzy context are provided. © 2007 Elsevier B.V. All rights reserved. Keywords: Fuzzy preference relations; Fuzzy social choice functions; Fuzzy strategy-proofness
1. Introduction A voting choice procedure is known to be subject to strategic manipulation when an individual reveals a non-sincere preference relation in order to change the social choice in his favor. Gibbard [11] and Satterthwaite [14] (henceforth G–S) proved independently that any non-dictatorial voting choice procedure is manipulable whenever the set of alternatives contains at least three elements. Since the publication of this negative result, researchers have been attempting to avoid it by relaxing its original assumptions [1,17]. This is the case when individuals have some difficulties to express clearly their preferences over the set of alternatives but they can, however, specify a preference degree between 0 and 1 for each ordered pair of alternatives [2,9,10,18]. Therefore, the choice of a single alternative requires the use of a fuzzy social choice function (FSCF). This paper intends to confirm the negative G–S’s result even when individuals are not crisp in their preferences over the set of alternatives. We are considering in our contribution, in opposition to Tang [17], weak fuzzy preferences and three types of transitivity of the preference relations. To our best knowledge, this is the first result that deals with FSCFs on fuzzy weak preference domains. ∗ Corresponding author. LARODEC, Institut Supérieur de Gestion, University of Tunis, 41, Rue de la liberté, 2000 Le Bardo, Tunisia.
Tel.: +216 961 3 962766; fax: +216 916 1 750214. E-mail addresses: [email protected] (F. Ben Abdelaziz), fi[email protected] (J.R. Figueira), [email protected] (O. Meddeb). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.06.015
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One cannot introduce fuzzy manipulation without first defining how alternatives must be compared and how the social choice must be interpreted. The concept of best alternative set, as proposed by Dutta et al. [8], is reintroduced to provide three possible answers to our concerns. Consequently, three new definitions of fuzzy manipulability are proposed. In each case, the strategic manipulation of an FSCF by an individual is possible if two conditions are fulfilled. The first condition is that the sincere social choice does not belong to the best alternative set of the manipulator. The second one is that there exists a fuzzy relation securing the manipulator an outcome at least as good as the sincere social choice. Moreover, we define the concept of a dictator for an FSCF as the individual who secures the social choice in his best alternative set in all situations. Under these new definitions, we state our main result which is that G–S’s negative result remains true. The paper is organized as follows. Section 2 presents the main concepts, definitions, and notation. The fundamental concept of best alternative set is also introduced. Section 3 presents the different definitions of fuzzy manipulability and dictatorship. An illustrative example is provided in each of the above sections. Section 4 establishes the impossibility results regarding the strategy-proofness of FSCFs. The last section provides concluding remarks. 2. Concepts: definitions and notations We introduce, in this section, three fuzzy preference relations and their best alternative sets. Consider a finite set of alternatives, X = {x, y, z, . . .} and a finite set of individuals N = {1, 2, . . . , i, . . . , n} with |X|3. The social choice problem consists of finding the best alternative in X according to the preferences of all individuals in N. The best alternative is also called the social choice. It is assumed here that individuals have difficulties to express their preferences, i.e. fuzzy rather than crisp preferences should be considered. Fuzzy binary relations (FBRs) can thus, be introduced to model the vagueness or the fuzzy aspect of the preferences. These fuzzy relations can be defined as fuzzy sets on X 2 = X × X with a membership function, R. The formal definitions of crisp and FBRs and some fundamental properties are introduced next [9–11,13]. Definition 1 (Crisp binary relation). A crisp binary relation (CBR) on X is a function R : X 2 → {0, 1}. It is said to be, (i) connected, if for all x, y ∈ X, R(x, y) = 1 or R(y, x) = 1; (ii) transitive, i.e. for all x, y, z ∈ X, if R(x, y) = 1 and R(y, z) = 1, then R(x, z) = 1; (iii) anti-symmetric, i.e. for all x, y ∈ X, if R(x, y) = 1 and R(y, x) = 1, then x = y. Definition 2 (Fuzzy binary relation). A fuzzy binary relation (FBR) on X is a function R : X2 → [0, 1]. It is said to be, (i) (ii) (iii) (iv)
reflexive, if for all x ∈ X, R(x, x) = 1; connected, if for all x, y ∈ X, R(x, y) + R(y, x)1; max–min transitive, if for all x, y, z ∈ X, R(x, z) min{R(x, y), R(y, z)}; weak max–min transitive, if for all x, y, z ∈ X, [R(x, y) R(y, x) ∧ R(y, z) R(z, y)] ⇒ [R(x, z) min{R (x, y), R(y, z)}].
There are several versions of transitivity for fuzzy preference relations [3–6]. Definition 3 (Fuzzy weak preference relation). A fuzzy weak preference relation (FWPR) on X is a reflexive FBR. Let R be an FWPR. For all x, y ∈ X, R(x, y), is interpreted as the degree to which alternative x is “at least as good as” alternative y [12]. Definition 4 (Crisp linear order). Let R be a CBR. R is a crisp linear order (CLO) if it is connected, anti-symmetric, and transitive. Let Ri be a CLO with respect to individual i ∈ N . The best alternative set for individual i coincides with the most preferred element. It can be stated in the following manner: P0 (X, Ri ) = {x ∈ X | ∀ y ∈ X, Ri (x, y) = 1}.
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Unlike the case of a CLO, there are different ways to define “at least as good as” based on an FWPR [8]. Thus, “the alternative x is at least as good as alternative y”, if: (1) Simple degree rule (SDR) Ri (x, y)Ri (y, x). The best alternative set can be defined as follows: P1 (X, Ri ) = {x ∈ X | ∀ y ∈ X, Ri (x, y) Ri (y, x)}. (2) Confidence threshold rule (CTR) Ri (x, y)
where ∈]0, 1/2].
The best alternative set can be defined as follows: P2 (X, Ri ) = {x ∈ X | ∀ y ∈ X, Ri (x, y) }. (3) Dominance degree rule (DDR) d(Ri , X)(x)d(Ri , X)(y), where, d(Ri , X)(x) = minz∈X {Ri (x, z)}, is a dominance degree of the alternative x with respect to all the remaining alternatives in X. The best alternative set can be defined as follows: P3 (X, Ri ) = x ∈ X | d(Ri , X)(x) = max{d(Ri , X)(y)} . y∈X
Let us introduce the following notation: • • • • •
Dc is the set of all connected FWPRs. D0 is the set of all CLOs on X. D1 is the set of all FWPRs in Dc satisfying max–min transitivity when adopting SDR. D2 is the set of all FWPRs in Dc satisfying weak max–min transitivity when adopting CTR. D3 is the set of all FWPRs in D1 when adopting DDR.
Remark 1 (Dutta et al. [8], Peleg [13]). Since any CLO satisfies reflexivity and the max–min transitivity implies the weak max–min transitivity, we have D0 ⊂ D1 = D3 ⊂ D2 .
(1)
The following result on the composition of the best alternative sets can be stated [8,12,15,16]. Proposition 1. Let A be a non-empty subset of X and R be an FWPR. (i) If R ∈ D , then P (A, R) = ∅, for all ∈ {0, 1, 2, 3}. (ii) If R ∈ D0 , then P0 (A, R) = P1 (A, R) = P2 (A, R) = P3 (A, R). Throughout the paper, RN = (R1 , R2 , . . . , Ri , . . . , Rn ) is a profile of individuals’ preference relations. Example 1. Consider the following illustrative example with X = {a, b, c} and N = {1, 2, 3, 4, 5}. The relations, Ri , for i ∈ {1, 2, 3, 4, 5} belong to D1 and are presented in the following tables: R1
a
b
c
R2
a
b
c
R3
a
b
c
a b c
1 0.4 0.4
0.7 1 0.4
0.8 0.6 1
a b c
1 0.6 0.65
0.7 1 0.65
0.65 0.6 1
a b c
1 0.7 0.6
0.65 1 0.6
1 0.9 1
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R4
a
b
c
R5
a
b
c
a b c
1 0.65 1
0.7 1 0.9
0.6 0.6 1
a b c
1 0.7 0.65
0.6 1 0.6
0.64 0.65 1
For example, according to R5 , R5 (b, a) > R5 (a, b) and R5 (b, c) > R5 (c, b). Thus, P1 (X, R5 ) = {b}. When a confidence threshold = 0.45, P2 (X, R1 ) = {a}. For individual 4, the alternative c has the maximal dominance degree, d(R4 , X)(a) = d(R4 , X)(b) = 0.6 and d(R4 , X)(c) = 0.9. Thus, P3 (X, R4 ) = {c}. 3. Fuzzy social choice functions This section introduces fundamental definitions on FSCFs. Three types of FSCFs and dictatorship are presented along with the corresponding manipulation procedures. Definition 5 (Fuzzy social choice function). An FSCF is a function that associates a single alternative (in X) to a profile of individuals’ preference relations. Definition 6 (-Fuzzy social choice function). Let ∈ {0, 1, 2, 3}. An -FSCF is an FSCF such that its domain is Dn . In other words, when each individual expresses an FWPR in D , an -FSCF defines a social choice. It should be noticed that a 0-FSCF can be viewed as a voting procedure. Example 2. Reconsider the previous example. Each individual’s FWPR, Ri , i ∈ {1, 2, 3, 4, 5}, belongs to D , for ∈ {1, 2, 3}. Apply the arithmetic mean function as a fuzzy social welfare function to obtain the fuzzy social relation R s . Then, consider P1 (X, R s ) = {a} as the social choice. Such an FSCF can be viewed as an -FSCF, for ∈ {1, 2, 3}. Rs
a
b
c
a b c
1 0.61 0.66
0.67 1 0.63
0.73 0.67 1
Now, consider the following notation: • • • •
(RN | Ri ) is the profile of individuals’ preference relations (R1 , . . . , Ri−1 , Ri , Ri+1 , . . . , Rn ). ) is the profile of individuals’ preference relations (R , . . . , R (RN | Ri , Ri+1 1 i−1 , Ri , Ri+1 , . . . , Rn ). (RN | R1 , . . . , Rk ) for k ∈ {1, . . . , n}, is the profile of individuals’ preference relations (R1 , . . . , Rk , Rk+1 , . . . , Rn ). R N is the profile of individuals’ preference relations (R1 , . . . , Ri , . . . , Rn ).
Consider the context where a profile of individuals’ preference relations has to be expressed in Dn , and the social choice is obtained by the use of an -FSCF, , for ∈ {0, 1, 2, 3}. Recall that when each individual i ∈ N expresses a preference relation, Ri , that belongs to D , he considers P (X, Ri ) as his best alternative set. Suppose that an individual m ∈ N, with a sincere preference relation, Rm , comes to know the (n−1) preference relations declared by the remaining individuals and the -FSCF, . Therefore, he can anticipate the outcome (RN ). The question to be asked by the individual m is whether or not the outcome (RN ) can be considered as a good social choice on the basis of his sincere preference relation, Rm . The answer to this question is to check if the outcome (RN ) belongs to the m’s best alternative set P (X, Rm ). In the case where the outcome (RN ) does not belong to the set P (X, Rm ), the individual m seeks for a social choice at least good as the outcome (RN ) on the basis of his preference relation, Rm . If there exists , in D , such that the outcome (R | R ) is better than the outcome (R ), the individual m a binary relation, Rm N N m . Consequently, the -manipulability of an -FSCF can formally be can manipulate the -FSCF, , by revealing Rm introduced as follows.
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Definition 7 (-Manipulability). Let be an -FSCF for ∈ {0, 1, 2, 3}. ∈ D such that (R | (i) The function 0 is said to be 0-manipulable if there exists m ∈ N , RN ∈ D0n , and Rm 0 0 N Rm ) = 0 (RN ) and Rm (0 (RN | Rm ), 0 (RN )) = 1. ∈ D such that there (ii) The function 1 is said to be 1-manipulable if there exists m ∈ N , RN ∈ D1n , and Rm 1 exists x ∈ X, such that Rm (x, 1 (RN )) > Rm (1 (RN ), x), 1 (RN | Rm ) = 1 (RN ), and Rm (1 (RN | ), (R ))R ( (R ), (R | R )). Rm m 1 N 1 N 1 N m ∈ D such that for a (iii) The function 2 is said to be 2-manipulable if there exists m ∈ N , RN ∈ D2n , and Rm 2 fixed ∈]0, 1/2], there exists x ∈ X, such that Rm (2 (RN ), x) < , 2 (RN | Rm ) = 2 (RN ) and Rm (2 (RN | ), (R )). Rm 2 N ∈ D there exists (iv) The function 3 is said to be 3-manipulable if there exists m ∈ N , RN ∈ D3n , and Rm 3 x ∈ X, such that d(Rm , X)(x) > d(Rm , X) (3 (RN )), 3 (RN | Rm ) = 3 (RN ), and d(Rm , X)(3 (RN | ))d(R , X)( (R )). Rm N m 3
Remark 2. The 0-manipulability of a 0-FSCF coincides with the one of G–S. In addition, for ∈ {1, 2, 3}, the -manipulability of an -FSCF implies its 0-manipulability when its domain is restricted to the set D0n . Definition 8 (-Strategy-proofness). Let be an -FSCF for ∈ {0, 1, 2, 3}. The function is said to be -strategyproof, if is not -manipulable. Example 3. Reconsider the previous example. (1) Individual 5 can 1-manipulate the FSCF. Indeed, the alternative a does not belong to his best alternative set P1 (X, R5 ) = {b}. Therefore, he can reveal the non-sincere fuzzy relation R5 to obtain b as the social choice. R5
a
b
c
a b c
1 0.95 0.9
0 1 0
0 1 1
(2) Let = 0.45 be a confidence threshold for all individuals. Alternative a belongs to every best alternative set P2 (X, Ri ), for all i ∈ {1, 2, 3, 4, 5}. Thus, no individual can 2-manipulate the FSCF in this situation. (3) Individual 4 can 3-manipulate the FSCF. Indeed, alternative a does not belong to his best alternative set P3 (X, R4 ) = {c}. Therefore, he can reveal the non-sincere fuzzy relation R4 to obtain c as the social choice. R4
a
b
c
a b c
1 0.9 0.95
0 1 1
0 0 1
In the following definition we introduce and generalize the G–S’s dictatorship [11,14] to the fuzzy context. For ∈ {0, 1, 2, 3}, an individual d ∈ N , having a preference relation, Rd in D , is considered to be an -dictator for the FSCF, , if the outcome (RN ) belongs to his best alternative set P (X, Rd ) for any RN = (R1 , R2 , . . . , Rd , . . . , Rn ). If an -dictator exists for an -FSCF, , then is considered to be -dictatorial. Consequently, the -dictatorship of an -FSCF can formally be introduced as follows. Definition 9 (-Dictatorship). Let be an -FSCF for ∈ {0, 1, 2, 3}. (i) The function 0 is said to be 0-dictatorial if there exists d ∈ N such that for every RN ∈ D0n , ∀x ∈ X, Rd (0 (RN ), x) = 1. (ii) The function 1 is said to be 1-dictatorial if there exists d ∈ N such that for every RN ∈ D1n , ∀x ∈ X, Rd (1 (RN ), x)Rd (x, 1 (RN )).
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(iii) The function 2 is said to be 2-dictatorial if there exists d ∈ N such that for every RN ∈ D2n , for a fixed ∈]0, 1/2], ∀x ∈ X, Rd (2 (RN ), x) . (iv) The function 3 is said to be 3-dictatorial if there exists d ∈ N such that for every RN ∈ D3n , ∀x ∈ X, d(Rd , X)(3 (RN ))d(Rd , X)(x). Remark 3. The 0-dictatorship of a 0-FSCF coincides with the G–S’s dictatorship. In addition, for ∈ {1, 2, 3}, the -dictatorship of an -FSCF implies its 0-dictatorship when its domain is restricted to the set D0n . 4. Impossibility results This section presents the strategy-proofness theorem of G–S and generalizes it to the -FSCFs for ∈ {1, 2, 3}. Theorem 1 (Gibbard [11], Satterthwaith [14]). Let 0 : D0n → X be a 0-FSCF such that 0 (D0n ) = X. If 0 is 0-strategy-proof, then it is 0-dictatorial. The following lemma will be useful for the proof of our main theorem. Lemma 1. Let A be a non-empty subset of X and R be a CBR on X. If R satisfies the following conditions: (i) R is a CLO on A; (ii) R(x, y) = 1, and R(y, x) = 0, for all x ∈ A, y ∈ X\A; (iii) R is a CLO on X\A; then R is a CLO on X and P0 (X, R) ⊆ A. R is a CLO on X, extended from A in such a way that the elements of A are preferred to the elements in X\A. Proof. • R is a CLO if it fulfills: ◦ Reflexivity (i) ⇒ for all x ∈ A, R(x, x) = 1, (iii) ⇒ for all x ∈ X\A, R(x, x) = 1. Thus, for all x ∈ X, R(x, x) = 1. ◦ Connectedness (i) ⇒ for all x, y ∈ A, R(x, y) + R(y, x)1, (ii) ⇒ for all x ∈ A, y ∈ X\A, R(x, y) + R(y, x) 1, (iii) ⇒ for all x, y ∈ X\A, R(x, y) + R(y, x)1. Thus, for all x, y ∈ X, R(x, y) + R(y, x)1. ◦ Transitivity: Let x, y, z ∈ X, such that R(x, y) = 1 and R(y, z) = 1 and show that R(x, z) = 1. Assume that z ∈ A and x ∈ X\A. If y ∈ A, then R(x, y) = 0. This contradicts the equality R(x, y) = 1. If y ∈ X\A, then R(y, z) = 0. This contradicts the equality R(y, z) = 1. Therefore, we have either x, y ∈ A, or x, z ∈ X\A or (x ∈ A and z ∈ X\A). In each of three cases, we have R(x, z) = 1. • Let x ∈ P0 (X, R). Suppose that x does not belong to A, i.e. x ∈ X\A. Thus, there exists y ∈ A, such that R(x, y) = 0. This leads to a contradiction with the assumption that x ∈ P0 (X, R). Thus, P0 (X, R) is included in A. In the following we show that the G–S’s result can be extended to the fuzzy context. We establish that any -strategyproof -FSCF is -dictatorial. The proof follows the main steps as in [17]. Theorem 2. Let be an -FSCF such that (D0n ) = X and ∈ {1, 2, 3}. If is -strategy-proof, then it is -dictatorial. Proof. Here, we detail only the proof for = 1 since it can be performed in almost the same way for each ∈ {1, 2, 3}.
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Consider a 1-strategy-proof 1-FSCF, 1 : D1n → X. Let 0 : D0n → X be a 0-FSCF such that for all RN ∈ D0n , 0 (RN ) = 1 (RN ). The 0-FSCF is 0-strategy-proof because of the 1-strategy-proofness of 1 . Thus, according to Theorem 1, 0 is 0-dictatorial. Let individual 1 be the 0-dictator for 0 . In the remainder we show that individual 1 is also 1-dictator for 1 . Let RN be any profile of individuals’ preference relations in D1n . Let x0 be 1 (RN ). Consider P1 (X, R1 ) = {x ∈ X | ∀ y ∈ X, R1 (x, y)R1 (y, x)} be the best alternative set of individual 1. Next, it will be proved that x0 belongs to P1 (X, R1 ). Let R N ∈ D0n be a profile of individuals’ CLOs such that: • For i = 1 ⎧ ⎨ Ri is a CLO on P1 (X, R1 ), R (x, y) = 1 and Ri (y, x) = 0 ⎩ i Ri is a CLO on X\P1 (X, R1 ).
if x ∈ P1 (X, R1 ), y ∈ X\P1 (X, R1 ),
• For all i = 1 ⎧ ⎨ Ri is a CLO on P1 (X, R1 ), R (y, x) = 1 and Ri (x, y) = 0 ⎩ i Ri is a CLO on X\P1 (X, R1 ).
if x ∈ P1 (X, R1 ), y ∈ X\P1 (X, R1 ),
According to Lemma 1, Ri is a CLO for all i ∈ N and P0 (X, R1 ) ⊆ P1 (X, R1 ). Suppose that xk = 1 (RN | R1 , R2 , . . . , Ri , . . . , Rk ) is the social choice when the k first individuals change their preference relations Ri into Ri in order to contradict individual 1. Note that k ∈ {0, 1, . . . , n}. If k = n, then xn = 1 (R N ) = 0 (R N ). Thus, xn belongs to P0 (X, R1 ) because of the dictatorship of 1 by 0 . Therefore, xn belongs to P1 (X, R1 ). Now, suppose that j denotes the least k in {0, 1, . . . , i, . . . , n} such that xk ∈ P1 (X, R1 ). To have x0 in P1 (X, R1 ), it is needed to show that j = 0. The proof is made by contradiction. Suppose that j 1. • If j = 1, then x1 = 1 (RN | R1 ) ∈ P1 (X, R1 ),
(2)
/ P1 (X, R1 ). x0 = 1 (RN ) ∈
(3)
(2) ⇒ ∀ y ∈ X, R1 (x1 , y)R1 (y, x1 ). Thus, R1 (x1 , x0 ) R1 (x0 , x1 ). (3) ⇒ There exists x ∈ X such that R1 (x, x0 ) > R1 (x0 , x). Consequently, 1 is 1-manipulable by individual 1 at RN . • If j > 1, then xj = 1 (RN | R1 , R2 , . . . , Rj ) ∈ P1 (X, R1 ), xj −1 = 1 (RN | R1 , . . . , Ri , . . . , Rj −1 ) ∈ / P1 (X, R1 ). Therefore, Rj (xj −1 , xj ) = 1, and Rj (xj , xj −1 ) = 0. Thus, Rj (xj −1 , xj ) > Rj (xj , xj −1 ). Consider the situation where (RN | R1 , R2 , . . . , Rj ) is the profile of individuals’ preference relations. If individual j declares a fuzzy preference Rj instead of a crisp relation Rj , then 1 (RN | R1 , R2 , . . . , Rj ) is changed in his favor. Consequently, 1 is 1-manipulable by individual j at (RN | R1 , R2 , . . . , Rj ). We conclude that j must be equal to 0. Thus, x0 = (RN ) ∈ P1 (X, R1 ), for any RN ∈ D1n . It follows that individual 1 is also a 1-dictator for 1 . 5. Conclusion This paper generalizes to the fuzzy context the well-known result of G–S on the manipulability of the crisp social choice function. The paper shows how an individual could manipulate a social choice even if the preferences of the
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individuals are ambiguous. Depending on the type of transitivity of the fuzzy preference relations, three definitions of strategy-proofness have been proposed. It has been proved that for all the three cases, the G–S’s theorem remains valid. Further researches should be carried out to examine the validity of the G–S’s theorem under other domains of FSCFs and to determine the minimum requirements for fuzzy strategy-proofness. Acknowledgments This work has benefited from the Luso–Tunisian bilateral cooperation (2004–2007). The authors are very grateful to the anonymous referees for their valuable comments and suggestions that helped to improve the quality of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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