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Remarks on Suzumura consistent collective choice rules
Mathematical Social Sciences 65 (2013) 40–47
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Remarks on Suzumura consistent collective choice rules Susumu Cato ∗ Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, Tokyo, 192-0397, Japan
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Article history: Received 29 May 2010 Received in revised form 18 June 2012 Accepted 21 June 2012 Available online 28 June 2012
abstract Suzumura consistency is a necessary and sufficient condition for the existence of a weak-order extension. This paper provides some remarks on collective choice rules that generate Suzumura consistent social preferences. We examine the properties of such collective choice rules by introducing a procedural condition on collective choice rules. As applications of the procedural condition, we first investigate the decisive structure of a Paretian collective choice rule, and then consider the assignment of individual rights. In our analysis, the concept of semi-decisiveness works effectively. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The seminal work of Suzumura (1976) addresses the problem of the existence of a Paretian Bergson–Samuelson social welfare function. As a technical tool, Suzumura introduces the ‘‘axiom of consistency’’, which is a weakening of transitivity. Suzumura consistency requires that a binary relation have no cycle involving at least one strict preference.1 It is stronger than acyclicity of strict preference, and is independent of quasi-transitivity. As shown by Suzumura (1976), a binary relation has a weak-order extension if and only if it is Suzumura consistent.2 Then, a Paretian Bergson–Samuelson social welfare ordering is constructable if and only if the Pareto unanimity relation is Suzumura consistent. Hence, Suzumura consistency serves as the logical foundation of Bergsonian welfare economics. However, in spite of its importance, Suzumura consistency has not received much attention in the context of social choice, and studies on this subject have recently become prominent (Bossert and Suzumura, 2008, 2009, 2010, 2012; Cato, forthcoming-a, b). The purpose of this paper is to provide several remarks on Suzumura consistent collective choice rules. We first provide a general impossibility result on Suzumura consistent collective choice rules, which is inspired by the recent work on acyclical collective choice rules by Schwartz (2007). We introduce a procedural condition that partially specifies a power structure
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1 Bossert and Suzumura (2010) provide a comprehensive argument on Suzumura consistency. Houy (2008) and Cato (2012c) are recent studies on Suzumura consistency. Bossert (2007) provides a survey on the applications of Suzumura consistency. 2 Suzumura’s theorem is a generalization of the well-known ordering extension theorem provided by Szpilrajn (1930). 0165-4896/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2012.06.006
of collective choice rules, and investigate Suzumura consistent collective choice rules. Next, we apply the general result to two traditional subjects. As the first application, we investigate the decisive structure behind Suzumura consistent collective choice rules. Many researchers are interested in the structure of decisive sets in the Arrovian framework (Kirman and Sondermann, 1972; Brown, 1973, 1974, 1975; Hansson, 1976). Their results imply that rationality requirements on social preferences play a crucial role on the decisive structure. The benchmark result on the decisive structure of acyclical social choice is established by Brown (1973, 1974, 1975). He shows that the family of decisive sets associated with an acyclical collective choice rule forms a prefilter. Since Suzumura consistency is stronger than acyclicity, the family of decisive sets associated with a Suzumura consistent collective choice rule forms a prefilter. The problem is whether or not we can obtain an additional property of the decisive structure by strengthening acyclicity to Suzumura consistency. To capture an implication of Suzumura consistency on the decisive structure, we focus on the family of semi-decisive sets. A semi-decisive set is a group that has veto power. It is shown that if a Suzumura consistent collective choice rule satisfies weak Pareto, then the family of semi-decisive sets forms a prefilter. As a corollary to this result, we obtain the following result: if the number of alternatives is larger than the number of individuals, then a Suzumura consistent collective choice rule has a nonempty intersection of all semi-decisive sets (weak collegium). As the second application, we consider the classical problem of the rights assignment.3 Sen’s minimal liberalism requires that there exist at least two individuals who have decisive power over some pairs. As the first step, we introduce the concept of minimal liberalism based on semi-decisiveness, weak minimal
3 Suzumura (2010) provides a comprehensive survey of this subject.
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liberalism, which was developed by Aldrich (1977) and Salles (2008). According to weak minimal liberalism, there exist at least two individuals who have semi-decisive power over some pairs. If we weaken Sen’s minimal liberalism to weak minimal liberalism, then a possibility result arises: there exists an acyclical collective choice rule that satisfies unrestricted domain, weak Pareto, and weak minimal liberalism. Our question is whether or not there exists a Suzumura consistent collective choice rule satisfying the three axioms. We show that the answer is negative. Our next task is to consider how we can resolve the negative result. We show that the Sen–Suzumura resolution procedure works effectively to address this problem. The rest of this paper is organized as follows. Section 2 introduces the notation and basic definitions. Section 3 establishes a general result on a Suzumura consistent collective choice rule. Sections 4 and 5 provide two applications of our general result. Section 6 concludes the paper. The Appendix briefly examines a necessity condition for the violation of Suzumura consistency in collective decision making.
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The set of alternatives is X with |X | ≥ 3. Let R ⊆ X × X be a binary relation on X . The symmetric and asymmetric parts of R are denoted by I (R) and P (R), respectively. That is, I (R) := {(x, y) ∈ X × X |(x, y) ∈ R and (y, x) ∈ R} and P (R) := {(x, y) ∈ X × X |(x, y) ∈ R and (y, x) ̸∈ R}.5 The diagonal relation on X is given by ∆ = {(x, y) ∈ X × X |x = y}. Let B be the set of all possible binary relations R on X . Let us now introduce several properties of a binary relation. Reflexivity: For all x ∈ X , (x, x) ∈ R.
Transitivity:
(x, z ) ∈ R.
be complete, transitive, etc., if for all R ∈ D , f (R) is complete, transitive, etc. We next introduce basic axioms on a CCR f . Unrestricted domain: D = RN . Weak Pareto: For all R ∈ D , and for all x, y ∈ X , if (x, y) ∈ i∈N P (Ri ), then (x, y) ∈ P (f (R)). The following lemma is the fundamental result on Suzumura consistency.6
Lemma 1 (Suzumura, 1976). A binary relation R on X has a weakorder extension if and only if it is Suzumura consistent. Let tc (R) be the transitive closure of R, i.e., tc (R) = {(x, y)|∃K ∈ N and x0 , . . . , xK ∈ X such that x = x0 , (xk−1 , xk ) ∈ R ∀k ∈ {1, . . . , K }, and xK = y}. It is well known that tc (R) is the smallest transitive binary relation containing R. The consistent closure of R, denoted as cc (R), is defined as follows: cc (R) = R ∪ {(x, y)|(y, x) ∈ R and (x, y) ∈ tc (R)}.
2. Preliminaries
Completeness: (y, x) ∈ R.
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For all x, y ∈ X such that x ̸= y, (x, y) ∈ R or For all x, y, z ∈ X , [(x, y) ∈ R and (y, z ) ∈ R] ⇒
P-acyclicity: For all K ≥ 1, and for all x0 , . . . , xK ∈ X ,
[(xk−1 , xk ) ∈ P (R) ∀k ∈ {1, . . . , K }] ⇒ (xK , x0 ) ̸∈ P (R). Suzumura consistency: For all K ≥ 1, and for all x0 , . . . , xK ∈ X ,
[(xk−1 , xk ) ∈ R ∀k ∈ {1, . . . , K }] ⇒ (xK , x0 ) ̸∈ P (R). A finite sequence {x0 , . . . , xK } in X (K ≥ 1) is called an SC-cycle of order K if [(xk−1 , xk ) ∈ R ∀k ∈ {1, . . . , K }] and (xK , x0 ) ∈ P (R). Suzumura consistency excludes an SC-cycle of any order. In general, Suzumura consistency is weaker than transitivity. When a binary relation satisfies completeness, however, the two conditions are equivalent. It is clear that Suzumura consistency implies P-acyclicity, but not vice versa. A reflexive, complete, and transitive binary relation is called a weak order. Let R be the set of all possible weak orders R on X . A binary relation R′ is said to be an extension of R if R ⊆ R′ and P (R) ⊆ P (R′ ). A binary relation R is a subrelation of R′ if and only if R′ is an extension of R. If an extension R′ of R is a weak order, then we call it a weak-order extension of R. Let N with |N | ≥ 2 be the (finite or infinite) set of individuals. Each individual i ∈ N has a weak order Ri ∈ R on X . A preference profile R = (Ri )i∈N ∈ RN is a list of individual weak orders on X . A subset D of RN is an admissible preference domain. A collective choice rule (CCR) is a function f : D → B that maps each profile R ∈ D to a social preference f (R) ∈ B . We call a CCR f to
4 For a set A, |A| denotes its cardinality. 5 By definition, I (R) is not required to be reflexive without the reflexivity of R.
The consistent closure was proposed by Bossert et al. (2005). Its fundamental property is stated as follows.7 Lemma 2 (Bossert et al., 2005). For any binary relation R on X , (i) cc (R) ⊆ tc (R); (ii) cc (R) is the smallest Suzumura consistent binary relation containing R. 3. A procedural condition for SC-cyclic social preference Schwartz (2007) introduces the concept of relative impotence in order to examine P-acyclical CCRs. A set D ⊆ N is impotent relative to D′ ⊆ N over (x, y) for f if and only if D ∩ D′ = ∅ and
(y, x) ∈ P (Ri ) for all i ∈ N \ (D ∪ D′ ) and (y, x) ∈ Ri for all i ∈ D′ ⇒ (y, x) ∈ P (f (R)), for all R ∈ RN . Schwartz’s procedural condition is as follows. Impotence partition: For some M ≥ 2, there exist G, G1 , . . . , GM ⊆ N and x1 , . . . , xM , y1 , . . . , yM ∈ X such that (I) (II) (III) (IV)
either {G, G1 , . . . , GM } or {G1 , . . . , GM } is a partition of N; each Gm is impotent relative to G over (xm , ym ) for f ; never xm = xℓ or ym = yℓ or Gm = Gℓ if m ̸= ℓ; and {xm |m ∈ S } ̸= {ym |m ∈ S } if S is a nonempty proper subset of {1, . . . , M }.
Schwartz (2007) shows that there exists no P-acyclical CCR f that satisfies unrestricted domain, strong Pareto, and impotence partition.8 His result is a generalization of the Condorcet majority rule and other well-known voting paradoxes. We aim to extend Schwartz’s result to a Suzumura consistent CCR. Now, we propose the concept of relative semi-impotence. A set D ⊆ N is semi-impotent relative to D′ ⊆ N over (x, y) for f if and only if D ∩ D′ = ∅ and
(y, x) ∈ P (Ri ) for all i ∈ N \ (D ∪ D′ ) and (y, x) ∈ Ri for all i ∈ D′ ⇒ (y, x) ∈ f (R), for all R ∈ RN .
6 For related results, see Cato (2012b). 7 For another property of the consistent closure, see Cato (2012a). 8 A CCR f satisfies strong Pareto if and only if (x, y) ∈ P ( i∈N Ri ) ⇒ (x, y) ∈ P (f (R)). Schwartz (2007) refers to this as the Pareto principle.
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Weak impotence partition: For some M ≥ 2, there exist G, G1 , . . . , GM ⊆ N and x1 , . . . , xM , y1 , . . . , yM ∈ X such that (I) either {G, G1 , . . . , GM } or {G1 , . . . , GM } is a partition of N; (II′ ) each Gm is semi-impotent relative to G over (xm , ym ) for f ; (III) never xm = xℓ or ym = yℓ or Gm = Gℓ if m ̸= ℓ; and (IV′ ) {xm |m ∈ S } ̸= {ym |m ∈ S } if S is a nonempty subset of {1, . . . , M }. Weak impotence partition differs from impotence partition in two respects. Impotence partition requires that Gm be impotent relative to G over some pair, while weak impotence partition requires that Gm be semi-impotent relative to G over some pair. Impotence partition requires that {xm |m ∈ S } ̸= {ym |m ∈ S } if S is a nonempty ‘‘proper’’ subset of {1, . . . , M }, while weak impotence partition additionally requires that {x1 , . . . , xM } ̸= {y1 , . . . , yM }. This change raises the minimum required number of alternatives from M to M + 1. By definition, (II) implies (II′ ), while (IV) is implied from (IV′ ). Hence, weak impotence partition and impotence partition are independent. Our sufficiency result is as follows. Theorem 1. There exists no Suzumura consistent CCR f that satisfies unrestricted domain, weak Pareto, and weak impotence partition. Proof. Let f be a Suzumura consistent CCR satisfying unrestricted domain, weak Pareto, and weak impotence partition. By weak impotence partition, there exist M ≥ 2, G, G1 , . . . , GM ⊆ N and x1 , . . . , xM , y1 , . . . , yM ∈ X such that (I), (II′ ), (III), and (IV′ ) are satisfied. By (I), either {G, G1 , . . . , GM } or {G1 , . . . , GM } is a partition of N. Note that G is allowed to be the empty set. First, we focus on the case where G is nonempty. Define a binary relation E on {1, . . . , M } by mE ℓ ⇔ xm = yℓ . First, we will show the following claim: for each m ∈ {1, . . . , M }, |{ℓ ∈ {1, . . . , M }|mE ℓ}| ≤ 1, and [mE ℓ ⇒ m ̸= ℓ]. By way of contradiction, suppose that (a) mEm or (b) (mE ℓ ∧ mE ℓ′ ) for some two distinct ℓ, ℓ′ . In case (a), we have xm = ym , which contradicts (IV′ ). In case (b), we have yℓ = yℓ′ , which contradicts (III). Hence, the claim is proved. Next, we will show that E is acyclic. Suppose, to the contrary, that E has a cycle. Without loss of generality, there exists m ∈ {1, . . . , M } such that 1E2E . . . Em and
mE1.
By the definition of E, it is clear that {x1 , . . . , xm } = {y1 , . . . , ym }, which contradicts (IV′ ). Thus, E does not has a cycle. These claims and (III) together imply that (a) for all m ∈ {1, . . . , M }, xm is equal at most to ym+1 , and (b) for some m ∈ {1, . . . , M }, xm ̸= ym+1 . Define T :=
{(xm , ym ), (ym , xm )}.
m∈{1,...,M }
Let R ∈ B N be such that
∀i ∈ G1 : Ri = {(x1 , y2 ), (y2 , x2 ), (x2 , y3 ), . . . , (xM , y1 )}, ∀i ∈ G2 : Ri = {(x2 , y3 ), (y3 , x3 ), (x3 , y4 ), . . . , (x1 , y2 )}, .. . ∀i ∈ Gm : Ri = {(xm , ym+1 ), (ym+1 , xm+1 ), (xm+1 , ym+2 ), . . . , (xm−1 , ym )}, .. . ∀i ∈ GM : Ri = {(xM , y1 ), (y1 , x1 ), (x1 , y2 ), . . . , (xM −1 , yM )}, and
∀i ∈ G : Ri = {(x1 , y2 ), (x2 , y3 ), (x3 , y4 ), . . . , (xm , y1 )} ∪ T .
From (a) and the construction of profile R, it follows that for all i ∈ N , Ri is Suzumura consistent. Lemma 1 implies that Ri has a weak-order extension R′i . Therefore, there exists R′ ∈ RN such that for each i ∈ N , R′i is a weak-order extension of Ri . In the rest of this proof, we write y1 as yM +1 . Weak Pareto implies that
[xm ̸= ym+1 ] ⇒ (xm , ym+1 ) ∈ P (f (R′ )), (1) ′ because (xm , ym+1 ) ∈ P ( i∈N Ri ) whenever xm ̸= ym+1 . From (b), it follows that
(xm , ym+1 ) ∈ P (f (R′ )) for some m ∈ {1, . . . , M }.
(2)
Note that
(ym , xm ) ∈ P (Ri ) for all i ∈ N \ (Gm ∪ G), and
(ym , xm ) ∈ Ri for all i ∈ G. By (II), Gm is semi-impotent relative to G for (xm , ym ), and thus, it follows that (ym , xm ) ∈ f (R′ ). Then, we have
(ym , xm ) ∈ f (R′ ) for all m ∈ {1, . . . , M }.
(3)
By (1)–(3), f (R ) has an SC-cycle for some order. This contradicts the Suzumura consistency of f . The same argument can be applied for the case where G is the empty set. ′
It is noteworthy that Suzumura’s extension lemma is applied to prove Theorem 1. Schwartz (2007) imposes strong Pareto, while we impose weak Pareto in Theorem 1. Many studies in social choice theory employ weak Pareto, and hence this modification is useful when we apply Theorem 1. In the following two sections, we present two applications to traditional subjects in social choice theory. 4. Semi-decisive sets and Suzumura consistency This section examines the decisive structure of Suzumura consistent CCRs through the concept of ‘‘semi-decisiveness’’. Definition 1. A set D ⊆ N is (i) decisive for f if for all x, y ∈ X and for all R ∈ D , (x, y) ∈ i∈D P (Ri ), then (x, y) ∈ P (f (R)); (ii) semi-decisive for f if for all x, y ∈ X and for all R ∈ D , (x, y) ∈ i∈D P (Ri ), then (x, y) ∈ f (R).
Let Ω (f ) denote the family of decisive sets for f , and Ω ∗ (f ) denote the family of semi-decisive sets for f . By definition, a set D ⊆ N is semi-decisive for f if it is decisive for f . This implies that for every CCR f , Ω (f ) ⊆ Ω ∗ (f ). It is clear that if a set D ⊆ N is semi-decisive for f , then every superset D′ of D is also semi-decisive for f . As an auxiliary step, we introduce additional concepts. Let G be a family of subsets of N. A family G is a prefilter on N if (f.1) N ∈ G and ∅ ̸∈ G; (f.2) ∀G, G′ ⊆ N , [G ∈ G and G ⊆ G′ ] ⇒ G′ ∈ G; (f.3) ∀ K ≥ 1, ∀G1 , . . . , GK ⊆ N , [Gk ∈ G for all k ∈ {1, . . . , K }] ⇒ k∈{1,...,K } Gk ̸= ∅. A family G is a filter on N if (f.1); (f.2); (f.4) ∀G, G′ ⊆ N , [G ∈ G and G′ ∈ G] ⇒ G ∩ G′ ∈ G. A family G is an ultrafilter on N if (f.1); (f.2); (f.4); (f.5) ∀G ⊆ N , [G ̸∈ G] ⇒ N \ G ∈ G. By definition, an ultrafilter is a filter, and a filter is a prefilter. It is well known that one of the causes of Arrow’s impossibility theorem is the postulate of collective rationality. Arrow (1963) requires that a social preference generated by a CCR must be a weak order. Under this postulate, the family Ω (f ) of decisive sets
S. Cato / Mathematical Social Sciences 65 (2013) 40–47
forms an ultrafilter on N.9 This has very negative implications because every ultrafilter has a singleton element whenever N is a finite set. However, we have an escape route from this dead end in the absence of completeness of social preference. Indeed, the family Ω (f ) of decisive sets may not form an ultrafilter when completeness is dropped. Suzumura consistency is equivalent to transitivity in the presence of completeness, but Suzumura consistency is weaker than transitivity in the absence of completeness. Thus, we have a natural weakening of transitivity when social preference is allowed to be incomplete. Arrow’s independence of irrelevant alternatives has also been attacked by many researchers. It excludes many reasonable CCRs, such as the Borda rule and the Copeland rule. To allow such rules, we also drop the independence of irrelevant alternatives. To sum up, we examine a Suzumura consistent CCR that satisfies unrestricted domain and weak Pareto. Since Suzumura consistency is a necessary and sufficient condition for the existence of a weakorder extension, such a CCR guarantees the constructability of a Paretian welfare ordering. Thus, the underlying aim of this section is to investigate the decisive structure behind the Bergsonian school of welfare economics. The seminal works of Brown (1975) and Banks (1995) clarify the power structure of Paretian P-acyclical CCRs: Brown (1975) shows that if a P-acyclical CCR satisfies unrestricted domain, weak Pareto, and independence of irrelevant alternatives, then Ω (f ) forms a prefilter on N. Banks (1995) points out that the same result holds without independence of irrelevant alternatives.10 Since a Suzumura consistent CCR is P-acyclical, the Brown–Banks theorem immediately implies that the family of decisive sets forms a prefilter on N. To capture additional properties, we employ the concept of semi-decisiveness.11 The following result implies the usefulness of semi-decisiveness. Proposition 1. For every CCR f , if Ω ∗ (f ) forms a prefilter on N, then Ω (f ) forms a prefilter on N. Proof. Since (f.1) and (f.2) are obviously satisfied, it suffices to check (f.3). Suppose, to the contrary, that there exist K ≥ 1 and D1 , . . . , DK ∈ Ω (f ) such that k∈{1,...,K } Dk = ∅. Since Ω (f ) ⊆ Ω ∗ (f ), it follows that D1 , . . . , DK ∈ Ω ∗ (f ). This ∗ implies that D1 , . . . , DK ∈ Ω (f ) and k∈{1,...,K } Dk = ∅. This is a contradiction. We distinguish two cases: (i) the case where N is a finite set and (ii) the case where N is an infinite set. We first consider a collective decision making with a finite population. In this case, if the number of individuals is strictly smaller than the number of alternatives, then the family of semi-decisive sets forms a prefilter for any Suzumura consistent CCR. Theorem 1 is employed to prove this result.
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N is decisive, and thus, the family Ω (f ) of decisive sets for f is nonempty. Since Ω (f ) ⊆ Ω ∗ (f ), the family Ω ∗ (f ) of semi-decisive sets for f is nonempty. By way of contradiction, suppose that there exist K ≥ 1 and D1 , . . . , DK ∈ Ω ∗ (f ) such that k∈{1,...,K } Dk = ∅. This implies that
D = ∅.
D∈Ω ∗ (f )
Hence, every individual i ∈ N has a semi-decisive set Di that does not include him. It is clear that Di ⊆ N \ {i}. Then N \ {i} is semidecisive for f . Let us consider the partition {{1}, {2}, . . . , {n}} of N. Take distinct alternatives z 1 , . . . , z n , w, ∈ X . Since N \ {i} is semidecisive for f , {1} is semi-impotent relative to ∅ over (z 1 , w), and {i} is semi-impotent relative to ∅ over (z k , z k−1 ) for k ∈ {2, . . . , n}. Hence, we can apply Theorem 1, and thus, the proof is complete. Theorem 2 and Proposition 1 imply the following result. Corollary 1. Suppose that N is a finite set and that a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If |X | > |N |, then the family Ω (f ) of decisive sets for f is a prefilter on N. Definition 2. For each CCR f , (i) a collegium is a nonempty intersection of all decisive sets for f ; (ii) a weak collegium is a nonempty intersection of all semidecisive sets for f . Remark 1. A CCR f has a collegium if it has a weak collegium. As a corollary to Theorem 2, we have the following result. Corollary 2. Suppose that N is a finite set and a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If |X | > |N |, f has a weak collegium. Remark 2. Banks (1995) proves that if |X | ≥ |N |, then every P-acyclical CCR f satisfying weak Pareto and unrestricted domain is collegial. Corollary 2 and Remark 1 imply that if |X | > |N |, then every Suzumura consistent CCR f satisfying weak Pareto and unrestricted domain is collegial. Because Suzumura consistency is stronger than P-acyclicity, if |X | ≥ |N |, then every Suzumura consistent CCR f satisfying weak Pareto and unrestricted domain is collegial.
Proof. Note that (f.1) follows from weak Pareto, and (f.2) immediately follows from the definition of semi-decisiveness. Thus, it suffices to show that (f.3) holds. Weak Pareto implies that
Next, we examine the case with an infinite population. Recent works on intergenerational equity assume infinite future generations, and some of these works apply Szpilrajn’s or Suzumura’s extension lemmas. Hence, the analysis on Suzumura consistent CCR with the infinite population case is of interest because it might provide a mathematical tool for such a subject. In this case, the family of semi-decisive sets forms a prefilter for any Suzumura consistent CCR whenever the set of alternatives is an infinite set. It is noteworthy that the following result holds under a/an (countably or uncountably) infinite population.
9 This result was independently proved by Kirman and Sondermann (1972) and Hansson (1976). 10 See also Packel (1984) and Blau and Brown (1989). Packel (1984) examines the
Theorem 3. Suppose that N is an infinite set and that a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If X is an infinite set, then the family Ω ∗ (f ) of semi-decisive sets for f is a prefilter on N.
Theorem 2. Suppose that N is a finite set and a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If |X | > |N |, then the family Ω ∗ (f ) of semi-decisive sets for f is a prefilter on N.
decisive structure of Paretian social choice rules based on ‘‘rejection’’ decisiveness. 11 There exists an article that examines the family of semi-decisive sets. Campbell and Kelly (2000) show that if a complete and transitive CCR satisfies unrestricted domain, weak Pareto, and weak independence of irrelevant alternatives, then Ω ∗ (f ) forms an ultrafilter on N.
Proof. Note that (f.1) follows from weak Pareto, and (f.2) immediately follows from the definition of semi-decisiveness. Thus, it suffices to show that (f.3) holds. By way of contradiction, suppose that there exist K ≥ 1 and D1 , . . . , DK ∈ Ω ∗ (f ) such that
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1 K k∈{1,...,K } Dk = ∅. Let z , . . . , z , w be distinct alternatives in X . Since X is an infinite set, we can choose them for any finite number K . Let us define
G1 := N \ D1 , Gk := D1 ∩ D2 ∩ · · · ∩ Dk−1 − Dk
Sen’s minimal liberalism is formally restated as follows. Minimal liberalism: There exists a symmetric rights system (Di )i∈N such that (i) for some distinct i, j ∈ N , Di , Dj ̸= ∅; (ii) a CCR f fulfills (Di )i∈N .
for k ∈ {2, . . . , K }.
Let k ∈ {2, . . . , K }. Since Dk is decisive for f and Dk ⊆ N \ Gk , N \ Gk is decisive for f . Hence, G1 is semi-impotent relative to ∅ over (z 1 , w), and Gk is semi-impotent relative to ∅ over (z k , z k−1 ) for each k ∈ {2, . . . , K }. This implies that weak impotence partition is satisfied. By Theorem 1, f generates a social preference with an SC-cycle in some order for some profile R ∈ RN . In the infinite population case, the infiniteness of X is necessary for Theorem 3.12 To clarify this point, we show that the family of semi-decisive sets may not form a prefilter if X is a finite set. Suppose that N = N and X is a finite set with |X | = s. Let H := {1, . . . , s, s + 1} ⊆ N. Define
Sen (1970) establishes the following theorem: there exists no P-acyclical CCR that satisfies unrestricted domain, weak Pareto, and minimal liberalism. This theorem indicates that individual liberty can conflict with the Pareto principle. It is known as the ‘‘impossibility of Paretian liberal’’ (or the liberal paradox). Sen’s paradox has been extended by many researchers. Among others, Gibbard (1974) clarifies the possibility that minimal liberalism can be self-contradictory. We now illustrate this problem. Consider a system of rights (Di )i∈N where there exist four distinct alternatives x, y, z , w ∈ X such that (x, y), (z , w) ∈ D1 and (y, z ), (w, x) ∈ D2 . Let R ∈ RN be such that
B(x, y; R) := {i ∈ H |(x, y) ∈ P (Ri )}.
(x, y) ∈ P (R1 ) and (z , w) ∈ P (R1 )
By construction, B(x, y; R) ≤ s + 1 for all x, y ∈ X and all R ∈ R . Define fˆ as follows: for all x, y ∈ X and all R ∈ RN , N
and
(y, z ) ∈ P (R2 ) and (w, x) ∈ P (R2 ).
(x, y) ∈ fˆ (R) ⇔ B(x, y; R) ≥ s .
If a CCR f fulfills this system of rights, it follows that
Social preferences generated by CCR fˆ are equivalent to those
{(x, y), (y, z ), (z , w), (w, x)} ⊆ P (f (R)).
generated by one of the S-rules on an H-individual society, which are introduced by Bossert and Suzumura (2008, 2010).13 As shown by Bossert and Suzumura (2008, 2010), every S-rule always generates a Suzumura consistent social preference, so that fˆ is Suzumura consistent. However, every subset H ′ of H with s members is semi-decisive, and thus, there exist K ≥ 1 and D1 , . . . , DK ∈ Ω ∗ (fˆ ) such that k∈{1,...,K } Dk = ∅. Therefore, Ω ∗ (fˆ ) is not a prefilter on N. This argument can be applied for any infinite set N. Theorem 3 and Proposition 1 imply the following result. Corollary 3. Suppose that N is an infinite set and that a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If X is an infinite set, then the family Ω (f ) of decisive sets for f is a prefilter on N. 5. Weak form of liberalism and Suzumura consistency In this section, we investigate the implication of Suzumura consistency in the problem of Paretian liberalism. According to Sen’s minimal liberalism, ‘‘[t]here are at least two individuals such that for each of them there is at least one pair of alternatives over which he is decisive, that is, there is a pair x, y, such that if he prefers x (respectively y) to y (respectively x), then society should prefer x (respectively y) to y (respectively x) (Sen, 1970, p. 153)’’. A system of rights is (Di )i∈N where Di ⊆ X × X \ ∆ for each i ∈ N. A system of rights (Di )i∈N is symmetric if for all i ∈ N, and x, y ∈ X ,
(x, y) ∈ Di ⇒ (y, x) ∈ Di . Given (Di )i∈N , a CCR f fulfills (Di )i∈N if for all i ∈ N, for all (x, y) ∈ Di , and for all R ∈ RN ,
(x, y) ∈ P (Ri ) ⇒ (x, y) ∈ P (f (R)).
The CCR may generate the cycle of a strict social preference. Hence, Sen-type liberalism and unrestricted domain are not compatible under some systems of rights. To avoid this problem, the assignment of rights has to be more restrictive. Suzumura (1978, 1983) examines under what condition there is a P-acyclical CCR fulfilling a system of individual claim rights. He shows that the coherency of the system of rights, originally by Farrell (1976), is a necessary and sufficient condition for the existence of a P-acyclical CCR satisfying minimal liberalism. We now consider a weak form of liberalism. Given (Di )i∈N , a CCR f weakly fulfills (Di )i∈N if for all i ∈ N, for all (x, y) ∈ Di , and for all R ∈ RN ,
(x, y) ∈ P (Ri ) ⇒ (x, y) ∈ f (R). Weak minimal liberalism: There exists a symmetric rights system (Di )i∈N such that (i) for some distinct i, j ∈ N , Di , Dj ̸= ∅; (ii) a CCR f weakly fulfills (Di )i∈N . This axiom is proposed by Aldrich (1977), and it has been recently investigated by Salles (2008). Weak minimal liberalism requires that there exist at least two individuals who have veto power over some pair (x, y) in both directions. Note that in the case of weak minimal liberalism, the coherency of the rights assignment is not problematic: for any system of rights, there always exists a P-acyclical CCR satisfying weak minimal liberalism. Moreover, we can obtain the following possibility result: there exists a quasitransitive CCR that satisfies unrestricted domain, weak Pareto, and weak minimal liberalism. Hence, there exists a P-acyclical CCR satisfying the three axioms. To see this fact, consider the following CCR f : for all x, y ∈ X and for all R ∈ RN ,
(x, y) ∈ f (R) ⇔ (y, x) ̸∈ P
Ri
.
i∈N
12 If X is an infinite set, one might think that the unrestricted domain is too strong because it contains a profile wherein individuals do not have a maximal element on X . Indeed, in some applications, it would be reasonable to restrict attention to profiles where each individual has a maximal alternative. All results in this paper hold under the domain assumption that all profiles where each individual has a maximal element are available. 13 See also Cato and Hirata (2010).
This CCR is the Pareto-extension rule. It is well-known that this rule is quasi-transitive. Under this CCR, Di = X × X for all i ∈ N. Thus, for any rights system, we have a CCR that weakly fulfills it. Aldrich (1977) shows that there exists no transitive and complete CCR satisfying the three axioms. Moreover, Salles (2008) proves that similar impossibility results hold for semi-transitivity
S. Cato / Mathematical Social Sciences 65 (2013) 40–47
or interval order property under a natural restriction on a rights system. These rationality requirements are stronger than quasitransitivity. A similar negative result holds on a Suzumura consistent CCR. There exists no Suzumura consistent CCR f satisfying unrestricted domain, weak Pareto, and weak minimal liberalism provided that |X | ≥ 4 and that there exist distinct i, j ∈ N and distinct x, y, z , w ∈ X such that (x, y) ∈ Di and (z , w) ∈ Dj in the definition of weak minimal liberalism.14 To prove this, it suffices to consider a partition {{i}, N \ {i}} of N. Since f weakly fulfills (Di )i∈N , {i} is semi-impotent relative to ∅ over (w, z ) and N \ {i} is semiimpotent relative to ∅ over (y, x). By Theorem 1, f generates a social preference with an SC-cycle in some order for some profile R ∈ RN . We now specify a necessary and sufficient condition on a rights system for the existence of a Suzumura consistent CCR satisfying unrestricted domain, weak Pareto, and weak liberalism. As an auxiliary step, we introduce a variant of weak minimal liberalism. Weak minimal liberalism∗ : There exists a rights system (Di )i∈N such that a CCR f weakly fulfills (Di )i∈N . We introduce an additional requirement on a rights system. A rights system (Di )i∈N is contagious if either (i) there exist distinct i, j ∈ N and distinct x, y, z , w ∈ X such that (x, y) ∈ Di and (z , w) ∈ Dj or (ii) there exist distinct i, j ∈ N and distinct x, y, z ∈ X such that (x, y) ∈ Di and (y, z ) ∈ Dj . Theorem 4. There exists no Suzumura consistent CCR f satisfying unrestricted domain, weak Pareto, and weak minimal liberalism∗ if and only if the corresponding rights system (Di )i∈N is contagious. Proof. ‘Only if’. It suffices to show that there exists a Suzumura consistent CCR f satisfying unrestricted domain, weak Pareto, and weak minimal liberalism if the corresponding rights system (Di )i∈N is not contagious. Define f as follows: for all x, y ∈ X , and for all R ∈ RN ,
f (R) ⇔
(x, y) ∈ P
i∈N
Ri
or (x, y) ̸∈ P
Ri
i∈N
and (x, y) ∈ Dj ∩ P (Rj ) for some j ∈ N
.
Note that P (f (R)) = f (R) for all R ∈ RN . We will show that this CCR f is Suzumura consistent. By way of contradiction, suppose not. Then, there exist R ∈ RN , K ≥ 1, and distinct x0 , . . . , xK such that
k−1 k (x , x ) ∈ f (R) for k ∈ {1, . . . , K } and (xK , x0 ) ∈ P (f (R)). By construction, (xK , x0 ) ∈ i∈N P (Ri ). We now say that individual j exercises his weak rights on pair (xk−1 , xk ) if (xk−1 , xk ) ̸∈ P ( i∈N Ri ) and (xk−1 , xk ) ∈ Dj ∩ P (Rj ). We distinguish two cases. Case (a): there exists no individual who exercises his weak rights on somepair in {(x0 , x1 ), . . . , (xK −1 , xK )}. In this case, −1 k (xk , x ) ∈ P ( i∈N Ri ) for all k ∈ {1, . . . , K }, so that (x0 , xK ) ∈ P ( i∈N Ri ). This contradicts the supposition that (xK , x0 ) ∈ P (f (R)). Case (b): there exists only one individual who exercises his weak rights on some pair in {(x0 , x1 ), . . . , (xK −1 , xK )}. Let j ∈ N be an individual who exercises his weak rights on some pair in {(x0 , x1 ), . . . , (xK −1 , xK )}. Define
Kj := {k ∈ {1, . . . , K }| j exercises his weak rights on (xk−1 , xk )}.
14 This restriction on a rights system is essentially the same as that introduced by Salles (2008).
45
By construction, (xk−1 , xk ) ∈ Dj ∩ P (Rj ) for all k ∈ Kj and (xk−1 , xk ) ∈ Rj for all k ∈ {1, . . . , K } \ Kj . Since individual j’s preference is a weak order, we have (x0 , xK ) ∈ P (Rj ). This contradicts the fact that (xK , x0 ) ∈ i∈N P (Ri ). Hence, either there exist distinct i, j ∈ N and distinct x, y, z , w such that (x, y) ∈ Di and (z , w) ∈ Dj or there exist distinct i, j ∈ N and distinct x, y, z such that (x, y) ∈ Di and (y, z ) ∈ Dj . This implies that (Di )i∈N is contagious. This is a contradiction. ‘If’. Let f be a Suzumura consistent CCR that satisfies weak Pareto and weak minimal liberalism with a contagious rights system (Di )i∈N . We distinguish two cases. Case (i): there exist distinct four alternatives x, y, z , w ∈ X such that (x, y) ∈ Di and (z , w) ∈ Dj . As discussed before, Theorem 1 implies that f generates a social preference with an SC-cycle for some profile R ∈ RN . Case (ii): there exist distinct three alternatives x, y, z ∈ X such that (x, y) ∈ Di and (y, z ) ∈ Dj . Since f weakly fulfills (Di )i∈N , {i} is semi-impotent relative to ∅ over (z , y), and N \ {i} is semiimpotent relative to ∅ over (y, x). From Theorem 1, f generates a social preference with an SC-cycle in some order for some profile R ∈ RN . Since a rights system (Di )i∈N that is not contagious is very restrictive, the message of Theorem 4 is negative: in most cases, we cannot construct a Suzumura consistent CCR on the unrestricted domain satisfying weak Pareto and weak minimal liberalism. We now consider how we can resolve the difficulty stated by Theorem 4. The approach that we use here was introduced by Sen (1976) and developed by Suzumura (1978, 1983).15 They examine the existence of a P-acyclical CCR satisfying minimal liberalism. They weaken the Paretian requirement and show that if there exists an individual who respects others’ rights (liberal individual), there exists a P-acyclical CCR satisfying the desireble axioms. We focus on a Suzumura consistent CCR and therefore reformulate the concept of a ‘‘liberal’’ individual. We now assume that for each profile R ∈ RN , there is a profile (R∗i )i∈N such that, for each i ∈ N, tc (R∗i ) = R∗i ,
R∗i ⊆ Ri ,
and
P (R∗i ) ⊆ P (Ri ).
Given a preference Ri of individual i ∈ N , R∗i is a transitive subrelation of Ri that individual i wants to count in collective choice (R∗i = Ri is possible). ∗ ∗ Conditional weak Pareto: For all x, y ∈ X and for all R = (Ri )i∈N , if [(x, y) ∈ i∈N P (R∗i )], then (x, y) ∈ P (f (R)). Define
Qi := Di ∩ Ri , and
S := cc
Qi
.
i∈N
It is noteworthy that (x, y) ∈ Qi if and only if (x, y) is a protected personal pair of individual i and he prefers x to y. Therefore, i∈N Qi represents individuals’ claims on their rights, but it can be cyclical without coherency of the rights system. This cyclicity can be resolved by the consistent closure. Recall that for each R, cc (R) = R ∪ {(x, y)|(y, x) ∈ R and (x, y) ∈ tc (R)}. By Lemma 2, cc ( i∈N Qi )is the smallest Suzumura consistent binary relation containing i∈N Qi . Note that for all i ∈ N, if (x, y) ∈ Di and (x, y) ∈ P (Ri ), then (x, y) ∈ S. By construction, it is possible that (x, y) ∈ Di , (x, y) ∈ P (Ri ), (x, y) ∈ S, and (y, x) ∈ S. Then, S weakly fulfills individual rights. Hence, S represents the ‘‘weak’’
15 See also Austen-Smith (1984).
46
S. Cato / Mathematical Social Sciences 65 (2013) 40–47
form of libertarian claims, which is Suzumura consistent. Since S is Suzumura consistent, it has a weak-order extension S¯ by Lemma 1. An individual i ∈ N is S-liberal if and only if for each profile R ∈ RN , R∗i is such that there exists a weak-order extension Ti of which R∗i is a subrelation and so is S. An S-liberal individual claims the parts of his preference to count which are compatible with the weak form of libertarian claims S. Since S is Suzumura consistent, such a weak-order extension always exists. It is noteworthy that the existence of an S-liberal individual is a strong restriction on a profile R∗ and the CCR f (·), not just on R∗i . In particular, a weak-order extension Ti respects the rights system (Di )i∈N that is generated from the CCR f (·). We now offer a possibility result on a Suzumura consistent CCR. If there exists at least one S-liberal individual, we can construct a Paretian Suzumura consistent CCR on the unrestricted domain that weakly fulfills any arbitrary given rights system.
of acyclical social choice by Brown (1973, 1974, 1975), Nakamura (1979) and Banks (1995), who focus on the structure of decisive sets (or winning coalitions).17 We then considered a weak version of Sen’s (1970) minimal liberalism. Weak liberalism requires that there exist at least two semi-decisive individuals (vetoers). A Sen-type negative result is proved: there exists no Suzumura consistent CCR satisfying unrestricted domain, weak Pareto, and weak liberalism. Moreover, we provided a resolution for this problem. Our results on two subjects suggest that semi-decisiveness works effectively for a Suzumura consistent CCR. Through two applications, we examined the possibility of democratic and liberal social welfare weak orders.
Theorem 5. Let (Di )i∈N be an arbitrary given rights system. If there exists at least one S-liberal individual, then there exists a Suzumura consistent CCR f that satisfies unrestricted domain and conditional weak Pareto and that weakly fulfills the rights system (Di )i∈N .
I thank Katsuhito Iwai, François Maniquet, and an anonymous referee of the journal for their valuable suggestions, and I thank Kotaro Suzumura for the helpful conversations over several years on related issues. This research was financially supported by Grant-in-Aids for Young Scientists (B) from the Japan Society for the Promotion of Science and the Ministry of Education, Culture, Sports, Science and Technology.
Proof. Without loss of generality, we can suppose that individual 1 is S-liberal. Define a CCR f as follows: for all x, y ∈ X , and for all R,
(x, y) ∈ P (f (R)) ⇔ (x, y) ∈
P (R∗i ) or (x, y) ∈ P (S ),
i∈N
(x, y) ∈ I (f (R)) ⇔ (x, y) ∈ I (S ). Conditional weak Pareto and weak minimal liberalism follow from construction of the CCR f . Thus, it suffices to show that the CCR f is Suzumura consistent. By way of contradiction, suppose that there exist K ≥ 1 and x0 , . . . , xK such that
(xk−1 , xk ) ∈ f (R) ∀k ∈ {1, . . . , K } and (xK , x0 ) ∈ P (f (R)). For convenience, we write (xK , x0 ) as (xK , xK +1 ). The binary relation S is Suzumura consistent, and thus, we have (xk−1 , xk ) ∈ ∗ k−1 , xk ) ∈ i∈N P (Ri ) for some k ∈ {1, . . . , K }. This implies that (x ∗ P (R1 ). However, there must exist a weak-order extension T1 of which R∗1 is a subrelation and so is S because individual 1 is Sliberal. This is a contradiction. To conclude this section, we comment on the underlying aim. The motivation of this section is to examine how we can incorporate liberalism into Bergsonian welfare economics. Most studies on the assignment of individual rights assume the acyclicity of strict social preference.16 However, P-acyclicity is insufficient to construct a social welfare ordering. Theorem 4 states the difficulty of a Paretian social welfare ordering respecting the weak form of liberalism, and Theorem 5 provides an escape route from this problem.
Acknowledgments
Appendix In Section 3, we showed that Schwartz’s (2007) sufficiency theorem can be extended to Suzumura consistent CCRs. Schwartz (2007) also provides a necessary theorem. He shows that under a weak background assumption, if a CCR satisfying what he call mild independence is P-cyclical, it satisfies impotence partition. This section briefly discusses an extension of the necessary theorem. The following axiom is a variant of Schwartz’s mild independence. Weak independence: For all R, R′ ∈ RN , if (x, y) ∈ P (R) ⇒ (x, y) ∈ P (R′ ) and (y, x) ∈ P (R′ ) ⇒ (y, x) ∈ P (R), then (x, y) ∈ f (R) ⇒ (x, y) ∈ f (R′ ). We present a necessary theorem on Suzumura consistency. Theorem 6. Suppose that a CCR f satisfies weak independence and there is R ∈ RN such that f (R) has an SC-cycle and (x, y) ∈ P (Ri ) for some i ∈ N and for some pair (x, y) in the cycle. Then, it satisfies weak impotence partition. Sketch of Proof. Following the same argument as in Schwartz (2007), there is a finite sequence {z1 , . . . , zM +1 } of distinct alternatives that is an SC-cycle and (zm , zm+1 ) ∈ P (Ri ) for some m and for some i ∈ N. Define
6. Concluding remarks
G := {i ∈ N |(zm−1 , zm ) ∈ I (Ri ) for all m},
This paper examined the implications of Suzumura consistency in a collective decision situation. By establishing a general result on Suzumura consistent CCRs, which is an extension of Schwartz (2007), we addressed two classical subjects: decisive structures and the assignment of individual rights. Throughout the paper, we applied the concept of semi-decisiveness: a semi-decisive set is a group that has veto power. We first considered the family of semi-decisive sets associated with a Suzumura consistent CCR, and proved that it forms a prefilter. This result is related to the studies
G1 := {i ∈ N |(z2 , z1 ) ∈ P (Ri )},
16 Exceptions are Aldrich (1977) and Salles (2008).
Gm := {i ∈ N |(zm+1 , zm ) ∈ P (Ri )} for all m ≥ 2. We can prove that {G, G1 , . . . , Gm } is a partition and Gm is semiimpotent relative to G. By letting (x1 , . . . , xm ) = (z2 , . . . , zm+1 ) and (y1 , . . . , ym ) = (z1 , . . . , zm ), we can derive weak impotence partition.
17 See also Blair and Pollak (1982), who investigate the existence of an individual who has veto power in acyclic social choice. Obviously semi-decisiveness relates to veto power: a semi-decisive set represents a coalition that has veto power.
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Cato, S., 2012c. A note on the extension of a binary relation on a set to the power set. Econom. Lett. 116, 46–48. Cato, S., 2012d. Quasi-decisiveness, quasi-ultrafilter, and social quasi-orderings. Soc. Choice Welf., (forthcoming-b) http://dx.doi.org/10.1007/s00355-0120677-z. Cato, S., Hirata, D., 2010. Collective choice rules and collective rationality: a unified method of characterizations. Soc. Choice Welf. 34, 611–630. Farrell, M.J., 1976. Liberalism in the theory of social choice. Rev. Econom. Stud. 43, 3–10. Gibbard, A., 1974. A Pareto-consistent libertarian claim. J. Econom. Theory 7, 388–410. Hansson, B., 1976. The existence of group preference functions. Pubic Choice 28, 89–98. Houy, N., 2008. A note on the Suzumura consistency. Math. Social Sci. 55, 90–95. Kirman, A.P., Sondermann, D., 1972. Arrow’s theorem, many agents, and invisible dictators. J. Econom. Theory 5, 267–277. Nakamura, K., 1979. The vetoers in a simple game with ordinal preferences. Internat. J. Game Theory 8, 55–61. Packel, E.W., 1984. Power structure and cardinality restrictions for Paretian social choice rules. Soc. Choice Welf. 1, 105–111. Salles, M., 2008. Limited rights as partial veto and Sen’s impossibility theorem. In: Pattanaik, P., Tadenuma, K., Xu, Y., Yoshihara, N. (Eds.), Rational Choice and Social Welfare: Theory and Applications Essays in Honor of Kotaro Suzumura. Springer, New York, pp. 11–23. Schwartz, T., 2007. A procedural condition necessary and sufficient for cyclic social preference. J. Econom. Theory 137, 688–695. Sen, A.K., 1970. Collective Choice and Social Welfare. Holden-Day, San Francisco. Sen, A.K., 1976. Liberty, unanimity and rights. Economica 43, 217–245. Suzumura, K., 1976. Remarks on the theory of collective choice. Economica 43, 381–390. Suzumura, K., 1978. On the consistency of libertarian claims. Rev. Econom. Stud. 45, 329–342. Suzumura, K., 1983. Rational Choice, Collective Decisions, and Social Welfare. Cambridge University Press, Cambridge. Suzumura, K., 2010. Welfarism, individual rights, and procedural fairness. In: Arrow, K.J., Sen, A.K., Suzumura, K. (Eds.), Handbook of Social Choice and Welfare, Volume 2. North-Holland, Amsterdam, pp. 605–685. Szpilrajn, S., 1930. Sur l’extension de l’ordre partiel. Fund. Math. 16, 386–389.
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Remarks on Suzumura consistent collective choice rules Susumu Cato ∗ Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, Tokyo, 192-0397, Japan
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Article history: Received 29 May 2010 Received in revised form 18 June 2012 Accepted 21 June 2012 Available online 28 June 2012
abstract Suzumura consistency is a necessary and sufficient condition for the existence of a weak-order extension. This paper provides some remarks on collective choice rules that generate Suzumura consistent social preferences. We examine the properties of such collective choice rules by introducing a procedural condition on collective choice rules. As applications of the procedural condition, we first investigate the decisive structure of a Paretian collective choice rule, and then consider the assignment of individual rights. In our analysis, the concept of semi-decisiveness works effectively. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The seminal work of Suzumura (1976) addresses the problem of the existence of a Paretian Bergson–Samuelson social welfare function. As a technical tool, Suzumura introduces the ‘‘axiom of consistency’’, which is a weakening of transitivity. Suzumura consistency requires that a binary relation have no cycle involving at least one strict preference.1 It is stronger than acyclicity of strict preference, and is independent of quasi-transitivity. As shown by Suzumura (1976), a binary relation has a weak-order extension if and only if it is Suzumura consistent.2 Then, a Paretian Bergson–Samuelson social welfare ordering is constructable if and only if the Pareto unanimity relation is Suzumura consistent. Hence, Suzumura consistency serves as the logical foundation of Bergsonian welfare economics. However, in spite of its importance, Suzumura consistency has not received much attention in the context of social choice, and studies on this subject have recently become prominent (Bossert and Suzumura, 2008, 2009, 2010, 2012; Cato, forthcoming-a, b). The purpose of this paper is to provide several remarks on Suzumura consistent collective choice rules. We first provide a general impossibility result on Suzumura consistent collective choice rules, which is inspired by the recent work on acyclical collective choice rules by Schwartz (2007). We introduce a procedural condition that partially specifies a power structure
∗
Tel.: +81 042 677 2312; fax: +81 042 677 2298. E-mail address: [email protected].
1 Bossert and Suzumura (2010) provide a comprehensive argument on Suzumura consistency. Houy (2008) and Cato (2012c) are recent studies on Suzumura consistency. Bossert (2007) provides a survey on the applications of Suzumura consistency. 2 Suzumura’s theorem is a generalization of the well-known ordering extension theorem provided by Szpilrajn (1930). 0165-4896/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2012.06.006
of collective choice rules, and investigate Suzumura consistent collective choice rules. Next, we apply the general result to two traditional subjects. As the first application, we investigate the decisive structure behind Suzumura consistent collective choice rules. Many researchers are interested in the structure of decisive sets in the Arrovian framework (Kirman and Sondermann, 1972; Brown, 1973, 1974, 1975; Hansson, 1976). Their results imply that rationality requirements on social preferences play a crucial role on the decisive structure. The benchmark result on the decisive structure of acyclical social choice is established by Brown (1973, 1974, 1975). He shows that the family of decisive sets associated with an acyclical collective choice rule forms a prefilter. Since Suzumura consistency is stronger than acyclicity, the family of decisive sets associated with a Suzumura consistent collective choice rule forms a prefilter. The problem is whether or not we can obtain an additional property of the decisive structure by strengthening acyclicity to Suzumura consistency. To capture an implication of Suzumura consistency on the decisive structure, we focus on the family of semi-decisive sets. A semi-decisive set is a group that has veto power. It is shown that if a Suzumura consistent collective choice rule satisfies weak Pareto, then the family of semi-decisive sets forms a prefilter. As a corollary to this result, we obtain the following result: if the number of alternatives is larger than the number of individuals, then a Suzumura consistent collective choice rule has a nonempty intersection of all semi-decisive sets (weak collegium). As the second application, we consider the classical problem of the rights assignment.3 Sen’s minimal liberalism requires that there exist at least two individuals who have decisive power over some pairs. As the first step, we introduce the concept of minimal liberalism based on semi-decisiveness, weak minimal
3 Suzumura (2010) provides a comprehensive survey of this subject.
S. Cato / Mathematical Social Sciences 65 (2013) 40–47
liberalism, which was developed by Aldrich (1977) and Salles (2008). According to weak minimal liberalism, there exist at least two individuals who have semi-decisive power over some pairs. If we weaken Sen’s minimal liberalism to weak minimal liberalism, then a possibility result arises: there exists an acyclical collective choice rule that satisfies unrestricted domain, weak Pareto, and weak minimal liberalism. Our question is whether or not there exists a Suzumura consistent collective choice rule satisfying the three axioms. We show that the answer is negative. Our next task is to consider how we can resolve the negative result. We show that the Sen–Suzumura resolution procedure works effectively to address this problem. The rest of this paper is organized as follows. Section 2 introduces the notation and basic definitions. Section 3 establishes a general result on a Suzumura consistent collective choice rule. Sections 4 and 5 provide two applications of our general result. Section 6 concludes the paper. The Appendix briefly examines a necessity condition for the violation of Suzumura consistency in collective decision making.
4
The set of alternatives is X with |X | ≥ 3. Let R ⊆ X × X be a binary relation on X . The symmetric and asymmetric parts of R are denoted by I (R) and P (R), respectively. That is, I (R) := {(x, y) ∈ X × X |(x, y) ∈ R and (y, x) ∈ R} and P (R) := {(x, y) ∈ X × X |(x, y) ∈ R and (y, x) ̸∈ R}.5 The diagonal relation on X is given by ∆ = {(x, y) ∈ X × X |x = y}. Let B be the set of all possible binary relations R on X . Let us now introduce several properties of a binary relation. Reflexivity: For all x ∈ X , (x, x) ∈ R.
Transitivity:
(x, z ) ∈ R.
be complete, transitive, etc., if for all R ∈ D , f (R) is complete, transitive, etc. We next introduce basic axioms on a CCR f . Unrestricted domain: D = RN . Weak Pareto: For all R ∈ D , and for all x, y ∈ X , if (x, y) ∈ i∈N P (Ri ), then (x, y) ∈ P (f (R)). The following lemma is the fundamental result on Suzumura consistency.6
Lemma 1 (Suzumura, 1976). A binary relation R on X has a weakorder extension if and only if it is Suzumura consistent. Let tc (R) be the transitive closure of R, i.e., tc (R) = {(x, y)|∃K ∈ N and x0 , . . . , xK ∈ X such that x = x0 , (xk−1 , xk ) ∈ R ∀k ∈ {1, . . . , K }, and xK = y}. It is well known that tc (R) is the smallest transitive binary relation containing R. The consistent closure of R, denoted as cc (R), is defined as follows: cc (R) = R ∪ {(x, y)|(y, x) ∈ R and (x, y) ∈ tc (R)}.
2. Preliminaries
Completeness: (y, x) ∈ R.
41
For all x, y ∈ X such that x ̸= y, (x, y) ∈ R or For all x, y, z ∈ X , [(x, y) ∈ R and (y, z ) ∈ R] ⇒
P-acyclicity: For all K ≥ 1, and for all x0 , . . . , xK ∈ X ,
[(xk−1 , xk ) ∈ P (R) ∀k ∈ {1, . . . , K }] ⇒ (xK , x0 ) ̸∈ P (R). Suzumura consistency: For all K ≥ 1, and for all x0 , . . . , xK ∈ X ,
[(xk−1 , xk ) ∈ R ∀k ∈ {1, . . . , K }] ⇒ (xK , x0 ) ̸∈ P (R). A finite sequence {x0 , . . . , xK } in X (K ≥ 1) is called an SC-cycle of order K if [(xk−1 , xk ) ∈ R ∀k ∈ {1, . . . , K }] and (xK , x0 ) ∈ P (R). Suzumura consistency excludes an SC-cycle of any order. In general, Suzumura consistency is weaker than transitivity. When a binary relation satisfies completeness, however, the two conditions are equivalent. It is clear that Suzumura consistency implies P-acyclicity, but not vice versa. A reflexive, complete, and transitive binary relation is called a weak order. Let R be the set of all possible weak orders R on X . A binary relation R′ is said to be an extension of R if R ⊆ R′ and P (R) ⊆ P (R′ ). A binary relation R is a subrelation of R′ if and only if R′ is an extension of R. If an extension R′ of R is a weak order, then we call it a weak-order extension of R. Let N with |N | ≥ 2 be the (finite or infinite) set of individuals. Each individual i ∈ N has a weak order Ri ∈ R on X . A preference profile R = (Ri )i∈N ∈ RN is a list of individual weak orders on X . A subset D of RN is an admissible preference domain. A collective choice rule (CCR) is a function f : D → B that maps each profile R ∈ D to a social preference f (R) ∈ B . We call a CCR f to
4 For a set A, |A| denotes its cardinality. 5 By definition, I (R) is not required to be reflexive without the reflexivity of R.
The consistent closure was proposed by Bossert et al. (2005). Its fundamental property is stated as follows.7 Lemma 2 (Bossert et al., 2005). For any binary relation R on X , (i) cc (R) ⊆ tc (R); (ii) cc (R) is the smallest Suzumura consistent binary relation containing R. 3. A procedural condition for SC-cyclic social preference Schwartz (2007) introduces the concept of relative impotence in order to examine P-acyclical CCRs. A set D ⊆ N is impotent relative to D′ ⊆ N over (x, y) for f if and only if D ∩ D′ = ∅ and
(y, x) ∈ P (Ri ) for all i ∈ N \ (D ∪ D′ ) and (y, x) ∈ Ri for all i ∈ D′ ⇒ (y, x) ∈ P (f (R)), for all R ∈ RN . Schwartz’s procedural condition is as follows. Impotence partition: For some M ≥ 2, there exist G, G1 , . . . , GM ⊆ N and x1 , . . . , xM , y1 , . . . , yM ∈ X such that (I) (II) (III) (IV)
either {G, G1 , . . . , GM } or {G1 , . . . , GM } is a partition of N; each Gm is impotent relative to G over (xm , ym ) for f ; never xm = xℓ or ym = yℓ or Gm = Gℓ if m ̸= ℓ; and {xm |m ∈ S } ̸= {ym |m ∈ S } if S is a nonempty proper subset of {1, . . . , M }.
Schwartz (2007) shows that there exists no P-acyclical CCR f that satisfies unrestricted domain, strong Pareto, and impotence partition.8 His result is a generalization of the Condorcet majority rule and other well-known voting paradoxes. We aim to extend Schwartz’s result to a Suzumura consistent CCR. Now, we propose the concept of relative semi-impotence. A set D ⊆ N is semi-impotent relative to D′ ⊆ N over (x, y) for f if and only if D ∩ D′ = ∅ and
(y, x) ∈ P (Ri ) for all i ∈ N \ (D ∪ D′ ) and (y, x) ∈ Ri for all i ∈ D′ ⇒ (y, x) ∈ f (R), for all R ∈ RN .
6 For related results, see Cato (2012b). 7 For another property of the consistent closure, see Cato (2012a). 8 A CCR f satisfies strong Pareto if and only if (x, y) ∈ P ( i∈N Ri ) ⇒ (x, y) ∈ P (f (R)). Schwartz (2007) refers to this as the Pareto principle.
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Weak impotence partition: For some M ≥ 2, there exist G, G1 , . . . , GM ⊆ N and x1 , . . . , xM , y1 , . . . , yM ∈ X such that (I) either {G, G1 , . . . , GM } or {G1 , . . . , GM } is a partition of N; (II′ ) each Gm is semi-impotent relative to G over (xm , ym ) for f ; (III) never xm = xℓ or ym = yℓ or Gm = Gℓ if m ̸= ℓ; and (IV′ ) {xm |m ∈ S } ̸= {ym |m ∈ S } if S is a nonempty subset of {1, . . . , M }. Weak impotence partition differs from impotence partition in two respects. Impotence partition requires that Gm be impotent relative to G over some pair, while weak impotence partition requires that Gm be semi-impotent relative to G over some pair. Impotence partition requires that {xm |m ∈ S } ̸= {ym |m ∈ S } if S is a nonempty ‘‘proper’’ subset of {1, . . . , M }, while weak impotence partition additionally requires that {x1 , . . . , xM } ̸= {y1 , . . . , yM }. This change raises the minimum required number of alternatives from M to M + 1. By definition, (II) implies (II′ ), while (IV) is implied from (IV′ ). Hence, weak impotence partition and impotence partition are independent. Our sufficiency result is as follows. Theorem 1. There exists no Suzumura consistent CCR f that satisfies unrestricted domain, weak Pareto, and weak impotence partition. Proof. Let f be a Suzumura consistent CCR satisfying unrestricted domain, weak Pareto, and weak impotence partition. By weak impotence partition, there exist M ≥ 2, G, G1 , . . . , GM ⊆ N and x1 , . . . , xM , y1 , . . . , yM ∈ X such that (I), (II′ ), (III), and (IV′ ) are satisfied. By (I), either {G, G1 , . . . , GM } or {G1 , . . . , GM } is a partition of N. Note that G is allowed to be the empty set. First, we focus on the case where G is nonempty. Define a binary relation E on {1, . . . , M } by mE ℓ ⇔ xm = yℓ . First, we will show the following claim: for each m ∈ {1, . . . , M }, |{ℓ ∈ {1, . . . , M }|mE ℓ}| ≤ 1, and [mE ℓ ⇒ m ̸= ℓ]. By way of contradiction, suppose that (a) mEm or (b) (mE ℓ ∧ mE ℓ′ ) for some two distinct ℓ, ℓ′ . In case (a), we have xm = ym , which contradicts (IV′ ). In case (b), we have yℓ = yℓ′ , which contradicts (III). Hence, the claim is proved. Next, we will show that E is acyclic. Suppose, to the contrary, that E has a cycle. Without loss of generality, there exists m ∈ {1, . . . , M } such that 1E2E . . . Em and
mE1.
By the definition of E, it is clear that {x1 , . . . , xm } = {y1 , . . . , ym }, which contradicts (IV′ ). Thus, E does not has a cycle. These claims and (III) together imply that (a) for all m ∈ {1, . . . , M }, xm is equal at most to ym+1 , and (b) for some m ∈ {1, . . . , M }, xm ̸= ym+1 . Define T :=
{(xm , ym ), (ym , xm )}.
m∈{1,...,M }
Let R ∈ B N be such that
∀i ∈ G1 : Ri = {(x1 , y2 ), (y2 , x2 ), (x2 , y3 ), . . . , (xM , y1 )}, ∀i ∈ G2 : Ri = {(x2 , y3 ), (y3 , x3 ), (x3 , y4 ), . . . , (x1 , y2 )}, .. . ∀i ∈ Gm : Ri = {(xm , ym+1 ), (ym+1 , xm+1 ), (xm+1 , ym+2 ), . . . , (xm−1 , ym )}, .. . ∀i ∈ GM : Ri = {(xM , y1 ), (y1 , x1 ), (x1 , y2 ), . . . , (xM −1 , yM )}, and
∀i ∈ G : Ri = {(x1 , y2 ), (x2 , y3 ), (x3 , y4 ), . . . , (xm , y1 )} ∪ T .
From (a) and the construction of profile R, it follows that for all i ∈ N , Ri is Suzumura consistent. Lemma 1 implies that Ri has a weak-order extension R′i . Therefore, there exists R′ ∈ RN such that for each i ∈ N , R′i is a weak-order extension of Ri . In the rest of this proof, we write y1 as yM +1 . Weak Pareto implies that
[xm ̸= ym+1 ] ⇒ (xm , ym+1 ) ∈ P (f (R′ )), (1) ′ because (xm , ym+1 ) ∈ P ( i∈N Ri ) whenever xm ̸= ym+1 . From (b), it follows that
(xm , ym+1 ) ∈ P (f (R′ )) for some m ∈ {1, . . . , M }.
(2)
Note that
(ym , xm ) ∈ P (Ri ) for all i ∈ N \ (Gm ∪ G), and
(ym , xm ) ∈ Ri for all i ∈ G. By (II), Gm is semi-impotent relative to G for (xm , ym ), and thus, it follows that (ym , xm ) ∈ f (R′ ). Then, we have
(ym , xm ) ∈ f (R′ ) for all m ∈ {1, . . . , M }.
(3)
By (1)–(3), f (R ) has an SC-cycle for some order. This contradicts the Suzumura consistency of f . The same argument can be applied for the case where G is the empty set. ′
It is noteworthy that Suzumura’s extension lemma is applied to prove Theorem 1. Schwartz (2007) imposes strong Pareto, while we impose weak Pareto in Theorem 1. Many studies in social choice theory employ weak Pareto, and hence this modification is useful when we apply Theorem 1. In the following two sections, we present two applications to traditional subjects in social choice theory. 4. Semi-decisive sets and Suzumura consistency This section examines the decisive structure of Suzumura consistent CCRs through the concept of ‘‘semi-decisiveness’’. Definition 1. A set D ⊆ N is (i) decisive for f if for all x, y ∈ X and for all R ∈ D , (x, y) ∈ i∈D P (Ri ), then (x, y) ∈ P (f (R)); (ii) semi-decisive for f if for all x, y ∈ X and for all R ∈ D , (x, y) ∈ i∈D P (Ri ), then (x, y) ∈ f (R).
Let Ω (f ) denote the family of decisive sets for f , and Ω ∗ (f ) denote the family of semi-decisive sets for f . By definition, a set D ⊆ N is semi-decisive for f if it is decisive for f . This implies that for every CCR f , Ω (f ) ⊆ Ω ∗ (f ). It is clear that if a set D ⊆ N is semi-decisive for f , then every superset D′ of D is also semi-decisive for f . As an auxiliary step, we introduce additional concepts. Let G be a family of subsets of N. A family G is a prefilter on N if (f.1) N ∈ G and ∅ ̸∈ G; (f.2) ∀G, G′ ⊆ N , [G ∈ G and G ⊆ G′ ] ⇒ G′ ∈ G; (f.3) ∀ K ≥ 1, ∀G1 , . . . , GK ⊆ N , [Gk ∈ G for all k ∈ {1, . . . , K }] ⇒ k∈{1,...,K } Gk ̸= ∅. A family G is a filter on N if (f.1); (f.2); (f.4) ∀G, G′ ⊆ N , [G ∈ G and G′ ∈ G] ⇒ G ∩ G′ ∈ G. A family G is an ultrafilter on N if (f.1); (f.2); (f.4); (f.5) ∀G ⊆ N , [G ̸∈ G] ⇒ N \ G ∈ G. By definition, an ultrafilter is a filter, and a filter is a prefilter. It is well known that one of the causes of Arrow’s impossibility theorem is the postulate of collective rationality. Arrow (1963) requires that a social preference generated by a CCR must be a weak order. Under this postulate, the family Ω (f ) of decisive sets
S. Cato / Mathematical Social Sciences 65 (2013) 40–47
forms an ultrafilter on N.9 This has very negative implications because every ultrafilter has a singleton element whenever N is a finite set. However, we have an escape route from this dead end in the absence of completeness of social preference. Indeed, the family Ω (f ) of decisive sets may not form an ultrafilter when completeness is dropped. Suzumura consistency is equivalent to transitivity in the presence of completeness, but Suzumura consistency is weaker than transitivity in the absence of completeness. Thus, we have a natural weakening of transitivity when social preference is allowed to be incomplete. Arrow’s independence of irrelevant alternatives has also been attacked by many researchers. It excludes many reasonable CCRs, such as the Borda rule and the Copeland rule. To allow such rules, we also drop the independence of irrelevant alternatives. To sum up, we examine a Suzumura consistent CCR that satisfies unrestricted domain and weak Pareto. Since Suzumura consistency is a necessary and sufficient condition for the existence of a weakorder extension, such a CCR guarantees the constructability of a Paretian welfare ordering. Thus, the underlying aim of this section is to investigate the decisive structure behind the Bergsonian school of welfare economics. The seminal works of Brown (1975) and Banks (1995) clarify the power structure of Paretian P-acyclical CCRs: Brown (1975) shows that if a P-acyclical CCR satisfies unrestricted domain, weak Pareto, and independence of irrelevant alternatives, then Ω (f ) forms a prefilter on N. Banks (1995) points out that the same result holds without independence of irrelevant alternatives.10 Since a Suzumura consistent CCR is P-acyclical, the Brown–Banks theorem immediately implies that the family of decisive sets forms a prefilter on N. To capture additional properties, we employ the concept of semi-decisiveness.11 The following result implies the usefulness of semi-decisiveness. Proposition 1. For every CCR f , if Ω ∗ (f ) forms a prefilter on N, then Ω (f ) forms a prefilter on N. Proof. Since (f.1) and (f.2) are obviously satisfied, it suffices to check (f.3). Suppose, to the contrary, that there exist K ≥ 1 and D1 , . . . , DK ∈ Ω (f ) such that k∈{1,...,K } Dk = ∅. Since Ω (f ) ⊆ Ω ∗ (f ), it follows that D1 , . . . , DK ∈ Ω ∗ (f ). This ∗ implies that D1 , . . . , DK ∈ Ω (f ) and k∈{1,...,K } Dk = ∅. This is a contradiction. We distinguish two cases: (i) the case where N is a finite set and (ii) the case where N is an infinite set. We first consider a collective decision making with a finite population. In this case, if the number of individuals is strictly smaller than the number of alternatives, then the family of semi-decisive sets forms a prefilter for any Suzumura consistent CCR. Theorem 1 is employed to prove this result.
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N is decisive, and thus, the family Ω (f ) of decisive sets for f is nonempty. Since Ω (f ) ⊆ Ω ∗ (f ), the family Ω ∗ (f ) of semi-decisive sets for f is nonempty. By way of contradiction, suppose that there exist K ≥ 1 and D1 , . . . , DK ∈ Ω ∗ (f ) such that k∈{1,...,K } Dk = ∅. This implies that
D = ∅.
D∈Ω ∗ (f )
Hence, every individual i ∈ N has a semi-decisive set Di that does not include him. It is clear that Di ⊆ N \ {i}. Then N \ {i} is semidecisive for f . Let us consider the partition {{1}, {2}, . . . , {n}} of N. Take distinct alternatives z 1 , . . . , z n , w, ∈ X . Since N \ {i} is semidecisive for f , {1} is semi-impotent relative to ∅ over (z 1 , w), and {i} is semi-impotent relative to ∅ over (z k , z k−1 ) for k ∈ {2, . . . , n}. Hence, we can apply Theorem 1, and thus, the proof is complete. Theorem 2 and Proposition 1 imply the following result. Corollary 1. Suppose that N is a finite set and that a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If |X | > |N |, then the family Ω (f ) of decisive sets for f is a prefilter on N. Definition 2. For each CCR f , (i) a collegium is a nonempty intersection of all decisive sets for f ; (ii) a weak collegium is a nonempty intersection of all semidecisive sets for f . Remark 1. A CCR f has a collegium if it has a weak collegium. As a corollary to Theorem 2, we have the following result. Corollary 2. Suppose that N is a finite set and a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If |X | > |N |, f has a weak collegium. Remark 2. Banks (1995) proves that if |X | ≥ |N |, then every P-acyclical CCR f satisfying weak Pareto and unrestricted domain is collegial. Corollary 2 and Remark 1 imply that if |X | > |N |, then every Suzumura consistent CCR f satisfying weak Pareto and unrestricted domain is collegial. Because Suzumura consistency is stronger than P-acyclicity, if |X | ≥ |N |, then every Suzumura consistent CCR f satisfying weak Pareto and unrestricted domain is collegial.
Proof. Note that (f.1) follows from weak Pareto, and (f.2) immediately follows from the definition of semi-decisiveness. Thus, it suffices to show that (f.3) holds. Weak Pareto implies that
Next, we examine the case with an infinite population. Recent works on intergenerational equity assume infinite future generations, and some of these works apply Szpilrajn’s or Suzumura’s extension lemmas. Hence, the analysis on Suzumura consistent CCR with the infinite population case is of interest because it might provide a mathematical tool for such a subject. In this case, the family of semi-decisive sets forms a prefilter for any Suzumura consistent CCR whenever the set of alternatives is an infinite set. It is noteworthy that the following result holds under a/an (countably or uncountably) infinite population.
9 This result was independently proved by Kirman and Sondermann (1972) and Hansson (1976). 10 See also Packel (1984) and Blau and Brown (1989). Packel (1984) examines the
Theorem 3. Suppose that N is an infinite set and that a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If X is an infinite set, then the family Ω ∗ (f ) of semi-decisive sets for f is a prefilter on N.
Theorem 2. Suppose that N is a finite set and a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If |X | > |N |, then the family Ω ∗ (f ) of semi-decisive sets for f is a prefilter on N.
decisive structure of Paretian social choice rules based on ‘‘rejection’’ decisiveness. 11 There exists an article that examines the family of semi-decisive sets. Campbell and Kelly (2000) show that if a complete and transitive CCR satisfies unrestricted domain, weak Pareto, and weak independence of irrelevant alternatives, then Ω ∗ (f ) forms an ultrafilter on N.
Proof. Note that (f.1) follows from weak Pareto, and (f.2) immediately follows from the definition of semi-decisiveness. Thus, it suffices to show that (f.3) holds. By way of contradiction, suppose that there exist K ≥ 1 and D1 , . . . , DK ∈ Ω ∗ (f ) such that
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S. Cato / Mathematical Social Sciences 65 (2013) 40–47
1 K k∈{1,...,K } Dk = ∅. Let z , . . . , z , w be distinct alternatives in X . Since X is an infinite set, we can choose them for any finite number K . Let us define
G1 := N \ D1 , Gk := D1 ∩ D2 ∩ · · · ∩ Dk−1 − Dk
Sen’s minimal liberalism is formally restated as follows. Minimal liberalism: There exists a symmetric rights system (Di )i∈N such that (i) for some distinct i, j ∈ N , Di , Dj ̸= ∅; (ii) a CCR f fulfills (Di )i∈N .
for k ∈ {2, . . . , K }.
Let k ∈ {2, . . . , K }. Since Dk is decisive for f and Dk ⊆ N \ Gk , N \ Gk is decisive for f . Hence, G1 is semi-impotent relative to ∅ over (z 1 , w), and Gk is semi-impotent relative to ∅ over (z k , z k−1 ) for each k ∈ {2, . . . , K }. This implies that weak impotence partition is satisfied. By Theorem 1, f generates a social preference with an SC-cycle in some order for some profile R ∈ RN . In the infinite population case, the infiniteness of X is necessary for Theorem 3.12 To clarify this point, we show that the family of semi-decisive sets may not form a prefilter if X is a finite set. Suppose that N = N and X is a finite set with |X | = s. Let H := {1, . . . , s, s + 1} ⊆ N. Define
Sen (1970) establishes the following theorem: there exists no P-acyclical CCR that satisfies unrestricted domain, weak Pareto, and minimal liberalism. This theorem indicates that individual liberty can conflict with the Pareto principle. It is known as the ‘‘impossibility of Paretian liberal’’ (or the liberal paradox). Sen’s paradox has been extended by many researchers. Among others, Gibbard (1974) clarifies the possibility that minimal liberalism can be self-contradictory. We now illustrate this problem. Consider a system of rights (Di )i∈N where there exist four distinct alternatives x, y, z , w ∈ X such that (x, y), (z , w) ∈ D1 and (y, z ), (w, x) ∈ D2 . Let R ∈ RN be such that
B(x, y; R) := {i ∈ H |(x, y) ∈ P (Ri )}.
(x, y) ∈ P (R1 ) and (z , w) ∈ P (R1 )
By construction, B(x, y; R) ≤ s + 1 for all x, y ∈ X and all R ∈ R . Define fˆ as follows: for all x, y ∈ X and all R ∈ RN , N
and
(y, z ) ∈ P (R2 ) and (w, x) ∈ P (R2 ).
(x, y) ∈ fˆ (R) ⇔ B(x, y; R) ≥ s .
If a CCR f fulfills this system of rights, it follows that
Social preferences generated by CCR fˆ are equivalent to those
{(x, y), (y, z ), (z , w), (w, x)} ⊆ P (f (R)).
generated by one of the S-rules on an H-individual society, which are introduced by Bossert and Suzumura (2008, 2010).13 As shown by Bossert and Suzumura (2008, 2010), every S-rule always generates a Suzumura consistent social preference, so that fˆ is Suzumura consistent. However, every subset H ′ of H with s members is semi-decisive, and thus, there exist K ≥ 1 and D1 , . . . , DK ∈ Ω ∗ (fˆ ) such that k∈{1,...,K } Dk = ∅. Therefore, Ω ∗ (fˆ ) is not a prefilter on N. This argument can be applied for any infinite set N. Theorem 3 and Proposition 1 imply the following result. Corollary 3. Suppose that N is an infinite set and that a Suzumura consistent CCR f satisfies unrestricted domain and weak Pareto. If X is an infinite set, then the family Ω (f ) of decisive sets for f is a prefilter on N. 5. Weak form of liberalism and Suzumura consistency In this section, we investigate the implication of Suzumura consistency in the problem of Paretian liberalism. According to Sen’s minimal liberalism, ‘‘[t]here are at least two individuals such that for each of them there is at least one pair of alternatives over which he is decisive, that is, there is a pair x, y, such that if he prefers x (respectively y) to y (respectively x), then society should prefer x (respectively y) to y (respectively x) (Sen, 1970, p. 153)’’. A system of rights is (Di )i∈N where Di ⊆ X × X \ ∆ for each i ∈ N. A system of rights (Di )i∈N is symmetric if for all i ∈ N, and x, y ∈ X ,
(x, y) ∈ Di ⇒ (y, x) ∈ Di . Given (Di )i∈N , a CCR f fulfills (Di )i∈N if for all i ∈ N, for all (x, y) ∈ Di , and for all R ∈ RN ,
(x, y) ∈ P (Ri ) ⇒ (x, y) ∈ P (f (R)).
The CCR may generate the cycle of a strict social preference. Hence, Sen-type liberalism and unrestricted domain are not compatible under some systems of rights. To avoid this problem, the assignment of rights has to be more restrictive. Suzumura (1978, 1983) examines under what condition there is a P-acyclical CCR fulfilling a system of individual claim rights. He shows that the coherency of the system of rights, originally by Farrell (1976), is a necessary and sufficient condition for the existence of a P-acyclical CCR satisfying minimal liberalism. We now consider a weak form of liberalism. Given (Di )i∈N , a CCR f weakly fulfills (Di )i∈N if for all i ∈ N, for all (x, y) ∈ Di , and for all R ∈ RN ,
(x, y) ∈ P (Ri ) ⇒ (x, y) ∈ f (R). Weak minimal liberalism: There exists a symmetric rights system (Di )i∈N such that (i) for some distinct i, j ∈ N , Di , Dj ̸= ∅; (ii) a CCR f weakly fulfills (Di )i∈N . This axiom is proposed by Aldrich (1977), and it has been recently investigated by Salles (2008). Weak minimal liberalism requires that there exist at least two individuals who have veto power over some pair (x, y) in both directions. Note that in the case of weak minimal liberalism, the coherency of the rights assignment is not problematic: for any system of rights, there always exists a P-acyclical CCR satisfying weak minimal liberalism. Moreover, we can obtain the following possibility result: there exists a quasitransitive CCR that satisfies unrestricted domain, weak Pareto, and weak minimal liberalism. Hence, there exists a P-acyclical CCR satisfying the three axioms. To see this fact, consider the following CCR f : for all x, y ∈ X and for all R ∈ RN ,
(x, y) ∈ f (R) ⇔ (y, x) ̸∈ P
Ri
.
i∈N
12 If X is an infinite set, one might think that the unrestricted domain is too strong because it contains a profile wherein individuals do not have a maximal element on X . Indeed, in some applications, it would be reasonable to restrict attention to profiles where each individual has a maximal alternative. All results in this paper hold under the domain assumption that all profiles where each individual has a maximal element are available. 13 See also Cato and Hirata (2010).
This CCR is the Pareto-extension rule. It is well-known that this rule is quasi-transitive. Under this CCR, Di = X × X for all i ∈ N. Thus, for any rights system, we have a CCR that weakly fulfills it. Aldrich (1977) shows that there exists no transitive and complete CCR satisfying the three axioms. Moreover, Salles (2008) proves that similar impossibility results hold for semi-transitivity
S. Cato / Mathematical Social Sciences 65 (2013) 40–47
or interval order property under a natural restriction on a rights system. These rationality requirements are stronger than quasitransitivity. A similar negative result holds on a Suzumura consistent CCR. There exists no Suzumura consistent CCR f satisfying unrestricted domain, weak Pareto, and weak minimal liberalism provided that |X | ≥ 4 and that there exist distinct i, j ∈ N and distinct x, y, z , w ∈ X such that (x, y) ∈ Di and (z , w) ∈ Dj in the definition of weak minimal liberalism.14 To prove this, it suffices to consider a partition {{i}, N \ {i}} of N. Since f weakly fulfills (Di )i∈N , {i} is semi-impotent relative to ∅ over (w, z ) and N \ {i} is semiimpotent relative to ∅ over (y, x). By Theorem 1, f generates a social preference with an SC-cycle in some order for some profile R ∈ RN . We now specify a necessary and sufficient condition on a rights system for the existence of a Suzumura consistent CCR satisfying unrestricted domain, weak Pareto, and weak liberalism. As an auxiliary step, we introduce a variant of weak minimal liberalism. Weak minimal liberalism∗ : There exists a rights system (Di )i∈N such that a CCR f weakly fulfills (Di )i∈N . We introduce an additional requirement on a rights system. A rights system (Di )i∈N is contagious if either (i) there exist distinct i, j ∈ N and distinct x, y, z , w ∈ X such that (x, y) ∈ Di and (z , w) ∈ Dj or (ii) there exist distinct i, j ∈ N and distinct x, y, z ∈ X such that (x, y) ∈ Di and (y, z ) ∈ Dj . Theorem 4. There exists no Suzumura consistent CCR f satisfying unrestricted domain, weak Pareto, and weak minimal liberalism∗ if and only if the corresponding rights system (Di )i∈N is contagious. Proof. ‘Only if’. It suffices to show that there exists a Suzumura consistent CCR f satisfying unrestricted domain, weak Pareto, and weak minimal liberalism if the corresponding rights system (Di )i∈N is not contagious. Define f as follows: for all x, y ∈ X , and for all R ∈ RN ,
f (R) ⇔
(x, y) ∈ P
i∈N
Ri
or (x, y) ̸∈ P
Ri
i∈N
and (x, y) ∈ Dj ∩ P (Rj ) for some j ∈ N
.
Note that P (f (R)) = f (R) for all R ∈ RN . We will show that this CCR f is Suzumura consistent. By way of contradiction, suppose not. Then, there exist R ∈ RN , K ≥ 1, and distinct x0 , . . . , xK such that
k−1 k (x , x ) ∈ f (R) for k ∈ {1, . . . , K } and (xK , x0 ) ∈ P (f (R)). By construction, (xK , x0 ) ∈ i∈N P (Ri ). We now say that individual j exercises his weak rights on pair (xk−1 , xk ) if (xk−1 , xk ) ̸∈ P ( i∈N Ri ) and (xk−1 , xk ) ∈ Dj ∩ P (Rj ). We distinguish two cases. Case (a): there exists no individual who exercises his weak rights on somepair in {(x0 , x1 ), . . . , (xK −1 , xK )}. In this case, −1 k (xk , x ) ∈ P ( i∈N Ri ) for all k ∈ {1, . . . , K }, so that (x0 , xK ) ∈ P ( i∈N Ri ). This contradicts the supposition that (xK , x0 ) ∈ P (f (R)). Case (b): there exists only one individual who exercises his weak rights on some pair in {(x0 , x1 ), . . . , (xK −1 , xK )}. Let j ∈ N be an individual who exercises his weak rights on some pair in {(x0 , x1 ), . . . , (xK −1 , xK )}. Define
Kj := {k ∈ {1, . . . , K }| j exercises his weak rights on (xk−1 , xk )}.
14 This restriction on a rights system is essentially the same as that introduced by Salles (2008).
45
By construction, (xk−1 , xk ) ∈ Dj ∩ P (Rj ) for all k ∈ Kj and (xk−1 , xk ) ∈ Rj for all k ∈ {1, . . . , K } \ Kj . Since individual j’s preference is a weak order, we have (x0 , xK ) ∈ P (Rj ). This contradicts the fact that (xK , x0 ) ∈ i∈N P (Ri ). Hence, either there exist distinct i, j ∈ N and distinct x, y, z , w such that (x, y) ∈ Di and (z , w) ∈ Dj or there exist distinct i, j ∈ N and distinct x, y, z such that (x, y) ∈ Di and (y, z ) ∈ Dj . This implies that (Di )i∈N is contagious. This is a contradiction. ‘If’. Let f be a Suzumura consistent CCR that satisfies weak Pareto and weak minimal liberalism with a contagious rights system (Di )i∈N . We distinguish two cases. Case (i): there exist distinct four alternatives x, y, z , w ∈ X such that (x, y) ∈ Di and (z , w) ∈ Dj . As discussed before, Theorem 1 implies that f generates a social preference with an SC-cycle for some profile R ∈ RN . Case (ii): there exist distinct three alternatives x, y, z ∈ X such that (x, y) ∈ Di and (y, z ) ∈ Dj . Since f weakly fulfills (Di )i∈N , {i} is semi-impotent relative to ∅ over (z , y), and N \ {i} is semiimpotent relative to ∅ over (y, x). From Theorem 1, f generates a social preference with an SC-cycle in some order for some profile R ∈ RN . Since a rights system (Di )i∈N that is not contagious is very restrictive, the message of Theorem 4 is negative: in most cases, we cannot construct a Suzumura consistent CCR on the unrestricted domain satisfying weak Pareto and weak minimal liberalism. We now consider how we can resolve the difficulty stated by Theorem 4. The approach that we use here was introduced by Sen (1976) and developed by Suzumura (1978, 1983).15 They examine the existence of a P-acyclical CCR satisfying minimal liberalism. They weaken the Paretian requirement and show that if there exists an individual who respects others’ rights (liberal individual), there exists a P-acyclical CCR satisfying the desireble axioms. We focus on a Suzumura consistent CCR and therefore reformulate the concept of a ‘‘liberal’’ individual. We now assume that for each profile R ∈ RN , there is a profile (R∗i )i∈N such that, for each i ∈ N, tc (R∗i ) = R∗i ,
R∗i ⊆ Ri ,
and
P (R∗i ) ⊆ P (Ri ).
Given a preference Ri of individual i ∈ N , R∗i is a transitive subrelation of Ri that individual i wants to count in collective choice (R∗i = Ri is possible). ∗ ∗ Conditional weak Pareto: For all x, y ∈ X and for all R = (Ri )i∈N , if [(x, y) ∈ i∈N P (R∗i )], then (x, y) ∈ P (f (R)). Define
Qi := Di ∩ Ri , and
S := cc
Qi
.
i∈N
It is noteworthy that (x, y) ∈ Qi if and only if (x, y) is a protected personal pair of individual i and he prefers x to y. Therefore, i∈N Qi represents individuals’ claims on their rights, but it can be cyclical without coherency of the rights system. This cyclicity can be resolved by the consistent closure. Recall that for each R, cc (R) = R ∪ {(x, y)|(y, x) ∈ R and (x, y) ∈ tc (R)}. By Lemma 2, cc ( i∈N Qi )is the smallest Suzumura consistent binary relation containing i∈N Qi . Note that for all i ∈ N, if (x, y) ∈ Di and (x, y) ∈ P (Ri ), then (x, y) ∈ S. By construction, it is possible that (x, y) ∈ Di , (x, y) ∈ P (Ri ), (x, y) ∈ S, and (y, x) ∈ S. Then, S weakly fulfills individual rights. Hence, S represents the ‘‘weak’’
15 See also Austen-Smith (1984).
46
S. Cato / Mathematical Social Sciences 65 (2013) 40–47
form of libertarian claims, which is Suzumura consistent. Since S is Suzumura consistent, it has a weak-order extension S¯ by Lemma 1. An individual i ∈ N is S-liberal if and only if for each profile R ∈ RN , R∗i is such that there exists a weak-order extension Ti of which R∗i is a subrelation and so is S. An S-liberal individual claims the parts of his preference to count which are compatible with the weak form of libertarian claims S. Since S is Suzumura consistent, such a weak-order extension always exists. It is noteworthy that the existence of an S-liberal individual is a strong restriction on a profile R∗ and the CCR f (·), not just on R∗i . In particular, a weak-order extension Ti respects the rights system (Di )i∈N that is generated from the CCR f (·). We now offer a possibility result on a Suzumura consistent CCR. If there exists at least one S-liberal individual, we can construct a Paretian Suzumura consistent CCR on the unrestricted domain that weakly fulfills any arbitrary given rights system.
of acyclical social choice by Brown (1973, 1974, 1975), Nakamura (1979) and Banks (1995), who focus on the structure of decisive sets (or winning coalitions).17 We then considered a weak version of Sen’s (1970) minimal liberalism. Weak liberalism requires that there exist at least two semi-decisive individuals (vetoers). A Sen-type negative result is proved: there exists no Suzumura consistent CCR satisfying unrestricted domain, weak Pareto, and weak liberalism. Moreover, we provided a resolution for this problem. Our results on two subjects suggest that semi-decisiveness works effectively for a Suzumura consistent CCR. Through two applications, we examined the possibility of democratic and liberal social welfare weak orders.
Theorem 5. Let (Di )i∈N be an arbitrary given rights system. If there exists at least one S-liberal individual, then there exists a Suzumura consistent CCR f that satisfies unrestricted domain and conditional weak Pareto and that weakly fulfills the rights system (Di )i∈N .
I thank Katsuhito Iwai, François Maniquet, and an anonymous referee of the journal for their valuable suggestions, and I thank Kotaro Suzumura for the helpful conversations over several years on related issues. This research was financially supported by Grant-in-Aids for Young Scientists (B) from the Japan Society for the Promotion of Science and the Ministry of Education, Culture, Sports, Science and Technology.
Proof. Without loss of generality, we can suppose that individual 1 is S-liberal. Define a CCR f as follows: for all x, y ∈ X , and for all R,
(x, y) ∈ P (f (R)) ⇔ (x, y) ∈
P (R∗i ) or (x, y) ∈ P (S ),
i∈N
(x, y) ∈ I (f (R)) ⇔ (x, y) ∈ I (S ). Conditional weak Pareto and weak minimal liberalism follow from construction of the CCR f . Thus, it suffices to show that the CCR f is Suzumura consistent. By way of contradiction, suppose that there exist K ≥ 1 and x0 , . . . , xK such that
(xk−1 , xk ) ∈ f (R) ∀k ∈ {1, . . . , K } and (xK , x0 ) ∈ P (f (R)). For convenience, we write (xK , x0 ) as (xK , xK +1 ). The binary relation S is Suzumura consistent, and thus, we have (xk−1 , xk ) ∈ ∗ k−1 , xk ) ∈ i∈N P (Ri ) for some k ∈ {1, . . . , K }. This implies that (x ∗ P (R1 ). However, there must exist a weak-order extension T1 of which R∗1 is a subrelation and so is S because individual 1 is Sliberal. This is a contradiction. To conclude this section, we comment on the underlying aim. The motivation of this section is to examine how we can incorporate liberalism into Bergsonian welfare economics. Most studies on the assignment of individual rights assume the acyclicity of strict social preference.16 However, P-acyclicity is insufficient to construct a social welfare ordering. Theorem 4 states the difficulty of a Paretian social welfare ordering respecting the weak form of liberalism, and Theorem 5 provides an escape route from this problem.
Acknowledgments
Appendix In Section 3, we showed that Schwartz’s (2007) sufficiency theorem can be extended to Suzumura consistent CCRs. Schwartz (2007) also provides a necessary theorem. He shows that under a weak background assumption, if a CCR satisfying what he call mild independence is P-cyclical, it satisfies impotence partition. This section briefly discusses an extension of the necessary theorem. The following axiom is a variant of Schwartz’s mild independence. Weak independence: For all R, R′ ∈ RN , if (x, y) ∈ P (R) ⇒ (x, y) ∈ P (R′ ) and (y, x) ∈ P (R′ ) ⇒ (y, x) ∈ P (R), then (x, y) ∈ f (R) ⇒ (x, y) ∈ f (R′ ). We present a necessary theorem on Suzumura consistency. Theorem 6. Suppose that a CCR f satisfies weak independence and there is R ∈ RN such that f (R) has an SC-cycle and (x, y) ∈ P (Ri ) for some i ∈ N and for some pair (x, y) in the cycle. Then, it satisfies weak impotence partition. Sketch of Proof. Following the same argument as in Schwartz (2007), there is a finite sequence {z1 , . . . , zM +1 } of distinct alternatives that is an SC-cycle and (zm , zm+1 ) ∈ P (Ri ) for some m and for some i ∈ N. Define
6. Concluding remarks
G := {i ∈ N |(zm−1 , zm ) ∈ I (Ri ) for all m},
This paper examined the implications of Suzumura consistency in a collective decision situation. By establishing a general result on Suzumura consistent CCRs, which is an extension of Schwartz (2007), we addressed two classical subjects: decisive structures and the assignment of individual rights. Throughout the paper, we applied the concept of semi-decisiveness: a semi-decisive set is a group that has veto power. We first considered the family of semi-decisive sets associated with a Suzumura consistent CCR, and proved that it forms a prefilter. This result is related to the studies
G1 := {i ∈ N |(z2 , z1 ) ∈ P (Ri )},
16 Exceptions are Aldrich (1977) and Salles (2008).
Gm := {i ∈ N |(zm+1 , zm ) ∈ P (Ri )} for all m ≥ 2. We can prove that {G, G1 , . . . , Gm } is a partition and Gm is semiimpotent relative to G. By letting (x1 , . . . , xm ) = (z2 , . . . , zm+1 ) and (y1 , . . . , ym ) = (z1 , . . . , zm ), we can derive weak impotence partition.
17 See also Blair and Pollak (1982), who investigate the existence of an individual who has veto power in acyclic social choice. Obviously semi-decisiveness relates to veto power: a semi-decisive set represents a coalition that has veto power.
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