Incomplete decision-making and Arrow’s impossibility theorem

Mathematical Social Sciences (

)

–

Contents lists available at ScienceDirect

Mathematical Social Sciences journal homepage: www.elsevier.com/locate/mss

Incomplete decision-making and Arrow’s impossibility theorem Susumu Cato *,1 Institute of Social Science, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Woodrow Wilson School of Public and International Affairs, Princeton University, Robertson Hall, 20 Prospect Ave, Princeton, NJ 08540, United States

highlights • • • •

This paper is concerned with social choice without completeness of social preference. We introduce the concept of minimal comparability. Complete silence should be avoided according to minimal comparability. There exists no normatively desirable aggregation rule satisfying minimal comparability.

article

info

Article history: Received 20 May 2016 Received in revised form 14 October 2016 Accepted 10 October 2017 Available online xxxx

a b s t r a c t This paper is concerned with social choice without completeness of social preference. Completeness requires that pairs of alternatives are perfectly comparable. We introduce the concept of minimal comparability, which requires that for any profile, there is some comparable pair of distinct alternatives. Complete silence should be avoided according to this condition. We show that there exists no normatively desirable aggregation rule satisfying minimal comparability. © 2017 Elsevier B.V. All rights reserved.

1. Introduction This paper examines incompleteness of preferences underlying choice. In a series of writings, Sen (1970, 1985, 1992, 2004) criticized imposing completeness on individual or social preferences. According to him, the postulate is too demanding because there exist cases where no decision/judgment is relevant (Sen, 2004). Several recent works focus on characterizations of incomplete preferences and corresponding choice behavior (Dubra et al., 2004; Eliaz and Ok, 2006; Evren and Ok, 2011; Mandler, 2004; Mandler, 2005; Mandler, 2009; Ok, 2002).2 These studies suggest that multi-utility representations can be obtained for incomplete preferences.3 A multi-utility representation is clearly related to the problem of collective decision-making. In collective decision models, multi-

*

Correspondence to: Institute of Social Science, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan. E-mail address: [email protected]. 1 Postdoctoral Fellow for Research Abroad of the Japan Society for the Promotion of Science. 2 A pioneering work is presented by Aumann (1962). 3 Dubra et al. (2004) provide a multi-utility representation for incomplete preferences over lotteries. Evren and Ok (2011) and Ok (2002) consider preferences over alternatives.

criteria or individual rankings are agglomerated into unified collective preferences. Here, incomplete preferences can be regarded as a consequence of preference aggregation. We conduct collective choice without completeness. The benchmark is Arrow’s impossibility theorem, which states that there exists no preference aggregation method satisfying unanimity, the independence of irrelevant alternatives, and nondictatorship (Arrow, 1963). In Arrow’s original framework, social preferences are required to be transitive and complete. Transitivity requires that social preferences be coherent, while completeness requires that all pairs of alternatives must be comparable for every preference profile. It is known that if the completeness of social preference is dropped, the impossibility does not hold: there exists an aggregation rule satisfying unanimity, the independence of irrelevant alternatives, and non-dictatorship.4 An example is the so-called Pareto rule, which makes a social judgment only if there is consensus among individuals (or multi-criteria). This rule is corresponding to the representation procedure employed by Dubra et al. (2004), Evren and Ok (2011), and Ok (2002). According to the Pareto rule, social preference is silent for a pair of alternatives over which there is a conflict of individual interests. Therefore, if one individual completely disagrees with other individuals, then 4 See Weymark (1984) and Bossert and Suzumura (2008).

https://doi.org/10.1016/j.mathsocsci.2017.10.002 0165-4896/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: Cato S., Incomplete decision-making and Arrow’s impossibility theorem. Mathematical Social Sciences (2017), https://doi.org/10.1016/j.mathsocsci.2017.10.002.

2

S. Cato / Mathematical Social Sciences (

no pair of alternatives is comparable under the Pareto rule. The function of the rule is stopped completely by a conflict between two individuals. In this paper, we drop the postulate of completeness, and instead impose a substantially weaker postulate: minimal comparability. This requires that some pair must be comparable for every preference profile. That is, no social judgment is prohibited under the axiom. Its requirement is relevant to social decision-making. We show that Arrow’s impossibility theorem is still valid, even if completeness is weakened to minimal comparability: there exists no transitive aggregation rule that satisfies unanimity, the independence of irrelevant alternatives, non-dictatorship, and minimal comparability. Minimal comparability has a similar meaning to nearcompleteness, which is introduced by Ok (2002). Near-completeness requires that a preference is not too incomplete in the sense that a measurement of incompleteness is not sufficiently large. Roughly speaking, a completely non-comparable subset of a preference relation should be finite. Both our minimal comparability and Ok’s near-completeness restrict the degree of incompleteness. Minimal comparability requires that only one pair be comparable. Intuitively, minimal comparability is weaker than near-completeness when the set of alternatives is infinite. However, it is stronger than near-completeness when the set of alternatives is finite. Moreover, we extend our analysis to cases with weaker coherence properties, such as semi-transitive properties and Suzumura consistency. We find that impossibility results are still valid under these properties. However, it is possible to construct an Arrovian aggregation rule that generates a complete social preference under quasi-transitivity. Therefore, the types of intransitivity matter. Now, we explain how our collective choice approach is different from the multi-utility representation approach of Dubra et al. (2004), Evren and Ok (2011), and Ok (2002). The crucial point is that the domain of an aggregation rule includes multiple profiles. In the multi-utility representation approach, a profile of multiple utility functions is unique and fixed. On the other hand, multiple profiles of individual presences are considered simultaneously in the collective choice approach, and their connection is required by the independence of irrelevant alternatives. The remainder of this paper is organized as follows. Section 2 formulates our setting and introduces Arrovian axioms. Section 3 presents our results. Section 4 concludes the paper. 2. Preliminaries 2.1. Arrovian framework Let X be the set of alternatives. We assume that there exist at least three alternatives. Let R be a binary relation on X , and thus, R is a subset of X × X . Let B be the set of binary relations on X . The symmetric part of R is defined as follows: I(R) = {(x, y) ∈ X × X : (x, y) ∈ R and (y, x) ∈ R}.

)

–

We mention several properties of binary relations.5 Transitivity: For all x, y, z ∈ X ,

[(x, y) ∈ R and (y, z) ∈ R] ⇒ (x, z) ∈ R. Quasi-transitivity: For all x, y, z ∈ X ,

[(x, y) ∈ P(R) and (y, z) ∈ P(R)] ⇒ (x, z) ∈ P(R). S-consistency:6 For all K ∈ N and for all x0 , x1 , . . . , xK ∈ X ,

[(xk−1 , xk ) ∈ R for all k ∈ {1, . . . , K }] ⇒ (xK , x0 ) ̸∈ P(R). Completeness: For all x, y ∈ X , (x, y) ∈ R or (y, x) ∈ R. Reflexivity: For all x ∈ X , (x, x) ∈ R. Antisymmetry: For all x, y ∈ X ,

[(x, y) ∈ R and (y, x) ∈ R] ⇒ x = y. Transitivity is stronger than quasi-transitivity and Suzumura consistency. Quasi-transitivity and Suzumura consistency are independent. Reflexivity follows from completeness. A binary relation R is called an ordering if it satisfies transitivity and completeness; a binary relation R is called a linear ordering if it satisfies transitivity, completeness, and antisymmetry. Let R be the set of orderings on X . Let N be the set of individuals in a society. We assume that N is finite. Each individual i ∈ N in the society has a preference ordering Ri on X . A preference profile (Ri ) = R ∈ RN is a list of individual preferences. A collective choice rule is a function f from RN to B.7 A group A ⊆ N is decisive over (x, y) for f if for all R ∈ RN , (x, y) ∈

⋂

P(Ri ) ⇒ (x, y) ∈ P(f (R)).

i∈A

A group A ⊆ N is decisive for f if it is decisive for all (x, y) ∈ X × X . A group A ⊆ N is almost decisive over (x, y) for f if for all R ∈ RN ,

[(x, y) ∈

⋂ i∈A

P(Ri ) and (y, x) ∈

⋂

P(Ri )] ⇒ (x, y) ∈ P(f (R)).

i∈N \A

Let Df (x, y) be the set of decisive groups over (x, y) for f and Df be the set of decisive groups for f . Let Df− (x, y) be the set of groups almost decisive over (x, y). 2.2. Arrow’s impossibility theorem In this section, we review Arrow’s impossibility theorem. First, we introduce axioms for f . Unanimity: For all x, y ∈ X , and for all R ∈ RN , (x, y) ∈

⋂

P(Ri ) ⇒ (x, y) ∈ P(f (R)).

i∈N

Independence of irrelevant alternatives: For all R, R′ ∈ RN , and for all x, y ∈ X , R|{x,y} = R′ |{x,y} ⇒ f (R)|{x,y} = f (R′ )|{x,y} .

The asymmetric part of R is defined as follows:

Non-dictatorship: There exists no i ∈ N such that for all x, y ∈ X , and for all R ∈ RN ,

P(R) = {(x, y) ∈ X × X : (x, y) ∈ R and (y, x) ̸ ∈ R}.

(x, y) ∈ P(Ri ) ⇒ (x, y) ∈ P(f (R)).

Note that R = I(R) ∪ P(R). The set of non-comparable pairs is defined as follows:

Transitivity: f (R) satisfies transitivity for all R ∈ RN . Completeness: f (R) satisfies completeness for all R ∈ RN .

N(R) = {(x, y) ∈ X × X : (x, y) ̸ ∈ R and (y, x) ̸ ∈ R}. The dual of R is defined as follows: d(R) = {(x, y) ∈ X × X : (y, x) ∈ R}. Let ∆ be the diagonal relation: ∆ = {(x, y) ∈ X × X : x = y}.

5 See Cato (2016) for general explanation of properties of binary relations. 6 S-consistency is introduced by Suzumura (1976). Bossert and Suzumura (2010) examine the properties and implications of S-consistency. 7 Aleskerov (1999) introduces a more general framework of preference aggregation. He shows various types of characterizations of aggregation procedures.

Please cite this article in press as: Cato S., Incomplete decision-making and Arrow’s impossibility theorem. Mathematical Social Sciences (2017), https://doi.org/10.1016/j.mathsocsci.2017.10.002.

S. Cato / Mathematical Social Sciences (

Then, Arrow’s theorem is stated as follows.

)

–

3

Proof. f P (R) = ∆ is equivalent to N(f P (R)) = X × X \ ∆. Note that

[

Theorem 1 (Arrow, 1963). There exists no collective choice rule f that satisfies unanimity, independence of irrelevant alternatives, nondictatorship, transitivity, and completeness. Usually, a collective choice rule satisfying transitivity and completeness is called a social welfare function. Thus, a more familiar statement of the theorem is as follows: there exists no social welfare function f that satisfies unanimity, independence of irrelevant alternatives, and non-dictatorship. The core of the proof of the theorem is known as the field expansion lemma.8 The lemma is valid under weaker rationality axiom, which requires social preference to be quasi-transitive. Quasi-transitivity: f (R) satisfies quasi-transitivity for all R ∈ RN . The statement of the lemma is given as follows. Lemma 1 (Sen, 1995). Suppose that a collective choice rule f satisfies unanimity, independence of irrelevant alternatives, and quasitransitivity. If A ∈ Df− (x, y) for some x, y ∈ X (x ̸ = y), then A ∈ Df . According to Lemma 1, the decisive power is neutral over alternatives. The point is that completeness is not imposed in Lemma 1.9 However, completeness is necessary for Arrow’s theorem. If we drop the axiom, we can obtain a possible result. The following rule, called the Pareto rule, satisfies all Arrovian axioms other than completeness: f P (R) =

⋂

Ri .

i∈N

Under the Pareto rule, x is socially better than y when x is at least as good as y for all individuals and x is better than y for some individual. That is, this rule makes a social judgment when there is a consensus among individuals. 3. Arrow’s impossibility theorem and incompleteness 3.1. The pareto rule and complete silence The difficulty of the Pareto rule is the frequent occurrence of silence over collective decisions. Suppose that individual 1 has a linear preference over alternatives and individual 2 has the opposite preference. Then,

[(x, y) ∈ R1 and (y, x) ∈ R1 ] ⇒ x = y, and R2 = d(R1 ). It is clear that f P (R) = ∆. For this class of profiles, the Pareto rule does not make a judgment for any pair of distinct alternatives: the rule does not work. Proposition 1. Given R ∈ RN , f P (R) = ∆ if and only if for all x, y ∈ X with x ̸ = y, there exists i, j ∈ N such that (x, y) ∈ P(Ri ) ∩ d(P(Rj )). 8 The complete proof can be found in Arrow (1963), Feldman and Serrano (2006), Sen (1995), and Suzumura (2000). Recently, many authors have provided alternative proofs of Arrow’s theorem without the field expansion lemma. See, for example, Barberá (1983), Cato (2010, 2013b), Geanakoplos (2005), Reny (2001), Ubeda (2003), and Yu (2012). 9 This observation is the key to establishing the oligarchy theorem or the existence of a vetoer (Mas-Colell and Sonnenschein, 1972; Gibbard, 2014).

(x, y) ∈ N(f P (R)) ⇔ (x, y) ̸ ∈

⋂

Ri and (y, x) ̸ ∈

i∈N

⋂ ] Ri . i∈N

There exists (y, x) ̸ ∈ Ri and (x, y) ̸ ∈ Rj if and only if ⋂ i, j ∈ N such that⋂ (x, y) ̸ ∈ i∈N Ri and (y, x) ̸ ∈ i∈N Ri . Therefore, because individual preferences are complete, we have (x, y) ∈ N(f P (R)) ⇔ (x, y) ∈ P(Ri ) and (y, x) ∈ P(Rj ) for some i.j ∈ N . Thus, we have the claim. ■ This proposition implies that when the number of individuals is sufficiently large, it is highly likely that f P (R) = ∆. 3.2. Arrow’s impossibility theorem under minimal comparability The following axiom requires that there exists a pair of alternatives that can be comparable for all preference profiles. The existence of a comparable pair: There exist distinct x, y ∈ X such that (x, y) ∈ f (R) ∪ d(f (R)) for all R ∈ RN . There exists a special pair under the existence of a comparable pair. A practical problem is choice of this special pair. That is, what factors affect the comparability of the pair. The point is that the pair should be comparable independently of individual preferences. Therefore, exogenous factors outside of preferences, such as historical context or procedures, are crucial for comparability. The existence of a comparable pair seems to be too demanding because it needs a special treatment of some pairs unless completeness is satisfied. We need more weak and natural axiom. The following axiom requires that for all preference profiles, there exists a comparable pair of alternatives. Minimal comparability: For all R ∈ RN , there exist distinct x, y ∈ X such that (x, y) ∈ f (R) ∪ d(f (R)). Another expression of minimal comparability is as follows: f (R) \ ∆ ̸ = ∅ for all R ∈ RN . The following result is intermediately obtained from definitions. Lemma 2. If f satisfies the existence of a comparable pair, then it satisfies minimal comparability. The following is our main result, which states that Arrow’s impossibility is still valid when completeness is weakened to minimal comparability. Theorem 2. There exists no collective choice rule that satisfies unanimity, independence of irrelevant alternatives, non-dictatorship, transitivity, and minimal comparability. Proof. We first prove that given any A ⊆ N, if A ∈ Df , then there exists a proper subset of A that is decisive for f . Take any B ⊊ A. Let R∗ be a linear ordering on X . That is, it is an antisymmetric ordering. Let R ∈ RN be such that

{ Ri =

R∗ for all i ∈ B; d(R∗ ) for all i ̸ ∈ B.

By minimal comparability, there exist distinct x∗ , y∗ ∈ X such that (x∗ , y∗ ) ∈ f (R) ∪ d(f (R)).

Please cite this article in press as: Cato S., Incomplete decision-making and Arrow’s impossibility theorem. Mathematical Social Sciences (2017), https://doi.org/10.1016/j.mathsocsci.2017.10.002.

4

S. Cato / Mathematical Social Sciences (

Since R∗ is a linear ordering, either (i) (x∗ , y∗ ) ∈ P(R∗ ) or (ii) (y∗ , x∗ ) ∈ P(R∗ ). Suppose that (i) is true. Take any z ̸ ∈ {x∗ , y∗ }. Let R′ ∈ RN be such that (x∗ , z) ∈

⋂

P(R′i ) and (z , y∗ ) ∈

i∈B

⋂

P(R′i ) and (y∗ , x∗ ) ∈

i∈A\B

(y∗ , x∗ ) ∈

P(R′i ),

P(R′i ) and (x∗ , z) ∈

i∈N \A

where K ∈ N. Since f P (R) is transitive, the existence of the cycle implies that

P(R′i ).

(x∗ , y∗ ) ∈ {(x0 , x1 ), (x1 , x2 ), . . . , (xK −1 , xK ), (xK , x0 )},

⋂ i∈N \A

Since R|{x∗ ,y∗ } = R′ |{x∗ ,y∗ } , the independence of irrelevant alternatives implies that (x∗ , y∗ ) ∈ f (R′ ) ∪ d(f (R′ )). First, suppose that (y∗ , x∗ ) ̸ ∈ f (R′ ). Since (y∗ , x∗ ) ∈ f (R′ ) ∪ d(f (R′ )), we have (y∗ , x∗ ) ∈ d(f (R′ )). Then, we have (x∗ , y∗ ) ∈ P(f (R′ )). The independence of irrelevant alternatives implies that B ∈ Df− (x∗ , y∗ ). By Lemma 1, B ∈ Df . ⋂ Second, suppose that (y∗ , x∗ ) ∈ f (R′ ). Since (z , y∗ ) ∈ i∈A P(R′i ) and A ∈ Df , we have (z , y∗ ) ∈ P(f (R′ )). By transitivity, we have (z , x∗ ) ∈ P(f (R′ )). The independence of irrelevant alternatives implies that A \ B ∈ Df− (z , x∗ ). By Lemma 1, A \ B ∈ Df . Thus, either B ∈ Df and A \ B ∈ Df . Then, the claim is proved for case (i). Suppose that (ii) is true. We can prove this case by taking any z ̸ ∈ {x∗ , y∗ } and R′′ ∈ RN such that (y∗ , z) ∈

⋂

P(R′′i ) and (z , x∗ ) ∈

i∈B

(z , x ) ∈

⋂

P(Ri ) and (x , y ) ∈ ′′

∗

∗

i∈A\B

(x∗ , y∗ ) ∈

P(R′′i ),

i∈B

⋂

∗

⋂

P(Ri ), ′′

i∈A\B

⋂

P(R′′i ) and (y∗ , z) ∈

i∈N \A

⋂

P(R′′i ).

i∈N \A

Now, we prove our theorem. Note that N is decisive for f by unanimity. From our claim, we can find a proper subset N1 of N such that N1 ∈ Df . By applying our claim again, there is a proper subset N2 of N1 such that N2 ∈ Df . By repeating this process, we have d ∈ N such that {d} ∈ Df since N is finite. ■ 3.3. Consistency and neutrality As argued by Bossert and Suzumura (2010), S-consistency is a natural weakening of transitivity because it is a necessary and sufficient condition for the existence of an ordering, which is compatible with the original binary relation. S-consistency per se is substantially weaker than transitivity. S-consistency as an axiom for f can be formulated as follows: S-consistency: f (R) satisfies S-consistency for all R ∈ RN . Is there a collective choice rule that satisfies unanimity, independence of irrelevant alternatives, non-dictatorship, S-consistency, and minimal comparability? The answer is yes. Example 1. The simple majority rule f M is defined as follows: for all R ∈ RN : (x, y) ∈ f M (R) ⇔ #{i ∈ N : (x, y) ∈ P(Ri )}

≥ #{i ∈ N : (y, x) ∈ P(Ri )}. Take two alternatives x∗ , y∗ ∈ X (x∗ ̸ = y∗ ). Define f ∗ as follows: f ∗ (R)|{x,y} =

{

Since x∗ , y∗ are always comparable by the simple majority rule, minimal comparability is satisfied. It suffices to show S-consistency. Suppose that S-consistency is violated. Then, there exists a preference cycle for some R ∈ RN . We can take a minimal preference cycle such that

P(R′i ),

i∈A\B

⋂

–

(xk−1 , xk ) ∈ f ∗ (R) for all k ∈ {1, . . . , K } and (xK , x0 ) ∈ P(f ∗ (R)),

i∈B

⋂

(z , y∗ ) ∈

⋂

)

f M (R)|{x,y} if {x, y} = {x∗ , y∗ } f P (R)|{x,y} otherwise.

By construction, f ∗ satisfies unanimity, independence of irrelevant alternatives, and non-dictatorship.

or (y∗ , x∗ ) ∈ {(x0 , x1 ), (x1 , x2 ), . . . , (xK −1 , xK ), (xK , x0 )}. Without loss of generality, we can assume that the former is true. First, we consider the case where (x∗ , y∗ ) = (xK , x0 ). Note that (x∗ , y∗ ) ∈ P(f M (R)). Since we take a minimal cycle, x0 , x1 , . . . , xK are distinct, and thus, it must be true that (xk , xk−1 ) ∈ f P (R) for all k ∈ {1, . . . , K }. Since x0 = y∗ and xK = x∗ , the transitivity of f P implies that (y∗ , x∗ ) ∈ f P (R), which implies that (y∗ , x∗ ) ∈ f M (R). This contradicts the assumption that (x∗ , y∗ ) ∈ P(f M (R)). Second, we consider the case where (x∗ , y∗ ) ̸ = (xK , x0 ). Then, (xℓ−1 , xℓ ) = (x∗ , y∗ ) for some ℓ ∈ {1, . . . , K }. Then, we have (xℓ−1 , xℓ ) ∈ f M (R). Note that (xk−1 , xk ) ∈ f P (R) for all k ∈ {ℓ + 1, . . . , K }, (xK , x0 ) ∈ P(f P (R)), (xk−1 , xk ) ∈ f P (R) for all k ∈ {1, . . . , ℓ − 1}. Then, transitivity implies that (xℓ , xℓ−1 ) ∈ P(f P (R)). This contradicts the assumption that (xℓ−1 , xℓ ) ∈ f M (R). Thus, S-consistency is satisfied. □ Now, we impose the axiom of neutrality, which is stronger than the independence of irrelevant alternatives. Neutrality: For all x, y, z , w ∈ X and for all R, R′ ∈ RN , if [(x, y) ∈ Ri ⇔ (z , w ) ∈ R′i and (y, x) ∈ Ri ⇔ (w, z) ∈ R′i ] for all i ∈ N, then

[(x, y) ∈ f (R) ⇔ (z , w) ∈ f (R′ ) and (y, x) ∈ f (R) ⇔ (w, z) ∈ f (R′ )]. Neutrality requires that alternatives are treated in a symmetric way. An immediate implication of neutrality is the following. Given neutrality, the existence of a comparable pair implies completeness. Proposition 2. If f satisfies neutrality and the existence of a comparable pair, then it satisfies completeness. Recall that in the presence of completeness, S-consistency implies transitivity. Therefore, there exists no collective choice rule that satisfies unanimity, neutrality, non-dictatorship, S-consistency, and the existence of a comparable pair. Note that the collective choice rule in Example 1 satisfies the existence of a comparable pair. Therefore, the possibility crucially depends on the absence of neutrality. We consider the case of minimal comparability. Theorem 3. There exists no collective choice rule f that satisfies unanimity, neutrality, non-dictatorship, S-consistency, and minimal comparability. Lemma 3. Suppose that a collective choice rule f satisfies unanimity, neutrality, S-consistency, and minimal comparability. If A ∈ Df− (x, y) for some x, y ∈ X , then A ∈ Df .

Please cite this article in press as: Cato S., Incomplete decision-making and Arrow’s impossibility theorem. Mathematical Social Sciences (2017), https://doi.org/10.1016/j.mathsocsci.2017.10.002.

S. Cato / Mathematical Social Sciences (

Proof. Suppose that A ∈ Df− (x, y) for some x, y ∈ X (x ̸ = y). Let R ∈ RN be such that (x, y) ∈

⋂

⋂

P(Ri ), (y, x) ∈

i∈A

P(Ri ), and (y, z) ∈

i∈N \A

⋂

P(Ri ).

Since A ∈ Df (x, y), we have (x, y) ∈ P(f (R)). Unanimity implies that (y, z) ∈ P(f (R)). Then, S-consistency implies that

–

5

Since A ∈ Df , we have (z , y∗ ) ∈ P(f (R′ )). By S-consistency, we have (x∗ , z) ̸ ∈ f (R′ ). Now, let R′′ ∈ RN be such that ′′

Ri =

i∈N

−

)

{

R∗ for all i ∈ A \ B; d(R∗ ) for all i ̸ ∈ A \ B.

By minimal comparability, there exist distinct a∗ , b∗ ∈ X such that (a∗ , b∗ ) ∈ f (R′′ ) ∪ d(f (R′′ )).

(z , x) ̸ ∈ f (R).

and

Either (x, z) ∈ f (R) or (x, z) ̸ ∈ f (R). Suppose that (x, z) ̸ ∈ f (R). Then, (x, z) ∈ N(f (R)). Let

{i ∈ N : (x∗ , z) ∈ P(R′i )} = {i ∈ N : (a∗ , b∗ ) ∈ P(R′′i )}; {i ∈ N : (z , x∗ ) ∈ P(R′i )} = {i ∈ N : (b∗ , a∗ ) ∈ P(R′′i )}.

A1 = {i ∈ N : (x, z) ∈ P(Ri )};

Neutrality implies that

A2 = {i ∈ N : (z , x) ∈ P(Ri )};

(x∗ , z) ∈ f (R′ ) ∪ d(f (R′ )).

A3 = {i ∈ N : (x, z) ∈ I(Ri )}.

Since (x∗ , z) ̸ ∈ f (R′ ), it follows that (z , x∗ ) ∈ P(f (R′ )). By neutrality, it follows that A \ B ∈ Df− (x∗ , z). Lemma 3 implies that A \ B ∈ Df . Thus, either B ∈ Df and A \ B ∈ Df . The claim is proved, and the rest of the proof of this theorem is the same as that of Theorem 2. ■

Let R∗ be a linear ordering on X and R′ ∈ RN be such that R′i = R∗ for all i ∈ A1 ; R′i = d(R∗ ) for all i ∈ A2 ; R′i = X × X for all i ∈ A3 . Take any a, b ∈ X (a ̸ = b). Since R∗ is a linear ordering, either (a, b) ∈ P(R∗i ) or (b, a) ∈ P(R∗i ). Without loss of generality, we can assume the former. By construction of R′ , we have A1 = {i ∈ N : (a, b) ∈ P(R′i )}, A2 = {i ∈ N : (b, a) ∈ P(R′i )}, and A3 = {i ∈ N : (a, b) ∈ I(R′i )}. Therefore, [(x, z) ∈ Ri ⇔ (a, b) ∈ R′i and (z , x) ∈ Ri ⇔ (b, a) ∈ R′i ]. Since (x, z) ∈ N(f (R)), neutrality implies that (a, b) ∈ N(f (R′ )). Since a, b are arbitrary (but they must be distinct), it follows that f (R′ ) ⊆ ∆. This contradicts minimal comparability. We have (x, z) ∈ P(f (R)). The individual ranking under R for i ∈ N \ A is not specified, and thus, A ∈ D(x, z). Neutrality implies that A ∈ Df . ■ Proof of Theorem 3. It suffices to show that given any A ⊆ N, if A ∈ Df , then there exists a proper subset of A that is decisive for f . Take any B ⊊ A. Let R∗ be a linear ordering on X . Let R ∈ RN be such that

{ Ri =

R∗ for all i ∈ B; d(R∗ ) for all i ̸ ∈ B.

⋂

P(R′i ) and (z , y∗ ) ∈

i∈B

⋂

(z , y ) ∈ ∗

⋂

⋂

P(Ri ) and (y , x ) ∈ ∗

∗

i∈A\B

(y∗ , x∗ ) ∈

P(R′i ),

i∈B

′

⋂

3.4. Extensions to semi-order properties In this section, we consider extensions to the semi-order properties, which are defined as follows. Semi-transitivity: For all x, y, z , w ∈ X ,

[(x, y) ∈ R, (y, z) ∈ P(R), and (z , w) ∈ P(R)] ⇒ (x, w) ∈ P(R), and

[(x, y) ∈ P(R), (y, z) ∈ P(R), and (z , w) ∈ R] ⇒ (x, w) ∈ P(R). Interval-order property: For all x, y, z , w ∈ X ,

Minimal comparability implies that there exist distinct x∗ , y∗ ∈ X such that (x∗ , y∗ ) ∈ f (R) ∪ d(f (R)). Either (i) (x∗ , y∗ ) ∈ P(R∗ ) or (ii) (y∗ , x∗ ) ∈ P(R∗ ). Suppose that (i) is true (the other case can be proved in a similar way). Take any z ̸ ∈ {x∗ , y∗ }. Let R′ ∈ RN be such that (x∗ , z) ∈

Some recent works conduct social choice with S-consistency (Bossert and Suzumura, 2008, 2012a, b; Cato, 2013a). Bossert and Suzumura (2008) provide a characterization of a class of Sconsistent collective choice rules. Bossert and Suzumura (2012a) and Cato (2013a, c) clarify the decisive structure underlying an Sconsistent collective choice rule. These studies show the difference between transitivity and S-consistency as collective rationality conditions because a resulting collective choice rule is far from the dictatorship. Theorem 3 demonstrates that there is no difference between transitivity and S-consistency under certain axioms.10

P(R′i ),

i∈A\B

P(R′i ) and (x∗ , z) ∈

i∈N \A

⋂

P(R′i ).

i∈N \A

Since R|{x∗ ,y∗ } = R′ |{x∗ ,y∗ } , neutrality implies that (x∗ , y∗ ) ∈ f (R) ∪ d(f (R)). First, suppose that (y∗ , x∗ ) ̸ ∈ f (R′ ). Since (x∗ , y∗ ) ∈ f (R′ ) ∪ d(f (R)), we have (x∗ , y∗ ) ∈ f (R′ ). Then, we have (x∗ , y∗ ) ∈ P(f (R)). By neutrality, it follows that B ∈ Df− (x∗ , y∗ ). Lemma 3 implies that B ∈ Df . Second, suppose that (y∗ , x∗ ) ∈ f (R′ ). Note that R|{x∗ ,y∗ } = R′ |{x∗ ,y∗ } and (z , y∗ ) ∈

⋂ i∈A

P(R′i ).

[(x, y) ∈ P(R), (y, z) ∈ R, and (z , w) ∈ P(R)] ⇒ (x, w) ∈ P(R). A binary relation is said to be a semi-order if it satisfies semitransitivity, the interval-order property, and completeness. Collective choice with semi-order properties is examined by Blair and Pollak (1979) and Blau (1979).11 According to their results, under completeness and each of semi-order properties, there exists no collective choice rule satisfying Arrow’s axioms if X contains at least four alternatives. Semi-transitivity (the interval-order property) generally does not imply quasi-transitivity.12 If reflexivity is satisfied, quasitransitivity follows from each of them. Lemma 4. Let R be a binary relation on X . (i) If R satisfies semi-transitivity and reflexivity, then it satisfies quasitransitivity; (ii) If R satisfies the interval-order property and reflexivity, then it satisfies quasi-transitivity. 10 To the best of our knowledge, Theorem 3 is the first theorem to establish the dictatorship under S-consistency. 11 Their analysis contains more general cases. 12 Assume that X = {x, y, z } and R = {(x, y), (y, z)}. Since ∆ ∩ R = ∅, R satisfies semi-transitivity and the interval-order property. However, it does not satisfy quasi-transitivity.

Please cite this article in press as: Cato S., Incomplete decision-making and Arrow’s impossibility theorem. Mathematical Social Sciences (2017), https://doi.org/10.1016/j.mathsocsci.2017.10.002.

6

S. Cato / Mathematical Social Sciences (

N

Reflexivity: f (R) satisfies reflexivity for all R ∈ R . The proof is obvious from the definitions, and thus, it is omitted here.

–

Take z , w ∈ X \ {x∗ , y∗ }. Let R′ ∈ RN be such that

Semi-transitivity: f (R) satisfies semi-transitivity for all R ∈ RN . Interval-order property: f (R) satisfies the interval-order property for all R ∈ RN .

)

(w, x∗ ) ∈

⋂

(y , z) ∈

⋂

P(R′i ), (x∗ , y∗ ) ∈

⋂

P(Ri ), (z , w ) ∈

⋂

(z , w ) ∈

′

i∈A\B

i∈A\B

⋂

⋂

P(R′i ), (w, x∗ ) ∈

i∈N

Theorem 4. If X contains at least four alternatives, then there exists no collective choice rule that satisfies unanimity, independence of irrelevant alternatives, non-dictatorship, semi-transitivity, reflexivity, and minimal comparability.

(y∗ , x∗ ) ∈ P(R) for all i ∈ A \ B; (x∗ , y∗ ) ∈ P(R) for all i ̸ ∈ A \ B, and (x∗ , y∗ ) ∈ f (R) ∪ d(f (R)). Take z , w ∈ X \ {x∗ , y∗ }. Let R′ ∈ RN be such that (x∗ , y∗ ) ∈

⋂

(y∗ , z) ∈

⋂

P(R′i ), (y∗ , z) ∈

⋂

P(R′i ), (z , w ) ∈

⋂

i∈B

(z , w ) ∈

P(R′i ), and (z , w ) ∈

⋂

i∈B

i∈A\B

i∈N

⋂

⋂

P(R′i ), (w, x∗ ) ∈

i∈N

P(R′i ),

i∈N

P(R′i ), and (w, x∗ ) ∈

P(R′i ),

i∈A\B

i∈N \A

⋂

P(R′i ).

i∈N \A

′

Since R|{x∗ ,y∗ } = R |{x∗ ,y∗ } , the independence of irrelevant alternatives implies that (x∗ , y∗ ) ∈ f (R′ ) ∪ d(f (R′ )). First, suppose that (x∗ , y∗ ) ∈ f (R′ ). Note that R|{x∗ ,y∗ } = R′ |{x∗ ,y∗ } , (y∗ , z) ∈

⋂ i∈A

P(Ri ), and (z , w ) ∈

⋂

P(Ri ).

i∈N

Since A ∈ Df , we have (y∗ , z) ∈ P(f (R′ )). Unanimity implies that (z , w ) ∈ P(f (R′ )). By semi-transitivity, we have (x∗ , w ) ∈ P(f (R′ )). Then, the independence of irrelevant alternatives implies that B ∈ − Df (x∗ , w ). By Lemma 1, B ∈ Df . Second, suppose that (x∗ , y∗ ) ̸ ∈ f (R′ ). Since (x∗ , y∗ ) ∈ f (R′ ) ∪ d(f (R′ )), we have (x∗ , y∗ ) ∈ d(f (R′ )). It follows that (y∗ , x∗ ) ∈ P(f (R′ )). The independence of irrelevant alternatives implies that A \ B ∈ Df− (y∗ , x). By Lemma 1, A \ B ∈ Df . Thus, either B ∈ Df and A \ B ∈ Df . The remaining process is the same as that of the previous theorems. ■ Theorem 5. If X contains at least four alternatives, then there exists no collective choice rule that satisfies unanimity, independence of irrelevant alternatives, non-dictatorship, the interval-order property, reflexivity, and minimal comparability. Proof. It suffices to show that, given any A ⊆ N, if A ∈ Df , then there exists a proper subset of A that is decisive for f . Take any B ⊊ A. By minimal comparability, we can find R ∈ RN such that, for some x∗ , y∗ ∈ X , (y∗ , x∗ ) ∈ P(R) for all i ∈ A \ B; (x∗ , y∗ ) ∈ P(R) for all i ̸ ∈ A \ B, and (x∗ , y∗ ) ∈ f (R) ∪ d(f (R)).

i∈B

P(Ri ), and (w, x ) ∈ ′

∗

⋂

P(R′i ),

i∈A\B

P(R′i ), and (x∗ , y∗ ) ∈

i∈N \A

⋂

P(R′i ).

i∈N \A

Since R|{x∗ ,y∗ } = R |{x∗ ,y∗ } , the independence of irrelevant alternatives implies that (x∗ , y∗ ) ∈ f (R) ∪ d(f (R)). First, suppose that (x∗ , y∗ ) ∈ f (R′ ). Note that

⋂

P(R′i ), R|{x∗ ,y∗ } = R′ |{x∗ ,y∗ } , and (y∗ , z) ∈

i∈N

⋂

P(Ri ).

i∈A

Since A ∈ Df , we have (y∗ , z) ∈ P(f (R′ )). Unanimity implies that (w, x∗ ) ∈ P(f (R′ )). By the interval order property, we have (w, z) ∈ P(f (R′ )). The independence of irrelevant alternatives implies that B ∈ Df− (w, z). By Lemma 1, B ∈ Df . Second, suppose that (x∗ , y∗ ) ̸ ∈ f (R′ ). Since (x∗ , y∗ ) ∈ f (R′ ) ∪ d(f (R′ )), we have (x∗ , y∗ ) ∈ d(f (R′ )). It follows that (y∗ , x∗ ) ∈ P(f (R′ )). The independence of irrelevant alternatives implies that A \ B ∈ Df− (y∗ , x). By Lemma 1, A \ B ∈ Df . Thus, either B ∈ Df and A \ B ∈ Df . The remaining process is the same as that of the previous theorems. ■ 4. Concluding remarks

⋂

P(R′i ), and (x∗ , y∗ ) ∈

P(R′i ),

′

(w, x∗ ) ∈ Proof. It suffices to show that, given any A ⊆ N, if A ∈ Df , then there exists a proper subset of A that is decisive for f . Take any B ⊊ A. By minimal comparability, we can find R ∈ RN such that, for some x∗ , y∗ ∈ X ,

⋂

i∈B

i∈B

∗

P(R′i ), and (y∗ , z) ∈

In this paper, we introduced the concept of minimal comparability, which is substantially weaker than completeness. As Theorems 2, 4, and 5 have shown, minimal comparability is incompatible with the Arrow axioms (unanimity, independence, and non-dictatorship) under transitivity (or semi-order properties), suggesting that a small amount of comparability definitely leads to impossibilities in collective decision processes with these coherence properties. The case of S-consistency is noteworthy. S-consistency allows the Arrow axioms to be satisfied under minimal comparability. Our example (Example 1) combines the simple majority rule and the Pareto rule. We can easily check that it also satisfies anonymity.13 However, it does not satisfy neutrality. Theorem 3 shows that an impossibility result arises when neutrality is imposed instead of independence. Bossert and Suzumura (2008) characterize a class of S-consistent collective choice rules satisfying anonymity and neutrality. According to them, the class includes collective choice rules that can resolve some conflict of interests of individuals, unlike the Pareto rule. Our theorem implies that such rules do not satisfy minimal comparability and cannot make a judgment for some cases. This suggests the presence of a tension between neutrality and comparability under S-consistency. In contrast to other coherence properties, we can easily to implement completeness under quasi-transitivity. Let us consider the Pareto extension rule f E that is defined as follows: (x, y) ∈ f E (R) ⇔ (y, x) ̸ ∈ P(f P (R)). It is easy to show that f E satisfies unanimity, neutrality, nondictatorship, transitivity, and completeness (see Sen (1969) for a detailed argument). Moreover, f E also satisfies anonymity and neutrality. Table 1 shows the possibilities and impossibilities found in this paper. Finally, we comment on extensions to economic environments, where given the concrete structure of a set of alternatives, individual preferences are required to be monotonic and continuous. 13 A collective choice rule satisfies anonymity if social preferences are invariant for any permutation over the set of individuals.

Please cite this article in press as: Cato S., Incomplete decision-making and Arrow’s impossibility theorem. Mathematical Social Sciences (2017), https://doi.org/10.1016/j.mathsocsci.2017.10.002.

S. Cato / Mathematical Social Sciences ( Table 1 Com = completeness, Tr = transitivity, MC = minimal comparability, S-con = S-consistency, ST = semi-transitivity, IOP = interval-order property, QT = quasitransitivity, Ref = reflexivity, ⊗ = impossibility. Rationality Com + Tr MC + Tr MC + S-con MC + S-con MC + ST + Ref MC + IOP + Ref Com + QT

Additional condition

Result

⊗ (Arrow’s theorem) ⊗ (Theorem 2) Example 1 Neutrality #X ≥ 4 #X ≥ 4 Neutrality

⊗ (Theorem 3) ⊗ (Theorem 4) ⊗ (Theorem 5) Pareto extension rule

Arrow’s theorem is basically robust under such an environment. We can consider the Arrovian social choice with minimal comparability. Our impossibility results are not robust under such economic environments. This is because the proofs of our results utilize a preference in which some individuals have a linear preference and the other individuals have the opposite. Such a preference is not available in an economic environment. A stronger comparability is necessary to get an impossibility result. Acknowledgments I thank Marc Fleurbaey, Maurice Salles, and an anonymous referee of this journal for helpful comments. This paper was financially supported by Grant-in-Aids for Young Scientists (B) from the Japan Society for the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology (26870477). This paper is also supported by Postdoctoral Fellowship for Research Abroad of JSPS. References Aleskerov, F.T., 1999. Arrovian Aggregation Models. Kluwer Academic Publishers, Dordrecht. Arrow, K.J., 1963. Social Choice and Individual Values, second ed. Wiley, New York. Aumann, R.J., 1962. Utility theory without the completeness axiom. Econometrica 30, 445–462. Barberá, S., 1983. Pivotal voters: a new proof of Arrow’s theorem. Econom. Lett. 6, 13–16. Blair, D.H., Pollak, R.A., 1979. Collective rationality and dictatorship: the scope of the Arrow theorem. J. Econom. Theory 21, 186–194. Blau, J.H., 1979. Semiorders and collective choice. J. Econom. Theory 21, 195–206. Bossert, W., Suzumura, K., 2008. A characterization of consistent collective choice rules. J. Econom. Theory 138, 311–320. Bossert, W., Suzumura, K., 2010. Consistency, Choice, and Rationality. Harvard University Press: Harvard University Press, Cambridge MA.

)

–

7

Bossert, W., Suzumura, K., 2012a. Quasi-transitive and Suzumura consistent relations. Soc. Choice Welf. 39 (2–3), 323–334. Bossert, W., Suzumura, K., 2012b. Product filters, acyclicity and Suzumura consistency. Math. Social Sci. 64, 258–262. Cato, S., 2010. Brief proofs of Arrovian impossibility theorems. Soc. Choice Welf. 35 (2), 267–284. Cato, S., 2013a. Quasi-decisiveness, quasi-ultrafilter, and social quasi-orderings. Soc. Choice Welf. 41, 169–202. Cato, S., 2013b. Alternative proofs of Arrow’s general possibility theorem. Econ. Theory Bull. 1 (2), 131–137. Cato, S., 2013c. Remarks on Suzumura consistent collective choice rules. Math. Social Sci. 65 (1), 40–47. Cato, S., 2016. Rationality and Operators: The Formal Structure of Preferences. Springer, Singapore. Dubra, J., Maccheroni, F., Ok, E.A., 2004. Expected utility theory without the completeness axiom. J. Econom. Theory 115, 118–133. Eliaz, K., Ok, E.A., 2006. Indifference or indecisiveness? Choice-theoretic foundations of incomplete preferences. Games Econom. Behav. 56, 61–86. Evren, Ö., Ok, E.A., 2011. On the multi-utility representation of preference relations. J. Math. Econom. 47, 554–563. Feldman, A.M., Serrano, R., 2006. Welfare Economics and Social Choice Theory, second ed. Springer, New York. Geanakoplos, J., 2005. Three brief proofs of Arrow’s impossibility theorem. Econom. Theory 26, 211–215. Gibbard, A.F., 2014. Social choice and the Arrow conditions. Econ. Philos. 30 (03), 269–284. Mandler, M., 2005. Incomplete preferences and rational intransitivity of choice. Games Econom. Behav. 50, 255–277. Mandler, M., 2009. Indifference and incompleteness distinguished by rational trade. Games Econom. Behav. 67, 300–314. Mandler, M., 2014. Status quo maintenance reconsidered: changing or incomplete preferences? Econ. J. 114 (499), F518–F535. Mas-Colell, A., Sonnenschein, H., 1972. General possibility theorems for group decisions. Rev. Econom. Stud. 39, 185–192. Ok, E.A., 2002. Utility representation of an incomplete preference relation. J. Econom. Theory 104, 429–449. Reny, P.J., 2001. Arrow’s theorem and the Gibbard-Satterthwaite theorem: a unified approach. Econom. Lett. 70, 99–105. Sen, A.K., 1969. Quasi-transitivity, rational choice and collective decisions. Rev. Econom. Stud. 36, 381–393. Sen, A.K., 1970. Interpersonal aggregation and partial comparability. Econometrica 38, 393–409. Sen, A.K., 1985. Commodities and Capabilities. North Holland, Amsterdam. Sen, A.K., 1992. Inequality Reexamined. Oxford University Press, Oxford. Sen, A.K., 1995. Rationality and social choice. Amer. Econ. Rev. 85 (1), 1–24. Sen, A.K., 2004. Incompleteness and reasoned choice. Synthese 140 (1), 43–59. Suzumura, K., 1976. Remarks on the theory of collective choice. Economica 43, 381–390. Suzumura, K., 2000. Welfare economics beyond welfarist-consequentialism. Jpn. Econ. Rev. 51, 1–32. Ubeda, L., 2003. Neutrality in arrow and other impossibility theorems. Econom. Theory 23, 195–204. Weymark, J.A., 1984. Arrow’s theorem with social quasi-orderings. Public Choice 42, 235–246. Yu, N.N., 2012. A one-shot proof of Arrow’s impossibility theorem. Econom. Theory 50, 523–525.

Please cite this article in press as: Cato S., Incomplete decision-making and Arrow’s impossibility theorem. Mathematical Social Sciences (2017), https://doi.org/10.1016/j.mathsocsci.2017.10.002.