The Possibility of Paretian Anonymous Decision-Making with an Infinite Population ∗
Susumu Cato
†November 20, 2018
Abstract
This paper considers the trade-off between unanimity and anonymity in collective decision- making with an infinite population. This efficiency-equity trade-off is a fundamental diffi- culty in making a normative judgment in a conflict between generations. In particular, it is known that this trade-off is quite sensitive in the formulation of unanimity axioms. In this study, we consider the trade-off in a preference-aggregation framework instead of the standard utility-aggregation framework. We show that there exists a social welfare function that satisfies I-strong Pareto, independence of irrelevant alternatives, and finite anonymity.
This contrasts with an impossibility result in the standard utility-aggregation framework, and this means that the trade-off is also sensitive for background frameworks of aggregations.
Keywords: Social choice; Intergenerational equity; Possibility theorem; Unanimity; Anonymity;
Ultrafilter
JEL classification: D63, D64, D71
∗I thank Marc Fleurbaey and Hannu Salonen for their helpful suggestions. This study was financially supported by JSPS KAKENHI (18K01501), and was also supported by the Postdoctoral Fellowship for Research Abroad of the JSPS.
†Institute of Social Science, the University of Tokyo. Tel: +81-3-5841-4904, Fax :+81-3-5841-4905, susumu.cato@gmail.com, Institute of Social Science, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113- 0033, Japan.
1 Introduction
This paper considers a tension between unanimity and anonymity in a society with an infinite population. According to unanimity, a society respects individuals’ interests in the sense that social preferences reflect their agreements. According to anonymity, society treats people equally, in the sense that social preferences are unchanged after permuting two individuals’ names. This subject is one of the main problems in the literature on intergenerational equity.1
This study examines an Arrovian framework with an infinite population.2 We consider a Paretian axiom that states that if alternativexis at least as good as alternativeyby all individuals and x is preferred to y by infinitely many individuals, then x is socially better than y. We call this axiom I-strong Pareto. This axiom was introduced by Sakai (2006), Crespo, Nunez, and Rincon-Zapatero (2009), and Lauwers (2010) in order to consider an intergenerational problem in a framework, in which each individual is assigned a numerical value of utility (not a utility function).3 This formulation of the Pareto principle is well-defiend only in a setting with an infinite population. Thus, I-strong Pareto is applicable only to infinite-population settings, and is a natural Paretian axiom for anonymous preference aggregation.
To see the relevance of I-strong Pareto, consider the case of a countably infinite population:
the set of individuals is the set of natural numbers,N. Let us suppose that only one individual is indifferent between two alternatives,xandy, and all individuals preferxtoy. Weak Pareto, which is a common formulation of the Pareto principle, states that an alternative is socially preferred to another alternative if the former is preferred to the latter by everyone. By definition, weak Pareto does not make any judgment on the above situation. However, the size of people with indifference is negligible in this case, and this particular person does not struggle with a choice of x over y.
Thus, it is natural to require that x to be preferred. Indeed, I-strong Pareto is compatible with this requirement.
Next, let us suppose that only one individual prefersx toy, and that all other individuals are indifferent between the two alternatives, x and y. Another formulation of the Pareto principle, strong Pareto, requires that an alternative is socially preferred to another alternative if the former is at least as good as the latter by everyone, and the former is preferred to the latter by someone.
Obviously,xis socially better than yunder strong Pareto in the current case, but I-strong Pareto does not make a judgment. The influence of one person is negligible, especially if anonymity is required and, thus, it is natural to decline to require that x be preferred to y. Indeed, strong Pareto and finite anonymity are not compatible under the Arrovian independence axiom, namely, the independence of irrelevant alternatives (IIA).4
Now, we consider F-strong Pareto, which requires that if finitely many individual are indifferent
1Asheim (2010) provides a survey of subjects of intergenerational equity. See also Fleurbaey (2009).
2In the case of a finite population, many works analyze Arrovian impossibility theorems: for example, see Iritani, Kamo, and Nagahisa (2013), Man and Takayama (2013), Ninjbat (2015), and Muto and Sato (2016).
3To the best of our knowledge, I-strong Pareto is originally proposed by Sakai (2006) under the name of “Strong Monotonicitty for Infinite Generations,” Crespo, Nunez, and Rincon-Zapatero (2009) call it “infinite Pareto,”, and Lauwers (2010) calls it “intermediate Pareto.”
4This is proved by Cato (2017).
between x and y and the other individuals prefers x to y, then x is socially preferred to y. Note that F-strong Pareto and I-strong Pareto do not make a difference in social judgements on the above-mentioned two cases. However, there is a big difference between them. Suppose thatxand y are indifferent for individuals in the set of odd numbers and x is preferred to y by individuals in the set of even numbers. I-strong Pareto requires thatxis socially preferred toy, but F-strong Pareto does not make a judgement. Note that the two sets have the same big size in the sense that each set is countably infinite and, thus, it is non-negligible. Then, one non-negligible set has a strict preference overx toy and the other non-negligible set are indifferent between them. Given the fact that any set of individuals is non-negligible in the finite population case, it is normatively natural to expect that x is socially preferred to y in this situation as long as we consider strong Pareto as a normatively relevant axiom in the finite-population case.
For these observations, I-strong Pareto can be regarded as a natural formulation of the Pareto principle in a setting with an infinite population. The problem is to determine whether or not a possibility holds for I-strong Pareto. We show the following possibility theorem: there exists a social welfare function that satisfies I-strong Pareto, IIA, and finite anonymity. Since finite anonymity is an equity principle requires that social preferences are invariance for finite permu- tations over the set of individuals, this implies that it is possible to construct a Paretian and equitable social judgement along with IIA. Two points are noteworthy. First, this result contrasts with an impossibility result for an aggregation of individual utility values: Sakai (2006) shows that there is no sustainable consumption path when I-strong Pareto, finite anonymity, and continuity are imposed, and Crespo, Nunez, and Rincon-Zapatero (2009) show that there exists no welfare representation that satisfies I-strong Pareto and finite anonymity. Second, this contrasts with an impossibility result for the finite-population case: even if I-strong Pareto is weakened to weak Pareto and finite anonymity is weakened to non-dictatorship, there is no way to make a collective decision (Arrow, 1951).
There are two streams of related works. First, this study is related to those on infinite utility streams. A fundamental difficulty was identified by Lauwers (1997a), who showed that there is no social ordering that satisfies strong Pareto and finite anonymity in the line of Diamond (1965).
In addition, Basu and Mitra (2003) prove that there is a welfare representation that satisfies versions of strong Pareto and finite anonymity. This efficiency-equity trade-off is a core issue of intergenerational equity.5
Second, this study is related to an extension of works on Arrovian social choice. Fishburn (1970) finds that there is a social welfare function that satisfies Arrow’s axioms in any setting with an infinite population. Since his work, many researchers have examined possibility and impossibility results in infinite settings. Mihara (1997) and Lauwers and Van Liedekerke (1995) are remarkable examples of earlier works that impose anonymity axioms in addition to the standard Arrow axioms. Recently, many works have tried to clarify various anonymity axioms, or to introduce some other requirements, such as continuity (Gomberg, Martinelli, and Torres, 2005;
Torres, 2005; Salonen and Saukkonen, 2005).
5See, for instance, Lauwers (1997b, 2010, 2012), Van Liedekerke and Lauwers (1997), Mitra and Basu (2007), Sakai (2010), and Kamaga and Kojima (2009, 2010).
Kirman and Sondermann (1972) and Hansson (1976) characterize the power structure behind social choice. They employ the concept of an ultrafilter, which is a useful mathematical tool in general topology, geometric group theory, and model theory. The family of decisive coalitions is an ultrafilter that is robust under any assumption on the population structure (finite or infinite set of individuals). Then, it can be applied directly to obtain Arrow’s impossibility theorem. The Kirman-Sondermann-Hansson theorem has been extended by other works, including Armstrong (1980), Noguchi (2011), Takayama and Yokotani (2014), and Cato (2012, 2013ab).
To prove the existence of a social welfare function, we need to employ a stronger concept, related to an ultrafilter. Because of I-strong Pareto, we need to introduce a more complicate structure on decisive coalitions. We employ the concept of an ultrafilter mapping, which can capture a hierarchy of ultrafilters. This concept is introduced by Takayama and Yokotani (2017) to analyze the structure of strongly Paretian social choice functions. Here, we apply it to construct a social welfare function satisfying I-strong Pareto.
The rest of this paper is organized as follows. Section 2 introduces our definitions. Section 3 shows the existence of a social welfare function. Section 4 provides an extension to the case where social preferences are quasi-orderings. Section 5 concludes this paper.
2 Setting
LetX be a set of alternatives: it includes at least three alternatives. A binary relation is a subset of X×X, which is denoted by R. Let
P(R) = {(x, y)∈X×X : (x, y)∈R and (y, x)∈/ R}, I(R) = {(x, y)∈X×X : (x, y)∈R and (y, x)∈R}.
Here, P(R) represents a strict preference and I(R) represents indifference. An ordering R is a binary relation that satisfies completeness and transitivity.6 LetR be the set of all orderings on X.
LetN be aninfinite set of individuals. It can be countable or uncountable. Each individual i has his/her ordering (rational preference) Ri on X. A list of individual orderings (Ri)i∈N ∈ RN is called apreference profile. We often write (Ri)i∈N asR. A social welfare function f is a mapping fromRN to R.
The following axiom restricts the use of information, which we maintain throughout this paper.
Independence of Irrelevant Alternatives (IIA): For all R,R′ ∈ RN, and for all x, y ∈X, if [(x, y)∈ Ri ⇔ (x, y) ∈ R′i and (y, x)∈ Ri ⇔(y, x) ∈R′i] for all i∈ N, then [(x, y) ∈f(R)⇔ (x, y)∈f(R′) and (y, x)∈f(R)⇔(y, x)∈f(R′)].
In Arrow’s original work, non-dictatorship is imposed to avoid a society with a person who has extreme power.7 In this study, we impose a stronger axiom, which requires that decision-making
6R is complete if ∀x, y ∈ X, (x, y) ∈ R or (y, x) ∈ R, and R is transitive if ∀x, y, z ∈ X, if (x, y) ∈ R and (y, z)∈R, then (x, z)∈R. Cato (2016) examines the implications of these conditions.
7Non-dictatorship requires that there exists noi∈N such that P(Ri)⊆P(f(R)) for allR∈ RN.
is impartial with respect to the names of people.8
Finite Anonymity: For all R,R′ ∈ RN, and for allj, k ∈N, ifRj =Rk′,Rk=Rj′, andRi =R′i for all i∈N \ {k, j}, then f(R) =f(R′).
There are many anonymity axioms in the setting with an infinite population.9 Finite anonymity is the weakest anonymity axiom among them.
Two types of the Pareto principle are quite well known.
Weak Pareto: ∩
i∈NP(Ri)⊆P(f(R)) for allR∈ RN. Strong Pareto: P(∩
i∈NRi)⊆P(f(R)) for all R∈ RN.
Weak Pareto requires that if everyone prefers x toy, then x is socially better thany. Strong Pareto requires that if someone prefersxtoyand everyone weakly prefersxtoy, thenxis socially better than y. Strong Pareto implies weak Pareto, which is employed in Arrow’s theorem.
Now, we introduce a concept of an ultrafilter, which is important for our analysis.10 LetH be an arbitrary set. A collection Ω of subsets ofH is said to be a filter onH when
(i)H ∈Ω and ∅∈/ Ω;
(ii) If A∈Ω and A⊆B, then B ∈Ω (monotonic property);
(iii) If A, B ∈Ω, thenA∩B ∈Ω (finite-intersection property).
A filter is said to befreeif∩
A∈ΩA=∅, and isprincipalif it is not free. A filter onHis called anultrafilter if the following is satisfied: (iv) Either A ∈Ω or (A)c ∈Ω for any A⊆ H (strong property).11 It is easy to show that every principal ultrafilter contains a singleton (i.e., {h} ∈ Ω for some h ∈ H).12 Since the collection of decisive coalitions forms an ultrafilter under Arrow’s axioms, the concept of an ultrafilter is quite important in the analysis of collective choice.13
3 Main Theorem
We start from the following observations:
• There is a social welfare function that satisfies IIA, weak Pareto, and finite anonymity (Lauwers and Van Liedekerke, 1995).
• There is no social welfare function that satisfies IIA, strong Pareto, and finite anonymity (Cato, 2017).
This situation shows that there is a big difference between the two Pareto axioms in terms of their consequences in the case where finite anonymity and IIA are desired: weak Pareto allows us to
8For other types of anonymity in this setting, see Cato (2017).
9See Cato (2017) for other anonymity axioms. In the context of intergenerational equity, a lot of anonymity axioms have been examined. See, for example, Kamaga and Kojima (2009, 2010).
10Willard (1970) and Aliprantis and Border (2006) provide concise arguments about a filter and related concepts.
11For a setA, (A)c (orAc) denotes the complement ofA.
12See Aliprantis and Border (2006).
13A coalitionAis decisive if∩
i∈AP(Ri)⊆P(f(R)) for allR∈ RN.
make a decision, while strong Pareto does not. This leads us to the following question: is there a Paretian axiom that falls between the two?
Consider the following axiom, which is introduced by Cato (2017).
F-strong Pareto: For all R ∈ RN, and for all x, y ∈X, if (x, y) ∈ P(∩
i∈NRi) and {i ∈ N : (x, y)∈I(Ri)} is finite, then (x, y)∈P(f(R)).
F-strong Pareto is clearly stronger than weak Pareto, but is weaker than strong Pareto. We can see that there is a social welfare function that satisfies IIA, F-strong Pareto, and finite anonymity.
Given a free ultrafilter Ω on N, we can define a social welfare function as follows:
fΩ(R) = ∪
A∈Ω
∩
i∈A
Ri.
This social welfare function is employed in many works, such as Kirman and Sondermann (1972), Hansson (1986), Armstrong (1980), Lauwers and Van Liedekerke (1995), and Cato (2017). It satisfies F-strong Pareto as well as the other axioms. To check this, let x, y ∈X and R∈ RN be such that (x, y) ∈P(∩
i∈N Ri) and {i∈ N : (x, y) ∈I(Ri)} is finite. Because Ω is an ultrafilter, there is no element that finite. Then, {i ∈ N : (x, y) ∈ I(Ri)} ∈/ Ω. This implies that the complement of {i ∈N : (x, y)∈ I(Ri)} is an element of Ω. Thus, {i ∈N : (x, y)∈ P(Ri)} ∈ Ω.
Then (x, y)∈P(f(R)), which implies that F-strong Pareto is satisfied.
Now, we introduce another Paretian axiom between F-strong Pareto and strong Pareto. The following axiom is proposed by Crespo, Nunez, and Rincon-Zapatero (2009) in the context of intergenerational equity.
I-strong Pareto: For all R ∈ RN, and for all x, y ∈ X, if (x, y) ∈ P(∩
i∈NRi) and {i ∈ N : (x, y)∈P(Ri)} is infinite, then (x, y)∈P(f(R)).
This axiom excludes an application of strong Pareto for the case where{i∈N : (x, y)∈P(Ri)} is finite. Note that
strong Pareto⇒I-strong Pareto⇒F-strong Pareto⇒weak Pareto, where “⇒” means “logically implies.”
We need to determine whether there is a social welfare function that satisfies IIA, I-strong Pareto, and finite anonymity. The standard construction of a social welfare function cannot be applied. Note thatfΩ does not satisfy I-strong Pareto. BecauseN is infinite, we take two infinite subsets,Aand B, such thatA∪B =N andA∩B =∅. Because Ω is an ultrafilter, eitherAorB is an element of Ω. Without loss of generality, we can assume thatA ∈Ω. Then, B /∈Ω. Given x, y ∈X, let R∈ RN be such that
{i∈N : (x, y)∈I(R)}=A and {i∈N : (x, y)∈P(R)}=B.
Then, we have (x, y)∈I(fΩ(R)), but I-strong Pareto requires (x, y)∈P(f(R)).
We need another way of constructing a social welfare function. Here is our main theorem.
Theorem 1. There exists a social welfare function f that satisfies IIA, I-strong Pareto, and finite anonymity.
Now, we prove this theorem. An ultrafilter is not sufficient to show the existence of the social welfare function. Thus, we need another concept. Let Mbe a non-empty collections of subsets of N. A correspondenceψ :M →2N is an ultrafilter mapping if ψ(M) is an ultrafilter onM for eachM ∈ M. Here, ψ is calledfree ifψ(M) is a free ultrafilter for every infinite setM ∈ M. An ultrafilter mapping onMis coherent if, for all M, M′ ∈ M such thatM′ ∈ψ(M),
ψ(M) ={M′′⊆M :M′′∩M′ ∈ψ(M′)}.
Coherency is introduced in Takayama and Yokotani (2017) who examine strongly Paretian social choice correspondences.
The following result is proved by Takayama and Yokotani (2017) using Zorn’s Lemma (or the Axiom of Choice).
Proposition 1 (Takayama and Yokotani, 2017, Proposition 6). Let MI be the collection of all infinite subsets of N. Then, there exists a coherent free ultrafilter mapping ψ :MI →2N.
Note that coherency implies a type of transitivity: if M′ ∈ ψ(M) and M′′ ∈ ψ(M′), then M′′ ∈ψ(M). The following result is also useful to proving our theorem.
Lemma 1. Let ψ be a coherent free ultrafilter mapping. For allM, M′, M′′ ⊆N, if M′ ∈ψ(M) and M′ ⊆M′′⊆M, then M′ ∈ψ(M′′).
Proof. Because M′ ∈ ψ(M) and M′ ⊆ M′′, the monotonic property implies that M′′ ∈ ψ(M).
If M′ ∈/ ψ(M′′), the strong property implies that M′′\M′ ∈ ψ(M′′). Coherency implies that M′′ \M′ ∈ ψ(M). Note that M′′\M′ ∈ ψ(M) and M′′ ∈ ψ(M). The intersection of M′′\M′ and M′′ is an element of ψ(M), by the finite-intersection property, but is empty. This is a contradiction.
Now, we prove the theorem.
Step 1: Construction of a Social Welfare Function: Given R∈ RN, let
NC(x, y;R) = {i∈N : (x, y)∈/ I(Ri)} and NU(x, y;R) ={i∈N : (x, y)∈I(Ri)}.
Here, NC(x, y;R) is the set of “concerned” individuals, and NU(x, y;R) is the set of unconcerned individuals. Note thatNC(x, y;R) =NC(y, x;R) and NU(x, y;R) =NU(y, x;R).
Take a coherent free ultrafilter mapping ψ on MI (here we apply Proposition 1). Definef as follows:
[#NC(x, y;R) = ∞and {i∈N : (x, y)∈P(Ri)} ∈ψ(NC(x, y))]⇔(x, y)∈P(f(R));
#NC(x, y;R)<∞ ⇔(x, y)∈I(f(R)).
Step 2: Showing that the properties are satisfied.
[IIA and I-strong Pareto]. It is clear that IIA is satisfied.
We now show that I-strong Pareto is satisfied. Let R∈ RN be such that (x, y)∈P(∩
i∈NRi) and {i∈N : (x, y)∈P(Ri)} is infinite. Then, #NC(x, y;R) =∞. Note that
NC(x, y;R) ={i∈N : (x, y)∈P(Ri)}.
Because ψ(NC) is an ultrafilter, it follows thatNC ∈ψ(NC). This implies that (x, y)∈P(f(R)).
Thus, I-strong Pareto is satisfied.
[Completeness]. Now, we show thatf(R) is complete. Takex, y ∈X. Suppose that (x, y)∈/ f(R). This implies that (x, y) ∈/ I(f(R)). Then, #NC(x, y;R) =∞. Because (x, y)∈/ P(f(R)), {i∈N : (x, y)∈P(Ri)}∈/ ψ(NC). This means that {i∈N : (y, x)∈P(Ri)} ∈ψ(NC). Then, we have (y, x)∈P(f(R)).
[Transitivity]. Next, we check transitivity. Suppose that (x, y) ∈ f(R) and (y, z) ∈ f(R).
There are four cases.
(i) (x, y)∈I(f(R)) and (y, z)∈I(f(R)). By construction,
#NC(x, y;R)<∞and #NC(y, z;R)<∞. This implies that
#NC(x, z;R)≤#NC(x, y;R) + #NC(y, z;R)<∞. By the construction off, we have (x, z)∈I(f(R)).
(ii) (x, y)∈I(f(R)) and (y, z)∈P(f(R)). Then, it follows that
#NC(x, y;R)<∞,#NC(y, z;R) = ∞, and {i∈N : (y, z)∈P(Ri)} ∈ψ(NC(y, z;R)).
BecauseNC(x, y;R)<∞,NU(x, y;R)∩NC(y, z;R) is infinite. BecauseNU(x, y;R)∩NC(y, z;R)⊆ NC(x, z;R), it follows that #NC(x, z;R) = ∞. The fact that NC(x, y;R) < ∞ implies that A /∈ψ(NC(x, z;R)) for anyA⊆NC(x, y;R). Thus,
NC(x, z;R)\NC(x, y;R)∈ψ(NC(x, z;R)).
Note that
NC(x, z;R)\NC(x, y;R) = G1 ∪G2, where
G1 :={i∈N : (x, y)∈I(Ri) and (x, z)∈P(Ri)}, G2 :={i∈N : (x, y)∈I(Ri) and (z, x)∈P(Ri)}.
Because ψ(NC(x, z;R)) is an ultrafilter, either G1 ∈ ψ(NC(x, z;R)) or G2 ∈ ψ(NC(x, z;R)).
Suppose that the latter is true. because G2 ⊆ G1 ∪G2 ⊆ NC(x, z;R), Lemma 1 implies that G2 ∈ψ(G1∪G2). Let G3 :={i∈N : (x, y)∈/ I(Ri) and (y, z)∈/ I(Ri)}. Note that
NC(y, z;R) =G1∪G2∪G3.
Note that G3 is finite. Thus, either G1 ∈ ψ(NC(y, z;R)) or G2 ∈ ψ(NC(y, z;R)). From the assumption that {i ∈N : (y, z)∈P(Ri)} ∈ ψ(NC(y, z;R)), it follows that G1 ∈ψ(NC(y, z;R)), because {i ∈ N : (y, z) ∈ P(Ri)} ∩G2 = ∅. Because G1 ⊆ G1 ∪G2 ⊆ NC(y, z;R), Lemma 1 implies that G1 ∈ ψ(G1 ∪G2). This is a contradiction. Thus, we have G1 ∈ ψ(NC(x, z;R)).
Because #NC(x, z;R) =∞ and (x, z)∈P(Ri) for all i∈G1, (x, z)∈P(f(R)).
(iii) (x, y)∈P(f(R)) and (y, z)∈I(f(R)). This case is proved in the same way as in (ii).
(iv) (x, y)∈P(f(R)) and (y, z)∈P(f(R)). By definition, #NC(x, y;R) = ∞, #NC(y, z;R) =
∞, and{i∈N : (x, y)∈P(Ri)} ∈ψ(NC(x, y;R)) and{i∈N : (y, z)∈P(Ri)} ∈ψ(NC(y, z;R)).
Let
H :={i∈N : (x, y)∈I(Ri) and (y, z)∈I(Ri)}. Note thatHc=NC(x, y;R)∪NC(y, z;R). Thus, #Hc=∞.
First, we show that {i ∈ N : (y, x) ∈ P(Ri)} ∈/ ψ(Hc). Suppose that {i ∈ N : (y, x) ∈ P(Ri)} ∈ ψ(Hc). because {i ∈ N : (y, x) ∈ P(Ri)} ⊆ NC(x, y;R) ⊆ Hc, it follows that {i ∈ N : (y, x) ∈ P(Ri)} ∈ ψ(NC(x, y;R)). This is a contradiction. Similarly, we show that {i ∈ N : (z, y) ∈ P(Ri)} ∈/ ψ(Hc). Because ψ(Hc) is an ultrafilter, Hc \({i ∈ N : (y, x) ∈ P(Ri)} ∪ {i∈N : (z, y)∈P(Ri)}) = {i∈N : (x, z)∈P(Ri)} ∈ψ(Hc).
Because ψ(Hc) is free, every set included in it is infinite. This implies that {i ∈N : (x, z)∈ P(Ri)} is infinite. Thus, NC(x, z;R) = ∞. Because NC(x, z;R) ⊆ Hc, Lemma 1 implies that {i∈N : (x, z)∈P(Ri)} ∈ψ(NC(x, z;R)).
[Finite Anonymity]. Finally, we show that finite anonymity is satisfied. Suppose that (x, y) ∈f(R). It suffices to show that finite anonymity is satisfied. Fix R ∈ RN. Take a, b∈ N.
LetR′ ∈ RN be such that
R′a=Rb, R′b =Ra,
R′i =Ri for all i∈N \ {a, b}.
It suffices to show that (i) (x, y) ∈ I(f(R)) ⇒ (x, y) ∈ I(f(R′)) and (ii) (x, y) ∈ P(f(R)) ⇒ (x, y) ∈ P(f(R′)). If (x, y) ∈ I(f(R)), then NC(x, y;R) < 0. It is clear that NC(x, y;R′) < 0.
Thus, (x, y)∈I(f(R′)). Suppose that (x, y)∈P(f(R)). We distinguish four cases.
(i) a, b∈ NU(x, y;R). Note that NC(x, y;R) = NC(x, y;R′) and {i ∈ N : (x, y) ∈ P(Ri)} = {i ∈ N : (x, y) ∈ P(R′i)}. Thus, {i ∈ N : (x, y) ∈ P(R′i)} ∈ ψ(NC(x, y;R′)). Then, we have (x, y)∈P(f(R′)).
(ii) a, b ∈ NC(x, y;R). Then, NC(x, y;R) = NC(x, y;R′). If {i ∈ N : (x, y) ∈ P(Ri)} = {i ∈ N : (x, y) ∈ P(R′i)}, we have (x, y) ∈ P(f(R′)), because {i ∈ N : (x, y) ∈ P(Ri)} ∈ ψ(NC(x, y;R′)). Suppose that {i∈N : (x, y)∈P(Ri)} ̸={i∈N : (y, x)∈P(R′i)}. Without loss of generality, we can assume that a∈ {i∈ N : (x, y)∈P(Ri)} and b ∈ {i∈N : (y, x)∈ P(Ri)}. Because ψ(NC(x, y;R)) is an ultrafilter, {i∈ N : (x, y) ∈P(Ri)} \ {a} ∈ ψ(NC(x, y;R)). Then, ({i∈N : (x, y)∈P(Ri)} \ {a})∪ {b} ∈ψ(NC(x, y;R)). Thus, we have (x, y)∈P(f(R′)).
(iii) a ∈ NC(x, y;R) and b ∈ NU(x, y;R). By assumption, {i ∈ N : (x, y) ∈ P(Ri)} ∈ ψ(NC(x, y;R)). Either a ∈ {i ∈ N : (x, y) ∈ P(Ri)} or a ∈ {i ∈ N : (y, x) ∈ P(Ri)}. First, we assume thata ∈ {i∈N : (x, y)∈P(Ri)}. Let
J1 = (NC(x, y;R)\ {a})∪ {b}.
Because ψ(J1) is an ultrafilter, one of {i ∈ N : (x, y) ∈ P(Ri)} \ {a}, {b}, or {i ∈ N : (y, x) ∈ P(Ri)} is an element of ψ(J1). Because ψ(J1) is free, it follows that {b} ∈/ ψ(J1). If {i ∈ N :
(y, x) ∈P(Ri)} ∈ψ(J1), Lemma 1 implies that {i∈N : (y, x)∈P(Ri)} ∈ψ(NC(x, y;R)\ {a}), because NC(x, y;R)\ {a} ⊆ J1. Because ψ(NC(x, y;R)) is a free ultrafilter, we have {i ∈ N : (x, y)∈P(Ri)} \ {a} ∈NC(x, y;R). However, because NC(x, y;R)\ {a} ⊆NC(x, y;R) and{i∈ N : (x, y)∈P(Ri)}\{a} ∈ψ(NC(x, y;R), Lemma 1 implies that{i∈N : (x, y)∈P(Ri)}\{a} ∈ ψ(NC(x, y;R)\ {a}). This is a contradiction. Therefore, {i∈N : (x, y)∈P(Ri)} \ {a} ∈ψ(J1).
Note thatNU(x, y;R′) =J1∪ {a}<∞. Then, we have (x, y)∈P(f(R)).
Second, we assume that a ∈ {i∈ N : (y, x)∈ P(Ri)}. Because ψ(J1) is an ultrafilter, one of {i∈N : (x, y)∈ P(Ri)}, {b}, or {i ∈N : (y, x)∈P(Ri)} \ {a} is an element of ψ(J1). Because ψ(J1) is free, it follows that {b} ∈/ ψ(J1). If {i ∈ N : (y, x) ∈ P(Ri)} \ {a} ∈ ψ(J1), Lemma 1 implies that{i∈N : (y, x)∈P(Ri)}\{a} ∈ψ(NC(x, y;R)\{a}), becauseNC(x, y;R)\{a} ⊆J1. BecauseNC(x, y;R)\ {a} ⊆NC(x, y;R) and{i∈N : (x, y)∈P(Ri)} ∈ψ(NC(x, y;R), Lemma 1 implies that{i∈N : (x, y)∈P(Ri)} ∈ψ(NC(x, y;R)\ {a}). This is a contradiction. Therefore, {i∈N : (x, y)∈ P(Ri)} \ {a} ∈ ψ(J1). Note that NU(x, y;R′) =J1∪ {a} <∞. Then, we have (x, y)∈P(f(R)).
From (i) to (iii),f(R)⊆f(R′). An analogous procedure yields thatf(R′)⊆f(R). Therefore, f(R) =f(R′). Thus, finite anonymity is satisfied.
4 Extension to Quasi-Orderings
In the previous section, we require a social preference to be transitive. In this section, we allow it to be quasi-transitive, which requires that
(x, y)∈P(R) and (y, z)∈P(R)⇒(x, z)∈P(R).
Quasi-transitivity is transitivity of the asymmetric part. It is substantially weaker than transitiv- ity. LetQ be the set of complete and quasi-transitive binary relations. A quasi social welfare function f is a mapping from RN toQ. Our theorem in the previous section implies that there exists a quasi social welfare function that satisfies IIA, I-strong Pareto, and finite anonymity. A difficulty is that its construction is completely dependent on Zorn’s lemma. This means that the social welfare function is a non-constructible object. This is quite problematic when we apply it.
Here, we show that a quasi-social welfare function with our axioms is possible with an ex- plicit construction. Since Hansson (1976), it has been known that Arrovian quasi-social welfare functions are associated with the concept of a filter.14 A particular filter is important:
FF ={M ⊆N :Mc is finite}.
This is called the Fr´echet filter. It is known that FF is the coarsest free filter. Now, we construct a filter version ofψ. GivenMa non-empty collections of subsets ofN, a correspondence ψ : M → 2N is a filter mapping if ψ(M) is a filter on M for each M ∈ M. Here, ψ is called freeif ψ(M) is a free filter for every infinite set M ∈ M. Coherency is defined as the same as that for an ultrafilter.
14Hansson (1976) shows that if a quasi-social welfare function satisfies weak Pareto and IIA, then the family of decisive coalitions forms a filter onN.
The following shows that the filter mapping that yields the Fr´echet filter for each infinite set is coherent. Thus, a coherent free filter mapping can be explicitly constructed.
Proposition 2. Let MI be the collection of all infinite subsets of N. Then, the following is a coherent free filter mappingψˆ:MI →2N: for each M ∈ MI,
M′ ∈ψ(Mˆ )⇔M \M′ is finite.
Proof. Since ˆψ is a free filter mapping, it suffices to show that for all M, M′ ∈ MI such that M′ ∈ψ(Mˆ ),
ψ(Mˆ ) ={M′′⊆M :M′′∩M′ ∈ψˆ(M′)}.
LetM ∈ MI and M′ ⊆M be such that M′ ∈ψ(Mˆ ). In order to show that ˆψ(M)⊆ {M′′ ⊆M : M′′∩M′ ∈ψ(Mˆ ′)}, suppose thatM′′ ∈ψ(Mˆ ). By the construction of ˆψ, bothM\M′ andM\M′′
are finite. This implies that (M \M′)∪(M \M′′) is also finite. Thus, M \(M′′∩M′) is finite, which implies that M′\(M′′∩M′) is finite. By definition, M′′∩M′ ∈ψ(Mˆ ′) sinceM′ ∈ MI. In order to show that ˆψ(M) ⊇ {M′′ ⊆ M : M′′∩M′ ∈ ψ(Mˆ ′)}, suppose that M′′∩M′ ∈ ψ(Mˆ ′).
Then,M′\(M′′∩M′) is finite. By our assumption,M\M′ is finite, and thus, so isM\(M′∪M′′).
Note that
M \M′′=M′\(M′′∩M′)∪M \(M′∪M′′).
Thus,M \M′′ is finite. By definition, M′′∈ψ(Mˆ ).
By using the Fr´echet filter, we can construct a quasi social welfare function that satisfies IIA, I-strong Pareto, and finite anonymity. Now, define fF as follows:
[#NC(x, y;R) =∞ ⇒[
{i∈N : (y, x)∈P(Ri)}∈/ψ(Nˆ C(x, y))⇔(x, y)∈fF(R) ]
;
#NC(x, y;R)<∞ ⇒(x, y)∈I(fF(R)).
It can be shown that fF is a quasi social welfare function that satisfies our axioms. IIA and I-strong Pareto are satisfied.15 Next, we show that fF(R) is quasi-transitive. Suppose that (x, y)∈P(fF(R)) and (y, z)∈P(fF(R)). Note that
(x′, y′)∈P(fF(R))⇔[
#NC(x′, y′;R) = ∞ and #{i∈N : (y′, x′)∈P(Ri)}<∞] .
Then, it follows that #NC(x, y;R) =∞,#NC(y, z;R) = ∞and
#{i∈N : (y, x)∈P(Ri)}<∞and #{i∈N : (z, y)∈P(Ri)}<∞. This implies that{i∈N : (y, x)∈P(Ri)} ∪ {i∈N : (z, y)∈P(Ri)} is finite. Note that
{i∈N : (z, x)∈P(Ri)} ⊆ {i∈N : (y, x)∈P(Ri)} ∪ {i∈N : (z, y)∈P(Ri)}.
Thus, #{i∈N : (z, x)∈P(Ri)}<∞. We have to show that #NC(x, z;R) = ∞. Note that both {i∈N : (x, y)∈P(Ri)} and {i∈N : (y, z)∈P(Ri)} are infinite. Since{i∈N : (y, x)∈P(Ri)} is finite, the following set is infinite:
H :={i∈N : (y, z)∈P(Ri)} \ {i∈N : (y, x)∈P(Ri)}.
15This point can be shown as in the case of a social welfare function (Theorem 1).
Since
H ⊆ {i∈N : (x, y)∈P(Ri)} ∪ {i∈N : (x, y)∈I(Ri)},
it follows that (x, z)∈P(Ri) for alli∈H. Then,{i∈N : (x, z)∈P(Ri)}is infinite, and thus, we have #NC(x, z;R) = ∞. Then, (x, z) ∈P(fF(R)) holds. It is easy to see that finite anonymity and completeness satisfied.16 Therefore, we can obtain a natural construction of a quasi social welfare function.
5 Concluding Remarks
This study examined social choice with an infinite population. We showed a possibility result, which demonstrates that anonymity and Paretian judgments are compatible. Strong Pareto is an appropriate formulation of the Pareto principle in the case of a finite population. However, it can be too demanding in the case of an infinite population. Therefore, in this study, we employ the I-strong Pareto, which is applied only when some alternative is better than another for an infinite number of individuals.
We utilized the concept of an ultrafilter mapping. The existence of this mapping is completely dependent on Zorn’s Lemma. This results in a serious problem for the structure of a social welfare function, because we cannot explicitly construct the concrete structure of the social welfare function, which we considered in this study. The social welfare function exists, but it is invisible to us. Thus, when we face applied issues, we need some other ways to make social welfare functions visible. One possible way is relaxing social rationality. However, it makes many pairs of alternatives indifferent.
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