Limiting Dictatorial Rules

Ricard Torres∗

ITAM and Universitat de Girona

December 2002This revision: May 2004

Abstract

We consider the preference aggregation problem in infinite societies. Inour model, there are arbitrarily many agents and alternatives, and admis-sible coalitions may be restricted to lie in an algebra. In this framework(which includes the standard one), we characterize, in terms of Strict Neu-trality, the Ultrafilter Property of preference aggregation rules. Based onthis property, we define the concept of Limiting Dictatorial rules, whichare characterized by the existence of arbitrarily small decisive coalitions.We show that in infinite societies which can be well approximated by fi-nite ones, any Arrovian rule is Limiting Dictatorial.JEL: D71, C69.Key words: Preference Aggregation, Arrow’s Theorem, Dictatorial Rules,Strict Neutrality, Ultrafilter Property, Finite Approximations.

∗Centro de Investigacion Economica, Instituto Tecnologico Autonomo de Mexico (ITAM),and Universitat de Girona. Address: Faculty of Economics and Business, Campus Montilivi,Universitat de Girona, E-17071 Girona, Spain. Phone: +34 972 418040. Fax: +34 972 418032.Email address: ricard.torres@udg.es. This paper was formerly circulated under the title“Smallness of Invisible Dictators.” I thank Subir Chattopadhyay, Andrei Gomberg, HeliosHerrera, Alan Kirman, Cesar Martinelli, Herve Moulin, Beatriz Rumbos, and Luis Ubedafor comments and suggestions. I thank as well the comments of two anonymous referees; areferee of this Journal, especially, made helpful suggestions. I am also grateful to participantsat the 12th European Workshop on General Equilibrium Theory, the 2003 North and LatinAmerican Summer Meetings of the Econometric Society, and seminars at various institutions.I appreciate the financial support of the Asociacion Mexicana de Cultura.

1 Introduction

Arrow [4] considered the problem faced by a given number of individuals thatare trying to set up a collective decision mechanism. This is the preferenceaggregation problem: find a collective preference among the available alterna-tives, given the preferences the different individuals have. Arrow tackled thisproblem by requiring any preference aggregation rule to satisfy the followingconditions: (1) Universal Domain (individuals can have any preference order-ing); (2) Unanimity (if all individuals strictly prefer one alternative to another,so should the society do); and (3) Pairwise Independence (in order to decide thesocial ranking between two alternatives, only the rankings between those twoalternatives by the individuals should matter). A preference aggregation rulethat obeys these requirements is called Arrovian. Arrow’s now classical resultis that any Arrovian rule is necessarily dictatorial : an individual (the dictator)is singled out, and no matter what the others’ preferences are, the dictator’sstrict preferences between alternatives are adopted by the collectivity.

In voting models, it is often convenient to assume the existence of infinitelymany individuals (see, e.g., Caplin and Nalebuff [10]). The same is true in mod-els with local public goods (see, e.g., Caplin and Nalebuff [11]). In Economics,there is a vast literature, originating in Aumann [5], where the competitiveparadigm is modeled by means of an infinite society in which the influence ofeach individual is negligible. These different trends have motivated the study ofmodels of social choice in which the society is assumed to be formed by an infi-nite number of individuals (see, e.g., Armstrong [2], Banks, Duggan and Le Bre-ton [7], Gomberg, Martinelli and Torres [14], and Mihara [18], [19]). However,Fishburn [12] showed1 that in this case there may exist Arrovian preferenceaggregation rules that are not dictatorial. Now infinite societies are, in general,a modeling device that is used to study the properties of large, but finite, soci-eties. In this case, a requirement of the model (see the discussion by Saari [21],section 2.3) should be “continuity at infinity:” the properties of the infinitesociety should correspond to the limit of the corresponding properties of finitesocieties, when the number of individuals in them tends to infinity. Fishburn’spaper suggests that, with regard to Arrow’s theorem, there is a discontinuityat infinity: in any finite society, an Arrovian rule gives rise to a dictator, whilethere are infinite societies with no dictator associated to the corresponding lim-iting Arrovian rule. Our aim in this paper is to show that, if one adopts theappropriate viewpoint, there is no such discontinuity.

To do this, we resort to a concept defined and used by Arrow, that of a deci-sive coalition: a set of individuals that are able to impose their strict preferenceson the society, no matter what the preferences of the remaining individuals are.In fact, a dictator is just a decisive coalition formed by a single individual; oneway to interpret this is to view a dictator as the smallest kind of decisive coali-tion. When there are infinitely many individuals, it may happen that there isno dictator (a smallest decisive coalition), but we show that there always are

1Fishburn attributes prior knowledge of this result to Julian Blau. See Hansson [16].

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arbitrarily small decisive coalitions: in other words, that there is no discontinu-ity at infinity. Therefore, the preference aggregation problem that, according toArrow’s theorem, finite societies have, is still present when the society is infinite.

The classical proof of Arrow’s theorem is based on showing that, given anyArrovian rule, there always exist decisive coalitions and, moreover, one can ac-tually obtain progressively smaller decisive coalitions, until the process stopswith a dictator. This intuitive idea is expressed mathematically by the Ultra-filter Property. This property has two aspects. On the one hand, it impliesthat, given any two decisive coalitions, there is always a third decisive coalitionwhich is smaller than each of them; this is its filtering side. On the other hand,the property implies that there are actually many decisive coalitions; this isits ultra side. Arrovian rules, whether in finite or in infinite societies, alwayssatisfy the Ultrafilter Property. This has been shown in the previous literature(see below), but here we give a complete characterization of this property (the-orem 1), and precisely clarify its relationship with Arrovian rules (theorem 2).Our result implies that there may exist non-Arrovian rules which satisfy the Ul-trafilter Property (in particular, that are dictatorial), but there always exists anArrovian rule which is equivalent, in some precise sense, to any such rule. Thischaracterization is an important step in our work, since the Ultrafilter Propertyis the tool we use to show that Arrovian rules have always associated arbitrarilysmall decisive coalitions.

We also extend our work to the case in which some coalitions are admissible,but some are not. This case has interest, because admissibility constraints ap-pear naturally in certain models of social choice. In other models, admissibilityconstraints are used so that the process of finding aggregate magnitudes hascontinuity properties. This case reinforces our previous view of dictatorship asa concept having to do with smallest decisive coalitions: even in a finite society,a dictator may well be a coalition formed by more than one individual, thoughsmallest in the sense that no subcoalition of it is admissible (example 1). Inan infinite society, there is even a more extreme occurrence: there may be anindividual such that a coalition is decisive if, and only if, it contains this individ-ual; but the individual itself is not an admissible coalition (example 3). Whenthis happens, it does not represent a problem from our conceptual viewpoint,because we can still refer to arbitrarily small decisive coalitions.

However, there may be models with admissibility constraints on coalitions,in which our concept of smallness of decisive coalitions does not apply, that is,we cannot affirm that there are arbitrarily small decisive coalitions (example 2).We show that this may never happen if our infinite agent set can be well approx-imated by large finite agent sets (theorem 4). Thus, we may argue, as Saari [21]suggests, that one should simply discard those models as inadequate, since theproperties of the infinite model do not correspond to the properties of approx-imating finite models with a large number of individuals. We display broadclasses of models for which finite approximations are possible (corollary 2), andgive examples of models in which it does not (example 4). Our results should bereassuring for Economists and Political Scientists that have been using modelswith infinitely many individuals, since all of the models we are aware of fall

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within the category of infinite models that are well approximated by finite ones.

1.1 Literature review and technical introduction

After Fishburn’s pioneering work, Kirman and Sondermann [17] suggested twodifferent lines to tackle the “discontinuity at infinity” problem. The first linewas measure-theoretic: the authors showed that if the agent set is an atomlessprobability measure space, then any non–dictatorial Arrovian rule has decisivesets of arbitrarily small measure. This approach was contested by Schmitz [22],who showed that, if the agent set has a σ-finite but infinite measure, then one canalways construct non–dictatorial Arrovian rules for which all decisive sets haveinfinite measure. The second line suggested by Kirman and Sondermann wastopological: restricting themselves to the case in which there are finitely manyalternatives, they showed that the agent set can be enlarged in such a way (theStone-Cech compactification) that the original profile of preferences determinesthe preferences of all agents in the larger set, and each Arrovian rule has adictator in the enlarged space. They referred to dictators in the larger spacethat do not belong to the original one as invisible dictators. The topologicalapproach initiated by Kirman and Sondermann was extended by Armstrong [2]to measurable structures and arbitrary sets of alternatives. In the topologicalapproach followed by Armstrong, one constructs a (discrete–like) topology2 onthe agent space and shows that the invisible dictators lie in its Stone-Cechcompactification. The problem with this approach is that, if the original agentspace is endowed with a topology that for some reason is relevant, this topologyis essentially ignored in the construction of the enlarged agent space. Thus, onecan always associate “invisible dictators” with non–dictatorial Arrovian rules,but in a manner that in some cases may seem somewhat arbitrary.

In this paper, we pursue a third, set-theoretic, approach to show that to non–dictatorial Arrovian rules there correspond arbitrarily small decisive coalitions.We define (definition 5) the concept of Limiting Dictatorial rules as preferenceaggregation rules in which this happens: there are arbitrarily small decisivecoalitions, in the sense that one can find a nested collection of decisive coalitionswhose limit is smallest (whether it is empty or not). Dictatorial rules are alwaysLimiting Dictatorial, but the opposite is not necessarily true.

We define (definition 6) a concept of approximation of an infinite society byfinite ones in which the admissibility constraints on coalitions are taken intoaccount. Our main result (theorem 4) states that, if the infinite society can bewell approximated by large finite societies, then any Arrovian rule is LimitingDictatorial.

Thus, we provide a means to classify infinite societies according to whetheror not they can be well approximated by finite ones. When this is the case, onecan affirm that there is no discontinuity with regards to preference aggregation.

2In this topology, the original agent set (after identifying points that are not separatedby the algebra) is viewed as a subspace of the Stone space that corresponds to the algebraof coalitions. The Stone space is a totally disconnected compact topological space. SeeSikorski [24].

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Kirman and Sondermann [17] consider a framework in which all coalitionsare admissible. These authors were the first3 to show the Ultrafilter Property ofArrovian rules: the collection of all decisive coalitions that correspond to anyArrovian rule is an ultrafilter. The importance of this property for an under-standing of Arrow’s theorem has been emphasized by Sen [23], who interprets itin terms of invariance conditions in the preference aggregation process. With noadmissibility constraints on coalitions, there are non–dictatorial Arrovian rulesif the intersection of all decisive coalitions is empty. When there are admissi-bility constraints (an algebra or σ-algebra of admissible coalitions), there arealso non–dictatorial Arrovian rules in which the intersection of all decisive setsis nonempty (example 3). We show examples of the different possibilities thatarise (examples 1, 2, 3, and 4), and precisely characterize the decisive sets in allof them (see subsection 8.4 in the appendix).

As we mentioned before, Schmitz [22] showed that, if the agent space hasan infinite σ-finite measure, then, given a free ultrafilter of decisive coalitions,there do not necessarily exist decisive coalitions of arbitrarily small measure. Asexamples of spaces satisfying his requirements, Schmitz cites the naturals witha counting measure, and Rn with its Borel subsets and Lebesgue measure. How-ever, in both of these examples the infinite societies are well approximated byfinite ones (see section 6), so our concept of solution (Limiting Dictatorial rules)applies, and it implies that it is still possible to refer to arbitrarily small decisivecoalitions, though in set-theoretic terms rather than in terms of measures.

The model we use is presented in section 2. In section 3, we characterize theUltrafilter Property in the general framework (arbitrary sets of individuals andalternatives, measurable structures) we are considering. We resort to the StrictNeutrality property which Geanakoplos [13] and Ubeda [25] have used recentlyto provide very efficient proofs of Arrow’s theorem in the finite case. We show(theorem 1) that, in general, the Ultrafilter Property holds if, and only if, thepreference aggregation rule satisfies Unanimity and Strict Neutrality. We alsostate (theorem 2) the exact relationship between Arrovian rules and the Ultra-filter Property: if a non-Arrovian rule satisfies the Ultrafilter Property, thenthere is an Arrovian rule that is equivalent to it, in the sense that it has exactlythe same decisive sets. To understand the meaning of this characterization, re-member that all dictatorial rules satisfy the Ultrafilter Property, regardless ofwhether or not they are Arrovian. In other words, this result implies in partic-ular the (known) fact that dictatorial rules are not necessarily Arrovian whenone moves out of the domain of linear orders.

In section 4 we define the concept of Limiting Dictatorial rules. We show byexample the different kinds of Limiting Dictatorial rules one can encounter, andprove (theorem 3) that Arrovian rules are always Limiting Dictatorial whenever

3See subsection 8.1 in the appendix for a formal definition of ultrafilters. This result wasimplicit in the proof given by Fishburn [12]. Essentially, it was earlier discovered by Guil-baud [15] as a consistency condition for the aggregation of preferences. (I thank Alan Kirmanfor pointing me in the direction that allowed me to find this earlier reference.) Hansson [16]also found it independently in a working paper circulated before the appearance of Kirmanand Sondermann’s article.

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there are no admissibility constraints on coalitions. We display (example 4)an example that shows that Arrovian rules need not be Limiting Dictatorial ingeneral. In section 5, we define our concept of finite approximations to infi-nite societies, and show (theorem 4) that in any finitely approachable societyArrovian rules are always Limiting Dictatorial. In section 6 we single out (corol-lary 2) two broad classes of infinite agent sets that are finitely approachable.In section 7 we conclude. We relegate technical results and definitions to anappendix.

2 The model

The (non-empty) set of individuals is denoted by N , which can be either finiteor infinite. Admissible coalitions of individuals are members of an algebra L ofsubsets of N . An algebra is a collection that contains N and is closed under(finite) unions and set difference. These are reasonable requirements for admis-sibility: on the one hand, we allow any two admissible coalitions to merge; onthe other hand, given a coalition and any admissible subcoalition, the membersthat are left off should form another admissible subcoalition. If we do not wishto impose any admissibility constraint on coalitions, then we set L = 2N , thecollection of all subsets of N .

The (non-empty) set of alternatives is denoted by X. This set can be ei-ther finite or infinite, but it must have at least three different elements. Eachindividual has a (weak) preference relation on X, i.e. a reflexive, transitive andcomplete binary relation (a complete preorder). Let R denote the set of allpreference relations on X. Given a preference relation R on X, we define itsindicator function, I(R) : X ×X → {−1, 0, 1}, by

I(R)(x, y) =

−1 if yRx and not xRy;0 if xRy and yRx;1 if xRy and not yRx.

A preference profile is a mapping ρ : N → R. Given a profile of preferencesρ, we define its indicator function, I(ρ) : X ×X → {−1, 0, 1}N , by

I(ρ)(x, y) =(I[ρ(i)](x, y)

)i∈N

We impose the following assumption:

(UD) For any x, y ∈ X, the mapping i 7→ I[ρ(i)](x, y), from N to {−1, 0, 1}, isL–measurable.

Assumption (UD) means that all rankings among any given finite subset ofalternatives are admissible (the letters UD stand for “universal domain”). Tech-nically, assumption (UD) imposes a measurability requirement on the profiles ofpreferences that are allowed. We denote by RN

L the set of all preference profilesthat satisfy assumption (UD).

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We will impose throughout our paper assumption (UD). Thus, in our contexta preference aggregation rule is a map

f : RNL → R

That is, with each preference profile ρ the rule f associates a social preferencerelation on X.

Definition 1 (Decisive Sets) Given a preference aggregation rule f , we saythat a coalition A ∈ L is f–decisive (for short, decisive) if

∀i ∈ A, I[ρ(i)](x, y) = 1 =⇒ I[f(ρ)](x, y) = 1.

N

Definition 2 (Dictators) Given a preference aggregation rule f , we say thatan individual i ∈ N is an f -Dictator (for short, a Dictator) if the coalition {i}is f–decisive. N

3 The Ultrafilter Property

In subsection 8.1 in the appendix, we define the concepts of an L–filter and anL–ultrafilter, which are used in the remainder of the paper. We have providedan intuitive interpretation of those concepts in the Introduction.

We are going to use the following properties a preference aggregation rule fmay satisfy:

• Unanimity or weak Pareto: The coalition N of all individuals is f–decisive.

• Independence of irrelevant alternatives or Pairwise Independence (PI):The social preference between x and y is determined exclusively by theindividual preferences between these alternatives:

I(ρ)(x, y) = I(ρ′)(x, y) =⇒ I[f(ρ)](x, y) = I[f(ρ′)](x, y).

• Strict Neutrality (SN): If all preferences are strict, then f treats in thesame manner any pairs of alternatives that have the same relative positionsin the individuals’ preferences, even in different profiles:

I(ρ)(x, y) = I(ρ′)(x′, y′) ∈ {−1, 1}N

=⇒ I[f(ρ)](x, y) = I[f(ρ′)](x′, y′) ∈ {−1, 1}.

• Dictatorship: There is an f–Dictator. In this case, we also say that f isDictatorial.

• Ultrafilter Property (UP): The set of all f–decisive coalitions is an ultra-filter.

Definition 3 (Arrovian Rules) A preference aggregation rule is called Arro-vian if it satisfies Unanimity and Pairwise Independence. N

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In our definition of what an Arrovian rule is, we separate the requirementsrelative to the aggregation procedure (the rule), from the domain requirements.In our work, we do require (a suitable variant of) Universal Domain, but we listit as an additional requirement, and not part of the rule itself.

We first show that the Ultrafilter Property is closely related to Dictatorship.Sen [23] interprets this property in terms of invariance requirements, and stressesits importance for an understanding of Arrow’s theorem.

Proposition 1 (Ultrafilters and Dictators) If a preference aggregation ruleis Dictatorial, then it satisfies the Ultrafilter Property. Conversely, suppose Nis finite and all coalitions are admissible (L = 2N ), then any rule that satisfiesthe Ultrafilter Property is Dictatorial.

Proof: Assume the preference aggregation rule f is Dictatorial. Let i ∈ N bethe f–Dictator. Then any coalition that contains i is f–decisive. If A ∈ L andi /∈ A, then by definition A may not be f–decisive (two disjoint sets may notbe f–decisive). Concluding, D is equal to the ultrafilter Li (see subsection 8.1in the appendix).

Assume now N is finite and L = 2N . If D is an ultrafilter, then it is of theform D = {S ⊂ N : S 3 i}, for some i ∈ N (see, e.g., Aliprantis and Border [1],section 2.5, or proposition 2 in this paper). This means that i is an f–Dictator.�

We consider in the remainder of the paper the case in which the UltrafilterProperty is satisfied, but N is infinite and/or there is an admissibility constrainton coalitions.

We now characterize the Ultrafilter Property in terms of the remaining prop-erties we have defined. We break the proof into a series of simple lemmas. Seealso Austen-Smith and Banks [6], section 2.4, for generalizations of the Ultra-filter Property in the finite case.

The first lemma is similar in spirit to the pivotal argument used by Geanako-plos [13] and Ubeda [25], whose origins are to be found in Barbera [8].

Lemma 1 Let (UD) hold, and assume the preference aggregation rule f satis-fies Unanimity and Strict Neutrality (SN). Let x, y ∈ X, x 6= y, and let S, T ∈ L

satisfy S ⊂ T ⊂ N . Let ρ0 and ρ1 be profiles in which:

∀i ∈ S, I[ρ0(i)](x, y) = 1, I[ρ1(i)](x, y) = 1;∀i ∈ T \ S, I[ρ0(i)](x, y) = −1, I[ρ1(i)](x, y) = 1;∀i ∈ N \ T, I[ρ0(i)](x, y) = −1, I[ρ1(i)](x, y) = −1.

Then I[f(ρ0)](x, y) = −1 and I[f(ρ1)](x, y) = 1 imply that T \ S is a decisivecoalition.

Proof: Let a, b ∈ X, with a 6= b, and assume that ρ is an arbitrary profile inwhich, for all i ∈ T \ S, I[ρ(i)](a, b) = 1.

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Let c ∈ X be such that c /∈ {a, b} (this is possible because X has at leastthree different elements). Let ρ′ be a profile in which

I(ρ′)(a, b) = I(ρ)(a, b),

and, additionally,

∀i ∈ S, I[ρ(i)](a, c) = 1, and I[ρ(i)](b, c) = 1;∀i ∈ T \ S, I[ρ(i)](a, c) = 1, and I[ρ(i)](b, c) = −1;∀i ∈ N \ T, I[ρ(i)](a, c) = −1, and I[ρ(i)](b, c) = −1.

Now I(ρ′)(b, c) = I(ρ0)(x, y) and I[f(ρ0)](x, y) = −1 imply, by (SN), thatI[f(ρ′)](b, c) = −1.

Analogously, I(ρ′)(a, c) = I(ρ1)(x, y) and I[f(ρ1)](x, y) = 1 imply, by (SN),that I[f(ρ′)](a, c) = 1.

By transitivity, I[f(ρ′)](a, b) = 1, and by (SN), I[f(ρ)](a, b) = 1. �

Lemma 2 Let (UD) hold, and assume the preference aggregation rule f satis-fies Unanimity and Strict Neutrality (SN). Let x, y ∈ X, x 6= y, let T ∈ L, andlet ρ be a profile in which:

∀i ∈ T, I[ρ(i)](x, y) = 1;∀i ∈ N \ T, I[ρ(i)](x, y) = −1.

Then I[f(ρ)](x, y) = 1 implies that T is a decisive coalition.

Proof: Let S = ∅, let ρ0 be the profile in which all individuals strictly prefery to x, and let ρ1 = ρ. Then use Unanimity and apply lemma 1. �

The next lemma (an easy consequence of the previous ones) is the resultcalled Equivalent Subsets (or excludability of the undecisive) in Sen [23].

Lemma 3 Let (UD) hold, and assume the preference aggregation rule satisfiesUnanimity and Strict Neutrality (SN). Suppose T ∈ L is decisive, and S ⊂ T ,S ∈ L. Then either S or T \ S (but not both) is decisive.

Proof: Let ρ0 and ρ1 be as in the statement of lemma 1.By (SN), we must have I[f(ρ0)](x, y) ∈ {−1, 1}.If I[f(ρ0)](x, y) = 1, lemma 2 implies that S is decisive.Since by assumption T is decisive, we know that I[f(ρ1)](x, y) = 1, so if

I[f(ρ0)](x, y) = −1, lemma 1 implies that T \ S is decisive. �

Lemma 4 Let (UD) hold, and assume the preference aggregation rule satisfiesUnanimity and Strict Neutrality (SN). Then the intersection of any two decisivecoalitions is a decisive coalition.

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Proof: Let S and T be decisive coalitions in L. Then T is the union of thedisjoint sets S ∩ T and T \ S. Now T \ S is disjoint from the decisive coalitionS, so by definition it cannot be decisive. Therefore, lemma 3 implies that S ∩Tis decisive. �

Theorem 1 (Ultrafilter Property) Let (UD) hold. Then a preference ag-gregation rule f satisfies Unanimity and Strict Neutrality (SN) if, and only if,the collection of all f–decisive sets forms an L–ultrafilter.

Proof: Assume f satisfies Unanimity and (SN). Let D be the collection ofall f–decisive coalitions. Unanimity implies that N ∈ D and ∅ /∈ D. We haveseen (lemma 4) that D is closed under (finite) intersections, and by definitionsupersets of elements of D are in D. This implies that D is a filter. Now, takingT = N in lemma 3, we obtain that D is an ultrafilter.

Assume now that f satisfies the Ultrafilter Property, that is, the collectionD of all f–decisive sets forms an L–ultrafilter. Since N ∈ D, Unanimity issatisfied. Let us show that (SN) holds as well. Assume that

I(ρ)(x, y) = I(ρ′)(x′, y′) ∈ {−1, 1}N .

Let S = {i ∈ N : I[ρ(i)](x, y) = 1}. If S ∈ D, then I[f(ρ)](x, y) = I[f(ρ′)](x′, y′) =1. Otherwise, N \ S ∈ D, which implies I[f(ρ)](x, y) = I[f(ρ′)](x′, y′) = −1.�

Theorem 1 gives a complete characterization of the Ultrafilter Property.It is important to notice that, in our domain of preferences, if f satisfies theUltrafilter Property, then f is not necessarily an Arrovian rule. This is only trueover the domain of linear orders, where indifference between distinct alternativesis not allowed. Actually, it is easy to construct simple (finite) examples inwhich a dictatorial rule does not satisfy Pairwise Independence. For example,it is possible for a dictatorial rule to have x strictly preferred to y and theopposite preference, for two different profiles in which individual preferencesbetween x and y do not change, as long as the dictator is indifferent betweenboth alternatives.

We next show the precise relationship between Arrovian Rules and the Ul-trafilter Property.

Geanakoplos [13] and Ubeda [25] have recently developed very efficient proofsof Arrow’s theorem in the finite case, by showing first that Arrovian rules satisfyStrict Neutrality. Their argument can be generalized to our setting.

Lemma 5 (Geanakoplos, Ubeda) Let (UD) hold. Then for any Arrovianpreference aggregation rule f we have:

(i) Strict individual preferences result in a strict social preference:

I(ρ)(x, y) ∈ {−1, 1}N =⇒ I[f(ρ)](x, y) ∈ {−1, 1}.

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(ii) f satisfies Strict Neutrality (SN).

Proof: Suppose thatI(ρ)(x, y) ∈ {−1, 1}N .

Since X has at least 3 different elements, there is z ∈ X such that z /∈ {x, y}.Assume, without loss of generality, that I[f(ρ)](x, y) ∈ {0, 1} (otherwise, reversethe roles of x and y).

Let ρ′ be a profile that satisfies I(ρ′)(x, y) = I(ρ′)(z, y) = I(ρ)(x, y).Let ρ′′ be a profile that satisfies I(ρ′′)(x, y) = I(ρ)(x, y), I(ρ′′)(z, y) =

I(ρ′)(z, y), and I(ρ′′)(z, x) ∈ {1}N .By unanimity, I[f(ρ′′)](z, x) = 1. By (PI), I[f(ρ′′)](x, y) ∈ {0, 1}, so by

transitivity of f(ρ′′), we must have I[f(ρ′′)](z, y) = 1. Apply again (PI) to getI[f(ρ′)](z, y) = 1.

Let now ρ′′′ be a profile that satisfies I(ρ′′′)(x, y) = I(ρ)(x, y), I(ρ′′′)(z, y) =I(ρ′)(z, y), and I(ρ′′′)(z, x) ∈ {−1}N .

By unanimity, I[f(ρ′′′)](z, x) = −1. By (PI), I[f(ρ′′′)](z, y) = 1, so bytransitivity of f(ρ′′′), we must have I[f(ρ′′′)](x, y) = 1.

(PI) then implies that I[f(ρ)](x, y) = 1. This concludes the proof of the firststatement.

Let us now consider Strict Neutrality. Assume

I(ρ)(x, y) = I(ρ′)(x′, y′) ∈ {−1, 1}N .

We have just shown that I[f(ρ)](x, y) ∈ {−1, 1}. Assume, without loss ofgenerality, that I[f(ρ)](x, y) = 1 (otherwise, reverse the roles of x and y, and x′

and y′).If (x, y) = (x′, y′), apply (PI) to obtain I[f(ρ′)](x′, y′) = I[f(ρ)](x, y).

Therefore, suppose in what follows that (x, y) 6= (x′, y′).Let ρ′′ be a profile that satisfies I(ρ′′)(x, y) = I(ρ)(x, y), I(ρ′′)(x′, y′) =

I(ρ′)(x′, y′), and, additionally,{x 6= x′ ⇒ I(ρ′′)(x, x′) ∈ {−1}N ⇒ I[f(ρ′′)](x, x′) = −1;y 6= y′ ⇒ I(ρ′′)(y, y′) ∈ {1}N ⇒ I[f(ρ′′)](y, y′) = 1.

Where the second implications follow because of unanimity. Also, (PI) impliesI[f(ρ′′)](x, y) = I[f(ρ)](x, y) = 1.

Now (x, y) 6= (x′, y′) means that either x 6= x′ or y 6= y′, and in either casetransitivity of f(ρ′′) implies I[f(ρ′′)](x′, y′) = 1. A final application of (PI)results in I[f(ρ′)](x′, y′) = 1. �

Definition 4 (Equivalence of rules) We say that two preference aggregationrules are d-equivalent if they share the same decisive sets. N

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Theorem 2 (Arrovian Rules and Ultrafilters) Let (UD) hold. Then anyArrovian preference aggregation rule satisfies the Ultrafilter Property. Con-versely, if the preference aggregation rule f satisfies the Ultrafilter Property,there is an Arrovian rule g which is d-equivalent to f .

Proof: If f satisfies Unanimity and Pairwise Independence, lemma 5 im-plies that f satisfies the assumptions of theorem 1, so it satisfies the UltrafilterProperty.

Assume now that f satisfies the Ultrafilter Property. In the appendix (sec-tion 8.3) we show how to define an Arrovian preference aggregation rule g thathas exactly the same decisive sets f has: the simple rule generated by thosedecisive sets. �

Theorem 2 and Proposition 1 imply Arrow’s classical result for the finitecase.

Corollary 1 (Arrow’s Theorem) Let (UD) hold, and assume N is finite andall coalitions are admissible (L = 2N ). Then any Arrovian rule is Dictatorial.

4 Limiting Dictatorial Rules

We now introduce our main concept: Limiting Dictatorial rules. Intuitively,these are rules that have associated arbitrarily small decisive coalitions. Wewill see later that this concept is intimately connected with the existence offinite approximations to the agent set.

To define the concept of Limiting Dictatorial rules, we will need to be ableto refer to smallest sets of agents. When all coalitions are admissible, a smallestset of agents is just a single agent (or the empty set). When not all coalitionsare admissible, it may happen that two given agents are “bunched,” in the sensethat they appear together in all admissible coalitions to which they belong; thatis, those agents are never separated by the collection L of admissible coalitions,so they form part of a single, but still smallest, unit from the viewpoint of ad-missibility. This intuition is formalized by the equivalence relation ∼L, definedin subsection 8.2 in the appendix.

We first show that, when f satisfies the Ultrafilter Property, the intersectionof all decisive sets is always smallest.

Proposition 2 Assume the preference aggregation rule f satisfies the UltrafilterProperty, and let D be the collection of all f–decisive sets. Let M = ∩{D : D ∈D}. Then M is a smallest set with respect to L, i.e. either M = ∅ or M is anequivalence class of ∼L.

Proof: When the society is infinite, it is easy to construct examples (see, e.g.,example 2 below) in which M = ∅. Assume therefore that M 6= ∅.

Let i ∈ M . This implies that D ⊂ Li, the ultrafilter of all sets in L thatcontain i (see subsection 8.1 in the appendix). Since D is also an ultrafilter by

11

hypothesis, we must have D = Li. Now j �L i implies that there is A ∈ L suchthat i ∈ A and j /∈ A, so j /∈ M . On the other hand, j ∼L i implies that j ∈ Afor each A ∈ Li, so j ∈ M . �

We are now ready to state our generalization of dictatorial rules. We recallthat a chain in a partially ordered set is a totally ordered subset, that is, asubset where all elements are comparable. In the definition that follows, weneed to impose condition (iii) to avoid artificially assuming power where thereis no such power; we explain later this necessity with the aid of the examplesthat follow the definition.

Definition 5 (Limiting Dictatorial Rules) We say that a preference aggre-gation rule f is Limiting Dictatorial if there exists a collection C of f–decisivecoalitions satisfying:

(i) C is a chain with respect to the partial order defined by set inclusion, thatis, for any C,C ′ ∈ C, C ∩ C ′ ∈ {C,C ′}.

(ii) The intersection K.=

⋂{C : C ∈ C} is smallest with respect to L, in the

sense that either K = ∅, or K is an equivalence class of ∼L.

(iii) The intersection K of all sets in C coincides with the intersection of allf–decisive sets.

When f is Limiting Dictatorial, we also say that it has a Limiting Dictator(even though when K = ∅ there is no agent in N which can be identified withsuch a dictator). N

Of course, any Dictatorial rule is Limiting Dictatorial (if i is the Dictator,let C contain only the coalition {i}). But, when the society is infinite, there mayexist Limiting Dictatorial rules that are not Dictatorial, as was first shown byFishburn [12]. We next display examples showing that there are three differentkinds of Limiting Dictatorial rules; the distinction depends on what kind of setis the intersection of all decisive sets. In the first case, the intersection of alldecisive sets is still a decisive set, although it is not necessarily a single–agentcoalition (this situation may only arise when there are admissibility constraintson coalitions). In the second case, the intersection of all decisive sets is empty(this situation may only arise in infinite societies). In the third case, the inter-section of all decisive sets is not empty, but it is not an admissible coalition,so it cannot be a decisive set (this situation may only arise when the societyis infinite and also there are admissibility constraints on coalitions). Adaptingthe terminology of Kirman and Sondermann [17], we can say that in the lasttwo cases the rules are Invisibly Dictatorial: there is no Dictator that can bedistinguished by the society, in the sense that it is an admissible coalition, butnevertheless there are arbitrarily small decisive coalitions, so the preferences ofthe larger part of the society are ignored (except for those cases in which theindividuals in a decisive coalition are indifferent between alternatives).4

4I am grateful to an anonymous referee for suggesting this explanation.

12

Example 1. In this example we show that, when admissibility constraints arepresent, a Dictator is not necessarily a single individual. Let N = {1, 2, 3},and let L = {∅, {1, 2}, {3}, {1, 2, 3}}. Let D = {{1, 2}, {1, 2, 3}}, and let f bethe simple rule generated by D (see subsection 8.3 in the appendix). Then thisfinite society has a Limiting Dictatorial rule, in which the coalition {1, 2} can beused as the collection required in the definition. Note that {1, 2} is a smallestunit of L. Note also that assumption (UD) implies that the preferences of 1and 2 are identical in any admissible profile, and therefore any strict preferencebetween two alternatives by any of these individuals becomes under f a socialstrict preference. This is the way in which a Dictatorial rule is often defined:the strict preferences of an individual are adopted by the society. In this sense(but not according to the definition used in this paper), one may affirm thatany individual in {1, 2} is a Dictator, and the rule is Dictatorial. 2

Example 2. Let N = {1, 2, 3, . . .} be the set of all natural numbers. Let L =2N : all coalitions are admissible. For each k ≥ 1, let Tk = {k, k + 1, k + 2, . . .}be the set of all numbers larger than k. If k ≥ k′, then Tk ⊂ Tk′ . This impliesthat the collection (Tk)k≥1 is closed under finite intersections. By the axiom ofchoice, there is an ultrafilter D containing this collection (see, e.g., Aliprantisand Border [1], section 2.5). The simple rule f generated by D is Arrovian (seesubsection 8.3 in the appendix). Now the intersection of all sets in C

.= (Tk)k≥1

is empty, so there is a Limiting Dictator, but no Dictator.In this case, we are leaving the power to coalitions that are further and

further off. Notice, by the way, that any such coalition contains the majorityof the society, in the sense that only finitely many individuals are left off. Theproblem is that any individual is eventually left out of such coalitions, so noindividual in the society can be identified with this Limiting Dictator. Thissituation of “power escaping to infinity” may only happen, of course, when thesociety is infinite. Another way of looking at this situation is the following: calla coalition f–negligible if it is disjoint from some f–decisive coalition. Then theentire society becomes f–negligible in the limit. 2

Example 3. Let N = [0, 1), and let L be the algebra generated by the intervalsof the form [a, b), with a < b (this algebra is formed by finite unions of suchintervals). Note that: (1) the algebra separates the points of [0, 1) (see subsec-tion 8.2 in the appendix); (2) single–agent coalitions are not in L, because allelements of the algebra are sets with positive length. Given any i ∈ [0, 1), letD = Li, all the elements of L that contain i; we show in subsection 8.1 in theappendix that this collection is an ultrafilter. The simple rule generated by D

is Arrovian (see subsection 8.3 in the appendix). Since the algebra separatesthe points of [0, 1), the intersection of all sets in D is precisely {i}. Considernow the collection C of all intervals of the form [i, i + (1 − i)/n), for n ≥ 1.The sets in this collection are decisive, and their intersection is {i}, so the rulef has a Limiting Dictator, but it does not have a Dictator, since {i} is not anadmissible coalition. 2

13

Consider a case in which the rule is Limiting Dictatorial and the intersectionof all decisive coalitions is empty, like in example 2. If we did not imposecondition (iii) in the definition, then it can be proved that one can actuallyconstruct nested collections of decisive coalitions whose intersection is any givenpoint (more precisely, equivalence class) in N .

Kirman and Sondermann [17] consider an infinite set of agents N , with noadmissibility constraints on coalitions (L = 2N ). With finitely many alterna-tives, they use a topological argument (the Stone–Cech compactification) toshow that the agent set can be enlarged in such a way that the preferences ofthe original agents determine the preferences of agents in the extended space,and that, in the larger space, any Arrovian rule is determined by a dictator; ifsuch an Arrovian rule has a dictator which does not belong to the original agentset, they refer to it as an Invisible Dictator. This terminology, which has alsobeen adopted by later authors (Armstrong [2], Mihara [19]), will only be usedhere in our informal discussion. Note, however, that in general our LimitingDictators are always Invisible Dictators in the sense defined by Armstrong [2],but not all of this author’s Invisible Dictators are Limiting Dictators. The rea-son for the difference will be clear later when we define finite approximations tothe infinite society.

We present now a result which shows that, if all coalitions are admissible,any Arrovian rule has associated a Limiting Dictator, that is, arbitrarily smalldecisive coalitions.

Theorem 3 (Limiting Dictators with no Admissibility Constraints) LetN be an arbitrary agent set, and assume that all coalitions are admissible:L = 2N . Then any Arrovian rule is Limiting Dictatorial.

Proof: Let f be an Arrovian rule. By theorem 2, we know that the set D

of all f–decisive coalitions forms an ultrafilter. Let M = ∩{D : D ∈ D}. Byproposition 2, we know that M is either empty or it is an equivalence class ofL. Since L = 2N , equivalence classes are singletons. If M = {i}, then i is aDictator, and therefore a Limiting Dictator. Assume from now on that M = ∅.

If we partially order D by (reverse) inclusion there can be no maximal ele-ment, since if one existed it would coincide with the intersection.

By (the contrapositive of) Zorn’s lemma, there must exist a chain C in D

which has no upper bound. Let B = ∩{C : C ∈ C}. Note first that B /∈ D,since if B were a member of D then it would be an upper bound of the chain.Since D is an ultrafilter, we have that N \B ∈ D. Therefore, for each C ∈ C, theset C \B = C ∩ (N \B) belongs to D. But then the chain C′ = {C \B : C ∈ C}has an empty intersection, and satisfies our definition of a Limiting Dictator.�

However, when there are admissibility constraints on coalitions, some infinitesocieties may have Arrovian preference aggregation rules that are not LimitingDictatorial. This is illustrated by the following example.

14

Example 4. Let N = [0, 1), and let L be the algebra formed by the finite setsand their complements. This is the smallest algebra that contains all single–agent coalitions. Consider the collection D of all sets whose complement is finite.The fact that L contains only finite sets and their complements, implies that D

is an L–ultrafilter. Let f be the simple rule generated by D (see subsection 8.3in the appendix); then f is Arrovian and D is the collection of all f–decisive sets.Suppose f has a Limiting Dictator, and let C be a collection of nested decisivesets satisfying the requirements in its definition. Let K be the intersection ofthis collection. Consider the collection formed by all complements of sets inC: this is an increasing collection of finite sets whose union is N . If such acollection existed, we could define a mapping from it to the natural numbersassociating with each set its cardinality. The mapping must be injective, becausethe collection is increasing (two different sets have different cardinality), andthis then implies that the collection can be at most countable, and hence itsunion must be at most a countable set, and therefore cannot coincide with theuncountable set N = [0, 1). In other words, there does not exist any LimitingDictator.

Note that, if we allowed L to be a σ–algebra instead (an algebra closed undercountable unions), then the argument we have made breaks, and there actuallyis a Limiting Dictator, as corollary 2 below shows. 2

In this example, a Limiting Dictator fails to exist because the collection ofadmissible coalitions is too poor with respect to the cardinality of the agent set.We will next see that, if we consider finite approximations of this society, thefact that there are too few admissible coalitions will prevent this infinite societyfrom being well approximated by finite ones. Therefore, the failure of a LimitingDictator to exist can be interpreted in the sense that we have an infinite societywhich is not the limit of finite ones.

5 Finite approximations of Infinite Societies

Our objective here is to characterize those infinite societies that are a goodapproximation of large, but finite, societies. In order to do this in a consis-tent way, we require that the admissibility constraints imposed on coalitions inthe limiting infinite society are respected by the approximating finite societies.Armstrong [2] followed a topological approach to tackle this problem. Our ap-proach here is different; it is weaker in the sense that we do away with topologicalconsiderations, but it is stronger in the sense that we are more demanding indefining what a good approximation is.

Our approach can be considered a generalization of the discussion in Mi-hara [20], section 4.15. There Mihara considers finite approximations like theones we use here, in the particular case in which the infinite society consistsof finitely many agents in an uncertain environment characterized by infinitelymany states of nature, so that the “individuals” that form N are agent–state

5I thank an anonymous referee for pointing out this fact to me.

15

pairs. Finite approximations arise when each agent has a finite partition ofthe states in such a way that her preferences are the same in each element ofthe partition. In each finite approximation, strategy–proofness takes the formof dictatorship, but in the infinite society there exists a coalitionally strategy–proof social choice funcion which is not dictatorial. This is closely related, in asocial choice function context, to our concept of limiting dictatorial rules. Withan intuition similar to ours, Mihara calls that social choice function oligarchic.

Our point of departure is an infinite society N , with an algebra L of admis-sible coalitions. The finite approximations we are going to consider are definedvia finite partitions of the agent set N . The intuitive idea is that we identify acertain coalition of agents in the infinite society with a single agent in a finiteapproximation; and this process will be continued, by providing increasinglyfiner approximations.

To be able to state our concept of approximation, we need to define previ-ously a mathematical concept: that of a finite partition compatible with L. Afinite L–partition of N is a finite collection π of coalitions in L satisfying: (i)∅ /∈ π; (ii) A,B ∈ π and A 6= B implies A∩B = ∅; and (iii) ∪{A : A ∈ π} = N .We denote by ΠL(N) the set of all finite L–partitions of N . If A ∈ L, we denoteby ΠL(A) the set of all finite partitions of A, when subcoalitions of A are con-strained to belong to L. If π ∈ ΠL(N) and i ∈ N , we denote by π(i) the elementof π that contains i. Given a partition π ∈ ΠL(N), we designate by Nπ thefinite agent set that identifies all agents that belong to the same element of thepartition; in this case, all coalitions of agents of Nπ are considered admissible.We will approximate (N,L) by finite agent sets of the form (Nπ, 2Nπ ), for finitepartitions π ∈ ΠL(N).

Let π, π′ ∈ ΠL(N); we say that π is a refinement of π′ if, for each A ∈ π′,there is β ∈ ΠL(A) such that β ⊂ π; we also say in this case that π′ is acoarsening of π. Given π, π′ ∈ ΠL(N), there always exists a coarsest π′′ ∈ΠL(N) that is a refinement of both π and π′; we denote this by π′′ = π ∨ π′,and name it the join (coarsest common refinement) of π and π′.

Definition 6 (Finitely Approachable Societies) Let N be an infinite agentset, and L an algebra of admissible coalitions of agents. We say that (N,L) isfinitely approachable if there is a collection (πτ )τ∈T of finite L–partitions of Nthat satisfy:

(i) (πτ )τ∈T is a chain with respect to the partial order defined by partitionrefinement, that is, for each τ, τ ′ ∈ T , πτ is either a coarsening or arefinement of πτ ′ .

(ii) For each i, j ∈ N such that i �L j, there is τ ∈ T such that πτ (i) 6= πτ (j).

The collection of finite partitions (πτ )τ∈T is said to define a finite approximationof (N,L), by means of the finite societies (Nπτ

, 2Nπτ ) that correspond to it. N

Let (UD) hold, and let π ∈ ΠL(N) be a finite L–partition. We say thata preference profile ρ : N → R is compatible with π if it is measurable with

16

respect to π, that is, if all the members of any given element of π are assignedthe same preference by ρ. A preference aggregation rule f : RN

L → R induces apreference aggregation rule fπ over the agent set Nπ, defined over all profiles ρthat are compatible with π. Conversely, any preference aggregation rule definedover (Nπ, 2Nπ ) is induced by some rule f : RN

L → R.We are now ready to state our main result.

Theorem 4 (Limiting Dictators and Finite Approximations) Let (UD)hold. Let N be an infinite agent set, and L an algebra of admissible coalitions. If(N,L) is finitely approachable, then any Arrovian rule is Limiting Dictatorial.

Proof: Let f be an Arrovian preference aggregation rule, and let (πτ )τ∈T

define a finite approximation of (N,L). For each τ ∈ T , let fτ denote theinduced preference aggregation rule over the finite agent set (Nπτ , 2Nπτ ). It isimmediate to verify that fτ is also Arrovian. In theorem 2 we have shown thatthe collection D of all f–decisive coalitions in L forms an ultrafilter. Therefore,given a finite L–partition πτ , there is precisely one element of the partitionwhich is a decisive set. This element will coincide a fortiori with the dictator offτ : call Cτ this dictator in the finite society (Nπτ , 2Nπτ ). Let τ, τ ′ ∈ T be suchthat πτ is a refinement of πτ ′ ; then Cτ ⊂ Cτ ′ . Hence, C = (Cτ )τ∈T is a nestedcollection of f–decisive sets.

Let K =⋂{Cτ : τ ∈ T}, and let M = ∩{D : D ∈ D}. Now C ⊂ D implies

that M ⊂ K. So, if K = ∅, then M = ∅, and C is Limiting Dictatorial.If M 6= ∅, then K 6= ∅, so let i ∈ K. If j �L i, then there is τ such that

πτ (j) 6= πτ (i); therefore j /∈ Cτ , and so j /∈ K. If j ∼L i, then j ∈ Cτ forall τ , so j ∈ K. Concluding, K equals the equivalence class of i. Since M isan equivalence class (proposition 2), this implies that K = M . That is, f isLimiting Dictatorial.

Suppose now that M = ∅ but K 6= ∅ (we show in example 5 below that thispossibility cannot be excluded). Let i ∈ K; then the same procedure we followabove shows that K coincides with the equivalence class of i. Since M = ∅,there is D ∈ D such that i /∈ D. For each τ ∈ T , let π′τ = πτ ∨ {D,N \ D}.Starting from the corresponding new finite approximation, proceed as above toconstruct a nested collection of f–decisive sets which, by construction, has anempty intersection. This shows that in this case also f is Limiting Dictatorial.�

Discussion: We mention in the introduction the intuitive concept of “continu-ity at infinity,” in the sense that the properties of an infinite society shouldcorrespond to the properties of large finite societies that approximate it. Inthe case of Arrovian rules and their relationship with dictatorship, the proof oftheorem 4 helps in making this intuition a bit more precise (albeit not formal).Fix an Arrovian rule in the infinite society, which we know induces an Arrovianrule in each society belonging to a finite approximation. Fix now one such finiteapproximation to the infinite society. The Arrovian rule associates with each

17

finite society its dictator, i.e. the intersection of all decisive coalitions (note thateach of those dictators corresponds to a decisive coalition in the infinite society).With the infinite society it associates the intersection of all decisive sets. Theintuitive idea of continuity means that the dictators in the finite societies shouldin some sense approximate the intersection of all decisive sets in the infinite so-ciety. But this precisely is the content of the proof of the theorem (with theproviso implied by the last paragraph in it, and illustrated in example 5).

A consequence of theorem 4 is that, if some infinite society has an Arrovianrule for which there is no Limiting Dictator, like in example 4, then this societycannot be finitely approximated. That is, it should be considered a suspiciousmodeling device to study the properties of large finite societies.

In the proof of the previous theorem, we have mentioned the fact that notevery finite approximation will provide a nested collection of decisive coalitionsable to fulfill the requirements in the definition of a Limiting Dictator. The nextexample illustrates why this may happen.

Example 5. Let N = [0, 1), and let L be the σ–algebra of Borel sets (the onegenerated by the intervals of the form [a, b), with a < b). Consider the sequenceof open intervals {(

12,12

+12n

): n ∈ N

}This collection of nonempty sets is closed under finite intersections, and thereforethe axiom of choice implies that there is an L–ultrafilter D containing it. Let fbe the simple rule generated by D.

Consider now the finite approximations to (N,L) defined from the dyadicintervals: for each n ∈ N, πn consists of all semiclosed intervals of the form[

k − 12n

,k

2n

), k = 1, 2, . . . , 2n

Note, in particular, that πn contains the semiclosed interval

Cn =[12,12

+12n

),

which is precisely the f–decisive set in πn.Let M = ∩{D : D ∈ D}, and K = ∩{Cn : n ∈ N}. Since the intersection of

all open intervals we used to define D is empty, we have M = ∅. On the otherhand, we have K = {1/2}, so the collection (Cn)n∈N does not define a LimitingDictator. But of course we can define a suitable collection by modifying thepartitions πn as we do in the proof of theorem 4. 2

18

6 Approachable Infinite Agent Sets

In the introduction, we have mentioned how the interest in studying models ofsocial choice with infinitely many individuals arose because of the use of modelswith infinite societies both in Economics and in Political Science. The resortto an infinite society is supposed to be a modeling device intended to capturethe properties of large, but finite, societies. In this section, we are going todisplay examples of infinite societies that are, at least from the viewpoint of ourdefinition, approachable by finite ones, and therefore are compatible with theprevious intuition.

We have seen above (theorem 4) that, in an infinite society (N,L) that canbe finitely approximated, any Arrovian rule is Limiting Dictatorial. We presentnow a corollary of this result which covers the most widely used examples ofinfinite sets of agents: countable agent sets, and (societies indexed by subsetsof) the real numbers.

We say that a σ–algebra L is countably generated if there is a countablecollection S of subsets of N , such that L is the smallest σ–algebra containingS. For instance, the Borel subsets of the real line form a countably generatedσ–algebra, since it is the smallest σ–algebra that contains all open intervals withrational endpoints. In general, the Borel subsets of any separable metric spaceform a countably generated σ–algebra.

Corollary 2 (Approachable Infinite Agent Sets) Let (UD) hold. Thenany Arrovian rule is Limiting Dictatorial in the following cases:

(i) N is a countable set, and L is any algebra of coalitions.

(ii) N is any infinite set, and L is a countably generated σ–algebra.

This corollary is a consequence of theorem 4, and of the following lemmas.

Lemma 6 (Countable Agent Sets) Given an infinite agent set N , assumeL is any algebra of subsets of N , and N/∼L

is countable. Then (N,L) is finitelyapproachable.

Proof: Enumerate the elements of N/∼L:

N/∼L= {a1, a2, . . . , an, . . .} .

Given n, let An ∈ L be such that an ⊂ An, and an+1 ⊂ N \ An. Let π′n ={An, N \An}. Define

πn = π′1 ∨ π′2 ∨ · · · ∨ π′n.

Then from (πn)n∈N we can generate a finite approximation to (N,L). �

Lemma 7 (Countably generated σ–algebras) Given an infinite set N , as-sume L is a countably generated σ–algebra of subsets of N . Then (N,L) isfinitely approachable.

19

Proof: Let S = (Sn)n∈N be a countable collection such that L = σ(S). Foreach n ∈ N, let π′n = {Sn, N \ Sn}. Define

πn = π′1 ∨ π′2 ∨ · · · ∨ π′n.

Suppose i �L j. Since L = σ(S), there is n such that i ∈ Sn and j ∈ N \ Sn,and hence πn(i) 6= πn(j). Therefore, from (πn)n∈N we can generate a finiteapproximation to (N,L). �

7 Conclusions

Since Fishburn [12] first showed the possibility of the existence of non–dictatorialArrovian preference aggregation rules in societies with infinitely many agents,there has been an interest in precisely characterizing those rules and knowingin what sense the passage to infinity allows the transition from dictatorial tonon–dictatorial rules.

In this paper, we contribute to this line of work by introducing the concept ofLimiting Dictatorial rules, which are characterized by arbitrarily small decisivecoalitions. We show that in infinite societies which can be well approximated bylarge finite societies, all Arrovian rules are Limiting Dictatorial, and thereforethere is no discontinuity with respect to preference aggregation in the passageto infinity.

Our central idea in the paper is the finite approximation of infinite societies.We have defined a natural way to approximate infinite societies in which therepossibly are admissibility constraints on coalitions. We have only consideredwhether there is continuity with respect to the property of preference aggrega-tion. It would also be interesting to consider whether or not there is continuitywith respect to other properties that have been studied in the literature.

8 Appendix

8.1 Ultrafilters

A (non-empty) collection C of non-empty subsets of N is (downward) filtering if foreach A, B ∈ C there exists C ∈ C such that A ⊃ C and B ⊃ C, that is, A ∩ B ⊃ C.A filter is the collection of all supersets of a filtering collection. In other words, acollection D of subsets of N is a filter (Bourbaki [9], I.6.1) if: (i) ∅ /∈ D; (ii) A, B ∈ D

implies A∩B ∈ D; and (iii) A ∈ D and A ⊂ B implies B ∈ D. A filter is an ultrafilterif there is no other filter that strictly contains it. Using this maximality property, itcan be shown (Bourbaki [9], I.6.4), that a filter D is an ultrafilter iff ∀A, precisely oneof A and N \A is in D. An ultrafilter D is called free if ∩{D : D ∈ D} = ∅.

Given an algebra L, a collection D of sets of L is an L–filter if: (i) ∅ /∈ D; (ii)A, B ∈ D implies A ∩B ∈ D; and (iii) A ∈ D, B ∈ L, and A ⊂ B implies B ∈ D. AnL–filter is an L–ultrafilter if there is no other L–filter that strictly contains it. The(ultra)filters not restricted to lie in an algebra can be identified with 2N–(ultra)filters.

20

Given any 2N–(ultra)filter F, the intersection F∩L is an L–(ultra)filter. Any collectionof nonempty sets closed under finite intersections is always contained in a 2N–filter,and the axiom of choice implies also that there is a 2N–ultrafilter containing it. Inparticular, given an L–filter D there is a 2N–filter F that contains it, and we have thatF ∩L ⊃ D. In particular, if D is an L–ultrafilter and F is any 2N–filter that containsit, we must have F ∩L = D. Concluding, D is an L–ultrafilter if, and only if, there isa 2N–ultrafilter U such that D = U ∩ L. That is, an L–filter D is an L–ultrafilter iff∀A ∈ L, precisely one of A and N \ A is in D. The theory of measurable filters andultrafilters has been developed in the context of the study of Boolean Algebras (seeSikorski [24]).

Given any i ∈ N , the collection Ui.= {A ⊂ N : A 3 i} is a 2N–ultrafilter, as it can

be readily verified by checking that it satisfies the four properties that characterizeultrafilters. Hence, given i ∈ N and an algebra L, the collection Li

.= Ui ∩ L = {A ∈

L : A 3 i} is an L–ultrafilter.

8.2 Separation of points

Let i, j ∈ N , i 6= j; we say that the algebra L separates i and j if there exists A ∈ L

such that i ∈ A and j ∈ N \A. We say that L separates the points of N if it separatesany two distinct points. We say that i is identified with j by L, and denote it i ∼L j,if L does not separate i and j, i.e. if [∀A ∈ L, i ∈ A ⇔ j ∈ A ]. The relation ∼L

thus defined is an equivalence relation (it is reflexive, symmetric and transitive), so itpartitions the set of agents into equivalence classes; two agents in a given equivalenceclass are not distinguishable from the admissibility viewpoint. We denote by N/∼L

the quotient space, the set of all equivalence classes. Note that equivalence classes arethe smallest sets defined by L, because no other set which L can distinguish is strictlysmaller.

8.3 Simple rules

Given a preference profile ρ and a nonempty collection D ⊂ L, define a binary relationP on X by xPy ⇔ ∃D ∈ D such that ∀i ∈ D, I[ρ(i)](x, y) = 1. Define also a binaryrelation R on X by xRy ⇔ not yPx. The next lemma is basically the proof of part iiof theorem 1 in Kirman and Sondermann [17], and proposition 3.1 in Armstrong [2].

Lemma 8 Let assumption (UD) hold. If D is an ultrafilter, then R is reflexive,transitive and complete, and P is asymmetric and negatively transitive. That is, R isa weak preference relation and P the corresponding strict preference. The preferenceaggregation rule thus defined satisfies unanimity and (PI), and D coincides with thecollection of its decisive sets.

Proof: We claim that xRy if, and only if, Rxy.=

˘i ∈ N : I[ρ(i)](x, y) ∈ {0, 1}

¯∈ D.

The proof is immediate from the definitions and the fact that D is an ultrafilter.Now the fact that R is reflexive follows from Rxx = N ∈ D. Transitivity follows

because the fact that Rxy and Ryz are in D implies that so is Rxz ⊃ Rxy ∩ Ryz ∈D. Finally, completeness follows because Rxy /∈ D implies N \ Rxy =

˘i ∈ N :

I[ρ(i)](x, y) = −1¯∈ D.

Since N ∈ D, the preference aggregation rule satisfies Unanimity. Pairwise Inde-pendence is satisfied because the social preference between each pair of alternativesdepends only on the individual preferences between those alternatives. By construc-tion, each D ∈ D is decisive with respect to the new rule. Since the aggregation rule

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is Arrovian, its collection of decisive sets is an ultrafilter. That is, the decisive setsform an ultrafilter that contains the ultrafilter D: by maximality of ultrafilters, bothmust coincide. �

If D is an ultrafilter, the (Arrovian) preference aggregation rule defined above iscalled the simple rule defined by D. Note that, among all rules that have D as theirdecisive sets, the simple rule is the one that gives rise to the smallest (as a subset ofX ×X) strict social preference.

8.4 Identifying Limiting Dictators

We have seen in section 4 that there are three different kinds of Limiting Dictators,depending on whether the intersection of all decisive sets is empty, a decisive coalition,or a non–admissible coalition. The proof of proposition 2 shows that, in the two lattercases, the collection of decisive sets is of the form Li, for some i ∈ N (see subsection 8.1in this appendix). These facts, together with the tools we have developed in this paper,allow us to tackle the following question: Given a society (N, L), identify all possibleLimiting Dictators it may have.

The strategy consists of the following three steps: first, given the characteristicsof the agent set and the algebra of coalitions, we proceed to the identification of thedifferent types of Limiting Dictators; second, once a candidate to Limiting Dictatoris identified, we characterize its decisive sets; third, we characterize the correspondingpreference aggregation rule as the simple rule associated with those decisive sets (seesubsection 8.3 in this appendix).

The identification in the case in which the intersection of all decisive sets is empty isillustrated by examples 2 and 5 above. Start from a given nested collection of nonemptyinfinite sets that is closed under finite intersections, but whose overall intersection isempty. Next appeal to the axiom of choice to justify the existence of an ultrafilter thatcontains this collection. So we know the simple rule corresponding to this ultrafilterof decisive sets will have among its decisive sets the collection we started from. Fromdifferent starting collections we can obtain different Limiting Dictators. For example,in [0, 1) with its Borel subsets, the Limiting Dictator defined from {(0, 1/n) : n ∈ N}is different from the one defined from {(1− 1/n, 1) : n ∈ N}.

The case in which the intersection of all decisive sets is itself a decisive set issimpler. We know that this decisive set must be an equivalence class, so it must bewhat is called an atom of L: a set A ∈ L such that, for any B ∈ L, B ⊂ A impliesthat B ∈ {A, ∅}. Given any atom of L, we can construct an Arrovian rule for whichany i ∈ A is the dictator by starting from Li and proceeding as outlined above. Andthis is the only possibility for this kind of Limiting Dictatorial rule.

Finally, if we want a Limiting Dictatorial rule in which the intersection of alldecisive sets is a non–admissible coalition, then we must start from an equivalenceclass of N/∼L which is not an admissible coalition, take any i in it, and take Li as itsdecisive sets.

Let us study a concrete example.

Example 6. In the space N of the natural numbers, consider the algebra L generatedby the collection of sets Ek = {2 k n : n ∈ N}, for a given k ∈ N. Note first ofall that this algebra contains no finite sets. The odd numbers belong to the sameequivalence class, and they are also admissible because they are the complement of

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the even numbers; therefore, they are an atom of the algebra. One can check thatthere is no other atom.

Since the odd numbers form the only atom, they are indistinguishable from theviewpoint of admissibility, and they are the only candidate for a Dictator (note thatall odd numbers have the same preferences under any admissible profile). So if westart from L1, all the sets of L that contain 1, we construct a Limiting Dictatorial rulefor which the intersection of all decisive sets is a decisive set, the odd numbers. Andthis is the only rule with such property. (Note also that L1 = Ln, for any n odd.)

Now the algebra separates any two even numbers, so that the equivalence classof any even number is a singleton. Since the algebra does not contain any finite set,the equivalence class of any even number is not in L. For any k ∈ N, the simple rulethat corresponds to L2k has the property that the intersection of its decisive sets isprecisely {2 k}. So any even number can give rise to a different Limiting Dictator.

Let now C2 = {Ek : k = 2n, n = 0, 1, 2, . . .}. This collection is nested, so it is closedunder finite intersections, and the axiom of choice implies that there is an ultrafilter D

containing it, from which we can define the corresponding Arrovian simple rule. Theintersection of all decisive sets is empty. This defines a Limiting Dictator which shouldbe expected to be different from the Limiting Dictator one obtains starting from thecollection C3 = {Ek : k = 3n, n = 0, 1, 2, . . .}, and, in general, from the correspondingcollection Cp, defined for any prime number p. 2

References

[1] Aliprantis, C. and Border, K., Infinite Dimensional Analysis: A Hitch-hiker’s Guide (2nd edition), Springer-Verlag, Berlin, 1999.

[2] Armstrong, T.E., Arrow’s Theorem with Restricted Coalition Algebras,Journal of Mathematical Economics (1980), 7: 55-75.

[3] Armstrong, T.E., Precisely Dictatorial Social Welfare Functions: Erratumand Addendum to ‘Arrow’s Theorem with restricted coalition algebras’,Journal of Mathematical Economics (1985), 14: 57 - 59.

[4] Arrow, K.J., Social Choice and Individual Values (2nd edition), Wiley, NewYork, 1963.

[5] Aumann, R., Markets with a Continuum of Traders, Econometrica (1964),32: 39–50.

[6] Austen-Smith, D. and Banks, J.S., Positive Political Theory I, Universityof Michigan Press, Ann Arbor, 1999.

[7] Banks, J.S., Duggan, J., and Le Breton, M., Social Choice and ElectoralCompetition in the General Spatial Model, mimeo, 2003.

[8] Barbera, S., Pivotal Voters. A New Proof of Arrow’s Theorem, EconomicsLetters (1980), 6: 13–16.

23

[9] Bourbaki, N., Elements de Mathematique. Topologie generale. Chapitres 1a 4, Diffusion C.C.L.S., Paris, 1971.

[10] Caplin, A., and Nalebuff, B., On 64% Majority Rule, Econometrica (1988),56: 787–814.

[11] Caplin, A., and Nalebuff, B., Competition among Institutions, Journal ofEconomic Theory (1997), 72: 306–342.

[12] Fishburn, P.C., Arrow’s Impossibility Theorem: Concise Proof and InfiniteVoters, Journal of Economic Theory (1970), 2: 103-106.

[13] Geanakoplos, J., Three Brief Proofs of Arrow’s Impossibility Theorem, Eco-nomic Theory, forthcoming.

[14] Gomberg, A., C. Martinelli and R. Torres, Anonymity in Large Societies,Social Choice and Welfare, forthcoming.

[15] Guilbaud, Georges, Les theories de l’interet general et le probleme logiquede l’aggregation, Economie Appliquee (1952), 5: 501-584.

[16] Hansson, B., The Existence of Group Preference Functions, Public Choice(1976), 28: 89-98.

[17] Kirman, A.P. and Sondermann, D., Arrow’s Theorem, Many Agents, andInvisible Dictators, Journal of Economic Theory (1972), 5: 267-277.

[18] Mihara, H.R., Anonymity and Neutrality in Arrow’s Theorem with Re-stricted Coalition Algebras, Social Choice and Welfare (1997), 14: 503-512.

[19] Mihara, H.R., Arrow’s Theorem, Countably Many Agents and More VisibleInvisible Dictators, Journal of Mathematical Economics (1999), 32: 267-287.

[20] Mihara, H.R., Existence of a coalitionally strategyproof social choice func-tion: A constructive proof, Social Choice and Welfare (2001), 18: 543-553.

[21] Saari, D., Informational geometry of social choice, Social Choice and Wel-fare (1997), 14: 211-232.

[22] Schmitz, N., A Further Note on Arrow’s Impossibility Theorem, Journalof Mathematical Economics (1977): 189-196.

[23] Sen, A., Information and invariance in normative choice, in Heller et al.,eds., Social Choice and Public Decision Making, Cambridge U.P., NewYork, 1986.

[24] Sikorski, R., Boolean Algebras (3rd edition), Springer-Verlag, Berlin, 1969.

[25] Ubeda, L., Neutrality in Arrow and Other Impossibility Theorems, Eco-nomic Theory (2003), 23: 195-204.

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