simple collective identity functions

DOI 10.1007/s11238-008-9098-yTheory and Decision (2010) 68:417–443 © Springer 2008

MURAT ALI CENGELCI and M. REMZI SANVER

SIMPLE COLLECTIVE IDENTITY FUNCTIONS

ABSTRACT. A Collective Identity Function (CIF) is a rule whichaggregates personal opinions on whether an individual belongs to a cer-tain identity into a social decision. A simple CIF is one which can beexpressed in terms of winning coalitions. We characterize simple CIFsand explore various CIFs of the literature by exploiting their ability ofbeing expressed in terms of winning coalitions. We also use our settingto introduce conditions that ensure the equal treatment of individuals asvoters or as outcomes.

KEY WORDS: collective identity functions, aggregation

JEL CLASSIFICATION: D71

1. INTRODUCTION

A first attempt to analyze the collective identity determina-tion problem through concepts of social choice theory is madeby Kasher and Rubinstein (1997) who, based on an explora-tion of Kasher (1993) about the Jewish identity, propose totreat the question as an aggregation problem. They considera society and some abstract concept of identity (referred toas “being a J”) to which every member of the society may ormay not belong. The individual opinions about who does andwho does not belong to this identity are aggregated into a col-lective opinion – hence the introduction of a collective iden-tity function (CIF). The model, while mathematically simple,incorporates a plethora of concepts related to collective iden-tity determination. Hence, it paved the way to a growing lit-erature, the pivots of which will be mentioned as a start forour analysis.

Liberalism appears as a central concept. Under the liberalCIF, an individual is socially conceived as belonging to some

418 MURAT ALI CENGELCI and M. REMZI SANVER

identity J if and only if he/she believes “to carry identity J”or “to be a J”, so to speak. A first axiomatic characteriza-tion of liberalism is given by Kasher and Rubinstein (1997).1

Recently, Samet and Schmeidler (2003) characterized a classof CIFs which they call consent rules. This class is parame-trized by the weights given to individuals in determining theirown identity. It contains liberalism at one extreme and major-itarianism2 at the other.

Another strand of the literature views a CIF as a recur-sive procedure. For example, the procedural CIF of Kasher(1993) suggests to determine an initial set J (0) of individu-als who are unanimously agreed to carry identity J . All indi-viduals who are considered to be J by at least one memberof J (0) are added to J (0), hence expanding the set of J s toJ (1). The procedure continues inductively until the set of J scannot be expanded anymore. Kasher and Rubinstein (1997)mention a variant of this procedure, where the initial set J (0)

consists of individuals who consider themselves as J s. Dim-itrov et al. (2007) characterize both procedures. As we discussin Section 3, the procedural view of CIFs is almost orthogo-nal to their conception as consent rules.

We propose to approach the collective identity determina-tion problem by considering CIFs that can be expressed interms of winning coalitions: Given a society N , we say that acoalition K of individuals is winning over individual i if andonly if i is socially qualified as a J , whenever all members ofK qualify i as a J while those in N\K do not. For CIFs thatcan be expressed in terms of winning coalitions, the informa-tion about the social opinion contained in the set of winningcoalitions is the same as that in the corresponding aggrega-tion rule. This brings us a new perspective in the explorationof the collective identity determination problem, as it allowsus to examine CIFs through their winning coalitions.3

In Section 2, we start by showing that the class of CIFswhich can be expressed in terms of winning coalitions coin-cides with the set of CIFs that satisfy an independence axiom.This rather strong axiom, also used by Samet and Schmeidler(2003), ensures that the social decision about a member of the

SIMPLE COLLECTIVE IDENTITY FUNCTIONS 419

society depends only on the individual opinions about this par-ticular member. Next, we impose further structure over win-ning coalitions by requiring every supercoalition of a winningcoalition to be winning as well. This additional requirementrestricts the class of CIFs to those which satisfy independenceand a particular monotonicity condition which ensures thatadditional opinions which qualify an individual i do not endup in i’s disqualification while he/she was previously qualified.We qualify independent and monotonic CIFs as simple. If thewinning coalitions of a simple CIF are such that a coalition andits complement cannot be both winning or both non-winning,then that CIF satisfies the self-duality condition of Samet andSchmeidler (2003), which imposes a certain kind of neutralitytowards the identity to be aggregated: While aggregating indi-vidual opinions about who is a J , we simultaneously aggre-gate the individual opinions about who is not a J . Self-dualityrequires a symmetric treatment of these two aggregations. Tobe more precise, self-duality requires that in case all individu-als reverse their opinions about who is a J , the social decisionmust also be reversed.

In Section 3, we use the findings of Section 2 to furtherexplore particular CIFs of the literature. Among other things,we give two new characterizations of liberalism; we show thatthe consent rules of Samet and Schmeidler (2003) are simpleCIFs which satisfy self-duality if and only if they are symmet-ric, i.e., the weight of a given individual in qualifying or dis-qualifying him/herself as a J is the same; and we establish ageneral incompatibility between procedural and simple CIFs:When the determination of the initial set of J ’s satisfies inde-pendence, all non-trivial procedural CIFs fail independence.

Section 4 is devoted to the equal treatment of individualsas voters and as outcomes. In the usual social choice frame-work where voters and outcomes form disjoint sets, this is anaturally arising distinction: The equal treatment of voters isusually called “anonymity” while the equal treatment of out-comes is typically referred to as “neutrality”. On the otherhand, in our environment where voters and outcomes coin-cide, we need a more subtle discrimination. So we adapt the


standard anonymity and neutrality conditions of social choicetheory to our model. We define an essentially anonymousCIF as one which decides on the identity of an individuali by equally treating all opinions but possibly the opinionof i about him/herself.4 Similarly, an essentially neutral CIFdecides on the identities of any two individuals i and j bythe same rule whenever voters and outcomes are disjoint, i.e.,at profiles where a winning coalition over i contains neitheri nor j . We characterize the class of independent, mono-tonic, self-dual, essentially anonymous and essentially neutralCIFs in terms of symmetric consent rules. This result, com-bined with the Samet and Schmeidler (2003) characterizationof symmetric consent rules in terms of independence, monoto-nicity, self-duality and symmetry shows that over the domainof simple and self-dual CIFs, the essential anonymity andessential neutrality conditions we propose successfully decom-pose the symmetry condition of Samet and Schmeidler (2003)which is an incorporation of both equal treatment properties.Section 5 concludes.

2. COLLECTIVE IDENTITY FUNCTIONS AND WINNINGCOALITIONS

We consider a society N which is a finite set of individualswith #N = n ≥ 2. The society is confronted to the problemof deciding on its members who belong to some “group” or“who are a J”, to use the original terminology. We write Gi ⊆N for the set of individuals whom i ∈N perceives as a memberof the group. We refer to Gi as the opinion of i. An (opinion)profile is an n-tuple (G1, . . .,Gn)∈� where � = (2N)n is the setof all profiles. A Collective Identity Function (CIF) is a map-ping F : � → 2N that assigns a subset of individuals to eachprofile. We let F stand for the set of all CIFs.

For each i ∈ N , we define a family ω(i) ⊆ 2N of subsets ofN , to which we refer as the set of winning coalitions over i.Every collection {ω(i)}i∈N of winning coalitions induces a(unique) CIF F ∈ F in the following natural way: Given any


G ∈ � and any i ∈ N , we have i ∈ F(G) ⇐⇒ {j ∈N : i ∈Gj

} ∈ω(i). So a coalition K ∈ 2N is winning over i ∈ N iff when-ever all members of K qualify i as a J while all membersof N\K do not qualify i as a J , i is socially qualified as aJ . CIFs which are induced by some collection {ω(i)}i∈N ofwinning coalitions can be characterized by the following inde-pendence axiom:

A CIF F ∈F satisfies independence iff for all i ∈N and forall G,H ∈ � with i ∈ Gj ⇐⇒ i ∈ Hj for all j ∈ N , we have i ∈F (G) ⇐⇒ i ∈F (H).

THEOREM 2.1. A CIF F ∈ F is induced by some collection{ω(i)}i∈N of winning coalitions iff F satisfies independence.

Proof. To see “if” part, take any CIF F which satisfiesindependence. For any i ∈ N and any K ⊆ N , let �i,K = {G ∈� : i ∈ Gk for all k ∈ K and i /∈ Gk for all k ∈ N\K} be the setof profiles where i is qualified only by the members of K ⊆N . As F satisfies independence, for all G,H ∈ �i,K , we havei ∈F (G) ⇐⇒ i ∈F (H). Now take any i ∈N and define ω(i)={K ⊆N : i ∈F (G) for all G∈�i,K

}. It is straightforward to see

that ω(i) is well-defined and {ω(i)}i∈N induces F . To see the“only if” part, take any CIF F induced by some {ω(i)}i∈N ,any i ∈ N and any G,H ∈ � with i ∈ Gj ⇐⇒ i ∈ Hj for all j ∈N . If i ∈F(G) then

{j ∈N : i ∈Gj

}∈ω(i). By the choice of H ,{j ∈N : i ∈Gj

}= {j ∈N : i ∈Hj

}∈ω(i), which implies i ∈F(H).A similar argument shows i /∈F(G)�⇒ i /∈F(H), establishingthe independence of F .

We have not imposed yet any particular structure over thecollection of winning coalitions {ω(i)}i∈N . So, for example,Theorem 2.1 also covers the CIF F defined for any G∈� andany i ∈N as i ∈ F (G) ⇐⇒ #

{j ∈N : i ∈Gj

}is odd. Remark that

under F , there exists K,K ′ ⊆ N with K ⊆ K ′ such that K iswinning over i ∈N while K ′ is not. In fact, F fails the follow-ing monotonicity axiom:


A CIF F ∈ F is said to be monotonic iff given any i ∈ N ,and any two profiles G,H ∈ � such that (Hj = Gj) or (Hj =Gj ∪{i}) for all j ∈N , we have i ∈F(G) �⇒ i ∈F(H).5

To rule out the failure of monotonicity, we consider certainrestrictions over winning coalitions. A collection {ω(i)}i∈N ofwinning coalitions is called regular iff for all i ∈N and for allK,K ′ ⊆N with K ⊆K ′, K ∈ω(i) implies K ′ ∈ω(i). We qualifya CIF that is induced by some regular collection {ω(i)}i∈N ofwinning coalitions as simple.6 Simple CIFs can be character-ized by the conjunction of independence and monotonicity:

THEOREM 2.2. A CIF F ∈F is simple iff F satisfies indepen-dence and monotonicity.

Proof. To see “if” part, take any CIF F which satisfiesindependence and monotonicity. By Theorem 2.1, there is acollection {ω(i)}i∈N of winning coalitions that induces F . Wewill show that {ω(i)}i∈N is regular. Take any individual i ∈ N

and any K,K ′ ∈ 2N with K ⊆ K ′ while K ∈ ω(i). Let G ∈ �

be such that i ∈ Gk for all k ∈ K and i /∈ Gk for all k ∈N\K.As F satisfies independence, we have i ∈ F (G). Now con-sider H ∈ � with Hj = Gj for all j ∈ (N\K ′) ∪ K and Hj =Gj ∪ {i} for all j ∈ K ′ \K. By monotonicity of F , we havei ∈ F (H), implying K ′ ∈ ω(i), showing the regularity of{ω(i)}i∈N . To see “only if” part, take any simple CIF F . Let{ω(i)}i∈N be the regular family of winning coalitions thatinduces F . By Theorem 2.1, F satisfies independence. To seethat F also satisfies monotonicity, take any i ∈N and any G,H ∈� such that (Hj = Gj) or (Hj = Gj ∪ {i}) for all j ∈ N . Let i ∈F(G). Hence

{j ∈N : i ∈Gj

}∈ω(i). Moreover{j ∈N : i ∈Gj

}⊆{j ∈N : i ∈Hj

}by the choice of G and H . As {ω(i)}i∈N is reg-

ular,{j ∈N : i ∈Hj

}∈ω(i), implying i ∈F(H), which establishesthe monotonicity of F .

From now on, we only consider simple CIFs. As a matterof fact, simple CIFs allow a coalition and its complement tobe both winning. Ruling this out leads to the following prop-erness condition7: We call a collection {ω(i)}i∈N of winning


coalitions proper iff {ω(i)}i∈N is regular while for all i ∈N andfor all K ⊆N , K ∈ω(i) implies N\K /∈ω(i).8 In a similar vein,simple CIFs admit a coalition and its complement to be bothnon-winning. To rule this out, we call a collection {ω(i)}i∈N ofwinning coalitions strong iff {ω(i)}i∈N is regular while for alli ∈N and for all K ⊆N , K /∈ω(i) implies N\K ∈ω(i).9

Remark that if {ω(i)}i∈N is proper and strong, then for eachi ∈N and each K ⊆N , we have either K ∈ω(i) or N\K ∈ω(i).

The characterization of simple CIFs induced by a properand/or strong family of winning coalitions utilizes the follow-ing self-duality axioms:

For any G∈�, we define G∈� as Gi =N\Gi for all i ∈N .A CIF F ∈F is said to satisfy positive self-duality iff for anyi ∈ N and for any G ∈ � with i ∈ F (G), we have i /∈ F

(G

). A

CIF F ∈F is said to satisfy negative self-duality iff for any i ∈N and for any G∈� with i /∈F (G), we have i ∈F

(G

). A CIF

F ∈F which satisfies both positive and negative self-duality iscalled self-dual.

THEOREM 2.3. A CIF F ∈ F is induced by a proper (resp.,strong) collection {ω(i)}i∈N of winning coalitions iff F satisfiesindependence, monotonicity and positive (resp., negative) self-duality.

Proof. We start by showing the “if” part. We know byTheorem 2.2 that any F satisfying independence and mono-tonicity is induced by a regular collection {ω(i)}i∈N of winningcoalitions. First let F satisfy positive self-duality. To see that{ω(i)}i∈N is proper, take any i ∈N and any K ⊆N with K ∈ω(i).Suppose, for a contradiction, that N\K ∈ω(i) as well. For anyi ∈ N and for any L ⊆ N , let �i,L = {G ∈ � : i ∈ Gj for all j ∈ L

and i /∈Gj for all j ∈N\L}. Take some G∈�i,K . As K ∈ω(i), wehave i ∈F(G). Now take G∈� such that Gi =N\Gi for all i ∈N .So G∈�i,N\K , implying i ∈F(G) as N\K ∈ω(i), thus contradict-ing the positive self-duality of F . Now let F satisfy negativeself-duality. To see that {ω(i)}i∈N is strong, take any i ∈N andany K ⊆ N with K /∈ ω(i). Suppose, for a contradiction, thatN\K /∈ω(i) as well. Take some G∈�i,K . As K /∈ω(i), we have


i /∈F(G). Now take G∈� such that Gi =N\Gi for all i ∈N . SoG∈�i,N\K , implying i /∈F(G) as N\K /∈ω(i), thus contradictingthe negative self-duality of F .

We now show the “only if” part. Take any CIF F inducedby a regular collection of winning coalitions {ω(i)}i∈N . By The-orem 2.2, F satisfies independence and monotonicity. First let{ω(i)}i∈N be proper. Take any i ∈N and any G∈� with i ∈F (G).So

{j ∈N : i ∈Gj

}∈ω(i). Now take G∈� such that Gi =N\Gi

for all i ∈ N . As {ω(i)}i∈N is proper,{j ∈N : i ∈Gj

}/∈ ω(i),

thus i /∈F(G), showing the positive self-duality of F . Now let{ω(i)}i∈N be strong. Take any i ∈N and any G∈� with i /∈F (G).So

{j ∈N : i ∈Gj

}/∈ω(i). Now take G∈� such that Gi =N \Gi

for all i ∈N . As {ω(i)}i∈N is strong,{j ∈N : i ∈Gj

}∈ω(i), thusi /∈F(G), showing the negative self-duality of F .

Theorem 2.3 leads to the following corollary:

COROLLARY 2.1. A (simple) CIF F ∈ F is induced by aproper and strong collection {ω(i)}i∈N of winning coalitions iffF satisfies independence, monotonicity and self-duality.

We close the section by noting the logical independence ofthe axioms we use. The CIF F which is defined for each G∈� as F(G) = N \{i ∈ N : i ∈ Gi} satisfies self-duality and inde-pendence but not monotonicity. To see that self-duality andmonotonicity do not imply independence, consider the CIF F

defined as follows: At each G∈� and for all i ∈N ,

• If i ∈Gi , then i ∈F(G) ⇐⇒ # {j ∈N \ {i} : j ∈Gi}≤ n−12

• If i /∈Gi , then i /∈F(G) ⇐⇒ # {j ∈N \ {i} : j /∈Gi}≤ n−12

A CIF which satisfies independence and monotonicity butfails either positive or negative self-duality (hence self-duality)can be exemplified through the consent rules of Samet andSchmeidler (2003), which we formally define in the next sec-tion: For N = {1,2,3}, the consent rules F 2,1 and F 1,2 satisfyindependence and monotonicity while violate negative andpositive self-duality, respectively.

We now extend our analysis to certain CIFs of the literature.


3. EXAMPLES

3.1. The consensus CIF

The consensus CIF C ∈F explored by Kasher and Rubinstein(1997) is defined for each G∈� as C(G)={i ∈N : i ∈Gj for allj ∈ N}. Although C satisfies independence, monotonicity andpositive self-duality, it fails negative self-duality. Hence by The-orem 2.3, C is a (simple) CIF induced by a proper collection ofwinning coalitions {ω(i)}i∈N where ω(i)={N} for each i ∈N .

3.2. Dictatorial CIFs

A dictatorial CIF Fd ∈ F (where some d ∈ N is the dicta-tor) which is also explored by Kasher and Rubinstein (1997)is defined for each G ∈ � as Fd(G) = Gd . Fd satisfies inde-pendence, monotonicity and self-duality – hence by Corollary2.1, is a (simple) CIF induced by a proper and strong col-lection of winning coalitions {ω(i)}i∈N where ω(i) = {K ∈ 2N :d ∈K} for each i ∈N .

3.3. The majoritarian CIF

Let n be odd. The majoritarian CIF M ∈ F which can befound in Samet and Schmeidler (2003) is defined as follows:For each G∈� and each i ∈N we have i ∈M(G), if and only if#{j ∈N : i ∈Gj } > n

2 . Again, M satisfies independence, monoto-nicity and self-duality – hence by Corollary 2.1, is a (simple)CIF induced by a proper and strong collection of winningcoalitions {ω(i)}i∈N where ω(i) = {K ∈ 2N : #K > n

2 } for eachi ∈N .

3.4. The liberal CIF

The liberal CIF L ∈ F introduced by Kasher and Rubinstein(1997) is defined for each G ∈ � as L(G) = {i ∈ N : i ∈ Gi}. L

satisfies independence, monotonicity and self-duality – henceby Corollary 2.1, is a (simple) CIF induced by a proper and


strong collection of winning coalitions {ω(i)}i∈N where ω(i)={K ∈2N : i ∈K} for each i ∈N .

We now present two new characterizations of the liberalCIF by exploiting the fact that it is simple. We start by con-sidering a specific profile G0 with G0

i ={i} for all i ∈N , whereeach individual only considers him/herself as a J . As shownbelow, the behavior of simple CIFs on G0 determines the setof individuals who have the right of self-determination.

LEMMA 3.1. Take any simple CIF F ∈F with a proper familyof winning coalitions {ω(i)}i∈N . For any i ∈N , we have ω(i)={K ∈2N : i ∈K} if and only if i ∈F(G0).

Proof. Let F be a simple CIF with a proper family of win-ning coalitions {ω(i)}i∈N . The “only if” part directly followsfrom definitions. To see the “if” part, take any i ∈ N withi ∈ F(G0). Thus {i} ∈ ω(i) which, by the monotonicity of F ,implies {K ∈ 2N : i ∈ K} ⊆ ω(i). Now take any K ∈ 2N with i /∈K. Since i ∈N \K, we have N\K ∈ω(i) and by properness of{ω(i)}i∈N we have K /∈ω(i), establishing ω(i)={K ∈2N :i ∈K}.

Liberalism is the assignment to each individual the rightof self-determination. So the behavior of CIFs over G0 iscritical in characterizing liberalism. We say that a CIF F ∈F satisfies the weak equal treatment property (WETP) if andonly if F(G0) ∈ {∅, N}. WETP is satisfied positively in caseF(G0)=N . The following theorem shows that among the classof simple CIFs with proper winning coalitions, the liberal CIFis the unique one that satisfies WETP positively.

THEOREM 3.1. A simple CIF F ∈F with a proper family ofwinning coalitions {ω(i)}i∈N satisfies the weak equal treatmentproperty positively if and only if F is the liberal CIF.

Proof. The “if” part immediately follows from the definitions.To see the “only if” part, take any simple CIF F that satisfies


WETP positively. By Lemma 3.1, we have ω(i)={K ∈ 2N : i ∈K}for each i ∈N , which means that F is the liberal CIF. �

Theorem 3.1 Paves the way to another characterization ofthe liberal CIF through a liberalism axiom introduced by Ka-sher and Rubinstein (1997) who say that a CIF F ∈F satisfiesthe liberal principle if and only if for each G∈�, ∃ i ∈N withi ∈Gi �⇒F(G) �=∅ and ∃ i ∈N with i /∈Gi �⇒F(G) �=N .

THEOREM 3.2. A simple CIF F ∈F with a proper family ofwinning coalitions {ω(i)}i∈N satisfies the weak equal treatmentproperty and the liberal principle if and only if F is the liberalCIF.

Proof. The “if” part can be seen by checking that the lib-eral CIF satisfies WETP and the liberal principle. To see the“only if” part, take any simple CIF F that satisfies WETPand the liberal principle. By WETP, we have F(G0) ∈ {∅, N}while the liberal principle rules F(G0)=∅ out. So F(G0)=N ,which means that F satisfies WETP positively and, by Theo-rem 3.1, F is the liberal CIF.

3.5. Consent rules

The consent rules of Samet and Schmeidler (2003) are param-etrized by two positive integers s and t with s + t ≤ n + 2. Aconsent rule (with consent quotas s and t) is a CIF F st ∈F suchthat given any G∈� and any i ∈N

• if i ∈Gi , then i ∈F st (G)⇐⇒#{j ∈N : i ∈Gj

}≥ s• if i /∈Gi , then i /∈F st (G)⇐⇒#

{j ∈N : i /∈Gj

}≥ t .

PROPOSITION 3.1. A consent rule F st ∈F is a (simple) CIFinduced by a proper (resp., strong) family {ω(i)}i∈N of winningcoalitions if and only if s ≥ t (resp., s ≤ t).

Proof. We start by showing the “if” part. We know fromTheorem 1 of Samet and Schmeidler (2003) that every con-sent rule F st satisfies independence and monotonicity, hence,


by our Theorem 2.2, there is a regular collection {ω(i)}i∈N ofwinning coalitions that induces F st . First let s ≥ t . To see that{ω(i)}i∈N is proper, take any i ∈ N and any K ⊆ N with K ∈ω(i). By definition of F st , either (#K ≥ s and i ∈K) or (#K >

n− t and i /∈K). If (#K ≥ s and i ∈K) then #N\K ≤n− s andi /∈N\K. As s ≥ t , we have #N\K ≤n− t, implying N\K /∈ω(i).If (#K >n− t and i /∈K) then #N\K <t and i ∈N\K. As s ≥ t ,we have #N\K < s implying N\K /∈ ω(i). So, N\K /∈ ω(i) inboth cases, showing that {ω(i)}i∈N is proper. Now let s ≤ t .To see that {ω(i)}i∈N is strong, take any i ∈N and any K ⊆N

with K /∈ω(i). By definition of F st , either (#K <s and i ∈K)

or (#K ≤n− t and i /∈K). If (#K <s and i ∈K) then #N\K >

n − s and i /∈ N\K. As s ≤ t , we have #N\K > n − t , imply-ing N\K ∈ω(i). If (#K ≤n− t and i /∈K) then #N\K ≥ t andi ∈N\K. As s ≤ t , we have #N\K ≥ s implying N\K ∈ω(i). So,N\K ∈ω(i) in both cases, showing that {ω(i)}i∈N is strong.

We now show the “only if” part. Take any consent ruleF st and let {ω(i)}i∈N be the regular collection of winningcoalitions that induces F st . Suppose for a contradiction that{ω(i)}i∈N is proper while s < t . Take some i ∈ N and someK ⊆ N with #K = s while i ∈ K. By definition of F st , wehave K ∈ω(i). Note that #N\K =n− s and i /∈N\K. As s < t ,we have #N\K > n − t implying N\K ∈ ω(i), contradictingthat {ω(i)}i∈N is proper. Now suppose for a contradiction that{ω(i)}i∈N is strong while t <s. Take some i ∈N and some K ⊆N with #K = s − 1 while i ∈K. By definition of F st , we haveK /∈ ω(i). Note that #N\K = n − s + 1 and i /∈ N\K. As t < s,we have #N\K <n− t +1, implying N\K /∈ω(i), contradictingthat {ω(i)}i∈N is strong.

Taking s = t is a case of particular interest where we callF st a symmetric consent rule (with quota s) and denote it F s .Remark that for symmetric consent rules, the quota variesbetween s =1 and s = �n

2�+1 where �n/2� is the highest inte-ger less than or equal to n/2. At one extreme where s =1, F s

coincides with the liberal CIF L. At the other extreme wheres =�n

2�+1, we go to majoritarianism.10


REMARK 3.1. When n is odd while s ∈ {1, . . ., �n2�+ 1} or n is

even while s ∈{1, . . . , n2 }, the winning coalitions of the symmetric

consent rule F s are defined for every i ∈N as ω(i)={K ∈2N : i ∈K and s ≤ #K ≤ n− s} ∪ {K ∈ 2N : #K >n− s}. So ω(i) consistsof coalitions

– Whose cardinality varies from s to n− s while they contain i11

– Whose cardinality exceeds n−s (independent of whether theycontain i or not).

On the other hand, when n is even and s = n2 +1, we have

ω(i)={K ∈2N : i /∈K and #K =n/2} ∪{K ∈2N :#K > n2 } for every

i ∈N .

The set of symmetric consent rules is the intersection of theset of consent rules with the set of (simple) CIFs induced bya proper and strong family {ω(i)}i∈N of winning coalitions, aswe state and show below:

THEOREM 3.3. A consent rule F st ∈F is symmetric, i.e., s = t

if and only if F st satisfies independence, monotonicity and self-duality.

Proof. Take any consent rule F st . If s = t , then, by Proposi-tion 3.1, F st is a simple CIF induced by a proper and strongfamily {ω(i)}i∈N of winning coalitions, hence satisfying inde-pendence, monotonicity and self-duality by Corollary 2.1. IfF st satisfies independence, monotonicity and self-duality, then,by Corollary 2.1, F st is a simple CIF induced by a proper andstrong family {ω(i)}i∈N of winning coalitions, implying s = t byProposition 3.1.12

3.6. Procedural CIFs

Take any CIF F 0 ∈ F . For any G ∈ � and any non-negativeinteger k let Fk+1(G) = Fk(G) ∪ {j ∈ N : j ∈ Gi for some i ∈Fk(G)}. Let k be the smallest integer for which Fk+1(G) =Fk(G). Define the CIF �F 0 ∈ F as �F 0

(G) = Fk(G) foreach G ∈ �. We call �F 0

the procedural CIF based on F 0.The consensus-start-respecting procedure proposed by Kasher


(1993) and the liberal-start-respecting procedure mentionedby Kasher and Rubinstein (1997) are particular proceduralCIFs based on the consensus CIF C and the liberal CIF L,respectively. Both of these procedural CIFs fail independence,hence by Theorem 2.1, procedural CIFs cannot be expressedin terms of a family of winning coalitions. In fact, the incom-patibility between procedural and simple CIFs is more gen-eral, as announced by the following proposition:

PROPOSITION 3.2. Let F 0 be a CIF satisfying independenceand non-degeneracy13, and let �F 0

be the procedural CIF basedon F 0. �F 0

fails independence, unless �F 0 =F 0.

Proof. Let F 0 and �F 0 �= F 0 be as in the statement of theproposition. Note that, by the definition of F 0 and �F 0

, wehave F 0 (H) ⊆ �Fo

(H) for all H ∈ �. Moreover, as �F 0 �= F 0,F 0 (H)⊂�Fo

(H) for some H ∈�. Now take some G∈� withi /∈F 0 (G) and i ∈�F 0

(G) for some i ∈N . Now consider someH ∈� such that

(i) Gj ∩ (N\F 0 (G))=Hj∩ (N\F 0 (G)) for all j ∈N

(ii) {k ∈N : j ∈Hk} /∈ω (j) for all j ∈F 0(G).

Remark that the satisfaction of this second condition is guar-anteed by the non-degeneracy of F 0. As F 0 satisfies indepen-dence, we have k /∈ F 0 (H) for all k /∈ F 0 (G). Moreover, wehave j /∈ F 0 (H) for all j ∈ F 0 (G). Hence F 0 (H) = ∅, imply-ing �F 0

(H)=∅. So i /∈�F 0(H) while i ∈�F 0

(G). However, byconstruction of H , we have i ∈ Gj ⇐⇒ i ∈ Hj for all j ∈ N ,showing that �F 0

is not independent. �

4. EQUAL TREATMENT OF INDIVIDUALS

The literature of social choice theory contains two well-known equal treatment conditions, one for voters (usuallycalled “anonymity”) and one for outcomes (usually called“neutrality”). In a framework where voters and alternativesform mutually exclusive sets, the conceptual discriminationbetween these two conditions is straightforward. On the other


hand, the matter is more complicated to handle when votersand outcomes coincide – as is the case in our model.

We first consider the equal treatment of voters. When vot-ers and alternatives are mutually exclusive, the usual ano-nymity condition of social choice theory endows all voterswith equal decision power over the social qualification of agiven alternative. Anonymity can be adapted to our modelsas follows: While deciding whether some individual i ∈ N isa J , all individuals must have equal decision power. How-ever, given the peculiarities of the collective identity determi-nation problem, it is possible to remain within the confines ofanonymity even when individuals are allowed to have partic-ular self-determination powers. In other words, one can qual-ify a CIF as essentially anonymous if while deciding whethersome individual i ∈N is a J , all individuals, with the possibleexception of i him/herself, have equal decision power. We nowformalize essential anonymity:

Given a bijection � : N → N over N , we write, by a slightabuse of notation, �(K) = {�(j) : j ∈ K} for any non-emptyK ⊆ N . Fixing some i ∈N , we let �−i :N →N stand for somebijection with �−i(i) = i. A simple CIF F ∈ F induced by acollection {ω(i)}i∈N of winning coalitions is essentially anony-mous if and only if K ∈ω(i)⇔�−i(K)∈ω(i) for all i ∈N , forall K ∈ 2N and for any bijection �−i . So under essential ano-nymity, a coalition K which does not contain individual i iswinning over i if and only if every coalition K ′ with #K ′ =#K

and which does not contain i is winning over i as well. Sim-ilarly, a coalition K which contains i is winning over i if andonly if every coalition K ′ with #K ′ = #K and which containsi is winning over i as well.

Essentially anonymous simple CIFs induced by a properand strong family of winning coalitions can be characterizedin terms of what we call generalized symmetric consent rules.Fix some n-tuple of positive integers s = (s1, . . . , sn) such thatsi ∈{1, . . ., �n

2�+1} for each i ∈N . A generalized symmetric con-sent rule (with quota s) is a CIF F s ∈ F such that given anyG∈� and any i ∈N


if i ∈Gi , then i ∈F s(G)⇐⇒#{j ∈N : i ∈Gj

}≥ si

if i /∈Gi , then i /∈F s(G)⇐⇒#{j ∈N : i /∈Gj

}≥ si

Note that symmetric consent rules of Samet and Schmeidler(2003) are particular cases of F s where s is such that si = sj

for all i, j ∈N .

THEOREM 4.1. A CIF F ∈F satisfies independence, monoto-nicity, self-duality and essential anonymity if and only if F is ageneralized symmetric consent rule F s with si ∈ {1, . . ., �n

2� + 1}for each i ∈N .

Proof. We leave the “if” part to the reader. To show the“only if” part, take any CIF F ∈ F which satisfies inde-pendence, monotonicity, self-duality and essential anonymity.Let {ω(i)}i∈N be the (proper and strong) family of win-ning coalitions that induces F . For each i ∈ N , let si =min {#K :K ∈w (i)}. First consider the case where n is odd.As {ω(i)}i∈N is proper and strong, either K ∈ ω(i) or N\K ∈ω(i) holds for each K ∈ 2N . Thus, si ∈ {1, . . ., �n

2� + 1}. More-over, there exists K ∈ω(i) with i ∈K and #K = si . To see this,suppose the contrary. In case si ∈ {1, . . ., �n

2�}, there exists, bythe essential anonymity of F , K,K ′ ∈ ω(i) such that #K =#K ′ = si while K ∩ K ′ = ∅, which contradicts that {ω(i)}i∈N

is proper and strong. In case si = �n2� + 1, there exists, as

{ω(i)}i∈N is proper and strong, K ∈ω(i) with i /∈K and #K =�n

2�, which contradicts the choice of si = min {#K :K ∈w (i)}.Now, as there exists K ∈ ω(i) with i ∈ K and #K = si , by theessential anonymity of F , we have K ∈ ω(i) for any K ∈ 2N

with i ∈K and #K = si . Moreover, as F satisfies monotonicity,we have K ∈ ω(i) for any K ∈ 2N with i ∈ K and #K ≥ si . So,{K ∈ 2N : i ∈K and #K ≥ si}⊆ω(i). As si = min {#K :K ∈w (i)},by the definition of a winning coalition, we have {K ∈ 2N :#K >n− si}⊆ω(i). By the fact that either K ∈ω(i) or N\K ∈ω(i) holds for each K ∈ 2N , we have ω(i)={K ∈ 2N : i ∈K and#K ≥ si} ∪ {K ∈ 2N : #K >n− si}. So, we conclude, by referringto Remark 3.1, that the decision over i is taken according toa symmetric consent rule with quota si .


Now consider the case where n is even. As for each K ∈2N ,either K ∈ ω(i) or N \K ∈ ω(i) holds, we have si ∈ {1, . . . , n

2 }.In case si �= n

2 , the arguments we used for the case where n

is odd show the existence of K ∈ω(i) with i ∈K and #K = si

and the essential anonymity and monotonicity of F similarlyestablishes that the decision over i is taken according to asymmetric consent rule with quota si . In case si = n

2 , we have,by the essential anonymity of F and the definition of a win-ning coalition, two possible mutually exclusive cases:

CASE 1: K ∈ω(i) for all K ∈2N with i ∈K and #K = n2 while

K /∈ω(i) for all K ∈2N with i /∈K and #K = n2 .

CASE 2: K ∈ω(i) for all K ∈2N with i /∈K and #K = n2 while

K /∈ω(i) for all K ∈2N with i ∈K and #K = n2 .

For CASE 1, by the monotonicity of F , we have ω(i)={K ∈2N : i ∈ K and #K = si} ∪ {K ∈ 2N : #K > n − si}. So we con-clude, by referring to Remark 3.1, that the decision over i

is taken according to a symmetric consent rule with quotasi = n

2 . For CASE 2, by the monotonicity of F , we haveω(i) = {

K ∈2N : i /∈K and #K =n/2}∪ {K ∈ 2N : #K > n

2 }. Sowe conclude, by referring to Remark 3.1, that the decisionover i is taken according to a symmetric consent rule withquota si +1= n

2 +1. �

Remark that under essential anonymity, while deciding onthe identity of some i ∈N , the opinion of i about him/herselfmay be favored, disfavored or equally treated compared to theother individuals’ opinions over i. The following three condi-tions classify simple and essentially anonymous CIFs accord-ing to this notion:

A simple and essentially anonymous CIF F ∈F with a fam-ily of winning coalitions {ω(i)}i∈N is self-favoring for i ∈ N ifand only if.

• There exists K ∈2N with i /∈K and K /∈ω(i) such that givenany j ∈K we have (K \{j})∪ {i}∈ω(i).

• For all K ∈ 2N with i /∈K, we have K ∈ω(i)�⇒ (K\{j})∪{i}∈ω(i) for all j ∈K


A simple and essentially anonymous CIF F ∈F with a fam-ily of winning coalitions {ω(i)}i∈N is self-disfavoring for i ∈ N

if and only if

• There exists K ∈2N with i ∈K and K /∈ω(i) such that givenany j ∈N\K we have (K\{i})∪ {j}∈ω(i).

• For all K ∈ 2N with i ∈ K, we have K ∈ ω(i) �⇒ (K\{i})∪{j}∈ω(i) for all j ∈N\K

A simple and essentially anonymous CIF F ∈F with a fam-ily of winning coalitions {ω(i)}i∈N is self-equal-treating for i ∈N if and only if given any K ∈2N with i ∈K and any j ∈N\K,we have K ∈ω(i)⇔ (K\{i})∪{j}∈ω(i).

Remark that if an essentially anonymous simple CIF isself-equal-treating for every i ∈N , then while deciding whethersome individual k ∈N is a J , all individuals have equal deci-sion power – hence the usual anonymity condition in socialchoice theory.

This treatment of self-opinions by generalized symmetricconsent rules depends on the quota si and the number of indi-viduals n in the society, as we remark below:

REMARK 4.1. A generalized symmetric consent rule F s ∈F is

(i) Self-favoring for i ∈N if and only if si ∈{1, . . ., �n2�}

(ii) Self-disfavoring for i ∈N if and only if n is even and si =n2 +1

(iii) Self-equal-treating for i ∈ N if and only if n is odd andsi =

⌊n2

⌋+1

Observe that almost all generalized symmetric consent rulesare self-favoring, except two cases: When n is even, it is self-disfavoring to determine the identity of i by a version ofmajoritarianism where a coalition K of cardinality n/2 is win-ning over an individual i if and only if i is not a memberof K.14 Similarly, when n is odd, determining individual iden-tities by (usual) majoritarianism is self-equal-treating. Thus,majoritarianism with an odd number of individuals is the onlysimple and self-dual CIF which satisfies the usual anonymitycondition in social choice theory.


We now consider the equal treatment of outcomes. As Sametand Schmeidler (2003) argue, neutrality can be adapted to ourmodel as “using the same rule to determine the qualification ofeach individual”.15 However, the meaning of this requirementis obscured by the fact that voters and outcomes coincide. Forexample, it is arguable whether the liberal CIF uses the samerule to determine the qualification of each individual. We pro-pose to overcome the ambiguity by imposing the usual neu-trality requirement only for cases where voters and outcomesdiffer. A simple CIF F ∈F induced by a collection {ω(i)}i∈N ofwinning coalitions is essentially neutral if and only if

(EN1) K ∈ ω(i) �⇒ K ∈ ω(j) for all i, j ∈ N and for allK ∈2N with i, j /∈K.

and

(EN2) Given any i, j ∈ N and any K ∈ ω(i), there exists L ∈ω(j) with #L=#K.

So for simple CIFs, EN1 ensures that a coalition K whichexcludes some i, j ∈N is winning over i if and only if K is win-ning over j – a requirement which incorporates the usual neu-trality idea to our model for cases where those who decide andthose over which the decision is made form disjoint sets. Onthe other hand, EN2 ensures that all individuals have smallestwinning coalitions of the same cardinality.16 Our next result isa characterization of CIFs satisfying independence, monotonic-ity, self-duality, essential anonymity and essential neutrality.

THEOREM 4.2. A CIF F ∈ F satisfies independence, mono-tonicity, self-duality, essential anonymity and essential neutral-ity if and only if F is a symmetric consent rule F s with s ∈{1, . . ., �n

2�+1}.

Proof. To show the “if” part, take any symmetric consentrule F s with s ∈ {1, . . . , �n

2� + 1}. As every symmetric consentrule is, by definition, a generalized consent rule, we knowby Theorem 4.1 that F s satisfies independence, monotonicity,self-duality and essential anonymity. It is straightforward to


check that symmetric consent rules are essentially neutral. Toprove the “only if” part, take any CIF F which satisfies inde-pendence, monotonicity, self-duality, essential anonymity andessential neutrality. By Theorem 4.1, F is a generalized sym-metric consent rule F s . Consider first the case where n is odd.As F s is essentially neutral, hence the smallest winning coali-tions of all individuals are of the same cardinality, we havesi = sj for all i, j ∈N , showing that F s is a symmetric consentrule. Now consider the case where n is even. As F s is essen-tially neutral, hence the smallest winning coalitions of all indi-viduals are of the same cardinality, we have si �= sj for somei, j ∈ N only if si, sj ∈ {n

2 , n2 + 1}. However EN1 ensures that

there exists no i, j ∈ N such that si = n2 and sj = n

2 + 1, show-ing that F s is a symmetric consent rule.

We now relate essential anonymity and essential neutralityto the weak equal treatment property (WETP) introduced inSection 3.

PROPOSITION 4.1. Within the class of simple CIFs withproper and strong winning coalitions

(i) All essentially neutral CIFs, but dictatoriality, satisfyWETP.

(ii) An essentially anonymous CIF F fails WETP if and onlyif F is a generalized symmetric consent rule F s with si = 1for some i ∈N and sj >1 for some j ∈N .

Proof. (i) Let F be a simple and essentially neutral CIFwith proper and strong winning coalitions. Suppose for acontradiction that F is not dictatorial and F

(G0

) = K forsome K ∈ 2N \ {∅,N}. Take some i ∈ K and j ∈ N\K. ByLemma 3.1 we have ω(i)= {

K ∈2N :i ∈K}. As F is essentially

neutral, hence satisfies EN2, ∃L∈ω(j) with #L=1. Moreover,as j /∈ F

(G0

), we have {j} /∈ ω(j), hence L �= {j}. If L = {i},

then for any k ∈ N \ {i}, EN1 implies {i} ∈ ω(k) while mono-tonicity ensures

{K ∈2N :i ∈K

} ⊆ ω(k) and the properness ofω establishes ω(k) = {

K ∈2N :i ∈K}, contradicting that F is


not dictatorial. If L = {k} for some k ∈ N \ {i, j}, then EN1implies {k} ∈ ω(i), contradicting the definition of ω(i). HenceF

(G0

)∈{∅,N}, i.e., F satisfies WETP. Checking that dictato-rial CIFs violate WETP is left as a straightforward exercise tothe reader.

(ii) To see the “if” part, let F be a generalized symmetricconsent rule F s with si =1 for some i ∈N and sj >1 for somej ∈N . By definition of F s , we have i ∈F

(G0

)and j /∈F

(G0

)

violating WETP. To see the “only if” part, let F be a simpleand essentially anonymous CIF with proper and strong win-ning coalitions. By Theorem 4.1, F is a generalized symmet-ric consent rule. If F fails WETP, then there exist i, j ∈N suchthat i ∈F

(G0

)and j /∈F

(G0

). As F is a generalized symmetric

consent rule, we have si =1 and sj >1.

Thus, neither essential neutrality nor essential anonym-ity implies WETP. Now let N = {1,2,3} and let F be theCIF induced by the proper and strong family of winningcoalitions ω(1)= {

K ∈2N :2∈K}, ω(2)= {

K ∈2N :3∈K}, ω(3)={

K ∈2N : 1∈K}. F satisfies WETP but fails essential ano-

nymity and essential neutrality. So essential anonymity andWETP, as well as essential neutrality and WETP are logicallyindependent.

We close the section by noting that our equal treatmentconditions are related to the symmetry condition used bySamet and Schmeidler (2003). A simple CIF F ∈ F inducedby a collection {ω(i)}i∈N of winning coalitions is Samet–Schmeidler symmetric if and only if given any permutation � :N →N , any i ∈N and any K ∈2N , we have K ∈ω(i)⇔�(K)∈ω(�(i)). Theorem 4.2 is related to Theorem 2 of Sametand Schmeidler (2003) which characterizes symmetric consentrules in terms of independence, monotonicity, self-duality andSamet–Schmeidler symmetry. It immediately follows from jux-taposing Theorem 2 of Samet and Schmeidler (2003) with ourTheorem 4.2 that over the class of independent, monotonicand self-dual CIFs, Samet–Schmeidler symmetry is equivalentto the conjunction of essential anonymity and essential neu-trality.17 So over the domain of simple CIFs with proper and


strong winning coalitions, the anonymity and neutrality con-ditions we propose successfully decompose Samet–Schmeidlersymmetry which is an incorporation of both equal treatmentproperties.

5. CONCLUSION

We consider a model of aggregating vectors of sets into a set.This mathematical structure is sufficiently rich to allow var-ious interpretations such as the qualification problem wherea set of objects is to be partitioned as “goods” and “bads”depending on individuals’ opinions18; electing committees byapproval balloting where voters may approve any set of can-didates19; a society which has to choose a subset of new mem-bers from a finite set of candidates20; deciding over acceptingor rejecting a set of issues21 or more generally the aggregationof individual choices into a social choice.22 The interpretationwe explore is the collective identity determination problem,proposed by Kasher and Rubinstein (1997), where individu-als opinions about “Who is a J” are aggregated into a socialdecision. We introduce and characterize the family of CIFswhich can be expressed in terms of winning coalitions. Thisbrings a new perspective in the analysis of CIFs and manyinteresting CIFs of the literature – with the notable exceptionof procedural CIFs – fall within the scope of our analysis. Itis also worth noting that we work in an environment wherevoters and outcomes coincide. This requires a finer distinctionbetween the standard anonymity and neutrality conditions ofsocial choice theory – a matter which we address and handle.

To sum up, we show that the class of simple CIFs withproper and strong winning coalitions exhibits the followingproperties:

• The family of essentially anonymous and essentially neu-tral CIFs coincides with the family of symmetric consentrules ranging from liberalism to majoritarianism.


• The family of essentially anonymous CIFs coincides withthe family of generalized symmetric consent rules. Amongthese, majoritarianism (with an odd number of individuals)is the only one which is self-equal-treating, hence satisfyingthe usual anonymity condition in social choice theory.

• All essentially neutral CIFs, but dictatoriality, satisfyWETP.

• An essentially anonymous CIF F fails WETP if and onlyif F is a generalized symmetric consent rule F swith si = 1for some i ∈N and sj >1 for some j ∈N .

• Liberalism is the unique CIF that satisfies WETP posi-tively.

The following figure summarizes our findings regardingsimple CIFs with proper and strong winning coalitions:

ACKNOWLEDGEMENTS

This paper, originally called “Embracing Liberalism for Col-lective Identity Determination”, is part of a project enti-


tled “Social Perception – A Social Choice Perspective”, sup-ported by Istanbul Bilgi University Research Fund. We arevery much indebted to William S. Zwicker for his carefulreading of the article and constructive suggestions. We alsothank Fuad Aleskerov, Goksel Asan, Emre Dogan, SemihKoray and two anonymous referees for their helpful com-ments. Remzi Sanver acknowledges the support of the Turk-ish Academy of Sciences Distinguished Young Scientist AwardProgram (TUBA-GEBIP). Of course, the authors are respon-sible from all possible errors.

NOTES

1. While Dimitrov and Sung (2005) show that the five axioms used byKasher and Rubinstein (1997) are logically dependent whereas threeof them suffice to establish the desired equivalence. More recently,Miller (2006) and Houy (2007) offered alternative characterizationsof liberalism in a setting where there are multiple identities and ina setting where there is a given number of classes in the society,respectively.

2. Where personal opinions about the identity of an individual areaggregated according to the majority rule.

3. A detailed discussion of this issue in a general social choice settingcan be found in Taylor and Zwicker (1999) and Austen-Smith andBanks (1999).

4. de Andres and Garcıa-Lapresta (2004) introduce the same anonym-ity condition in a more general framework where individuals’ non-dichotomous opinions belong to the [0, 1] interval.

5. Our monotonicity condition is logically equivalent to Kasher andRubinstein (1997)’s monotonicity while it is weaker than the mono-tonicity used by Samet and Schmeidler (2003). Nevertheless, underindependence, our monotonicity and Samet and Schmeidler (2003)’smonotonicity turn out to be logically equivalent.

6. Remark that when {ω(i)}i∈N is regular, for each i ∈N , we have eitherω(i)=∅ or N ∈ω(i). To be sure, the regularity of {ω(i)}i∈N allowsω(i)=2N for some i ∈N . As a result, simple CIFs can fail the fol-lowing non-degeneracy axiom of Samet and Schmeidler (2003): ACIF F is non-degenerate iff for each i ∈ N , there exists G ∈ � withi ∈F(G) and there exists H ∈� with i /∈F(H).

As a particular example of a degenerate CIF, we have F which isdefined for each G∈� as F(G)=K where K ⊆N is previously fixed.


7. In many other aggregation frameworks, non-properness would preventaggregation rules to be well-defined. In fact, Austen-Smith andBanks (1999) qualify an aggregation rule as “simple” if it is inducedby a proper collection of winning coalitions. Since non-propernessand well-definedness are compatible in our model, we adopt thenomenclature of Taylor and Zwicker (1999) who qualify voting rulesas simple whenever they can be expressed in terms of some collec-tion of winning coalitions which satisfies a monotonicity conditionthat has same spirit as our’s.

8. When {ω(i)}i∈N is proper, we have K ∩K ′ �=∅ for any i ∈N and anyK,K ′ ∈ω(i). This rules out ω(i)=2N . See Footnote 6.

9. When {ω(i)}i∈N is strong, we have K,K ′ /∈ ω(i) �⇒ K ∪ K ′ �= N forany i ∈N and any K, K ′ ∈ 2N , which rules out ω(i)=∅. See Foot-note 6.

10. This has two subcases which is worth distinguishing. When n is odd,F �n/2�+1 coincides with the majoritarian CIF M. When n is even,we have two versions of majoritarianism depending on the choice ofs ∈{n

2 , n2 +1}. When s =n/2, a coalition K of cardinality n/2 is win-

ning over an individual i if and only if i is a member of K. Onthe other hand, when s = n

2 + 1, a coalition K of cardinality n/2 iswinning over an individual i if and only if i is not a member of K.

11. There is no such coalition when n is odd and s =�n2 �+1.

12. Theorem 3.3 can also be shown through Samet and Schmeidler (2003)whose Theorem 1 establishes that all consent rules satisfies independenceand monotonicity and Proposition 2 shows that a consent rule satisfiesself-duality if and only if it is a symmetric consent rule.

13. See Footnote 6.14. See Footnote 1015. See Section 4.1, p.225 of Samet and Schmeidler (2003).16. EN1 and EN2 are logically independent. To see that EN1 does not

imply EN2, let N ={1,2,3} and consider the generalized symmetricconsent rule F s with s = (1,1,2). To see that EN2 does not implyEN1, consider the generalized symmetric consent rule F s with s =(2,2,2,3) used in the society N = {1,2,3,4}. We also wish to notethat essential anonymity and essential neutrality are logically inde-pendent.

17. As noted in Footnote 5, although the Samet and Schmeidler (2003)monotonicity condition is stronger then ours, the two monotonicitiescoincide under independence.

18. Such as the analysis made by Dimitrov et al. (2004) and Ju (2005a)19. Examples of which can be found in Brams et al. (2007) and Kilgour

et al. (2006)20. See Barbera et al. (1991), Berga et al. (2004).21. See Ju (2005b)


22. Aleskerov (1999) and Aizerman and Aleskerov (1995) give an excel-lent treatment of the choice aggregation problem. We wish to saythat the plethora of results they establish in an abstract frame-work can certainly bring further insights to particular applications ofaggregating choices, such as the one we consider in this article. Seealso Lahiri (2001) and Quesada (2003).

REFERENCES

Aizerman, M. and Aleskerov, F. (1995), Theory of Choice, North Holland.Aleskerov, F. (1999), Arrovian Aggregation Models, Kluwer Academic Pub-

lishers.Austen-Smith, D. and Banks, J.S. (1999), Positive Political Theory I: Col-

lective Preference, The University of Michigan Press.Barbera, S., Sonnenschein, H. and Zhou, L. (1991), Voting by committees,

Econometrica 59(3), 595–609.Berga, D., Bergantinos, G., Masso, J. and Neme, A. (2004), Stability and

voting by committees with exit, Social Choice and Welfare 23(2), 229–247.Brams, S.J., Kilgour, D.M. and Sanver, M.R. (2007), A minimax procedure

for electing committees. Public Choice 132(3–4), 401–420.de Andres, R. and Garcıa-Lapresta, J.L. (2004), Choosing qualified mem-

bers of a society with non dichotomous assesment. Presented in Societyfor Economic Design.

Dimitrov, D., Born, P. and Hendrickx, R. (2004) Good and bad objects:The symmetric difference rule, Economics Bulletin 4(11), 1–7.

Dimitrov, D. and Sung, S.C. (2005), On the axiomatic characterization of“who is a j?”, Logique et Analyse 189–192, 101–112.

Dimitrov, D., Sung, S.C. and Xu, Y. (2007), Procedural group identifica-tion. Mathematical Social Sciences 54(2), 137–146.

Houy, N. (2007), “I Want to Be a J!”: Liberalism in group identificationproblems, Mathematical Social Sciences 54(1), 59–70.

Ju, B.-G. (2005a), An efficiency characterization of plurality social choiceon simple preference domains, Economic Theory 26(1), 115–128.

Ju, B.-G. (2005b). Individual powers and social consent: An axiomaticapproach. Working Paper.

Kasher, A. (1993), Jewish collective identity, in Goldberg, D.T. andKrausz, M. (eds.), Jewish Identity, Temple University Press, Philadelphia.

Kasher, A. and Rubinstein, A. (1997), On the question of “Who is a J?”:A social choice approach, Logique et Analyse 160, 385–395.

Kilgour, D.M., Brams, S.J. and Sanver, M.R. (2006), How to elect arepresentative committee using approval balloting, in Simeone, B. andPukelsheim, F. (eds.), Mathematics and Democracy: Recent Advances inVoting Systems and Collective Choice. Springer, Berlin-Heidelberg.


Lahiri, S. (2001), Axiomatic characterization of voting operators, Mathe-matical Social Sciences 41(2), 227–238.

Miller, A.D. (2007), Group identification. Games and Economic Behavior.Forthcoming.

Quesada, A. (2003), Dictatorial voting operators, Review of EconomicDesign 8(3), 347–358.

Samet, D. and Schmeidler, D. (2003), Between liberalism and democracy.Journal of Economic Theory 110(2), 213–233.

Taylor, A.D. and Zwicker, W.S. (1999), Simple Games: Desirability Rela-tions, Trading, Pseudoweightings. Princeton University Press Princeton,New Jersey.

Address for correspondence: Murat Ali Cengelci, M. Remzi Sanver, Depart-ment of Economics, Istanbul Bilgi University, Inonu Cad. No. 28, Kustepe,80310, Istanbul 34387, Turkey E-mail: [email protected]

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