Dimitrov,D.

Xu,Y.

English

Research Papers in Economics

Self-Supporting Liberals and Their Cliques:
An Axiomatic Characterization∗
Dinko Dimitrov
Center for Interdisciplinary Research (ZiF), University of Bielefeld
Wellenberg 1, D-33615 Bielefeld, Germany
Email: dinko.dimitrov@uni-bielefeld.de
Yongsheng Xu
Department of Economics, Andrew Young School of Policy Studies
Georgia State University, Atlanta, GA 30303, U.S.A.
Email: yxu3@gsu.edu
June 2002
Abstract
In this paper we axiomatically characterize a recursive procedure
for deﬁning a social group. The procedure starts with the set of all in-
dividuals who deﬁne themselves as members of the social group. This
initial set is then expanded by adding individuals who are considered
to be appropriate group members by someone in the initial set, and
continues inductively until there is no possibility of expansion any
more. Journal of Economic Literature Classiﬁcation Numbers: D63,
D71.
Key Words: clique, liberalism, procedure, social identity
∗This paper was written while the authors were research fellows at the ZiF (Bielefeld)
for the project “Procedural Approaches to Conﬂict Resolution”, 2002. We thank our hosts
for their hospitality.
11 Introduction
The problem of group identiﬁcation serves as a background in many social
and economic contexts. For example, when one examines the political princi-
ple of self-determination of a newly formed country, one would like to deﬁne
the extension of a given nationality. Or when a newly arrived person in
Atlanta chooses where to live, the person is interested in ﬁnding out a res-
idential neighborhood that would suit her: “Are they my kind of people?
Do I belong to this neighborhood?” In all those contexts, it is typically as-
sumed that there is a well-deﬁned group of people who share some common
values, beliefs, expectations, customs, jargon, or rituals. Consequently, the
questions like “how to deﬁne a social group” or “who belongs to the social
group” arise.
In a recent paper, Kasher and Rubinstein (1997) provide an answer to
the above questions from a social choice perspective. They view that each
individual of a society has an opinion about every individual, including one-
self, whether the latter is a member of a group to be formed. The collective
identity of the group to be formed is then determined by aggregating opinions
of all the individuals in the society. For this purpose, they provide, among
others, an axiomatic characterization of a “liberal” aggregator whereby the
members of the group consist of those and only those who each of them views
oneself a member of the group.
The purpose of this paper is to extend the “liberal” aggregator charac-
terized by Kasher and Rubinstein (1997) by adding a procedural view to the
study of the group identiﬁcation problem. This procedural view allows us
to see a collective as “a family of groups, subcollectives, each with its own
view of who is a member of the collective, its own sense of tradition and
2its own underlying conceptual realm, but each bearing some resemblance to
the other ones” (Kasher (1993, p. 70)). More speciﬁcally, we axiomatically
characterize a recursive procedure for determining “who is a member of a
social group”. The recursive procedure starts with the set of all individuals
who deﬁne themselves as members of the social group. This initial set is then
expanded by adding individuals who are considered to be appropriate group
members by someone in the initial set, and continues inductively until there
is no possibility of expansion any more.
The rest of the paper is organized as follows. In Section 2, we present
the basic notation and deﬁnitions. Section 3 introduces the axioms used for
characterization. The main result is contained in Section 4, and Section 5
contains some concluding remarks.
2 Basic Notation and Deﬁnitions
Let N = {1,...,n} denote the set of all individuals in the society. Each
individual i ∈ N forms a set Gi ⊆ N consisting of all society members that
in the view of i have the social identity G.F o ra l li ∈ N,w h e ni ∈ Gi,w e
also say that i considers himself as a G.A proﬁle of views is an n−tuple
of vectors (G1,...,G n) where Gi ⊆ N for all i ∈ N.L e t G be the set of
all proﬁles of views. A Collective Identity Function (CIF) assigns to each
proﬁle (G1,...,G n) ∈ G a set G(G1,...,G n) ⊆ N of socially accepted Gs.
For the purpose of simplicity, we will often write G instead of G(G1,...,G n).
For any (G1,...,G n) ∈ G,d e ﬁne L(0)(G1,...,G n)={i ∈ N : i ∈
Gi}.T h u s , L(0)(G1,...,G n) consists of all individuals in the society who
consider themselves as Gs. For any (G1,...,G n) ∈ G, with the help of
L(0)(G1,...,G n),w en o wd e ﬁne a CIF being self-supporting-liberals-and-
3their-cliques,t ob ed e n o t e db yL(G1,...,G n), as follows: for each positive
integer t,l e tL(t)(G1,...,G n)=L(t − 1) ∪ {i ∈ N : i ∈ Gk for some
k ∈ L(t − 1)(G1,...,G n)};a n di ff o rs o m et ≥ 0, L(t)(G1,...,G n)=L(t +
1)(G1,...,G n),t h e nL(G1,...,G n)=L(t)(G1,...,G n).
To illustrate the above procedure of deﬁning a CIF, consider the following
example. Let N = {1,2,3} and consider the proﬁle G1 = {1,2},G 2 = {3}
and G3 = ∅. Then, for this proﬁle, L(0) = {1}, L(1) = L(0)∪{2} = {1}∪{2},
L(2) = L(1) ∪ {3} = {1,2,3}, L(3) = L(2). Therefore, for the given proﬁle
of views, we have L = {1,2,3}.
The CIF L deﬁned above is discussed in Kasher and Rubinstein (1997).
It starts with L(0) which consists of all members of the society who view
themselves as Gs. Thus, the set L(0) reﬂects a weak notion of self determi-
nation: if one considers oneself a member of G, then one should be a member
of G collectively. In the liberal tradition, those individuals may be called self-
supporting liberals1. The collective identity function L now expands the set
L(0) by the following procedure. If, according to some individual i ∈ L(0),
an individual k ∈ N is viewed as a G,t h e nk should be a G collectively.
By adding all such kst oL(0), we obtain the set L(1). We then repeat the
above process with L(1) by adding those individuals who are considered as
Gs by some individuals in L(1) to L(1) to obtain L(2).S i n c en is ﬁnite, at
certain step t,w em u s th a v eL(t)=L(t +1 ) : the set L(t) can no longer be
expanded. The intuition behind each step of the expansion is in line with
Kasher’s argument (1993): every socially accepted G as being newly added
brings a possibly unique new view of being a G collectively with him, and
a collective identity function is supposed to aggregate those views and must
1 In Kasher and Rubinstein (1997), the individuals in L(0) are called lib-
erals.
4pay attention to this new individual’s G−concept in order to cover the whole
diversity of views in the society about the question “what does it mean to
be a G”.
3A x i o m s
In order to present our axiomatic characterization of the CIF L,w ei n t r o d u c e
the following axioms for a CIF to satisfy.
AC I Fs a t i s ﬁes
Consensus (C) iﬀ,f o ra l l(G1,...,G n) ∈ G, [j ∈ Gi for all i ∈ N] ⇒ j ∈
G(G1,...,G n),a n d[j/ ∈ Gi for all i ∈ N] ⇒ j/ ∈ G(G1,...,G n).
Symmetry (SYM) iﬀ, for all (G1,...,G n) ∈ G, for all j,k ∈ N,i f( i )
Gj − {j,k} = Gk − {j,k}; (ii) ∀i ∈ N − {j,k}, j ∈ Gi iﬀ k ∈ Gi; (iii)
j ∈ Gj iﬀ k ∈ Gk;( i v )j ∈ Gk iﬀ k ∈ Gj,t h e nj ∈ G(G1,...,G n) ⇔
k ∈ G(G1,...,G n).
Independence (I) iﬀ, for all proﬁles (G1,...,G n) and (G0
1,...,G 0
n) in G,
and all i ∈ N,i f[ f o re v e r yk 6= i, k ∈ G(G1,···,G n) if and only if
k ∈ G(G0
1,...,G 0
n)], and [for all k ∈ N, i ∈ Gk if and only if i ∈ G0
k],
then i ∈ G(G1,...,G n) ⇔ i ∈ G(G0
1,...,G 0
n).
Weak monotonicity (WM) iﬀ, for all (G1,...,G n),(G0
1,...,G 0
n) ∈ G and
all i,k ∈ N,i f[ i/ ∈ G(G1,...,G n)] and [(G0
1,...,G 0
n) is a proﬁle
identical to(G1,...,G n) except that i ∈ Gk and i/ ∈ G0
k], then i/ ∈
G(G0
1,...,G 0
n).
Equal treatment of insiders’ views (ETIV) iﬀ,f o ra l l(G1,...,G n),
(G0
1,...,G 0
n) ∈ G,a n da l li,k,m ∈ N,i f[ Gh = G0
h for all h ∈ N −
5{i,k}], [m ∈ Gk, m/ ∈ Gi], [G0
k = Gk−{m}], and [G0
i = Gi∪{m}], then
[k ∈ G(G1,...,G n)&i ∈ G(G0
1,...,G 0
n)] ⇒ [m ∈ G(G1,...,G n) iﬀ
m ∈ G(G0
1,...,G 0
n)].
Irrelevance of an outsider’s view (IOV) iﬀ, for all i,k ∈ N with i 6= k and
all (G1,...,G n),(G0
1,...,G 0
n) ∈ G,i f[Gh = G0
h for all h ∈ N−{i}] and
[G0
i = Gi − {k}],t h e n[i/ ∈ G(G1,...,G n)] ⇒ [k ∈ G(G1,...,G n) ⇔
k ∈ G(G0
1,...,G 0
n)].
The ﬁrst three axioms are introduced and discussed in Kasher and Ru-
binstein (1997). Weak monotonicity is a weaker version of the monotonicity
condition introduced in Kasher and Rubinstein (1997). (WM) requires that
if an individual i is not collectively recognized as a G and someone changes
his view about i from being a G t ob e i n gan o n −G,t h e ni can not be a G
after this change.
(ETIV) requires that if two individuals i,k in the society have opposite
views about a society member m with m ∈ Gk and m/ ∈ Gi in a given proﬁle,
a n di ft h e ys w i t c ht h e i rv i e w sc o n c e r n i n gm in a new proﬁle and nothing else
has changed, then, when k is a G collectively in the original proﬁle and i is
a G collectively in the new proﬁl e ,i tm u s tb et r u et h a tm is a G collectively
in the original proﬁle if and only if m is a G collectively in the new proﬁle.
This axiom essentially requires that a CIF should treat the views of all the
m e m b e r sw h oa r ec o n s i d e r e dt ob eGs collectively equally.
Finally, (IOV) stipulates that if someone is collectively deﬁned as a non−G,
then this person’s view about any society member is not relevant in decid-
ing the collective identity of G. The axiom is in the spirit of the exclusive
self-determination axiom introduced in Samet and Schmeidler (forthcoming).
It should be noted that our axiom is much weaker than the exclusive self-
6determination axiom used in that paper.
4T h e R e s u l t
In this section, we give an axiomatic characterization of a CIF being L deﬁned
in Section 2.
Theorem 1 A CIF Gsatisﬁes the axioms (C), (SYM), (WM), (I), (ETIV),
and (IOV) if and only if G = L.
Proof. It can be veriﬁed that the CIF Lsatisﬁes the axioms (C), (SYM),
(WM), (I), (ETIV) and (IOV). We now show that if a CIF G satisﬁes these
six axioms, we must have G = L.L e taCIF G satisfy the six axioms. For
all (G1,...,G n) ∈ G,l e tG = G(G1,...,G n) and L = L(G1,...,G n) for
short. We have to show that, for all (G1,...,G n) ∈ G,
(i) [i ∈ Gi for some i ∈ N] ⇒ [i ∈ G];
(ii) [i ∈ G and k ∈ Gi for any k ∈ N] ⇒ [k ∈ G];
(iii) [i ∈ L and k ∈ Gi for any k ∈ N] ⇒ [k ∈ G];
(iv) G = L.
(i) The proof of this part is very close to the proof of Theorem 1(a) in
Kasher and Rubinstein (1997). Assume that there is a proﬁle (G1,...,G n)
in which i ∈ Gi but i/ ∈ G. By applying (WM) several times, we arrive
at a proﬁle (G0
1,...,G 0
n) that is identical to (G1,...,G n) with the possible
exception that for all k 6= i, i/ ∈ G0
k so that i/ ∈ G0 = G(G0
1,...,G 0
n).L e t
(G00
1,...,G 00
n) be the proﬁle where G00
j = {j} for all j ∈ G0 ∪ {i} and G00
j = ∅
for any other j.B y ( C ) , G00 = G(G00
1,...,G 00
n) does not contain any of the
members of N − G0 − {i}. By (SYM), the CIF classiﬁes all members of
7G0 ∪ {i} identically. Therefore, there are two possibility: (1) G00 = ∅;o r( 2 )
G00 = G0 ∪ {i}. Suppose G00 = ∅.L e tN00 = {i ∈ N : G00
i = {i}}.C o n s i d e r
the proﬁle G000
i = N00 for all i ∈ N.S i n c e G00 = ∅,b yt h er e p e a t e du s eo f
(IOV), we obtain that G000 = ∅. On the other hand, by (C), G000 = N00,a
contradiction. Therefore, it is impossible that G00 = ∅.T h u s ,G00 = G0 ∪{i}.
Finally, we get a contradiction to (I) because G0 and G00 are identical with
the exception of member i,a n dm e m b e ri is treated equally by all members
of N; nevertheless, i ∈ G00 and i/ ∈ G0.
(ii) Assume that there is a proﬁle (G1,...,G n) where i ∈ G, k ∈ Gi.
Note ﬁrst that if k ∈ Gk, then, according to (i), k ∈ G.F o rt h ec a s ew h e r e
k/ ∈ Gk,l e t(G0
1,...,G 0
n) be a proﬁle that is identical to (G1,...,G n) with
the possible exception that G0
i = Gi − {k}, G0
k = Gk ∪ {k}.S i n c ek ∈ G0
k,
from (i), it follows that k ∈ G0 = G(G0
1,...,G 0
n).T h e r e f o r e ,k ∈ G follows
from (ETIV).
(iii) This assertion follows from (i) and (ii) directly.
(iv) Let the proﬁle (G1,···,G n) ∈ G be given. We need to show that
G(G1,...,G n)=L (= L(G1,...,G n)).C o n s i d e rﬁrst the proﬁle (G0
1,...,G 0
n)
such that, for all i ∈ L,G0
i = Gi and for all i ∈ N − L,G0
i = ∅.I t i s c l e a r
that L(G0
1,...,G 0
n)=L.B y( C ) ,G0 ⊂ L, and from (iii), L ⊂ G0.T h e r e f o r e ,
G0 = L. Now, starting with the proﬁle (G0
1,...,G 0
n) and by appropriately
changing G0
i for all i ∈ N − L, we can obtain (G1,...,G n). Noting that, at
each stage of the change, we can invoke (IOV). Then, by the repeated use of
(IOV), we obtain G = G0 = L.
85C o n c l u d i n g R e m a r k s
In this paper, we have axiomatically characterized the procedure that deﬁnes
the collective identify function L in the framework proposed by Kasher and
Rubinstein (1997). Though it is not our intention to advocate the CIF L,
we note some interesting features of L. It includes all those individuals
L(0) who consider themselves as members of a social group as well as all
other individuals who are considered as members of the social group by some
member of the social group. In other words, L consists of all those individuals
who are self-supporting liberals (self-claimed members of a social group) and
their cliques (who are viewed as members of the social group by some member
of the social group). Therefore, L reﬂects a liberal view2 of collective identity.
References
[1] A. Kasher, Jewish Collective Identity, in ” J e w i s hI d e n t i t y ”( D .T .G o l d -
berg and M. Krausz, Eds.), pp. 56-78, Temple University Press, Philadel-
phia, 1993.
[2] A. Kasher and A. Rubinstein, On the Question ”Who is a J?”: A Social
Choice Approach, Logique & Analyse 160 (1997), 385-395.
2 See, for example, Kasher and Rubinstein (1997), and Samet and Schmei-
dler (forthcoming). If the determination of the membership of a social group
is a personal matter, there is indeed some reason to call individuals in L(0)
as (self-supporting) liberals (see Sen (1970)).
9[3] D. Samet and D. Schmeidler, Between Liberalism and Democracy, J.
Econ. Theory,f o r t h c o m i n g .
[4] A. K. Sen, The Impossibility of a Paretian Liberal, J. Political Econ. 78
(1970), 152-157.
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Self-supporting liberals and their cliques : an axiomatic characterization

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