Inequalities and Identities

Abstract

We introduce concepts and measures relating to inequality between identity groups. We define and discuss the concepts of Representational Inequality, Sequence Inequality and Group Inequality Comparison. Representational Inequality captures the extent to which an attribute is shared between members of distinct groups. Sequence Inequality captures the extent to which groups are ordered hierarchically. Group Inequality Comparison captures the extent of differences between groups-. The concepts have application in interpreting segregation, clustering and polarization in societies. There exists a mapping from familiar inequality measures to the measures we identify, making them empirically applicable.

Full text

Inequalities and Identities
Sanjay G. Reddy∗ and Arjun Jayadev+
November 23rd, 2011
Abstract
We introduce concepts and measures relating to inequality between identity groups. We define
and discuss the concepts of Representational Inequality, Sequence Inequality and Group
Inequality Comparison. Representational Inequality captures the extent to which an attribute is
shared between members of distinct groups. Sequence Inequality captures the extent to which
groups are ordered hierarchically. Group Inequality Comparison captures the extent of
differences between groups-. The concepts have application in interpreting segregation,
clustering and polarization in societies. There exists a mapping from familiar inequality
measures to the measures we identify, making them empirically applicable.
∗ New School for Social Research ,Economics Department.and School of International and Public Affairs, Columbia
University, reddys1@newschool.edu
+ Economics Department, University of Massachusetts, Boston arjun.jayadev@umb.edu

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"Civil& paths& to& peace& also& demand& the& removal& of& gross& economic& inequalities,& social&
humiliations& and& political& disenfranchisement,& which& can& contribute& to& generating&
confrontation& and& hostility.& Purely& economic& measures& of& inequality& do& not& bring& out& the&
social& dimension& of& the& inequality& involved.& For& example,& when& the& people& in& the& bottom&
groups& in& terms& of& income& have& different& non‐economic& characteristics,& in& terms& of& race&
(such& as& being& black& rather& than& white),& or& immigration& status& (such& as& being& recent&
arrivals& rather& than& older& residents),& then& the& significance& of& the& economic& inequality& is&
substantially& magnified& by& its& "coupling"& with& other& divisions,& linked& with& non‐economic&
identity&groups."&
&
Amartya&Sen,&The$Guardian,&Friday&November&9th,&2007&
&
Introduction
Do differences in the economic and social achievements of distinct groups merit attention? Sen’s
remarks above suggest that the salience of interpersonal differences in welfare can be increased
when these are correlated with certain other differences among individuals. Such a conclusion
can be justified from at least two perspectives. First, inter-group differences may possess an
intrinsic significance from the standpoint of assessments of justice and fairness in the distribution
of goods and opportunities.1 Second, the fact that there exist distinct groups in society and that
these groups exhibit inter-group differences may have instrumental significance from the
standpoint of their impact on social goods such as peace, stability or economic growth.
The concern with the intrinsic significance of inter-group differences has centered on the degree
to which ‘morally irrelevant’ characteristics of a person (such as belonging to a given race, sex,
caste or other group as a result of birth) should be permitted to determine her or his life chances2.
Such a motivation is distinct from one based on the idea that social goods or ‘bads’ may be
generated by inter-group differences in economic and social achievements, and that inter-group
differences may be relevant for that reason. A long standing body of literature in economics and
other social sciences has empirically explored this instrumental concern.3 Both concerns have led
to the development of a growing literature which has identified and empirically examined such
concepts as ‘horizontal inequality’, segregation, polarization and related ideas about differences
between groups.
That a multitude of concepts concerning inter-group difference has been proposed is not entirely
surprising in light of the fact that such differences can be understood as occurring in more than
one way. For example, studies on segregation focus on the degree to which members of different
groups share a location, occupation or other attribute while studies on horizontal inequality focus
1 See Appendix One for a brief discussion.
2 For a review of these debates, see e.g. Roemer (1996) and Sen (1992). Arguments that societies should be
organized so as to limit the consequences of the “brute luck” of being born into a particular position include those of
“luck egalitarians” such as Arneson (1989), Cohen (1989), Dworkin (2000), Rawls (1971) and Roemer (1996).
Egalitarians of other kinds may come to similar conclusions for different reasons (see e.g. Anderson (1999)).
3 For some recent examples see e.g. Stewart (2001), Alesina et al (2003), Alesina and La Ferrara (2000, 2002),
Montalvo and Reynal Querol (2005), Miguel and Gugerty (2005) and Østby (2008).

2
on the extent of difference in the income or other achievements of separate groups. In both cases
however, the subject of interest is the degree of unevenness or inequality in the possession of
attributes between groups. The goal of our paper is to elucidate some distinct ways in which
inter-group differences can be conceived, which encompass but are not restricted to the concerns
of these existing approaches.
A common underlying concern in analyses of inter-group differences is the degree to which
distinct groups are systematically over-or under-represented in their possession of various
attributes (levels of income or health, club membership or political office etc.). In this paper we
introduce the concept of Representational Inequality (RI) as a way to capture this concern. This
concept describes the extent to which a given attribute (for instance, a level of income or health,
or right or left handedness) is shared by members of distinct groups. It can be used to measure
the degree of ‘segregation’ of distinct identity groups in the attribute space.4
When individuals can be ordinally ranked in relation to an attribute (such as income or health but
not right or left handedness) we may be interested not only in how segregated or separated each
identity group is in terms of their achievements, but in some measure of their relative positions in
the ranking. Sequence Inequality (SI), understood as the degree to which members of one group
are placed higher in a given hierarchy than those from another, captures this concern. Such a
concept provides an intuitive framework for understanding the degree of ‘clustering’ of various
identity groups in distinct sections of a hierarchy.5
When individuals’ level of achievement can also be cardinally identified for an attribute (as for
income but not for right or left handedness) the distance between groups’ attribute levels may be
of interest. We may identify a distinction between two different concepts, which we term
respectively Group Inequality Comparison (I) and Group Inequality Comparison (II) and
abbreviate as GIC (I) and GIC (II). The concept of Group Inequality Comparison (I) involves a
comparison of counterfactuals. Specifically, it is derived by comparing the inequality arising in
a society in which all of the members of a group are assigned a representative income for that
group and the total interpersonal inequality in a society. This concept is concerned with
identifying the extent to which between-group inequality ‘accounts for’ overall inequality in
society. Group Inequality Comparison (II) by contrast measures only the inequality arising in the
first situation, i.e. that in a society in which all of the members of a group are assigned a
representative income for that group. This latter concept is concerned with the absolute
magnitude of the inequality generated by between-group inequality.
Our purpose in this paper is two-fold. We seek not only to clarify the concepts described above
but to show that combining them can provide a way to understand conjoint concepts of group
differences. In particular, ‘polarization’,6 understood to involve the collection of like elements
and the separation of such collections of like elements from one another, can be fruitfully
4 Segregation is defined by the Oxford English Dictionary, inter alia, as “The separation of a portion of portions of a
collective or complex unity from the rest; the isolation of particular constituents of a compound or mixture”.
5 A cluster is defined by the Oxford English Dictionary as, inter alia, “A collection of things of the same
kind…growing closely together; a bunch… a number of persons, animals, or things gathered or situated close
together; an assemblage, group, swarm, crowd.”
6 The Oxford English Dictionary defines the verb “polarize” as “To accentuate a division within (a group, system,
etc.); to separate into two (or occas. several) opposing groups, extremes of opinion, etc.”

3
described as involving the simultaneous presence of between-group differences of different
kinds. The combination of Representational Inequality with Sequence Inequality alone provides
a measure of what might be termed ‘Ordinal Polarization.’ Combining Group Inequality
Comparison (of either type I or type II) with these other two indices can provide a richer index of
Polarization applicable to the case in which the attribute is cardinally measurable as well. Our
purpose is not to provide a unique characterization of a single measure of polarization, but
rather to show that a broad class of measures of polarization can be derived from a simple set of
unexceptionable axioms concerning different types of between-group differences and their
combination.
The concept of polarization that we employ here is distinct from that developed in the
preponderance of the existing literature in that it draws on information about the identity groups
to which those who possess distinct attributes belong. In contrast, the existing frameworks
generally employ a ‘collapsed’ framework in which the level of the attribute (typically income)
defines the identity group (Esteban and Ray (1994), Duclos, Esteban and Ray (2004)). In these
frameworks, polarization of an income distribution is understood to involve ‘identification’
between individuals possessing a certain level of income and ‘alienation’ between those
individuals and others possessing different incomes. In our framework, in contrast, polarization
of an income distribution is understood to involve segregation of individuals belonging to
distinct identity groups at certain levels of income and the separation of these groupings of
individuals in the income space from other groupings of individuals possessing distinct
identities.
Part I: Concepts of Group Inequality
One approach to evaluating inter-group differences is to construct a measure of overall group
advantage or disadvantage for each group prior to assessing the differences in these overall
measures.7 Although there can be advantages to such an approach, it can obscure the diverse
aspects of inter-group difference (by reducing inter-group differences to inequalities in a single
dimension). We accordingly explicitly identify here three distinct concepts of inter-group
difference, and a fourth which builds upon them.
Representational Inequality:
We define a situation of representational inequality as occurring when, for some attribute and
some identity group, the proportion of the group possessing the attribute is either greater or less
than the proportion of the group in the overall population. To provide some graphical intuition
for this idea, consider the distribution of income among different groups in a society that consists
of fifty percent whites and fifty percent blacks. Figure 1 depicts the situation in which there is no
representational inequality. The location of each bar on the horizontal axis represents an income
level ordered from lowest to highest and the proportion of persons possessing that income of
either group is represented through shading. At all levels of income, blacks and whites are
7 See Jayaraj and Subramanian (2006) for an example of such an approach.

4
represented in equal proportion to their share of the population as a whole (i.e. one half each).
Any deviation from such equi-proportionality leads to a situation of representational inequality.
Such a situation is depicted in Figure 2, in which at certain levels of income blacks or whites
comprise a larger or smaller proportion of the individuals possessing that level of income than
they do in the population as a whole.
While the situation depicted in Figure 2 is one of representational inequality, both groups are
represented at all the incomes. In contrast, Figure 3 depicts a situation in which at each level of
income there is complete segregation, in the sense that at each level of income there is one and
only one identity group represented. It may be noted that although this is a situation of complete
segregation the incomes at which whites and blacks appear are evenly interspersed. We depict
this example in order to make sharp the distinction between segregation and clustering as we use
the terms. The former refers to a situation in which those possessing a specific attribute (in this
case an income level) belong disproportionately to a particular group. The latter refers to a
situation in which the attributes disproportionately possessed by members of a particular group
are grouped together in a certain part of an attribute hierarchy (in this case the income spectrum).
The concept of representational inequality clearly need not be restricted to a scenario in which
the attribute is cardinally orderable. Thus, for example, we can apply the principle in an equally
straightforward manner to unordered attributes such as location of residence, or membership in
distinct clubs or legislatures. If instead of income brackets, each bar referred to a distinct
legislature in a federal country, the figures we have discussed here would depict the degree of
inequality in political representation.
Sequence Inequality
The distinction between ‘complete segregation’ and ‘complete clustering’ can be seen by
comparing Figure 3 and Figure 4. Figure 4 depicts the situation that results from a transfer of
incomes such that all the whites move to the richer half of society while all the blacks move to
the poorer half of society. This situation is one in which each sub-group is concentrated in a
different part of the income distribution. Such a situation can plausibly be described as one of
‘complete clustering’ of groups8. In both cases, there is complete segregation and thus maximal
representational inequality. However, in Figure 3, whether an individual is black or white
provides very little information on his or her rank in society. By contrast, in Figure 4, whether an
individual is black or white provides a great deal of information. One simple way to capture the
distinction between Figure 3 and Figure 4 is through the concept of sequence inequality, which
together with representational inequality captures the clustering of the income distribution. This
concept is linked to the position in the overall societal ranking possessed by individuals
belonging to distinct groups in the hierarchy.
An individual (weakly) rank-dominates another if that individual is ranked equal to or higher
than the other in the possession of the attribute. For any population partitioned into given identity
groups, there are a fixed number of between-group pair-wise comparisons between individuals
from different identity groups. The share of the total number of such between group pair-wise
8 Massey and Denton (1988) make reference to equivalent concepts

5
comparisons involving a given group in which a member of the group rank-dominates a member
of some other group is called its level of group rank dominance. Group rank dominance is an
indicator of the position the group occupies in the ordinal hierarchy of attribute levels. Another
way to understand the difference between Figure 3 and Figure 4 is simply that the average rank
of the whites and the blacks is different. This is clearly a necessary condition for distinct groups
to be clustered in different parts of the attribute space. We establish in Appendix Two that a
monotonic relationship exists between the concepts of group rank dominance and of average
rank. Both of these could be seen to be indicators of the placement of groups in the attribute
hierarchy (in the extreme complete clustering of groups) and will thus be referred to as indicators
of a group’s rank sequence position.
The level of inequality in different groups’ rank sequence position (whether as measured by
group rank dominance or by average rank) indicates the extent to which a population is clustered.
We refer to this concept of inequality as sequence inequality (SI). Some reflection will suffice to
show that this is an unambiguous criterion even when group sizes differ. In any situation
sequence inequality is minimal when the groups are evenly interspersed or symmetrically placed
around the median member(s).
It is clear from this discussion that while Figure 3 and Figure 4 depict two groups with equal
representational inequality, the two groups possess different levels of group rank dominance and
average rank. In Figure 4, whites have 100% of the available instances of rank domination and
higher average rank.
While sequence inequality and representational inequality are related, they are also distinct
concepts. A simple example which makes this distinction transparent is provided in Figures 5
and 6. In Figure 5, both groups possess the same level of group rank dominance and average
rank. The black group has two of the possible four instances of rank domination as does the
white group, and their average rank is the same. Thus there is no sequence inequality between
the groups. In the second, both groups again share equally in levels of group rank domination
(both have two of the potential four instances once again) and have the same average rank. The
situation once again is one in which there is no sequence inequality. However, in the first case
there is complete representational inequality and in the second case there is zero representational
inequality. In neither case is group membership always associated with higher rank, yet the cases
differ in the degree to which income levels are shared by members of distinct groups.
Group Inequality Comparison
Figure 4 depicts a situation of maximal representational inequality and maximal sequence
inequality. It could perhaps be thought of as a situation of polarization in the sense that each
group is concentrated at a given pole of the income distribution. However, this is true only in an
ordinal sense. Both the situations depicted in Figure 4 and in Figure 7 are identical from the
standpoints of representational inequality and sequence inequality since neither concept takes
note of cardinal information, which alone accounts for the difference between the two situations
described. To take account of cardinal information (for instance, concerning the distance
between distinct clusters), it is necessary to introduce an additional concept.

6
A common way to account for such information is to take note of the distance between the
means of distinct sub-populations, for example by using measures of inequality between group
means. This, indeed is the conception behind Group Inequality Comparison (II). However such
an approach ignores relevant information on within group inequality. Consider a two-group
society in which all members of each group originally respectively possess the mean incomes of
their groups. Suppose that both groups experience within-group transfers leading to intra-group
inequality. The extent of inequality in the society must be judged to have increased if the
measure of inequality employed obeys the Pigou-Dalton Transfer Principle (ensuring that such
transfers between persons are deemed to increase overall inequality). However, between-group
inequality (understood in terms of inequality between mean incomes of groups) is unchanged.
Between-group inequality must be deemed to have become relatively less substantial in
comparison with total interpersonal inequality.
An approach to inter-group inequalities which is based on between-group inequalities in isolation
rather than on the contribution of between-group inequality to overall inter-personal inequality
(i.e. Group Inequality Comparison (II)) will fail to contrast situations that might be
distinguished. Consider Figure 8 which depicts a two group society in which all members of
each group originally possess mean income A and B respectively. Both groups now experience
within-group transfers which increase inequality and their distributions are now depicted by
densities A’ and B’ respectively. Assume further that the transfers are such that the span
between the means is
Δ
and the span between the richest and poorest members of each group is
also
Δ.
We might plausibly consider inter-group differences to have become less significant after
the transfer since no member of the richer group is further away from some member of the
poorer group than before the transfer, and all but the very richest member of the richer group is
closer to some member of the poorer group.
On the other hand, Group Inequality Comparison (I) can have the disadvantage of ignoring
information relevant for understanding the extent to which inter-group differences generate
overall inequality. To see this, consider what would happen if in Figure 8, the original
populations A and B were made arbitrarily closer to each other while maintaining their
separation. According to Group Inequality Comparison (I), there would be no difference
between the two situations. If we employed instead the concept of Group Inequality Comparison
(I) the degree to which between group differences generate inequality will have fallen. It can be
seen that there are potentially good reasons to choose wither approach.
Group Inequality Comparison need not be measured, of course, in terms of differences in means
and could potentially be understood in other ways -- for instance in terms of differences in
medians, generalized means, or other measures of central tendency. Indeed, still other ways of
viewing group differences can be envisioned, for example involving comparison of higher
moments of the group-specific distributions of incomes, examination of the extent of ‘non-
overlap’ between distributions etc. For a wide-ranging discussion of methods of defining group
separation, see Anderson (2004, 2005). We limit our further discussions of the concept however
to the case where it is measuring mean differences, for expositional simplicity.

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Combining Concepts: Polarization
We have introduced above three concepts relating to inter-group inequalities: representational
inequality, sequence inequality and group inequality comparison. How are these concepts
related to polarization? Polarization is a concept which has been used in many different ways in
the literature, for example, to mean the absence of ‘middleness’ in a distribution (Wolfson,
1994), the distance between the average achievements of groups (Østby, 2008) and the presence
of distinct sizable groupings in the income distribution (Esteban and Ray, 1994). Many of these
approaches do not explicitly rely on the identification of individuals by identity groups
(understood as being distinct from attributes). A contrasting approach understands the level of
polarization of a distribution in terms of the extent of inter-group differences in the possession of
an attribute. If polarization is defined in this way, it becomes clear that each one of the concepts
of inter-group inequality defined above is itself a measure of polarization. However, taken
individually each may prove to be an unsatisfactory measure of polarization, because of the
information to which each is individually indifferent. Thus the relative ranking of the situations
depicted in Figure 3, Figure 4 and Figure 7 according to the extent of polarization depends on the
expansiveness of the approach used. In particular, all the figures depict maximal polarization as
judged according to RI, whereas Figures 4 and 7 depict maximal polarization according to both
RI and SI, and Figure 7 depicts a higher degree of polarization than does Figure 4 according to
GIC (taking the figures to possess the same income scale on the horizontal axis).
The fact that our judgments regarding the polarization of society may depend on more than one
concept suggests the value of combining measures of inter-group differences to construct
orderings of social situations according to the extent of their polarization. Such orderings can be
partial and based on dominance of the vectors (2-tuples or 3-tuples) defined by the individual
measures of inter-group differences, or can be complete if based on some method of aggregation
of these measures.
This said, orderings based on combining only a pair of the concepts we have defined (and not all
three) will be indifferent to some important considerations that may be deemed relevant in any
assessment of polarization. We have already seen that in the two group case, combining
representational inequality and sequence inequality will be sufficient to give us a measure of
ordinal polarization. Such a combination however will be indifferent to cardinality and will be
unable to distinguish, for example, between the situations depicted in Figure 4 and Figure 8
respectively.
A measure combining sequence inequality and group inequality comparison is not indifferent to
cardinal information on the achievements of individuals but it is indifferent to the degree of
clustering of identity groups in any specific income bracket. In order to see this, consider Figures
5 and 6 again. Let us assume that, by construction, the mean income of both blacks and whites is
the same in both groups in both situations. If this is the case, the index of group inequality
comparison is the same in both figures (i.e. zero) and sequence inequality is the same, but
representational inequality is different. We may argue that in Figure 5 there is no clustering of
identity groups in distinct parts of the income spectrum, as there is no representational inequality.
In Figure 6, however, blacks are clustered at the top and bottom ends of the income spectrum,
and indeed there is complete segregation between the two groups. Note further that we could

8
increase the distance between the blacks at the ends and the whites in the middle, keeping the
means of both groups the same (so that the blacks at each end are very distant from the whites at
the center) and yet record the same level of polarization defined according to such a measure.
Finally, combining representational inequality and group inequality comparison (I) alone leads to
an approach that is indifferent to the sequencing of individuals from distinct identity groups in
the income spectrum. Consider the distinction between Figure 9a and Figure 9b. Both depict
cases of complete segregation. However in Figure 9b, some population of blacks has been moved
to a higher income than all of the whites, thereby increasing within group inequality for the
blacks and total inter-personal inequality. We can further imagine that every white has been
given a higher income in such a way that within-group inequality among whites is unchanged
and the ratio of between-group inequality to total inequality (which would otherwise have fallen)
is restored to its level prior to the initial movement of blacks. In other words, the index of group
inequality comparison (I)remains the same by construction, as does representational inequality.
However, the sequencing of blacks and whites in the income distribution (and thus sequence
inequality) is different. An analogous argument can be made for group inequality comparison
(II) by moving the blacks and whites so as to keep mean incomes of the groups the same.
Any approach to polarization based on a pair of the group inequality concepts we have defined
will capture certain judgments about social situations and neglect others. Only by combining all
three concepts can an approach to polarization which takes account of the considerations
reflected in each of the concepts be constructed.
A variant of group inequality comparison (I) has been proposed as a stand-alone measure of
polarization (Kanbur and Zhang, 2001). However, such a measure, while attractive in its
simplicity can violate some intuitions. Consider Figure 10 in which two completely segregated
and clustered groups A and B experience within-group progressive transfers which reduce
within-group inequality. Further suppose that they also experience a reduction of between-group
inequality through progressive transfers between the members of the two groups in such a way
that the ratio of between-group inequality to overall inequality remain unchanged and the groups
( whose densities are now depicted by A’ and B’) overlap. If we utilize group inequality
comparison (I) alone as our measure of polarization, a social configuration with A and B is
viewed as being exactly as polarized as a situation with A’ and B’, which seems to conflict with
our intuitions. If we, however, combine it with some measure of sequence inequality and /or
representational inequality (both of which are lower when the groups overlap), the first situation
is unambiguously more polarized than the second.
It should be noted that the regressive transfers considered above led to a decrease in the index of
group inequality comparison (I) and therefore their impact was in the opposite direction from
that which would normally be expected of an inequality measure ( i.e. to obey the Pigou-Dalton
principle of responding to a regressive transfer with an increase in measured inequality). It
follows that any measure of polarization which increases when the index of group inequality
comparison (I) increases would similarly potentially violate the Pigou-Dalton principle9.
9 This view corresponds to the findings of Esteban and Ray (1994) among others that polarization and inequality are
distinct concepts and that measures of polarization need not therefore be expected to obey the Pigou-Dalton
principle.

9
Part II: From Concepts to Measures
Formalizing Concepts
Our purpose in this section is to formalize the concepts relating to group differences which we
have introduced above and develop measures of them10.
We begin by supposing a `social configuration’ ( ) in which there is a population, S0, of
individuals {i} of size N partitioned11 into K distinct identity groups (S1, S2....SK). The individuals
possess an attribute (let us say y), drawn from an attribute set, Y. The attributes are not
necessarily ordered. For example, the attribute may be a level of income (ordered and cardinally
measured), a quality of health (ordered but not cardinally measured) or a club to which a person
may belong (distinguished from one another, but not ordered). We employ a superscript to
distinguish the information associated with distinct social configurations. For simplicity, we
assume (although nothing depends on this other than notation) that the number of elements, l, in
the set Y, is finite.
More specifically, the individuals {i} each belong to a distinct identity group so
that
, for some J , with
and .
Our assumptions imply there are at least two identity groups which are each smaller than the
population as a whole and non-empty. Let the number of persons in group J be denoted by nJ.
The proportion of persons of a group J in the society is defined by
for .
Each individual i has attribute . The same attribute may be shared by more than one
individual.
10 These measures can be readily implemented using a Stata module that we have developed and which will be made
publicly available in due course. For a ready example of how these measures can be and have been applied to actual
data, see Reddy and Jayadev (2009).
11 We do not consider currently the case of societies in which individuals belong to more than one identity group
simultaneously and in which the identity groups do not form a partition of the society into mutually exclusive
categories. Generally, a partition of a society can be constructed on the basis of the Cartesian product of the identity
groups in the society. This solution may not be deemed appropriate, however, in every situation. For example, a
mixed race group in a society otherwise divided into two races may be deemed to belong to both of the races rather
than to neither, and it may be thought that this characterization is relevant to our judgments regarding inter-group
differences.

10
Define the membership function for group J by: , for .
Moreover define the complementary membership function for group J by .
In other words, the membership function specifies the number of persons in group J who possess
attribute y while the complementary membership function specifies the number of persons not in
group J who possess attribute y.
Representational Inequality
A simple way to capture the degree to which each identity group is disproportionately
represented among those who share a given attribute would be to describe the ratio of the number
of the persons possessing a given attribute who belong to each group, J, to their overall number
in society for any given attribute (y): . In other words, refers to the
proportion of persons who possess a given attribute who belong to group J. This information can
be captured in what we call the representational inequality (RI) Lorenz curve (Figure 11). As we
shall see, this framework allows for a simple way of presenting information concerning these
proportions and for analyzing this information using familiar tools12.
In order to construct the RI Lorenz Curve for each group, J, we first create a rank ordering, RJ,
such that
,
where reflects the ordering of the attributes according to the
proportion of the population in the attributes belonging to group J. The ordering starts from the
attribute for which the proportion of the population consisting of members of group J is the
lowest and proceeds to the attribute for which the proportion of the population consisting of
members of group is the highest.
Clearly, in the case in which the attribute can itself be ordered (e.g. income) the sequence in
which the appear in the ordering RJ will not necessarily be from lowest to highest.
Define:
and where and .
12 In spirit, this approach is similar to that adopted by Duncan and Duncan (1955) and later, inter alia, by Silber
(1989, 1991, 1992), and Hutchens (1991, 2004). Other references include Flückiger and Silber (1994). Boisso et
al., (1994), and Reardon and Firebaugh, (2002). Silber notes that various information structures (for example
involving the frequencies with which distinct groups possess an attribute such as membership in an occupation) can
be analyzed using ‘measures of dissimilarity’ which are analogous to measures of inequality. Our approach builds
upon this insight but differs from all of the authors above in explicitly going beyond the two-group case in a distinct
mannerand aggregating information derived from the concentration curves of different groups.

11
The RI Lorenz Curve for group J, , can be defined by the following rule, which creates a
piecewise linear curve:
When , for integer values then and, when is such that
, [ ] then
where .
In using this definition, we follow the procedure described by Shorrocks (1983), p.5.
This gives rise to a curve as in Figure 11 below. By construction, the RI Lorenz curve must, in
the familiar way, begin at (0,0) and end at (1,1), as well as slope upwards, with the slope
increasing as one moves to the right, since each addition to the total cumulative population of
others is associated with an addition of a larger proportion of group J. Note that the 45 degree
line here has the interpretation of being the line of equiproportionate representation (analogous to
the line of perfect equality in the case of an ordinary Lorenz curve). That is, all along this line,
the members of identity group J are represented at every attribute in the same proportion as they
are represented in the population as a whole.
Any deviation from the line of equiproportionality represents a situation in which members of
the group are disproportionately represented’ in the possession of certain attributes, leading them
to be ‘over-represented’ in the possession of certain attributes and ‘under-represented’ in the
possession of others. The RI-Lorenz curve therefore contains information on the extent of
segregation of a population in relation to the attributes possessed. Having defined it, we can
draw on the analogy between the RI-Lorenz curve and the ordinary Lorenz curve to suggest
further useful concepts.
Consider for instance what might correspond to the familiar idea of a progressive transfer. Just as
a progressive transfer in an income distribution involves a transfer from a person with higher
income to a person with lower income, in the context of representational inequality a progressive
transfer could be defined as a transfer of a person from the set of persons who possess an
attribute in which his or her identity group is represented more to one in which it is represented
less. However, since we are dealing with proportions of identity groups possessing different
attributes, a transfer of a single person will change the overall population that possesses each
attribute involved, affecting the ‘denominator’ used to assess population proportions for the
groups possessing these attributes. We overcome this problem and maintain an unchanged
denominator by instead employing the concept of a ‘balanced bilateral population transfer’13:
13 This concept of a balanced bilateral population transfer is related to that of a ‘disequalizing movement’ between
groups used by Hutchens (2004) in his discussion of a two group case. However, the latter concept is insufficient in
a multi-group case and necessitates the use of the alternative concept which we develop and employ. The concept is
also intimately related to the idea of a “marginal preserving swap" which has appeared in the statistical literature
(see, for example, Tchen, 1980, Schweizer and Wolff, 1981, and Bartolucci et al., 2001). However, to the best of our
knowledge, no one has shown that in the absence of perfect representational equality, there always exists the

12
Definition: Balanced Bilateral Population Transfers
Suppose and SP, SQ such that
and
with P≠Q and i ≠j.
Then a progressive (regressive) balanced bilateral population transfer is one in which population
mass (i.e. some number of persons; we abstract from integer problems here) of group P is
shifted from yi to yj and equal population mass of group Q is shifted from yj to yi, thereby
lowering (raising) and while raising (lowering) and .
A balanced bilateral progressive population transfer results in two upward shifts in the RI Lorenz
curves for the identity groups (and corresponding downward shifts for regressive transfers). An
example of the latter is provided in Figures 12a and 12b. The RI Lorenz curve that results from a
progressive (regressive) balanced population transfer dominates (is dominated by) the RI Lorenz
Curve that preceded the transfer.14 We note further that:
Lemma 1: There exists a pair of identity groups and a pair of attributes (yi,yj) for which a
progressive balanced bilateral population transfer can take place if all groups are not equi-
proportionately represented in the possession of every attribute.
Proof: See Appendix Three
An RI Lorenz curve weakly dominates an RI Lorenz curve if and only if
for all . An implication of this framework is that any Lorenz consistent
measure of inequality, for which inequality never decreases when is replaced by L(x), i.e.
all income inequality measures used in practice, can also be applied to measure representational
inequality. It is also well known in the literature on income distribution that it is possible to shift
from an income distribution that possesses a Lorenz curve to another that possesses the
Lorenz curve where if and only if there exists a corresponding sequence of
progressive transfers. Equivalently, in our case, it is possible to shift from a situation for which
each group possesses a Lorenz curve to another in which each group possesses a Lorenz
curve where if and only if there exists a corresponding sequence of
balanced bilateral progressive population transfers. For this reason a balanced bilateral
progressive population transfer can be deemed to decrease overall representational inequality.
possibility of achieving a balanced bilateral transfer (as we do in Appendix Two) and that this can be given a natural
interpretation in terms of Lorenz curves.
14 For the relevant reasoning, see Shorrocks (1983).

13
The consequence is a striking parallel between inequality measures in the income space and
inequality measures in the representation space. Table (I) provides a map of the isomorphism
between corresponding concepts introduced so far.15
15 The concepts of the generalized Lorenz curve and dominance of generalized Lorenz curves do not possess
straightforward and interpretatively useful analogs in the area of representational inequality since the concept of an
income mean does not possess a straightforward analog in this realm.

14
Table (I)
Correspondences between Conventional Inequality and Representational Inequality Concepts
Conventional Inequality Concept
Representational Inequality Concept
Inequality
Over or Under Representation
Pigou-Dalton Transfers
Balanced Bilateral Population Transfers
(First order) Lorenz Dominance
(First order) RI Lorenz Dominance
Suppose that we apply Lorenz-consistent inequality measure to assess representational
inequality for group J and denote the resulting vector of measured inequality for all groups in the
society by and its individual components by , . Then, an overall
measure of representational inequality in the society is given by
where . One simple version of such an
aggregation function, f, is the mean of the group-specific representational inequality measures. It
may seem attractive for a measure of overall representational inequality to take into account
subgroup sizes and respond to unequally sized groups differently. Indeed, it will be argued below
that there can be sound reason for such weighting. We may define a population weighted overall
representational inequality measure of the form:
where refers to the population weight of subgroup J.
Such a measure can be offered some justification through axiomatic underpinnings which we
consider in the next section.
Any empirical application of the concept of segregation, and thus of representational inequality,
requires by its very nature the partitioning of the attribute space in some way. Representational
inequality concerns the extent to which particular attributes (whether income levels, occupations,
or locations of residence) are shared by members of different groups. It is evident that this
determination will depend on how these attributes are defined. For example, in an analysis of
residential racial segregation in a city, defining the neighborhood of residence in the broadest
way (to encompass the entire city) will lead to the conclusion that there is no racial segregation
at all, since all races are represented in the same way that they are represented in the city as a
whole. At the opposite extreme, defining the neighborhood of residence to be the individual
household may lead to the conclusion that there is almost complete racial segregation if
individuals in households are overwhelmingly from a single race. The appropriate way to define
the neighborhood will lie between these extremes and will depend on the form in which data are

15
available as well as the interests and purposes of the researcher. The fact that judgments as to
the appropriate ‘bin size’ are needed in empirical work is not therefore a detriment, and rather is
intrinsic to the exercise.16
Sequence Inequality
As noted in the discussion of the previous section, representational inequality is a measure of
group differences which is indifferent to the ordering of attributes as well as to their cardinal
properties. To operationalize our concept of sequence inequality therefore, we now assume that
the attributes can be ordered17.
Considering first the concept of group rank dominance, we define a pair-wise individual rank
domination function, , for a given pair of individuals and as follows:
. We can now define the group rank domination quotient for
group J as follows: . It can be seen that possesses the interpretation of the
proportion of possible instances of pair-wise domination involving members of group J and
members of other groups in which such domination actually occurs. It is evident that this
quotient varies between a minimum of 0 and a maximum of 1 for any group. The size of the
group plays no direct role in determining the value of the group rank domination quotient. Rather
it is the placement of members of the group relative to members of other groups that determines
the quotient. Sequence inequality could be treated simply as the measured inequality in
across groups. It is common for individuals from a given group to express pride or shame at the
achievements or failures of other members from that group. Such a psychological interpretation
can provide justification for treating the group rank domination quotient as defining the
experience of each individual in that group and measure inequality across all individuals in
possession of that experience18.
A seemingly puzzling asymmetry is implied by our approach to sequence inequality. Consider
two populations, consisting of one white individual and ten black individuals each. In the first
population, the individuals are ordered in the income space from lowest to highest as (w, b, b,
b,....b), and in the second, the individuals are ordered from lowest to highest as (b, b, b, b.....w).
In the first instance, all 10 black members possess a domination quotient of 1, while the white
individual possesses a domination quotient of zero. The inequality in domination quotient is
16 It is possible to conceive of several different ways of determining bin size. For instance, bin size may be fixed in
absolute terms (e.g. in terms of some number of dollars or years lived), fixed in proportional terms (e.g. as one
percent of the highest possible income) or fixed in relation to the size distribution of the attributes (e.g. at the income
thresholds corresponding to successive deciles of the population).
17 There is a small nascent literature on the measurement of ordinal inequality. Some key references include Allison
and Foster (2004), Reardon (2008) and Abul Naga and Yalcin (2009)
18 One way to interpret sequence inequalities is in terms of an analogy to a society wherein each group practices
radical egalitarianism. In such a society, an even distribution of each group’s share of the social assets, in this case
instances of rank domination, results among the individuals belonging to the group.

16
therefore inequality in a population having scores (0, 1,1,1,1....1). In the second case all 10 black
members possess a domination quotient of 0, while the white individual possesses a domination
quotient of 1. The inequality in domination quotients is therefore the inequality in a population
having scores (0,0,0,0,0....1). It is clear that more sequence inequality will be recorded in the first
case than in the second, even though all that has been done is to change the placement of the
white from being at the bottom to being at the top of the income spectrum. While this may
initially appear puzzling, it is perhaps appropriate to treat these cases asymmetrically. By the
psychological interpretation, in the first instance most people in society do not experience a
relative deprivation. By contrast, in the second, most do.
As we noted above, the average rank of a group (call it , ) is also an indicator of
group rank sequence position. In fact, it is linked in a direct and monotonic fashion to group rank
dominance. It is easily shown that the relation between them, for a perfectly segregated
population is :
In the case of populations which are not perfectly segregated, appropriate changes to the
definition of a rank maintains this relationship (see Appendix Two):
The inequality in group rank sequence position across groups can be assessed either in terms of
the inequality of group rank dominance quotients or that of average group ranks.19 In either case,
if a member of a group (the ‘beneficiary’) exchanges his or her attribute with another person in a
different group who has a higher level of the attribute, then the indicator of group rank sequence
position is increased for the group to which the beneficiary belongs and is decreased for the other
group. We assume henceforth in this section that we are specializing to the case of group rank
dominance quotients, although the concepts we present can equally be applied to average ranks.
The group rank dominance quotients achieved by members of distinct groups can be captured by
what we call the group rank dominance (GRD) Lorenz curve. The GRD Lorenz curve relates the
cumulative proportion of the total of the group rank domination quotients to the cumulative
population of groups, when the identity groups are ordered from lowest group rank domination
quotient to highest. It captures the degree of inequality in group rank domination quotients. Any
symmetric arrangement of identity groups in the attribute space (i.e. one in which for any
instance in which a member of a given group rank dominates a member of another group, a
distinct pair can be found in which the opposite is true) is one of perfect equality in group rank
domination quotients, and will give rise to a GRD Lorenz curve which is on the forty five degree
line.
19 For a given group, although the ordinal ranking of social configurations according to the group’s rank sequence
position does not depend on the choice between these indicators (or indeed any other monotonic transformation
thereof) the cardinal level of the indicator does depend on it. As a result, the choice of indicator can be
consequential for determining the measured sequence inequality

17
We can now define coordinates of the GRD Lorenz curve associated with each group added as
follows:
and where and .
We can now define the GRD Lorenz curve as a whole, , as follows:
When , for integer values then and, when is such that
,, then
where .
An example of such a curve is shown in Figure 13. Since the Lorenz curve is defined for
sequence inequality analogously to income inequality, with income corresponding to the group
rank domination quotient of the groups to which individuals belong, the properties of the GRD
Lorenz curve are analogous to those of the ordinary Lorenz curves. Once again, therefore, any
Lorenz consistent measure of inequality will suffice to capture the level of sequence inequality.
Group Inequality Comparison
Group Inequality Comparison (I) refers to the degree to which between-group inequalities
contribute to overall inequality. Typically, measures which are “additively separable” (such as
members of the generalized entropy class) have been utilized for this purpose (see e.g.
Shorrocks, 1980, Foster and Shneyerov (1999) and Zhang and Kanbur (2001)), although such a
restriction is not required. In particular, if the between-group inequality is defined as the
inequality that arises when every member of the population is assigned a representative level of
attribute (mean, generalized mean, median or other measure of central tendency) of the group to
which they belong, then the ratio of between-group inequality to total interpersonal inequality
can serve as an index of Group Inequality Comparison (I). This measure has the advantage of
always lying between zero and one and responding in an appropriate way to intra-group
transfers. More generally, any indicator that the distributions associated with different groups
are different can potentially serve as a measure of Group Inequality Comparison.
Polarization
Polarization as we have defined it above aggregates the three concepts concerning group
differences which we have defined. The range of polarization measures which could be used is
very wide indeed since any such measure could involve any form of aggregation of a three-tuple
(RI, SI, GIC), and in turn each element of this three-tuple could be defined in various ways.
Further, any measure of polarization which is positively responsive to all three will only be
maximized in a situation where all three are maximized.

18
An empirical examination which involves these four concepts can, as we have noted, be achieved
through the use of almost any commonly used measure of inequality. The choice of measure will
naturally bring in additional implications and properties. Given this flexibility, an analyst can
choose which measure to utilize in order to satisfy the additional properties he or she thinks
important. Thus, for example a researcher who wishes to treat sequence inequality as being
decreased more in a situation where an exchange of ranks happens between members of different
groups, each of whom has lower ranks to begin with, can choose an inequality measure which
shows the required form of transfer sensitivity (e.g. a generalized entropy index with
appropriately chosen parameters). Whether the measure of polarization can be normalized in a
specific way will also depend on the choice of the underlying measures of inequality.
Part III: Axiomatic Framework
We define below some requirements that may reasonably be imposed on measures of each of the
concepts defined above, considering each of them in turn. We also identify some classes of
measures which satisfy these requirements.
Axioms (Representational Inequality):
We begin by suggesting some requirements which may be imposed on an overall
representational inequality measure RI when it is viewed as a function of the information in a
social configuration
ζ.
We write RI=RI(
ζ
) to reflect this dependence.
Axiom (RI1): Lorenz Consistency
Let (
ζ1, ζ2)
refer to two different social configurations and (I,J) refer to two different
identity groups. If (
ζ1, ζ2)
are such that , , and
then RI(
ζ1)
RI(
ζ2).
In other words, all else remaining equal, a social configuration which is at least as segregated
according to the criterion of Lorenz dominance of representational inequality Lorenz curves is
one which is at least as representational unequal. It may be noted that just as there is an
equivalence between Lorenz consistency of an inequality measure and that measure’s respect for
the Pigou-Dalton transfer principle, there is an equivalence between Lorenz consistency of a
representational inequality measure as defined here and the requirement that the representational
inequality measure respond to a progressive balanced bilateral transfer by registering a decrease.
Axiom (RI2): Within Group Anonymity
If represents the attribute of person i belonging to group J
(and

19
if (
ζ1, ζ2)
are such that where is a permutation operator applied
to then RI(
ζ1)
=RI(
ζ2).
In other words, a measure of overall representational inequality is invariant to permutations of
the attributes assigned to individuals within an identity group.
Axiom (RI3): Group Identity Anonymity
If represents the attribute of person i belonging to group J
(and if (
ζ1, ζ2)
are such that and , ,where is a
permutation operator applied to then RI(
ζ1)
=RI(
ζ2).
In other words, a measure of overall representational inequality is invariant to permutations of
the group identities with which distinct sets of individual attributes are associated. This axiom
incorporates the idea that all of the information relevant to assessing representational inequality
is taken into account by noting the partition of the society into groups and the attributes of the
members of these groups. The axiom embodies the idea that there is no need to take independent
account of any other features of groups. This approach disallows the incorporation of judgments
that group identities are additionally relevant (e.g. by reason of past histories or present injustices
not already reflected in the information described by the social configuration)20.
Axiom (RI4): Minimal Representational Inequality
Let be the RI Lorenz curve corresponding to even representation (i.e. the line of
equiproportionate representation). If then RI = 0.
In other words, minimal overall representational inequality is achieved when all identity groups
are represented in the same proportion as their share of the population for all attributes, and has
measure zero.
Axiom (RI5): Maximal Representational Inequality
The maximum level of Representational Inequality is 1.
This is a normalization axiom which may be imposed for interpretative convenience. It may be
dispensed with if it is desired to employ an unbounded inequality measure (such as a measure of
the additively decomposable generalized entropy class).
Axiom (RI6): Positive Population Share Responsiveness of Overall Representational Inequality
20 See Loury, (2004) for an extensive discussion on the merits of the anonymity axiom as applied to groups.

20
Suppose that a measure of overall representational inequality is a function of the vector of
measures of representational inequality of groups, . Suppose further that the population share
for group J is increased and that for group H is decreased, and the set of measures of
representational inequality of groups remains unchanged as do the population shares for any
remaining groups. Suppose further that , i.e. that the group-specific representational
inequality of group J is greater than that of group H. Then the measure of overall
representational inequality must increase.
This axiom can be motivated in different ways. We might for example believe that a group
which is very small in the population but which is highly unequally represented simply because
it is a small group in a society where there is unequal representation should not affect overall
representational inequality in the same manner as a group which is much larger.
We may note that the measure of overall representational inequality defined above,
, satisfies these axioms as long as the measure used to assess
representational inequality for each group, , is Lorenz consistent, which will be the case if it
has the form of any standard inequality measure, for example the Gini coefficient.
From another perspective, it may not be appropriate disproportionately to disvalue the unequal
representation of smaller groups. If one is interested in the experience of groups as opposed to
the experience of individuals within groups, it should make no difference whether the group is
small or large. Following this intuition, there is no reason to promote a population weighted
overall measure and one should instead adopt a measure which weights every group equally.
This alternative may seem especially compelling if one views polarization as an attribute of the
society as opposed to the individuals who belong to it. Such a measure can satisfy all the other
axioms.
Axioms (Sequence Inequality):
In what follows we shall use to refer to the indicator of group rank sequence position (which
may be either the group rank domination quotient or the average rank) of group J. Let SI refer to
the measure of overall sequence inequality. Some reasonable axioms are as follows:
Axiom (SI1): Lorenz Consistency
Let (
ζ1, ζ2)
refer to two different social configurations and (I,J) refer to two different
identity groups. Further, let refers to the Lorenz curve describing inequality across
groups in the indicator of group rank sequence position, . If (
ζ1, ζ2)
are such that
, then SI(
ζ1)
SI(
ζ2)
.
Axiom (SI2): Within Group Anonymity

21
If represents the attribute of person i belonging to group J
(and
if (
ζ1, ζ2)
are such that where is a permutation operator applied
to then SI(
ζ1)
=SI(
ζ2).
Axiom (SI3): Group Identity Anonymity
If represents the attribute of person i belonging to group J
(and if (
ζ1, ζ2)
are such that and , ,where is a
permutation operator applied to then SI(
ζ1)
=SI(
ζ2).
Axiom (SI4): Sequence Inequality Limits
Let be the Lorenz curve (describing inequality in the indicator of group rank sequence
position, ) that corresponds to even group rank sequence position (i.e. the case in
which is the same for all groups). If then SI = 0.
Axiom (SI5): Maximal Sequence Inequality
The maximum level of sequence inequality is 1. As with Axiom RI5 above, this is a
normalization axiom which may be imposed for interpretative convenience. It may be dispensed
with if it is desired to employ an unbounded inequality measure (such as a measure of the
additively decomposable generalized entropy class).
Axioms (Group Inequality Comparison):
Some reasonable axioms may be as follows, assuming that members of each group, , are
assigned a representative income, , and also possesses an individual income, .
Axiom (GIC1): Between Group Synthetic Population Lorenz Consistency
Let (
ζ1, ζ2)
refer to two different social configurations. Assume that a synthetic
population is constituted in which every member of a group, , is assigned the same
representative income for its group, . Consider the Lorenz curve, , for the
resulting synthetic population in each social configuration. If (
ζ1, ζ2)
are such that

22
and (i.e. the overall Lorenz curves for the actual population remain unchanged)
then GIC(
ζ1)
GIC(
ζ2)
.
This axiom states that between-group regressive transfers which do not change the overall
interpersonal distribution must have an appropriate directional effect (non-decreasing) on the
measure of GIC. Thus, for example, an exchange of incomes between individuals of different
incomes belonging to two different groups that results in an increase in inequality in the
synthetic population must increase the measure of GIC.
Axiom (GIC2): Within Group Anonymity
If represents the attribute of person i belonging to group J
(and
if (
ζ1, ζ2)
are such that where is a permutation operator applied
to then GIC(
ζ1)
=GIC(
ζ2).
Axiom (GIC3): Group Identity Anonymity
If represents the attribute of person i belonging to group J
(and if (
ζ1, ζ2)
are such that and , , where is
a permutation operator applied to then GIC(
ζ1)
=GIC(
ζ2).
Axiom (GIC4): Within Group Lorenz Consistency
Let (
ζ1, ζ2)
refer to two different social configurations. Further, let and refer to the
Lorenz curves describing inequality within each group, , in the respective social
configurations. If (
ζ1, ζ2)
are such that , but then GIC((
ζ1)
GIC((
ζ2)
.
This axiom states that within-group (weakly) regressive transfers of income must have an
appropriate directional effect (non-increasing) on the measure of GIC, holding the representative
incomes of groups constant. Clearly, since Group Inequality Comparison (II) does not rely on
any information about within group inequality, imposing this axiom will exclude its use.
It may be readily checked that a measure of GIC of the form B/T, where B represents the
inequality measure for the synthetic population in which each member of the society is assigned
the representative income of its group and T represents the total interpersonal inequality of the
society, satisfies all of the axioms above. Such a measure would capture the concept of Group
Inequality Comparison (I). In contrast, employing B alone as the measure of GIC would capture
the concept of Group Inequality Comparison (II). Such a measure would satisfy Axioms (GIC1)-
(GIC3) alone.

23
Axioms (Polarization):
We have proposed above to define polarization as a function of the other concepts of group
difference we have defined. In this section we assume that the attribute of concern is cardinally
measurable so that all three concepts have a role to play. Without loss of generality, we shall
assume that the attribute is income. However, it would be sufficient to assume that the attribute
was ordinally measurable in which case polarization can be conceived as a function of RI and SI
alone, with appropriate amendments to the axioms presented below. Adopting this approach to
the construction of a polarization measure, we specify a few possible desiderata for such a
measure, as follows. These may be selectively drawn upon and combined as desired.
Axiom (P1): Arguments of the Polarization Function
Polarization is a function of RI, SI, and GIC and of these arguments alone.
Axiom (P2): Zero Product Rule
If
ζ
is such that RI=0 or SI=0 or =0 then P(
ζ)
=0
In other words, if there is no representational inequality in society or there is no sequence
inequality or the index of group inequality comparison is zero then there is zero polarization. The
rationale for this axiom is discussed more extensively below.
Axiom (P3): Conjoint Effects on Polarization
If (
ζ)
is such that RI=1 and SI=1 then P(
ζ )
=GIC
If (
ζ)
is such that SI=1 and GIC=1 then P(
ζ )
=RI
If (
ζ)
is such that RI=1 and GIC=1 then P(
ζ )
=SI
In other words, if any two of the measures have reached their maximum values then the degree
of polarization of the society is determined by the third measure and is equal to it. This axiom
may be applied if the underlying inequality measures used to calculate RI and SI are respectively
bounded but not otherwise.
Axiom (P4): Maximal Polarization is one and Minimum Polarization is zero.
This is a normalization axiom imposed for interpretative convenience. It may be applied if the
underlying inequality measures used to calculate RI and SI are respectively bounded but not
otherwise.
Axiom (P5): Positive Responsiveness of Polarization to its Arguments
If (.
ζ1, ζ2)
is such that (RI1, SI1, GIC1) >(RI2, SI2, GIC2 ) then P(
ζ1)
>P(
ζ2)
In other words, if any argument of the polarization function is increased without changing other
arguments of the polarization function, then polarization increases.

24
Given these axioms, we can specify an appealingly simple example of a polarization measure
which satisfies Axioms (P1)-(P5), where the inequality measure used to calculate RI and SI is
any which is bounded and normalized to vary between zero and one:
This is not the only measure that satisfies the axioms above although it is an attractive one.
However, it can be shown that it is the unique measure which satisfies the axioms above and
additional ones. For example, it is the only measure which satisfies axioms P1, P3, P4 and P5
and is of the CES functional form (as shown in Appendix Four). This highly specific result is
surprising since the CES functional form is very flexible indeed, and accommodates a wide
range of judgments concerning the tradeoffs that are appropriate to make between the different
components of a measure. The requirement that RI and SI are bounded and normalized to vary
between zero and one excludes certain inequality measures, such as the additively decomposable
members of the generalized entropy class.
It is interesting to note that the simple polarization measure above also satisfies additional
axioms which may be imposed, such as
Axiom (P6): Conjoint Responsiveness of Polarization to its Arguments
If the polarization measure is twice differentiable, then:
for such that .
In other words, the impact of an increase in any argument of the polarization measure is greater
when any other argument of the polarization function is higher (if remaining arguments are
unchanged). The merits of such an axiom are open to discussion.
In the case in which the attribute of interest is cardinally measurable, the following axiom is also
satisfied by the polarization measure (and indeed, mutatis mutandis, by RI, SI and GIC
individually):
Axiom (P7): Scale Invariance
If two social configurations (
ζ1, ζ2)
are such that and the
social configurations are otherwise identical then P(
ζ1)
=P(
ζ2).
In the case where the attribute is only ordinally measurable, P may be defined simply as
and can be supported by a similar set of axioms.
We have required above that polarization takes a value of zero when any one of its arguments is
zero. If representational inequality is zero, this means that at every attribute level, individuals

25
from all groups are represented in exactly the same proportion as they are represented in the
population as a whole. As such it is difficult to characterize such a configuration as possessing
any polarization. Equally, when group rank sequence position is zero, this means that the
distribution of members is perfectly symmetric (i.e. every instance of domination by a member
of a group vis-à-vis a member of another group is balanced by a reciprocal instance of
domination by a member of the other group vis-à-vis a member of the first group. Once again,
while there may be inequality in the society, it would be hard to characterize this situation as one
in which there is any polarization between groups. Finally in the accustomed approach to
assessing group inequality comparison, if the means of all groups coincide then this contribution
is zero, and polarization is equal to zero.
When is polarization maximized? RI is maximized when there is complete segregation of groups
in the achievement space. Assuming this condition is satisfied, what additional conditions are
required to maximize SI? It is easily seen that the maximum value of SI is approached in the
limit in which the social configuration is such that one group is as large as possible relative to all
others and is disadvantaged relative to all others (specifically in the sense that every member of
the disadvantaged group possesses lower achievements than every member of every other
group). Assuming this condition too is satisfied, what additional conditions are required to
maximize GIC? If the concept of GIC employed is of Type I then the within-group inequalities
in the population must be as small as possible. This will be attained when there are no intra-
group differences in achievements. Any distribution which satisfies these properties will suffice
to approach the maximum level of polarization (for example, that shown in Figure 14a, which
depicts three groups with differing population sizes and achievement levels).
If the concept of GIC employed is of Type II then the maximum value of GIC is approached in
the limit in which the relatively disadvantaged group possesses the lowest possible level of the
achievement and the members of other groups possess levels of the achievement which are as
high as possible compatible with complete segregation. Thus, if there are two groups one of
them must possess the lowest level of the achievement (e.g. zero income) and the other must
possess the highest level of the achievement (e.g. all of the income, evenly divided among its
members). If there are more than two groups, all groups other than the most disadvantaged must
possess distinct achievement levels which are as close as possible to the attainable maximum.
Any distribution which satisfies these properties will suffice to approach the maximum possible
level of polarization (for example, that shown in Figure 14b, which depicts three groups with
differing population sizes and achievement levels in a feasible range).
The conclusion as to the circumstance in which polarization is maximized differs from that
reached by others. As opposed to Duclos, Esteban and Ray (2004), for example, in which
polarization is maximized only when there are two equal sized groups, in our analysis it can be
maximized regardless of the number of groups and it is necessary that the most disadvantaged
group is as large as possible relative to the others. For polarization to approach its maximum it is
required that the poorer group is large relative to the richer groups, there is complete segregation
and there is no within-group inequality.
Part IV: Conclusion

26
This paper has sought to clarify how one may assess social situations according to the extent to
which attributes are disproportionately possessed by different social groups. The measures we
have developed capture the various different ways in which experiences of members of distinct
groups may differ. Thus, social situations can differ in the extent to which members of a group
share experiences with members of other groups (representational inequality), experience the
same or different relative positions (sequence inequality) and experience differences in the extent
to which interpersonal inequalities are accounted for by inter-group differences (group inequality
comparison). These concepts are distinct but complexly interrelated. They each integrate
empirical observations and evaluative judgments. Judgments concerning the relative importance
to be attached to different aspects of inter-group difference are also involved when they are
combined (for example to form a measure of polarization).21 These measures have an intuitive
appeal and can have widespread application in social science.
There appear to deep-seated tendencies for societies to exhibit segregation, clustering and
polarization of identity groups. This observation has important empirical implications and may
also give rise to normative concern. For either reason, the concepts and measures we have
discussed may prove useful.
21 The concepts we have discussed can be understood as “thick ethical concepts”, on which see Putnam (2004).

27
Acknowledgments
We would like to thank a range of persons and institutions for their invaluable assistance in
preparing this paper.22
22 We would like to thank for their useful comments or suggestions (without implicating them in errors and
imperfections) James Boyce, Rahul Lahoti, Hwok-Aun Lee, Glenn Loury, Yona Rubinstein, Peter Skott, Rajiv
Sethi, Joseph Stiglitz, S. Subramanian, Roberto Veneziani and other participants at seminars in the Dept. of
Economics at Brown University, the University of Massachusetts at Amherst, the Jerome Levy Institute at Bard
College, Queen Mary, University of London and the Brooks World Poverty Institute at the University of
Manchester.

28
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31

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Figures

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34

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40
APPENDICES
Appendix One: Moral Reasoning and Inter-Group Differences

41
Should inter-group differences be disvalued? We have argued above that there are both intrinsic
and instrumental reasons that might be appealed to in answering this question. The instrumental
reasons need not depend on any direct concern with the relative well-being of groups since they
can be purely derivative of a concern for individual advantage (insofar as such advantages are
causally affected, through various mechanisms, by their distribution across the different groups
in society to which individuals belong). The moral justification for such concern is consequently
dependent only on there being moral reason to favor a particular distribution of individual
advantages with regard to which inter-group differences play a causal role.
In contrast, the perspective in which inter-group differences are intrinsically disvalued is more
challenging to justify. This perspective appears to contrast with that advocated in standard
"individualist" approaches to social assessment, which are embodied in social choice theory and
in moral philosophy. For example, the "anonymity axiom" in social choice theory is typically
interpreted to demand that permutations of individual advantages across persons, whatever
groups they may each belong to, should not have any effect on assessments of the desirability of
social states of affairs. Such an approach appears to rule out any direct concern for inter-group
differences (on which, see for example Loury, 2004).
A thought experiment may be helpful in investigating the appropriate role that group related
considerations should play in social assessment. We may imagine an "original position" of the
general type described by Rawls (1971) or other theorists in the modern contractarian tradition of
moral philosophy. Imagine that the parties to this original position (perhaps trustees choosing
alternative social arrangements on behalf of specific individuals who will ultimately inhabit
them, along the lines defined by Rawls (2001)) engage in bargaining with each other about future
social arrangements in one of two different situations.
In the first situation, the trustees know only that the individual who they represent will eventually
occupy a particular position in a well-defined distribution of individual advantages, where that
distribution depends on the particular choice of social arrangements made by the trustees and
where a particular position in society is assigned to each individual through a "lottery". In the
second situation, the trustees know that the individual that they represent will be assigned to a
particular rank in a specific social group in a "first round lottery", and that each social group will
be assigned to occupy a particular set of positions in a social distribution of advantages in a
"second round lottery"23.
Suppose that the expected final distribution of individual advantages (disregarding group
membership) is expected to be identical in the two situations, given a specific choice of social
arrangements by the trustees. Suppose further that the second situation is such that as a result of
the second round lottery systematic differences in the possession of individual advantages
between members of distinct groups are expected to result. No such systematic differences
would normally be expected to result from the lottery implemented in the first situation. By
construction, the outcomes resulting from the social arrangements put in place in the first
situation will be ones in which there are no systematic inter-group differences of the kind that are
present in the outcomes resulting from the second situation, although the distribution of
23 We abstract from the issue of relative sizes of groups here.

42
individual advantages is identical. Should any moral distinction be made between these two
cases?
It seems clear that the parties to the original position would have no reason to prefer one
situation over the other if they are concerned only about the individual advantages that are
distributed, since the individuals who they represent have an identical chance in either case of
occupying any given position. If there is reason to prefer the first situation over the second it
must have to do either with the concern that systematic inter-group differences are likely over
time to have causal effects which merit moral attention which have not otherwise been
considered (for example by leading to internalized stigma or other psychological effects on
individuals, or the entrenchment over generations of social inequalities between groups) or with
a view that group membership is a morally salient feature to be taken account of in the
assessment of the distribution of individual advantages quite apart from its effect on individual
lives. An attempt to justify the latter perspective might employ a conception in which such non-
individualistic considerations play a role (on which see e.g. Parfit, 1997, on “telic” and “deontic”
approaches to the assessment of inequality).
Of course, the device of the original position may only have partial relevance to the moral
assessment of actual empirical cases. In practice, distributions of individual advantage derive
from historical processes, some of which may have involved systematic and perhaps grievous
harm done by members of a given group toward members of another group. In such a case,
considerations of historical reparation and other moral concerns related to the rectification of
historical injustice may play a role in the justification of moral concern for inter-group
differences.

43
Appendix Two (Rank Domination Quotient and Average Rank)
As we noted above, the average rank of a group (call it , ) is also an indicator of
group rank sequence position. In fact, it is linked in a direct and monotonic fashion to group rank
dominance. As before, we understand rank as referring to the position in which an individual
appears when incomes are sequenced from lowest to highest (the ascending order of values).
When individuals from the same group share an income we shall assign them a rank equal to the
average position in which an individual appears when incomes are sequenced from lowest to
highest. We shall consider subsequently the rule to be applied in assigning ranks when
individuals from different groups share an income.
Consider at the outset, for simplicity, a perfectly segregated population in which there is no more
than one individual in each income bracket. In such a population, the total number of instances of
pair-wise rank domination that members of group J enjoy vis-à-vis others can be understood as a
function of the ranks of members of group J in the population. The lowest ranked member of
group J, having rank r1 dominates (r1-1) persons belonging to other groups. The second lowest
ranked member of group J, having rank r2 dominates (r2-2) persons belonging to other groups
(i.e. (r2-1) persons belonging to all groups -1 person belonging to the same group)). Extending
this logic, the total number of instances of pair-wise domination by members of group J is:
The rank domination quotient correspondingly is
from which it follows that:
.
It is easy to see that this formula also applies in the case in which there may be more than one
person in an income bracket but all persons who share an income bracket are always from the
same group. In contrast, in the most general case of populations in which there may exist some
income brackets which contain members of distinct groups there can be ties in the income ranks
assigned to members of different groups, which will imply that this formula will no longer hold
exactly unless the ranks are assigned appropriately to individuals in the same income bracket.
Specifically, if strict domination is the concept that is employed then this relationship will hold
exactly if individuals in the same income bracket are assigned a rank equal to the lowest of their
positions in the ascending order of values. Correspondingly, if weak domination is the concept
that is employed then this relationship will hold exactly if each individual in the income bracket
is assigned a value equal to the sum of the lowest of the positions of the individuals sharing the

44
income bracket (in the ascending order of values) and the number of individuals from other
groups with whom they share the income bracket.
The correspondence we have derived between and holds also in the case of continuous
distributions, as can be shown through limit properties. In this case, the average rank of members
of a group, J, is defined by
and the rank domination quotient for the group is defined by:
Where is the cumulative distribution function for incomes of the entire population, is
the density function for incomes of members of the group, J, and the integrals are calculated over
the domain of all possible incomes.

45
Appendix Three (Proof of Lemma 1)
Without loss of generality, we shall assume that the attributes can be understood as income
levels. Let A refer to a matrix of size K by n with K identity groups and n income levels. Each
element in the matrix . We say that the ith identity group is ‘under-
represented’ at the jth income level if the proportion of persons from group i at income j is less
than the proportion of persons of group i in the population as a whole. We denote the statement
that the ith identity group is ‘under-represented’ at the jth income level by . We say further
that that the ith identity group is ‘over-represented’ if the proportion of persons from group i at
income j is greater than the proportion of persons of group i in the population as a whole. We
denote the statement that the ith identity group is ‘over-represented’ at the jth income level by
. If the ith identity group is represented at the jth income level in the same proportion as it
is represented in the population as a whole then we say that it is “equiproportionally represented”
and we denote this by .
Thus A is a matrix in which every element is 0,1 or 2. We may further note that if any row or
any column contains a zero then it must contain a one and vice versa. This requirement captures
the necessity that if an identity group is overrepresented at an income level, it must be
underrepresented at another income level and that if a group is overrepresented at an income
level then another group is under-represented at that same income level.
A balanced bilateral transfer is always possible if an identity group is represented to a greater
extent at one income level (call it y1) than it is at another (call it y2) and another identity group is
represented to a lesser extent at y1 than it is at y2. This condition is satisfied as long as it is
possible to identify two rows (i and j) and two columns (l and m) of the matrix A such that they
form a matrix A~=which is of one of the following forms, or which can be constructed
from one of the following forms by permuting either their rows or their columns:
, , , , .
The lemma is therefore equivalent to the statement that there exists a matrix A~ for any matrix A
which contains at least a single one or zero. Suppose that the lemma is false. Then, it is possible
to construct an A such that there is no A~ associated with it.
We now try to construct such a matrix A. Without loss of generality consider the case in which A
contains at least one zero (i.e. an identity group is under-represented at a particular level of
income). We can present this as occurring at the top left corner (a11)of the matrix, without loss of
generality, as below:

46
A=
This however means that there must be at least one level of income in the first column and in the
first row in which there is over-representation of an identity group. Without loss of generality, let
us say that this occurs at a12 and a21 respectively so that
A=
Now if a22 = 0 or 2 then A~ exists. If a22 0 or 2 then a22 = 1. That is
A=
Consider row 2 and column 2 now. Since for the already fixed elements, there is over-
representation, there must be elements in row 2 and in column 2 respectively that have value
zero (reflecting under-representation). Without loss of generality, let these occur at a23 and a32
respectively so that
A= =
But this in turn fixes a13, a31 and a33 to be 0 since if any of these are 1 or 2, we can construct
matrix A~ . This in turn implies that there exist elements elsewhere in row 3 and column 3 with
value 1 (indicating over-representation), which we can place without loss of generality at a34 and
a43, respectively. It can readily be seen that this in turn fixes a41, a42 , a44, a24 and a14 to be 1
since if any of these are 0 or 2, we can construct matrix A~ . Thus we may construct a matrix A
such that aij= aji =0 if i is odd and and aij=1 otherwise.
Let us now consider the matrix where the row (k-1) is odd. This means that A has the form

47
A=
This in turn implies that ak-1,n =1 and ak,n-1 =1. It may be verified that for A~ not to exist all
elements in row k and in column n must equal 1. However, this violates the requirements on a
matrix A.
Consider now the matrix where the row (k-1) is even. This means that A has the form
A=
This in turn implies that ak-1,n =0 and ak,n-1 =0. It may be verified that for A~ not to exist all
elements in row k and in column n must equal 0. However, this violates the requirements on a
matrix A.
Thus it is not possible to construct a matrix A such that A~ does not exist.
A~ must exist, thereby proving the lemma.
QED

48
Appendix Four: The Multiplicative Polarization Function in the CES Family
Proposition: The only Polarization function which satisfies Axioms P1, P3, P4, and P5 and
belongs to the CES family of functions is the multiplicative form , where RI
and SI are calculated using an inequality measure which has a range between zero and one
Proof: By Axiom P1 it is a function of all three arguments and only those. Since the Polarization
measure is a member of the CES family it can be taken to have one of the forms:
i. with
β
> 0.
ii.
iii.
It is easy to check that functions of form (i) fail Axiom P3, and functions of form (iii) fail Axiom
P5. What about functions of form (ii)? By Axiom P4 , the coefficient A must have value 1 and
by Axiom P3, the exponents must also have value 1. Thus, form (ii) collapses to the pure
multiplicative form . It may be checked that this function satisfies all of the
axioms.
QED
It may be noted that if Axiom P3 is dropped then the function
is admissible. This may be attractive from the viewpoint that the values of the polarization index
which are produced are in of the same order of magnitude as the component parts of the index.
If Axiom P3 is dropped then functions of form (i) cannot be excluded. However, if it is required
that Axiom P2 be satisfied then functions of form (i) can be excluded. It follows that functions
of the form are the only members of the CES family which satisfy Axioms
P1, P2, P4, and P5, where RI and SI are calculated using an inequality measure which has a
range between zero and one.
If Axiom P3 is dropped and additionally Axiom P4 is adjusted so that there is no requirement
that the index of polarization take on a maximum value then a broader range of inequality
measures can be used to calculate RI and SI, such as the additively decomposable generalized
entropy measures (which are unbounded above).