Similarity of Options and the Measurement of Diversity
| Date | 01 October 2003 |
| Author | Prasanta K. Pattanaik,Walter Bossert,Yongsheng Xu |
| Published date | 01 October 2003 |
| DOI | 10.1177/0951692803154004 |
| Subject Matter | Conference |
SIMILARITY OF OPTIONS AND THE
MEASUREMENT OF DIVERSITY
Walter Bossert, Prasanta K. Pattanaik and Yongsheng Xu
ABSTRACT
This article analyzes the measurement of the diversity of sets based on the dis-
similarity of the objects contained in the set. We discuss axiomatic approaches
to diversity measurement and examine the considerations underlying the appli-
cation of speci®c measures. Our focus is on descriptive issues: rather than
assuming a speci®c ethical position or restricting attention to properties that
are appealing in speci®c applications, we address the foundations of the
measurement issue as such in the context of diversity.
KEY WORDS .diversity .measurement .similarity
1. Introduction
Does the newly created, left-wing political party increase the diversity of poli-
tical options available to the voters in a country? Will the addition of several
blue-colored passenger trains to the MARTA system increase the diversity of
transportation modes in the city of Atlanta? Would the extinction of giant
pandas reduce the diversity of species on the planet? How does the preserva-
tion of a native American language add to the diversity of world cultures?
Has the increase in concentration in the Canadian print and television
media industry that occurred over the last few years reduced the diversity
of opinions and viewpoints to which the general public gets exposed? The
answer to each of these questions requires the measurement of diversity. It is
thus clear that the problem of measuring diversity may arise in the context of
a very broad array of issues which often ®gure in public discussions and
debates. While diversity may be desirable in many contexts, it is not dif®cult
to think of situations where greater diversity is not necessarily bene®cial. For
instance, adding a new member with very similar positions to a coalition of
Journal of Theoretical Politics 15(4): 405±421 Copyright &2003 Sage Publications
0951±6298[200310]15:4; 405±421; 035950 London, Thousand Oaks, CA and New Delhi
An earlier version of this article was presented at the International Conference on the Analysis
and Measurement of Freedom in Palermo, September 2001. We thank Marc Fleurbaey and Bob
Russell for comments and the Social Sciences and Humanities Research Council of Canada for
®nancial support.
agents may improve the strength and cohesion of the group, which may
enable it to pursue its objectives more effectively and, consequently, a reduc-
tion in diversity may be considered a good thing from the viewpoint of the
group members.
In a more abstract context, in recent years, the measurement of diversity
has become an increasingly important issue in the literature on the ranking
of opportunity sets in terms of freedom of choice, where opportunity sets
are interpreted as sets of options available to a decision-maker and alterna-
tive opportunity sets are assumed to re¯ect possibly different amounts of
freedom of choice for the decision-maker.
The purpose of this article is to present an integrated approach and a dis-
cussion of some measures of diversity that have been proposed in the litera-
ture in contributions such as Weitzman (1992), Pattanaik and Xu (2000) and
Bossert et al. (2001); see also Weikard (1998) and Nehring and Puppe (2002)
for discussions.
Consider the ®rst question we posed at the beginning of this article: does
the newly created, left-wing political party increase the diversity of political
options for the voters in a country? For convenience, let us call this left-wing
political party l. We consider two alternative political scenarios regarding the
situation before the creation of the new party. In the ®rst scenario, to be
called situation ,there are initially ®ve political parties a,b,c,dand e;of
which parties ato dcan all reasonably be described as rightist parties with
only slight differences between their platforms, and eis a centrist party. In
the second scenario, to be called situation ,there are initially ®ve parties
f,g,h,iand j. Assume that the ®rst four of those are leftist parties with
only minor differences between their platforms, that the difference between
the platform of land that of each of f,g,hand iis also very minor and,
®nally, that jis a centrist party. Since lis a leftist party and is very dissimilar
to the existing parties in situation , it seems to make sense to postulate that
when we add lto the set of already existing parties in this situation, the addi-
tion signi®cantly increases the diversity of political ideologies available to the
voters. In contrast, since lis very similar to the parties that are already
present in , when we add lin this situation, such an addition does not
appear to change signi®cantly the diversity of political platforms available to
the voters. For similar reasons, it also seems that the new set fa;b;c;d;e;lgin
situation is characterized by a greater degree of diversity than the new set
ff;g;h;i;j;lgin situation .
The gist of this simple example is that, when assessing the diversity of a set
of options, the extent to which the options in the set under consideration are
similar to each other should be a relevant factor. To reiterate this point, let us
consider another of the questions posed earlier. In assessing the impact of
preventing a native American language from becoming extinct on the diver-
sity of cultures, it seems clear that the degree of diversity preserved depends
406 JOURNAL OF THEORETICAL POLITICS 15(4)
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