www.elsevier.nl / locate / econbase
On diversity and freedom of choice
a b,c ,*
Prasanta K. Pattanaik , Yongsheng Xu a
Department of Economics, University of California, Riverside, CA 92521, USA b
Department of Economics, Andrew Young School of Policy Studies, Georgia State University, Atlanta, GA30303, USA
c
School of Economics, University of Nottingham, Nottingham, NG7 2RD, UK
Received September 1998; received in revised form June 1999; accepted July 1999
Abstract
We discuss how information about diversity of alternatives may affect our judgments about an individual’s freedom of choice as reflected in the available set of alternatives. We provide an axiomatic characterization of a similarity-based rule for ranking alternative opportunity sets in terms of the individual’s freedom of choice. 2000 Elsevier Science B.V. All rights reserved.
Keywords: Diversity; Similarity; Freedom of choice; Opportunity set; Ranking
JEL classification: D00; D11; D63; D71
1. Introduction
Consider an agent who may be faced with different sets of alternatives. Given a set of alternatives, he has to choose exactly one alternative from the set. Each possible set of alternatives offers him some freedom of choice. How does one rank the different sets in terms of the degrees of freedom that they offer to the agent? What are the types of information that may be relevant for one’s judgment about the degrees of freedom that the individual enjoys from the different sets and what may be the implications of using alternative informational bases for such judgments about freedom? Some aspects of these problems have been discussed in earlier contributions of Arrow (1995), Dutta and Sen (1996), Klemisch-Ahlert (1993), Jones and Sugden (1982), Pattanaik and Xu (1990), Pattanaik and Xu (1998), Puppe (1996), Sen (1985), Sen (1991), Sen (1993), Steiner (1983), Sugden (1998) and Suppes (1987) (see also Bossert et al. (1994) for a
*Corresponding author. Tel.:11-404-651-2769; fax:11-404-651-4958. E-mail address: yxu3@gsu.edu (Y. Xu)
discussion of a related problem). In this paper, we concentrate on one specific issue which has not so far received much attention in the formal analysis of the freedom of choice: we argue that the extent of diversity (or, similarity) among the alternatives in the opportunity set is an important consideration in judging an agent’s freedom of choice and we suggest an axiomatic structure that reflects this consideration.
In Section 2, we lay down the basic notation and definitions. Section 3 presents an intuitive discussion of the relevance, for judgments about an individual’s freedom, of the degree of diversity / similarity that may exist among alternatives in the set(s) under consideration. In Section 4, we discuss a model which explicitly incorporates in-formation about similarities of alternatives. We conclude in Section 5.
2. The basic notation and definitions
Let X be the universal set of alternatives, assumed to be finite. One can think of several interpretations of the alternatives in X. For example, these alternatives can be thought of as ordinary commodity bundles. They may also be interpreted as bundles of relevant characteristics of the commodities, in the sense of Lancaster (1966) and Gorman (1959), or as bundles of functionings in the sense of Sen (1985) and (1987). Finally, the alternatives may have dimensions which refer to non-economic aspects of the individual’s life — his religion, expression of a specific political belief, etc.
At any given time, the set of all alternatives available to the individual will be a non-empty subset of X, and he has to choose exactly one alternative from this set of available alternatives. The individual’s freedom is seen as his freedom to choose from this set. Let Z be the set of all non-empty subsets of X. The elements of Z are the alternative feasible sets with which the agent may be faced. Let K be a reflexive and
transitive, but not necessarily connected, binary relation defined over Z. For all A, B[Z, [AKB] is to be interpreted as ‘the degree of freedom offered by the feasible set A is at
least as great as the feasible set B’. The asymmetric and symmetric parts ofKwill be
denoted by s and |, respectively.
3. Similar alternatives and freedom of choice
individual enjoys when he / she is faced with the problem of choosing an alternative from that set.
To see one of the reasons why the cardinality of the set of alternatives may be a bad index of the degree of freedom, consider the following example. Suppose a, b, c, d, e, f and g are political parties. Further suppose that a, b, c, d and e are all leftist parties with only ‘slight’ differences in their platforms, f is a rightist political party and, lastly, g is a centrist political party.
Then, intuitively we feel that, in ranking the two feasible sets,ha,b,c,d,ejandha, f, gj, in terms of freedom of choice, we should take into account the fact that the diversity of political ideologies represented in the setha, f, gjis greater than the diversity represented inha,b,c,d,ej. In general, it seems intuitively plausible to argue that the extent to which the alternatives in the set under consideration are similar to (or, different from) each other should be a relevant factor in judging the degree of freedom offered by that set. How does one interpret the notion of similarity in this context? In this paper, we view similarity of options as a matter of social judgment or norms. Thus, the issue of whether two options are similar is decided by appealing to the prevailing social norms rather than to the opinion of the individual whose freedom is under consideration. It is perfectly possible that the individual may not view traveling by a red bus to be ‘similar’ to traveling by a blue bus, but, if the prevailing social norms consider these as similar options, then they are treated as similar in our approach.
Finally, it may be worth noting that our formulation of the notion of similarity allows for only two different ‘levels’ of similarity: two options are either similar or dissimilar. An informationally richer formulation could be in terms of many different (ordinal) levels of similarity, so that one could, for example, say that the similarity between x and
y is greater than the similarity between z and w and the similarity between z and w is
greater than the similarity between a and b. However, in this paper, we do not pursue this approach.
4. Characterization of the simple similarity-based ordering
In this section, we present a model that incorporates information about similarity of alternatives. Let S be a reflexive, symmetric, but not necessarily transitive binary relation defined over X. For all x, y[X, xSy is to be interpreted as ‘x is similar to y’ and¬xSy
]]]]
is to be interpreted as ‘x is not similar to y’ or ‘x is dissimilar to y.’ Since we are not]]]]] ]]]] assuming S to be transitive, it is possible that x is similar to y and y is similar to z, but x is not similar to z. Thus ‘being similar to’ is not necessarily an equivalence relation. For
9 9
all A[Z, we say that A is homogeneous iff, for all a, a [A, aSa . For all x[X and all
]]]]
A[Z, we write xSA iff xSa for all af [A . Note that, if A is homogeneous, then xSA iffg
A<hxjis homogeneous. For all A[Z, a similarity-based partition of A is defined as a
]]]]]]]
class hA , . . . , A1 mj such that: (1) A , . . . , A1 m are all non-empty subsets of A; (2)
A <? ? ?<A 5A; (3) A , . . . , A are pairwise disjoint; and (4) for all k[
1 m 1 m
h1, . . . ,mj, A is homogeneous. The similarity-based partition will be denoted byk f(A),
9 09
Let F(A) be the set of all similarity-based partitions f(A) of A such that, for every similarity-based partition f9(A) of A, [f9(A)$[f(A). Thus, F(A) is the set of all smallest similarity-based partitions of A. An example may be helpful in clarifying our notation. Let A5hx, y,zj, and let [xSy and ySz and ¬xSz]. Then F(A)5
mimic B, every smallest similarity-based partition of A<B will have a larger number of sets than every smallest similarity-based partition of B.
Definition 4.1.K satisfies:
(4.1.1) Indifference between No-choice Situations (INS) iff ;x, y[X,hxj|hyj.
]]]]]]]]]]]]
(4.1.2) S-Monotonicity (SM) iff, for all A[Z such that A is homogeneous, and, for
]]]]]
Remark 4.2. INS is introduced in Jones and Sugden (1982) and Pattanaik and Xu (1990). It says that all singleton sets offer the individual the same amount freedom. The
intuition of INS is that, since a singleton opportunity set does not offer any choice at all,
if one is interested exclusively in the intrinsic value of freedom, then there is little to
distinguish between one singleton opportunity set and another. See Sen (1991) and (1993) for a critique of this axiom.
Remark 4.3. SM explicitly takes into account information about similarity of
alter-natives in the simple cases involving freedom comparisons of an existing set A in which all the elements in A are similar to each other and an enlarged set A<hxj where x is outside of A. In choosing different modes of transport, if the option of traveling by a red
bus is similar to the option of traveling by a blue bus, then it is plausible to argue that
the sethblue bus, red busj offers the same amount freedom as the sethred busj; and if
traveling by a red bus is not similar to traveling by a red train, then it is reasonable to
argue that the sethred bus, red trainj offers more freedom than the sethred busj. SM
thus formally requires that, given a homogeneous set A and x[X2A, if x is similar to
all the elements in A, then the addition of x to A does not change the degree of freedom
already offered by the opportunity set A, and if x is dissimilar to at least one element in
A, then, adding x to A will actually increase the degree of freedom.
Remark 4.4. SC is a weaker version of an axiom proposed by Sen(1991). Sen’s axiom
requires that, given A>C5B>D55, if [AKB and CKD], then [A<CKB<D],
and, if [AsB and CKD], then [A<CsB<D]. However, in our context, a
modification of Sen’s axiom seems to be warranted. Suppose A5haj, B5hbj, C5hcj
¬bSd, i.e. a and c are similar but b and d are dissimilar. In view of this, we can
justifiably feel that adding c to the set haj does not significantly increase the degree of freedom while adding d to the sethbjdoes. In that case, we may be unwilling to accept
that [A<CKB<D] even if [AKB and CKD]. Our axiom SC deals with this problem
of Sen’s axiom by restricting the applicability of the axiom to the case where both C and
D are homogeneous, and C does not mimic A.
Definition 4.5. K will be called the simple similarity-based ordering iff for all A,
]]]]]]]]]
B[Z, AKB iff [f(A)$[f(B ) for all f(A)[F(A) and allf(B )[F(B ).
Under the simple similarity-based ordering, opportunity sets are ranked according to the cardinalities of their smallest similarity-based partitions.
We now characterize the simple similarity-based ordering.
Theorem 4.6. K is the simple similarity-based ordering iff K satisfies indifference
between no choice situations, S-monotonicity and S-composition.
Proof. The necessity part of the theorem is obvious; we prove only sufficiency. LetK
satisfy INS, SM and SC. We first show that
(4.1) For all C, D[Z, if C and D are homogeneous, then C|D.
the repeated use of SM and the transitivity ofK,hc j|C. Similarly, we can show that,
By (4.2), A1<Am11|A1<ham11j. Noting A1<ham11jsA , by the transitivity of1
By the repeated use of (4.4) and the transitivity ofK, (4.5) can be proved very easily.
This, together with (4.2), completes the proof. h
Remark 4.7. In Theorem 4.4, we do not assume K to be connected, though
connectedness of K follows as a consequence of our assumptions.
Remark 4.8. If, for all distinct x, y[X,¬xSy (that is, if all alternatives are dissimilar
to each other), then the simple similarity-based ordering coincides with the following ranking rule:
(4.6) for all A, B[Z, AKB iff [A$[B.
The ordering given by(4.6) was originally characterized by Pattanaik and Xu (1990)
using a framework where similarity did not play any role in judgments about freedom.
On the other hand, if for all x, y[X, xSy (that is, if all alternatives in X are similar to
each other), then the simple similarity-based ordering is such that, for all A, B[Z,
A|B.
Remark 4.9. If S is assumed to be transitive, in addition to being reflexive and
symmetric, then for all A[Z, the smallest similarity-based partition of A is unique. In
this case, it can be shown that the simple similarity-based ordering is characterized by
INS and the following axioms:
(4.7) for all distinct x, y[X, xSy⇒hx, yj|hxjand ¬xSy⇒hx, yjshxj.
(4.8) ;A, B[Z, and ;x[X2(A<B ), if [xSa and xSb for some a[A and some
b[B] or [¬xSz for all z[A<B], then AsB iff A<hxjsB<hxj.
Remark 4.10. It may be helpful to have a convenient algorithm to identify the
cardinality of smallest similarity-based partitions of a set. It is possible to provide such
an algorithm. For all A[Z with [A$2 and all a[A, let d(a, A2haj):51, if haj
does not mimic A2haj; and let d(a, A2haj):50, otherwise. Now, for all A[Z, define
V(A) as follows:
and
V(A):5mina[AhV(A2haj)1d(a, A2haj)j, if [A.1.
It can be checked that, for all A[Z, V(A)5[f(A), where f(A) is a smallest
similarity-based partition of A.
5. Concluding remarks
In this paper, we have discussed the relevance of the diversity / similarity of alternatives for the evaluation of freedom, and, we have axiomatically characterized the simple similarity-based ordering of different sets of available alternatives in terms of freedom of choice. The simple similarity-based ordering is a natural extension of the simple cardinality-based ordering (Pattanaik and Xu, 1990) for ranking opportunity sets when the information about similarities of alternatives is available.
Though we have focused on how information about the diversity / similarities of alternatives may affect our judgments about freedom of choice, we do not intend to suggest that this is the only information that we should take into account in assessing an agent’s freedom of choice offered by alternative opportunity sets. For discussions of other types of information and their relevance in evaluating freedom of choice, see contributions by Pattanaik and Xu (1998), Puppe (1996), Sen (1991) and Sugden (1998).
Acknowledgements
We are grateful to Kunal Sengupta for many helpful discussions. After we derived the result in this paper, we learned that Professor Sengupta also had independently explored problems relating to similarity of alternatives and the freedom of choice. Several improvements are due to a referee.
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