4.1 Introduction

It often becomes impossible to gain sufficient information on achievement levels in different dimensions of well-being. This, therefore, may make the poverty status of a person quite ambiguous. To understand this, consider a person who has sufficiently high income so that he can certainly be regarded as income-rich. It is likely that he will remain rich if his income reduces by a tiny amount. But if this tiny reduction process continues, he will be poor after some time. In the process of these sequential reductions, we may classify him as “marginally poor” or “borderline poor” at some stage. However, there may not be a clear borderline or line of demarcation that will unambiguously partition the population into sets of poor and rich. Therefore, the predicate “poor” may often involve vagueness. Often, the respondents may be reluctant to supply correct data on achievement levels, particularly, on income and wealth holdings. There can be a wide range of threshold limits for achievement levels, which may be socially acceptable. The possibility that there is lack of relevant information on achievement quantities indicates that there is a degree of unclearness in the concept of poverty.

Now, if there is some vagueness in a concept, then vagueness is preserved by a clear-cut representation of that ambiguous concept. Fuzzy set theory has been introduced an equipment for tackling problems in which uncertainty resulting from a sort of unclearness plays a major role (see Zadeh, 1965). Consequently, given that the notion of poverty itself is dubious, the poverty position of a person is intrinsically fuzzy. This in turn demonstrates that the fuzzy set approach to the measurement of poverty can be advocated logically. Given the multidimensionality of poverty, we can regard the problem as one of fuzzy capability deprivation. This contention is addressed in this chapter of the book.

For each dimension, there is no unique threshold limit that can identify whether the person is deprived or nondeprived in the dimension. Thus, there may be ambiguity in identifying the status of a person uniformly in a dimension. There is an interval such that the positions of a person are unambiguous when his achievement in the dimension lies above the upper limit and below the lower limit. More precisely, the person is unambiguously deprived or nondeprived depending on whether his achievement in the dimension lies below or above the respective threshold limit. Equivalently, if we arrange the individual achievement levels in the dimension nondecreasingly, then there is a partitioning of the population into subgroups with respect to dimensional achievement with clear presumption about the statuses of individual deprivations in the dimension in the first and third subgroups. In the second subgroup, the extent of deprivation of a person in the specified dimension is determined by employing a fuzzy membership function. This clear partitioning of the population shows that one of the innovative features of the fuzzy set approach lies at the stage of identifying the poor. By providing a systematic treatment of vagueness arising in the context of identification problem, the fuzzy set approach establishes its clear merit. Standard poverty measurement methodologies that rigidly dichotomize the population into the poor and nonpoor using threshold limits disregard any ambiguity that may emerge in connection with identification of the poor and hence cause loss of information.

Application of fuzzy sets to the measurement of poverty, which has already gained considerable popularity, was pioneered by Cerioli and Zani (1990). Further contributions along this line came from Chiappero-Martinetti (1994, 1996, 2000, 2008), Cheli and Lemmi (1995), Makdissi and Wodon (2004), Lelli (2001), Deutsch and Silber (2005, 2006), Baliamoune-Lutz and MacGillivray (2006), Bérenger and Celestini (2006), Betti et al. (2006a,b), Chakravarty (2006), Molnar et al. (2006), Miceli (2006), Panek (2006), Qizilbash (2006), Vero (2006), Bérenger and Verdier-Chouchane (2007), Betti and Verma (2008), Roche (2008), Amarante et al. (2010), Belhadj and Matoussi (2010), Clark and Hulme (2010), D'Ambrosio et al. (2011), Belhadj and Limam (2012), and Zheng (2015), and others. While some of these contributions deal with theoretical issues, the concern of others is empirical analysis across methodologies. The contribution by Makdissi and Wodon (2004) deals with a suggestion for fuzzy targeting applied to Chile. A recent survey by Alkire et al. (2015) provides a critical evaluation of some methodological aspects.

It may be worthwhile to mention here that apart from the measurement of poverty, the theory of fuzzy sets has been applied to many areas of research, including, learning (Wee and Fu, 1969), philosophy (Goguen, 1969), control systems (Zadeh, 1965), linguistics (Lee and Zadeh, 1969), management science (Bellman and Zadeh, 1970), income inequality (Basu, 1987 and Ok, 1996), and social choice theory (Mordeson et al., 2015).

The next section of the chapter presents an analytical discussion on the fuzzy membership function that quantifies the degree of belongingness of a person to the poverty population. As we will observe, there can be many choices of membership functions. It is, therefore, necessary to justify a particular choice using intuitively reasonable arguments. This matter is also a subject of discussion of the section. The concern of Section 4.3 is the set of axioms for a fuzzy multidimensional poverty index. These axioms are fuzzy twins to their nonfuzzy sisters. Section 4.4 analyzes some examples of fuzzy multidimensional poverty indices. It may be necessary to verify if these indices satisfy these properties. As we will observe, satisfaction of the basic properties by the proposed indices will narrow down the choice of membership functions. While the general functional form scrutinized in this section is insensitive to a correlation increasing switch, we also analyze an index using the Bourguignon–Chakravarty aggregation criterion that clearly indicates sensitivity to such aswitch.

Often, it may be worthwhile to investigate whether one population is characterized by no more poverty compared to another for a given membership function. In other words, for a given membership function, the objective is to establish dominance conditions under which one distribution does not have higher poverty compared to another for all fuzzy poverty indices. In such a case, for this membership function, it is not necessary to calculate the values of the poverty indices to rank distributions in terms of poverty. Once the dominance conditions hold, we can clearly infer if one distribution has no more poverty compared to another. A similar line of investigation is to order distributions in terms of fuzzy poverty for a class of membership functions when the poverty index is given. A worthwhile exercise here is poverty ranking of distributions when both membership functions and poverty indices are allowed to vary. Following Zheng (2015), we investigate these issues in Section 4.5 when income is taken as the only dimension of well-being.

4.2 Fuzzy Membership Function

Before presenting an analytical discussion on the fuzzy membership function, the degree of membership of a person of the poor population, we give some simple examples to illustrate how vagueness arises often in different situations. Consider a heap of very small stone chips. If one chip is taken away from this pile, it still remains a heap. If we continue this “taking away” job with patience, then at the last but one step, only one chip will be left, which definitely does not constitute a heap. Evidently, at some stage of this “taking away” assignment, the pile became a “borderline” heap. But precise formulation of this “borderline” notion may not be unambiguous, which clearly indicates the presence of imprecision here. As pointed out in Keefe and Smith (1996) and Qizilbash (2006), such a vague predicate possesses three characteristics: it includes “borderline” positions, where one cannot decide emphatically between “heap” and “not heap”; a sharp boundary between “heap” and “not heap” cannot be drawn; and the presence of Sorties paradox – reducing the size of the heap by one chip keeps the pile “heap” but repeating the same step many times will make the pile “heap” or “not heap.” For the linguistic variable “tall,” we cannot assuredly distinguish between “tall” and “not tall” in borderline situations; a clear barrier between “tall” and “not tall” cannot be built; and the Sortie paradox holds. This clearly demonstrates that “tall” is a vague predicate. Similar remarks apply also to the linguistic variable “bald.” Human qualities such as “being nice,” “politeness,” “generosity” can as well be regarded as vague predicates.

In the “income-rich” example we considered in Section 4.1, since the notion of “poverty” fulfils all the three criteria discussed earlier, it is a vague predicate (Qizilbash, 2006). Vagueness is also relevant to the predicate “poor” in the multidimensional framework. We assume that there is no ambiguity with respect to definition of the identification of the poor. To illustrate the situation, let us assume that there are only two dimensions of well-being, income and nourishment. As Sen (1981, p. 13) argued, nutritional requirement is a rather imprecise concept. Following Chiappero-Martinetti (2006), we assume that for the dimension “nourishment,” possible specifications of “being nourished” ranked in increasing order are as follows: (i) starvation, (ii) very seriously undernourished, (iii) malnourishment, (iv) almost achievement of minimum calorie intake, (v) achievement of minimum calorie intake without fully balanced diet, (vi) achievement of minimum calorie intake with quite balanced diet, and (vii) achievement of nutritional level intake and balanced diet. It should be clear that a sharp difference between two consecutive specifications can barely be appropriated using some threshold. Thus, here also we recognize the presence of vagueness.

To formulate and discuss the membership function rigorously, we follow the notation adopted in Chapters 1 and 3. As in Chapters 1, 2, and 3, the number of persons in the society is denoted by and the number of dimensions of well-being is . We denote the set of all dimensions by . The achievement of any person i in dimension j, denoted by , is the entry of an achievement matrix , where is arbitrary and (see Chapter 1). Person is a member of a crisp population set for dimension if his achievement in the dimension falls below the corresponding threshold limit, that is, if he is deprived in the dimension. Recall from Chapter 3 that in such a case, we can partition , the domain of definition of achievement in dimension , as , where, as before, is the threshold limit for the dimension. The deprivation status of a person i in dimension j can be indicated using a two-valued characteristic function defined as follows:

Thus, while for a person who is deprived in dimension j, the characteristic function assigns the value 1; for a nondeprived person, the assigned value is 0. Since person i is arbitrary in the population set, it follows that the characteristic function explicitly dichotomizes the entire population into two subgroups that are respectively deprived and nondeprived with respect to dimension j. In the case of income poverty, the characteristic function partitions the population into income-poor and -rich subgroups.

As we have argued, in the multidimensional framework, the job of making distinction between deprived and nondeprived in a dimension often becomes difficult. Unambiguity in poverty measurement can be captured using a poverty membership function, which assigns the value 0 or 1 when there is complete information that the person is nondeprived or deprived in the dimension. However, when there is lack of perfect information arising from uncertainty, on a person's deprivation status in the dimension, the membership function counts him as partially deprived by assigning a number lying between 0 and 1.

Note that the magnitudes of vary over the interval , and doubtfulness regarding deprivation position of a person in the dimension emerges for denominations of for which he is neither deprived nor nondeprived unambiguously. Then, given that deprivation is a decreasing function of , it must be the situation that there exists an interval , a nonempty subset of , such that the person is unequivocally deprived or nondeprived for quantities not above or not below . In other words, we need to specify two achievement quantities and such that any person with achievement not more than will be counted as certainly deprived in dimension j and any person with achievement not less than will be unambiguously treated as nondeprived in the dimension. For any quantity in the interval , a partial degree of certainty about the stature of a person in terms of deprivation is specified by assigning a number between 0 and 1, and in such a case, we say that person i is partially deprived in dimension j. This numbering assignment is done by the fuzzy membership function, which is at the core of the fuzzy poverty measurement. The specification of the membership function relies on the source of fuzziness. We can refer to and as respectively the lower and upper fuzzy boundaries for dimension j. The vector of ordered pairs of lower and upper fuzzy boundaries taken together for all the dimensions is denoted by , which is assumed to be an element of , a finite set of vectors of d ordered pairs, where each number of a pair is positive and the second number is greater than the first number. That is, .

Since the interval , which we refer to as fuzzy poverty region for dimension j, is open, we have . Otherwise, , in which case, we have the standard multidimensional framework characterized by a unique threshold limit for each dimension. The domain of quantities can now be represented in terms of the union of three nonoverlapping sets as .

We can summarize the earlier discussion on the values of an arbitrary membership function in the following definition:

Evidently, the function h is the restriction of on , that is, for . Thus, for person i who is partially deprived in dimension j, . By expressing h as a function of , we recognize that for any achievement , the membership grade will depend on the achievement quantity itself and the lower and upper bounds and of the set . Continuity of the membership function ensures that negligible observational errors on achievement denominations will not give rise to steep changes in the value of the membership function. Decreasingness of the membership function reflects the view that as the quantity of achievement increases, the extent of deprivation decreases. Clearly, h is continuous and decreasing on . By continuity, as , and as , . The function is homogeneous of degree 0 over the domain . Clearly, Definition 4.1 enables us to say that specifies the degree of belongingness of person i in the deprivation range of dimension j, where the deprivation range for any dimension is given by , the range of the function .

For each value of h, there corresponds a different membership function. The choice of the value of h is a matter of value judgment. Observe that we maintain the extreme conditions and as two defining conditions. This is not unrealistic because we have a clear idea about the extent of deprivation in these extreme cases, whereas our idea about the same is quite unclear in the intermediate region . Corresponding to each membership function, there will be a different fuzzy poverty index. Each index will be directly related to its associated membership function.

Two major concerns for poverty indices based on membership functions are the lower and upper bounds and . Selection of these bounds of the fuzzy poverty region is a challenging job in this situation. Following Zheng (2015), we discuss here an illustrative example put forward for income poverty. In the European Income and Living Conditions survey, a household is requested to report “the very lowest net monthly income” that it would require for making “ends meet.” Suppose that this monthly income, denoted by , is interpreted as the income poverty line that each household regards. Let be the cumulative proportion of voters in the society who believe that the income poverty line is less than or equal to z. Hence, is the proportion of voters who opine that the income poverty line should be at least z. Then the minimum value of z that has been elected by the voters can be taken as , the lower bound of fuzzy income poverty region. On the other hand, , the upper bound of the fuzzy income poverty territory, can be taken as the maximum value of z chosen by the voters.

In some rich countries, such as in Western European countries, the income poverty line is taken as some proportion of the median income. In a society, individuals with incomes less than the median income may regard the median as a reference income, an income level with which they compare their own incomes and suffer from depression from a feeling that their incomes are lower. (The median of a nondecreasingly ordered n-person income distribution with odd number of observations is the observation, the middle-most income. If the number of incomes is even, then the median is defined as the arithmetic mean of the and incomes.¹)

All individuals whose incomes belong to some small income range around the median are said to constitute the middle class of the society, where is the median income and is small. It is highly likely that a poor person's reference group in a society is the middle class of the society. Now, a rich and large middle class is highly beneficial to the society in view of the high extent of its contribution to society's economic growth, education, public goods through provision of higher taxes, and so on. Therefore, a person with income less than the median income may regard the middle class as his targeted group since living conditions will be better then. The lower and upper limits of this income range, and , may be considered as the lower and upper bounds and , of fuzzy income poverty domain. (See Chakravarty (2015, Chapter 2), for further discussion.) These discussions can be extended to the other dimensions of well-being under appropriate modifications.

In the general functional form, specified in (4.1), if and , then the inequalities and have to be replaced with the perfect equalities and . In such a case, the ambiguous destitution domain for dimension j becomes .

Since estimation of fuzzy poverty indices relies on the choice of the membership functions, it is certainly necessary to justify the use of specific membership functions. We briefly discuss here some suggestions made by Chiappero-Martinetti (2006) along this line. As we have argued earlier, in the case of (4.1), the choice of an explicit membership function will depend on the value of . Such a choice is a matter of value judgment. For instance, if we assume that reduction in its value resulting from an increase in achievement is independent of the initial magnitude of achievement, then it implicitly represents the value judgment that is linear. Membership functions may be obtained from empirical evidence. One such example is the totally fuzzy and relative function, which we discuss later in greater length. This function, introduced by Cheli and Lemmi (1995), has its membership values dependent on the relative positions of theindividuals in the distribution of the achievements in the dimension. A third way of choosing a membership function is to take opinions of external experts. For instance, doctors and dieticians may be requested to classify nutritional levels. The subjective questionnaire method, similar to that adopted by the European Income and Living Conditions survey, can also be an alternative move for selecting grades of a membership function.

We may explain the role of a membership function further using literacy as a dimension of well-being. In a household, a person is either literate or illiterate, where a literate person is assumed to be one who can read and write with grasp in some language. Often, an illiterate in a family can get literacy benefit if there are one or more literates in the family. For instance, such a situation arises if the illiterate is required to fill in some form. Following Basu and Foster (1998), we can refer to such a person as a proximate illiterate. On the other hand, if there is no literate person in a family, then a member of the family is called isolated illiterate. If we assign the number 1 or 0 depending on whether a person in a household is isolated illiterate or literate, then we can indicate the illiteracy grade of a proximate illiterate by a number a lying between 0 and 1, that is, . The value of the pure positive fraction a depends on several external factors, including attitude and availability of literates in the family to help illiterates literally. As these external factors become more favorable to him, the value of should be lower for him. This indicates that the concept of proximate illiterate involves some factors, which themselves are dubious. In view of this, we can regard it as a fuzzy concept. The corresponding membership function takes on the value 0 or 1 depending on whether a person is literate or isolated illiterate. If the person is a proximate illiterate, then the membership grade is a number lying between 0 and 1. The fuzzy deprivation range for the dimension literacy is, therefore, given by .

De Luca and Termini (1972) raised the interesting question of measuring the extent of fuzziness of a fuzzy set. Smithson (1982) argued that some transformation of a measure of variability of membership grades can be employed for measuring the level of fuzziness. A unified approach to this issue has been developed by Chakravarty and Roy (1985). Since in this chapter we are dealing with fuzzy multidimensional poverty, we only need to be concerned with the relevance of a membership function to the poverty measurement problem.

We now present some illustrations of membership functions that drop out as particular cases of the general membership function given by (4.1). For , reduces to the form suggested by Chakravarty (2006), where is a constant. The explicit form of this membership function is given by

4.2

As we will see next, the parametric restriction can be justified in terms of satisfaction of several fuzzy poverty axioms. For , (4.2) coincides with the Cerioli and Zani (1990) trapezoidal function given by

4.3

While is linear in achievement quantities over the uncertain poverty space, is strictly convex there. Thus, although for (4.2), the reduction in the membership grade resulting from an increase in achievement denomination in the territory takes place at an increasing rate, for (4.3), the corresponding change occurs at a constant rate.

An alternative of interest arises from the stipulation , where . In such a case, the resulting membership function turns out to be

4.4

One similarity of with is its strict convexity on the fuzzy poverty territory in the achievement extent. The last of the three membership functions (4.2)–(4.4) expresses membership grades over the fuzzy region in terms of the excess of achievement over the lower fuzzy limit as a fraction of the length of the fuzzy region . This ensures homogeneity of degree 0 of the membership function over the entire domain. This common property of the three functions guarantees that the membership grades remain invariant if we change the unit of measurement of achievement quantities and the lower and upper fuzzy boundaries. Thus, if we convert the unit of energy consumption from calorie into joule, the membership grades of the dimension “energy” does not alter. Another characteristic of these membership functions is that they are translatable of degree 0 or translation invariant, which says that for an equal absolute change in the achievements and the fuzzy boundaries, the membership grades do not change. Hence, if each person's income and fuzzy income limits reduce by $0.4, then the membership grades remain unaltered.

For an appropriate choice of in the fuzzy region , (4.1) generates the following specification of the membership function, which is a member of the class constructed by Dombi (1990, Theorem 2):

4.5

The function is unambiguously decreasing over the hazy slot , but it initially decreases at a decreasing rate and then decreases at an increasing rate. In other words, there exists a number , such that it is strictly concave over and strictly convex over . Equivalently, we say that this function has an inverted S-shape.

In the totally fuzzy and relative approach, advocated by Cheli and Lemmi (1995), the membership function for dimension , , is defined as the shortfall of the dimension's distribution function from its maximum attainable value so that it equals 1 for the poorest and 0 for the richest. Formally,

4.6

Thus, in this case, . Since a distribution function may not be increasing, although nondecreasing, is nonincreasing, if not decreasing, over the sphere . In addition, need not be continuous, since a monotone function has at most a countable number of points of discontinuity. If , then the fuzzy sphere here should be .

As we have argued earlier, the selection of a particular membership function may contemplate some value judgments implicit in the theoretical notion that we wish to delineate. One way of judging the choice of a particular form of the membership function is to develop an axiomatic characterization of the function so that the axioms represent the underlying subjective evaluation. An attempt along this line was made in Chakravarty (2006), where the linear membership function (4.3) has been characterized axiomatically.

4.3 Axioms for a Fuzzy Multidimensional Poverty Index

The objective of this section is to propose and analyze some desirable properties for fuzzy set theory–based poverty indices. Most of these properties, which we refer to as fuzzy poverty axioms, are fuzzy dittos to the multidimensional poverty axioms considered in Chapter 3.

Given that there are d dimensions of well-being and represents the fuzzy membership function for dimension , we write for the vector of membership functions when all the dimensions are considered together. Let stand for the set of d dimensional vectors of membership functions, that is, . For any achievement matrix X, let denote the set of persons who are only partially deprived in X_. Formally, . Similarly, stands for the set of persons who are only certainly deprived in X. The sets of partially and certainly deprived persons in X corresponding to dimension j are respectively and . The set of dimensions in which person i is certainly deprived is given by . Then indicates the number of dimensions in which person i is certainly deprived in X.

We follow the union method of identification of the poor. That is, a person i will be called fuzzy multidimensionally poor if he is deprived in at least one dimension, certainly or partially. Accordingly, the set of fuzzy poor persons in the society with the distribution matrix X is the union of the nonoverlapping sets and . More precisely, . Consequently, the number of fuzzy poor persons in X is . If for some pair , there is at least one dimension j such that , which is the same as the requirement that , and for no dimension, person i is certainly deprived, then we can call him strictly fuzzy poor, where is arbitrary. We can also refer to this person as possibly multidimensionally poor, but not certainly poor. Clearly, . The fuzzy head-count ratio in , the proportion of fuzzy poor persons in , is then given by . If all the deprived persons' achievements are in the corresponding fuzzy spaces, then so that reduces to its strict fuzzy version , the strict fuzzy head-count ratio. Then and stand respectively for the dimension adjusted fuzzy head-count and its strict version.² We can also define here the fuzzy head-count ratio in dimension j as and its strict form as . Their dimension-adjusted varieties are respectively and . Obviously, for any , , , , . Similarly, for any , , and , and .

In order to distinguish the fuzzy approach from the standard approach, unless specified, we assume, throughout the chapter, that at least a person in the society is strictly fuzzy multidimensionally poor, that is, at least one person i is partially deprived in at least one dimension j. In other words, there exists at least one pair such that indicating that . Person i may be unambiguously deprived in one or more dimensions, but if none of his achievements is in the fuzzy space of a deprived dimension, then he is definitely poor but not strictly fuzzy poor. Therefore, if achievements in deprived dimensions of all the poor are below the corresponding fuzzy lower boundaries, the problem becomes one of standard poverty measurement.

By a fuzzy multidimensional poverty index , we mean a nonconstant nonnegative real-valued function defined on the Cartesian product . More precisely, , where is the nonnegative part of the set of real numbers . For any , , , determines the extent of poverty associated with the distribution matrix X, the vector of ordered pairs of fuzzy boundaries , and the vector of membership functions . By nonconstancy, is assumed to take at least two different values. Otherwise, it becomes insensitive to achievement levels of the poor. A constant index will assign the same value to all distribution matrices with the same dimensions, irrespective of whether the achievements are equal or not equal across individuals and dimensions.

The selection of a particular index of poverty will depend on the purpose we have in mind. For instance, suppose that we are interested in investigating how a poverty index changes under an increase in an achievement of a person in the fuzzy region of the corresponding dimension. This increase in achievement may be a consequence of improvement in some positive external factors that influence the membership grades. Similarly, it may be worthwhile to check how a poverty index responds following an increase in the membership grades in the fuzzy region of a dimension. Such changes in the membership grades may represent a policy evaluator's opinion on susceptibility about fuzziness in the dimension. In all these cases, we need some particular poverty indices that we have not analyzed in earlier chapters. We may also be interested in intersociety poverty comparisons for given membership functions and fuzzy boundaries and identifying subgroups and or dimensions that affect fuzzy poverty more.

As in Chapters 1, 2, and 3, following Chakravarty and Lugo (2016), we will subdivide the presentation of the relevant axioms into several subsections. First, we analyze the invariance postulates.

4.3.1 Invariance Axioms

The first invariance axiom, we discuss, is fuzzy ratio-scale invariance, which demands that the extent of poverty remains unaltered if there are changes in the units of measurement of achievement quantities.

4.3.1.1 Fuzzy Strong Ratio-Scale Invariance

For all , , , , where , for all i.

The fuzzy strong ratio-scale invariance axiom requires that the distribution matrix, upon postmultiplication by a diagonal matrix accompanied by the condition that the fuzzy demarcation lines are also multiplied by the corresponding proportionality factors, does not change the extent of fuzzy poverty. In other words, fuzzy poverty remains invariant under positive proportional changes in achievements and fuzzy boundaries. As a result, if we change the units of measurements of dimensional quantities and fuzzy limits, the degree of poverty does not change. The scales of proportionalities need not be the same across dimensions because different achievements and hence fuzzy boundaries may have different units of measurements. This axiom is the fuzzy twin of the corresponding inequality axiom. A weaker form of this axiom is fuzzy ratio-scale invariance, which requires equality of proportionality factors across dimensions (see Chapter 2). A fuzzy poverty index satisfying the fuzzy ratio-scale invariance axiom is called a relative index.

In order to illustrate this axiom, let us assume that there are three dimensions of well-being, namely daily energy consumption by an adult male, per capita income, and life expectancy, measured respectively in calories, dollars, and years. With these three dimensions of well-being, we consider the following matrix as an example of a social matrix in a four-person economy:

The first entry in row i of the given matrix shows person i's daily calorie intake. On the other hand, the second and third entries of the row exemplify respectively the person's life expectancy and income.

For illustrative purpose, let us assume that any calorie consumption level below 2000 will be referred to as “too low,” whereas a consumption quantity in the interval (2000, 2700) may be regarded as “not too low,” which is a fuzzy concept. On the other hand, the consumption magnitude 2700 is the optimum figure, which, for simplicity, is assumed to be the maximum. The fuzzy space of malnutrition arising from low calorie intake is, therefore, given by , where the subscript “EC” stands for energy consumption. The upper fuzzy boundary value is . The fuzzy space for life expectancy is taken as , where the subscript “LE” symbolizes life expectancy. Finally, for income, this region is assumed to be , where the subscript “IN” appears for income.

Thus, although person 2's energy consumption is not too low, he does not have any feeling of destitution with respect to life expectancy and income. In contrast, person 4 has no sensuality of deprivation in any dimension. Similar interpretations hold for the other entries of the matrix.

For the three dimensions we have considered earlier, recall that for energy absorption dimension the fuzzy domain is . Since 1 calorie = 4.18 joules, the entries in the first column of , starting from the first row, get transformed into 11286, 10450, 11286, and 11286 joules, respectively (see Chapter 2). Under this transformation, the fuzzy space (2000, 2700) gets transformed into (8360, 11286) and any becomes , where . Next, suppose that we measure life expectancy in months and incomes in cents so that the entries in the second and third columns get transformed appropriately. This means that we are postmultiplying by the diagonal matrix to obtain a new social matrix , where

The fuzzy ratio-scale invariance property demands that is fuzzy poverty equivalent to .

4.3.1.2 Fuzzy Strong Translation-Scale Invariance

For all , , , , where A is an matrix with identical rows given by such that and for all .

This axiom needs that the fuzzy poverty index does not change if there are equal absolute additions/subtractions to all achievements and their fuzzy boundaries in a dimension. It parallels the strong translation invariance axiom in Chapter 2. Since we do not rule out the possibility that for one or more , achievements in some of the dimensions and their fuzzy limits may remain unaltered under this operation. A weaker version of the axiom, fuzzy translation invariance axiom, requires constancy of the amounts across dimensions. A fuzzy poverty index satisfying this weaker postulate is called an absolute index. If a fuzzy poverty index is both relative and absolute, we call it a compromise index.

The next postulate, the fuzzy weak focus axiom, involves a change in the achievement in a dimension of a person who does notsuffer from deprivation in any dimension, and the change does not make him deprived in the dimension.

4.3.1.3 Fuzzy Weak Focus

For all , , , if for some , for all , for some pair , , where c is a scalar such that and for all , then .

Since the change affects only a person who is nondeprived in all dimensions, it is similar to the Bourguignon–Chakravarty weak focus axiom considered in Chapter 3. (The person is neither certainly nor partially deprived in any dimension.) However, here in the formulation, we note the existence of membership functions and fuzzy lines of demarcation, which are assumed to be given. These characteristics, which are absent in the standard structure, enable us to refer to the property as fuzzy weak focus axiom instead of weak focus axiom.

To illustrate this axiom, suppose that income of person 4 in increases by 50. The resulting social matrix is given by

Since person 4 is nondeprived in all the dimensions, giving him more income should not affect global poverty assessment in any way.

The third postulate deals with an augmentation/diminution of achievement in a nondeprived dimension of a poor or a nonpoor person that does not make him deprived in the dimension. Formally,

4.3.1.4 Fuzzy Strong Focus

For all , , , if for some pair , , where , c is a scalar such that and for all , then .

Note that here we do not rule out the possibility that person i is fuzzy poor. This axiom parallels the Bourguignon–Chakravarty strong focus axiom analyzed in Chapter 3.

The matrix , given as follows, is obtained from by increasing only person 2's income, a nondeprived dimension for the person. The person is, however, deprived in energy consumption. The fuzzy strong focus axiom insists that this augmentation does not change the evaluation of overall poverty.

The additional income is unable to lift person 2 out of deprivation in calorie intake.

The next invariance postulate, which we refer to as fuzzy symmetry because it provides an impartial treatment of persons, can be stated as follows:

4.3.1.5 Fuzzy Symmetry

For all , , , if , where is any permutation matrix of order n, then .

Clearly, in this case, we consider reordering of all the individuals, not necessarily of those who have achievements in the fuzzy regions. However, as in the focus axioms, we recognize the presence of membership functions and fuzzy boundaries. Precisely, because of this, we refer to this axiom as fuzzy symmetry.

For cross-population comparison of fuzzy poverty, we assume that the fuzzy boundaries and membership functions are the same in the two populations under comparison. Otherwise, the comparison is not meaningful. We can formally state this axiom as follows:

4.3.1.6 Fuzzy Population Replication Invariance

For all , , , , where is any k-fold replication of , being any finite integer.

Under ceteris paribus assumptions, we consider replications of the entire population. Because of the presence of fuzziness in the structure, we call this property the axiom of fuzzy population replication invariance.

The following subsection concentrates on the analysis of desirable directional movements of a fuzzy poverty index under some sensible transformation in a social matrix.

4.3.2 Distributional Axioms

Given that we are dealing with fuzzy poverty measurement here, it is rational to assume that a scaling down of the achievement in the fuzzy region of adeprived dimension of a person should enlarge fuzzy poverty. Note that if the contraction takes place in the space where the person is certainly deprived, then the membership function remains insensitive to such a reduction.

The following axiom addresses the impact of debasement in the attainment lying in the fuzzy domain of a deprived dimension of a poor.

4.3.2.1 Fuzzy Monotonicity

For all , , , if for some pair , , where , for all and , then .

Here the reduced achievement will be either in the fuzzy region or in the certainly deprived region of the dimension.

The achievement matrix given as follows is obtained from by a cutback of person 1's life expectancy, a deprived dimension for the person with its achievement in the fuzzy territory, by 0.1.

Fuzzy monotonicity axiom then demands that .

Since existence of poverty can be regarded as withdrawal of human rights from the affected people, from policy point of view, it is appealing that reductions (respectively, increments) in achievements should raise (respectively, cutback) poverty by higher quantities, the poorer the affected persons are.

This plausible view is presented analytically in the following axiom.

4.3.2.2 Fuzzy Monotonicity Sensitivity

For all , , , if for some pair , , for all , and for some pair , , and for all , where , and , then .

We assume that attainments of multidimensionally poor persons i and h in dimension j are in the fuzzy space of the dimension, and person h has a higher deprivation here. Then increase in poverty resulting from shrinkage in the dimensional achievement that applies identically to both i and h will be higher whenever it corresponds to person h who is more deprived in the dimension. Note that a decrement in the achievements of the two poor persons who are certainly deprived in the corresponding dimensions does not influence the value of a membership function. That is why we have restricted attention to the fuzzy dominion of the dimension.

Consider the achievement matrices

In , both persons 1 and 3 are poor, and they are deprived in life expectancy whose values are in the fuzzy space. Person 3 had originally lower deprivation compared to person 1 in the dimension. We obtain and from the initial matrix by curtailing the life expectancy of persons 1 and 3, respectively, by 0.1. The fuzzy monotonicity sensitivity axiom appeals that .

The next axiom is concerned with the effect of increasing the number of deprived dimensions of a fuzzy poor. It requires that when a nondeprived dimension of a fuzzy poor person, who is not deprived in all dimensions, becomes partially or certainly deprived, then fuzzy poverty should increase. Formally,

4.3.2.3 Fuzzy Dimensional Monotonicity

For all , , , if is obtained from X such that for some pair , , where person i is partially deprived in X in at least one dimension different from j and for all pairs , then .

Person i, who is fuzzy poor and nondeprived in dimension j in X, comes to be deprived in the dimension in Y. Nevertheless, all other dimensional achievements for all persons in are the same in both X and Y. Now, under the change considered in the axiom, in Y, person i will be either partially or certainly deprived in dimension j, a nondeprived dimension for the person in X. Hence, fuzzy poverty should increase. This axiom may be treated as the fuzzy analog to the Alkire and Foster (2011) dimensional monotonicity axiom.

In our achievement matrix , calorie intake is a nondeprived dimension for person 1, a fuzzy poor person in . If this calorie intake now reduces from 2700 to 2450, person 1 becomes deprived in this dimension in the transformed matrix given as follows:

Fuzzy dimensional monotonicity axiom demands that .

Let be the identical set of dimensions in which persons i and h are partially deprived, that is, , where are arbitrary. Assume further that all partial deprivations of h are higher than the corresponding deprivations of i. Formally, for all , .

According to condition (i), there is at least one element in the identical set of partially deprived dimensions of the persons h and i, where h has higher deprivations than i in this set. Condition (ii) declares that all individuals except persons h and i have identical achievements in all the dimensions in both X and Y. Part (a) of condition (iii) claims that achievements of each of the persons h and i remain the same in all the dimensions that are outside the set . In part (b) of the condition, it is indicated that we derive and from and , respectively, by transfers of achievements from person h to person i for dimensions in ,where the size of the transfer is nonnegative for any dimension in the set and positive for at least one dimension. Finally, part (b) of condition (iii) confirms that the size of the transfer in any dimension does not allow the recipient to cross the upper fuzzy boundary or to be even at the boundary itself. Since for any , can be no more than , it is ensured that . In the posttransfer distributions, both persons h and i may remain partially deprived in a dimension affected by the transfer so that . Partial deprivation of the persons in dimension j shows that fuzzy environment exists for both the persons in the posttransfer situation as well. However, situations such as (c) and (d) are also possible. While in situation (c) of the posttransfer circumstance, person h's achievement in the dimension is exactly at the lower boundary of the fuzzy space of the dimension, in (d), this achievement is even lower. Note that the transfer activity applies only to the set of partially deprived dimensions of individuals i and h.

In the distribution matrix X₁, and in each of the two dimensions in this identical set of deprived dimensions of persons 1 and 3, person 3 has higher achievement than person 1. Now, a regressive transfer of 5 units of income from person 1 to person 3, who are both partially deprived in the dimension, transforms the matrix into , given by

We then say that is deduced from by a fuzzy Pigou–Dalton bundle of regressive transfers. The following postulate for a multidimensional fuzzy poverty index can now be stated.

4.3.2.4 Fuzzy Transfer

For all , , , if is obtained from by a fuzzy Pigou–Dalton bundle of regressive transfers, then .

For the illustrative example, where we derive from by a fuzzy Pigou–Dalton regressive transfer, the fuzzy transfer axiom demands that .

In Definition 4.2, we can alternatively assume that person h has higher deprivations than person i in their identical set of deprived dimensions, and in at least one dimension, person h is partially deprived. Then a Pigou–Dalton bundle of regressive transfers from h to i takes place in a fuzzy environment if the size of transfer of achievement from at least one partially deprived dimension is positive. Evidently, our formulation of the Pigou–Dalton bundle of regressive transfers presented in Definition 4.2 is quite simple and easy to understand.

The final axiom of the subsection is applicable only to multidimensional poverty in a fuzzy environment and depends on the association between deprivations.

In Definition 4.3, conditions (i) and (ii) accompanied by condition (iv) for and specify that person p who had lower achievement in dimension j and higher achievement in dimension q than person i in X have higher achievements in both the dimensions in Y. It is also given that all the remaining dimensional achievements of person i are not higher than the corresponding dimensional achievements of person p. In condition (iii), it is asserted that achievements for the remaining individuals in dimension j are the same in both the social matrices X and Y. Finally, condition (iv) stipulates that achievements of all persons in all the dimensions except j are the same in the two social matrices. We deduce Y from X by an interchange of achievements in dimension j between persons i and p. In the postswitch situation, person p does not possess lower achievement than person i in each dimension and possesses strictly more achievement in at least onedimension. The swap of achievements in the fuzzy region of dimension j, defined by (i) and (ii), increases the correlation between dimensions. That is why, we refer to the switch as a correlation-increasing switch in a fuzzy poverty setting. The switch does not alter the total of the achievements in the dimension on which it applies.

If the two dimensions are substitutes, then one neutralizes the shortness of the other. Observe that one person (person p), who was originally richer than the other person (person i) in the fuzzy space of dimension q, is becoming richer in the fuzzy space of the other dimension (dimension j) as well after the switch. Since the two dimensions represent akin features of well-being, the switch should increase fuzzy poverty. For the other person (person i), the incapability to outweigh the deficiency in one dimension by the other now goes up since he is now poorer in both the dimensions.

The aforementioned discussion enables us to state the following axiom:

4.3.2.5 Increasing Fuzzy Poverty under Correlation-Increasing Switch

For all , , , if Y is obtained from X by a correlation-increasing switch in a fuzzy poverty environment, then if the dimensions involved in the switch are substitutes.

The corresponding postulate when the dimensions are complements requires fuzzy poverty to decrease under a correlation-increasing switch. The switch will not affect fuzzy poverty at all if the two dimensions are independents. These axioms are fuzzy versions of the corresponding Bourguignon–Chakravarty axioms for standard multidimensional poverty indices stated in Chapter 3.

To exemplify the aforementioned property, note that in , the first person has more achievement than the third person in the third dimension, and the opposite inequality holds in the second dimension. The matrix is derived from by a switch of achievements in dimension 2 between the two persons. In the postswitch social matrix , achievements of person 1 in dimensions 2 and 3 are higher than the corresponding quantities for person 3, and their achievements in dimension 1 are the same. Observe also that the achievements of each of the persons 1and 3 in dimensions 2 and 3 are in the corresponding fuzzy regions. As a result, we can claim that has been obtained from by a correlation-increasing switch.

In consequence, , if the second and third dimensions are regarded as substitutes. The reverse inequality or exact equality holds if the two dimensions are complements or independents.

The concern of the first axiom of the next set of two axioms is the relationship between overall poverty and poverty levels of two or more population subgroups that are formed by breaking down the population using some homogeneous characteristic. The second axiom deals with the connection between overall poverty and dimensional poverty levels. These two axioms consider respectively partitions of the population and dimensions keeping other characteristics such as membership functions and fuzzy limits unchanged.

4.3.3 Decomposability Axioms

4.3.3.1 Fuzzy Subgroup Decomposability

For any , , , where , is obtained by placing matrices from above to below for , and for all , , where .

This axiom, which is a fuzzy variant of subgroup decomposability postulate for standard poverty indices, demands that global fuzzy poverty is the population share weighted average of subgroup fuzzy poverty levels. Here , is fuzzy poverty of subgroup i weighted by the corresponding population fraction. Aggregation of all such weighted numbers across population subgroups gives us the comprehensive fuzzy poverty. Following our arguments in Chapter 3, we can say that each of these numbers becomes important from policy perspective for mitigating fuzzy poverty.

Suppose that the social distribution is broken down into two submatrices and with population sizes 3 and 1, respectively, where

Then subgroup decomposability requires that .

4.3.3.2 Fuzzy Factor Decomposability

For any , , , , where is the weight assigned to dimension j and .

This axiom, which is a fuzzy sister of the factor decomposability axiom introduced by Chakravarty et al. (1998), claims that total fuzzy poverty is the weighted average of dimension-wise fuzzy poverty levels, where the nonnegative weights add up to 1. As in the standard case, this axiom has also interesting policy applications.

This axiom empowers us to express the fuzzy poverty quantity of as

The axioms presented so far assume the existence of fuzzy zone in each dimension of well-being. However, we did not analyze how poverty changes when we replace the existing membership function by a newer one that assigns higher values to deprivations in one or more fuzzy belts. This is the subject of our next subsection.

4.4 Fuzzy Multidimensional Poverty Indices

The fuzzy-set-based axiomatic approach to multidimensional poverty measurement involves indices that should fulfill axioms suggested in the earlier section. We will assume subgroup decomposability at the outset because of its interesting policy applications and discuss some indices possessing this property.

Repeated applications of subgroup decomposability shows that we can write an index satisfying this axiom as

4.7

where , , are arbitrary. Here is the individual fuzzy poverty function. Following Chakravarty (2006), we define , the average of grades of memberships of person i in different dimensions. Since gives us the extent of deprivation of person i in dimension j, this definition of individual fuzzy poverty function is quite sensible. The choice of dimensional weights in the individual poverty function makes in (4.7) symmetric in dimensions; any reordering of dimensions does not change poverty.

Substituting this definition of individual poverty function into (4.7), we obtain

4.8

where , , are arbitrary. The expression in (4.8) means that the sum is taken over membership grades of all individuals who are partially deprived in dimension j, and as before, is the total number of dimensions in which person i is certainly deprived. In (4.8) at the outset, membership grades are aggregated across individuals. At the second stage, the individual averages derived at the first stage are aggregated across dimensions.

Given any membership function for dimension j, there corresponds a particular fuzzy poverty index of the type (4.8). It is given by the sum of grades of memberships of all persons in the society divided by , the maximum number of dimensions in which all the persons are certainly deprived. These indices will differ in the way we stipulate the membership grades in the dimension. For any arbitrary given by (4.1), the general index in (4.8) treats any two dimensions as independents. For satisfaction of the fuzzy transfer axiom, we need strict convexity of over the corresponding fuzzy region. Strict convexity also ensures verification of the fuzzy monotonicity sensitivity axiom. The general index meets all other axioms unambiguously for any arbitrary of the type (4.1).

We can now state the following proposition:

An explicit specification of the general family (4.8) can be provided using the membership function (4.2). In this case, (4.8) reduces to the following fuzzy version of the multidimensional extension of the Foster et al. (1984) index suggested by Chakravarty et al. (1998) and Bourguignon and Chakravarty (2003):

4.9

where , , are arbitrary. Here reflects different attitudes toward poverty. Satisfaction of the transfer axiom by this compromise index in dimension j requires , the necessary and sufficient condition for strict convexity of . For for all , becomes , the fuzzy head-count ratio. If for all j, then becomes

4.10

The quantity is the average of membership grades of persons who are partially deprived in dimension j, where the grading is done using the Cerioli–Zani membership function given by (4.3). A weighed average of these grade averages across dimensions gives us the first term in the third bracketed expression on the right-hand side of (4.10), where the weights are the respective strict dimension-adjusted fuzzy head-count ratios in the dimensions. The second term is simply the fraction of dimensions in which an average person is certainly deprived. The sum of these two terms gives us the poverty index. For a given , an increase in , say, following a reduction in , increases the index. This index satisfies the monotonicity axiom but not its sensitivity version. It is a transgressor of the transfer axiom as well.

A second member of the general index in (4.8) is obtained by using (4.3) as the membership function. The affiliated compromise index turns out to be

4.11

where , , are arbitrary and . This index is a fuzzy translation of the multidimensional extension of the Chakravarty (1983) index suggested by Chakravarty, Mukherjee, and Ranade (1998). For a given X, for all , it is increasing in , as , its value approaches the proportion of dimension in which an average person is certainly deprived and for index coincides with the particular case of for , . For , , the index satisfies the transfer axiom and sensitivity version of the monontonicity axiom.

The general family (4.8) we have considered earlier is insensitive to a correlation-increasing switch. Given subgroup decomposability, the axiom that leads to such insensitivity property is factor decomposability. This can be avoided by using some suitable transformation of the membership functions employed in the aggregation. One such example is a Bourguignon and Chakravarty (2003) aggregation criterion. To see this more explicitly, assume that in given by (4.2) for all . Then the following compromise index may be regarded as a fuzzy sister of a member of the Bourguignon–Chakravarty family of indices in the standard situation:

4.12

where , , are arbitrary. Here means that the sum is taken over all dimensional shortfalls as fractions of the lengths of the corresponding fuzzy regions for all those dimensions in which person is partially deprived and is a parameter. The parameter is the same as in (4.9), assuming that for all . This index takes into account the joint distribution of achievements in an explicit manner. It increases or decreases in the fuzzy region under a correlation-increasing switch if or . For , the index bears similarity with the one given in (4.9) and becomes insensitive to a switch. However, satisfaction of the transfer and monotonicity sensitivity axioms is ensured in all cases.

We now systematically compare our approach with an alternative approach suggested in the literature. This latter proposal mostly aggregates first dimensional deprivation membership values advocated by Cerioli and Zani (1990) and Cheli and Lemmi (1995). In terms of our notation, this aggregated membership value for person i is given by , where is the weight assigned to dimension j. As a fuzzy poverty index, Cerioli and Zani (1990) suggested the use of the arithmetic average of the individual membership values. More precisely, their index is given by . If we employ and for defining individual membership values, then the resulting indices becomeviolators of the fuzzy transfer axiom. There is a possibility that index based on will not satisfy the fuzzy monotonicity axiom. It is a violator of the fuzzy subgroup decomposability property as well since it relies on relative frequency distributions and categorical rank orders (see Alkire et al., 2015).

As Chiappero-Martinetti (1996, 2000) argued, a more general aggregation for membership will be the use of weighted generalized mean. In fact, employs a closely related aggregation. Evidently, the process of arriving at (4.8) has a different logic than the Cerioli–Zani suggestion. We consider subgroup decomposability at the outset and then substitute a particular form of the individual poverty function to obtain (4.8) (see Chakravarty, 2006).

4.5 Fuzzy Poverty Orderings

In this section, following Zheng (2015), we assume that income is the only dimension of well-being. The objective of the section is to obtain the dominance conditions, which ensure that for a given poverty index, one income distribution is not characterized as more poverty stricken than another for all membership functions. We will also analyze the dominance conditions under variability of poverty indices for a fixed membership function. Finally, it will be worthwhile to investigate dominance when both membership functions and poverty indices are variable.

We begin by assuming that each income v is an element of , that is, . The lower and upper fuzzy boundaries for income are given respectively by and . A person with income below will definitely be counted as income-poor while a person with income above will definitely be treated as nonpoor. Since in this section we will assume continuous-type income distributions, it will be necessary to calculate the value of a fuzzy poverty index, obtained after some integration operation, at the limiting points and . Hence, anybody with income lying in the fuzzy interval will be counted as fuzzy poor. Thus, the domain of income can now be written as . The fuzzy membership function is denoted by .

Given that income distributions are defined on the continuum, let be the cumulative distribution function of income. Then gives the proportion of persons with income less than or equal to . is nondecreasing, and for some .

An additively decomposable income crisp poverty index with the unique poverty line is

4.13

where is the individual income poverty function with for and for all . Further, is nonincreasing in , that is, an increase in a poor person's income does not increase poverty, and is convex in , which means that a transfer of income from a poor person to a poorer person does not increase poverty, where the transfer does not reverse the ranks of the donor and the recipient. In the remainder of this section, we will assume that poverty indices are of additively decomposable type.

As Shorrocks and Subramanian (1994) proved, under ceteris paribus conditions, every crisp poverty index can be extended uniquely to a fuzzy poverty index

4.14

Zheng (2015) refers to for all and for any and , as the density function of the membership function . Hence, we can rewrite as

4.15

For some of the dominance conditions, it will be necessary to consider poverty indices of two varieties: relative and absolute. An additively decomposable income poverty index given by (4.13) is of relative type if

4.16

and of absolute type if

4.17

for some individual poverty function .

The following four subclasses of poverty indices will be used for presenting some of the dominance conditions:

The following proposition of Zheng (2015) identifies the partial ordering desiderata for all membership functions belonging to a general family.

According to Proposition 4.2, of two income distributions and , has at least as high fuzzy poverty as for all membership functions in if and only if for any crisp poverty index , similar poverty ranking between and holds over the fuzzy area . We refer to this as fuzzy-membership ordering because it is independent of the specific form of the membership function in the set . This is a first order condition. Zheng (2015) also derived a second order dominance condition using inverted S-shape membership functions. But this ordering is not relevant to our context, since use of such a membership function will lead to a clear violation of the fuzzy transfer axiom.

We can also derive fuzzy poverty-measure ordering for a given membership function. For this given membership function, we wish to look for the dominance conditions that will ensure that one distribution does not have higher poverty compared to another for all fuzzy poverty indices. In this context, Zheng (2015) established the following proposition for members of and .

The following proposition of Zheng (2015) can be regarded as the absolute mate of Proposition 4.3:

Finally, Zheng's (2015) following propositions establish dominance conditions under variability of both poverty indices and membership functions.

These orderings are referred to as fuzzy poverty-membership-measure orderings because of variability of both the poverty indices and membership functions. Clearly, we can arrive at this orderings using first poverty-measure ordering for a given membership function and then allowing membership function to vary or first looking at membership ordering for a given poverty index and later on considering all poverty indices.

4.6 Concluding Remarks

One issue that remains to be explored is to extend Zheng (2015)-type analysis to the multidimensional framework. We may examine the possibility of broadening the Duclos et al. (2006a,b) and Bourguignon and Chakravarty (2009) multidimensional poverty orderings to the fuzzy situation. Another issue of relevance here is assignment of membership grades to achievements in a dimension when the dimension is of ordinal nature (see Chapter 3). In a situation of this type, we can assign real numbers to fuzzy boundaries and membership grades using some specific ranking criterion. This numbering procedure is arbitrary in the sense that we need to ensure that the ranks of the fuzzy boundaries and grades should always be preserved. In such a case, formulation of some appropriate axioms and aggregation of membership grades across dimensions and individuals are our concerns. Information invariance assumption requires that the aggregated value remains invariant under renumbering of original boundaries and membership grades.

References

1. Alkire, S. and J.E. Foster. 2011. Counting and Multidimensional Poverty Measurement. Journal of Public Economics 95: 476–487.
2. Alkire, S., J.E. Foster, S. Seth, M.E. Santos, J.M. Roche, and P. Ballon. 2015. Multidimensional Poverty Measurement and Analysis. New York: Oxford University Press.
3. Amarante, V., R. Arim, and A. Vigorito. 2010. Multidimensional Poverty among Children in Uruguay. Research on Economic Inequality 18: 31–53.
4. Baliamoune-Lutz, M. and M. MacGillivray. 2006. Fuzzy Well-being Achievement in Pacific Area. Journal of the Asia Pacific Economy 11: 168–177.
5. Basu, K. 1987. Axioms for a Fuzzy Measure of Inequality. Mathematical Social Sciences 14: 275–288.
6. Basu, K. and J.E. Foster. 1998. On measuring Literacy. Economic Journal 108: 1733–1749.
7. Belhadj, B. and M. Limam. 2012. Unidimensional and Multidimensional Fuzzy Poverty Measures. Economic Modelling 29: 995–1002.
8. Belhadj, B. and M.S. Matoussi. 2010. Poverty in Tunisia: A Fuzzy Measurement Approach. Swiss Journal of Economics and Statistics 146: 431–450.
9. Bellman, R.E. and L.A. Zadeh. 1970. Decision-Making in a Fuzzy Environment. Management Science 17: BI41–BI64.
10. Bérenger, V. and F. Celestini. 2006. French Poverty Measures using Fuzzy Set Approaches. In A. Lemmi and G. Betti (eds.), 139–154.
11. Bérenger, V. and Verdier-Chouchane, A. 2007. Multidimensional Well-being: Standard of Living and Quality of Life across Countries. World Development 35: 1259–1276.
12. Betti, G., B. Cheli, A. Lemmi, and V. Verma. 2006a. Multidimensional and Longitudinal Poverty. In A. Lemmi and G. Betti (eds.), 115–137.
13. Betti, G., A. D'agostino, and L. Neri. 2006b. Modelling Fuzzy and Multidimensional Poverty Measures in the United Kingdom with Variance Components Panel Regression. In A. Lemmi and G. Betti (eds.), 257–275.
14. Betti, G. and V. Verma. 2008. Fuzzy Measures of the Incidence of Relative Poverty and Deprivation: A Multi-Dimensional Perspective. Statistical Methods and Applications 17: 225–250.
15. Bourguignon, F. and S.R. Chakravarty. 2003. The Measurement of Multidimensional Poverty. Journal of Economic Inequality 1: 25–49.
16. Bourguignon, F. and S.R. Chakravarty. 2009. Multidimensional Poverty Orderings: Theory and Applications. In K. Basu and R. Kanbur (eds.) Arguments for a Better World: Essays in Honor of Amartya Sen ,Volume 1: Ethics, Welfare and Measurement. Oxford: Oxford University Press, 337–361.
17. Cerioli, A. and Zani, S. 1990. A Fuzzy Approach to the Measurement of Poverty. In C. Dagum and M. Zenga (eds.) Income and Wealth Distribution, Inequality and Poverty. New York: Springer, 272–284.
18. Chakravarty, S.R. 1983. A New Index of Poverty. Mathematical Social Sciences 6: 307–313.
19. Chakravarty, S.R. 2006. An Axiomatic Approach to Multidimensional Poverty Measurement via Fuzzy Sets. In A. Lemmi and G. Betti (eds.), 49–72.
20. Chakravarty, S.R. 2015 Inequality, Polarization and Conflict: An Analytical Study. New York: Springer.
21. Chakravarty, S.R. and M.A. Lugo. 2016. Multidimensional Indicators of Inequality and Poverty. In M.D. Adler and M. Fleurbaey (eds.) Oxford Handbook of Well-Being and Public Policy. New York: Oxford University Press, 246–285.
22. Chakravarty, S.R., D. Mukherjee, and R. Ranade. 1998. On the Family of Subgroup and Factor Decomposable Measures of Multidimensional Poverty. Research on Economic Inequality 8: 175–194.
23. Chakravarty, S.R. and T. Roy. 1985. Measurement of Fuzziness: A General Approach. Theory and Decision 19: 163–169.
24. Cheli, B. and A. Lemmi. 1995. A “Totally” Fuzzy and Relative Approach to the Multidimensional Analysis of Poverty. Economic Notes 24: 115–133.
25. Chiappero-Martinetti, E. 1994. A New Approach to Evaluation of Well-Being and Poverty by Fuzzy Set Theory. Giornale degli Economisti e Annali di Economia 53: 367–388.
26. Chiappero-Martinetti, E. 1996. Standard of Living Evaluation Based on Sen's Approach: Some Methodological Suggestions. Notizie di Politeia 12: 37–53.
27. Chiappero-Martinetti, E. 2000. A Multidimensional Assessment of Well-Being Based on Sen's Functioning Approach. Revista Internazionale de Sciencze Sociali 108: 207–239.
28. Chiappero-Martinetti, E. 2006. Capability Approach and Fuzzy Set Theory: Description, Aggregation and Inference Issues. In A. Lemmi and G. Betti (eds.), 93–113.
29. Chiappero-Martinetti, E. 2008. Complexity and Vagueness in the Capability Approach: Strengths or Weaknesses? In F. Comin, M. Qizilbash, and S. Alkire (eds.) The Capability Approach: Concepts, Applications and Measurement. Cambridge: Cambridge University Press.
30. Clark, D.A. and D. Hulme. 2010. Poverty, Time and Vagueness: Integrating the Core Poverty and Chronic Poverty Frameworks. Cambridge Journal of Economics 34: 347–366.
31. D'Ambrosio, C., J. Deutsch, and J. Silber. 2011. Multidimensional Approaches to Poverty Measurement: An Empirical Analysis of Poverty in Belgium, France, Germany, Italy and Spain. Applied Economics 43: 951–961.
32. De Luca, A. and S. Termini. 1972. A Definition of Non-probabilistic Entropy in the Setting of Fuzzy Sets Theory. Information and Control 20: 301–312.
33. Deutsch, J. and J. Silber. 2005. Measuring Multidimensional Poverty: An Empirical Comparison of Various Approaches. Review of Income and Wealth 51: 145–174.
34. Deutsch, J. and J. Silber. 2006. The “Fuzzy Set” Approach to Multidimensional Poverty Analysis using the Shapley Decomposition to Analyze the Determinants of Poverty in Israel. In A. Lemmi and G. Betti (eds.), 155–174.
35. Dombi, J. 1990. Membership Functions as an Evaluation. Fuzzy Sets and Systems 35: 1–21.
36. Duclos, J.V., D.E. Sahn, and S.D. Younger. 2006a. Robust Multidimensional Poverty Comparisons. Economic Journal 116: 943–968.
37. Duclos, J.V., D.E. Sahn, and S.D. Younger. 2006b. Robust Multidimensional Spatial Poverty Comparisons in Ghana, Madagascar and Uganda. World Bank Economic Review 20: 91–113.
38. Foster, J.E., J. Greer, and E. Thorbecke. 1984. A Class of Decomposable Poverty Measures. Econometrica 52: 761–766.
39. Goguen, J.A. 1969. The Logic of Inexact Concepts. Synthese 19: 325–373.
40. Keefe, R. and P. Smith (eds.) 1996. Vagueness: A Reader. Cambridge, MA: MIT Press.
41. Lee, E.T. and L.A. Zadeh. 1969. Note on Fuzzy Languages. Information Sciences I: 421–434.
42. Lelli, S. 2001. Factor Analysis versus Fuzzy Sets Theory: Assessing the Influence of Different Techniques On Sen's Functioning Approach. Center of Economic Studies Discussion Paper 01.21.
43. Lemmi, A. and G. Betti (eds.) 2006. Fuzzy Set Approaches to Multidimensional Poverty Measurement. New York: Springer.
44. Makdissi, P. and Q. Wodon. 2004. Fuzzy Targeting Indices and Ordering. Bulletin of Economic Research 56: 41–51.
45. Miceli, D. 2006. Multidimensional and Fuzzy Poverty in Switzerland. In A. Lemmi and G. Betti (eds.), 195–209.
46. Molnar, M., F. Panduru, A. Vasile, and V. Duma. 2006. Multidimensional Fuzzy Set Approach to Poverty Estimates in Romania. In A. Lemmi and G. Betti (eds.), 175–194.
47. Mordeson, J.N., D.S. Malik, and T.D. Clark. 2015. Applications of Fuzzy Logic to Social Choice Theory. New York: CRC Press.
48. Mukherjee, M.N. and S.P. Sinha. 1990. On Some Strong Forms of Fuzzy Continuous Mappings on Fuzzy Topological Spaces. Fuzzy Sets and Systems 38: 375–387.
49. Ok, E. 1996. Fuzzy Income Inequality Measurement. Economic Theory 7: 513–530.
50. Panek, T. 2006. Multidimensional Fuzzy Relative Poverty Measures in Poland. In A. Lemmi and G. Betti (eds.), 233–255.
51. Qizilbash, M. 2006. Philosophical Accounts of Vagueness, Fuzzy Poverty Measures and Multidimensionality. In A. Lemmi and G. Betti (eds.), 9–28.
52. Roche, J.M. 2008. Monitoring Inequality among Social Groups: A Methodology Combining Fuzzy Set Theory and Principal Component Analysis. Journal of Human Development and Capabilities 9: 427–452.
53. Sen, A.K. 1981. Poverty and Famines: An Essay on Entitlement and Deprivation. New York: Oxford University Press.
54. Shorrocks, A.F. and S. Subramanian. 1994. Fuzzy Poverty Indices, Mimeographed, University of Essex.
55. Smithson, M. 1982. Applications of Fuzzy Set Concepts to Behavioral Sciences. Mathematical Social Sciences 2: 257–274.
56. Vero, J. 2006. A Comparison of Poverty According to Primary Goods, Capabilities and Outcomes: Evidences from French School Leavers' Surveys. In A. Lemmi and G. Betti (eds.), 211–231.
57. Wee, W.G. and K.S. Fu. 1969. A Formulation of Fuzzy Automata and its Application as a Model of Learning Systems. IEEE Transactions on Systems Science and Cybernetics SSC-5: 215–223.
58. Zadeh, L.A. 1965. Fuzzy Sets. Information and Control 8: 338–353.
59. Zheng, B. 2015. Poverty: Fuzzy Measurement and Crisp Ordering. Social Choice and Welfare 45: 203–229.