metadata: authors: 0: author: Seth,Suman
1: author: Yalonetzky,Gaston
count: Bangladesh
docna: 0: docna: Assessing Deprivation with an Ordinal
            Variable : Theory and Application to Sanitation Deprivation
            in Bangladesh
docty: Policy Research Working Paper
subtopic: Inequality,Poverty Assessment,Poverty Impact Evaluation,Poverty Monitoring & Analysis,Poverty Lines,Small Area Estimation Poverty Mapping,Poverty Diagnostics,Educational Sciences,Water and Human Health,Health and Sanitation,Environmental Engineering,Sanitary Environmental Engineering,Water Supply and Sanitation Economics,Town Water Supply and Sanitation,Sanitation and Sewerage,Small Private Water Supply Providers,Engineering,Health Care Services Industry
teratopic: Poverty Reduction,Education,Water Resources,Health, Nutrition and Population,Environment,Water Supply and Sanitation,Science and Technology Development,Industry
abstracts: cdata!: The challenges associated with poverty
            measurement with a cardinal variable have received due
            attention during the last four decades, but there is a
            dearth of literature studying how to meaningfully assess
            poverty with an ordinal variable. This paper first proposes
            a class of simple, intuitive and policy-relevant poverty
            measures for ordinal variables. These measures are sensitive
            to depth of deprivations, unlike the headcount ratio.
            Moreover, under appropriate restrictions, the measures
            ensure that priority is given to the poorest among the poor
            while targeting, monitoring and evaluating poverty
            alleviation programs. To assess the robustness of poverty
            comparisons to alternative choices of parameters, various
            stochastic dominance tests are developed (some of which are
            themselves novel contributions to the stochastic dominance
            literature). The empirical illustration documenting changes
            in sanitation deprivation in Bangladesh showcases the
            measures' ability to identify instances in which
            overall sanitation deprivation improved while leaving the
            poorest behind.
geo_reg_key: 517198,517191,517212
repnme: repnme: Assessing Deprivation with an Ordinal
            Variable : Theory and Application to Sanitation Deprivation
            in Bangladesh
display_title: Assessing Deprivation with an
            Ordinal Variable : Theory and Application to Sanitation
            Deprivation in Bangladesh
listing_relative_url: /research/2020/01/31722875/assessing-deprivation-ordinal-variable-theory-application-sanitation-deprivation-bangladesh
topicv3: Sanitation m1326453 2000,Poverty m1326687 695,Poverty Measurement and Analysis m1326941 707,Education m1327048 470,Health m1504661 3480
publishtoextweb_dt: 2020-01-23T00:00:00Z
historic_topic: Poverty Reduction,Education,Water Resources,Health, Nutrition and Population,Environment,Water Supply and Sanitation,Science and Technology Development,Industry
origu: DEC-Knowledge and Strategy (DECKS)
no_of_pages: 51
author_keys: 0: author_key: 1155370
model_extractions: Policy Research Working Paper 9120 Assessing Deprivation with an Ordinal Variable Theory and Application to Sanitation Deprivation in Bangladesh _Suman Seth Gaston Yalonetzky_ Development Economics Knowledge and Strategy Team January 2020
Policy Research Working Paper 9120

# **Abstract**

The challenges associated with poverty measurement with a cardinal variable have received due attention during the last four decades, but there is a dearth of literature studying how to meaningfully assess poverty with an ordinal variable. This paper first proposes a class of simple, intuitive and policy-relevant poverty measures for ordinal variables. These measures are sensitive to depth of deprivations, unlike the headcount ratio. Moreover, under appropriate restrictions, the measures ensure that priority is given to the poorest among the poor while targeting, monitoring and evaluating poverty alleviation programs. To assess the robustness of poverty comparisons to alternative choices of parameters, various stochastic dominance tests are developed (some of which are themselves novel contributions to the stochastic dominance literature). The empirical illustration documenting changes in sanitation deprivation in Bangladesh showcases the measures’ ability to identify instances in which overall sanitation deprivation improved while leaving the poorest behind.

This paper is a product of the Knowledge and Strategy Team, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at S.Seth@leeds.ac.uk and G.Yalonetzky@leeds.ac.uk.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
Assessing deprivation with an ordinal variable: theory and application to sanitation deprivation in Bangladesh Suman Seth<sup>_∗_</sup> Gaston Yalonetzky<sup>_†_</sup>

# **1. Introduction**

Around four decades ago, in an influential article titled ‘Poverty: An Ordinal Approach to Measurement’, Nobel laureate Amartya Sen proposed an axiomatically derived poverty measure to avoid some shortcomings of the traditionally used headcount ratio (Sen, 1976). Sen’s approach was ordinal in the sense that his poverty measures assigned an ordinal-rank weight to each poor person’s income, an otherwise cardinal variable. Since then, this seminal article has influenced a well-developed literature on poverty measurement involving cardinal variables within an axiomatic framework (Thon, 1979; Clark et al., 1981; Chakravarty, 1983; Foster et al., 1984; Foster and Shorrocks, 1988a,b; Ravallion, 1994; Shorrocks, 1995).

- _∗_ Leeds University Business School, University of Leeds, UK and Oxford Poverty & Human Development

Initiative, University of Oxford, UK, S.Seth@leeds.ac.uk.

> _†_ Leeds University Business School, University of Leeds, UK, G.Yalonetzky@leeds.ac.uk.
Distances between the values of cardinally measurable variables are meaningful; whereas ordinal variables merely consist of ordered categories. 1 When numeral scales are assigned to these categories according to their order, the implied cardinal distances are hard to interpret. Moreover, poverty orderings are inconveniently sensitive to changes in the choice of these scales. Let us consider a simple example with a five-category ordinal variable, where the four worst categories reflect deprivations. Suppose the population distributions across the five ordered categories in two countries are _A_ = (0 _._ 1 _,_ 0 _._ 3 _,_ 0 _._ 2 _,_ 0 _._ 2 _,_ 0 _._ 2) and _B_ = (0 _._ 2 _,_ 0 _._ 1 _,_ 0 _._ 1 _,_ 0 _._ 2 _,_ 0 _._ 4), respectively. If we use the numeral scales of (1 _,_ 2 _,_ 3 _,_ 4 _,_ 5) for the five categories and use the well-known poverty gap measure with the numeral scale of the fourth category (i.e., the second best category) as the poverty line, then the former society (A) reflects higher poverty than the latter (B).<sup>2</sup> However, if we simply change the numeral scales to (1 _,_ 5 _,_ 6 _,_ 7 _,_ 8), then the poverty ordering reverses.

Yet, the practice of using ordinal variables has been on the rise, in both developed and developing countries alike, due to the recent surge of interest in studying deprivation in non-monetary indicators, which are often ordinal in nature (e.g. access to basic facilities

> 1Based on the classification of measurement scales by Stevens (1946), whenever numeral scales are assigned to different ordered categories of an ordinal variable according to the orders or ranks of these categories, any ‘order-preserving’ or monotonic transformation should leave the scale form invariant. See Roberts (1979) for further in-depth discussions. In this paper, by ordinal variables we simply refer to variables with ordered categories, where numeral scales may not have necessarily been assigned to the categories.

> 2Let _xi_ be the attainment of individual _i_ and _z_ be the poverty line. Then the poverty gap measure is the average attainment shortfall from the poverty line as a proportion of the latter, attributing a zero shortfall to non-poor people. That is, if _N_ stands for the population size and _q_ stands for the number of poor people, then the poverty gap measure ( _PG_ ) is: _PG_ = _N_<sup><u>1</u></sup> � _qi_ =1 _<u>z−zxi</u>_ (Foster et al., 2013).

2
of different quality).<sup>3</sup> Moreover, there may be instances where ordinal categories of an otherwise cardinally measurable variable could have more policy relevance. For example, in some cases we might want to focus on ordered categories of income, nutritional status, or years of education completed, rather than these indicators’ cardinal values.

How should poverty be meaningfully assessed with an ordinal variable? One straightforward way may be to dichotomise the population into a group of deprived and a group of nondeprived people, and then use the _headcount ratio_ . However, this index is widely accused of ignoring the depth of deprivations (Foster and Sen, 1997). In our illustration in Section 5, for instance, in Sylhet province of Bangladesh, between 2007 and 2011, the proportion of population with inadequate sanitation facilities went down from around 70% to nearly 63%; whereas, during the same period, the proportion of people with the worst form of sanitation deprivation (‘open defecation’) increased significantly, from around 2% to more than 12% (see Table 3).

How can one reasonably capture the depth of deprivations for an ordinal variable? One approach may be to use an aggregate poverty measure that is sensitive to depth of deprivations. Alternatively one could simply either consider separately the relative frequencies of the population experiencing deprivation within each ordered category, or equivalently compute and compare headcount ratios (i.e. cumulative frequencies) for each possible poverty line. The latter two approaches seem reasonable especially when the variable under consideration

> 3For example, as part of the first Sustainable Development Goals, the United Nations has set the target to not only eradicate _extreme monetary poverty_ , but also to reduce _poverty in all its dimensions_ by 2030. See http://www.un.org/sustainabledevelopment/poverty/ (accessed in April 2017).

3
has only a few deprivation categories and also when the required number of populationcomparisons is relatively small. Otherwise they may become cumbersome: even with only four deprivation categories used in our illustration, an analysis of deprivation dynamics across six provinces of Bangladesh over three years would involve comparing a staggering number of 72 data points. Nonetheless, and in order to avoid a multitude of comparisons, one may choose to prioritise and focus on those experiencing the most severe category of deprivation; but this route comes at the expense of ignoring deprivations in other categories. Thus, in this paper, we pursue the former approach of using an aggregate poverty measure that is sensitive to all depths of deprivations within a variable.

The challenges associated with measuring well-being and inequality using an ordinal variable in an axiomatic framework have received due attention during the last few decades (e.g., Mendelson, 1987; Allison and Foster, 2004; Apouey, 2007; Abul Naga and Yalcin, 2008; Zheng, 2011; Kobus and Milos, 2012; Permanyer and D’Ambrosio, 2015; Kobus, 2015; Lazar and Silber, 2013; Yalonetzky, 2013; Gravel et al., 2015). Yet, when assessing poverty, such efforts have not been sufficiently thorough. Bennett and Hatzimasoura (2011), in a rare attempt, showed that indeed we can measure poverty with ordinal variables sensibly, but implicitly ruled out entire classes of well-suited measures (as shown by Yalonetzky, 2012). Moreover, their assessment of depth-sensitivity was restricted to the ordinal version of Pigou-Dalton transfers, thereby missing many other options including the burgeoning use of Hammond transfers (e.g. see Ebert, 2007; Gravel et al., 2015; Cowell et al., 2017; Gravel et al., 2018; Oui-Yang, 2018).<sup>4</sup>

> 4We refer to the unidimensional context here. The issue of ordinality has certainly been examined thoroughly

4
Our paper contributes theoretically to the poverty measurement literature in three ways. First, we axiomatically characterise a class of ordinal poverty measures under a minimal set of desirable properties. Our class consists of measures that are weighted sums of population proportions in deprivation categories, where these weights are referred to as _ordering weights_ because their values depend on the order of the categories. Our proposed measures are sensitive to the depth of deprivations (unlike the headcount ratio), additively decomposable, and bounded between zero and one.

Second, an adequately designed poverty measure should also ensure that policy makers have proper incentive to prioritise those poorer among the poor in the design of poverty alleviation policies so that _the poorest are not left behind_ .<sup>5</sup> In a novel attempt, we operationalise the concept of _precedence to poorer people_ by incorporating a _new_ form of _degree of poverty aversion_ in the ordinal context, reflecting a prioritarian point of view rather than an egalitarian one. Although grounded on prioritarianism, our new form of poverty aversion encompasses, as limiting cases, both previous attempts at sensitising ordinal poverty indices to the depth of deprivations (e.g., Bennett and Hatzimasoura, 2011; Yalonetzky, 2012) as well as current burgeoning approaches to distributional sensitivity in ordinal frameworks based on Hammond transfers (Hammond, 1976; Gravel et al., 2015). We define a range of properties based on this new form of degree of poverty aversion and characterise the corresponding subclasses in the context of multidimensional poverty measurement (Alkire and Foster, 2011; Bossert et al., 2013; Dhongde et al., 2016; Bosmans et al., 2017). However, even in the multidimensional context, ordinal variables are often dichotomised in empirical applications (see, Alkire and Foster, 2011; Bossert et al., 2013; Dhongde et al., 2016), thereby ignoring the depth of deprivations within indicators.

5 Poverty measures may affect the incentives of policy makers during poverty alleviation (Zheng, 1997).

5
of ordinal poverty measures. Within our framework, different degrees of poverty aversion merely require setting different restrictions on the ordering weights, preserving the measures’ additive decomposability property.

Third, since each of our classes and subclasses admits a large number of poverty measures, we develop related _stochastic dominance conditions_ whose fulfilment guarantees the robustness of poverty comparisons to alternative functional forms and measurement parameters. Although some of these conditions turn out to be the ordinal-variable analogue of existing dominance conditions for cardinal variables (Foster and Shorrocks, 1988b), several others are novel methodological contributions to the literature on stochastic dominance with ordinal variables (to the best of our knowledge).

To demonstrate the efficacy of our approach, we present an empirical illustration studying the evolution of sanitation deprivation in Bangladesh using Demographic Health Survey datasets. Interestingly, our measures are able to discern the instances where the improvements in overall sanitation deprivation did not necessarily include the poorest. Furthermore, we apply the stochastic dominance conditions to test the robustness of poverty comparisons over time.

The rest of the paper proceeds as follows. After providing the notation, we present and then axiomatically characterise the class of depth sensitive ordinal poverty measures in Section 2. Section 3 introduces the concept of precedence to poorer people, states the properties related to the degrees of poverty aversion, and characterises the subclass of relevant poverty indices. Then section 4 develops stochastic dominance conditions for the characterised class and subclasses of poverty measures. Section 5 provides an empirical illustration analysing sanitation deprivation in Bangladesh. Section 6 concludes.

6
# **2. A class of depth-sensitive poverty measures for an ordinal variable**

Suppose, there is a _social planner_ whose objective is to assess a hypothetical society’s poverty in some well-being dimension, which is measured with a set of ordered categories. For instance, self-reported health status may only include response categories, such as ‘good health’, ‘fair health’, ‘poor health’, and ‘very poor health’. Similarly, there are also instances where the ordinal categories of an otherwise cardinal variable, such as the _years of schooling completed_ , have more policy relevance.

Formally, suppose, there is a fixed set of _S ∈_ N _\ {_ 1 _}_ ordered categories _c_ 1 _, . . . , cS_ , where N is the set of positive integers. The ordered categories are such that _cs−_ 1 _≻D cs_ for all _s_ = 2 _, . . . , S_ , where _≻D_ , which reads as “is more deprived than”, is a binary and transitive relation whereby category _cs−_ 1 represents a worse-off situation than category _cs_ . Thus, _cS_ is the category reflecting least deprivation and _c_ 1 is the state reflecting highest deprivation. Suppose, for example, a society’s well-being is assessed by the education dimension and the observed ordered categories are ‘no education’, ‘primary education’, ‘secondary education’, and ‘higher education’, such that ‘no education’ _≻D_ ‘primary education’ _≻D_ ‘secondary education’ _≻D_ ‘higher education’. Then, _c_ 4 = ‘higher education’ and _c_ 1 = ‘no education’. We denote the set of all _S_ categories by **C** = _{c_ 1 _, c_ 2 _, . . . , cS}_ and the set of all categories excluding the category of least deprivation _cS_ by **C** _−S_ = **C** _\ {cS}_ .

Each individual in society must experience only one of the _S_ categories. We denote the proportion of population experiencing category _cs_ by _ps_ for all _s_ = 1 _, . . . , S_ . Clearly, _ps ≥_ 0 for all _s_ and<sup>�</sup><sup>_S_</sup> _s_ =1<sup>_ps_=1.Theproportionsofpopulationinthesocietyissummarisedby</sup> 7
the vector: **p** = ( _p_ 1 _, . . . , pS_ ). Note that **p** is nothing but the discrete probability (or relative frequency) distribution of the society’s population across the _S_ categories. We denote the set of all possible discrete probability distributions over _S_ categories by P.

It is customary in poverty measurement to define a poverty threshold for identifying the poor and the non-poor populations (Sen, 1976). Suppose, the social planner decides that category _ck_ for any 1 _≤ k < S_ and _k ∈_ N be the _poverty threshold_ , so that people experiencing categories _c_ 1 _, . . . , ck_ are identified as _poor_ ; whereas, people experiencing categories _ck_ +1 _, . . . , cS_ are identified as _non-poor_ . We assume that at least one category reflects the absence of poverty, as this restriction is both intrinsically reasonable and is required for stating certain properties in Section 2.1. When _k_ = 1, only category _c_ 1 reflects poverty and, in this case, _p_ 1 is the proportion of the population identified as poor. For any _ck ∈_ **C** _−S_ , we denote the _proportion of poor population_ , also known as the _headcount ratio_ , by _H_ ( **p** _, ck_ ) =<sup>�</sup><sup>_k_</sup> _s_ =1<sup>_ps_.</sup> We define a _poverty measure P_ ( **p** _, ck_ ) as _P_ : P _×_ **C** _−S →_ R+. In words, a poverty measure is a mapping from the set of probability distributions and the set of poverty thresholds to the real line (note that the set of categories remains fixed).

# **2.1. Desirable properties**

Now, as in many other areas of well-being measurement, the social planner needs to make certain assumptions to perform interpersonal comparisons and meaningfully aggregate the information available on the ordered deprivation categories in order to obtain a _numerical poverty measure_ . We present these assumptions in terms of properties or axioms. By relying on discrete probability distributions, we already implicitly assume that poverty measures 8
satisfy _anonymity_ and _population principle_ . Anonymity requires that merely shuffling the individual deprivation levels within a society should not alter the society’s poverty level; whereas, the population principle requires that a mere duplication of the number of individuals experiencing each deprivation level within a society should not affect the society’s poverty level. The population principle allows comparing poverty levels of societies with different population sizes. The social planner makes the following four additional assumptions.

The first property is _ordinal monotonicity_ , which requires that if the living standard of a poor person improves so that the person moves to a category of less deprivation, then societal poverty should be lower. Formally, the property requires that if a poor person or a group of poor people moves from a category _ct_ reflecting poverty (i.e. _t ≤ k_ ) to a less deprived category _cu_ (i.e. _ct ≻D cu_ ), while the deprivation levels of everybody else in the society remain unchanged, then poverty should fall. In terms of probability distributions, we set the requirement that if a fraction of the poor population is moved from category _ct_ to category _cu_ , while the proportion of population in other categories remain unchanged, then poverty should fall:

**Ordinal Monotonicity (OMN)** For any **p** _,_ **q** _∈_ P and for any _ck ∈_ **C** _−S_ , if _qt < pt_ for some _t ≤ k_ and _t < u_ but _ps_ = _qs ∀s_ = _{t, u}_ , then _P_ ( **q** _, ck_ ) _< P_ ( **p** _, ck_ ).

The second property is _single-category deprivation_ . The property requires that whenever there is only one category reflecting poverty (i.e. _c_ 1), then the poverty measure should be equal to the headcount ratio _H_ ( **p** _, c_ 1) = _p_ 1. In other words, we assume that whenever there is only one category reflecting poverty and the others reflect an absence of poverty, then 9
the headcount ratio becomes a sufficient statistic for the assessment of poverty. In fact, in this situation, any functional transformation of the headcount ratio would not add any meaningful information to the poverty assessment while being inferior in terms of intuitive interpretation.

**Single-Category Deprivation (SCD)** For any **p** _∈_ P and _c_ 1 _∈_ **C** _−S_ , _P_ ( **p** _, c_ 1) = _p_ 1.

The third property, _focus_ , is essential for a poverty measure. It requires that, _ceteris paribus_ , a change in a non-poor person’s situation should not alter societal poverty evaluation as long as the non-poor person remains in that status. In terms of probability distributions, we set the requirement that as long as the proportion of poor population within each of the _k_ categories reflecting poverty remains unchanged, the level of poverty should be the same. Note that the proportions of non-poor people may remain unchanged or may be different across the _S − k_ categories _not_ reflecting poverty, but this should not matter for poverty evaluation.

**Focus (FOC)** For any **p** _,_ **q** _∈_ P and for any _ck ∈_ **C** _−S_ , if _qs_ = _ps ∀s ≤ k_ , then _P_ ( **q** _, ck_ ) = _P_ ( **p** _, ck_ ).

Finally, the social planner may be interested in exploring the relationship between the overall poverty evaluation and the subgroup poverty evaluation, where population subgroups can be geographical regions, ethnic groups, etc. Suppose society is partitioned into _M ∈_ N _\{_ 1 _}_ mutually exclusive and collectively exhaustive population subgroups. We denote the population share in subgroup _m_ by _πm_ , such that _πm ≥_ 0 _∀m_ = 1 _, . . . , M_ and<sup>�</sup><sup>_M_</sup> _m_ =1<sup>_πm_=1.</sup> 10
We further denote the probability distribution across _S_ categories within subgroup _m_ by **p**<sup>_m_</sup> = ( _p_<sup>_m_</sup> 1<sup>_, . . . , pm_</sup> _S_<sup>)</sup><sup>_∈_Pforevery</sup><sup>_m_=1</sup><sup>_, . . . , M_,suchthat</sup><sup>**p**=�</sup><sup>_M_</sup> _m_ =1<sup>_πm_</sup><sup>**p**</sup><sup>_m_.Thefinal</sup> property, _subgroup decomposability_ , requires overall societal poverty to be expressible as a population-weighted average of subgroup poverty levels:

**Subgroup Decomposability (SUD)** For any _M ∈_ N _/{_ 1 _}_ , for any **p** _∈_ P such that **p** = � _Mm_ =1<sup>_πm_</sup><sup>**p**</sup><sup>_m_where(i)</sup><sup>**p**</sup><sup>_m∈_P</sup><sup>_∀m_= 1</sup><sup>_, . . . , M_,(ii)</sup><sup>_πm≥_0,and(iii)�</sup><sup>_M_</sup> _m_ =1<sup>_πm_= 1,andfor</sup> any _ck ∈_ **C** _−S_ ,

# **2.2. The class of poverty measures**

The four properties outlined in Section 2.1 lead to the class of poverty measures that we present in Theorem 2.1:

**Theorem 2.1** A poverty measure _P_ satisfies properties OMN, SCD, FOC and SUD if and only if

where _ω_ 1 = 1, _ωs−_ 1 _> ωs >_ 0 for all _s_ = 2 _, ..., k_ whenever _k ≥_ 2, and _ωs_ = 0 for all _s > k_ .

**Proof.** See Appendix A1.

We denote our class of poverty measures in Theorem 2.1 by _P_ . A poverty measure in our proposed class is a weighted sum of the population proportions in **p** , where the weights (i.e. _ωs_ ’s) are _non-negative_ for all categories, _strictly positive_ for the deprived categories, 11
and _unity_ for the most deprived category. We refer to weights _ωs_ ’s as _ordering weights_ and to **_ω_** = ( _ω_ 1 _, . . . , ωS_ ) as the _ordering weighting vector_ .<sup>6</sup> The ordering weights increase with deprived categories representing higher levels of deprivation. In practice, the ordering weights may take various forms. For example, Bennett and Hatzimasoura (2011) make the ordering weight for each deprivation category depend on the latter’s relative deprivation rank. Category _s_ is assigned an ordering weight equal to _ωs_ = [( _k − s_ + 1) _/k_ ]<sup>_θ_</sup> for all _s_ = 1 _, . . . , k_ and for some _θ >_ 0. Thus, the least deprived category _ck_ receives an ordering weight of _ωk_ = 1 _/k_<sup>_θ_</sup> ; whereas, the most deprived category _c_ 1 receives an ordering weight of _ω_ 1 = 1.

The class of poverty measures in Equation 2.1 bears several policy-relevant features. First, an appealing feature of the measures in our class is their _simplicity_ . Second, unlike the headcount ratio, the poverty measures in our class are _sensitive to the depth of deprivations_ as they assign larger weights to the more deprived categories. Thus, unlike the headcount ratio, the proposed measures are sensitive to changes in deprivation status among the poor even when they do not become non-poor owing to those changes. Third, the proposed poverty measures are _additively decomposable_ , which has two crucial policy implications. One is that the society’s overall poverty measure may be expressed as a population-weighted average of the population subgroups’ poverty measures, whenever the entire population is di-

> 6Kobus and Milos (2012, Theorem 3) also showed that a subgroup decomposable inequality measure for ordinal variables that is sensitive to spreads away from the median is a monotonic transformation of the _weighted sum of population proportions_ . However, our ordering weights are significantly different both in terms of their restrictions and interpretation vis-a-vis those involved in ordinal inequality measurement. Moreover, our SCD property (see below) imposes a more stringent restriction on the permissible functional transformations.

12
vided into mutually exclusive and collectively exhaustive population subgroups. The other is that additively decomposable measures are convenient for cross-sectional and inter-temporal econometric analysis as well as impact evaluation exercises. Fourth, the poverty measures are conveniently _normalised_ between zero and one. They are equal to zero only in a society where nobody is poor; whereas, they are equal to one only whenever everybody in the society experiences the worst possible deprivation category _c_ 1. Note that the normalisation behaviour of our measures between zero and one, especially owing to the restrictions on _ω_ ’s, is not an axiomatic assumption, but a logical conclusion from the foundational properties. Fifth, the poverty measure boils down to the _headcount ratio_ either when the poverty threshold is represented by the most deprived category or whenever the underlying ordinal variable has merely two categories.

Each poverty measure in class _P_ may also have the following alternative interpretation: when a policy maker only observes the population’s relative frequencies across ordered deprivation categories, then, based on the four assumptions, the policy maker assigns particular deprivation values in the form of _ωs_ ’s to these individuals. Each poverty measure in our proposed class is an average of these assigned deprivation values.<sup>7</sup>

# **3. Precedence to the poorer among the poor**

Poverty alleviation is a gradual process, where it is imperative to ensure that the poorest of the poor are not _left behind_ . Although all poverty measures in Equation (2 _._ 1) are sensitive to

> 7For an alternative version of the axiomatic characterisation of the measures in class _P_ using anonymity and population principle alongside the aforementioned properties, see Seth and Yalonetzky (2018).

13
the depth of deprivations, not all of them ensure that the poorest among the poor population receive precedence over the less poor population during a poverty alleviation process. For that purpose, we introduce an intuitive concept of giving _precedence to poorer people_ in the _ordinal_ framework, in tune with the _prioritarian view_ which holds that ‘benefiting people matters more the worse off these people are’ (Parfit, 1997, p. 213).<sup>8</sup> Furthermore, our notion of precedence is presented in a general framework where the _degree of precedence_ to poorer people can vary between a minimum and a maximum.<sup>9</sup> Let us introduce the concept using the example in Figure 1. Suppose, there are ten ordered deprivation categories, where _c_ 1 is the most deprived category and _c_ 10 is the least deprived category. Denoted by black circles, categories _c_ 1 _, . . . , c_ 7 reflect poverty; whereas the other three categories, denoted by gray circles, do not. Thus, _c_ 7 is the poverty threshold category, which is highlighted by a large black circle in each distribution. The original distribution is **p** , where each individual experiences one of the ten categories.

Suppose the policy maker has the following two competing options: either (a) obtain distribution **p**<sup>_′_</sup> from **p** by assisting a fraction _ε ∈_ (0 _,_ 1) of poor people to move from category _c_ 1 to category _c_ 2 ; or (b) obtain distribution **q** from **p** by assisting a fraction _ε_ of poor people to move from category _c_ 2 to category _c_ 3. Which option should lead to a larger reduction in poverty? One way of giving precedence to poorer people is to require that the move from

> 8For an application of the prioritarian concept to the multidimensional context, see Bosmans et al. (2017). For further discussion, comparing the prioritarian view with the egalitarian view see Fleurbaey (2015, p. 208).

> 9The concept is analogous to the degree of poverty or inequality aversion in the monetary poverty measurement literature (Clark et al., 1981; Chakravarty, 1983; Foster et al., 1984), but not technically identical.

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Figure 1: Precedence to poorer and the degree of precedence

|_c_10<br>_c_10<br>_c_10|_c_9<br>_c_9<br>_c_9|_c_8<br>_c_8<br>_c_8|_ck_=7<br>_ck_=7<br>_ck_=7|_c_6<br>_c_6<br>_c_6|_c_5<br>_c_5<br>_c_5|_c_4<br>_c_4<br>_c_4|_c_3<br>_c_3<br>_c_3|_c_2<br>_c_2<br>_c_2|_c_1<br>_c_1<br>_c_1|**p**<br>**p**<sup>_′_</sup><br>**q**|
|---|---|---|---|---|---|---|---|---|---|---|
|||||||||||<sup>_′_</sup>|
|_c_10|_c_9|_c_8|_ck_=7|_c_6|_c_5|_c_4|_c_3|_c_2|_c_1|**q**|
|_c_10|_c_9|_c_8|_ck_=7|_c_6|_c_5|_c_4|_c_3|_c_2|_c_1|**q**<sup>_′′_</sup>|

**p** to **p**<sup>_′_</sup> should lead to a larger reduction in poverty than the move from **p** to **q** . It is a minimal criterion for giving precedence to poorer people which we call _minimal precedence to poorer people_ (PRE-M). This property requires that, _ceteris paribus_ , moving a fraction of poorer people to an adjacent less deprived category leads to a larger reduction in poverty than moving a fraction of less poor people to a respectively adjacent less deprived category.<sup>10</sup>

The PRE-M property presents a minimal criterion for giving precedence to poorer people. Yet what happens when the policy maker faces the alternatives of improving the situation of a fraction of poorer people by one category and improving the situation of a similar fraction of less poor people by several categories? To ensure that the policy maker still chooses to improve the situation of the poorer people in these cases, we introduce the property of _greatest precedence to poorer people_ (PRE-G). This property requires that, _ceteris paribus_ , moving a

> 10We have defined only the strict versions of these properties, requiring poverty to be strictly lower in the aftermath of specific pro-poorest distributional change. Consequently the ensuing results impose strict inequality restrictions on weights. However, these strict restrictions may be relaxed with alternative versions if the latter only require that poverty does not rise due to the same pro-poorest distributional change.

15
fraction _ε_ of poorer people to a less deprived category leads to a larger reduction in poverty than moving a fraction _ε_ of less poor people to _any number_ of less deprived categories. Note here that the improvement is not restricted to a particular number of adjacent categories. For example, if PRE-G held, a move from **p** to **p**<sup>_′_</sup> in Figure 1 should lead to a larger reduction in poverty than even a move from **p** to **q**<sup>_′′_</sup> .

Conceptually, the PRE-G property is analogous to the notion of a Hammond transfer (Hammond, 1976; Gravel et al., 2015), which essentially involves, simultaneously, an improvement in a poor person’s situation and a deterioration of a less poor person’s situation, such that their deprivation ranks are not reversed (in the case of poverty measurement). Also, an ordinal poverty measure satisfying property PRE-G also satisfies property PRE-M, but the reverse is not true. A policy maker supporting property PRE-G over property PRE-M should be considered more poverty averse.

We could also consider intermediate forms of preference between the minimal (PRE-M) and the greatest (PRE-G) forms of precedence. For example, instead of the greatest forms of precedence, the policymaker may be satisfied with requiring, that, _ceteris paribus_ , moving a fraction _ε_ of poorer people to an adjacent less deprived category leads to a larger reduction in poverty than moving a fraction _ε_ of less poor people to, say, two adjacent less deprived categories. In such case, a move from **p** to **p**<sup>_′_</sup> in Figure 1 should lead to a larger reduction in poverty than a move from **p** to **q**<sup>_′_</sup> . We refer to this case as _precedence to poorer people of order two_ (PRE-2).

Likewise, we may obtain the policy maker’s preferred degree _α_ of giving precedence to poorer people. Thus, we introduce the general property of _precedence to poorer people of order α_ , 16
which requires that, _ceteris paribus_ , moving a fraction _ε_ of poorer people to an adjacent less deprived category leads to a larger reduction in poverty than moving a fraction _ε_ of less poor people up to an _α_ ( _≥_ 1) number of adjacent less deprived categories. A formal general statement of the property, which includes PRE-M and PRE-G as _limiting cases_ , is the following:

**Precedence to Poorer People of Order** **_α_ (PRE-** **_α_ )** For any **p** _,_ **p**<sup>_′_</sup> _,_ **q**<sup>_′_</sup> _∈_ P, for any _k ≥_ 2, for any _ck ∈_ **C** _−S_ , and for some _α ∈_ N such that 1 _≤ α ≤ k−_ 1, for some _s < t ≤ k < S_ and for some _ε ∈_ (0 _,_ 1), if (i) **p**<sup>_′_</sup> is obtained from **p** such that _p_<sup>_′_</sup> _s_<sup>=</sup><sup>_ps−ε_while</sup><sup>_p′_</sup> _u_<sup>=</sup><sup>_pu∀u_=</sup><sup>_{s, s_+</sup> 1 _}_ , and (ii) **q**<sup>_′_</sup> is obtained from **p** such that _qt_<sup>_′_=</sup><sup>_pt−ε_while</sup><sup>_q_</sup> _u_<sup>_′_=</sup><sup>_pu∀u_=</sup><sup>_{t,_min</sup><sup>_{t_+</sup><sup>_α, S}}_,</sup> then _P_ ( **p**<sup>_′_</sup> _, ck_ ) _< P_ ( **q**<sup>_′_</sup> _, ck_ ).

Note that PRE-1 is essentially the PRE-M property. This is the case where the social planner is least poverty averse. As the value of _α_ increases, the social planner’s poverty aversion rises. In this framework, the social planner’s poverty aversion is highest at _α_ = _k −_ 1. We shall subsequently show that PRE- _α_ for _α_ = _k −_ 1 leads to the same subclass of ordinal poverty measures as the PRE-G property. The PRE- _α_ property imposes further restrictions on the class of measures in Theorem 2.1. In Theorem 3.1, we present the subclass of measures _Pα_ that satisfy the general PRE- _α_ property:

**Theorem 3.1** For any _k ≥_ 2 and for some _α ∈_ N such that 1 _≤ α ≤ k −_ 1, a poverty measure _P ∈P_ (Theorem 2.1) satisfies property PRE- _α_ if and only if

a. _ωs−_ 1 _− ωs > ωs − ωs_ + _α ∀s_ = 2 _, . . . , k − α_ and _ωs−_ 1 _>_ 2 _ωs ∀s_ = _k − α_ +1 _, . . . , k_ whenever _α ≤ k −_ 2.

17
b. _ωs−_ 1 _>_ 2 _ωs ∀s_ = 2 _, . . . , k_ whenever _α_ = _k −_ 1.

**Proof.** See Appendix A2.

In a novel effort, Theorem 3.1 presents various subclasses of indices based on the degree of poverty aversion _α_ , which we denote as _Pα_ . In order to give precedence to poorer people, additional restrictions must be imposed on the ordering weights. Corollary 3.1 presents the limiting case of _P_ 1, featuring the least poverty averse social planner:

**Corollary 3.1** For any _k ≥_ 2, a poverty measure _P ∈P_ (Theorem 2.1) satisfies property PRE-M (i.e. PRE-1) if and only if _ωs−_ 1 _− ωs > ωs − ωs_ +1 _∀s_ = 2 _, ..., k −_ 1 and _ωk−_ 1 _>_ 2 _ωk_ .

**Proof.** The result follows directly from Theorem 3.1 by setting _α_ = 1.

To give precedence to poorer people in the spirit of property PRE-M, the ordering weights must be such that the difference _ωs−_ 1 _− ωs_ is larger than the subsequent difference _ωs − ωs_ +1, in addition to the restrictions imposed by Theorem 2.1. Suppose, we summarise the ordering weights by: **_ω_** = ( _ω_ 1 _, . . . , ωS_ ). Let us consider an example involving five categories and two ordering weight vectors: **_ω_**<sup>_′_</sup> = (1 _,_ 0 _._ 8 _,_ 0 _._ 5 _,_ 0 _,_ 0) and **_ω_**<sup>_′′_</sup> = (1 _,_ 0 _._ 5 _,_ 0 _._ 2 _,_ 0 _,_ 0), where _k_ = 3. The ordering weights in **_ω_**<sup>_′_</sup> fulfill all properties presented in Theorem 2.1, but the largest reduction in poverty is obtained whenever a poor person moves from the least poor category to the adjacent non-poor category. By contrast, ordering weights in **_ω_**<sup>_′′_</sup> require that the largest reduction in poverty be obtained whenever a poor person moves from the poorest category to the adjacent second poorest category. Thus, unlike the ordering weights in **_ω_**<sup>_′_</sup> , the ordering weights in **_ω_**<sup>_′′_</sup> make sure that poorer people receive precedence.

18
Next we present the subclass of poverty measures satisfying property PRE-G, i.e. _PG_ :

**Proposition 3.1** For any _k ≥_ 2, a poverty measure _P ∈P_ (Theorem 2.1) satisfies property PRE-G if and only if _ωs−_ 1 _>_ 2 _ωs ∀s_ = 2 _, . . . , k_ .

**Proof.** See Appendix A3.

The additional restriction on the ordering weights in Proposition 3.1 effectively prioritises the improvement in a poorer person’s situation over improvement of any extent in a less poor person’s situation. Let us consider an example involving five categories and two ordering weight vectors: **_ω_**<sup>_′_</sup> = (1 _,_ 0 _._ 6 _,_ 0 _._ 3 _,_ 0 _._ 1 _,_ 0) and **_ω_**<sup>_′′_</sup> = (1 _,_ 0 _._ 48 _,_ 0 _._ 23 _,_ 0 _._ 1 _,_ 0), where _k_ = 4. Clearly, both sets of weights in **_ω_**<sup>_′_</sup> and **_ω_**<sup>_′′_</sup> satisfy the restriction in Corollary 3.1 that _ωs−_ 1 _− ωs > ωs − ωs_ +1 for all _s_ = 2 _, . . . , k_ . However, the ordering weights in **_ω_**<sup>_′_</sup> do not satisfy the restriction in Proposition 3.1, since _ω_ 1<sup>_′<_2</sup><sup>_ω_</sup> 2<sup>_′_;whereastheorderingweightsin</sup><sup>**_ω_**</sup><sup>_′′_dosatisfy</sup> the restriction in Proposition 3.1 as _ωs_<sup>_′′_</sup> _−_ 1<sup>_>_2</sup><sup>_ω_</sup> _s_<sup>_′′_forall</sup><sup>_s_= 2</sup><sup>_, . . . , k_.</sup> An interesting feature of the set of weights satisfying property PRE-G is that, for _k ≥_ 3, any deprivation category up to the third least detrimental deprivation category (i.e. _k −_ 2), should receive a weight greater than the sum of weights assigned to all categories reflecting lesser deprivation, i.e. _ωs >_<sup>�</sup><sup>_k_</sup> _l_ = _s_ +1<sup>_ωl_forall</sup><sup>_s_= 1</sup><sup>_, . . . , k −_2.</sup> Finally, it is worth pointing out that, remarkably, the subclasses _PG_ (Proposition 3.1) and _Pk−_ 1 (Theorem 3.1 when _α_ = _k −_ 1) are identical; even though the distributional changes involved in the PRE- _α_ property are only specific cases of those involved in PRE-G. Besides being of interest in itself, this perfect overlap between the subclasses of indices will prove useful in Section 4 because by deriving the dominance conditions for the subclasses _Pα_ , we 19
will also obtain the relevant dominance conditions for subclass _PG_ .<sup>11</sup>

# **4. Poverty dominance conditions**

Stochastic dominance conditions come in handy whenever we want to ascertain the robustness of a poverty ranking of distributions to alternative reasonable comparison criteria, e.g. selection of poverty lines, choice of different functional forms, etc. (Atkinson, 1987; Foster and Shorrocks, 1988b; Fields, 2001). Moreover, often stochastic dominance conditions reduce an intractable problem of probing the robustness of a comparison across an infinite domain of alternative criteria to a finite set of distributional tests (Levy, 2006). In Sections 2 and 3, we introduced the class of poverty measures _P_ and its subclasses _Pα_ . The main parameters for these measures are the set of ordering weights _{ω_ 1 _, . . . , ωk}_ and the poverty threshold category _ck_ . It is thus natural to inquiry into the circumstances under which poverty comparisons based on ordinal variables are robust to alternative ordering weights as well as alternative poverty threshold categories. In this section, first we introduce the first-order dominance conditions relevant to _P_ , followed by the second-order dominance conditions for _Pα_ for all _α_ .

> 11Noteworthy is the expected resemblance between the weighting restriction identified in Proposition 3.1 and that in the class of welfare functions for ordinal variables derived in Gravel et al. (2015, Lemma 2). The latter characterises welfare functions that increase both when someone moves to a better category (so-called increments) and in the aftermath of Hammond transfers. Setting _αk_ = 0 and changing the inequality sign in Gravel et al. (2015, Lemma 2) to interpret their _α_ functions (not to be confused with our _α_ parameter) as ordering weights for poverty measurement yields the weight restriction in Proposition 3.1.

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# **4.1. Poverty dominance conditions for all measures in** _P_

Theorem 4.1 provides poverty dominance conditions that are relevant to all measures in class _P_ , but for a _given poverty threshold category ck ∈_ **C** _−_ **S** :

**Theorem 4.1** For any **p** _,_ **q** _∈_ P and a given _ck ∈_ **C** _−_ **S** , _P_ ( **p** _, ck_ ) _< P_ ( **q** _, ck_ ) _∀P ∈P_ if and only if<sup>�</sup><sup>_s_</sup> _l_ =1<sup>(</sup><sup>_pl−ql_)</sup><sup>_≤_0</sup><sup>_∀s ≤k_withatleastonestrictinequality.</sup>

**Proof.** See Appendix A4.

Once a particular poverty threshold category _ck ∈_ **C** _−_ **S** is chosen, Theorem 4.1 states that poverty in distribution **p** _∈_ P is strictly lower than that in distribution **q** _∈_ P for all measures in _P_ if and only if all partial sums of the probabilities up to category _ck_ in **p** are nowhere higher and at least once strictly lower than the respective partial sums in **q** . The result in Theorem 4.1 may also be presented in terms of the headcount ratio _H_ ( **p** _, ck_ ) as follows: the poverty comparison for a particular poverty threshold category _ck_ is robust to all poverty measures in _P_ if and only if _H_ ( **p** _, cs_ ) _≤ H_ ( **q** _, cs_ ) for all _s ≤ k_ and _H_ ( **p** _, cs_ ) _< H_ ( **q** _, cs_ ) for at least one _s ≤ k_ .

Corollary 4.1 extends the first-order dominance condition obtained in Theorem 4.1 to any measure _P ∈P_ and _all poverty threshold categories_ in **C** _−_ **S** :

**Corollary 4.1** For any **p** _,_ **q** _∈_ P and all _ck ∈_ **C** _−_ **S** , _P_ ( **p** _, ck_ ) _< P_ ( **q** _, ck_ ) for any _P ∈P_ and for all _ck ∈_ **C** _−_ **S** if and only if<sup>�</sup><sup>_s_</sup> _l_ =1<sup>(</sup><sup>_pl−ql_)</sup><sup>_≤_0</sup><sup>_∀s_= 2</sup><sup>_, . . . , S −_1and</sup><sup>_p_1</sup><sup>_< q_1.</sup>

**Proof.** The sufficiency part is straightforward and follows from Equation A1. We prove the necessary condition as follows. First, consider _k_ = 1. Then, _P_ ( **p** _, c_ 1) _< P_ ( **q** _, c_ 1) only if

21
_p_ 1 _< q_ 1. Subsequently, the requirement that<sup>�</sup><sup>_s_</sup> _l_ =1<sup>_pl≤_�</sup><sup>_s_</sup> _l_ =1<sup>_ql∀s_=2</sup><sup>_, . . . , S−_1follows</sup> from Theorem 4.1.

Interestingly, in terms of headcount ratios, poverty in distribution **p** is lower than poverty in distribution **q** for any _P ∈P_ and for all possible poverty threshold categories if and only if _H_ ( **p** _, cs_ ) _≤ H_ ( **q** _, cs_ ) for all _s ≤ k_ and _H_ ( **p** _, c_ 1) _< H_ ( **q** _, c_ 1). Thus, the results in Theorem 4.1 and Corollary 4.1 are the ordinal versions of the headcount-ratio orderings for continuous variables derived by Foster and Shorrocks (1988b).

# **4.2. Poverty dominance conditions for all measures in** _Pα_

We now present poverty dominance conditions that are relevant to all measures in subclass _Pα_ for some _α ∈_ [1 _, k −_ 1]. We refer to the poverty dominance for a particular value of _α_ as PRE- _α_ dominance. For the presentation of our results in this section, we define the following additional notation. First, by _⌊b⌋_ for any _b ∈_ R++, we denote the largest possible non-negative integer that is not greater than _b_ (for example, if _b_ = 5 _._ 2 then _⌊b⌋_ = 5). Secondly, for some _k ≥_ 2, we denote the _r_<sup>th</sup> ordinal weighting vector by **_ω_**<sup>_r_</sup> = ( _ω_ 1<sup>_r, . . . , ω_</sup> _S_<sup>_r_)</sup> for all _r_ = 1 _, . . . , k_ . In Theorem 4.2, we show that one needs to evaluate poverty orderings across two distributions at _k_ distinct ordinal weighting vectors to test PRE- _α_ dominance and we explicitly derive the weights _ωs_<sup>_r_forall</sup><sup>_s_= 1</sup><sup>_, . . . , S_andforall</sup><sup>_r_= 1</sup><sup>_, . . . , k_.</sup> Theorem 4.2 presents the PRE- _α_ dominance conditions for a _given poverty threshold category ck ∈_ **C** _−_ **S** and _k ≥_ 2:

**Theorem 4.2** For any **p** _,_ **q** _∈_ P, for some _k ≥_ 2, and for some _α ∈_ [1 _, k −_ 1] _⊆_ N, _P_ ( **p** _, ck_ ) _<_

22
_P_ ( **q** _, ck_ ) _∀P ∈Pα_ for a given _ck ∈_ **C** _−_ **S** if and only if, with at least one strict inequality,

**Proof.** See Appendix A5.

For a given _ck ∈_ **C** _−_ **S** and for a given _α ∈_ [1 _, k −_ 1], according to Theorem 4.2, poverty in distribution **p** is lower than that in distribution **q** for all _P ∈Pα_ (i.e., **p** PRE- _α_ dominates **q** ) if and only if<sup>�</sup><sup>_S_</sup> _s_ =1<sup>_ω_</sup> _s_<sup>_rps≤_�</sup><sup>_S_</sup> _s_ =1<sup>_ω_</sup> _s_<sup>_rqs∀r_=1</sup><sup>_, . . . , k_withatleastonestrictinequality,</sup> where the values for all _ωs_<sup>_r_’saredeterminedbyEquation4.1.Thefirstconditionwithin</sup> Equation 4.1 requires that _ωs_<sup>_r_=0whenever</sup><sup>_s>r_;whereas,thesecondconditionrequires,</sup> based on Theorem 2.1, that _ω_ 1<sup>_r_=1forall</sup><sup>_r_=1</sup><sup>_, . . . , k_.Forexample,for</sup><sup>_k_=3,</sup><sup>**_ω_**1=</sup> ( _ω_ 1<sup>1</sup><sup>_, ω_</sup> 2<sup>1</sup><sup>_, ω_</sup> 3<sup>1)=(1</sup><sup>_,_0</sup><sup>_,_0).ThethirdconditionwithinEquation4.1requiresthatforvaluesof</sup> _r_ ranging between 2 and _α_ + 1 for a given value of _α_ , _ωs_<sup>_r_=21</sup><sup>_−s_forall</sup><sup>_s_=1</sup><sup>_, . . . , r_,dueto</sup> the restriction _ωs−_ 1 _>_ 2 _ωs ∀s_ = 2 _, . . . , r_ whenever _α_ = _r −_ 1 or _r_ = _α_ + 1 as in part b. of Theorem 3.1. The fourth and fifth conditions within Equation 4.1 follow from the second 23
condition and the first condition, respectively, of part a. of Theorem 3.1.

For any chosen value of _α_ , _k_ different conditions must be tested for concluding dominance in a poverty comparison.<sup>12</sup> Let us provide an example with _S_ = 6 and _k_ = 5. In this case, _α_ may take four different values: _α_ = 1 _,_ 2 _,_ 3 _,_ 4. Moreover, since _S > k_ , _ω_ 6<sup>_k_=0for</sup> all _k_ . For each value of _α_ , the condition<sup>�6</sup> _s_ =1<sup>_ω_</sup> _s_<sup>_rps≤_�6</sup> _s_ =1<sup>_ω_</sup> _s_<sup>_rqs_,withonestrictinequality,</sup> must be satisfied for the five ordinal weighting vectors **_ω_**<sup>_r_</sup> corresponding to _r_ = 1 _, . . . ,_ 5, as presented in Table 1. Each column of the table presents the five ordering weighting vectors for a given value of _α_ ; whereas, the _k_<sup>th</sup> row of the table reports the _k_<sup>th</sup> ordering ̄ weights vector for different values of _α_ . The corresponding values of _r_ are also reported where they are relevant. For instance, if _α_ = 2, then poverty comparisons should be checked at the following five ordering weighting vectors: **_ω_**<sup>1</sup> = (1 _,_ 0 _,_ 0 _,_ 0 _,_ 0 _,_ 0), **_ω_**<sup>2</sup> = �1 _,_<sup><u>1</u></sup> 2<sup>_,_0</sup><sup>_,_0</sup><sup>_,_0</sup><sup>_,_0</sup> �, **_ω_**<sup>3</sup> = �1 _,_ 2<sup><u>1</u></sup><sup>_,_</sup><sup><u>1</u></sup> 4<sup>_,_0</sup><sup>_,_0</sup><sup>_,_0</sup> �, **_ω_**<sup>4</sup> = �1 _,_<sup><u>4</u></sup> 7<sup>_,_</sup><sup><u>2</u></sup> 7<sup>_,_</sup><sup><u>1</u></sup> 7<sup>_,_0</sup><sup>_,_0</sup> �, and **_ω_**<sup>5</sup> = �1 _,_ 12<sup><u>7</u></sup><sup>_,_</sup> 12<sup><u>4</u></sup><sup>_,_</sup> 12<sup><u>2</u></sup><sup>_,_</sup> 12<sup><u>1</u></sup><sup>_,_0</sup> �. We also provide simplified versions of the dominance conditions presented in Theorem 4.2 for the two extreme cases of minimal precedence (i.e. _α_ = 1) as in Corollary 3.1 and of greatest precedence (i.e. _α_ = _k −_ 1) as in Proposition 3.1. Whenever _α_ = 1, we must test _k_ restrictions on the linear combinations of partial sums of probabilities as in Corollary 4.2:

**Corollary 4.2** For any **p** _,_ **q** _∈_ P and for some _k ≥_ 2, _P_ ( **p** _, ck_ ) _< P_ ( **q** _, ck_ ) _∀P ∈P_ 1 for _l_ a given _ck ∈_ **C** _−_ **S** if and only if<sup>�</sup><sup>_s_</sup> _l_ =1 � _j_ =1<sup>(</sup><sup>_pj−qj_)</sup><sup>_≤_0</sup><sup>_∀s≤k_withatleastonestrict</sup> inequality.

> 12This approach to dominance testing is analogous to that used in robustness tests for composite indices with respect to alternative weights (see, Seth and McGillivray, 2018; Foster et al., 2012).

24
~~<u>Le)</u>~~
with at least one strict inequality and (b)<sup>�</sup><sup>_S_</sup> _s_ =1<sup>_ω_</sup> _s_<sup>_r_(</sup><sup>_ps−qs_)</sup><sup>_≤_0</sup><sup>_∀r_=3</sup><sup>_, . . . , S−_1,where</sup> **_ω_**<sup>_r_</sup> = ( _ω_ 1<sup>_r, . . . , ω_</sup> _S_<sup>_r_)</sup><sup>_∀r_= 3</sup><sup>_, . . . , S −_1areobtainedfromEquation4.1ofTheorem4.2.</sup>

**Proof.** First consider the case when _k_ = 2. Then from Equation 4.1 of Theorem 4.2, it follows that _p_ 1 _≤ q_ 1 and 2 _p_ 1 + _p_ 2 _≤_ 2 _q_ 1 + _q_ 2, with at least one strict inequality. Therefore, conditional (a) is both necessary and sufficient. Whenever, _k ≥_ 3, part (b) follows from Theorem 4.2 by setting _k_ = _S −_ 1.

In summary, the robustness of poverty comparisons for various classes and subclasses of ordinal measures introduced in Sections 2.1 and 3 can be assessed with a battery of dominance tests based on the theorems and corollaries presented in this section.

# **5. Empirical illustration: Sanitation deprivation in Bangladesh**

We now present an empirical illustration in order to showcase the efficacy of our proposed measurement method. In the current global development context, both the United Nations through the Sustainable Development Goals<sup>13</sup> and the World Bank through their Report of the Commission on Global Poverty (World Bank, 2017) have acknowledged the need for assessing, monitoring, and alleviating poverty in multiple dimensions besides the monetary dimension. In practice, most non-income dimensions are assessed by ordinal variables. In this section, we show how our measurement tools may be applied to analyse inter-temporal sanitation deprivation in Bangladesh.

For our analysis, we use the nationally representative Demographic Health Survey (DHS)

> 13Available at https://sustainabledevelopment.un.org/sdgs.

26
datasets of Bangladesh for the years 2007, 2011, and 2014. We compute the point estimates and standard errors incorporating the sampling weights as well as respecting the survey design.<sup>14</sup> Excluding the non-usual residents, we were able to consider the information on 50,215 individuals from 10,398 households in the 2007 survey, 79,483 individuals from 17,139 households in the 2011 survey, and 77,680 individuals from 17,299 households in the 2014 survey.

Table 2: The five ordered categories of access to sanitation facilities

|Category|Description|
|---|---|
|Open defecation|Human faeces disposed of in felds, forests, bushes, open bodies of water, beaches or <br>open spaces or disposed of with solid waste|
|Unimproved|Pit latrines without a slab or platform, hanging latrines and bucket latrines|
|Limited|Sanitation facilities of an otherwise acceptable type shared between two or more house|
|Basic unsafe|A basic sanitation facility which is not shared with other households, but excreta ar<br>disposed safely, such as fushed but not disposed to piped sewer system, septic tank <br>latrine|
|Improved|Sanitation facility which is not shared with other households and where excreta are <br>disposed in situ or treated of-site and includes fush/pour fush to piped sewer system,<br>tank or pit latrine, ventilated improved pit latrine, composting toilet or pit latrine w<br>slab|

One target of the United Nations’ sixth Sustainable Development Goal (whose aim is to ‘ensure availability and sustainable management of water and sanitation for all’) is ‘by 2030, [to] achieve access to adequate and equitable sanitation and hygiene for all and end open defecation.’ In order to hit the target, the Joint Monitoring Programme (JMP) of the World Health Organisation and the UNICEF proposes using ‘a _service ladder approach_ to benchmark and track progress across countries at different stages of development”, building

> 14See NIPORT et al. (2009, 2013, 2016) for details about the survey design.

27
on the existing datasets.<sup>15</sup> We pursue this service ladder approach and apply our ordinal poverty measures to study the improvement in sanitation deprivation in Bangladesh. We classify households’ _access to sanitation_ in the five ordered categories presented in Table 2. The five categories are ordered as ‘open defecation’ _≻D_ ‘unimproved’ _≻D_ ‘limited’ _≻D_ ‘basic unsafe’ _≻D_ ‘improved’. We consider all persons living in a household deprived in access to sanitation if the household experiences any category other than ‘improved’.

Table 3: Change in population distribution across sanitation categories in Bangladesh

||B|anglade|sh||Dhaka||R|ajshah|i||Sylhet||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
||2007|2011|2014|2007|2011|2014|2007|2011|2014|2007|2011|2014|
|Open defecation|7.5|4.2|3.3|7.5|4.0|2.2|13.8|3.9|3.2|2.1|12.5|9.4|
||(0.8)|(0.3)|(0.5)|(1.4)|(0.7)|(0.8)|(2.3)|(0.8)|(0.9)|(0.4)|(1.5)|(1.3)|
|Unimproved|47.1|38.3|25.7|44.3|35.9|22.6|45.3|36.4|28.3|57.2|34.3|23.0|
||(1.1)|(0.9)|(1.3)|(1.9)|(1.7)|(2.8)|(2.6)|(3.1)|(2.7)|(3.3)|(2.0)|(2.6)|
|Limited|13.4|16.7|20.9|14.4|18.0|26.1|14.7|20.7|20.2|10.1|17.7|22.3|
||(0.5)|(0.6)|(0.8)|(1.1)|(1.5)|(2.0)|(1.2)|(1.4)|(1.3)|(1.9)|(0.9)|(1.8)|
|Basic|3.5|4.3|2.3|8.6|10.5|5.4|0.2|0.2|0.3|0.6|0.1|0.3|
||(0.4)|(0.5)|(0.4)|(1.0)|(1.6)|(1.0)|(0.1)|(0.1)|(0.1)|(0.2)|(0.1)|(0.2)|
|Improved|28.5|36.6|47.8|25.2|31.6|43.7|26.0|38.8|48.0|30.1|35.4|45.0|
||(1.0)|(0.9)|(1.1)|(1.9)|(1.9)|(2.5)|(1.9)|(2.3)|(2.3)|(2.3)|(1.9)|(1.7)|

Sources: Authors’ own computations. Standard errors are reported in parentheses.

Table 3 shows how the estimated population shares in different deprivation categories have evolved over time in Bangladesh. Clearly, the estimated percentage in the ‘improved’ category has gradually increased (statistically significantly) from 28.5% in 2007 to 36.6% in 2011 to 47.8% in 2014. Thus, the proportion of the population in deprived categories has gone down over the same period. Changes within the deprived categories are however mixed.

> 15The JMP document titled _WASH Post-2015: Proposed indicators for drinking water, sanitation and hygiene_ was accessed in November 2019 at https://www.who.int/water ~~s~~ anitation ~~h~~ ealth/monitoring/ coverage/wash-post-2015-rev.pdf.

28
Although the estimated population shares in the two most deprived categories (‘open defecation’ and ‘unimproved’) have decreased (statistically significantly) systematically between 2007 and 2014, the population shares in the other two deprivation categories have not. Table 4: Change in sanitation deprivation by ordinal poverty measures in Bangladesh and its divisions

|||_H_|||_PI_|||_PM_|||_PG_||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
||2007|2011|2014|2007|2011|2014|2007|2011|2014|2007|2011|2014|
|Barisal|66.1|60.5|46.8|47.7|44.0|32.6|35.0|32.6|23.4|25.2|23.5|16.7|
||(2.7)|(2.1)|(3.3)|(1.9)|(1.7)|(2.9)|(1.4)|(1.4)|(2.4)|(1.1)|(1.1)|(1.8)|
|Chittagong|67.1|59.2|44.9|47.0|38.9|29.2|34.8|27.3|20.2|26.2|19.6|14.9|
||(2.9)|(2.1)|(3.1)|(2.8)|(1.7)|(2.9)|(2.7)|(1.5)|(2.7)|(2.6)|(1.2)|(2.5)|
|Dhaka|74.8|68.4|56.3|50.1|42.6|33.5|36.6|29.4|21.8|27.8|21.6|15.4|
||(1.9)|(1.9)|(2.5)|(1.7)|(1.3)|(1.8)|(1.6)|(1.1)|(1.6)|(1.4)|(1.0)|(1.3)|
|Khulna|69.0|61.4|50.3|48.9|41.8|33.0|35.7|29.4|22.7|25.9|20.9|16.1|
||(1.8)|(1.6)|(2.3)|(1.4)|(1.2)|(1.7)|(1.1)|(1.0)|(1.4)|(1.0)|(0.7)|(1.0)|
|Rajshahi|74.0|61.2|52.0|55.2|41.6|34.7|43.0|29.6|24.2|34.1|21.6|17.6|
||(1.9)|(2.3)|(2.3)|(1.8)|(1.9)|(1.8)|(1.9)|(1.6)|(1.5)|(1.9)|(1.3)|(1.3)|
|Sylhet|69.9|64.6|55.0|50.1|47.1|37.9|36.8|36.2|27.9|26.5|28.9|21.9|
||(2.3)|(1.9)|(1.7)|(1.9)|(1.6)|(1.7)|(1.5)|(1.4)|(1.6)|(1.1)|(1.4)|(1.4)|
|Bangladesh|71.5|63.4|52.2|50.4|42.3|33.5|37.5|30.1|23.1|28.5|22.2|16.8|
||(1.0)|(0.9)|(1.1)|(0.9)|(0.7)|(0.9)|(0.9)|(0.6)|(0.8)|(0.8)|(0.5)|(0.7)|

Sources: Authors’ own computations. Standard errors are reported in parentheses.

Has this estimated reduction pattern been replicated within all divisions? Table 3 also presents the changes in the discrete probability distributions of three divisions: Dhaka, Rajshahi, and Sylhet.<sup>16</sup> The estimated population shares in the ‘improved’ category have increased (statistically significantly) gradually in all three regions (Table 3), and so the shares

> 16A new division called Rangpur was formed in 2010, which was a part of the Rajshahi Division. The new division did not exist during the 2007 DHS, so we had to combine this division with the Rajshahi division in the 2011 and 2014 DHS in order to preserve comparability over time. Likewise, our selected surveys were not affected by the carving out of northern Dhaka to form the Mymesingh division in 2015.

29
of deprived population have gone down. We need, however, to point out two crucial aspects.

First, let us compare the reduction patterns in Dhaka and Rajshahi. The population share in the ‘improved’ category is higher in Rajshahi in 2011 and 2014 and statistically indistinguishable in 2007, implying that sanitation deprivation in Dhaka is never lower than sanitation deprivation in Rajshahi. However, the estimated population shares in the two most deprived categories (‘open defecation’ and ‘unimproved’) are higher in Rajshahi than in Dhaka in 2007 and 2014 and statistically indistinguishable in 2011. Second, like Dhaka and Rajshahi in Table 3, sanitation deprivation in Sylhet has also improved gradually. However, the estimated population share in the poorest category (‘open defecation’) is significantly higher in 2011 and in 2014 than in 2007. A simple headcount measure, which only captures the proportion of the overall deprived population, would always overlook these substantial differences.

Table 4 presents four different poverty measures for Bangladesh and for its six divisions (as per the pre-2010 administrative map). We assume the poverty threshold category to be ‘basic unsafe’. The first poverty measure is the headcount ratio ( _H_ ), which, in this context, is the population share experiencing any one of the four deprivation categories. The second measure is _PI_ , such that _PI ∈P \ {Pα}_ for _α ≥_ 1, and is defined by the ordering weights **_ω_**<sup>_I_</sup> = (1 _,_ 0 _._ 75 _,_ 0 _._ 5 _,_ 0 _._ 25 _,_ 0). The third measure is _PM ∈P_ 1 _\ {Pα}_ for _α ≥_ 2 with ordering weights **_ω_**<sup>_M_</sup> = (1 _,_ 0 _._ 75<sup>2</sup> _,_ 0 _._ 5<sup>2</sup> _,_ 0 _._ 25<sup>2</sup> _,_ 0), i.e. respecting the restrictions in Corollary 3.1, but not respecting, for instance, the restrictions in Proposition 3.1 or the restrictions in Theorem 3.1 for _α ≥_ 2; whereas, the fourth measure is _PG ∈PG_ with ordering weights **_ω_**<sup>_G_</sup> = (1 _,_ 0 _._ 4 _,_ 0 _._ 15 _,_ 0 _._ 05 _,_ 0), i.e. respecting the restrictions in Proposition 3.1. Note that measures _PM_ and _PG_ give precedence to those who are in the poorer categories. All four 30
measures lie between zero and one, but we have multiplied them by one hundred so that they lie between zero (lowest deprivation) and 100 (highest deprivation).

Comparisons of these measures provide useful insights, especially into the two crucial aspects that we have presented in Table 3. The headcount ratio estimate in Dhaka is statistically indistinguishable from the headcount ratio estimate in Rajshahi for 2007, despite deprivation in the two poorest categories being higher in Rajshahi. However, this crucial aspect is captured by the latter three measures, which show statistically significantly higher poverty estimates in Rajshahi than in Dhaka. Similarly, the headcount ratio estimate is higher in Dhaka than in Rajshahi for 2011, but the difference vanishes when poverty is assessed by the other three ordinal measures. Meanwhile, Table 4 also shows that, without accounting for the depth of deprivations, the headcount ratio ranks Dhaka as poorer than Sylhet in 2011. However, this ranking reverses as soon as we use deprivation-sensitive measures from the class _P_ .

Since most of the information in Table 4 points to experiences of poverty reduction, we further conduct pair-wise dominance tests in order to verify whether the comparisons over time are robust to all measures in class _P_ , and in sub-classes _P_ 1 and _PG_ . Note, in this case, that _S_ = 5 and _k_ = 4. The dominance test for _P_ is based on Theorem 4.1, the dominance test for _P_ 1 is based on Corollary 4.2 (i.e. _α_ = 1), and the dominance test for _PG_ is based on Corollary 4.3 (i.e. _α_ = _k −_ 1 = 3 ). Checking pair-wise dominance for _P_ 1 requires comparing poverty levels at the following four extreme points: (1 _,_ 0 _,_ 0 _,_ 0 _,_ 0), (1 _,_ 2<sup><u>1</u></sup><sup>_,_0</sup><sup>_,_0</sup><sup>_,_0),</sup> (1 _,_ 3<sup><u>2</u></sup><sup>_,_</sup> 3<sup><u>1</u></sup><sup>_,_0</sup><sup>_,_0)and(1</sup><sup>_,_</sup><sup><u>3</u></sup> 4<sup>_,_</sup> 4<sup><u>2</u></sup><sup>_,_</sup><sup><u>1</u></sup> 4<sup>_,_0).Meanwhile,checkingpair-wisedominancefor</sup><sup>_PG_requires</sup> comparing poverty levels at the following four extreme points: (1 _,_ 0 _,_ 0 _,_ 0 _,_ 0), (1 _,_<sup><u>1</u></sup> 2<sup>_,_0</sup><sup>_,_0</sup><sup>_,_0),</sup> 31
(1 _,_<sup><u>1</u></sup> 2<sup>_,_</sup> 4<sup><u>1</u></sup><sup>_,_0</sup><sup>_,_0)and(1</sup><sup>_,_</sup><sup><u>1</u></sup> 2<sup>_,_</sup><sup><u>1</u></sup> 4<sup>_,_</sup><sup><u>1</u></sup> 8<sup>_,_0).</sup> Table 5: Pair-wise dominance tests for changes in poverty over time in Bangladesh and its divisions

|||2007–2011|||2011–2014||
|---|---|---|---|---|---|---|
||_P_|_P_1|_PG_|_P_|_P_1|_PG_|
|Barisal|No|No|No|Yes|Yes|Yes|
|Chittagong|Yes**|Yes**|Yes**|No|No|No|
|Dhaka|Yes**|Yes**|Yes**|Yes**|Yes**|Yes**|
|Khulna|Yes*|Yes*|Yes*|Yes|Yes|Yes|
|Rajshahi|Yes***|Yes***|Yes***|Yes|Yes|Yes|
|Sylhet|No|No|No|Yes*|Yes*|Yes*|
|Bangladesh|Yes***|Yes***|Yes***|Yes*|Yes*|Yes*|

Sources: Authors’ own computations. A “Yes” implies reduction of the poverty levels within each region over time evaluated at the four extreme weights; a “No” implies otherwise. The levels of statistical significance for the dominance tests are: *** for 1%, ** for 5% and * for 10%.

Table 5 presents the pair-wise dominance tests over time for Bangladesh and its divisions.

Each cell of the table reports whether a _reduction_ or _increase_ in poverty within each region over time in Table 4 is robust (or not) to alternative ordering weighting vectors within the class (i.e., _P_ ) and subclasses (i.e., _P_ 1 and _P_ 2) to which the ordering weighting vectors used in Table 4 belong.<sup>17</sup> For example, it is evident from Table 4 that, according to _PM_ , poverty in Bangladesh has fallen from 37.5 points in 2007 to 30.1 points in 2011. So we could ask: is this reduction between 2007 and 2011 robust to all _P ∈P_ 1? The corresponding dominance test reported in Table 5 shows that the reduction is certainly robust with 1% significance

- 17 We implemented a standard intersection-union test (IUT) whose alternative hypothesis is that poverty levels at the four extreme points are jointly lower in year-region “A” (e.g. Bangladesh in 2011) vis-`a-vis “B” (e.g. Bangladesh in 2007). We reject the null (i.e. at least one poverty level in “A” is equal or higher than in “B”) in favour of this alternative only if every poverty level is lower in “A” than in “B” with a given significance level _α_ . As explained by Berger (1997, p. 226), the IUT’s overall significance level is also _α_ and no correction for multiple comparisons (e.g. Bonferroni, etc.) is required.

32
level. If a cell in Table 5 reports either “No” or “Yes”, but without appropriate statistical significance level, then we conclude that the relevant comparison in Table 4 is not robust. For instance, poverty reduction in Dhaka has been robust for the whole class _P_ at the 5% level of significance throughout both time periods (2007-2011 and 2011-2014); whereas, the poverty reduction for neither Barisal nor Chittagong for the period 2011-2014 reported in Table 4 can be claimed to be robust for any of the three (sub)classes of indices considered.

# **6. Concluding remarks**

There is little doubt that poverty is a multidimensional concept and the current global development agenda correctly seeks to ‘reduce poverty in all its dimensions’. To meet this target, it is indeed important to assess poverty from a multidimensional perspective. However, one should not discount the potential interest in evaluating the impact of a targeted program in reducing deprivation in a single dimension such as educational or health outcomes and access to public services, which may often be assessed by an ordinal variable with multiple ordered deprivation categories. The frequently used headcount ratio, in this case, is ineffective as it overlooks the depth of deprivations, i.e. any changes within the ordered deprivation categories.

Our paper has thus posed the question: ‘How should we assess poverty when variables are ordinal?’ Implicitly, the companion question is ‘Can we meaningfully assess poverty beyond the headcount ratio when we have an ordinal variable?’ Drawing on six reasonable axiomatic properties, our answer is: ‘Poverty can be measured with ordinal variables through weighted averages of the discrete probabilities corresponding to the ordered categories.’ We refer to 33
these weights as ordering weights, which need to satisfy a specific set of restrictions in order to ensure the social poverty indices fulfil these key properties. Our axiomatically characterised class of social poverty indices has certain desirable features, such as additive decomposability and being bounded between zero (when none experiences any deprivation) and one (when everyone experiences the most deprived category).

In contrast to previous attempts in the literature on poverty measurement with ordinal variables, we have gone fruitfully further in the direction of operationalising different concepts of ‘precedence to the poorer people among the poor’, which ensures that the policymaker has an incentive to assist the poorer over the less poor. We have shown that it is possible to devise reasonable poverty measures that prioritise welfare improvements among the most deprived when variables are ordinal. We have axiomatically characterised a set of subclasses of ordinal poverty measures based on different notions of precedence to those poorer among the poor. Each subclass is defined by an additional restriction on the admissible ordering weights. The precedence-sensitive measures have proven useful in the illustration pertaining to sanitation deprivation in Bangladesh by highlighting those provinces where the overall headcount improvement did not come about through reductions in the proportion of the population in the most deprived categories.

Since several poverty measures are admissible within each characterised class and subclass, we have also developed stochastic dominance conditions for each subclass of poverty measures. Their fulfilment guarantees that all measures within a given class (or subclass) rank the same pair of distributions robustly. While some of these conditions represent the ordinal-variable analog of existing conditions for continuous variables in the poverty dominance literature 34
(Foster and Shorrocks, 1988b); others are, to the best of our knowledge, themselves a novel methodological contribution to the literature on stochastic dominance with ordinal variables.

There has been a recent surge in the literature on multidimensional poverty measurement, especially within the counting framework. In this framework, however, it is still a usual practice to dichotomise deprivations within each dimension when using existing counting measures, ignoring the depth of deprivation across ordered categories. Future research could focus on the development of counting measures that incorporate both the depth of deprivations within dimensions and notions of precedence to the poorest among the poor.

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# **Appendices**

# **Appendix A1. Proof of Theorem 2.1**

It is straightforward to check that each poverty measure in Equation 2.1 satisfies the four properties: OMN, SCD, FOC and SUD.

We now prove the necessary part, namely, if a poverty measure satisfies these four properties, then it takes the functional form in Equation 2.1. First, we define _S ∈_ N _\ {_ 1 _}_ basis vectors:

**e**<sup>_s_</sup> = ( _e_<sup>_s_</sup> 1<sup>_, . . . , es_</sup> _S_<sup>)</sup><sup>_∈_P</sup><sup>_∀s_=1</sup><sup>_, . . . , S_,suchthat</sup><sup>_es_</sup> _s_<sup>=1</sup><sup>_∀s_and</sup><sup>_et_</sup> _s_<sup>=0</sup><sup>_∀s_=</sup><sup>_t_.Foragiven</sup> _ck ∈_ **C** _−S_ , denote _P_ ( **e**<sup>_s_</sup> _, ck_ ) = _ωs_ for some _ωs ∈_ R+ for every _s_ = 1 _, . . . , S_ .

40
Next, suppose a society with probability distribution **p** = ( _p_ 1 _, . . . , pS_ ) _∈_ P is divided into _M_ = _S_ mutually exclusive and collectively exhaustive sub-groups, such that the probability distribution of each subgroup is **p**<sup>_s_</sup> = ( _p_<sup>_s_</sup> 1<sup>_, . . . , ps_</sup> _S_<sup>)=</sup><sup>**e**</sup><sup>_s_forevery</sup><sup>_s_=1</sup><sup>_, . . . , S_(=</sup><sup>_M_).As</sup> each subgroup contains the population in each category, naturally, the population share of each subgroup is _πs_ = _ps_ for every _s_ = 1 _, . . . , S_ (= _M_ ). Then, by SUD: Now, consider any two **p** _,_ **q** _∈_ P, such that _ps_ = 1 for some _s ≤ k_ and _qs′_ = 1 for some _s_<sup>_′_</sup> _> s_ . Naturally, _pt_ = _qt_ = 0 for all _t_ = _{s, s_<sup>_′_</sup> _}_ . Clearly, by OMN, we have _P_ ( **q** _, ck_ ) _< P_ ( **p** _, ck_ ). Combined with Equation A1, we then obtain: The relationship in Equation A2 holds for any _s_ , such that _s ≤ k_ and _s < s_<sup>_′_</sup> . In other words, _ωs−_ 1 _> ωs > ωs′_ for all _s_ = 2 _, ..., k_ and even for any _s_<sup>_′_</sup> _> k_ , whenever _k ≥_ 2. When _S_ = 2, then _k_ = 1 and so _ω_ 1 _> ω_ 2.

We next use property SCD. Suppose, _k_ = 1. Then, property SCD leads to _P_ ( **p** _, ck_ ) = _p_ 1 and Equation A1 yields Note, by definition, that 0 _≤ p_ 1 _≤_ 1 and so 0 _≤ P_ ( **p** _, ck_ ) _≤_ 1. Consider some **p** _∈_ P, such that _p_ 1 = 1 and _ps_ = 0 for all _s_ = 1. Clearly, from Equation A3, _ω_ 1 = 1. Moreover, from 41
Equation A2, it follows that 1 _> ωs > ωs′_ for all _s_ = 2 _, . . . , k_ and for any _s_<sup>_′_</sup> _> k_ whenever _k ≥_ 2.

In order to complete the proof, we need to show that _ωs_ = 0 for all _s > k_ . For this purpose, consider some **p** _,_ **q** _∈_ P, such that _pt_ = _qu_ = 1 for some _t > u > k_ . Certainly, _ps_ = _qs_ = 0 _∀s ≤ k_ . By property FOC, we then require _P_ ( **p** _, ck_ ) = _P_ ( **q** _, ck_ ). Thus, from Equation A1, we obtain _P_ ( **p** _, ck_ ) = _ωt_ = _ωu_ = _P_ ( **q** _, ck_ ) for _t > u > k_ . Since, _ps_ = _qs_ = 0 _∀s ≤ k_ , it follows that _p_ 1 = _q_ 1 = 0. Consider _k_ = 1. Then, by property SCD, we must have _P_ ( **p** _, ck_ ) = _P_ ( **q** _, ck_ ) = 0. Hence, it must be the case that _ωs_ = 0 for all _s > k_ , which completes our proof.

# **Appendix A2. Proof of Theorem 3.1**

The sufficiency part is straightforward. We prove the necessity part as follows.

Suppose _k ≥_ 2 and _α ∈_ N such that 1 _≤ α ≤ k −_ 1. Now, suppose, **p**<sup>_′_</sup> and **q**<sup>_′_</sup> are obtained from **p** _∈_ P as follows. Consider some _s_<sup>_′_</sup> _< t ≤ k < S_ . Now, **p**<sup>_′_</sup> is obtained from **p** , such that _p_<sup>_′_</sup> _s_<sup>_′_=</sup><sup>_ps′ −ε_while</sup><sup>_p′_</sup> _u_<sup>=</sup><sup>_pu∀u_=</sup><sup>_{s′, s′_+ 1</sup><sup>_}_.Naturally,</sup><sup>_p′_</sup> _s_<sup>_′_</sup> +1<sup>=</sup><sup>_ps′_+1 +</sup><sup>_ε_.Similarly,</sup><sup>**q**</sup><sup>_′_is</sup> obtained from **p** , such that _qt_<sup>_′_=</sup><sup>_pt−ε_while</sup><sup>_q_</sup> _u_<sup>_′_=</sup><sup>_pu∀u_=</sup><sup>_{t, t′}_for some</sup><sup>_t′_= min</sup><sup>_{t_+</sup><sup>_α, S}_.</sup> Naturally, again, _qt_<sup>_′′_=</sup><sup>_pt′_+</sup><sup>_ε_.</sup> By property PRE- _α_ , we know that 42
Combining Equation 2.1 and Equation A1, we get Substituting _t_ = _s_<sup>_′_</sup> + 1 = _s_ for any _s_ = 2 _, . . . , k_ , we obtain First, suppose _t_<sup>_′_</sup> = _s_ + _α ≤ k < S_ or _s ≤ k − α_ . Then, _ωt′_ = _ωs_ + _α >_ 0 by Theorem 2.1 and Equation A2 can be expressed as _ωs−_ 1 _− ωs > ωs − ωs_ + _α_ for all _s_ = 2 _, . . . , k − α_ . Second, suppose _t_<sup>_′_</sup> = min _{s_ + _α, S} > k_ or _s > k − α_ . We know that _ωs_ = 0 for all _s > k_ by Theorem 2.1 and so Equation A2 can be expressed as _ωs−_ 1 _− ωs > ωs_ or _ωs−_ 1 _>_ 2 _ωs_ for all _s_ = _k − α_ + 1 _, . . . , k_ . This completes the proof.

# **Appendix A3. Proof of Proposition 3.1**

Let us first prove the sufficiency part. Suppose _k ≥_ 2. We already know from Theorem 2.1 that _ωs−_ 1 _> ωs >_ 0 _∀s_ = 2 _, . . . , k_ and _ωs_ = 0 _∀s > k_ . Suppose additionally that _ωs−_ 1 _>_ 2 _ωs ∀s_ = 2 _, . . . , k_ . Alternatively, _ωs−_ 1 _− ωs > ωs ∀s_ = 2 _, . . . , k_ .

For any **p** _,_ **p**<sup>_′_</sup> _,_ **q**<sup>_′_</sup> _∈_ P, for any _ck ∈_ **C** _−S_ and for some _ε ∈_ (0 _,_ 1), let _k ≥ t ≥ v_ + _α_ and suppose **p**<sup>_′_</sup> is obtained from **p** , such that _p_<sup>_′_</sup> _v_<sup>=</sup><sup>_pv−ε_and</sup><sup>_p′_</sup> _v_ + _α_<sup>=</sup><sup>_pv_+</sup><sup>_α_+</sup><sup>_ε_, while</sup><sup>_p′_</sup> _u_<sup>=</sup><sup>_pu∀u_=</sup><sup>_{v, v_+</sup><sup>_α}_;</sup> and **q**<sup>_′_</sup> is obtained from **p** , such that _qt_<sup>_′_=</sup><sup>_pt−ε_and</sup><sup>_q_</sup> _t_<sup>_′_</sup> + _β_<sup>=</sup><sup>_pt_+</sup><sup>_β_+</sup><sup>_ε_forsome</sup><sup>_β∈_N,while</sup> _qu_<sup>_′_=</sup><sup>_pu∀u_=</sup><sup>_{t, t_+</sup><sup>_β}_.</sup> With the help of Equation 2.1, we get: 43
By assumption of the sufficiency part: _ωs−_ 1 _− ωs > ωs ∀s_ = 2 _, . . . , k_ . Combining this assumption with the weight restrictions from Theorem 2.1 we can easily deduce that ( _ωv − ωv_ + _α_ ) _>_ ( _ωv − ωv_ +1) _> ωv_ +1 _> ωv_ + _α_ . Hence, ( _ωv − ωv_ + _α_ ) _> ωv_ + _α_ . Since _v_ + _α ≤ t ≤ k_ and _ωs−_ 1 _> ωs >_ 0 _∀s_ = 2 _, . . . , k_ , it also follows that _ωv_ + _α ≥_ ( _ωt − ωt_ + _β_ ). Hence, ( _ωv − ωv_ + _α_ ) _>_ ( _ωt − ωt_ + _β_ ) and _P_ ( **p**<sup>_′_</sup> _, ck_ ) _< P_ ( **q**<sup>_′_</sup> _, ck_ ).

Next we prove the necessity part starting with Equation A1. By property PRE-G, we know that _P_ ( **p**<sup>_′_</sup> _, ck_ ) _< P_ ( **q**<sup>_′_</sup> _, ck_ ). Thus, Now the inequality in Equation A2 must hold for any situation in which _t ≥ v_ + _α_ , including the comparison of the minimum possible improvement for the poorer person, given by _ωv − ωv_ +1 (i.e. with _α_ = 1), against the maximum possible improvement for the less poor person, given by _ωt − ωt_ + _β_ with _t_ = _v_ + 1 and _t_ + _β > k_ . Inserting these values into Equation A2, bearing in mind that _ωt_ + _β_ = 0 when _t_ + _β > k_ , yields Substituting _v_ = _s −_ 1 for any _s_ = 2 _, . . . , k_ yields _ωs−_ 1 _− ωs > ωs_ . Hence, _ωs−_ 1 _>_ 2 _ωs_ for all _s_ = 2 _, . . . , k_ .

# **Appendix A4. Proof of Theorem 4.1**

We denote the _difference operator_ by ∆and use the following handy definitions: For some

**p** _,_ **p**<sup>_′_</sup> _∈_ P, define ∆ _Pk ≡ P_ ( **p** _, ck_ ) _− P_ ( **p**<sup>_′_</sup> _, ck_ ), ∆ _ps ≡ ps − p_<sup>_′_</sup> _s_<sup>,and∆</sup><sup>_Fs≡_�</sup><sup>_s_</sup> _l_ =1<sup>∆</sup><sup>_pl_.</sup>

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We first prove the sufficiency part. From Theorem 2.1, we know that _ωs_ = 0 for all _s > k_ . Thus, Equation 2.1 may be presented using the difference operator as ∆ _Pk_ =<sup>�</sup><sup>_k_</sup> _s_ =1<sup>_ωs_∆</sup><sup>_ps_.</sup> Using summation by parts, also known as Abel’s lemma (Guenther and Lee, 1988) or formula (Fishburn and Lavalle, 1995, p. 518), it follows that We already know from Theorem 2.1 that _ωk >_ 0 and _ωs − ωs_ +1 _>_ 0 _∀s_ = 1 _, . . . , k −_ 1. Therefore, clearly from Equation A1, the condition that ∆ _Fs ≤_ 0 _∀s ≤ k_ and ∆ _Fs <_ 0 for at least one _s ≤ k_ is sufficient to ensure that ∆ _Pk <_ 0 _∀P ∈P_ and for a given _ck ∈_ **C** _−_ **S** .

We next prove the necessity part by contradiction. Consider the situation, where ∆ _Ft >_ 0 for some _t ≤ k_ , ∆ _Fs ≤_ 0 for all _s ≤ k_ but _s_ = _t_ , and ∆ _Fs <_ 0 for some _s ≤ k_ but _s_ = _t_ . For a sufficiently large value of _ωt − ωt_ +1 in Equation A1, it may always be possible to find that ∆ _Pk >_ 0. Or, consider the situation ∆ _Fs_ = 0 for all _s ≤ k_ . In this case, ∆ _Pk_ = 0. Hence, the necessary condition requires both ∆ _Fs ≤_ 0 for all _s ≤ k_ and ∆ _Fs <_ 0 for some _s ≤ k_ . This completes the proof.

# **Appendix A5. Proof of Theorem 4.2**

Consider some _k ≥_ 2 and some _α ∈_ [1 _, k −_ 1] _⊆_ N. Let Ω<sup>_k_</sup> _α_<sup>bethesetofall</sup><sup>_S_-dimensional</sup> 45
ordinal weighting vectors corresponding to subclass _Pα_ (see Theorems 3.1 and 2.1), so that: Now define Ω<sup> ̄</sup><sup>_k_</sup> _α_<sup>as:</sup> For given values of _k_ and _α_ , Ω<sup> ̄</sup><sup>_k_</sup> _α_<sup>isboundedbyasetoflinearconstraintsanditisaconvex</sup> hull of _k_ extreme points or extreme ordering weighting vector, denoted by **_ω_** ̄<sup>1</sup> _, . . . ,_ ̄ **_ω_**<sup>_k_</sup> . For the _r_<sup>th</sup> extreme point **_ω_** ̄<sup>_r_</sup> , where _r ∈{_ 1 _,_ 2 _, . . . , k}_ , _maximum_ feasible positive weights are assigned to the first _r_ elements and _minimum_ feasible weights are assigned to the rest of the elements (where feasibility refers to satisfying the linear constraints defining the set Ω<sup> ̄</sup><sup>_k_</sup> _α_<sup>),</sup> such that _ωs_ = _ω_ ̄ _s_<sup>_r>_0forall</sup><sup>_s_=1</sup><sup>_, . . . , r_and</sup><sup>_ωs_=</sup><sup>_ω_ ̄</sup> _s_<sup>_r_=0forall</sup><sup>_s_=</sup><sup>_r_+ 1</sup><sup>_, . . . , S_.18Any</sup> **_ω_** _∈_ Ω<sup> ̄</sup><sup>_k_</sup> _α_<sup>isaconvexcombinationofthese</sup><sup>_k_extremeorderingweightingvectors,suchthat</sup> **_ω_** =<sup>�</sup><sup>_k_</sup> _r_ =1<sup>_θr_ ̄</sup><sup>**_ω_**</sup><sup>_r_,where�</sup><sup>_k_</sup> _r_ =1<sup>_θr_= 1and</sup><sup>_θr≥_0forall</sup><sup>_r_= 1</sup><sup>_, . . . , k_.</sup> The restrictions on the aforementioned _θr_ ’s may be verified as follows. Since, _ω_ 1 = 1 and _ω_ ̄1<sup>_r_=1forall</sup><sup>_r_byTheorem2.1,thenpluggingthesevaluesin</sup><sup>_ω_1=�</sup><sup>_k_</sup> _r_ =1<sup>_θrω_ ̄</sup> 1<sup>_r_yields</sup> 18The proof is similar in spirit to the proof of Proposition 1 in Seth and McGillivray (2018).

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_k_ � _r_ =1<sup>_θr_= 1.Next,define:</sup> Since, _ωs_ =<sup>�</sup><sup>_k_</sup> _r_ =1<sup>_θrω_ ̄</sup> _s_<sup>_r_forall</sup><sup>_s_and</sup><sup>_ω_ ̄</sup> _s_<sup>_r_= 0forall</sup><sup>_s > r_,weobtain:</sup> Rearranging the right hand side of _us_ yields: where the third component on the right hand side of Equation A1 exists only when _s_ + _α_ +1 _≤_ _k_ . Given that the extreme weighting vectors satisfy the equality restrictions in Ω<sup> ̄</sup><sup>_k_</sup> _α_<sup>, the second</sup> component and the third component (whenever in existence) on the right-hand side must be equal to zero. Hence, _us_ = _θsω_ ̄ _s_<sup>_s_forall</sup><sup>_s_,whichisunrelatedtotherestrictionsofTheorem</sup> 3.1. Given that _us ≥_ 0 for all _s_ = 1 _, . . . , k_ by the definition of Ω<sup> ̄</sup><sup>_k_</sup> _α_<sup>andalso</sup><sup>_ω_ ̄</sup> _s_<sup>_s>_0(i.e.,</sup> _s_ = _r_ ) for all _s_ = 1 _, . . . , k_ by the definition of extreme weighting vectors, then, substituting _r_ = _s_ , it must be the case that _θr ≥_ 0 for all _r_ = 1 _, . . . , k_ .

The primary difference between Ω<sup>_k_</sup> _α_<sup>and ̄Ω</sup><sup>_k_</sup> _α_<sup>isthattheelementsof ̄Ω</sup><sup>_k_</sup> _α_<sup>satisfythenon-strict</sup> versions of the inequality restrictions prescribed by Theorem 3.1. Clearly, Ω<sup>_k_</sup> _α_<sup>_⊂_ ̄Ω</sup><sup>_k_</sup> _α_<sup>and</sup> therefore any **_ω_** _∈_ Ω<sup>_k_</sup> _α_<sup>isalsoaconvexcombinationof</sup><sup>**_ω_** ̄1</sup><sup>_, . . . ,_ ̄</sup><sup>**_ω_**</sup><sup>_k_,i.e.,</sup><sup>**_ω_**= �</sup><sup>_k_</sup> _r_ =1<sup>_θr_ ̄</sup><sup>**_ω_**</sup><sup>_r_,such</sup> that<sup>�</sup><sup>_k_</sup> _r_ =1<sup>_θr_=1.However,thestrictinequalityconstraintsinΩ</sup><sup>_k_</sup> _α_<sup>requirethat</sup><sup>_us>_0for</sup> all _s_ = 1 _, . . . , k_ , which implies that _θr >_ 0 for all _r_ = 1 _, . . . , k_ for Ω<sup>_k_</sup> _α_<sup>.</sup> 47
For any **p** _∈_ P, let us denote the poverty level evaluated at some **_ω_** as _P_ ( **p** _, ck_ ; **_ω_** ) and, given additive decomposability, we must have _P_ ( **p** _, ck_ ; **_ω_** ) =<sup>�</sup><sup>_k_</sup> _r_ =1<sup>_θrP_(</sup><sup>**p**</sup><sup>_, ck_; ̄</sup><sup>**_ω_**</sup><sup>_r_), where �</sup><sup>_k_</sup> _r_ =1<sup>_θr_=</sup> 1 and _θr >_ 0 for all _r_ = 1 _, . . . , k_ . Clearly, for some **p** _,_ **q** _∈_ P, if _P_ ( **p** _, ck_ ; ̄ **_ω_**<sup>_r_</sup> ) _≤ P_ ( **q** _, ck_ ; ̄ **_ω_**<sup>_r_</sup> ) for all _r_ = 1 _, . . . , k_ , with one strict inequality, then it follows directly that _P_ ( **p** _, ck_ ; **_ω_** ) _< P_ ( **q** _, ck_ ; **_ω_** ) for any **_ω_** _∈_ Ω<sup>_k_</sup> _α_<sup>or,equivalently,forall</sup><sup>_P∈Pα_foragiven</sup><sup>_ck_.Thisproves</sup> sufficiency.

In order to prove necessity, suppose _P_ ( **p** _, ck_ ; ̄ **_ω_**<sup>_r_</sup> ) _> P_ ( **q** _, ck_ ; ̄ **_ω_**<sup>_r_</sup> ) for some _r ∈{_ 1 _,_ 2 _, . . . , k}_ . Then, for a sufficiently large value of _θr_ (i.e., for _θr →_ 1), it is always possible to have _P_ ( **p** _, ck_ ; **_ω_** ) _> P_ ( **q** _, ck_ ; **_ω_** ). Furthermore, in case _P_ ( **p** _, ck_ ; ̄ **_ω_**<sup>_r_</sup> ) = _P_ ( **q** _, ck_ ; ̄ **_ω_**<sup>_r_</sup> ) for all _r_ = 1 _, . . . , k_ , then certainly _P_ ( **p** _, ck_ ; **_ω_** ) = _P_ ( **q** _, ck_ ; **_ω_** ) for all **_ω_** _∈_ Ω<sup>_k_</sup> _α_<sup>.Hence,theconditions,</sup> _P_ ( **p** _, ck_ ; ̄ **_ω_**<sup>_r_</sup> ) _≤ P_ ( **q** _, ck_ ; ̄ **_ω_**<sup>_r_</sup> ) for all _r_ = 1 _, . . . , k_ , with one strict inequality, are also necessary.

Finally, the solutions for the _k_ extreme points are obtained as follows. By Theorem 2.1, _ω_ ̄1<sup>_r_= 1 for all</sup><sup>_r_= 1</sup><sup>_, . . . , k_.From above, we already know that</sup><sup>_ω_ ̄</sup> _s_<sup>_r_= 0 for all</sup><sup>_s_=</sup><sup>_r_+1</sup><sup>_, . . . , S_</sup> ̄ and for every _r_ . Therefore, for every _r ∈{_ 1 _, . . . , k}_ , _ωs_<sup>_r_= 0forall</sup><sup>_s > r_.Wealsoknowthat</sup> _ω_ ̄ _s_<sup>_r>_0forall</sup><sup>_s≤r_.Now,forany</sup><sup>_α_andforevery</sup><sup>_r∈{_2</sup><sup>_, . . . , k}_,weobtainthefollowing</sup> ̄ system of ( _r −_ 1) equations: _ω_ 1<sup>_r_= 2 ̄</sup><sup>_ω_</sup> 2<sup>_r−ω_ ̄</sup> 2+<sup>_r_</sup> _α_<sup>,</sup><sup>_ω_ ̄</sup> 2<sup>_r_= 2 ̄</sup><sup>_ω_</sup> 3<sup>_r−ω_ ̄</sup> 3+<sup>_r_</sup> _α_<sup>,. . . ,</sup><sup>_ω_ ̄</sup> _r_<sup>_r_</sup> _−α−_ 1<sup>= 2 ̄</sup><sup>_ω_</sup> _r_<sup>_r_</sup> _−α_<sup>_−ω_ ̄</sup> _r_<sup>_r_,</sup> _ω_ ̄ _r_<sup>_r_</sup> _−α_<sup>=2 ̄</sup><sup>_ω_</sup> _r_<sup>_r_</sup> _−α_ +1<sup>,. . . ,</sup><sup>_ω_ ̄</sup> _r_<sup>_r_</sup> _−_ 2<sup>=2 ̄</sup><sup>_ω_</sup> _r_<sup>_r_</sup> _−_ 1<sup>,</sup><sup>_ω_ ̄</sup> _r_<sup>_r_</sup> _−_ 1<sup>=2 ̄</sup><sup>_ω_</sup> _r_<sup>_r_.Thereare(</sup><sup>_r −_1)unknowns:</sup><sup>_ω_ ̄</sup> 2<sup>_r, . . . ,_ ̄</sup><sup>_ω_</sup> _r_<sup>_r_,</sup> since _ω_ ̄1<sup>_r_= 1.Solvingthesystemofequations,weobtain</sup><sup>_ω_</sup> _s_<sup>_r_=</sup><sup>_ω_ ̄</sup> _s_<sup>_r_forall</sup><sup>_s_= 2</sup><sup>_, . . . , r_andfor</sup> all _r_ = 1 _, . . . , k_ in Equation 4.1. This completes the proof of the theorem.

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# **Appendix A6. Proof of Corollary 4.2**

We use the difference operators defined in Appendix A4. Summing by parts the first component on the right-hand side of Equation A1 and then rearranging the terms, we obtain: We know from Corollary 3.1 that [ _ωs − ωs_ +1] _−_ [ _ωs_ +1 _− ωs_ +2] _>_ 0 _∀s_ = 1 _, ..., k −_ 1. Likewise, by Theorem 2.1, _ωk >_ 0. Therefore<sup>�</sup><sup>_s_</sup> _l_ =1<sup>∆</sup><sup>_Fl≤_0forall</sup><sup>_s_= 1</sup><sup>_, ..., k_withatleastonestrict</sup> inequality is sufficient to ensure that, for any **p** _,_ **q** _∈_ P, _P_ ( **p** _, ck_ ) _< P_ ( **q** _, ck_ ) _∀P ∈P_ 1. On the contrary, suppose<sup>�</sup><sup>_s_</sup> _l_ =1<sup>∆</sup><sup>_Fl>_0forsome</sup><sup>_s ∈{_1</sup><sup>_, . . . , k}_.Now,notethatthereisnofurther</sup> restriction on whether any of the weight functions in Equation A1 is strictly greater than the others. Then, attaching a sufficiently large weight to this component (i.e.<sup>�</sup><sup>_s_</sup> _l_ =1<sup>∆</sup><sup>_Fl>_0</sup> for some _s ∈{_ 1 _, . . . , k}_ ) may result in _P_ ( **p** _, ck_ ) _> P_ ( **q** _, ck_ ) for some _P ∈P_ 1. Hence, the conditions are also jointly necessary, which completes the proof.

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