Policy Research Working Paper 11113 A Quick-Fix for Perverse Incentives Inherent in Mainstream Multidimensional Poverty Measures Benoit Decerf Development Economics Development Research Group May 2025 Policy Research Working Paper 11113 Abstract Most multidimensional poverty measures used in practice, the same properties as the adjusted headcount ratio, except including the global Multidimensional Poverty Index, are for Dimensional Breakdown. The paper argues that this based on the adjusted headcount ratio. This paper shows is not a sufficient reason to discard this index, by pro- that this poverty index provides perverse incentives. Poli- viding two examples illustrating key limitations of the cies that minimize this index prioritize targeting the least decomposition across dimensions permitted by Dimen- intensely poor individuals, rather than the most intensely sional Breakdown. This decomposition does not provide poor individuals. This paper proposes a quick-fix solution the necessary information to find optimal policies. More that tweaks the adjusted headcount ratio without affecting importantly, this decomposition may mislead policy makers the identification of the poor. The resulting index satisfies on the underlying sources of progress. This paper is a product of the Development Research Group, 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 author may be contacted at bdecerf@worldbank.org. 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 A Quick-Fix for Perverse Incentives Inherent in Mainstream Multidimensional Poverty Measures.∗ Benoit Decerf JEL: I32, D63. Keywords: Multidimensional Poverty, Global MPI, Adjusted Head-count Ratio, Dimensional Breakdown. ∗ Acknowledgments : I thank Kimberly Blair Bolch, Jakob Dirksen, Bilal Malaeb and Ricardo Nogales for their helpful comments and suggestions. All errors remain mine. The findings, interpretations, and conclusions expressed in this paper are entirely those of the author and should not be attributed in any manner to the World Bank, to its affiliated organizations, or to members of its Board of Executive Directors or the countries they represent. 1 Introduction Prioritarianism suggests that social protection policy efforts should target in priority worse off individuals. Therefore, prioritarian poverty measures should not provide incentives to target the least intensely poor individuals before targeting the most intensely poor individuals. This is one of the two main criticisms raised by Sen (1976) against the headcount ratio. In this paper, we show that the adjusted headcount ratio (Alkire and Foster, 2011) provides such perverse incentives and we propose a quick- fix solution. Such solution could prove useful as multidimensional poverty measures are increasingly used in academic studies and policy analysis,1 and most of these practical applications rely on the adjusted headcount ratio.2 This is for instance the case of the global Multidimensional Poverty Index (MPI), monitored by UNDP-OPHI, and of the World Bank’s Multi- dimensional Poverty Measure.3 These perverse incentives may thus affect policies. For instance, Santos et al. (2023) recently proposed an algorithm searching for the policy that minimizes the adjusted headcount ratio un- der a budget constraint. The perverse incentives inherent to the adjusted headcount ratio thus push their algorithm to target the deprivations of the least intensely poor individuals.4 The problem arises under the adjusted headcount ratio because an indi- vidual’s poverty contribution “jumps” when the individual escapes poverty. Our quick-fix solution consists in decreasing the poverty contributions of poor individuals by an amount equal to this “jump”. The solution we pro- pose thus preserves the Alkire-Foster poverty identification method but tweaks the adjusted headcount ratio index. Also, we show that this solu- tion gets rid of this perverse incentive and does not affect the comparisons made by the adjusted headcount ratio when no individual escapes poverty. Our solution yields an index that satisfies the same properties as the ad- justed headcount ratio except for Dimensional breakdown. We argue that the violation of Dimensional breakdown needs not be a sufficient reason to discard our solution. Dimensional breakdown implies that, after the iden- tification of poor individuals has taken place, the index can be decomposed between the respective contributions of the various dimensions considered (Alkire and Foster, 2016, 2019). If such decomposition seems useful in practice, we show by means of examples that the “after identification” pro- viso has important consequences. More precisely, this property need not be helpful when looking for the policy that minimizes the adjusted head- 1 See for instance the website mppn.org, which documents official national and inter- national multidimensional poverty measures. 2 A rich literature has proposed a large variety of multidimensional poverty measures that are not based on the adjusted headcount ratio, including proposals by Bourguignon and Chakravarty (2003) or Decancq et al. (2019). 3 See Decerf (2024) for a review of other limitations of mainstream multidimensional poverty indicators. 4 We observe that Santos et al. (2023), which is published as OPHI’s working paper No. 144, are unaware of this perverse incentive. They write about the adjusted head- count ratio that “using this measure avoids the perverse incentives of prioritizing the least intensely poor, which is in line with Sen (1976) warning for the unidimensional case”. 2 count ratio, which satisfies this property. Also, when monitoring poverty over time, this property can be misleading about the underlying sources of progress. Thus, when improperly understood, this property may induce a false sense of simplicity and could lead to misguided policymaking. Our paper speaks to the literature that studies prioritarianism in the context of multidimensional poverty measurement. Several authors con- sider theoretical settings where all variables capture continuous cardinal or ordinal achievements, like Bosmans et al. (2018), Datt (2019) or Decancq et al. (2019). In contrast, we focus on the practically relevant case for which variables are categorical (binary). A few papers consider theoreti- cal settings with categorical variables, like Seth and Yalonetzky (2018) or Alkire and Foster (2016, 2019). In contrast to Alkire and Foster, we do not restrict prioritarianism to inequality among the poor, an approach that is blind to the perverse incentives considered in this paper. Our solution can be understood as a new functional form for the (abstract) class of poverty measures defined by Eq. (2) in Seth and Yalonetzky (2018). The contribution of this paper is threefold. First, we formalize the perverse incentives inherent to the adjusted headcount ratio under binary variables, the most relevant case for applications. Datt (2019) already shows that indices in the M α class (Alkire and Foster, 2011) are affected by a similar issue. One difference is that his formalization relies on a version of the strong transfer property that requires continuous variables and is thus silent under binary variables, e.g., those entering the definition of the global MPI. Second, and more importantly, we provide a new ready-to-use solution to the perverse incentives problem. The solution proposed by Datt (2019) is to identify the poor using the so-called union approach. However, the union approach tends to yield implausibly high headcount ratios when the number of dimensions considered grows. Our solution preserves the identification method proposed by Alkire and Foster (2011), but slightly changes the contribution of poor individuals. Third, we document two practical limitations of Dimensional Breakdown, a property that is hailed in the literature (Alkire and Foster, 2019). Some authors already point to some of the limitations of this property. Pattanaik and Xu (2018) observe that the decomposability across dimensions of the adjusted headcount ratio implies that this index is not sensitive to the redistribution of deprivations across poor individuals.5 Datt (2019) regrets that this property provides a decomposition that is “post-identification”, but he does not study the practical limitations entailed. To the best of our knowledge, this paper is the first to document such limitations. The remainder is organized as follows. Section 2 presents the frame- work. Section 3 explains the perverse incentives inherent to the adjusted headcount ratio. Section 4 presents our quick-fix solutions. Section 5 presents two limitations of Dimensional Breakdown. Section 6 concludes. 5 The adjusted headcount ratio is sensitive to the redistribution of deprivations across individuals when their poverty statuses change, which leads to the perverse incentives problem considered in this paper. 3 2 Basic framework We introduce the minimal notation necessary for our purposes. There are n individuals, indexed by i and d dimensions, indexed by j . In each dimen- sion, individuals are either deprived or not deprived. Let g 0 = [gij 0 ] denote 0 the n · d deprivation matrix whose typical element gij = 1 if i is deprived in 0 dimension j , otherwise gij = 0. Let w = (w1 , . . . , wd ) be a vector of weights such that wj ∈ [0, 1] and j wj = 1. The deprivation score si denotes the 0 weighted sum of individual i’s deprivations, i.e., si := j wj gij . Alkire and Foster (2011) suggest identifying the poor by comparing deprivation scores to a fixed poverty cutoff k ∈ [0, 1].6 Formally, individual i is identified as poor when si ≥ k . The identification vector r = (r1 , . . . , rn ) is such that ri = 1 when i is identified as poor, otherwise ri = 0. The number of poor individuals is denoted by q := i ri . A poverty measure requires both a poverty identification method and a poverty index. A multidimensional poverty measure based on an index that is decom- posable in population subgroups can be defined as n 1 0 0 M (g ) = p( g i ) n i=1 where the poverty contribution function p : {0, 1}d → [0, 1] defines the amount that each individual i contributes to the poverty measure given 0 her vector of deprivations gi . For instance, the headcount ratio is defined as H = q/n. More importantly, the global MPI, as well as the World Bank’s MPM, are defined based on the adjusted headcount ratio defined as n 0 1 M0 (g ) = si ri , n i=1 0 where the contribution function is thus p(gi ) = si ri . Index M0 can also be written as M0 (g ) = H (g ) × A(g ) where A(g 0 ) = 1 0 0 0 q i si ri is the av- erage deprivation score among the poor. The Global MPI monitored by OPHI and the UNDP has d = 10 and k = 1/3 (see Table 1 for a definition). Table 1: Dimensions and weights of the Global MPI. School attendance Years of schooling Child mortality Drinking water Cooking fuel Sanitation Electricity Nutrition Housing Assets Dimension 1 1 1 1 1 1 1 1 1 1 Weight 6 6 6 6 18 18 18 18 18 18 Note: See Alkire et al. (2022) for a complete definition of the Global MPI. 6 Alternative linear methods to identify the multidimensionally poor have been pro- posed by Ravallion (2011) and Pattanaik and Xu (2018). Decerf and Fonton (2023) use an adjusted Alkire-Foster identification method. Decerf (2023) proposes a preference- based theory to compare and rank these alternative poverty identification methods. 4 3 Perverse incentives In the case of monetary poverty, prioritarian poverty measures satisfy the Pigou-Dalton transfer principle. That is, a regressive income trans- fer should not reduce the poverty measure. Sen (1976) raised this criticism against the headcount ratio. Indeed, an income transfer from the poorest individual toward a poor individual whose income is just below the poverty line may in fact reduce the headcount ratio, at least if the transfer implies that the income of the latter crosses the poverty line. The problem origi- nates in the fact that an individual’s poverty contribution discontinuously “jumps” when her income crosses the poverty line. As we show below, the same problem arises in the multidimensional case for the adjusted head- count ratio. In the multidimensional case, a regressive transfer can be captured by an association-increasing rearrangement (Atkinson and Bourguignon, 1982), in which two individuals switch their deprivation status in a given dimension. Such rearrangement is regressive when the resulting deprivation vectors of these individuals is ranked by vector dominance. That is, one in- dividual is deprived in all the dimensions in which the other is deprived and ′ the former has strictly more deprivations. Formally, g 0 is obtained from ′ g 0 by an association-increasing rearrangement if gi 0 = gi 0 for all i = h, m, 0′ 0 0′ 0 ′ 0′ 0 0′ 0 ghj = ghj and gmj = gmj for all j = j , ghj ′ = gmj ′ = 0, gmj ′ = ghj ′ = 1 0′ 0′ ′ and gmj ≥ ghj for all j (thus individual m is poorer than h under g 0 ). One intuition why this rearrangement can be interpreted as a regressive trans- fer can be grasped by observing that this rearrangement generates more extreme deprivation scores: s′h < sh , sm < s′m , s′h < sm and sh < s′m . A prioritarian measure should satisfy Strong Transfer . ′ Poverty axiom 1 (Strong Transfer ). If g 0 is obtained from g 0 by an ′ association-increasing rearrangement, then M(g 0) ≤ M(g 0 ). The particularity of Strong Transfer is that it still applies even when the individual that benefits from the rearrangement escapes poverty.7 To be fair, some authors argue that transfer properties should be restricted to cases where no individuals escape poverty. In his authoritative review of poverty indices properties, Zheng (1997) observes that Sen changed his mind and argued in favor of adding this restriction, but only after he re- alized that his proposed poverty index violates transfer absent this restric- tion. Zheng argues against this restriction based on a continuity argument, i.e., on the grounds that an individual’s poverty contribution should not “jump” when the individual escapes poverty. We argue that prioritarian policy makers would reject this restriction. Indeed, all poverty standards 7 We could further restrict the definition of Strong Transfer so that it only applies to individuals who are identified as poor before the association-increasing rearrangement takes place (but not necessarily identified as poor after ). The conclusions would not change. 5 are to some extent arbitrary. This is certainly the case for the value selected for the poverty cutoff k . As a result, there seems to be no convincing reason to grant priority to individuals who happen to be located in the immediate neighborhood of an arbitrarily selected threshold. The global MPI violates Strong Transfer . To see this, assume that individual h is deprived in Electricity, whose weight is wElec = 1/18. Indi- vidual h also suffers from other deprivations so that she has a deprivation score sh = k = 1/3. Assume also that individual m is not deprived in Electricity but has all the other deprivations of h together with additional deprivations, such that sm = 2/3. Both h and m are thus identified as poor and their poverty contributions correspond to their respective deprivation scores. Assume an association-increasing rearrangement takes place such that h is no longer deprived in Electricity while m becomes deprived in Electricity. This rearrangement is such that the new deprivation scores are s′h = 1/3 − 1/18 and s′m = 2/3 + 1/18. The problem is that the global MPI is reduced because the poverty contribution of h is reduced by more than the amount by which the poverty contribution of m is increased. Indeed, the poverty contribution of m is increased by 1/18, which corresponds to the weight attributed to Electricity. In turn, the poverty contribution of h ′ is reduced by 1/3. The reason is that individual h is no longer poor (rh = 0) ′ 0′ 0 because sh < 1/3, which implies that p(gj ) = 0, while p(gj ) = 1/3. The problem is illustrated in Figure 1, where we plot poverty contribu- tions as a function of deprivation scores. A poor individual whose depriva- tion score crosses value k becomes non-poor and her poverty contribution falls to zero. This reduces her poverty contribution by an amount k , which can be much larger than the reduction in her deprivation score. This creates a perverse incentive for policy makers looking for the policy most efficient at reducing M0 . This policy typically involves removing deprivations to individuals who are least intensely poor, i.e., whose deprivation score is no less but close to k . Such policy deviates from a prioritarian policy, which would target individuals with the largest deprivation scores. p(si) 1 k= 3 “jump” si non-poor k poor Figure 1: The poverty contribution associated to M0 “jumps” when indi- viduals escape poverty. Observe that the same perverse incentive is also present for the head- count ratio H . In fact, H is affected to a larger degree than M0 by perverse 6 incentives. Under H , the contribution of a poor individual is not even re- duced when her deprivation score is reduced, at least not until she is lifted out of poverty. Alkire and Foster (2011) define a larger Mα class of poverty indices, whose members require continuous variables (except M0 ). Datt (2019) shows that these indices violate a version of strong transfer defined for continuous variables. Alkire and Foster (2016) further propose another Mγ class of poverty indices, whose members can accommodate binary vari- ables. These indices also feature a discontinuous “jump” in their poverty contribution function. As a result, they also provide the perverse incentives discussed above, at least for some values of parameters k , d and w . 4 A quick fix: Tweaking M0 One simple solution involves keeping index M0 but changing the value of the poverty cutoff to k ′ = minj wj . There is no discontinuous “jump” in poverty contribution when using poverty cutoff k ′ , as illustrated in Figure 2.a. In that case, the identification corresponds to the union approach, i.e., all individuals who are deprived in some dimension are identified as poor. This is the solution proposed by Datt (2019). This author studies a class of distribution-sensitive multidimensional poverty measures based on the union approach. Unfortunately, this natural solution loses a key added value of the Alkire-Foster identification method, namely the ability to yield reasonable headcount ratios. As observed by Rippin (2010), headcount ratios tend to 100% under the union approach when using a large number of dimensions. For many applications, e.g., official poverty measurement, it can prove helpful to measure and communicate the incidence of poverty. For applications for which a reasonable headcount ratio is useful, we propose another simple solution. This second solution leaves identifica- tion unaffected but slightly changes the adjusted headcount ratio index. The proposed solution changes the poverty contribution of poor individu- als. Their new poverty contribution is what we call their multidimensional “poverty gap”, rather than their deprivation score si . The multidimensional poverty gap is similar to the unidimensional poverty gap. In the unidimensional monetary case, the poverty gap of poor indi- vidual i is defined as z − yi , where z denotes the monetary line and yi denotes the income of i. In the multidimensional case, the poverty gap of poor individual i is defined as si − sz , where sz denotes the largest de- privation score for which an individual can be considered non-poor. The multidimensional poverty gap of i is thus a gap in the space of deprivation scores, i.e., the minimal reduction in i’s deprivation score such that i is lifted out of poverty. Observe that sz is different from the poverty cutoff k . The reason is that an individual whose deprivation score is equal to k is considered poor (under the Alkire-Foster identification method). sz is close to the notion of a monetary line z in the sense that an individual whose income yi is equal to the monetary line z is considered non-poor (under the 7 monetary identification method). Formally, we have sz = max 0 ] such that r =0} si , {[gij i where sz is thus a constant that is slightly smaller than k . For the global MPI, we have sz = k − minj wj = 1/3 − 1/18 = 5/18. Our proposed solution consists in replacing the poverty contribution 0 function p(gi ) = si ri used by M0 by the following contribution function 0 p′ ( g i ) = ( si − sz ) ri , poverty gap where each poor individual’s contribution corresponds to her multidimen- sional poverty gap. Observed that the new contribution function p′ is ob- tained from the initial contribution function p by subtracting the constant sz from the contribution of all poor individuals.8 We thus propose using index n ′ 1 M0 (g 0 ) = 0 p′ ( g i ). n i=1 ′ Like M0 , index M0 can be expressed as the product of the headcount ra- tio and a factor capturing the intensity of poverty among the poor. Indeed, ′ M0 can be written as ′ 0 M0 (g ) = H (g 0) × A′ (g 0 ), where A′ (g 0 ) = 1 q z ′ i (si − s )ri has a simple interpretation, namely A is the average poverty gap among the poor. There is a simple mathematical relationship between A and A′ , namely A′ = A − sz . As illustrated in Figure 2.b, the new poverty contribution function does not have a discontinuous “jump”. There is thus no perverse incentive to target least intensely poor ′ individuals when searching for the policy that minimizes M0 . ′ Proposition 1. M0 violates Strong Transfer. M0 satisfies Strong Transfer. Proof. The proof straightforwardly follows from the above discussion. There is another interesting implication of the fact that, for poor in- dividuals, their poverty contribution p′ under M0 ′ is a translation of their ′ poverty contribution p under M0 . This implication is that M0 compares in the same way as M0 any two deprivation matrixes that share the same iden- ′ tification vector. Formally, if g 0 and g 0 have equal identification vectors (r ′ = r ), then9 ′ ′ M0 (g 0 ) ≤ M0 (g 0 ) ⇔ ′ M0 ′ 0 (g 0 ) ≤ M0 (g ). 8 When k < minj wj , any individual who suffers a deprivation is automatically con- sidered poor. Thus, the identification corresponds to the union approach. In that case, we have sz = 0 and thus p′ = p. Observe that there is no need to tweak M0 because M0 does not have the perverse incentive in that case. 9 ′ 0 The proof for this statement is a follows. We have by definition that M0 (g ) = ′ ′ ′ ′ 0 q z ′ 0 0 q z q q M0 (g ) − n s and M0 (g ) = M0 (g ) − n s . But we must have n = n because r′ = r. ′ 0 ′ 0′ ′ This shows that M0 (g ) = M0 (g 0 ) − c and M0 (g ) = M0 (g 0 ) − c for the same constant c, which directly yields the desired result. 8 ′ (a) k ′ and M0 (b) k and M0 p(si) p′(si) k minj wj si si k′ poor k poor non-poor k poor Figure 2: The poverty contribution of two simple solutions without per- ′ verse incentives: k ′ and M0 (a), k and M0 (b). ′ In other words, the poverty comparisons with M0 only differ from those with M0 because the former avoids the perverse incentives of M0 . There- fore, prioritarian policy makers looking for the policy minimizing multidi- ′ mensional poverty should base their analysis on M0 rather than on M0 . We note that our proposed solution also removes any perverse incentives for the other members of the M γ class with γ > 1. However, our proposed solution does not remove the perverse incentives for H (which corresponds to γ = 0). Also, our solution is not valid for the members of the M α class, which rely on continuous variables. Finally, we acknowledge that our proposed solution is not sensitive to the redistribution of deprivations across poor individuals and thus violates the Cross-Dimensional Convexity property proposed by Datt (2019). Our solution is a pragmatic quick-fix for policy makers who consider using the adjusted headcount ratio. 5 Two limitations of Dimensional Breakdown ′ ′ Using M0 instead of M0 comes at a cost. Indeed, index M0 satisfies the same properties as M0 except for Dimensional Breakdown (Alkire and Fos- ter, 2016, 2019). This should not come as a surprise: these authors show that Dimensional Breakdown is hard to reconciliate with transfer prop- ′ erties. In this last section, we argue that the fact that M0 violates Di- mensional Breakdown needs not necessarily be a reason serious enough to prefer M0 for policymaking purposes. As we show below, this property is not directly helpful when searching for the policy minimizing M0 and can even be misleading when monitoring progress as measured by M0 . We do not claim that Dimensional Breakdown has no policy relevance whatsoever, but we document two of its serious limitations. Index M0 satisfies Dimensional Breakdown because its mathematical expression can alternatively be written as d 0 1 M0 (g ) = w j Hj , (1) d j =1 9 where the censored headcount ratio Hj is the fraction of individuals that are simultaneously identified as poor and suffer a deprivation in dimension j . The value of M0 can thus be decomposed into the respective contribu- tions of each dimension taken separately. In the words of Alkire and Foster (2016), “multidimensional poverty can be expressed as a weighted sum of dimensional components”. However, as these authors acknowledge, this de- composition is made “after identification has taken place and the poverty status of each person has been fixed”. As we show below, this “after iden- tification” proviso is consequential. It implies that the decomposition in Eq. (1) is informative on the impact that a change in Hj has on M0 only if this change does not affect poverty statuses. Clearly, this is a big “if”. We illustrate below two problems that may arise with Eq (1) when this condition is not met. In search of the policy minimizing M0 First, we start with an example showing that Dimensional Breakdown is not directly helpful when looking for the policy minimizing M0 . We base our analysis on the global MPI, whose definition is reminded in Table 2. Consider the deprivation matrix gˆ0 with two individuals whose definition is provided in Table 2. The two individuals are identified as poor. Also, for each dimension, one of the two individuals is deprived while the other is not. Using the decomposition associated to Dimensional Breakdown, we get that Hj = 1/2 for any dimension j . From Eq. (1), any dimension j with weight 1/6 thus contributes an amount wj Hj = 1/12 to M0 (ˆ g 0 ) and any dimension j with weight 1/18 thus contributes an amount wj Hj = 1/36 to g 0 ). M0 (ˆ Table 2: Dimensional breakdown’s decomposition omits key information relevant for finding M0 -minimizing policy. School attendance Years of schooling Child mortality Drinking water Cooking fuel Sanitation Electricity Nutrition Housing Assets Dimension si ri Weight wj 1/6 1/6 1/6 1/6 1/18 1/18 1/18 1/18 1/18 1/18 0 1/3 ˆ1 g 1 0 0 0 1 1 1 0 0 0 1 0 2/3 ˆ2 g 0 1 1 1 0 0 0 1 1 1 1 0 1/3 ˜1 g 0 1 0 0 0 0 0 1 1 1 1 0 2/3 ˜2 g 1 0 1 1 1 1 1 0 0 0 1 Note: See Alkire et al. (2022) for a complete definition of the Global MPI. We now emphasize that this decomposition is not helpful when looking for the policy minimizing M0 .10 To place ourselves in the easiest case pos- 10 In their conclusion, Alkire and Foster (2016) suggest the need for future research to study a multidimensional version of the optimal budgeting exercises of Kanbur (1987). 10 sible, assume that the cost of removing any deprivation j is exactly equal to its weight wj . In that case, this decomposition suggests that policies minimizing M0 based on a fixed budget need not favor any particular di- mension. Indeed, any amount spent on any dimension j seems to have an impact on M0 (ˆ g 0 ) equal to this amount, regardless of the dimension targeted. This is, however, far from true. Indeed, individual 1 is least intensely poor (s1 = 1/3) while individual 2 has a larger deprivation score (s2 = 2/3). With a budget equal to 1/18 = minj wj , the policy minimiz- ing M0 consists in removing for individual 1 one deprivation among the following three: Cooking fuel, Sanitation or Drinking water. Such policy lifts individual 1 out of poverty because her new deprivation score becomes s′1 = 5/18, just below the poverty cutoff k = 1/3. As a result, her poverty contribution to M0 is reduced by an amount k = 1/3 even though the cost is only 1/18. The alternative policy that removes for individual 2 one de- privation among Electricity, Housing or Assets has the same cost (1/18) but only reduces the poverty contribution of individual 2 by 1/18. We emphasize that the example provided in Table 2 also works under alternative assumptions on the costs of removing deprivations or with larger budgets. First, consider alternative costs structures. Assume for instance that the cost of removing one deprivation among Electricity, Housing, and Assets is three times cheaper than the cost of removing one deprivation among Cooking Fuel, Sanitation, and Drinking. Say this cost is equal to wj /3 for the former while it is wj for the latter. With a budget equal to 1/18, the policy that removes one deprivation in Cooking Fuel for individ- ual 1 reduces M0 more than the policy that removes one deprivation in Electricity plus one deprivation in Housing plus one deprivation in Assets for individual 2. The reason is again that the former policy lifts individual 1 out of poverty, while the latter does not lift individual 2 out of poverty (it merely reduces 2’s deprivation score by 3/18). Second, consider a larger budget equal to 1/3. In that case, the policy minimizing M0 consists in (i) removing for individual 1 one deprivation among the following three: Cooking fuel, Sanitation or Drinking water and (ii) removing for individ- ual 2 two deprivations among the following three: Electricity, Housing or Assets. Such policy lifts individual 1 out of poverty, which reduces her poverty contribution by 1/3, and also reduces the poverty contribution of individual 2 by 2/18. Observe that any alternative policy that reduces the deprivation score of individual 1 below 5/18 reduces her poverty contri- bution by the same amount 1/3 and reduces the poverty contribution of individual 2 by an amount strictly smaller than 2/18. The problem we wish to emphasize with this example is not merely that the optimal policy targets the least poor individual 1. Our goal is to emphasize that the decomposition does not provide the necessary informa- tion to find the policy that minimizes M0 . In our example, minimizing M0 involves reducing deprivation score s1 below the poverty cutoff k = 1/3 (but not below 5/18 as any further reduction in s1 has no impact on the poverty contribution of individual 1). However, the decomposition does They wonder whether Dimensional Breakdown would play a pivotal role in the derivation of the poverty-minimizing policy. Our illustration shows it is not the case. 11 not provide any information on how different deprivations cumulate into the deprivation scores s1 and s2 of the poor individuals. In fact, the gist of Dimensional Breakdown is precisely that such information is ignored in the decomposition. The decomposition of M0 (ˆ g 0 ) into the respective contribu- tions of the various dimensions does not help identifying the optimal policy. Another way to make the same point is to consider the second deprivation matrix g ˜0 presented in Table 2. Matrix g ˜0 is obtained from matrix g ˆ0 by exchanging a few deprivations between the two individuals without chang- ing their deprivation scores. As a result, the policies minimizing M0 under ˜0 are different from the policies minimizing M0 under g g ˆ0.11 However, the decomposition of M0 (˜ g 0 ) is identical to the decomposition of M0 (ˆ g 0 ). Monitoring progress with M0 Second, we continue with an example showing that Dimensional Breakdown can be misleading when monitoring progress as defined by M0 . Assume that between an initial time t and a later time t′ , a society has its deprivation ′ matrix evolve from g 0 to g 0 , which are defined in Table 3. We show that the evolution of the dimensional components associated to Eq. (1), from Hj ′ to Hj , provides a misleading view of the progress achieved in the different dimensions. Table 3: Dimensional breakdown’s decomposition may generate mislead- ing assessment of the respective contributions of each dimension to M0 ’s trend. School attendance Years of schooling Child mortality Drinking water Cooking fuel Sanitation Electricity Nutrition Housing Assets Dimension si ri Weight wj 1/6 1/6 1/6 1/6 1/18 1/18 1/18 1/18 1/18 1/18 0 1/3 g1 1 0 0 0 1 1 0 0 0 1 1 0 1/3 g2 1 0 0 0 1 0 1 0 0 1 1 0 1/3 g3 1 0 0 0 1 0 0 1 0 1 1 0 0 5/18 g4 = g5 0 0 0 0 0 1 1 1 1 1 0 H j (g 0 ) 3/5 0 0 0 3/5 1/5 1/5 1/5 0 3/5 0′ 5/18 g1 1 0 0 0 1 0 0 0 0 1 0 0′ 5/18 g2 1 0 0 0 1 0 0 0 0 1 0 0′ 5/18 g3 1 0 0 0 1 0 0 0 0 1 0 0′ 0′ g4 = g5 0 0 0 0 1 1 1 1 1 1 1/3 1 ′ H j (g 0 ) 0 0 0 0 2/5 2/5 2/5 2/5 2/5 2/5 Note: See Alkire et al. (2022) for a complete definition of the Global MPI. 11 For instance, with a budget equal to 1/18 = minj wj , the policy minimizing M0 under g˜0 consists in removing for individual 1 one deprivation among the following three: Electricity, Housing or Assets. 12 For the sake of this example, assume that ministry C is in charge of Cooking fuel and ministry SDE is in charge of Sanitation, Drinking wa- ′ ter and Electricity. By contrasting g 0 and g 0 , we see that over the pe- riod from t to t′ , the number of individuals deprived in Cooking fuel in- creased (individuals 4 and 5), while the numbers of individuals deprived in Sanitation (individual 1), Drinking water (individual 2) or Electricity (individual 3) all decreased. No other dimension experienced any change in the number of individuals affected over the same period. One should conclude that ministry C performed much worse than ministry SDE. The ′ problem is that the decomposition of M0 (g 0) and M0 (g 0 ) through Eq. (1), which is illustrated in Figure 3, suggests the exact opposite conclu- sion. Indeed, ministry C seems to have made a positive contribution as HCook = 35 >5 2 ′ = HCook . In contrast, ministry SDE seems to have made a negative contribution as HSan = 1 5 < 25 ′ = HSan , HDrink = 1 5 2 < 5 ′ = HDrink and HElec = 15 <5 2 ′ = HElec . M0 Nutri 12 90 Housing 9 Assets 90 Assets 6 San 90 San Drink Drink 3 Elec Elec 90 Cook Cook g0 g0 ′ t Figure 3: Example for which Dimensional Breakdown is misleading about the dimensions driving progress. ′ 18 Note: By Dimensional Breakdown, we have M0 (g 0 ) = j wj Hj = 90 and M0 (g 0 ) = ′ ′ 12 0 0 j wj Hj = 90 , where deprivation matrixes g and g are defined in Table 3. ′ Reading: The decomposition suggests that, when moving from g 0 to g 0 , progress has been achieved in Cooking, while Sanitation, Drinking water, and Electricity deteriorated. This view is incorrect has deprivations in Cooking have increased while deprivations in Sanitation, Drinking water and Electricity have all been reduced. The decomposition in Eq. (1) may incorrectly suggest that M0 was reduced because of the positive action taken by Ministry C and in spite of ′ the negative action taken by Ministry SDE. Indeed, we have HCook > HCook ′ ′ and at the same time we have HSan < HSan , HDrink < HDrink and HElec < ′ HElec . However, this view would be wrong. The number of individuals 13 deprived in Sanitation, Drinking water and Electricity has been reduced, while the number of individuals deprived in Cooking fuel has increased. Observe also that the decomposition illustrated in Figure 3 also suggests that progress has been achieved in terms of Nutrition and Assets, while Housing deteriorated. This view would also be wrong, as no change is recorded in any of these deprivations. Again, the problem arises because the decomposition is made after poverty identification takes place. Alkire and Foster (2016, 2019) defend Dimensional Breakdown as an important property for policy analysis. They write that this property “fa- cilitates policy coordination” and “could help in the allocation of resources across sectors and the design of specific or multisectoral policies to address poverty; monitoring progress dimension by dimension can help clarify the underlying sources of progress”.12 As the two examples provided above illus- trate, when improperly understood, this property could also have the exact opposite effect. It could lead to suboptimal allocation of resources across sectors and mislead policy makers on the underlying sources of progress. The risk is that the decomposition allowed by Dimensional Breakdown may induce a false sense of simplicity and misguide policy makers. To be sure, we do not mean that the censored headcount ratio Hj is irrelevant for policymaking. Our discussion merely questions the added- value of the decomposition of M0 into a weighted sum of censored headcount ratios. If this decomposition has limited added-value for policymaking ′ purposes, then index M0 seems a perfectly suitable quick fix to the perverse incentive associated to M0 . 6 Conclusion In their conclusion, Alkire and Foster (2016) suggest exploring optimal bud- geting exercises, i.e., finding the policy minimizing M0 poverty on a budget constraint, as done by Kanbur (1987) in the monetary poverty case. They explain why such exercises would be more complex in the multidimensional case, namely because one would need to identify not only which individ- uals to target, but also which deprivation to alleviate. This paper shows that such optimal policy would target the least intensely poor individu- als rather than the most intensely poor individuals. We have proposed a simple tweak to the adjusted headcount ratio that does not provide these perverse incentives. This solution preserves the Alkire-Foster identification method. However, the tweaked index violates Dimensional Breakdown. As we show by means of examples, the decomposition of the adjusted head- count ratio permitted by Dimensional Breakdown need not necessarily play an unambiguously useful role in such exercise. 12 They add in a footnote that “the translation from measure to policy response requires additional analysis, as deprivations are often interconnected”. 14 References Alkire, S. and Foster, J. (2011). Counting and multidimensional poverty measurement. Journal of Public Economics, 95(7-8):476–487. Alkire, S. and Foster, J. (2016). Dimensional and distributional contribu- tions to multidimensional poverty. OPHI working paper No. 100. Alkire, S. and Foster, J. (2019). The role of inequality in poverty measure- ment. OPHI working paper No. 126. Alkire, S., Kanagaratnam, U., and Suppa, N. (2022). The global multidi- mensional poverty index (mpi) 2022 disaggregation results and method- ological note. Atkinson, A. B. and Bourguignon, F. (1982). The comparison of multi- dimensioned distributions of economic status. The Review of Economic Studies, 49(2):183–201. Bosmans, K., Lauwers, L., and Ooghe, E. (2018). Prioritarian poverty comparisons with cardinal and ordinal attributes. The Scandinavian Journal of Economics, 120(3):925–942. Bourguignon, F. and Chakravarty, S. R. (2003). The measurement of mul- tidimensional poverty. Journal of Economic Inequality, 1(1):25–49. Datt, G. (2019). Distribution-sensitive multidimensional poverty measures. The World Bank Economic Review, 33(3):551–572. Decancq, K., Fleurbaey, M., and Maniquet, F. (2019). Multidimensional poverty measurement with individual preferences. The Journal of Eco- nomic Inequality, 17(1):29–49. Decerf, B. (2023). A preference-based theory unifying monetary and non-monetary poverty measurement. Journal of Public Economics, 222:104898. Decerf, B. and Fonton, K. (2023). Reconceptualizing Global Multidimen- sional Poverty Measurement, with Illustration on Nigerian Data. World Bank. Decerf, B. M. A. (2024). Multidimensional well-being measurement prac- tices: A review focused on improving global multidimensional poverty indicators. Policy Research Working Paper Series, (10800). Kanbur, R. (1987). Transfers, targeting and poverty. Economic Policy, 2(4):111–136. Pattanaik, P. K. and Xu, Y. (2018). On measuring multidimensional de- privation. Journal of Economic Literature, 56(2):657–672. Ravallion, M. (2011). On multidimensional indices of poverty. The Journal of economic inequality, 9:235–248. 15 Rippin, N. (2010). Poverty severity in a multidimensional framework: the issue of inequality between dimensions. Courant Research Centre: Poverty, Equity and Growth-Discussion Papers. Santos, M. E., Lustig, N., and Zanetti, M. M. (2023). Counting and ac- counting: Measuring the effectiveness of fiscal policy in multidimensional poverty reduction. Oxford Poverty and Human Development Initiative. Sen, A. (1976). Poverty: an Ordinal Approach to Measurement. Econo- metrica, 44(2):219–231. Seth, S. and Yalonetzky, G. (2018). Assessing deprivation with ordinal variables: Depth sensitivity and poverty aversion. OPHI working paper No. 123. Zheng, B. (1997). Aggregate Poverty Measures. Journal of Economic Sur- veys, 11(2):123–162. 16