1 Introduction
Well-being consists of many components and cannot be measured by income alone (e.g., Fleurbaey,
2009; Fleurbaey & Blanchet,
2013, among many others). This realization has led to the inclusion of new dimensions beyond income and different frameworks to measure multidimensional well-being, especially in the context of imperfect markets and public provision of some well-being components (e.g., provision of public health and education) (e.g., Sen,
1985,
1987). The recent political agenda also considers improvements in many well-being dimensions beyond GDP when assessing social progress.
1 A different set of well-being dimensions are put together into composite well-being (sustainability) indices to allow policymakers to measure and monitor the overall societal well-being (Ness et al.,
2007). These composite well-being indices are calculated using simple aggregation of different well-being (sustainability) dimensions. For instance, the Human Development Index (HDI) of the United Nations’ Development Programme (UNDP) is based on the geometric average of the health, income and education dimensions (UNDP,
2010). Similarly, the Environmental Performance Index (EPI) is based on the weighted average of a set of indicators.
2 This paper proposes a methodology to obtain different sets of composite well-being and sustainability indices and uses the particular case of the HDI to demonstrate its application and its benefits. The methodology has the potential to be adapted to a range of other indices.
Most composite indices are aggregated through weighted averages where each dimension is given a relative weight suggesting its intrinsic importance (Alkire & Santos,
2014). This is based on the assumption that there is no interaction among dimensions and that there are constant marginal rates of substitution among its dimensions to maintain the composite index unchanged (see Decancq & Lugo,
2013, for a detailed analysis on the issue). However, explicit tradeoffs among dimensions (i.e., the weights attached to dimensions) are not in place implicitly, and implicit tradeoffs across dimensions can be different from the explicit ones (Pinar et al.,
2013,
2015; Ravallion,
2012). Ravallion (
2012) offered an alternative aggregation function based on the generalized aggregation formula by Chakravarty (
2003,
2011), which led to more sensible tradeoffs across the dimensions compared to the geometric mean (see Pinar,
2019 for the recent application of the generalized aggregation method to the OECD’s regional well-being index). Additionally, multiple studies analyzed the robustness of allocating weights to HDI dimensions using linear programming tools to assess the precision of rankings with alternative weights (Athanassoglou,
2015; Cherchye et al.,
2008; Foster et al.,
2013; Pinar et al.,
2017,
2020; Rogge,
2018).
3 Even though the existing literature focused on the impact of alternative weight allocation across the dimensions of the HDI on the ranking precision, the literature mentioned above did not consider the potential interaction among the dimensions of the HDI. This paper aims to fill this gap by allowing positive interactions (synergies) among the dimensions of the HDI with the use of the Choquet integral aggregation method. In a related approach developed by Mazziotta and Pareto (
2016), they obtain a non-compensatory index by penalizing the unbalanced values of the indicators through standard deviation across indicators. Another extreme case scenario for penalizing the unbalanced achievements is to use a minimum operator (i.e., index outcome results in minimum achieved dimension). However, both Mazziotta and Pareto (
2016) methodology and minimum operator do not determine the level of interaction among the pairs of dimensions, which is also possible with the Choquet integral as it allows one to choose a different degree of interaction across the pairs of the dimensions chosen.
There is an extensive set of literature that suggests that there is a synergy (or positive interaction) across the dimensions of the HDI. For example, education and health are considered to be part of the production function leading to different levels of income per capita (see e.g., Mankiw et al.,
1992 and Glaeser et al.,
2004 for education’s effect on economic development and growth; and also see Bhargava et al.,
2001; Bloom et al.,
2004; Behrman et al.,
2004 for health’s effect on economic growth and aggregate output through productivity). Additionally, higher educated societies tend to have better outcomes in health and healthy behaviors (Brunello et al.,
2013; Kemptner et al.,
2011; Soares,
2007). In particular, women’s educational level is profoundly important in improving health outcomes in the developing world (Chen & Li,
2009). Similarly, healthier students receive higher levels of schooling (Weil,
2007) and make better educational and occupational choices (Vogl,
2014). The above-mentioned theoretical and empirical literature highlights that the HDI dimensions are complements to some degree. Therefore, it is essential to incorporate these synergies across HDI dimensions while obtaining composite HDI scores, especially when it comes to informing governments and policies aiming at improving citizens’ well-being. In other words, one should choose an aggregation method that is flexible enough to take into account the positive interactions across the HDI dimensions.
To capture the synergies (or penalize the unbalanced achievements) across the dimensions of the HDI, the UNDP shifted from the arithmetic mean (AM hereafter) aggregation method to the geometric mean (GM) one since 2010 so that “poor performance in any dimension is now directly reflected in the HDI, and there is no longer perfect substitutability across dimensions. This method captures how well-rounded a country’s performance is across the three dimensions it recognizes that health, education and income are all important” (UNDP,
2010, p.15). In other words, the UNDP adopted a GM method to penalize countries with relatively unbalanced achievements across the three dimensions, where countries with unbalanced achievements across three dimensions obtain lower composite scores with the GM method compared to the AM. However, the GM aggregation method still has shortcomings in capturing the complementariness or synergies among the three dimensions as it does not penalize the unbalanced achievements across dimensions strongly enough. Firstly, there is a set of countries that have unbalanced achievements across three dimensions of the HDI but still achieve relatively similar composite scores with the GM and AM method (see Sect.
3 for some of these examples). Secondly, the GM aggregation method does not differentiate alternative synergies among different dimensions as relatively weak performance in either dimension is reflected similarly in the achieved composite score. Yet, the aggregation method should be flexible enough to allow more interaction among some dimensions compared to other sets of dimensions. To allow differing synergies across different dimensions of the HDI and also choose parameters of aggregation method that captures the unbalanced achievements across the dimensions of the HDI, this paper offers an alternative aggregation method, which is flexible enough to capture a various set of interactions across the HDI dimensions.
This study proposes an aggregation methodology based on the Choquet integral (CI hereafter), which considers interactions across dimensions as it evaluates all possible sets of dimensions, rather than evaluating single dimensions (Grabisch et al.,
2009). This permits taking into account how balanced (or unbalanced) the achievements across dimensions are and reflecting these differences in the composite score. The CI is a general method that allows interactions across dimensions while allocating different relative importance to dimensions (see e.g., Grabisch and Labreuche (
2010) for review of CI’s use in multicriteria decision making). The application of the CI for obtaining multivariate indices has increased in recent years. For example, Meyer and Pontheire (
2011) used the CI and demonstrated that individual preferences could not be captured by an additive model (i.e., weighted average aggregation methods) because of complementarities and redundancies between well-being dimensions (see also Angilella et al. (
2016) Oppio et al. (
2018), and Gálvez Ruiz et al. (
2018) for non-additive models for assessing urban quality). Carraro et al. (
2013) applied the CI aggregation method to capture interactions across different sustainability indicators, with other authors using the CI to construct indices that allow for different interactions among indicators (see also Merad et al.,
2013; Bertin et al.,
2018; Bottero et al.,
2015,
2018; Branke et al.,
2016; Campagnolo et al.,
2018). Most of the literature that utilized the Choquet integral aggregation method used expert elicitation to identify the weights (see, e.g., Grabisch et al. (
2008) for a review of methods used for identification of weights). However, in this paper, we use five hypothetical weights that allow different degrees of positive interactions across the dimensions of the HDI to illustrate the use of the Choquet integral for obtaining composite HDI scores. Furthermore, as a robustness analysis, we also simulate 500 weights for the aggregation, which allows different levels of interaction and offers a range of composite HDI scores obtained by countries. The simulation exercise also allows us to obtain a feasible range of composite index outcomes for countries when an interaction index varies between lower and upper values set by the decision-makers. We also obtain the HDI index outcomes with the minimum operator (i.e., perfect complementarity across the three dimensions of the HDI) to compare this extreme case scenario with the index outcomes obtained with the Choquet integral. It should be noted that the hypothetical weights and simulated weights are chosen by this paper is to show-case the usefulness of the Choquet integral in penalizing the unbalanced achievements across dimensions of the HDI and take into account varying interactions; however, they are not the ‘actual’ or ‘true’ interactions among the dimensions.
4 The remaining of the paper is organized as follows. Section
2 introduces the CI aggregation method and illustrates its characteristics. Section
3 shows an application of this methodology to obtain HDI scores by allowing different degrees of interaction across HDI dimensions, followed by a set of concluding remarks in Sect.
4.
2 The Choquet integral as an aggregation methodology
The CI aggregation method (Choquet,
1953) has a general expression that uses a diversity of inputs from policymakers and the public. This allows considering a wide range of political and personal choices. For example, this paper shows an application of the CI method to measuring well-being by allowing different degrees of interaction among HDI dimensions, with dimensions still being equally important.
Let
\(\{ x_{1} ,x_{2} , \ldots ,x_{d} \}\) be the index values of the well-being dimensions described by a set
\(D = \left\{ {1,2, \ldots ,d} \right\}\) of dimensions. Capacities are a set of functions where
\(2^{D}\) is the all possible subsets of the criteria, which assigns a weight (measure), ranging between 0 and 1, to every subset of dimensions. The set function,
\(\mu\), has to satisfy border and monotonicity conditions:
(i)
\(\mu \left( \emptyset \right) = 0,\mu \left( D \right) = 1,\)
(ii)
for any \(S,T \subseteq D\), \(S \subseteq T \subseteq D\mathop \Rightarrow \limits_{{}} \mu \left( S \right) \le \mu \left( T \right) \le 1.\)
The first condition represents scenarios (i.e.,
\(\mu \left( \emptyset \right) = 0\;{\text{and}}\;\mu \left( D \right) = 1\), which suggests that all dimensions are unsatisfactory (i.e., achievements in all dimensions are zero) and satisfactory (i.e., achievements in all dimensions are full), respectively. The second condition suggests that the value of
\(~\mu \left( T \right)\) represents the capacity (weight) of dimensions belonging to the subset
T for any subset
\(T \subseteq D\). This can be interpreted as the weight (importance) that one assigns to the fully satisfactory performances of the dimensions belonging to the subset
T, and with fully unsatisfactory performances by the remaining dimensions. For example, if a subset has two out of three HDI dimensions (e.g., health and education), then
\(\mu \left( {\left\{ {Health,Education} \right\}} \right)\) would represent the weight attached to the scenario where health and education achievements are fully satisfactory, and income is fully unsatisfactory. Since
\(\mu \left( \emptyset \right) = 0,\,\mu \left( D \right) = 1\) (i.e., when all dimensions are unsatisfactory and fully satisfactory, respectively), one needs to assign weights
\(\mu \left( S \right)\) for all other
\(2^{d} - 2\) subsets
S of
D, where d = card(D). The CI
\(x:\{ x_{1} ,x_{2} , \ldots ,x_{d} \}\) with respect to a capacity
\(\mu \) on
D is defined by:
$$ C_{\mu } \left( x \right) = \mathop \sum \limits_{{i = 1}}^{d} \left( {x_{{\tau \left( i \right)}} - x_{{\tau \left( {i - 1} \right)}} } \right)\mu \left( {\left\{ {\tau \left( i \right), \ldots ,\tau \left( d \right)} \right\}} \right), $$
where
\(\tau\) is a permutation on
D such that
\(x_{{\tau \left( 1 \right)}} \le \ldots \le x_{{\tau \left( d \right)}}\) and
\(x_{{\tau \left( 0 \right)}} = 0\) for all
\(i \in \left\{ {1, \ldots ,d} \right\}\). A convenient way of presenting the CI is by using Möbius values. Given a weight
\(\mu\) on
\(2^{D}\), its Möbius presentation (Rota,
1964; Shafer,
1976) is a function
\(m:2^{D} \to \Re\) such that, for all
\(S \subseteq D\) $$\mu \left( S \right) = \sum _{{T \subseteq S}} m\left( T \right) , $$
we have that.
\(m\left( S \right) = \sum _{{T \subseteq S}} \left( { - 1} \right)^{{s - t}} \mu \left( T \right)~\forall S \subseteq D , \)where
\(S = card\left( S \right)\) and
\(t = card\left( T \right)\). The following boundary and monotonicity conditions must be met:
$$ \left\{ {\begin{array}{*{20}c} {m\left( \emptyset \right) = 0,~~\mathop \sum \limits_{{T \subseteq D}} m\left( T \right) = 1,} \\ {\mathop \sum \limits_{{\begin{array}{*{20}c} {T \subseteq S} \\ {T \ni i} \\ \end{array} }} m\left( T \right) \ge 0,\forall S \subseteq D,\forall i \in S,} \\ \end{array} } \right. $$
where the Möbius value for the scenario where achievements in all dimensions are zero would be zero (i.e.,
\(m\left( \emptyset \right) = 0)\), and the sum of the whole Möbius values would be one (i.e.,
\(\sum _{{T \subseteq D}} m\left( T \right) = 1)\).
The CI can now be expressed in terms of the Möbius representation
m of the weight μ as follows:
$$ C_{\mu } \left( x \right) = \sum _{{T \subseteq D}} m\left( T \right)\Lambda _{{i \in T}} x_{i} $$
, where the symbol ^ denotes the minimum operator and
\(m\left( T \right)\) are the Möbius coefficients (Grabisch et al.,
2009).
2.1 Characteristics of the Choquet Integral
The CI aggregation method can be used to obtain general preferences in multidimensional well-being analysis. Three important characteristics of the CI are included below to illustrate the flexibility of the methodology to incorporate decision-makers’ preferences on multidimensional well-being.
2.1.1 Relative Importance Index
The relative importance of well-being dimensions can be estimated by the Shapley value (Shapley,
1953) of each dimension. This is calculated by comparing the weights in every set that includes that dimension against every set that does not include it. Therefore, the overall importance of dimension
\(i\in D\) can be obtained by averaging marginal contributions (Grabisch,
1995,
1996) as follows:
$$ v_{\mu } \left( i \right) = \sum _{{T \subseteq D\backslash i}} \frac{{\left( {d - t - 1} \right)!t!}}{{d!}}\left[ {\mu \left( {T \cup i} \right) - \mu \left( T \right)} \right] $$
where
\(d = card\left( D \right)\) and
\(t = card\left( T \right)\) represent the cardinality of the subset of
D and
T, respectively. For instance, to obtain the importance of the health dimension in the calculation of the HDI, one can compare the weights assigned to subsets that include the health dimension with the subsets that do not have the health dimension. This would consist of four comparisons: (i) weight attached to a subset that has health dimension only vs. weight attached to an empty subset; (ii) weight attached to a subset that includes health and education dimensions vs. weight attached to a subset that only includes education dimension; (iii) weight attached to a subset that includes health and income dimensions vs. weight attached to a subset that only includes income dimension; (iv) weight attached to a subset that includes all dimensions vs. weight attached to a subset that includes education and income dimensions. In terms of the Möbius representation of
\(\mu \), the Shapley value of dimension
i can be rewritten as:
$$ v_{\mu } \left( i \right) = \sum _{{T \subseteq D\backslash i}} \frac{1}{{t + 1}}\left[ {m\left( {T \cup i} \right)} \right] $$
Furthermore, the relative importance of the dimensions (i.e., Shapley values) sums to one (i.e.,
\(\sum _{{i = 1}}^{d} v_{\mu } (i) = 1)\), and higher Shapley values represent higher relative importance.
2.1.2 Orness Index
The Choquet integral aggregation also allows one to examine whether the choice of the weights by the decision-maker is
optimistic or
pessimistic (see Marichal,
2004 for the detailed discussion on the orness index). In other words, the orness index measures whether a decision-maker thinks that a good performance in one dimension compensates another one or not. The orness index ranges between 0 and 1, and higher (lower) values of this index represent that the decision-maker thinks that the dimensions are substitutes (complements) of each other. For instance, if orness index equals to 1, then the decision-maker is considered to be
fully compensative (i.e., the dimensions are
perfect substitutes to each other), and in this case, Choquet integral aggregation will be equal to the
maximum operator (i.e., an index outcome would be the maximum value amongst the dimensions). On the other hand, if orness index is equal to 0, then the decision-maker is considered to be
fully non-compensative, and the Choquet integral corresponds to the
minimum operator (i.e., the dimensions are
perfect complements), and the index outcome would be the lowest value amongst the dimensions. The
orness index is computed as follows:
$$ Orness\left( i \right) = \frac{1}{{d - 1}}\sum _{{T \subseteq D}} \frac{{d - t}}{{t + 1}}m\left( T \right) $$
where
\(d = card\left( D \right)\) and
\(t = card\left( T \right)\) represent the cardinality of the subset of
D and
T, respectively.
2.1.3 Interaction Index
A key reason for using the CI to construct a composite well-being index is to take into account the interaction between well-being dimensions. Consider two well-being dimensions of
i and
j. If
\(\mu \left( {\left\{ {i,j} \right\}} \right)\) is greater than the sum of
\(\mu \left( {\left\{ i \right\}} \right)\) and
\(\mu \left( {\left\{ j \right\}} \right)\) (i.e.,
\(\mu \left( {\left\{ {i,j} \right\}} \right) > \mu \left( {\left\{ i \right\}} \right) + \mu \left( {\left\{ j \right\}} \right)\)), this would suggest a
complementary (mutual-strengthening) effect between dimensions
i and
j. In terms of Möbius representation, this would suggest
\(m\left( {\left\{ {i,j} \right\}} \right) > 0\). On the other hand, if
\(\mu \left( {\left\{ {i,j} \right\}} \right)\) is less than the sum of
\(\mu \left( {\left\{ i \right\}} \right)\) and
\(\mu \left( {\left\{ j \right\}} \right)\) (i.e.,
\(\mu \left( {\left\{ {i,j} \right\}} \right) < \mu \left( {\left\{ i \right\}} \right) + \mu \left( {\left\{ j \right\}} \right)\) or
\(m\left( {\left\{ {i,j} \right\}} \right) < 0\), this would suggest a
redundancy (mutual-weakening) effect between dimensions
i and
j. Finally, if
\(\mu \left( {\left\{ {i,j} \right\}} \right) = \mu \left( {\left\{ i \right\}} \right) + \mu \left( {\left\{ j \right\}} \right)\) or
\(m\left( {\left\{ {i,j} \right\}} \right) = 0\), this would suggest that dimensions
i and
j do not interact. For instance, if the health and education dimensions of the HDI are mutually-strengthening (mutually-weakening), then the weight given to the subset that includes both dimensions should be larger (smaller) than the sum of the weights given to subsets that only includes health and education dimensions. Obviously, these two dimensions can join other subsets and therefore, an index of interaction between dimensions
i and
j should take into account all forms:
\(\mu \left( {T \cup i} \right)\),
\(\mu \left( {T \cup j} \right)\) and
\(\mu \left( {T \cup ij} \right)\) with
\(T \subseteq D\backslash ij\). Therefore, an average interaction index between the two dimensions
\(i\) and
\(j\) is calculated as follows (Murofushi & Soneda,
1993):
$$ I_{\mu } \left( {ij} \right) = \sum _{{T \subseteq D\backslash ij}} \frac{{\left( {d - t - 2} \right)!t!}}{{\left( {d - 1} \right)!}}\left[ {\mu \left( {T \cup ij} \right) - \mu \left( {T \cup i} \right) - \mu \left( {T \cup j} \right) + \mu \left( T \right)} \right] $$
where
\(d = card\left( D \right)\) and
\(t = card\left( T \right)\) represent the cardinality of subsets of
D and
T, respectively.
The quantity
\(I_{\mu } \left( {ij} \right)\) can be interpreted as a measure of the
average marginal interaction between
i and
j. An important property is that
\(I_{\mu } \left( {ij} \right) \in \left[ { - 1,1} \right]\) for all
\(ij \subseteq D\), the value 1 (respectively − 1) corresponding to maximum complementarity (respectively substitutivity) between
i and
j (see Grabisch,
1997). In terms of the Möbius representation of
\(I_{\mu } \left( {ij} \right)\), the interaction index between the two dimensions
\(i\) and
\(j\) can be rewritten as:
$$ I_{\mu } \left( {ij} \right) = \mathop \sum \limits_{{T \subseteq D\backslash i}} \frac{1}{{t + 1}}\left[ {m\left( {T \cup ij} \right)} \right] $$
4 Conclusions
Most well-being and sustainability composite indices are based on either arithmetic weighted averages or geometric weighted averages of sub-dimensions (i.e., AM or GM when indicators consisting of composite indices are given equal weights). When weighted averages are used in the aggregation of indicators, decision-makers (e.g., policymakers, index users) can choose the importance of indicators that make up the composite index. For instance, the OECD’s Better Life Index has a user-friendly interactive website that enables users to decide on all indicators’ relative importance (weights) to produce a composite index. However, one of the main issues associated with obtaining composite indices based on arithmetic and geometric weighted averages is that these methods do not consider the potential interactions among the indicators. Decision-makers often assume that certain pairs of indicators have a mutual-weakening (mutual-strengthening) effect and can substitute (complement) each other. The methodology presented in this article (CI) offers a flexible approach that allows considering these potential interactions, with benefits to decision-making, including the possibility to capture different levels of interactions across pairs of well-being indicators.
The HDI case, one of the best known composite indices, was analyzed in this paper to illustrate the use of the CI as an aggregator that considers potential interactions among well-being indicators. Prior to 2010, HDI scores were officially calculated based on the AM of the three HDI dimensions (income, health, and education). The UNDP changed its aggregation method to the GM after 2010 to promote well-rounded achievements across dimensions and acknowledge the literature exploring the mutual-strengthening of these three dimensions. In this paper, rather than using the expert elicitation (i.e., a method that is used by most of the existing literature using the CI aggregation) to identify the weights, we rely on the theoretical and empirical literature and use five hypothetical weight sets that allow synergies among the three dimensions of the HDI to avoid potential expert-selection bias. We also simulated 500 weight sets that would enable different levels of interaction between pairs of the HDI dimensions to demonstrate the CI aggregation use.
Overall, this article demonstrates that: (1) the GM adopted after 2010 actually results in HDI scores that are similar to those generated by the AM; (2) the GM allocates similar HDI scores to countries that show a relatively significant variation in achievements across dimensions, suggesting that this is not successful at accounting for positive interactions among dimensions; (3) the use of GM as an aggregation method does not promote well-rounded performances across the three HDI dimensions, because countries can still improve their composite scores without addressing unbalanced achievements.
The CI aggregation method offers several benefits over the AM and GM methods, including:
(1)
accounting for different degrees of positive interactions (i.e., mutual-strengthening effect) among pairs of dimensions (Sect.
3.3);
(2)
recognizing unbalanced achievements across dimensions by generating lower composite scores, hence encouraging countries to work in the direction of balanced achievements in education, health, and income (in the case of HDI);
(3)
allowing decision-makers to identify poor performance in dimensions (e.g., Table
5 includes countries ranking in high positions with the GM because of good achievements in only one or two dimensions, but poor achievements in the third; the same countries ranked in lower positions using the CI because this method allows measuring how well-rounded achievements across dimensions are);
(4)
temporal changes in HDI scores based on the GM (or AM) method did not explain whether changes in aggregate achievement levels across dimensions were in the direction of balanced achievements or not (Sect.
3.6). The CI method was again, however, successful at explaining whether temporal improvements were in the direction of rounded achievements across the three dimensions or not;
(5)
showing how the minimum operator is also successful in penalizing unbalanced achievements across the three dimensions similar to those of CI cases 1 and 2. However, the minimum operator does not allow for varying positive interactions among the pairs of the dimensions, which is possible with the CI method (see Appendix Tables
8,
9,
10,
11,
12,
13,
14,
15,
16 for comparisons between CI cases 3, 4 and 5, and minimum operator).
(6)
obtaining a feasible range of HDI scores for countries (minimum and maximum HDI scores presented in Tables
17) when interaction levels among the pairs of dimensions are allowed to vary between two values that could be chosen by the decision-makers.
The analyses presented in this paper concentrate on the positive interactions across the three dimensions based on the intention of UNDP policymakers (
2010 report) and existing literature. The Choquet methodology allows, however, multiple choices to reflect the preferences of policymakers and the public. This flexibility has important benefits because it offers the possibility of adopting different sets of constraints, including a variety of interaction levels across pairs of dimensions (e.g., by considering some of the well-being dimensions to be substitutes and some others to be complements) and different relative importance to dimensions (i.e., higher or lower Shapley values).
One of the limitations of this study was to choose weights a priori rather than obtaining weights through expert elicitation. However, the weights simulated in this study were to illustrate how Choquet integral—a flexible aggregation method—can take into account varying positive interactions to penalize the unbalanced achievements across the dimensions. Therefore, it should be noted that the weights chosen by this paper are neither the ‘best’ nor the ‘most democratic’ ones but were only chosen to illustrate the usefulness of the Choquet aggregation methodology in capturing varying interactions across the HDI dimensions. The future research venues are to identify the ‘true’ interactions among well-being indicators and use expert elicitation to identify more democratic weights to obtain a more representative measurement of well-being indices.
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