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An automatic-democratic approach to weight setting for the new human development index


Perhaps the most difficult aspect of constructing a multi-dimensional index is that of choosing weights for the components. This problem is often bypassed by adopting the ‘agnostic’ option of equal weights, as in the human development index. This is an annual ranking of countries produced by the United Nations Development Programme based on life expectancy, education, and per capita gross national income. These three dimensions are now aggregated multiplicatively. Whatever weights (exponents) are chosen for these dimensions, some nations will feel disadvantaged. To avoid the use of arbitrary weights, we propose for consideration a two-step approach: (1) find the most advantageous set of weights for each nation in turn, and (2) regress the associated optimal scores on the underlying indicators to find a single weight set. This approach has the properties of non-subjectivity, fairness, and convenience. The result is that the highest weight is placed on the life expectancy dimension.

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Correspondence to Chris Tofallis.

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Appendix 1: Explanation of DEA using a diagram

  1. 1.

    Explaining DEA using a two-dimensional scatter graph

For the purposes of illustration, we shall restrict attention to just two dimensions; the principles apply equally well in three dimensions. Suppose we plot the life expectancy and years of schooling for a number of countries as points on a graph as in Fig. 1.

Fig. 1
figure 1

The frontier according to DEA

DEA identifies the observed best practice—without having to pre-specify weights of importance. Instead, it uses a dominance argument: in the diagram below, country P is dominated by country B because the latter has a higher life expectancy and a higher level of schooling. Thus whatever weights are attached to the two criteria would cause B to have a higher score than P.

Countries A, B, and C are not dominated by any other country and so form the best-practice frontier or envelope. These are then assigned the maximum score of unity. Countries behind the frontier have their score calculated by the proportion of their distance to the frontier; thus, for P, the score would be given by the ratio OP/OP (where P is the point where the ray from the origin through P meets the frontier). So if P is three quarters of the way to the frontier, then its score would be 3/4 = 0.75.

  1. 2.

    Going beyond DEA to obtain a single set of weights

The slope of line AB implies one set of weights—these would favour countries A and B. The slope of line BC represents another set of weights—these would favour countries B and C. The method in this paper uses regression to identify an intermediate set of weights that does not favour any one country but rather allows every country to influence the outcome whilst keeping as close as possible to the scores found in the DEA analysis.

Appendix 2: The DEA model

We solve a separate optimisation problem for each country. Let H o denote the score of the particular country being assessed. We wish to find an individual set of weights (K o, A o, B o, C o) which maximises the score for that country:

$$ {\begin{array}{@{}l} {\mbox{Maximise}\;H_\mathrm{o} =K_\mathrm{o} \times L^{A\rm{o}}\times E^{B\rm{o}}\times Y^{C\rm{o}},} \\ {\mbox{subject to the constraints}} \\ {\qquad A_\mathrm{o} +B_\mathrm{o} +C_\mathrm{o} =1,} \\ {\qquad K_\mathrm{o} ,A_\mathrm{o} ,B_\mathrm{o} ,C_\mathrm{o} \ge 0\;\left( {\mbox{all weights are nonnegative}} \right),} \\ \end{array} } $$


$$ H_\mathrm{i} \le 1\;\quad \mbox{for}\;i=1\;\mbox{to}\;n\left( {n\;\mbox{is the number of countries}} \right), $$

i.e. none of the country scores exceed 100 % using these weights.

The above problem is converted to a linear programming problem by taking the logarithms: Use K′ to denote ln(K), L′ to denote ln(L), etc. The problem then becomes

$$ {\begin{array}{@{}l} {\mbox{Maximise}\;H^\prime_\mathrm{o} =K^\prime_\mathrm{o} +A_\mathrm{o} {L}^\prime_\mathrm{o} +B_\mathrm{o} E^\prime_\mathrm{o} +C_\mathrm{o} Y^\prime _\mathrm{o} ,} \\ {\mbox{subject to the constraints}} \\ {\qquad A_\mathrm{o} +B_\mathrm{o} +C_\mathrm{o} =1,} \\ {\qquad H^\prime_i \le 0\;\left( {\mbox{because}\;\ln\left(1\right)=0} \right),\;\mbox{for}\;i=1\;\mbox{to}\;n,} \\ {\qquad \mbox{i.e.}\;K^\prime_\mathrm{o} +A_\mathrm{o} L^\prime_i + B_\mathrm{o} E^\prime_i +C_\mathrm{o} Y^\prime_i \le 0,\;\mbox{for}\;i=1\;\mbox{to}\;n,} \\ {\qquad \mbox{and}\;A_\mathrm{o} ,B_\mathrm{o} ,C_\mathrm{o} \ge 0.} \\ \end{array} } $$

Once the optimal value is found in this way, it can be converted back using anti-logs:

$$ H=\exp \;H^\prime $$

Note that the optimal value of H arising from the above linear programming problem is unique, although it may be possible to achieve the same score with different weights—alternative optima. We do not use the weights in the regression stage, only the unique H values.

The scores can also be obtained using a different formulation which normalises the score of the country being assessed to unity (which means the log is zero) and then minimises the maximum score across all countries. The score of the assessed unit is then the reciprocal of the largest score:

$$ {\begin{array}{@{}l} {\mbox{Minimise}\;h^\prime_\mathrm{o} ,} \\ {\mbox{subject to the constraints}} \\ {\qquad A_\mathrm{o} +B_\mathrm{o} +C_\mathrm{o} =1,} \\ {\qquad K^\prime_\mathrm{o} +A_\mathrm{o} L^\prime_\mathrm{o} +B_\mathrm{o} E^\prime_\mathrm{o} +C_\mathrm{o} Y^\prime_\mathrm{o} =0,} \\ {\qquad K^\prime_\mathrm{o} +A_\mathrm{o} L^\prime_i +B_\mathrm{o} E^\prime _i +C_\mathrm{o} Y^\prime_i \le h^\prime_\mathrm{o} ,\;\mbox{for}\;i=1\;\mbox{to}\;n,} \\ {\qquad \mbox{and}\;A_\mathrm{o} ,B_\mathrm{o} ,C_\mathrm{o} \ge 0.} \\ \end{array} } $$

We applied both formulations and found identical scores for all countries.

Appendix 3:

Table 1 Table of results

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Tofallis, C. An automatic-democratic approach to weight setting for the new human development index. J Popul Econ 26, 1325–1345 (2013).

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  • Human development index
  • Index construction
  • Social composite index
  • Aggregation
  • Ranking
  • Data envelopment analysis