1. Social choice theory and composite indicators: In defense of linearity

2. 2 Overview Composite indicators vs MD social choice Axioms & results for MD social choice Implications for composite indicators

3. 3 CI versus MD social choice Illustration: we want to measure performance of  3 European countries (be,nl,lu)  1 benchmark country (us) via 2 performance dimensions (only)  GDP/h: GDP per hour worked  SSR: Schooling Success Rate

4. 4 CI versus MD social choice Composite indicators allow us to compare performance of countries, but not of groups of countries ↔ MD social choice allows both GDP/h SSR lu nl be : 2005 : 2006 us

5. 5 Axioms for MD social choice For simplicity we stick to the previous example assuming a fixed number of countries & equal population size Purpose of MD social choice: find attractive rule to judge whether one situation X is better or worse than another, say Y But what is attractive? introduce axioms:  create simple imaginary situations X and Y in which it is (relatively) easy to judge whether one situation is better than the other. All simple axioms together leads to a rule (or a family of rules) which also allow(s) us to judge more complex real-world situations MD social choice axioms might also impose structure on CI’s

6. 6 Three technical axioms Completeness: either X is at least as good as Y, or Y is at least as good as X (or both) Transitivity: if X is at least as good as Y and Y is at least as good as Z, then also X must be at least as good as Z Continuity: (technical) small changes in a situation X cannot lead to large changes in its comparison with other situations Result 1 (Debreu, 1954) If a rule satisfies Completeness, Transitivity as well as Continuity then there exists a continuous function f s.t. X is at least as good as Y if and only if f(X) ≥ f(Y).

7. 7 Separability GDP/h SSR lu nl be : 2005 : 2006 Separability: countries with the same performance in two situations X and Y do not matter when evaluating X and Y Result 2 (Debreu, 1954; Blackorby, Donaldson & Auersperg, 1981; Tsui, 1995) If a rule satisfies Separability in addition to Completeness, Transitivity and Continuity then there must exist continuous functions g be, g nl and g lu s.t. X is at least as good as Y if and only if g be (x be )+g nl (x nl )+g lu (x lu ) ≥ g be (y be )+g nl (y nl )+g lu (y lu )

8. 8 Monotonicity & Anonymity GDP/h SSR lu nl be : 2005 : 2006 Monotonicity: if all countries perform at least as good in X compared to Y (& some better), then X is better than Y Anonymity: the name of a country does not matter lu be SSR GDP/h nl : 2005 : 2006 Result 3 If a rule satisfies Monotonicity and Anonymity in addition to Separability, Completeness, Transitivity and Continuity then there must exist a strictly increasing & continuous function g s.t. X is at least as good as Y if and only if g(x be )+g(x nl )+g(x lu ) ≥ g(y be )+g(y nl )+g(y lu ) be lu g is the implicit CI-function of our rule which measures the performance of countries!

9. 9 Pigou-Dalton lu nl be GDP/h SSR : 2005 : 2006 Result 4 (Bosmans, Lauwers and Ooghe, 2006) If a rule satisfies Pigou-Dalton in addition to Separability, Completeness, Transitivity, Continuous Differentiability, Monotonicity and Anonymity then there exist weights w GDPh,w SSR > 0 and a function h with h’ > 0 and h” < 0 s.t. the CI-function g equals

10. 10 Implications for CI’s GDP/h SSR : 2005 : 2006 lu nl be Perfect Substitutability between dimensions

11. 11 Conclusion Composite indicators vs MD social choice If we want to be able to compare groups of countries EU versus benchmark group EU over time Old EU versus new EU members and if we care about convergence of countries, then the implicit CI should be linear, i.e., a weighted sum of the performance in the different dimensions.