Social choice theory and composite indicators:

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

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

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. : 2005 SSR : 2006 nl lu us be GDP/h 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

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. 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 situationX cannot lead to large changes in its comparison with other situations Three technical axioms 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. Separability:countries with the same performance in two situations X and Y do not matter when evaluating X and Y : 2005 SSR : 2006 nl lu be GDP/h Separability Result 2 (Debreu, 1954; Blackorby, Donaldson & Auersperg, 1981; Tsui, 1995) If a rule satisfies Separabilityin addition to Completeness, Transitivity and Continuity then there must exist continuous functions gbe, gnland glu s.t. X is at least as good as Y if and only if gbe(xbe)+gnl(xnl)+glu(xlu) ≥ gbe(ybe)+gnl(ynl)+glu(ylu)

8. 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 : 2005 : 2005 : 2006 : 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(xbe)+g(xnl)+g(xlu) ≥ g(ybe)+g(ynl)+g(ylu) SSR SSR lu nl nl lu be lu be be GDP/h GDP/h Monotonicity & Anonymity g is the implicit CI-function of our rule which measures the performance of countries!

9. : 2005 SSR : 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 wGDPh,wSSR > 0 and a function h with h’ > 0 and h” < 0 s.t. the CI-function g equals nl lu be GDP/h Pigou-Dalton

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

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.