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

CI versus MD social choice

Composite indicators allow us to compare performance of countries, but not of groups of countries ↔ MD social choice allows bothGDP/h

SSR

lunl

be

: 2005

: 2006

us

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

lunlbe

lunlbelunlbe

SSRSSRSSR

GDPhGDPhGDPh

xxx

xxxxxxX

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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).

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Separability

GDP/h

SSR

lunl

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 gbe, gnl and 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 & Anonymity

GDP/h

SSR

lunl

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(xbe)+g(xnl)+g(xlu) ≥ g(ybe)+g(ynl)+g(ylu)

be

lu

g is the implicit CI-function of our rule which measures the performance of countries!

9

Pigou-Dalton

lunl

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 wGDPh,wSSR > 0 and a function h with

h’ > 0 and h” < 0 s.t. the CI-function g equals

SSRSSRGDPhGDPhSSRGDPh xwxwhxxg ,:

10

Implications for CI’s

GDP/h

SSR : 2005

: 2006

lunl

be

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.