Social choice theory and composite indicators:
In defense of linearity
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Overview Composite indicators vs MD social choice
Axioms & results for MD social choice
Implications for composite indicators
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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
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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)
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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!
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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 ,:
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Implications for CI’s
GDP/h
SSR : 2005
: 2006
lunl
be
Perfect Substitutability between dimensions
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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.
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