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Social choice theory and
composite indicators:
In defense of linearity
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
SSR
: 2005
: 2006
nl
lu
us
be
Composite indicators allow us to compare
performance of countries, but not of groups of
GDP/h
countries ↔ MD social choice
allows both
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Axioms for MD social choice
 For simplicity we stick to the previous example

X  x be
x nl
x lu

 x be
GDPh

 be
 x SSR

nl
x GDPh
x nlS S R
x luGDPh 

lu 
x SSR 
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
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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
 Separability: countries with the same performance in two
situations X and Y do not matter when evaluating X and Y
SSR
: 2005
: 2006
Result 2 (Debreu, 1954; Blackorby, Donaldson
&
Auersperg, 1981; Tsui, 1995)
nl
If a rule satisfies
Separability
in addition to
lu
Completeness, Transitivity and Continuity then there
be
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)
GDP/h
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Monotonicity & Anonymity
 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
Result 3
SSRIf a rule satisfies Monotonicity
SSR and Anonymity in
: 2005
: 2005
: 2006Completeness, Transitivity and
: 2006
addition to Separability,
nl
lu
Continuity
then there must exist a
lu
be
strictly
nl
increasing &
continuous
as
lu good as Y if
be function g s.t. X is at least be
and only if g(xbe)+g(xnl)+g(xlu) ≥ g(ybe)+g(ynl)+g(ylu)
g is the implicitGDP/h
CI-function of our rule which
GDP/h
measures the performance of countries!
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Pigou-Dalton
Result
2006)
SSR 4 (Bosmans, Lauwers and Ooghe,: 2005
If a rule satisfies Pigou-Dalton in : 2006
addition to
Separability, Completeness, Transitivity, Continuous
Differentiability, Monotonicity
and Anonymity then
nl
lu
there exist weights
wGDPh,wSSR > 0 and a function h
with h’ > 0 and h” <be0 s.t. the CI-function g equals
g : xGDPh , xSSR   hwGDPh xGDPh  wSSR xSSR 
GDP/h
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Implications for CI’s
Perfect Substitutability between dimensions
SSR
: 2005
: 2006
nl
lu
be
GDP/h
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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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