Posetic Tools in the Social Sciences: A Tutorial Exposition
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status; ownership of goods. . . ). In such cases, there is no common unit of measure
among attributes and it is often forcing to look for one. Moreover, poverty data
are often, if not always, of an ordinal kind, so that aggregative procedures are
technically unfeasible.
Poset theory offers a natural solution to this kind of problems, allowing for multidimensional comparisons among profiles (i.e. set of achievements on attributes) to
be performed, without any preliminary attribute aggregation. This not only solves a
technical issue, but is also more consistent with the nature of the evaluation problem
itself; it is more natural to assess the level of poverty (to take the same example,
as above) of a subject by comparing his/her achievements to one or more poverty
benchmark profiles, rather than pretending to compress them into an “absolute”
deprivation score, to be compared with some numerical threshold. Moreover,
poverty is a multifaceted trait that may assume many different shapes; this is easily
accounted for in a poset setting, where different incomparable profiles can be chosen as structurally different poverty benchmarks, while aggregative/compensative
procedures are forced to identify just a single threshold level. In other words, the
posetic approach to evaluation is much more “complexity preserving” than the
composite indicator one, being capable to extract information directly from the
multidimensional comparison system (i.e. from the partial order relation) and not
from a “compressed” unidimensional reduction of the input MIS.
As detailed in Fattore (2016), the posetic evaluation of multidimensional ordinal
traits can be performed as follows:
1. Take the set of all possible profiles generated by the attributes of the input MIS
and structure them as a poset π .
2. Identify one or more reference profiles, to be used as benchmarks in the evaluation process. These benchmarks represent “one or more alternative reference
forms of deprivation”, identified based on socio-economic considerations. As
such, they must constitute an antichain τ , otherwise they would identify different
levels of deprivation, rather than its “border”. This antichain is the multidimensional ordinal analogue of the threshold level in aggregative procedures.
3. Given the antichain τ , poset elements can be partitioned into three disjoint
subsets: U , comprising elements of π that are ordered above all of the elements
of τ ; D, comprising τ itself and all of the poset elements that are ordered below at
least one element of τ and I , comprising poset elements which neither belong to
U nor to D. Referring again to the poverty example, U comprises non-deprived
profiles, D comprises deprived profiles (consistently with the interpretation of τ
as a poverty threshold) and I comprises profiles which are “partly” (in a fuzzy
sense) deprived.
4. Two evaluation functions can then be computed. The first, called identification
function and written idn(·), measures to which degree a poset element belongs to
D (in our reference example, this means measuring to what extent a profile can
be classified as poor). This function takes value 0 on U , value 1 on D and values
in (0, 1) on I . The second function, called severity function and written svr(·),
computes the “intensity” of the trait, for elements belonging to D or to I . The
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status; ownership of goods. . . ). In such cases, there is no common unit of measure
among attributes and it is often forcing to look for one. Moreover, poverty data
are often, if not always, of an ordinal kind, so that aggregative procedures are
technically unfeasible.
Poset theory offers a natural solution to this kind of problems, allowing for multidimensional comparisons among profiles (i.e. set of achievements on attributes) to
be performed, without any preliminary attribute aggregation. This not only solves a
technical issue, but is also more consistent with the nature of the evaluation problem
itself; it is more natural to assess the level of poverty (to take the same example,
as above) of a subject by comparing his/her achievements to one or more poverty
benchmark profiles, rather than pretending to compress them into an “absolute”
deprivation score, to be compared with some numerical threshold. Moreover,
poverty is a multifaceted trait that may assume many different shapes; this is easily
accounted for in a poset setting, where different incomparable profiles can be chosen as structurally different poverty benchmarks, while aggregative/compensative
procedures are forced to identify just a single threshold level. In other words, the
posetic approach to evaluation is much more “complexity preserving” than the
composite indicator one, being capable to extract information directly from the
multidimensional comparison system (i.e. from the partial order relation) and not
from a “compressed” unidimensional reduction of the input MIS.
As detailed in Fattore (2016), the posetic evaluation of multidimensional ordinal
traits can be performed as follows:
1. Take the set of all possible profiles generated by the attributes of the input MIS
and structure them as a poset π .
2. Identify one or more reference profiles, to be used as benchmarks in the evaluation process. These benchmarks represent “one or more alternative reference
forms of deprivation”, identified based on socio-economic considerations. As
such, they must constitute an antichain τ , otherwise they would identify different
levels of deprivation, rather than its “border”. This antichain is the multidimensional ordinal analogue of the threshold level in aggregative procedures.
3. Given the antichain τ , poset elements can be partitioned into three disjoint
subsets: U , comprising elements of π that are ordered above all of the elements
of τ ; D, comprising τ itself and all of the poset elements that are ordered below at
least one element of τ and I , comprising poset elements which neither belong to
U nor to D. Referring again to the poverty example, U comprises non-deprived
profiles, D comprises deprived profiles (consistently with the interpretation of τ
as a poverty threshold) and I comprises profiles which are “partly” (in a fuzzy
sense) deprived.
4. Two evaluation functions can then be computed. The first, called identification
function and written idn(·), measures to which degree a poset element belongs to
D (in our reference example, this means measuring to what extent a profile can
be classified as poor). This function takes value 0 on U , value 1 on D and values
in (0, 1) on I . The second function, called severity function and written svr(·),
computes the “intensity” of the trait, for elements belonging to D or to I . The