Abstract
The paper presents a general framework and an operative procedure for the evaluation of multidimensional deprivation with ordinal attributes. Evaluation is addressed in terms of multidimensional comparisons among achievement profiles, rather than through attribute score aggregations. This makes it unnecessary to scale ordinal attributes into numerical variables, overcoming the limitations of aggregative procedures and counting approaches. The evaluation procedure is fuzzy in nature, accounts for both vagueness and intensity of deprivation and produces a comprehensive set of synthetic indicators for policy-makers.
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Notes
For the sake of simplicity, in the following elements of X partially ordered by \(\le\) will be referred directly as elements of P.
We are assuming that deprivation attributes are measured in such a way that better achievements correspond to higher scores.
In a linear order, there is no need to introduce the concept of complete deprivation, since there is no classification ambiguity.
Linear extensions are graphs, so that graph distance applies. If \(\varvec{p}\) and \(\varvec{q}\) are elements of a linear extension \(\ell\), their distance is by definition the number of edges in the chain connecting them. More generally, the distance between a profile \(\varvec{p}\) and a subset of \(\ell\) is defined as the minimum of the distances between \(\varvec{p}\) and elements of the subset.
The rank of a profile \(\varvec{p}\) in linear extension \(\ell\) is 1 plus the number of profiles below \(\varvec{p}\) in \(\ell\).
The profiles it attains its minimum on, instead, depend upon the structure of the achievement poset and the choice of the threshold.
It is easy to check that lexicographic linear orders over \(v_1,\ldots ,v_k\) are indeed linear extensions of \({\varPi }\).
For example, consider that an antichain with n elements has n! linear extensions.
Reported computations have been performed on an Intel® Core™2 Duo CPU E8400@3.00 GHz \(\times\) 2, RAM 1.9 GB, equipped with Linux 32 bit operative system.
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Appendix
Appendix
In this appendix we prove that lexicographic linear extensions generate the basic achievement poset.
Proposition 2
The intersection of lexicographic linear extensions of \({\varPi }\) equals \({\varPi }\), i.e.:
Proof
From \(Lex({\varPi })\subseteq {\varOmega }({\varPi })\), it follows \({\varPi }=\cap {\varOmega }({\varPi }) \subseteq \cap Lex({\varPi })\), so it is sufficient to prove the opposite inclusion. Since lexicographic linear extensions are indeed extensions of \({\varPi }\), this is equivalent to prove that if \(\varvec{p}\) and \(\varvec{q}\) are incomparable in \({\varPi }\), they are also incomparable in \(\cap Lex({\varPi })\). To show this, notice that if \(\varvec{p},\varvec{q}\in {\varPi }\) are incomparable, then they have at least two conflicting scores. Without loss of generality, suppose that \(p_i<q_i\) and \(q_j<p_j\). Then necessarily \(\varvec{p}\) is ranked below \(\varvec{q}\) in linear extensions lexicographically ordered with \(v_j\, \trianglelefteq_{{\varLambda }} v_i\); viceversa, \(\varvec{p}\) is ranked above \(\varvec{q}\) when in the corresponding attribute rankings \(v_i\trianglelefteq_{{\varLambda }} v_j\). Therefore \(\varvec{p}\) and \(\varvec{q}\) are incomparable in the intersection of lexicographic linear extensions. \(\square\)
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Fattore, M. Partially Ordered Sets and the Measurement of Multidimensional Ordinal Deprivation. Soc Indic Res 128, 835–858 (2016). https://doi.org/10.1007/s11205-015-1059-6
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DOI: https://doi.org/10.1007/s11205-015-1059-6