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.

In this paper, we present a multidimensional fuzzy analysis of the levels and the patterns of poverty and social fragility of migrants' families, in the Italian region of Lombardy, in year 2014. Migrants' poverty emerges as a complex trait, better described as a stratification of nuanced patterns than in black and white terms; Lombard migrants are in fact affected, to different extents, by "a diffused sharing of deprivation facets" and cannot be trivially split into deprived and non-deprived. The paper employs innovative data analysis tools from the Theory of Partially Ordered Sets; compared to mainstream monetary approaches, this leads to more realistic estimates of poverty diffusion and eliminates some well-known biases of standard evaluation procedures, providing strong support to the use of partial order concepts and tools in social evaluation studies.

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a b s t r a c tIn the axiomatic approach to composite index numbers, a list of properties is given that both price and quantity indices should satisfy in order to ensure consistent comparisons. Usually, the price index is selected first and its cofactor is consequently adopted as the (implicit) quantity index. Unfortunately, even if the price index has good axiomatic properties, its cofactor need not, so the implicit quantity comparison may be axiomatically inconsistent. In this paper, we give a comprehensive study of a family of price indices sharing good axiomatic properties (proportionality, commensurability, and homogeneity) together with their cofactors. This family, called geo-logarithmic, is relevant also because of the empirical circumstance that all known price indices sharing such properties with their cofactors belong to it or can be obtained from geo-logarithmic index numbers through simple transformations. Thus, the geo-logarithmic family seems to play a central role when the joint consistency of price and quantity comparisons is concerned.

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