Conventional poverty gap measures require cardinal data --- variables where the distance between values is meaningful. Household income, measured in dollars, is cardinal: the gap between ten thousand and twenty thousand is the same as between fifty thousand and sixty thousand. But many indicators used in multidimensional poverty assessment are ordinal. Education level, housing quality, access to sanitation --- these have a clear ranking but no natural metric. The “distance” between primary and secondary education is not the same kind of object as the distance between two income levels. A new positional poverty gap measure addresses this by assigning deprivation scores based on each individual's position within the empirical distribution of each indicator.
The method compares the distribution of deprived individuals against the distribution of the most disadvantaged reference group. A person's deprivation score reflects how far down the distribution they sit relative to those worst off --- a positional rather than absolute measure. This is built on fuzzy set theory: instead of drawing a hard line between poor and not-poor, each person receives a continuous membership value in the set of the deprived.
What matters is what this framework preserves. The identification and aggregation structure of the counting approach --- the standard method for multidimensional poverty that classifies individuals as poor or not-poor based on how many deprivations they face --- remains intact. The axiomatic properties carry over. The measure simply adds depth information to a framework that previously could only register incidence.
The broader point is that ordinal data is not impoverished data --- it is data with a different geometry. The appropriate response is not to force cardinal metrics onto ordinal scales but to develop measures native to the ordinal structure. Rank carries information that magnitude does not, and sometimes rank is all you have.