Measuring the depth of multidimensional poverty with ordinal data

This paper proposes a positional poverty gap measure of multidimensional poverty within the Alkire-Foster counting framework. The measure captures the depth of deprivations even when indicators are ordinal, unlike the standard poverty gap, which requires cardinal variables. The proposed method draws on the fuzzy set literature and introduces a distribution-based measure of deprivation depth using the empirical cumulative distribution of each indicator, with the most deprived group as the benchmark. For each deprived individual, the method assigns a score based on the individual's relative position in the distribution. Depth is thus expressed as a difference in distributional positions, motivating the label positional poverty gap. The paper demonstrates that this measure preserves the identification and aggregation structure of the counting approach and satisfies its axiomatic properties when the reference distribution remains fixed over time. The framework remains flexible because it accommodates different identification rules, deprivation cutoffs, and variable types. Overall, it offers a simple, meaningful, and theoretically grounded way to incorporate depth into multidimensional poverty measurement with ordinal data.