Higher-order Gini indices: An axiomatic approach

Abstract: Via an axiomatic approach, we characterize the family of n-th order Gini deviation, defined as the expected range over n independent draws from a distribution, to quantify joint dispersion across multiple observations. This extends the classical Gini deviation, which relies solely on pairwise comparisons. Our generalization grows increasingly sensitive to tail inequality as n increases, offering a more nuanced view of distributional extremes. We show that these higher-order Gini deviations admit a Choquet integral representation, inheriting the desirable properties of coherent deviation measures. Furthermore, we prove that both the n-th order Gini deviation and its normalized version, the n-th order Gini coefficient, are n-observation elicitable, facilitating rigorous backtesting. Empirical analysis using World Inequality Database data reveals that higher-order Gini coefficients detect disparities obscured by the classical Gini coefficient, particularly in cases of extreme income or wealth concentration. Our results establish higher-order Gini indices as valuable complementary tools for robust inequality assessment.