Higher-order Gini indices: An axiomatic approach

2025-08-14

AI Summary

Classical Gini coefficients, relying solely on pairwise comparisons, lack sensitivity to extreme inequality in the distributional tails. Method: This paper axiomatically constructs higher-order Gini indices—namely, the *n*-th order Gini deviation and its standardized coefficient—extending inequality measurement to the joint dispersion of *n* observations. The proposed indices admit a Choquet integral representation, satisfy *n*-observability, and achieve analytical tractability via expected range modeling. Contribution/Results: Empirical analysis using the World Inequality Database demonstrates that higher-order Gini coefficients robustly uncover top-end income and wealth concentration obscured by conventional Gini measures, exhibiting superior discriminatory power and tail sensitivity under extreme inequality. This framework provides a theoretically grounded, tail-sensitive, and high-dimensional tool for inequality measurement.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI
• Web Mining and Content Analysis: Web measurements

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.

Problem

• Extends Gini deviation to quantify joint dispersion via n-th order measures
• Enhances sensitivity to tail inequality for nuanced extreme distribution analysis
• Validates elicitable properties for robust backtesting of inequality metrics

Innovation

• Extends Gini deviation via n-th order comparisons
• Uses Choquet integral for coherent deviation measures
• Enables n-observation elicitable rigorous backtesting