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Measuring AI harms with multidimensional Lorenz Zonoids
Paolo Giudici, Jose' Maria Sarabia, Sofia Vei
TL;DR
AI risk governance lacks harm-centered methods that account for severity alongside likelihood, particularly when harm data are ordinal and multidimensional. The paper extends Lorenz Zonoids and Gini indices to multiple dimensions and computes them for MIT incident data. Environmental, infrastructure, property, physical, and democracy-related harms show the strongest joint severity-frequency concentration and may warrant closer examination in mitigation prioritization.
Problem
AI risk management remains limited by provider-centric, compliance-oriented approaches and by ordinal, multidimensional harm data that complicate severity-based prioritization.
Method
The paper develops computable multidimensional Lorenz Zonoids and related Gini indices for ordinal AI-harm data across direct, indirect, and inferred pathways.
Results
Environmental, infrastructure, property, physical, and democracy-related harms attain the highest values under both multidimensional Gini indices.
Takeaways & Limitations
These categories exhibit the strongest concentration in joint severity-frequency structures and may warrant closer examination when mitigation priorities are considered.
Takeaways & Limitations
The analysis treats MIT severity ratings as ordinal: zero denotes no positive recorded severity, while levels 1–5 form ordered groups rather than cardinal scores.
Abstract
from arXiv · showhide
While AI systems increasingly shape high-stakes societal domains, their governance is limited by the lack of risk management methods that operate on real harms, taking their severity, and not only their likelihood, into account. As a consequence, AI risk management models remain compliance-driven and provider-centric, offering limited insight into how harms are dangerous, and on what should be the priority of intervention. The problem is amplified by the nature of harm data which are typically ordinal and multidimensional. To solve the problem, and offer an effective risk assessment methodology, in this paper we propose to model harm data by means of Lorenz Zonoids and Gini indices. To this aim we propose to extend them in a multidimensional setting, and show how to practically calculate them for a real AI incident data repository, provided by the Massachusetts Institute of Technology. The empirical findings indicate that environmental, infrastructure, property, physical, and democracy-related harms attain the highest values under the two multidimensional Gini indices and therefore exhibit the strongest concentration in their joint direct, indirect, and inferred severity-frequency distributions. These concentration patterns may help identify categories that warrant closer examination when mitigation priorities are determined.
1 Introduction
AI harm governance needs methods that assess realized harms by severity as well as likelihood, despite incident data being ordinal and multidimensional. The paper proposes computable multidimensional Lorenz Zonoids and Gini indices, applying them to MIT incident data to identify harm categories with the strongest concentration.
- Existing AI risk frameworks are often compliance-oriented and provider-centric, while coarse severity schemes can under-represent affected stakeholders and blur minor versus high-stakes failures.
- Incident repositories commonly provide ordinal severity because numerical scores are noisy, inconsistently assigned, or affected by annotator disagreement and epistemic uncertainty.
- The paper asks how harms can be prioritized when precise numerical severity data are unavailable or unreliable, requiring an ordinal measure that is also multidimensional.
- The method preserves the joint structure of direct, indirect, and inferred severity pathways without using projections, supporting closer examination of categories when mitigation priorities are considered.
- The proposed multidimensional Lorenz Zonoids extend the Gini index to multiple dimensions and make the previously theoretical construct explicit and computable.
- Environmental, infrastructure, property, physical, and democracy-related harms attain the highest values under the two multidimensional Gini indices in the MIT dataset.
2 Extending the Lorenz curve to higher dimensions
Multivariate Lorenz methods extend concentration analysis beyond one dimension, but defining higher-dimensional quantiles and Lorenz surfaces is challenging. The paper situates Lorenz Zonoids among copula-based and optimal-transport approaches and presents the geometric extension used here.
- Higher-dimensional Lorenz extensions are difficult partly because no generally convincing quantile-function definition exists for dimensions greater than or equal to two.
- The paper calls higher-dimensional Lorenz-curve extensions Multivariate Lorenz Surfaces and traces early proposals to Taguchi and Lunetta.
- The Lorenz Zonoid provides a geometric extension of the Lorenz curve to dimensions p ≥ 1, with the one-dimensional case lying between the Lorenz and reverse Lorenz curves.
- Alternative Multivariate Lorenz Surface methods use either marginal Lorenz inverses combined through copulas or multivariate quantiles constructed with optimal transportation theory.
- Optimal-transport approaches define multivariate quantiles through properties including invertibility, uniform inverse distribution, cyclic monotonicity, or center-outward quantile regions.
3 Methodological Issues
The paper presents an empirical construction of multidimensional Lorenz zonoids and introduces multivariate Gini indices for measuring concentration in multidimensional harm data. It illustrates the construction numerically and establishes bounds for comparing the indices.
- 3.1 Empirical Lorenz zonoids: The method constructs an empirical Lorenz zonoid as a convex set from all subset combinations of observations, using population shares and corresponding vector totals.Subsets G_k contain k observations, with k/n representing the population share; their candidate points generate the zonoid.
- 3.1 Empirical Lorenz zonoids: The empirical construction considers every subset size from 0 through n, with the empty subset contributing the origin and the total number of candidate points independent of dimension.The construction includes G_0 = ∅ and all subsets G_k for k = 0, 1, ..., n.
- 3.1 Empirical Lorenz zonoids: For a bivariate example with n = 5, Table 1 supplies the groups and zonoid points, while Figure 1 shows their convex hull and Figure 2 shows the marginal zonoids.The marginal zonoids correspond to the first and second data components.
- 3.2 Multivariate Gini indices: The paper introduces multivariate Gini indices, including distance-based and volume-based measures, to quantify concentration in multidimensional empirical distributions.The distance formulation uses Euclidean distances, while the volume-Gini index is based on the expanded volume of the lift zonoid.
- 3.2 Multivariate Gini indices: Upper bounds are established for the distance- and volume-based quantities to enable comparison between the two multivariate Gini indices.The cited bounds are described as sharp, and one index can be used directly while the other requires correction by its final bound.
4 MIT Incident Data
The MIT AI Incident Tracker export contains incident-level records with severity annotations across ten harm dimensions and three pathways. The analysis treats ratings as ordinal, preserving ordering without assuming equal distances between levels.
- The export contains 1498 AI-related incidents with descriptive fields and incident-level harm annotations.Table 2 presents illustrative raw rows before preprocessing; descriptive fields are retained for context.
- Each incident receives severity ratings for ten harm dimensions across direct, indirect, and inferred pathways, producing 30 severity columns per incident.The analytical structure contains 44,940 incident–harm–pathway severity observations before aggregation.
- The ratings use an observed 0–5 ordinal scale, with 0 treated as absence of positive recorded severity and levels 1–5 retained as ordered positive severities for zonoid analysis.Ordinal labels establish ordering but do not justify equal distances between adjacent levels.
5 Empirical results
The empirical analysis first evaluates ordinal severity separately by pathway and then combines direct, indirect, and inferred ratings with multidimensional Lorenz zonoids and concentration indices. The two scale-free indices produce similar rankings, identifying environmental, infrastructure, property, physical, and democracy-related harms as most concentrated jointly across pathways.
- 5.1 Univariate analysis: pathway-specific AIH: Pathway-specific AIH rankings differ across direct, indirect, and inferred pathways: malicious content leads direct and indirect harms, whereas financial harm leads inferred harms.Direct AIH values are highest for malicious content at 0.3588, while inferred financial harm reaches 0.3678.
- 5.1 Univariate analysis: pathway-specific AIH: Figures 3–6 visualize pathway-specific Lorenz-type curves, category-specific bivariate zonoids, and pooled marginal zonoids across the three pathways.Figures 4 and 5 show Privacy and Physical harms across all three pathway pairs, while Figure 6 summarizes pooled margins.
- 5.2 Multivariate analysis: Lorenz zonoids and Gini indices: The analysis constructs five-by-three severity-frequency matrices and computes empirical Lorenz zonoids to preserve each harm category’s joint pathway structure.Rows represent positive severity levels 1–5, while columns represent direct, indirect, and inferred pathways.
- 5.2 Multivariate analysis: Lorenz zonoids and Gini indices: Environmental, infrastructure, property, physical, and democracy-related harms attain the highest values under both scale-free multivariate concentration indices.These categories have the strongest concentration in joint severity-level frequency distributions across direct, indirect, and inferred pathways; the indices do not measure absolute severity or prevalence.
- 5.2 Multivariate analysis: Lorenz zonoids and Gini indices: The two scale-free indices yield broadly similar harm-category rankings, while unnormalized MD(FA) is reported only as a scale-dependent reference.The comparable indices are RD(FA) and normalized f MV(FA).
- 5.1 Univariate analysis: pathway-specific AIH: AIH measures ordered severity concentration separately by pathway, whereas multivariate indices evaluate the joint direct–indirect–inferred frequency structure.Consequently, malicious content can have the highest direct AIH while receiving one of the lowest multivariate concentration-index values.
6 Conclusions
The paper demonstrates how multidimensional Lorenz zonoids and multivariate Gini indices can be calculated for ordinal AI-harm data. Applied to the MIT AI Incident Tracker, the method identifies harm categories with the strongest joint severity-frequency concentration.
- The methodology is applied to ten AI-harm categories with ordinal severity recorded across Direct, Indirect, and Inferred pathways.
- Environmental, infrastructure, property, physical, and democracy-related harms attain the highest values under both multidimensional Gini indices.These categories therefore exhibit the strongest concentration in their joint severity-frequency structures.
- The multivariate indices characterize joint frequency structure across pathways, complementing pathway-specific AIH analysis of ordinal severity distributions.Categories with high pathway-specific AIH values do not necessarily have high multivariate concentration values.
- Multidimensional Lorenz zonoids and Gini indices provide a transparent way to examine joint AI-harm annotations and can contribute to mitigation planning alongside prevalence and stakeholder analysis.