# Multi-Sensitive Distributional Fairness: Joint Wasserstein Audits and Shapley Allocation

arXiv: https://arxiv.org/abs/2610.19252 · HTML: https://mlsanity.com/paper/2610.19252/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 28 references that are in the map.
- First posted on arXiv 13 Oct 2026.

## Builds on (most cited first, 12 of 12)

- [Retiring Adult: New Datasets for Fair Machine Learning](https://mlsanity.com/paper/2108.04884/) (2021)
- [Preventing Fairness Gerrymandering: Auditing and Learning for Subgroup Fairness](https://mlsanity.com/paper/1711.05144/) (2017)
- [An Intersectional Definition of Fairness](https://mlsanity.com/paper/1807.08362/) (2018)
- [Fair Regression with Wasserstein Barycenters](https://mlsanity.com/paper/2006.07286/) (2020)
- [Bounding the Estimation Error of Sampling-based Shapley Value Approximation](https://mlsanity.com/paper/1306.4265/) (2013)
- [Auditing and Achieving Intersectional Fairness in Classification Problems](https://mlsanity.com/paper/1911.01468/) (2019)
- [Fairness in Clustering with Multiple Sensitive Attributes](https://mlsanity.com/paper/1910.05113/) (2019)
- [Wasserstein-based fairness interpretability framework for machine learning models](https://mlsanity.com/paper/2011.03156/) (2020)
- [Cohort Shapley value for algorithmic fairness](https://mlsanity.com/paper/2105.07168/) (2021)
- [Fairness seen as Global Sensitivity Analysis](https://mlsanity.com/paper/2103.04613/) (2021)
- [A Sequentially Fair Mechanism for Multiple Sensitive Attributes](https://mlsanity.com/paper/2309.06627/) (2023)
- [Machine Learning with Multitype Protected Attributes: Intersectional Fairness through Regularisation](https://mlsanity.com/paper/2509.08163/) (2025)

## Abstract (arXiv)

Fairness audits with several sensitive attributes are often performed one coordinate at a time. This can miss purely intersectional disparities: a score may be independent of each sensitive coordinate while depending on their joint profile. We study distributional demographic parity for a real-valued score using one-dimensional Wasserstein geometry. We distinguish two complementary constructions. Conditional and sequential audits characterize joint parity through their zero sets, but their positive values do not generally decompose a fixed joint Wasserstein disparity. By contrast, full-cell conditional quantile functions define a Hilbert-valued random variable whose variance is the joint disparity. Conditional expectations of this quantile process induce a monotone information game: every reveal order yields an exact martingale decomposition of the same joint disparity, while Shapley values provide an order-free attribution across sensitive attributes. Under independent sensitive attributes, the game reduces to a Hoeffding-ANOVA decomposition and Shapley splits each interaction energy equally among its members; under dependence, orthogonality is lost and explicit formulas reveal suppression and redundancy effects that marginal audits can hide. We give fixed-grid estimators and permutation calibration, and illustrate the framework on synthetic examples and ACSIncome. The resulting framework separates three questions that are often conflated: whether joint distributional parity holds, how a fixed joint disparity decomposes along an information path, and how that disparity can be attributed across sensitive coordinates without privileging an ordering.