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

Agathe Fernandes Machado; Charlotte Jamotton; Arthur Charpentier

stat.MLOct 13, 2026

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

Authors: Agathe Fernandes Machado, Charlotte Jamotton, Arthur Charpentier

Abstract

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.

Figures & tables

Figure 1: Empirical rejection rates at nominal

5\%

(

\pm 1.96

Monte Carlo s.e.). The marginal statistic is stronger for the main-effect alternative at moderate

n

, whereas it stays near size under the pure interaction while the joint audit gains power.

Figure 2: ACSIncome . The joint statistic is far outside the permutation null in every state. Shapley shares allocate the retained-support barycentric disparity descriptively; they are coding- and grouping-dependent, not causal attributions.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Quantity	General $\rho$	Independence ( $\rho=0$ )
$U_{\mathrm{joint}}$	$13-4\rho^{2}$	$13$
$U_{1}$	$4+5\|\rho\|$	$4$
$U_{2}$	$9$	$9$
$U_{\mathrm{marg}}=U_{1}+U_{2}$	$13+5\|\rho\|$	$13$
$U_{2\mid 1}$	$13(1-\rho^{2})$	$13$
$U_{1\mid 2}$	$4(1-\rho^{2})$	$4$

Appendix

Table 1: Four-cell example with dependent sensitive attributes. The second column gives the expressions for

\rho=\operatorname{Corr}(S_{1},S_{2})\in[-1,1]

, while the last column recovers the independent case

\rho=0

Quantity	Value
$U_{\mathrm{joint}}$	$13$
$U_{1}$	$4$
$U_{2}$	$9$
$U_{2\mid 1}$	$13$
$U_{1\mid 2}$	$4$
$U_{1}+U_{2\mid 1}$	$17$

Appendix

Table 2: Summary of the four-cell example. Sequential

U

-audits use actual conditional mixture laws, whereas

V

and its increments are based on barycentric projections of the full-cell quantile process.

Figure 3: Gaussian location benchmark. Population coalition values and conditional projection increments illustrate marginal cancellation under correlated sensitive coordinates.

Task	Leading cost
Sort scores within full cells	$O(\sum_{s}n_{s}\log n_{s})$
Extract $K$ grid quantiles	$O(mK)$ after sorting
Compute all coalition values	$O(2^{d}mK)$
Exact Shapley from cached coalitions	$O(d2^{d})$
All order profiles	$O(d!\,d)$ after caching
Monte Carlo Shapley, $B$ permutations	$O(BdmK)$ without full coalition caching

Appendix

Table 3: Leading computational costs of the finite-grid implementation. Exact coalition enumeration is intended for moderate

d

; permutation Shapley is the natural approximation for larger sensitive profiles. After caching all coalition values, the marginal lookup cost is

O(Bd)

, but this requires the preceding

O(2^{d}mK)

preprocessing step.

$d$	$n$	True $U$	Mean $\widehat{U}$	Bias	RMSE	max strict $\widehat{V}$	$\phi$ RMSE	last share
3	500	1.000	1.006	0.006	0.048	0.0142	0.0135	0.990
3	1000	1.000	1.003	0.003	0.030	0.0070	0.0084	0.995
3	5000	1.000	1.002	0.002	0.015	0.0013	0.0039	0.999
3	10000	1.000	0.999	-0.001	0.010	0.0006	0.0027	1.000
4	500	1.000	1.017	0.017	0.051	0.0297	0.0111	0.977
4	1000	1.000	1.009	0.009	0.033	0.0149	0.0069	0.989

Appendix

Table 4: Pure highest-order interaction. The plug-in estimator recovers the joint disparity, strict lower-order coalition values vanish with

n

, and the last martingale increment approaches the full joint signal. The corresponding Shapley values sum to

\widehat{U}

to numerical precision.

	$(\beta_{1},\beta_{2})=(-1,1)$			$(\beta_{1},\beta_{2})=(-1.5,1)$
$\rho$	$\mathcal{V}(\{2\})$	$H_{2\mid 1}$	ratio	$\mathcal{V}(\{2\})$	$H_{2\mid 1}$	ratio
0.00	1.0000	1.0000	1.0	1.0000	1.0000	1.0
0.50	0.2500	0.7500	3.0	0.0625	0.7500	12.0
$2/3$	0.1111	0.5556	5.0	0.0000	0.5556	$\infty$
0.80	0.0400	0.3600	9.0	0.0400	0.3600	9.0
0.90	0.0100	0.1900	19.0	0.1225	0.1900	1.55

Appendix

Table 5: Closed-form suppression calculations. The ratio is

H_{2\mid 1}/\mathcal{V}(\{2\})

. In the stronger-suppression case the marginal signal cancels exactly at

\rho=2/3

and then changes sign before squaring, so the ratio is not monotone beyond the cancellation point.

Experiment	Configurations	Mean sec.	Median sec.	90% sec.	Max sec.
Null	96	0.01	0.01	0.02	0.02
Pure interaction	12	0.01	0.01	0.02	0.02
Gaussian suppression	200	0.24	0.19	0.50	0.50

Appendix

Table 6: Average computation time per Monte Carlo replicate in the long-form simulation study. Times are machine-dependent but confirm feasibility for moderate sensitive dimension and finite grids.

State	$n$	cells	min cell	median cell	$\mathsf{U}_{\mathrm{joint}}$ [bootstrap interval]
CA	9000	32	22	219.0	0.0173 [0.0163, 0.0192]
NY	9000	32	26	135.5	0.0116 [0.0109, 0.0129]
IL	8938	28	42	91.0	0.0142 [0.0134, 0.0156]
GA	8969	29	20	155.0	0.0122 [0.0114, 0.0135]
TX	9000	32	21	96.0	0.0120 [0.0113, 0.0132]
OH	8929	26	21	100.0	0.0132 [0.0125, 0.0145]

Appendix

Table 7: ACSIncome : held-out sample sizes, retained full cells, cell-size diagnostics, and joint Wasserstein disparity. In every state, the 999-permutation p-value is

\leq 0.001

; all 99% permutation thresholds are below

0.0011

State	$\mathsf{U}_{\mathrm{worst}}/\mathsf{U}_{\mathrm{joint}}$	order sens. (%)	Sex (%)	Race (%)	Age (%)
CA	4.42	0.18	2.0	22.1	75.8
NY	7.38	0.76	3.2	17.1	79.7
IL	5.39	0.21	4.6	15.7	79.8
GA	4.82	0.90	5.9	14.3	79.8
TX	5.87	0.93	5.2	13.4	81.4
OH	5.23	0.36	3.5	10.2	86.4

Appendix

Table 8: ACSIncome : concentration, relative sequential order sensitivity, and Shapley shares. The three Shapley columns sum to

100\%

up to rounding and allocate the fixed

\mathsf{U}_{\mathrm{joint}}

descriptively, not causally.

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Many fairness goals are defined at a population level that misaligns with siloed data collection, which remains unsharable due to privacy regulations. Horizontal federated learning (FL) enables collaborative modeling across clients with aligned features without sharing raw data. We study federated auditing of demographic parity through score distributions, measuring disparity as a Wasserstein--Frechet variance between sensitive-group score laws, and expressing the population metric in federated form that makes explicit how silo-specific selection drives local-global mismatch. For the squared Wasserstein distance, we prove an ANOVA-style decomposition that separates (i) selection-induced mixture effects from (ii) cross-silo heterogeneity, yielding tight bounds linking local and global metrics. We then propose a one-shot, communication-efficient protocol in which each silo shares only group counts and a quantile summary of its local score distributions, enabling the server to estimate global disparity and its decomposition, with

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