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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