Aumann-SHAP: The Geometry of Counterfactual Interaction Explanations in Machine Learning

Adam Belahcen; Stéphane Mussard

cs.LGMar 14, 2026

Aumann-SHAP: The Geometry of Counterfactual Interaction Explanations in Machine Learning

Authors: Adam Belahcen, Stéphane Mussard

Organizations: AIRESS & UM6P

Abstract

We introduce Aumann-SHAP, an interaction-aware framework that decomposes counterfactual transitions by restricting the model to a local hypercube connecting baseline and counterfactual features. Each hypercube is discretized into a grid to construct an induced micro-player cooperative game in which elementary grid-step moves become players. Shapley and LES values on this TU-micro-game yield geometry-aware within-pot attributions that converge to the diagonal Aumann--Shapley / Integrated Gradients limit under grid refinement, and recover equal-split Shapley as the degenerate $m=1$ special case. An exact grid-state closed form gives polynomial-time computation for fixed interaction order. On a synthetic benchmark with known ground truth, equal-split Shapley carries an irreducible bias while Aumann-SHAP converges to the correct decomposition. On German Credit, interaction geometry changes feature priority rankings in $12.3\%$ of instances. On UCI Heart Disease, equal-split misattributes a cholesterol suppressor as a positive contributor, which is a sign error Aumann-SHAP corrects. On MNIST, game-theoretic attribution reaches target confidence with $3.5\times$ fewer edits than magnitude-based ordering, with micro-game Shapley achieving the best efficiency across all budgets.

Figures & tables

Method	CF	Geo	Int	IG lim	Poly
SHAP [ 3 ]	$\times$	$\times$	$\times$	$\times$	$\times$
Integrated Gradients [ 6 ]	$\times$	$\sim$	$\times$	$\checkmark$	$\checkmark$
Shapley–Taylor [ 7 ]	$\times$	$\times$	$\checkmark$	$\times$	$\times$
Equal-split CF Shapley [ 14 ]	$\checkmark$	$\times$	$\checkmark$	$\times$	$\checkmark$
Aumann-SHAP	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$

TABLE I : Comparison of attribution methods on key properties. CF = designed for counterfactual transitions; Geo = geometry-aware within-pot split; Int = interaction pot decomposition; IG lim = converges to Aumann–Shapley/IG limit; Poly = polynomial closed-form computation.

Symbol	Meaning
$N=\{1,\dots,d\}$	full feature index set
$u\subseteq N$ , $\|u\|=k$	coalition of features under study
$T\subseteq u$	sub-coalition (corner of local cube)
$\mathbf{x}^{0},\mathbf{x}^{1}\in\mathcal{X}$	baseline and counterfactual instance
$\mathbf{x}^{S}$	mixed input ( cf. Eq. ( 1 ))
$V(S)=g(\mathbf{x}^{S})-g(\mathbf{x}^{0})$	finite-change coalition value

TABLE II : Notation used throughout the paper.

Panel of Figure 3 — Fig. 1 : Local slider cube for $u=\{i,j,k\}$ (a) and the sub-cube from $(0,0,0)$ to $(t_{i},t_{j},t_{k})$ (b), illustrating corner evaluations and the inclusion–exclusion construction.

Fig. 2 : Convergence to analytical ground truth. Micro-game Shapley (solid) converges to the exact $1/3$ – $2/3$ split (dashed) as $m$ increases. Equal-split Shapley (dotted red) remains at $0.5$ for all $m$ — a permanent, irreducible bias. Model: $g=x_{1}x_{2}^{2}+x_{3}$ , $u=\{1,2\}$ , $\mathbf{x}^{0}=(0,0,0)\to\mathbf{x}^{1}=(1,1,1)$ .

Panel of Figure 5 — Fig. 3 : Solid curves with markers provide runtimes: grid-state computation (blue) and, when feasible, full micro-coalition enumeration (orange). Dashed curves are fitted scaling laws: $\sim c\,k(m{+}1)^{k}$ for grid-state (green) and $\sim c\,n2^{n}$ with $n{=}km$ for full enumeration (red).

Fig. 4 : Medical misattribution risk. Equal-split (gray) assigns positive credit to chol , implying it contributes to the beneficial interaction. Aumann-SHAP (blue) reveals chol actively suppresses the interaction (negative share). A geometry-agnostic explanation could misdirect clinical intervention toward the wrong feature.

Pot $T$	Feat.	$\phi^{\mathrm{eq}}$	$S^{\mathrm{micro}}$	$\phi^{\mathrm{eq}}-S^{\mathrm{micro}}$
$X_{8}+X_{13}$	$X_{13}$	0.0659	0.0176	-0.0484
$X_{8}+X_{13}$	$X_{8}$	0.0659	0.1143	0.0484
$X_{8}+X_{14}$	$X_{14}$	-0.0004	-0.0366	-0.0362
$X_{8}+X_{14}$	$X_{8}$	-0.0004	0.0359	0.0362

TABLE VI : Growing Spheres: within-pot reallocations (TU-micro-game vs. equal split).

Fig. 5 : MNIST $1\to 7$ counterfactual: attribution maps and redistribution. Top row: baseline $x^{0}$ , endpoint $x^{1}$ (only the $k$ changed pixels are moved), and Top-25 masks for equal-split and TU-micro-game Shapley (red). Bottom row: signed heatmaps, redistribution $D=\text{micro}-\text{equal}$ , and an overlay of the largest re-allocations on $x^{1}$ .

Fig. 6 : Global MNIST $1\to 7$ mean explanations (nearest-neighbor pairing). We sample $200$ test-set baselines $x^{0}$ of digit $1$ and pair each with its nearest neighbor $x^{1}$ among test-set $7$ ’s (Euclidean distance), then compute local attributions on the changed-pixel set $\{(r,c):|x^{1}_{rc}-x^{0}_{rc}|>\varepsilon\}$ with $\varepsilon=0.05$ and average the resulting maps over pairs.

Method	$[email protected]$	$[email protected]$	AUC
Micro-game Shapley	4	6	0.8928
Equal-split Shapley	5	6	0.8916
Equal Surplus	5	7	0.8761
$\|x^{1}-x^{0}\|$ magnitude	14	21	0.5823
Random (mean of 200)	—	—	0.1094

TABLE VII : MNIST patch test efficiency metrics. Smaller

K@\tau

and larger AUC are better.

Feature Attribution

Feature attribution estimates which input features or internal components contribute to a model’s predictions.

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Beyond Shapley: Efficient Computation of Asymmetric Shapley Values

Ezequiel Companeetz, Santiago Cifuentes, Sergio Abriola

Departamento de Computación, Facultad de Ciencias Exactas y Naturales, UBA · Instituto de Ciencias de la Computación (ICC), CONICET

Cooperative Game Theory Feature Attribution Causal Attribution Shapley Value Attribution Causal Interpretability

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We address the problem of explainability in machine learning models through feature attribution methods. In particular, we consider a variant of Shapley values known as Asymmetric Shapley Values (ASV), which enables the incorporation of causal knowledge into model-agnostic explanations through the use of a causal graph. We show that in certain contexts in which the computation of SHAP is

\#P

-hard, the exact computation of ASV can be done in polynomial time. To extend this algorithmic result, we introduce a notion of equivalence classes over the topological orderings of the underlying causal graph, which is useful to reduce the time to compute ASV. In particular, we present a polynomial-time algorithm (in the number of equivalence classes) to compute it whenever the causal graph is a rooted directed tree. Finally, we develop an algorithm for approximating ASV in arbitrary causal DAGs which relies on a procedure to sample topological orderings uniformly at random. To implement this sampling mechanism we leverage known algorithms as well as simpler alternatives. Our experimental results demonstrate the practical viability of the proposed approach in realistic causal structures.

Abstract

Feature Attribution

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