Shapley Interaction Index Overview

• The Shapley Interaction Index is defined via a weighted sum over discrete derivatives, capturing the excess value of coalitions beyond individual contributions.
• Recent algorithmic advances, such as SHAP-IQ and KernelSHAP-IQ, enable efficient estimation despite the index’s inherent #P-hard computational complexity.
• SII plays a vital role in explainable AI, data valuation, and cooperative game theory by providing a structured measure for high-order interactions and fair attribution.

The Shapley Interaction Index (SII) is a cardinal interaction measure in cooperative game theory and explainable machine learning that generalizes the classical Shapley value to quantify interactions among groups of players or features, rather than assigning importance solely to individuals. The SII is defined through the discrete derivatives of a set function, capturing the excess value produced by a coalition beyond what is expected from the sum of its parts. It is governed by key axioms including linearity, symmetry, and dummy, and variants differ in their treatment of efficiency and recursive properties. Exact calculation of SII is computationally hard in general (#P-hard), but algorithmic advances have enabled its estimation in practical domains such as feature attribution for black-box models, graph neural network explainability, and data valuation.

1. Formal Definition and Cardinal Properties

The Shapley Interaction Index for a subset $S \subset N$ (typically $|S|=k$) is defined via a weighted sum over discrete derivatives:

$$\mathcal{I}_S^{\mathrm{SII}}(v) = \sum_{T \subset N \setminus S} \omega_{|S|, |T|} \; \Delta_S v(T),$$

where $$\Delta_S v(T) = \sum_{W \subset S} (-1)^{|S|-|W|} v(T \cup W)$$ is the discrete derivative for coalition $S$ at base $T$, and the weights $\omega_{|S|, |T|}$ correspond to the chosen cardinal index. For the classical SII (Grabisch–Roubens), these are combinatorial coefficients: $\omega_{k, t} = \frac{(n-t-k)! t!}{(n-k+1)!}$. Linearity, symmetry, and dummy are always satisfied, but efficiency may not hold for all SII variants.

The SII captures interaction by assigning zero to groups that contribute nothing jointly beyond additivity, and positive (or negative) values when synergy (or redundancy) is detected. Pairwise interaction ($|S|=2$) is especially important in applications, representing how two players or features jointly affect outcome over what is predicted by their individual effects.

2. Algorithmic Complexity and Computation

Exact computation of SII is infeasible for general set functions ($2^n$ complexity), reflecting its #P-hardness. The complexity landscape has been formalized for related power indices: e.g., calculating and comparing Shapley–Shubik indices is #P–many–one–complete and associated comparison problems are PP–complete (0801.4585). This suggests exact SII computation inherits comparable complexity. When set functions are explicitly defined on the coalitions (e.g., KNN data valuation, connectivity games), recursive or combinatorial reductions may lower computation for specific model classes (Belaid et al., 2023).

Recent advances in sampling-based and regression-based approximators have made practical estimation possible:

• SHAP-IQ presents a unified, unbiased sampling scheme for any cardinal interaction index, recycling model evaluations and providing variance bounds (Fumagalli et al., 2023).
• SVARM-IQ stratifies the coalition sampling, dramatically reducing variance (and query budget) by grouping discrete derivatives according to coalition cardinality and subset intersection (Kolpaczki et al., 24 Jan 2024).
• KernelSHAP-IQ characterizes SII as the optimal solution to a weighted least-squares problem, extending KernelSHAP from univariate to high-order interactions and achieving state-of-the-art performance in feature attribution tasks (Fumagalli et al., 17 May 2024).

These methods allow consistent and efficient estimation for SII and its variants in LLMs, vision tasks, tabular data, and synthetic cooperative games.

3. Axiomatic Foundations and Comparison to Related Indices

The axioms for the SII—with linearity, symmetry, dummy—do not single out a unique index for $|S| > 1$. Classic SII uses a recursive axiom; Shapley–Taylor introduces an efficiency axiom, distributing the total value among main effects and interaction effects, plus an interaction distribution axiom that prevents lower-order effects from absorbing pure interaction (Dhamdhere et al., 2019). Faith-Shap identifies a unique index by requiring the interaction index be the most faithful polynomial approximation subject to natural axioms—extending Shapley’s optimality to interactions (Tsai et al., 2022). Efficiency has proven difficult to enforce without loss of interpretability in some frameworks.

A comparison of formal properties:

Index	Linearity	Symmetry	Dummy	Efficiency	Recursive	Approximator
Classical SII	✓	✓	✓	×	✓	SHAP–IQ, SVARM–IQ
Shapley–Taylor	✓	✓	✓	✓	×	SHAP–Taylor
Faith–Shap	✓	✓	✓	✓	×	Weighted regression
KernelSHAP–IQ	✓	✓	✓	✓ (for SV, 2-SII)	×	Weighted regression
Myerson–Taylor	✓	✓	✓	✓ (comp. eff.)	×	Structure-aware

4. Methodological Innovations: Regression, Sampling, and Order-Based Formulations

A central development is the representation of SII (and related interaction indices) as solutions to weighted least-squares (WLS) regression over model evaluations (Fumagalli et al., 17 May 2024). For $k$-order interactions, one constructs a design matrix $X_k$ using combinatorial coefficients $\lambda(k, \ell)$ and targets residual responses $y_k$ from prior-order approximations, solving $\phi^* = (X_k^T W_k X_k)^{-1} X_k^T W_k y_k$. This unifies the allocation of contributions and interactions over all coalitions.

Order-based sampling, as formalized for fuzzy measures (Gutiérrez et al., 7 Feb 2024), interprets SII as averages over permutations, enabling estimation with polynomial complexity. Stratified sampling, as in SVARM-IQ, updates multiple interaction strata with a single query, increasing sample efficiency for high-dimension or large model classes.

5. Applications and Domain-Specific Adaptations

SII and its variants are deployed in diverse settings:

• Explainable AI: quantifying high-order feature interactions to illuminate predictions of LLMs (e.g., BERT), vision transformers, tabular classifiers, and regression models (Bordt et al., 2022, Fumagalli et al., 2023, Kolpaczki et al., 24 Jan 2024). SII reveals interaction motifs, e.g., “not” and “bad,” “never” and “forget,” whose joint contributions dominate univariate attributions.
• Data Valuation: STI-KNN leverages the linearity of the KNN model to compute exact pairwise SII for training samples, identifying redundant points, outliers, or adversarial examples (Belaid et al., 2023, Huang et al., 16 Aug 2025).
• Graph-based Models: The Myerson–Taylor index incorporates graph connectivity, attributing motifs and substructures (motifs in molecular graphs, motifs in social networks) by restricting coalitions to connected sets (Bui et al., 23 May 2024). Motif search optimization then maximizes the explanatory content of subgraph selections.
• Cooperative Game Theory and Fuzzy Decision Analysis: SII formalizes complementarity in payoff allocation, synergies in bargaining games, and the evaluation of group prospects (see Shapley group value and average complementarity) (Flores et al., 2014).
• Performance Attribution in Computer Systems: Shapley values (and by extension, SII) decompose throughput and quantify the contribution of internal events, providing efficiency-preserving attributions with theoretical error bounds (Alpay et al., 23 Sep 2025).

6. Theoretical Guarantees and Empirical Results

State-of-the-art sampling and regression methods deliver unbiased estimators, non-asymptotic mean squared error bounds (via Hoeffding and Chebyshev inequalities), and sample complexity guarantees. SVARM-IQ and KernelSHAP-IQ demonstrably outperform permutation sampling in MSE and Precision@K in language, vision, and synthetic domains (Kolpaczki et al., 24 Jan 2024, Fumagalli et al., 17 May 2024).

Empirical studies on SOUM synthetic games validate closed-form ground truths for SII: $$\phi^{\mathrm{SII}}_{\nu_R}(S) = \mathbb{1}(S \subset R) / (|R| - |S| + 1)$$. Real–world tasks (DistilBERT sentiment, ImageNet classifiers) show SII captures interactions missed by base Shapley attribution.

7. Extensions, Limitations, and Future Directions

While regression-based interaction estimation provides computational advances, exact SII calculation remains #P–hard. Accelerated stratified and order-based sampling methods, as well as efficient recursive schemes, are needed for very high-dimensional domains. Graph–restricted and structure-aware generalizations—such as Myerson–Taylor—enable tailored explanations for structured models, but at the cost of more complex combinatorial bookkeeping.

Further work is underway to rigorously extend optimality and efficiency properties to arbitrary interaction order ($k > 2$), unify the axiomatic frameworks, and validate conjectured closed-form solutions for high-order SII (Fumagalli et al., 17 May 2024). Research into leveraging CXp–Forests and logic-based explanation frameworks integrates robustness against adversarial examples with SII-based attribution (Huang et al., 16 Aug 2025).

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In conclusion, the Shapley Interaction Index serves as a formal, axiomatized, and computationally challenging yet information-rich measure of coalitional synergy. Recent work unifies sampling, regression, and order-based methods for SII estimation, extending the paradigm of fair attribution from individuals to feature interactions and group effects in cooperative games, machine learning, decision analysis, and networked dynamical systems.