Shapley Interaction Indices (SIIs)

• Shapley Interaction Indices (SIIs) quantify how joint feature interactions contribute to a model’s output by assigning attributions to both individual features and their combinations.
• They build on axioms such as linearity, symmetry, and the dummy axiom to extend the classical Shapley value into higher-order interaction effects, resembling a Taylor expansion.
• Advanced computational methods like Monte Carlo sampling and order-based representations make SIIs viable for sensitivity analysis, feature attribution, and risk assessment in complex systems.

Shapley Interaction Indices (SIIs) are a class of functionals originating in cooperative game theory that quantify the effect of interactions among subsets of players (or, equivalently, features or variables in a model) on a model’s output or a game’s value. Unlike the classical Shapley value, which allocates payoff to individuals, SIIs decompose the total value into attributions not only to single elements but also to interacting groups, allowing the nuanced identification and measurement of synergistic, redundant, or antagonistic effects within coalitions. SIIs are foundational to modern sensitivity analysis, feature attribution in machine learning, multi-criteria aggregation, and collective decision-making frameworks. Their computation, theoretical properties, axiomatic characterizations, and practical estimation strategies have been the subject of intensive research, resulting in a variety of indices, approximators, and application algorithms.

1. Mathematical Foundations and Core Definitions

SIIs generalize the Shapley value by assigning indices $I_T(f)$ to every non-empty subset $T \subseteq N$ (the set of players or features), capturing the additional value attributable specifically to the interaction among elements of $T$. For a set function $f: 2^N \rightarrow \mathbb{R}$, the most studied SII, introduced by Grabisch and Roubens, is defined by:

$$I_T(f) = \sum_{S \subseteq N \setminus T} \frac{(n - |S| - |T|)! \, |S|!}{(n - |T| + 1)!} \left[ \Delta_T f(S) \right]$$

where $$\Delta_T f(S) = \sum_{K \subseteq T} (-1)^{|T|-|K|} f(S \cup K)$$ is the discrete derivative or finite difference w.r.t. the subset $T$.

SIIs obey three foundational axioms for any cardinal interaction index: linearity (the index is linear in $f$), symmetry (labels of elements are exchangeable), and the dummy axiom (indices for coalitions containing dummy elements vanish). Under these axioms, the SII is uniquely determined (Fumagalli et al., 2023), with alternative forms corresponding to different weightings or combinatorial averaging schemes.

A significant insight is that the Shapley value for a singleton recovers the main effect; higher-order indices correspond to interaction among subsets, allowing a Taylor-type expansion of $f$ over its subsets (Dhamdhere et al., 2019).

2. Axiomatic Extensions and Generalizations

While the standard Shapley interaction index satisfies linearity, symmetry, and dummy axioms, it does not enforce efficiency, i.e., the sum of all attributions does not necessarily equal the total function difference $f(N) - f(\varnothing)$ (Dhamdhere et al., 2019). The Shapley–Taylor index remedies this by adding an interaction distribution axiom, resulting in an index akin to a truncated Taylor expansion of the multilinear extension of $f$:

• For $|S| < k$, the index for subset $S$ is the discrete derivative at $\varnothing$.
• For $|S| = k$, the index takes the average over all orderings, recapitulating the Lagrange remainder term of a Taylor series (Dhamdhere et al., 2019).

Alternatively, the Faithful Shapley Interaction Index (Faith–Shap) is defined as the unique solution to a weighted least squares problem over polynomial approximations, enforcing efficiency while eschewing less natural recursion or interaction distribution axioms, yielding a principled allocation of effects across interaction orders (Tsai et al., 2022).

Joint Shapley values (Harris et al., 2021) and n-Shapley values (Bordt et al., 2022) provide further generalizations; joint Shapley values assign scores to coalitions as units, and n-Shapley values interpolate classical Shapley explanations and full additive decompositions, with explicit recovery of generalized additive model terms.

3. Computation and Algorithmic Advances

Exact computation of SIIs involves evaluating $f$ on all $2^n$ subsets and is exponential in $n$ (Fumagalli et al., 2023). Key algorithms and approximators have been developed:

• Möbius Inverse Formula: Reformulates the SII using the Möbius coefficients of $f$, allowing expressions such as:

$I_T(f) = \sum_{S \supseteq T} a(S) / |S|$

with $a(S)$ being Möbius coefficients (Plischke et al., 2020). This yields significant computational savings (from $k! \cdot k$ to $2^k$ for $k$ inputs).

• Sampling Algorithms: Monte Carlo methods leverage sampling over random permutations or subsets to estimate SIIs, often using explicit variance bounds and unbiasedness assertions (Fumagalli et al., 2023, Kolpaczki et al., 24 Jan 2024, Gutiérrez et al., 7 Feb 2024).
  • SVARM-IQ uses a stratified representation of the coalition space, partitioned by intersection size and coalition cardinality, enabling maximal sample reuse and rapid convergence (Kolpaczki et al., 24 Jan 2024).
  • SHAP-IQ provides a unified, unbiased, and variance-controlled approximation applicable to any cardinal interaction index; it closely matches or outperforms permutation-based and kernel least-squares methods.
• Weighted Approximations: In sensitivity analysis contexts, SIIs can be computed via weighted least squares fits, as for the Banzhaf or Shapley interaction indices, which are interpretable as averages (centers of mass) over suitably defined probability spaces for coalition formation (Marichal et al., 2010).

In specific models, such as KNN classifiers, structure-exploiting algorithms (e.g., STI-KNN) reduce complexity from $O(2^n)$ to $O(t n^2)$ by exploiting linearity and ordering symmetries within the valuation function (Belaid et al., 2023).

4. SIIs in Sensitivity Analysis and Statistical Modeling

SIIs have been integrated into global sensitivity analysis as “Shapley effects” (Iooss et al., 2017, Benoumechiara et al., 2018). In this vein:

• Variance Decomposition: SIIs allocate output variance to groups of input variables in models where standard ANOVA (Sobol indices) may be ill-defined, e.g., with correlated inputs.
  • For linear Gaussian models, closed-form computation is possible via analytical conditional variances (Broto et al., 2018).
  • In nonlinear or empirical scenarios, Gaussian linear approximations provide provably convergent estimators under mild assumptions (Broto et al., 2020).
• Reliability-Oriented Sensitivity Analysis: Target Shapley effects assign sensitivity scores not to continuous outputs but to binary or thresholded events (e.g., failure probabilities in risk models), preserving efficiency and interpretability in multivariate risk allocation frameworks (Idrissi et al., 2021).
• Practical Algorithms: Efficient estimation leverages metamodels (e.g., kriging surrogates) and resampling for computationally intensive simulations (Iooss et al., 2017, Benoumechiara et al., 2018).

In GSA, SIIs facilitate factor prioritization/fixing and provide robustness to statistical dependence, outperforming classical Sobol indices in interpretability for correlated settings.

5. Interpretability, Feature Attribution, and Applications in AI

SIIs are central to feature attribution in explainable AI:

• Attribution Decomposition: SIIs describe not only individual feature effects but also quantify joint interactions (e.g., synergy, redundancy, antagonism) across any order (Dhamdhere et al., 2019, Bordt et al., 2022). For instance, the Shapley–Taylor index recovers key interaction patterns in sentiment models, regression, and context-dependent question answering tasks.
• Model Alignment: n-Shapley Values and similar indices reveal the extent to which a machine learning model is functionally decomposable into generalized additive components or requires explicit interaction modeling. This supports precise diagnostic and visualization tools (Bordt et al., 2022).
• Data Valuation: In data valuation, pairwise SIIs for training points can be computed efficiently via problem-specific algorithms, helping with summarization, acquisition, and outlier detection (Belaid et al., 2023).
• Joint Feature Importance: Joint Shapley values directly address the contribution of coalitions (as opposed to the distributed interaction terms in other indices), providing novel insights into group-level impact, particularly with binary features or structured attribution tasks (Harris et al., 2021).
• Fuzzy Measures and Aggregation Operators: SIIs and their sampling-based estimators underpin advanced aggregation techniques (e.g., the Choquet integral, fuzzy measures aggregation), capturing non-additive dependencies among criteria (Gutiérrez et al., 7 Feb 2024).

6. Developments in Approximation and Computational Efficiency

Recent research has yielded multiple efficient, theoretically supported approximation strategies:

• k-Additive Surrogate Games: SVA$_{k_{\text{ADD}}}$ fits a surrogate game where only interactions up to order $k$ are allowed, reducing parameterization to polynomial in $n$ and enabling rapid least squares-based Shapley value estimation (Pelegrina et al., 7 Feb 2025).
• Stratification and Optimized Sampling: SVARM-IQ and related methods exploit variance reduction via stratified coalition partitioning, often achieving state-of-the-art estimation error with dramatically fewer function evaluations (Kolpaczki et al., 24 Jan 2024).
• Order-Based Representations: Polynomial-time estimators using order (permutation)-averaged marginal contributions generalize the classical Shapley value and interaction computation, permitting unbiased, variance-controlled estimation even in high-dimensional settings (Gutiérrez et al., 7 Feb 2024).

These approaches have made feasible the application of SIIs in real-world industrial, ML, and statistical modeling scenarios where model evaluations are costly or the underlying combinatorics prohibit exhaustive calculation.

7. Conceptual Integration and Current Research Directions

Ongoing work investigates the axiomatic tradeoffs among linearity, symmetry, dummy, efficiency, and recursive or distributional interaction axioms, leading to diverse indices (Shapley–Taylor, Faith–Shap, classic SII). There is active exploration of:

• Efficient estimation in high dimensions,
• Variable order interaction indices,
• Extensions for reliability-oriented, probabilistic, or decision-theoretic sensitivity analysis,
• Integration with GAM-based decomposition and functional analysis (Bordt et al., 2022),
• Comparative performance across model classes and data-generating processes.

A plausible implication is that further conceptual unification, computational innovation, and empirical validation of SIIs will continue to advance their role in interpretable modeling, robust sensitivity analysis, and equitable resource attribution within complex systems.