Cardinal Interaction Indices (CIIs)

• Cardinal Interaction Indices (CIIs) are quantitative measures that assess the importance, synergy, and redundancy of criteria in decision models and AI explainability.
• They extend classical indices like Shapley and Banzhaf by incorporating axioms such as linearity, symmetry, and the dummy axiom for precise and interpretable evaluations.
• CIIs are computed through discrete derivative, Fourier, and variance-based methods, enabling both local sensitivity analysis and global variance decomposition in complex models.

Cardinal Interaction Indices (CIIs) are a class of quantitative measures designed to rigorously characterize the importance and interplay of criteria, features, or agents—across domains such as multicriteria decision analysis (MCDA), cooperative game theory, statistical sensitivity analysis, and explainable artificial intelligence (XAI). CIIs generalize foundational indices including the Shapley value and the Banzhaf interaction index, extending their axiomatic basis to any interaction order, supporting multilevel criterion models, and providing both local (derivative-based) and global (variance-based) perspectives on interaction. CIIs unify measures of marginal contribution, synergy, and redundancy, often leveraging principles such as linearity, symmetry, and the dummy axiom, and admitting representations via Fourier transforms, discrete derivatives, or weighted aggregation of model evaluations.

1. Axiomatic Foundations

Cardinal Interaction Indices are defined by a baseline set of axioms that ensure their mathematical consistency and interpretability:

• Linearity: For any two models (or games) $v$ and $w$, and any scalar $\alpha$, CIIs satisfy $I^{\alpha v + w}(S) = \alpha I^v(S) + I^w(S)$ for each subset $S$ of criteria.
• Symmetry: Permuting the criteria or features leaves the index unchanged; $I^{\sigma \circ v}(\sigma(S)) = I^v(S)$ for any permutation $\sigma$.
• Dummy (Null) Property: If a criterion $i$ does not affect the model's output regardless of other criteria, then $I^v(T) = 0$ for any coalition $T$ containing $i$.
• Efficiency and Recursivity are often invoked, ensuring that attributions sum to the total marginal contribution and that interactions can be decomposed recursively (as generalized by recursions for multichoice games (Ridaoui et al., 2018)).
• Invariance: The value depends only on the differential effect induced by criteria.

These axioms uniquely determine the form of many CIIs, including the Shapley, Banzhaf, and their higher-order interaction extensions.

2. Representations and Computational Formulations

CIIs admit several equivalent representations, which facilitate their computation and interpretation:

Discrete Derivative Representation

For multichoice games (criteria may have several participation levels), CIIs are constructed via recursive discrete derivatives: $$\Delta_i v(x) = v(x + 1_i) - v(x), \quad \Delta_T v(x) = \sum_{A \subseteq T} (-1)^{|T| - |A|} v(x + 1_A)$$ The interaction index for coalition $T$ is a weighted sum over these derivatives (Ridaoui et al., 2018): $I^v(T) = \sum_{x: x_T < k_T} b_x^T \Delta_T v(x)$ where $b_x^T$ are symmetry-invariant weights.

Fourier and Variance-Based Formulation

For models extended via multilinear or Choquet integrals, CIIs connect to the Fourier transform of a capacity $\mu$: $$\hat{\mu}(S) = \frac{1}{2^n} \sum_{T \subseteq N} (-1)^{|S \cap T|} \mu(T)$$ Sobol' indices, quantifying variance contribution of each criterion group, satisfy (Grabisch et al., 2016): $$\text{Var}[(f^{Ow}_\mu)_S] = \frac{1}{3^{|S|}} \left(\hat{\mu}(S)\right)^2$$ and the Banzhaf interaction index relates by $$\hat{\mu}(S) = \left(-\frac{1}{2}\right)^{|S|} I_B^\mu(S)$$, establishing that Sobol' indices (variance-based) are proportional to the square of the Banzhaf coefficients.

Unified Model Evaluation Representation

Any CII satisfying linearity, symmetry, and dummy axiom can be written as a weighted sum over model evaluations (Fumagalli et al., 2023): $$I^m(S) = \sum_{T \subseteq D} \nu_0(T) \cdot \gamma_s^m(t, |T \cap S|)$$ with weights $\gamma_s^m$ capturing the dependence on subset sizes and intersection with $S$, and $\nu_0(T)$ representing centered model evaluations.

3. Extensions: Multichoice Games and Continuous Models

While classical CIIs, including the Shapley and Banzhaf indices, focus on binary inclusion, the framework extends to multichoice games where each agent or criterion may assume multiple levels (from $0$ up to $k$):

• The discrete multichoice formulation characterizes interaction indices with extended axioms, preserving linearity, symmetry, dummy, and recursive properties.
• In the continuous setting, such as criteria aggregated via the Choquet integral over $[0,k]^N$, the interaction index lifts as an average of partial derivatives (Ridaoui et al., 2018): $$I^v_s(T) = \int_{[0,k]^N} \frac{\partial^{|T|} \mathcal{C}_v(z)}{\partial z_T} dz$$ This represents the mean local effect of varying the criteria in $T$ throughout the domain.

The equivalence between discrete and continuous expressions ensures that interaction indices quantify synergy and redundancy among criteria even when feature values, participation, or contributions are not binary.

4. Statistical and Combinatorial Connections

CIIs interface with classical statistical approaches to sensitivity and uncertainty quantification:

• Sobol' Indices: Decompose output variance additively by the contribution of groups of features or criteria, allowing direct comparison of single-agent effects versus interactions.
• Fourier Transform and Spectral Analysis: CIIs such as Banzhaf are viewed as spectral coefficients; their squares yield the variance contributions, elucidating the importance of each interaction order.
• Variance-Based versus Derivative-Based Indices: Sobol' indices provide global (variance) measures, while the interaction and Banzhaf indices quantify average directional sensitivities.

This duality allows one to select between local sensitivity (marginal impacts) and global importance (variance contributions), according to the needs of a specific MCDA or XAI application.

5. Practical Computation and Efficient Approximations

Calculating CIIs is intractable for large feature sets due to the exponential subset structure. Recent advances propose efficient stochastic and algebraic approximations:

Monte Carlo Integration (SHAP-IQ):

SHAP-IQ (Fumagalli et al., 2023) exploits a novel representation, recycling each model evaluation $\nu_0(T)$ to estimate all interaction scores $I^m(S)$ simultaneously using weights $\gamma_s^m$. The estimator is unbiased and consistent: $$\hat{I}^m_{k_0}(S) = c_{k_0}(S) + \frac{1}{K} \sum_{k=1}^K \nu_0(T_k) \frac{\gamma_s^m(t_k, |T_k \cap S|)}{p_{k_0}(T_k)}$$ Experiments demonstrate improved empirical efficiency compared with permutation-based (PB) and kernel-based (KB) alternatives, especially for estimation of all-order interactions in complex models.

Special Case—Shapley Value:

For $s=1$, the estimator aligns with Unbiased KernelSHAP (U-KSH), allowing computation of single-feature attributions as simple weighted sums rather than full least squares.

6. Applications and Interpretational Perspectives

CIIs serve in domains requiring quantification of the marginal and joint contributions of agents, criteria, or features:

• MCDA: CIIs support decision models accounting for non-additive, synergistic, or redundant criteria—integrating both importance (singletons) and interaction (groups).
• Cooperative Games: Synergy among coalitions is described via CIIs generalizing Shapley and Banzhaf concepts to multilevel participation.
• XAI: Feature interaction attributions for model interpretability leverage CIIs, particularly through SHAP-IQ, supporting transparent AI system explanations.
• Uncertainty Reduction: Variance decomposition by Sobol' indices enables targeted intervention on those criteria or interactions dominating outcome uncertainty.

Empirical studies indicate high correlation (Pearson $r \approx 0.994$) between certain CIIs and percentile-rank based metrics, emphasizing their precise capture of intensity (e.g., exact citation counts) versus merely quantity (publication counts) (Zhou et al., 2012).

7. Distinguishing Features, Limitations, and Theoretical Significance

• Nonlinear versus Linear Perspective: CIIs, formulated via exact function evaluations or derivatives rather than ranks, avoid issues of over/undervaluation and inconsistency—especially in small datasets.
• Document-Type Normalization: Practical indices such as the Combined Impact Indicator (CII) normalize for document type differences, support inclusion of both cited and uncited items (with scalable weight), and outperform percentile-rank approaches regarding interpretational clarity and stability (Zhou et al., 2012).
• Bridging Sensitivity and Interaction: The linkage between discrete sensitivity measures (derivatives or margins), spectral decomposition (Fourier/Banzhaf), and global variance-based measures (Sobol') cements the conceptual significance of CIIs as a unifying framework.

A plausible implication is that advances in efficient approximation frameworks—such as SHAP-IQ—foster scalable model interpretability, enabling broad application of CIIs in high-dimensional, real-world settings.

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