Multiperturbation Shapley-Value Analysis (MSA)

Multiperturbation Shapley-value Analysis (MSA) is a model-agnostic, game-theoretic framework for attributing the contributions of components—such as variables, units, or agents—to the function, output, or performance of complex systems, particularly in neural networks, sensitivity analysis, and portfolio allocation. Unlike classical input attribution methods, MSA systematically quantifies the effect of perturbing multiple components simultaneously and dissects both individual contributions and higher-order interactions. The method generalizes the Shapley value from cooperative game theory by averaging the marginal effects of component groups across all possible perturbation contexts, enabling granular, unbiased, and causally interpretable multi-factor attributions for high-dimensional outputs.

1. Mathematical Foundations of MSA

MSA extends the Shapley value to analyse multiperturbation settings by considering the value or performance impact of any subset of components, treating each subset as a "coalition" in game-theoretic terms. For a system with $n$ components $N = \{m_1, m_2, ..., m_n\}$ and a set function $V(S)$ (output when only components in $S \subseteq N$ are present), the marginal contribution of a component $m_i$ to coalition $S$ is: $\Delta_i(S) = V(S \cup \{m_i\}) - V(S)$ The Shapley value $\gamma(m_i)$ (or "Shapley Mode" for multidimensional outputs) is then defined by averaging $\Delta_i(S)$ over all possible orderings: $$\gamma(m_i) = \frac{1}{n!} \sum_{R \in \mathcal{R}} \Delta_i(S_i(R))$$ where $\mathcal{R}$ enumerates all permutations of $N$ and $S_i(R)$ is the set of units preceding $m_i$ in ordering $R$. For vector-valued outputs (such as images or logits), all operations are element-wise, producing a contribution map for each component with the same dimensionality as $V$.

This formulation satisfies Shapley’s axioms of efficiency, symmetry, dummy, and additivity, guaranteeing fair and interpretable attributions even in high-order interaction contexts.

2. Methodological Implementation

MSA operates through systematic "lesioning" or perturbations of internal components:

1. Set Selection: Select components (variables, neural units, or subsets) as "players."
2. Perturbation/Lesioning: For each sampled subset (coalition), deactivate ("lesion") or vary the chosen components.
3. Model Evaluation: Compute the system's response or output for each configuration.
4. Marginal Contribution Calculation: Measure the performance change when each component is added to each subset.
5. Combinatorial Averaging: Aggregate marginal effects across all (or sampled) contexts using the Shapley formula.

Exact computation requires evaluating all $2^n$ or $n!$ configurations, making it exponential in $n$. Monte Carlo or stratified sampling schemes (e.g., Layered Shapley, multilinear extension, order-of-addition designs) are widely used for large $n$ to achieve tractable, unbiased estimations (Plischke et al., 2020 , Okhrati et al., 2020 , Watson et al., 2022 , Yang et al., 2023 ).

For models with special structure (e.g., the variance game), the computation can be reduced to quadratic complexity via closed-form formulas for the Shapley value in terms of pairwise covariances (Colini-Baldeschi et al., 2016 ).

3. Computational Strategies and Practical Considerations

A central challenge in MSA is computational tractability. To address this, several strategies have been established:

• Structural Simplification: Partition the system into independent or weakly interacting blocks, thus reducing coalition space and enabling parallel computation (Lupia et al., 2017 ).
• Sampling Acceleration: Employ stratified sampling, order-of-addition experimental designs, or efficient surrogate models (e.g., $k$-additive games) to fill the permutation space more effectively and reduce variance (Okhrati et al., 2020 , Yang et al., 2023 , Pelegrina et al., 7 Feb 2025 ).
• Möbius Inverse and Group Effects: Algorithms based on the Möbius inverse enable efficient calculation of single and groupwise (Shapley-Owen) effects in $2^k$ time, relevant for detecting higher-order interactions in complex models (Plischke et al., 2020 ).

These approaches facilitate MSA’s application to large-scale problems such as neural networks with thousands of units, social or biological networks, and massive allocation settings.

4. Applications Across Domains

MSA has been applied in diverse contexts, with several major use cases:

• Deep Neural Networks: Systematically lesioning combinations of units in MLPs, GANs, or mixture-of-expert LLMs attributes each unit's causal contribution to outputs such as digits, pixels, or language tokens (Dixit et al., 24 Jun 2025 ). Shapley Modes produced by MSA allow full output dimensionality attribution, enabling interpretation of distributed versus localized computation, identification of hub units, and specialized domain experts within networks.
• Sensitivity and Variance Analysis: In the context of Markowitz portfolio optimization, the variance contributed by each asset can be allocated exactly via its covariance with the total outcome, providing both computational efficiency and fairness in correlated scenarios (Colini-Baldeschi et al., 2016 ). For nonlinear models or heterogeneous systems, MSA recovers both main and interaction effects, supporting sensitivity audits in engineering, biology, or finance (Plischke et al., 2020 ).
• Network and Feature Analysis: MSA has been used to identify central nodes in biological [C. elegans connectome] or social networks (9/11 terrorist network), revealing component criticality with improved precision and efficiency over simple random sampling (Yang et al., 2023 ).

Applications further include robust model explanation under data manifold restrictions (Taufiq et al., 2023 ), differentially private data valuation (Watson et al., 2022 ), and risk attribution in portfolios with non-orthogonal risk factors (Hagan et al., 2021 ).

5. Interaction Effects, Externalities, and Theoretical Extensions

MSA natively captures not only individual contributions but also synergy, redundancy, and high-order interactions:

• Group Shapley Effects: Generalizations such as Shapley-Owen indices measure joint effects of variable groups, crucial for disentangling collaborative or antagonistic phenomena within a system (Plischke et al., 2020 ).
• Externalities and Partition Functions: For games with external effects (partition function form), the MPW solution uniquely extends the Shapley framework using the expected worth over random coalitional partitions, preserving the null player property and ensuring fairness for complex multiperturbation scenarios with externalities (Casajus et al., 1 Feb 2024 ).
• Coalition Path View and Hodge Theory: Formulations based on stochastic path integrals over coalition graphs and Hodge-theoretic decompositions allow attribution not just to the grand coalition but at all partial perturbation states, providing a rigorous and unified methodology for cooperative networks and high-dimensional systems (Lim, 2022 ).

6. Algorithmic and Practical Trade-offs

MSA introduces several trade-offs that guide methodology selection:

Choice of algorithm depends on the number of components, system structure, interaction order, and desired trade-off between computational efficiency and exactness.

7. Significance and Impact

MSA enables interpretation, editing, and optimization of complex systems at unprecedented resolution, offering several unique capabilities:

• Causal, unbiased attribution for both internal and external components, applicable across model types and output dimensions.
• Robustness to structural and data issues through manifold restrictions and diagnosis of redundancy or criticality.
• Modular, model-agnostic framework supporting pruning, auditing, and targeted interventions.
• Axiomatic theoretical guarantees for fairness, efficiency, and interpretability, preserved even with nonlinear interactions and externalities.
• Scalable methodologies, with recent advances bringing sub-exponential and even linear scaling to large, structured, or sparsely interacting systems.

By aligning cooperative game theory with multiperturbation analysis, MSA has become foundational in explainable AI, sensitivity analysis, network science, and domains where fairness and transparency in attributions are essential.