Consistency

Consistency states that if a model changes such that a feature's marginal contribution increases or stays the same across all subsets, the feature's SHAP value must not decrease. Formally, if f_x'(S) - f_x'(S\{i}) ≥ f_x(S) - f_x(S\{i}) for all subsets S, then φ_i(f', x) ≥ φ_i(f, x). This ensures the explanation method respects the underlying model's behavior.

While Local Accuracy (Efficiency) ensures the sum of attributions equals the prediction difference, Consistency governs how attributions change between models. Local Accuracy is a per-instance constraint; Consistency is a cross-model constraint. A method can satisfy Local Accuracy but violate Consistency, leading to counterintuitive explanations when models are retrained or updated.

Consistency, combined with Efficiency, Symmetry, and Dummy (Missingness), uniquely defines the Shapley value solution concept. No other additive feature attribution method satisfies all four axioms simultaneously. This uniqueness is why SHAP is considered the theoretically principled choice for model explanation—any method violating Consistency will produce logically inconsistent feature rankings across model iterations.

When a model is retrained and a feature becomes genuinely more important, Consistency guarantees its SHAP value reflects this. Without Consistency, a feature's attributed importance could paradoxically drop even as its true impact grows. This is critical for model monitoring and drift detection in production ML systems where feature importance trends must be tracked reliably over time.

Consider a model that relies heavily on feature A. If the model is simplified to ignore feature A entirely, Consistency requires feature A's SHAP value to drop to zero. A method violating Consistency might still assign a non-zero value, or even increase it. This demonstrates why Consistency is essential for feature selection and model pruning workflows where features are systematically removed.

Consistency is closely related to the concept of monotonicity in attribution. If a feature's functional contribution increases monotonically across all coalitions, its attribution must increase monotonically. This prevents explanation methods from exhibiting erratic, non-monotonic behavior where adding more evidence for a feature's importance paradoxically reduces its attributed score.

The Missingness property requires that features not present in the original input must receive an attribution of zero.

• φᵢ = 0 when xᵢ is missing or unknown
• Prevents artificial importance from being assigned to absent features
• Critical for handling sparse data and missing values
• Works in tandem with the baseline value to define the null state

The Dummy property guarantees that a feature which never changes the model's prediction—regardless of which other features are present—receives a SHAP value of zero.

• If f(S ∪ {i}) = f(S) for all subsets S, then φᵢ = 0
• Identifies truly irrelevant features
• Prevents noise features from receiving spurious importance
• Complements the Marginal Contribution calculation

The Linearity axiom ensures that if a model is a linear combination of two sub-models, the SHAP values of the combined model equal the same linear combination of the individual SHAP values.

• φᵢ(αM₁ + βM₂) = αφᵢ(M₁) + βφᵢ(M₂)
• Enables SHAP to handle ensemble models consistently
• Critical for explaining Random Forests and Gradient Boosted Trees
• Underpins the computational efficiency of TreeSHAP