Efficiency Property

Efficiency mandates that the explanation model's output perfectly matches the original model's prediction for a specific instance. This is formally expressed as: f(x) = φ₀ + Σᵢ φᵢ, where f(x) is the model's prediction, φ₀ is the baseline expected value, and φᵢ are the individual Shapley values for each feature. This property ensures that the total prediction is fully decomposed into additive feature contributions with no unexplained residual.

Also known as the Local Accuracy property, Efficiency ensures that the explanation is faithful to the model at the individual prediction level. When you sum all SHAP values for a single instance and add the baseline, you recover the exact model output. This is critical for high-stakes auditing where every unit of the prediction must be accounted for, such as in credit denial or medical diagnosis explanations.

The Efficiency property explicitly incorporates the baseline value φ₀, which represents the model's expected output across the background dataset. This baseline acts as the starting point from which features push the prediction higher or lower. Without Efficiency, the relationship between the baseline and the final prediction would be ambiguous, undermining the additive nature of the explanation.

Efficiency works in concert with the other Shapley axioms but serves a distinct role:

• Efficiency: Guarantees the sum of attributions equals the prediction difference
• Missingness: Ensures absent features receive zero attribution
• Consistency: Ensures that if a feature's contribution increases, its attribution does not decrease Together, these three properties uniquely define the Shapley value solution.

In practice, you can verify the Efficiency property by summing the SHAP values for any instance and comparing to the model output. For a prediction of 0.82 with a baseline of 0.50, the sum of all SHAP values must equal exactly 0.32. Libraries like the official shap package enforce this property by construction in both KernelSHAP and TreeSHAP implementations.

Efficiency is the defining characteristic of the Additive Feature Attribution class of explanation models. Any method that satisfies this property expresses a prediction as a linear sum of feature effects. This additive structure is what makes SHAP explanations intuitive for human interpretation, as each feature's impact can be understood independently and then combined to reconstruct the full prediction.

The axiom requiring that the sum of all feature attributions plus the baseline value exactly equals the model's original prediction for a specific instance. This guarantees the explanation is a faithful, additive decomposition of the output.

• Mathematical form: f(x) = φ₀ + Σᵢ φᵢ
• Ensures no prediction magnitude is lost or created during explanation
• Works in tandem with Efficiency to provide a complete accounting of the model's decision

The axiom stating that features not present in the original input must receive an attribution of exactly zero. This prevents absent features from being credited or blamed for a prediction they did not influence.

• Enforces that x'ᵢ = 0 implies φᵢ = 0
• Critical for handling sparse inputs and missing data
• Distinguishes SHAP from methods that may assign non-zero importance to absent features

The property ensuring that if a model changes so a feature's marginal contribution increases (or stays the same) across all subsets, its SHAP value must not decrease. This guarantees attributions track true feature importance.

• Formalized as: if f'ₓ(S ∪ i) - f'ₓ(S) ≥ fₓ(S ∪ i) - fₓ(S) for all S, then φᵢ(f', x) ≥ φᵢ(f, x)
• Prevents counterintuitive attribution reversals
• Ensures SHAP values are monotonic with respect to feature impact

The expected model output computed across the background dataset, representing the prediction when no feature information is known. The Efficiency Property uses this as the starting point from which all feature contributions are measured.

• Denoted as φ₀ = E[f(X)]
• Acts as the intercept term in the additive explanation model
• Choice of background dataset significantly impacts SHAP value interpretation
• Observational vs. Interventional: baseline computation differs based on whether feature correlations are preserved

The difference in prediction when a feature is added to a specific subset of other features. This is the atomic unit of Shapley value computation, measuring a feature's incremental impact.

• Formula: Δᵢ(S) = f(S ∪ {i}) - f(S)
• Computed across all 2ⁿ⁻¹ possible subsets for exact Shapley values
• Efficiency Property ensures the weighted average of all marginal contributions sums to the total prediction difference
• Forms the basis for both KernelSHAP and TreeSHAP estimation algorithms