REVIEW 2 major objections 5 minor 1 cited by
The many Shapley values for model explanation
T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Shapley's uniqueness theorem does not select a single explanation method; Baseline Shapley gets its own uniqueness theorem.
desk verdict Sharp conceptual critique of Shapley-based attribution, but the advertised uniqueness theorem for BShap is misstated and needs an added axiom. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the set function that turns a model into a cooperative game. BShap defines $v(S) = f(x_S; x'_{N\setminus S})$, meaning features outside $S$ are pinned to baseline values, and then applies the usual Shapley average of marginal contributions. The argument's engine is Theorem 4.1, which shows that any attribution method obeying Linearity and Affine Scale Invariance reduces to a difference of two cost-sharing problems: the model is decomposed as $f_1 - f_2$ with both parts non-decreasing, using the infimum of its partial derivatives, provided that infimum exists (bounded derivatives). That reduction imports the uniqueness theorem for Shapley–Shubik sharing, giving Corollary 4.3, and the analogous theorem for Aumann–Shapley, giving Corollary 4.4. A second mechanism is Lemma 4.8, identifying CES over an independent distribution with Random Baseline Shapley, which localizes CES's failures to the dependence and sparsity of the chosen distribution.
What would settle it
The sharpest test is to search for an attribution rule that satisfies Linearity, Dummy, ASI, Demand Monotonicity, and Symmetry but differs from BShap on a simple problem like $\min(x_1, x_2)$; its existence would refute Corollary 4.3 as stated. A faster negative check is to compute the infimum of partial derivatives for a model like $f(x) = e^x$; because it is $-\infty$, the paper's decomposition into a difference of non-decreasing functions does not exist, so the theorem's domain is narrower than the general attribution problem.
Extended reading notes
Core claim
The paper's central claim is that Shapley's uniqueness theorem, often cited to justify attribution methods, does not select a single explanation method because the Shapley value can be applied to a model in several non-equivalent ways. The conditional-expectation operationalization (CES) is shown to be fragile: when continuous features have unique values in the empirical distribution, it hands out equal attributions to every feature, including ones the model does not use, and it can violate linearity even for simple two-variable functions. Against this, the paper proves that Baseline Shapley (BShap), defined by replacing absent features with baseline values, is the unique attribution method satisfying Linearity, Dummy, Affine Scale Invariance, Demand Monotonicity, and Symmetry, by reducing any attribution problem with bounded derivatives to a difference of two cost-sharing problems and importing the known uniqueness of Shapley–Shubik sharing. Integrated Gradients gets a parallel uniqueness theorem with Proportionality in place of Demand Monotonicity. The paper also shows that BShap is itself a limiting case of CES under a specially chosen distribution, so the practical difference between BShap and empirical CES is largely a difference in how the feature distribution is handled.
Load-bearing premise
The proof's reduction assumes bounded partial derivatives and silently uses an Efficiency condition that is not listed among the axioms; without both, the claimed uniqueness for BShap is not established.
Editorial extensions
If this is right
- BShap assigns zero attribution to dummy features and respects linearity, so it avoids the counterintuitive CES behavior the paper documents on the diabetes case study.
- Any use of CES must specify the feature distribution; with the empirical distribution on continuous data, it can collapse to equal attributions across all features.
- IG and BShap are both axiomatically grounded but differ; IG's proportionality forces smooth interpolation along the path from baseline to explicand, while BShap mixes baseline and explicand feature values directly.
- When feature distributions are independent, CES and RBShap coincide, so independence assumptions, not conditional expectation per se, drive axiom satisfaction.
- The baseline in BShap is a modeling choice that can encode the explanation context, such as fixing immutable features to the explicand's values for loan decisions.
Reading between the lines
- The reduction theorem suggests a general recipe: for any attribution method that is linear and scale-invariant, the search for a uniqueness theorem can be reframed as choosing a continuous cost-sharing rule, so the space of defensible explanation methods is larger than the two corollaries enumerate.
- The bounded-derivative condition means the uniqueness result is formally silent for models such as deep networks with unbounded gradients; BShap is still defined for them, but its axiomatic backing is not proven.
- Lemma 4.6 could be inverted as a diagnostic: by constructing the two-point distribution that makes CES approximate BShap, a practitioner can quantify how much of the disagreement between explanation methods is purely distributional choice.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper argues that the Shapley value's classical uniqueness theorem does not resolve the ambiguity of model explanation, because the value can be operationalized through different set functions (CES, BShap, RBShap, IG) that reference the model, the feature distribution, and the explanation context in different ways, leading to very different and sometimes counterintuitive attributions. It studies CES in detail, showing failures of Dummy and Linearity, and proposes BShap as a baseline-based method. The paper reduces model explanation to cost-sharing and claims uniqueness results for BShap and IG via known cost-sharing theorems, with an illustrative diabetes-prediction experiment.
Significance. The paper makes a valuable conceptual contribution by clarifying that existing uniqueness results for the Shapley value do not select a unique attribution method, because the choice of set function is left open. The concrete counterexamples to CES (e.g., dummy features receiving nonzero attributions in §3.3) are correct and practically important. The reduction from attribution to cost-sharing in §4.2 is an elegant and potentially useful bridge. However, the advertised uniqueness results are currently stated without an Efficiency axiom while their proofs rely on uniqueness theorems that include it, and the affine transformation in the reduction theorem has a sign error. Both issues are load-bearing for the paper's central claim but are fixable with reasonable changes.
major comments (2)
- [§4.2, Corollaries 4.3 and 4.4, and Appendix D.4] Both uniqueness results omit the Efficiency axiom from their stated axiom lists, yet their proofs invoke uniqueness theorems (Theorem 4.2 and Theorem 3 of [7]) whose hypotheses include Efficiency. The five listed axioms do not imply Efficiency: for any λ∈(0,1), the method M_λ = λ·BShap satisfies Linearity, Dummy, ASI, Demand Monotonicity, and Symmetry, but its attributions sum to λ(f(x)-f(x')), not f(x)-f(x'). Thus Corollary 4.3 is false as stated, and the same construction (λ·IG) refutes Corollary 4.4. The proof in Appendix D.4 silently uses Efficiency when it applies Theorem 4.2. The fix is to add Efficiency to the axiom list (matching Theorem 4.2) and to argue that the induced cost-sharing method inherits it; the present statement gives a uniqueness claim that is not true.
- [§4.2, Theorem 4.1 proof] The affine transformation step in the proof of Theorem 4.1 chooses c = 1 when the original explicand value x_i is non-negative and c = -1 otherwise, with d = -x'_i·c. The transformed explicand value is then c·x_i + d = c·(x_i - x'_i). This is non-negative if and only if c has the sign of x_i - x'_i, not the sign of x_i. For example, with x_i = 1 and x'_i = 2, the paper's rule gives c = 1 and transformed value -1 < 0, so the reduced problem is not a cost-sharing problem as claimed. The proof should set c = sign(x_i - x'_i) and d = -c·x'_i. As written, the reduction, and hence the transfer of the cost-sharing uniqueness result to all attribution problems, is not established.
minor comments (5)
- [§2.6, ASI definition] The equation defining Affine Scale Invariance is easy to misread: the relationship between f1, f2, c, d, and the explicand/baseline values should be stated in a way that makes clear which function is expressed in which coordinate system. Please rewrite it to avoid ambiguity.
- [§3.2, Remark 3.2] The sentence 'the first feature gets attribution f(x)-E[f] while all the other features get zero. Therefore all the variables get equal attributions' is confusing because equal attributions only arise after averaging over permutations; please clarify that the equality is the Shapley-value average.
- [§4.2, Theorem 4.1] The theorem relies on a 'minor technical condition that the derivatives are bounded.' This condition should be stated precisely (e.g., uniformly bounded partial derivatives on the relevant domain), and its restrictiveness should be discussed, since common models such as polynomials on unbounded domains do not satisfy it.
- [Appendix A] There is a typo: 'We introdoce' should be 'We introduce.'
- [§4.2, Corollary 4.3] For self-containedness, please state the technical conditions from [7] that are being suppressed, or at least indicate which conditions are needed for the transfer via Theorem 4.1.
Circularity Check
No circularity: BShap uniqueness is imported from an external cost-sharing theorem after an independent reduction; self-citations are descriptive, not load-bearing.
full rationale
The paper's central derivation chain is not circular. The claimed uniqueness of BShap is obtained by first proving Theorem 4.1, which reduces any attribution method satisfying Linearity and ASI to a difference of two cost-sharing problems, and then invoking Friedman and Moulin's external uniqueness theorem for Shapley-Shubik. This is a genuine reduction, not an equation of the conclusion with an input: the proof transforms the baseline to zero, decomposes the function into monotone parts using the bounded-derivative infimum, and applies Linearity. The stated Corollary 4.3 omits Efficiency from its axiom list even though the proof uses Theorem 4.2, whose uniqueness hypothesis includes Efficiency; this is a correctness flaw in the theorem as stated, and scaled-BShap counterexamples show the stated claim is false. But this does not make the argument circular, since supplying the missing axiom restores the intended external result. The paper's self-citations to Integrated Gradients and prior baseline-attribution work are descriptive references to existing methods, not load-bearing justifications for the uniqueness claim. Lemma 4.6 and Lemma 4.8 are constructive equivalences, not tautologies. No fitted parameter is relabeled as a prediction, and no uniqueness theorem is imported from the authors' own prior work. The bounded-derivative caveat is an explicit technical condition, not a hidden assumption of the conclusion. Accordingly, the paper receives score 0 for circularity, with the noted correctness concern about Corollary 4.3 left to the correctness assessment.
Assumptions & free parameters
free parameters (1)
- smoothing parameter for CES =
0.1 and 0.2
assumptions (4)
- standard math Shapley value definition and its uniqueness as the value satisfying Efficiency, Linearity, Dummy, and Symmetry for cooperative games (Shapley 1953).
- domain assumption Friedman-Moulin uniqueness theorems for Shapley-Shubik and Aumann-Shapley in cost-sharing.
- domain assumption Bounded partial derivatives of the model function (stated as 'minor technical condition' in Theorem 4.1).
- domain assumption Attribution methods under study satisfy the desirability axioms (Dummy, Efficiency, Linearity, Symmetry, ASI, Demand Monotonicity, Proportionality) as formal properties.
Cite this review
Pith. "Pith review of The many Shapley values for model explanation." pith.science (2026). https://pith.science/paper/C7CCEK5F
@misc{pith2026190808474,
author = {Pith},
title = {Pith review of: The many Shapley values for model explanation},
year = {2026},
howpublished = {\url{https://pith.science/paper/C7CCEK5F}},
note = {Machine review of arXiv:1908.08474}
}
read the original abstract
The Shapley value has become a popular method to attribute the prediction of a machine-learning model on an input to its base features. The use of the Shapley value is justified by citing [16] showing that it is the \emph{unique} method that satisfies certain good properties (\emph{axioms}). There are, however, a multiplicity of ways in which the Shapley value is operationalized in the attribution problem. These differ in how they reference the model, the training data, and the explanation context. These give very different results, rendering the uniqueness result meaningless. Furthermore, we find that previously proposed approaches can produce counterintuitive attributions in theory and in practice---for instance, they can assign non-zero attributions to features that are not even referenced by the model. In this paper, we use the axiomatic approach to study the differences between some of the many operationalizations of the Shapley value for attribution, and propose a technique called Baseline Shapley (BShap) that is backed by a proper uniqueness result. We also contrast BShap with Integrated Gradients, another extension of Shapley value to the continuous setting.
Figures
Forward citations
Cited by 1 Pith paper
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Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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