The Explanation Game -- Rekindled (Extended Version)

Joao Marques-Silva; Olivier Letoffe; Xuanxiang Huang

REVIEW 3 major objections 5 minor 1 cited by

The Explanation Game -- Rekindled (Extended Version)

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Replacing SHAP's averaging rule with a logical sufficiency test makes irrelevant features score zero.

desk verdict The WAXp-based SHAP fix is real, but the zero-score guarantee is proven only for the model-based v_a, not the sample-based estimator the experiments use. read the letter →

arxiv 2501.11429 v2 pith:YUBCUJJ2 submitted 2025-01-20 cs.AI

Joao Marques-Silva , Xuanxiang Huang , Olivier Letoffe This is my paper · ORCID

classification cs.AI MSC 91A12

keywords ExplainableAIShapleyvaluesSHAPscoresAbductiveexplanationsFeatureattributionrelevanceSample-basedLogic-basedexplainability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Standard SHAP scores mislead because their characteristic function averages the model's output over perturbed inputs, so regions far from the instance can distort importance. This paper claims the fix is to redefine the game: a set of features is worth 1 exactly when fixing those features already forces the prediction (a weak abductive explanation), and 0 otherwise. With that function, Shapley values provably give zero importance to every feature irrelevant to the prediction, and positive marginal contributions mean exactly that a feature is critical for making a set sufficient. The paper also shows the scores can be estimated with a sampling algorithm that has formal error guarantees, using data-sample checks of sufficiency for tractability, and reports that the resulting rankings are mostly uncorrelated with the SHAP tool's rankings across tabular and image benchmarks. The upshot, if the paper is right, is that Shapley values need not be abandoned in explainable AI; the characteristic function they are built on is what has to change.

What carries the argument

The load-bearing object is the WAXp-indicator characteristic function $v_a(S) = 1$ if fixing the features in $S$ to the instance's values makes the prediction invariant (so $S$ is a weak abductive explanation, a WAXp), and 0 otherwise. It converts a machine-learning model into a monotone simple game, so a feature's marginal contribution is 1 exactly when the feature is critical: the coalition $S$ is not yet sufficient, but $S \cup \{i\}$ is. Proposition 1, inherited from quantified relevance results, states that an irrelevant feature is never critical, which is what forces its Shapley value to zero. Estimation is carried by the CGT sampling algorithm, which averages marginal contributions over random permutations and comes with $\epsilon,\alpha$ guarantees; for scalability, sufficiency is decided by Algorithm 2, a linear-time scan of a dataset that checks whether any sample row with the fixed feature values contradicts the prediction.

What would settle it

Feed nuSHAP a model in which feature 2 is irrelevant, for example a binary classifier predicting 1 exactly when $x_1 = 1$, with target instance $((0,0),0)$, and give it a dataset containing only the row $(1,1)$. The sample-based sufficiency test judges $\{2\}$ to be a weak abductive explanation while $\emptyset$ is not, so feature 2 receives a positive estimated score even though it is irrelevant; that would show the sample-based implementation does not inherit the paper's central guarantee, and a reader could run this check with any SHAP-style sampling explainer.

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Extended reading notes

Core claim

The paper's central discovery is that the known failures of SHAP scores are not a failure of Shapley values but of the expected-value characteristic function $v_e$, and that replacing it with the WAXp-indicator function $v_a(S) = 1$ iff $S$ is a weak abductive explanation for the instance yields theoretical SHAP scores, called $\nu$SHAPT, immune to those failures. By Proposition 1, an irrelevant feature leaves $v_a$ unchanged for every coalition, so its Shapley value is 0; by Proposition 2 every marginal contribution is 0 or 1, and a 1 occurs exactly when the feature is critical, i.e. when adding it turns a non-sufficient set into a sufficient one. The paper further claims that these scores can be rigorously estimated by a sampling method with probabilistic error bounds, and that in practice the WAXp test can be carried out on a dataset sample in $O(mn)$ time, yielding the prototype nuSHAP. Experimental comparisons show the nuSHAP rankings and the SHAP tool's rankings are mostly uncorrelated, which the paper reads as further evidence that current SHAP scores are misleading while the new ones are not.

Load-bearing premise

The paper proves the zero-score property for a characteristic function that queries the model itself, but its tool defines the same function over a dataset sample; it assumes, without proof, that sample-based sufficiency judgments preserve the property, so scores produced by nuSHAP may not be the novel scores the theory describes.

Editorial extensions

If this is right

Irrelevant features receive Shapley value 0 under $v_a$, so no ranking can place a formally irrelevant feature above a relevant one, a failure documented for standard SHAP.
Feature attribution and feature selection become unified: a positive marginal contribution is exactly a criticality judgment of the kind used in abductive and contrastive explanations.
Because $v_a$ is monotone with $\{0,1\}$ marginal contributions, the CGT estimator keeps its probabilistic accuracy guarantee while remaining polynomial-time per permutation when WAXp tests are efficient.
Data-based sufficiency checks make the method model-agnostic and scalable; the reported rank-overlap comparisons indicate that the SHAP tool's rankings disagree with the new scores on most tested instances.
The practical conclusion is not to abandon Shapley values but to replace the characteristic function used by SHAP-style explainers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The sample-based implementation can diverge from the model-based guarantee: a dataset that omits the single row that exposes a feature's irrelevance can make an irrelevant feature look critical, so nuSHAPE is not automatically shielded by Proposition 1.
The low rank correlation with the SHAP tool is descriptive, not a certificate of correctness for nuSHAP; a direct comparison against ground-truth feature relevance, such as planted features or formal AXps, would test whether the new rankings are actually better rather than merely different.
The same $v_a$ game could support other game-theoretic indices, for example the Banzhaf index, or alternative feature-importance measures consistent with formal explanations, since the paper's reduction shows attribution reduces to counting critical coalitions.
A natural next experiment is to feed nuSHAP the exact sample used by the SHAP tool and compare its top-k features against known formal explanations on small models; the paper's theory predicts exact agreement at the model level and sample-induced exceptions in practice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the bottom line: the paper's core construction—replacing SHAP's expected-value game with the WAXp-indicator game v_a—is sound, and the CGT estimator provably assigns zero to irrelevant features when v_a is model-based. That matters. But the guarantee does not transfer to what nuSHAP actually computes. The implementation swaps model-based WAXp queries for sample-based sbWAXp checks over a finite dataset (Eq. 11, Algorithm 2), and Section 4.3 concedes these are data-accurate, not model-accurate. No theorem, bound, or sample-size argument closes that gap. So the abstract's claim that the earlier SHAP failures 'cannot occur' is established only for the theoretical νSHAP_T, not for the estimated νSHAP_E scores used in the experiments. The stress-test note is on target; the reader's weakest assumption is exactly right.

Credit where it's due. Algorithm 2 is a clean O(mn) decision procedure for sbWAXp, and the experimental design—reusing SHAP's own samples so both methods see the same data, then comparing top-5 rankings with RBO—is fair and useful. Propositions 1 and 2 are correct for the model-based v_a, and the critical-element view bridges voting power and abductive explanations nicely. The heavy self-citation isn't a sin here; it's honest about where v_a came from, but it also means the novelty is in the estimator and the empirical pipeline, not the definition.

The soft spots, in order. One: the sample-based gap is load-bearing. With a finite sample, an irrelevant feature can look critical if every sampled counterexample happens to differ on it, and a relevant feature can vanish if the witnessing point isn't sampled. The paper gives no handle on how likely these errors are. Two: the experiments conclude SHAP is misleading almost everywhere, but the only evidence is ranking disagreement with nuSHAP. That presumes nuSHAP's rankings are the correct standard, which is exactly what the missing transfer theorem would need to justify. RBO near zero proves divergence, not deception. Three: the zero-score property is built into v_a, not tested by the experiments; the boolean-function section just illustrates the definition.

Who should read this: anyone working on formal XAI or feature attribution should know the nuSHAP construction and the sbWAXp shortcut; it's a serious attempt to connect abductive explanations to Shapley values. It deserves a serious referee, but with major revision: prove or bound the sample-based transfer, release the code, and soften the 'misleading' language to 'rankings disagree.' I'd send it out, not desk-reject.

Referee Report

3 major / 5 minor

Summary. The paper proposes replacing the standard SHAP characteristic function v_e (the conditional expected prediction) with a new characteristic function v_a that is 1 exactly when a set of features is a weak abductive explanation (WAXp). Theoretical SHAP values based on v_a are called nuSHAP scores; the paper argues that, unlike SHAPT scores based on v_e, these scores assign zero importance to formally irrelevant features and therefore cannot produce the misleading rankings documented in earlier work. The paper pairs this definition with the Castro et al. (2009) sampling algorithm (CGT) for Shapley-value estimation, and for scalability replaces model-based WAXp checks with sample-based weak abductive explanations (sbWAXp) computed over a dataset (Algorithm 2). Experiments compare the resulting nuSHAP tool with the SHAP tool on tabular and image datasets, reporting rank-biased overlap (RBO) values that are often low and concluding that SHAP's rankings are misleading while nuSHAP's are not.

Significance. If the theoretical and practical claims were both fully established, the paper would make a useful contribution: it would connect feature-attribution scores with logic-based abductive explanations, provide a Shapley-based score with a formal irrelevant-feature guarantee, and give a scalable estimation procedure. The paper also deserves credit for using an estimation algorithm (CGT) with explicit error and probability guarantees, and for being explicit that sample-based explanations are data-accurate rather than model-accurate. However, the central guarantee is proven only for the model-based characteristic function, while the implemented and experimentally evaluated method uses a sample-based function; no theorem transfers the irrelevant-feature zero-score property to the estimated scores. The experimental sections treat nuSHAP's rankings as ground truth, so the comparison with SHAP shows disagreement but does not by itself establish that SHAP is misleading or that nuSHAP is not. These gaps are load-bearing for the paper's headline claims.

major comments (3)

[Sections 4.2 and 4.3 (Proposition 1, Eq. (9), Eq. (11), Algorithm 2)] Proposition 1 proves the zero-score property only for the model-based characteristic function v_a defined in Eq. (9). The practical nuSHAP implementation defines its characteristic function using sample-based sbWAXps, Eq. (11), decided by Algorithm 2. No theorem shows that Proposition 1 transfers to this sample-based function. For a finite dataset, a model-irrelevant feature can become critical in the sample: if a row with a different prediction differs from v on that feature and on no other feature in the tested set S, then S∪{i} is declared an sbWAXp while S is not, yielding a nonzero sample-based Δ_i and hence a nonzero Shapley estimate. Conversely, a model-relevant feature can be missed if the witnessing sample point is absent. The paper's own statement in Section 4.3 that sample-based explanations are "data-accurate, but not model-accurate" directly undercuts the Section 4.2 claim that the issues with SHAPT scores "cannot occur" for the CGT/v_a combination, since the experiments use the sample-based v_a, not the theoretical v_a.
[Section 5 (Table 4 and surrounding text)] The experiments compare nuSHAP rankings with SHAP rankings using RBO and interpret low RBO as evidence that SHAP is misleading and nuSHAP is "guaranteed not to mislead." This interpretation lacks a ground truth: the RBO comparison only shows that the two methods disagree. It does not establish which features are actually relevant for each model and instance, and it cannot validate the sample-based nuSHAP scores, which may violate the theoretical zero-score property for the reasons given above. To support the claim, the paper would need to compare against formal model-based relevance (e.g., AXp/CXp membership) or use synthetic models with known relevance, or supply a sample-size or coverage guarantee under which the sample-based scores converge to the model-based nuSHAP scores.
[Section 4.1 (Eq. (9)), Section 1 and Abstract] The paper's headline claim that the proposed SHAP scores "overcome" the flaws of SHAPT scores should be scoped carefully. The zero-score property is a direct consequence of defining irrelevance through WAXp membership and then defining v_a as the WAXp indicator: irrelevant features by construction cannot change WAXp status. This is a valid theorem, but it is a definitional property of the theoretical v_a, not an empirical discovery. The abstract and Section 1 present the result as if it were an experimentally confirmed property of the implemented tool. The paper should state clearly that the guarantee is for the model-based theoretical score, and that the experiments are preliminary evidence about a distinct sample-based estimator.

minor comments (5)

[Section 4.3: Algorithm 2, running time] The claim that deciding an sbWAXp runs in O(m n) is imprecise: the loops are over W and the dataset rows, so the cost is O(|W| n), which is at most O(m n) but should be stated as O(|W| n) for accuracy.
[Section 4.2: CGT error parameters] The paper states that r is estimated from epsilon and alpha but does not give the formula or a reference for the bound; a page-equation pointer to Castro et al. (2009) would help the reader verify the claimed guarantee.
[Section 5: RBO parameters] The RBO results depend on the chosen persistence (0.5) and depth (5); the paper reports only these settings. Since RBO values are sensitive to depth, the conclusions about "no correlation" should at least acknowledge this dependence, or report a sensitivity check.
[Section 6 / Discussion] The phrase "conclusive practical evidence" in the introduction and the claim that the experiments "demonstrate" that SHAP is misleading are stronger than what a disagreement-based RBO comparison can support; please temper the language to match the evidence presented.
[General notation] The acronyms SHAPE, nuSHAPE, SHAPT, and nuSHAPT are introduced in Table 1 but the experimental section sometimes refers to "SHAP scores" without specifying whether theoretical or estimated values are meant; a consistent qualifier would prevent ambiguity.

Circularity Check

2 steps flagged · score 6.0 of 10

nuSHAP's zero-score guarantee is definitionally built into v_a, and the sample-based implementation drops the model-level object of that proof; experiments use nuSHAP as its own standard.

self definitional [Section 2 (feature relevance), Section 4.1 (Eq. 9), Section 4.2 (Prop. 1 and the paragraph after Algorithm 1)]
"A feature is relevant if it is included in at least one AXp; otherwise it is irrelevant. ... v_a(S) := (1 if WAXp(S), 0 otherwise) ... Let i ∈ F be an irrelevant feature. Then, ∀(S⊆(F\{i})).[v_a(S∪{i}) = v_a(S)] ... Because of Proposition 1, Δ_i is never incremented by Algorithm 1 when i is irrelevant. As a result, the SHAP score for irrelevant features will always be 0 ... Thus, the issues with SHAPT scores reported in earlier work cannot occur if the characteristic function is v_a and the estimate is computed with CGT."

The zero-score guarantee is not an independent prediction: v_a is defined as the indicator of WAXp, and 'irrelevant' is defined as membership in no AXp (a subset-minimal WAXp). A feature can switch v_a from 0 to 1 only when S∪{i} is a WAXp and S is not; any minimal WAXp witnessing this contains i, making i relevant. Hence Prop. 1 is an immediate consequence of the two definitions, and the asserted 'cannot occur' conclusion is built into the construction of v_a rather than tested or derived from external facts. The passage even concedes that relevant features can receive zero, so the stronger claim that all earlier SHAP flaws are excluded does not follow from Prop. 1 alone.
other [Section 4.3 (Eq. 11, Algorithm 2) and Section 5 (nuSHAP experimental procedure)]
"For scalability, instead of employing standard model-based explanations, the characteristic function v_a is defined in terms of sample-based explanations, building on recent work [Cooper and Amgoud 2023]. ... As noted earlier, sample-based explanations are data-accurate, but not model-accurate."

Proposition 1 and the zero-score claim are proven for v_a built on model-level WAXp over the full feature space (Eq. 9). nuSHAP's actual estimator replaces WAXp with sbWAXp over a finite dataset (Eq. 11, Algorithm 2), yet no theorem shows that the irrelevant-feature-zero property transfers to this sample-based v_a. For a finite sample, a model-irrelevant feature can become critical in the sample (and vice versa), so the estimated νSHAPE scores are not the scores for which the guarantee was proven. The paper then uses these estimates as the standard for concluding that SHAP rankings are misleading, so the experimental 'confirmation' is circular: the criterion of correctness is the definition itself, not an external benchmark or falsifiable prediction.

full rationale

The paper's mathematical result, Proposition 1, is true: with v_a defined as the WAXp indicator and 'irrelevant' defined as 'in no AXp', an irrelevant feature can never be critical, so its exact Shapley value is 0. That is exactly why the result is not an independent prediction: it is the definition of v_a and of relevance unpacked. The headline statement that the earlier SHAP failures 'cannot occur' is a claim about this constructed characteristic function, not about the sample-based estimator used in the experiments. The implementation (Section 4.3) substitutes sbWAXp over a finite dataset for WAXp, and the paper concedes such explanations are 'data-accurate, but not model-accurate'; no theorem transfers the zero-score property or relevance judgments to this finite-sample v_a. Experiments then compare SHAP to nuSHAP and use nuSHAP as the standard, so they cannot independently confirm the built-in guarantee. Self-citations ([Letoffe et al. 2025], [Huang et al. 2023]) supply the characteristic function and the corollary, but the reduction is definitional, so the main circularity is the self-definitional guarantee plus the unproven sample-based transfer, giving a partial circularity score of 6 rather than 8 or 10.

Assumptions & free parameters 3 free parameters · 7 assumptions · 1 invented entities

The main dependencies are the normative choice of v_a, the uniform/discrete domain assumptions, the correctness of the CGT estimator, and the unproved substitution of sample-based explanations for model-based ones. No parameters are fitted to data; epsilon and alpha are user-chosen error bounds.

free parameters (3)

epsilon (CGT error bound) = 0.0015
Chosen by the authors for all nuSHAP experiments; affects the number of permutation samples and therefore the estimate quality.
alpha (CGT probability bound) = 0.015
Chosen by the authors for all nuSHAP experiments; controls the confidence in the error bound.
RBO persistence and depth = 0.5 and 5
Chosen for the ranking comparison; affects the RBO values and the conclusion about correlation.

assumptions (7)

domain assumption A uniform probability distribution over each feature and independence of features are assumed.
Stated in Eq. (3) and used implicitly throughout; needed for the expected value in the old characteristic function, less so for v_a, but the paper's framing relies on it.
domain assumption A user-specified similarity relational operator ≈ exists for each feature and for the prediction.
Defined in Section 2; all explanations and the characteristic function v_a depend on this choice, and the paper gives no concrete default.
domain assumption Feature domains are finite and discrete; real-valued features are discretized.
Stated in Section 2; needed for well-defined probabilities and for finite sample spaces.
domain assumption The ML model is non-constant.
Assumed at the start of Section 2 so that v_a(∅)=0 and the Shapley value is defined.
standard math Correctness of the CGT sampling algorithm for Shapley value estimation.
Adopted from [Castro et al. 2009] (Algorithm 1); the paper relies on its error guarantees without reproving it.
ad hoc to paper Irrelevant features should receive zero importance is the normative criterion for a non-misleading attribution method.
This value judgment is built into the choice of v_a and underlies the claim that nuSHAP cannot mislead; it is not an empirical finding.
ad hoc to paper Sample-based explanations computed on a dataset are an adequate substitute for model-based abductive explanations.
Introduced in Section 4.3; no theorem links sbXp-based v_a to the model-based v_a, yet the experimental nuSHAP relies on it.

invented entities (1)

nuSHAP scores (characteristic function v_a)
purpose: Feature attribution based on weak abductive explanations
The new quantity is defined in this line of work; it makes a normative claim about importance but has no falsifiable external prediction. The tool nuSHAP is a prototype implementation, not released with the paper.

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Pith. "Pith review of The Explanation Game -- Rekindled (Extended Version)." pith.science (2026). https://pith.science/paper/YUBCUJJ2

@misc{pith2026250111429,
  author       = {Pith},
  title        = {Pith review of: The Explanation Game -- Rekindled (Extended Version)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YUBCUJJ2}},
  note         = {Machine review of arXiv:2501.11429}
}

read the original abstract

Recent work demonstrated the existence of critical flaws in the current use of Shapley values in explainable AI (XAI), i.e. the so-called SHAP scores. These flaws are significant in that the scores provided to a human decision-maker can be misleading. Although these negative results might appear to indicate that Shapley values ought not be used in XAI, this paper argues otherwise. Concretely, this paper proposes a novel definition of SHAP scores that overcomes existing flaws. Furthermore, the paper outlines a practically efficient solution for the rigorous estimation of the novel SHAP scores. Preliminary experimental results confirm our claims, and further underscore the flaws of the current SHAP scores.

Figures

Figures reproduced from arXiv: 2501.11429 by the authors.

**Figure 2.** Simple ML model M2, with instance ( (1, 1), 1), and 𝛼 ≠ 0. The expected values are computed for all possible sets of features. Clearly, A2 = C2 = { {1} }. Sc𝑒 (1) Sc𝑒 (2) Sc𝑒 (3) Sc𝑒 (4) Rank 𝜅1 0.000 0.000 -0.042 -0.208 ⟨4, 3, 1:2⟩ (a) Scores for E1 Sc𝑒 (1) Sc𝑒 (2) Rank 𝜏2 0 𝛼 ⟨2, 1⟩ (b) Scores for E2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** SHAPT scores for E1 and E2. These are the values that the tool SHAP [Lundberg and Lee 2017] approximates. insight can then be used to identify different issues where misleading information is clearly unsatisfactory. As a consequence, the tool SHAP [Lundberg and Lee 2017] aims to approximate values that can be misleading. Although these observations have gone unnoticed in the many thousands of publications that buil… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Comparison of RBO values. Blue (resp. green) shows comparison with (resp. absolute) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Dimensions of Power: A Systematic Guide to Power Indices for Explainable AI
cs.GT 2026-08 accept novelty 6.0 of 10

The paper maps power indices for explainable AI onto single-player, set-based, and cardinality-based dimensions, and adds new cardinality-based Banzhaf and Owen variants with an axiom-by-axiom comparison.

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