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REVIEW 3 major objections 5 minor 27 references

Beyond Shapley Values: Cooperative Games for the Interpretation of Machine Learning Models

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that Shapley values are not a privileged way to explain machine learning predictions.

desk verdict Clean synthesis of Weber/Harsanyi allocations for XAI, but the 'efficiency is all you need' thesis is undercut by an internal inconsistency: the paper's own example sums to f(x)−E[f(X)], not f(x). read the letter →

arxiv 2506.13900 v1 pith:J6U2TWYQ submitted 2025-06-16 stat.ML cs.AIcs.LG

Marouane Il Idrissi , Agathe Fernandes Machado , Arthur Charpentier This is my paper
classification stat.MLcs.AIcs.LG MSC 91A12
keywords cooperativegametheoryfeatureattributionShapleyvaluesWebersetHarsanyiexplainableAIvaluefunctionefficiency
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

This paper argues that the default status of Shapley values in machine-learning interpretability is not justified by their axioms. Once a value function is fixed, the paper claims, Shapley values are just one efficient allocation among many, and any allocation in the Weber or Harsanyi families legitimately decomposes the quantity of interest. The load-bearing choice is the value function: it must make $v(D)$ equal to the quantity under study, and it determines whether an attribution is faithful to the model. The authors therefore offer a three-step blueprint — choose a quantity of interest, choose a value function, choose an allocation — and illustrate it with prediction and variance decomposition examples. If the paper is right, attribution research should shift from defending axioms to designing value functions.

What carries the argument

The Weber set is the collection of allocations of the form $\phi_v(j)=\mathbb{E}_p[v(\pi_j)-v(\pi_j\setminus\{j\})]$, where $p$ is a probability distribution over player orderings; it turns attribution into the choice of a random order. The Harsanyi set is the collection of allocations $\phi_v(j)=\sum_{A\ni j}\lambda_j(A)\varphi_v(A)$, where $\varphi_v(A)$ are the Harsanyi dividends computed by Möbius inversion and $\lambda$ is a row-stochastic weight system over coalitions. Both families contain exactly efficient allocations, and they position Shapley values as one point in a large space: uniform orderings in the first case, egalitarian dividend sharing in the second. The paper uses these families to separate the value function from the aggregation rule and to justify custom allocations such as proportional marginal effects.

What would settle it

Give a two-feature model and the value function $v(A)=\mathbb{E}[f(X)\mid X_A=x_A]$, and use the degenerate Weber allocation that puts all probability on one ordering, so the whole prediction goes to the last feature in that ordering. If stakeholders judge that attribution to be misleading even though it is efficient, that would contradict the claim that efficiency is the only critical axiom; observing such a rejection in a controlled user study would settle the question.

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Extended reading notes

Core claim

The central claim is that the only allocation-level property that matters for feature attribution is efficiency, meaning that the attribution sums to $v(D)-v(\emptyset)$ and therefore redistributes the chosen quantity among the features. The paper shows that every element of the Weber set, expected marginal contributions under a random order distribution, is efficient, and so is every element of the Harsanyi set, a redistribution of recursive dividends under a weight system. Shapley values are recovered in these families when the random order is uniform or when each coalition's dividend is shared equally among its members, which the authors read as evidence that Shapley values carry no special interpretive authority. Consequently, axioms such as symmetry or linearity govern only the aggregation step, and the decisive step in attribution design is selecting a value function whose value at the full feature set matches the quantity of interest.

Load-bearing premise

The framework's conclusion stands on the normative premise that efficiency is the relevant criterion for a feature attribution: that any efficient redistribution of the chosen quantity is acceptable, regardless of how shares are distributed among features.

Editorial extensions

If this is right

  • Shapley values lose their status as the default attribution; uniform random order or egalitarian dividend sharing is one legitimate choice, not the uniquely justified one.
  • A value function with $v(D)$ equal to the chosen quantity is what connects an allocation to an interpretation, so a poor value function cannot be repaired by any allocation rule.
  • Attribution fidelity can be improved by choosing projection-based value functions: in the paper's two-feature example, oblique projections recover the model's analytical structure rather than a correlation-mixed expression.
  • Custom allocations are attainable by constraining the random order distribution or the dividend weight system, for example to encode causal structure or to suppress spurious features under dependence.
  • Validation of an attribution method should check whether it efficiently redistributes the target quantity, rather than whether it satisfies Shapley axioms.

Reading between the lines

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

  • If efficiency is the whole story, then comparing attribution methods with the same value function is really comparing summaries of the same $2^d$ coalition values, which suggests benchmarks should hold the value function fixed before judging allocation rules.
  • The Harsanyi weight system, read as a sparse row-stochastic matrix, invites new allocations via optimal transport or sparsity constraints, a direction the paper mentions but does not develop.
  • The same blueprint applies to any decomposable model quantity, such as loss, variance, or information-theoretic metrics, so the argument generalizes beyond prediction explanations to model diagnostics.
  • A strict reading implies that two attributions are equally valid whenever they are efficient and share the same value function, even if one assigns all credit to a single feature; this is a strong consequence the paper does not embrace explicitly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper argues that Shapley values should not be treated as uniquely privileged in feature attribution. It reviews two broad families of efficient allocations, the Weber set and the Harsanyi set, and uses them to argue that the choice of value function is more important than the choice of allocation. The authors propose a three-step blueprint: choose a quantity of interest, choose a value function v such that v(D) equals that quantity, and choose an efficient allocation. They illustrate the framework with analytic examples for prediction decomposition and variance-based sensitivity analysis, and they argue that proportional marginal effects can outperform Shapley values for detecting spurious features.

Significance. If the paper's central claims are correct, it would be a useful conceptual contribution to the XAI literature: it clearly separates the value-function choice from the allocation rule, explains the Weber and Harsanyi sets in accessible terms, and productively widens the space of admissible attribution methods. The analytic examples are pedagogically valuable, and the paper correctly emphasizes that axiomatic characterizations of Shapley values do not, by themselves, justify their use in interpretability. However, the central normative claim that efficiency is the only critical allocation-level property is asserted rather than derived, and there is a concrete technical inconsistency in how efficiency is defined and applied. These issues affect the main blueprint and must be addressed before the paper can be recommended for publication.

major comments (3)
  1. [Section 2 and Section 3.1, Step 2] The blueprint is internally inconsistent with the paper's own definition of efficiency. Efficiency is defined in Section 2 as sum_j phi_v(j) = v(D) - v(emptyset), but Step 2 of the blueprint (Section 3.1) requires only that v(D) equals the chosen quantity of interest and never imposes v(emptyset) = 0. As a result, every efficient allocation redistributes v(D) - v(emptyset), not the chosen quantity. This is not a minor presentational issue: the claim that efficiency ensures the quantity of interest is "fully redistributed among the features" (end of Section 3.1) is false without an explicit normalization step, such as replacing v by v(A) - v(emptyset) for all A, or redefining efficiency to mean sum_j phi_v(j) = v(D). The authors should either add this normalization to the blueprint or revise the definition and all downstream statements accordingly.
  2. [Section 3.2, example after the displayed Shapley values] The example used to illustrate the failure of conditional-expectation value functions is computed incorrectly relative to the paper's efficiency definition. With v(A) = E[f(X) | X_A = x_A], one has v(emptyset) = E[f(X)] = rho for f(x) = x1 + x2 + x1*x2 under standard bivariate normality. The two displayed Shapley values therefore sum to x1 + x2 + x1*x2 - rho, not f(x). The sentence "this decomposition correctly redistributes the model output" is consequently wrong under the paper's own efficiency definition. This matters for the comparison with the oblique-projection value functions that follows: those values sum to f(x) precisely because they are normalized, and the claimed superiority in "purity" is partly conflated with the normalization gap. The example should be recomputed with a normalized game (or the baseline should be accounted for explicitly).
  3. [Section 2.3] The paper's central normative claim, that "the critical property (or axiom) that an allocation must satisfy for XAI remains efficiency," is asserted rather than derived. The paper does not provide a formal account of what makes an explanation correct, so the conclusion that any efficient allocation is acceptable once v(D) is chosen well does not follow. If a stakeholder cares about other properties, such as monotonicity, sparsity, or fidelity to local model behavior, then efficiency alone is not sufficient. This is not a circularity problem, but it is a load-bearing gap because the three-step blueprint in Section 3.1 depends on this premise. The authors should either justify efficiency as a necessary and sufficient correctness criterion, or state the claim more modestly as a design principle, e.g., efficiency is necessary for decomposing the quantity of interest, while other properties determine which efficient allocation is appropriate.
minor comments (5)
  1. [Section 3.3, PME example] The proportional marginal effects (PME) allocation is used to support an important claim, but its definition is not given in this paper. Readers must consult Herin et al. 2024 to know what is actually being computed. Please include the PME formula (or a precise self-contained definition) so that the example is reproducible from the manuscript.
  2. [Theorem 1 statement] The phrase "parametrize by the random order distribution" should be "parameterized by the random order distribution." This is a small language issue, but the sentence is currently ungrammatical.
  3. [Appendix] The appendix contains sketches for Propositions 1 and 2 but no sketch for Theorem 1 or Theorem 2. Since the paper is aimed at an XAI audience that may be unfamiliar with game theory, a one-sentence derivation or a more precise pointer to the exact statements in Weber 1988 and Harsanyi 1963 would improve readability.
  4. [References] In the Shapley 1951 reference, "Researach Memorandum" should be "Research Memorandum." Also, the authors' names in the Strumbelj and Kononenko reference appear with inconsistent diacritics; please standardize them.
  5. [Section 1.1] The authors say they "question the importance of axiomatic characterizations," but the paper never explicitly defines or lists the usual Shapley axioms (efficiency, symmetry, dummy, additivity) before discussing their relevance. A short explicit statement of these axioms would make the later discussion more accessible and precise.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; self-citations are illustrative, not load-bearing; the efficiency/emptyset mismatch is a correctness issue, not a circular reduction.

full rationale

Circularity pass: no step in the paper reduces a claimed result to its own input by construction. The core game-theoretic machinery (Weber set, Harsanyi dividends, Shapley characterizations, Propositions 1-2, Theorems 1-2) is quoted from external works (Weber 1988, Harsanyi 1963, Vasil'ev & van der Laan 2001, Rota 1964), not from the authors' prior papers. The authors' self-citations (Il Idrissi et al. 2023; Il Idrissi et al. 2025; Herin et al. 2024) appear only as illustrative examples of quantities, oblique-projection value functions, and proportional marginal effects; removing them would not alter the three-step blueprint or the efficiency argument. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported from the authors, and no ansatz is smuggled in via self-citation. I therefore weight the circularity score low. I do flag a consistency gap (not circularity): Section 2 defines efficiency as sum_j phi_v(j) = v(D) - v(emptyset), while Section 3.1/3.2 treats 'efficiency' as fully redistributing the chosen quantity v(D); for the conditional-expectation prediction game v(emptyset) = E[f(X)] != 0, so the displayed Shapley values sum to f(x) - E[f(X)], not f(x). This is a missing-normalization/baseline issue in the blueprint and should be corrected, but it is an arithmetic-consistency problem rather than a circular definition.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No data are fitted in this paper: the numeric examples are analytic. The load-bearing items are the normative premise that efficiency is the right target and the standard game-theoretic machinery. The free parameters p and lambda are exactly the design degrees of freedom the paper wants readers to exploit.

free parameters (2)
  • Random order distribution p = Uniform (1/d!) for Shapley; otherwise user-defined
    Eq. (1) defines the Weber set parameterized by a probability mass function over permutations. The paper presents this as a design choice, not a fitted quantity, but the value changes the resulting attribution.
  • Harsanyi dividend weight system lambda = Egalitarian lambda_i(A)=1/|A| for Shapley; otherwise user-defined
    Eq. (4) defines the Harsanyi set parameterized by a weight system. The paper treats this as a free modeling choice for tailoring allocations to interpretability goals.
assumptions (5)
  • domain assumption Transferable-utility cooperative game with players D and value function v is the right abstraction for feature attribution
    Section 2 sets up all attributions as TU game allocations; this is a modeling assumption about what an explanation should be.
  • domain assumption Efficiency is the only allocation-level property that matters for XAI
    Section 2.3 asserts this normative claim without proof; it is load-bearing for the blueprint and for dismissing other axioms.
  • domain assumption The value function must satisfy v(D) equals the selected quantity of interest
    Step 2 of Section 3.1 imposes this requirement so that efficient allocations decompose the target quantity.
  • standard math Mobius inversion over powersets, via Rota's formula, is valid for dividends
    Used in Appendix A in the proof sketch of Proposition 2.
  • domain assumption The model f and feature distribution P_X are given and fixed for post-hoc analysis
    The paper restricts to post-hoc interpretability of a learned model; the attribution is conditional on these inputs.

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Cite this review

Pith. "Pith review of Beyond Shapley Values: Cooperative Games for the Interpretation of Machine Learning Models." pith.science (2026). https://pith.science/paper/J6U2TWYQ

@misc{pith2026250613900,
  author       = {Pith},
  title        = {Pith review of: Beyond Shapley Values: Cooperative Games for the Interpretation of Machine Learning Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6U2TWYQ}},
  note         = {Machine review of arXiv:2506.13900}
}
read the original abstract

Cooperative game theory has become a cornerstone of post-hoc interpretability in machine learning, largely through the use of Shapley values. Yet, despite their widespread adoption, Shapley-based methods often rest on axiomatic justifications whose relevance to feature attribution remains debatable. In this paper, we revisit cooperative game theory from an interpretability perspective and argue for a broader and more principled use of its tools. We highlight two general families of efficient allocations, the Weber and Harsanyi sets, that extend beyond Shapley values and offer richer interpretative flexibility. We present an accessible overview of these allocation schemes, clarify the distinction between value functions and aggregation rules, and introduce a three-step blueprint for constructing reliable and theoretically-grounded feature attributions. Our goal is to move beyond fixed axioms and provide the XAI community with a coherent framework to design attribution methods that are both meaningful and robust to shifting methodological trends.

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