{"id":"4d335494-dd36-46c5-8ee9-d7c5f12aee79","arxiv_id":"1908.08474","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper demonstrates that Shapley-based feature attributions are non-unique, exposes flaws in conditional-expectation Shapley, and provides a uniqueness result for Baseline Shapley.","lead":"This paper shows that the Shapley value, often used to explain model predictions, is not unique because several different ways of applying it give different answers. The authors propose Baseline Shapley, a version backed by a uniqueness theorem, and show that popular alternatives can give misleading attributions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4.3 omits Efficiency from BShap's uniqueness axioms although the proof uses it; scaled-BShap methods satisfy the stated five axioms but fail Efficiency, so the theorem as written is false.","rationale":"Read in good faith, the paper's thesis is that Shapley uniqueness is not meaningful until the operationalization of the set function is fixed, and that BShap is the operationalization with a clean axiomatic foundation. The demonstrations of CES failures (Examples 3.3-3.4 and Appendix C) are concrete and useful. The uniqueness theorem is the part that must carry the 'proper uniqueness result' claim in the abstract, and it is the least secure link: Corollary 4.3 states five axioms, but its proof depends on a cited theorem whose hypothesis includes a sixth axiom, Efficiency. Since scaled BShap satisfies the five, the theorem as written is not merely incomplete; it is false. This is a real correctness risk in the central claim. However, it is repairable by adding Efficiency to the axiom list, and BShap does satisfy Efficiency, so the appropriate verdict is conditional rather than REJECT. I do not see a need to challenge the CES analysis, the IG axiomatization, or the empirical case study. The reader's conditional verdict is therefore preserved without escalation.","tokens_in":120,"tokens_out":19028,"duration_ms":323378,"concrete_test":"Check the counterexample family: define M_λ(f,x,x') = λ·BShap(f,x,x') for λ=1/2 and verify it satisfies each axiom listed in Corollary 4.3. Since BShap is linear, dummy, ASI, symmetric, and demand-monotone, M_λ inherits all five properties, but its attributions sum to (f(x)-f(x'))/2, contradicting Efficiency. This disproves the uniqueness claim as stated. Then repeat the Appendix D.4 proof with Efficiency added to the corollary's axiom list; with that amendment, Theorem 4.2 applies and the corrected uniqueness claim holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central load-bearing step is Corollary 4.3, which claims BShap is the unique method satisfying Linearity, Dummy, ASI, Demand Monotonicity, and Symmetry for all attribution problems. The proof in Appendix D.4 invokes Theorem 4.2, whose uniqueness hypothesis includes Efficiency. Efficiency is not derived from the five listed axioms; it is an independent axiom in Section 2.6. Without it the uniqueness claim is false: for any λ in (0,1), the method M_λ = λ·BShap satisfies all five listed axioms but its attributions sum to λ(f(x)-f(x')), not f(x)-f(x'). Thus there is a continuum of methods satisfying the stated axioms, not a unique one. The bounded-derivative caveat in Theorem 4.1 is a stated technical restriction and is less concerning; the Efficiency omission changes the truth value of the theorem. Adding Efficiency to the corollary preserves the intended result, so the paper's other contributions are not undermined.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15045,"tokens_out":11527,"duration_ms":110599,"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":[{"comment":"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.","section":"§4.2, Corollaries 4.3 and 4.4, and Appendix D.4"},{"comment":"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.","section":"§4.2, Theorem 4.1 proof"}],"minor_comments":[{"comment":"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.","section":"§2.6, ASI definition"},{"comment":"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.","section":"§3.2, Remark 3.2"},{"comment":"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.","section":"§4.2, Theorem 4.1"},{"comment":"There is a typo: 'We introdoce' should be 'We introduce.'","section":"Appendix A"},{"comment":"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.","section":"§4.2, Corollary 4.3"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth your time: it shows that the Shapley uniqueness theorem does not carry over to attribution because there are multiple operationalizations, and it documents real pathologies in conditional-expectation SHAP. The CES failures (Dummy, Linearity) are correct and important; the counterexample to Lundberg-Lee's redundancy claim is clean. The connection to Friedman-Moulin cost-sharing is genuinely new and gives a principled way to discuss BShap and IG.\n\nThe soft spot is Corollary 4.3. As written it claims BShap is the unique method satisfying Linearity, Dummy, ASI, Demand Monotonicity, and Symmetry. The proof invokes Theorem 4.2, whose uniqueness hypothesis includes Efficiency. Efficiency does not follow from the five listed axioms. For any λ in (0,1), λ·BShap satisfies all five axioms but sums to λ(f(x)-f(x')). So the theorem as stated is false, not merely underproved. Adding Efficiency to the corollary's axiom list restores the intended result, so the paper's substantive contribution survives. This is a one-line fix but it matters: the abstract advertises BShap as 'backed by a proper uniqueness result,' and the current statement, as a mathematical claim, is wrong.\n\nThe bounded-derivative condition in Theorem 4.1 is a real limitation for some models, but the authors flag it, and it does not undermine the conceptual point. The paper also overstates the practical difference between IG and BShap only slightly; the examples are honest.\n\nWho should read it: anyone working on Shapley-based explainability, especially practitioners deciding between SHAP variants. The axiomatic framework will be useful to researchers. It deserves a serious referee; the right outcome is conditional acceptance after Corollary 4.3 is corrected and the reduction theorem's conditions are stated carefully.\n\nRecommendation: engage with it. It is a solid conceptual paper with one misstated theorem.","headline":"Sharp conceptual critique of Shapley-based attribution, but the advertised uniqueness theorem for BShap is misstated and needs an added axiom.","tokens_in":15533,"tokens_out":1677,"would_cite":true,"duration_ms":16077,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A12","91A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"Shapley's uniqueness theorem does not select a single explanation method; Baseline Shapley gets its own uniqueness theorem.","keywords":["Shapley value","model explanation","axiomatic attribution","Baseline Shapley","Integrated Gradients","conditional expectations","cost-sharing","uniqueness"],"falsifier":"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.","tokens_in":14660,"feed_emoji":"🎲","tokens_out":13119,"duration_ms":116159,"temperature":0.7,"pith_summary":"The paper argues that the textbook uniqueness of the Shapley value does not carry over to machine-learning explanation, because the value can be applied to the model in several nonequivalent ways. It shows that the conditional-expectation version (CES) can violate basic desiderata: it can give nonzero credit to features the model ignores and can break linearity, especially when continuous features rarely repeat in the training data. The paper then proposes Baseline Shapley (BShap), which models a feature's absence by substituting a baseline value, and proves that BShap is the unique attribution method satisfying a specific set of axioms. Integrated Gradients receives a parallel uniqueness theorem. The practical point is that a model-explanation tool should state which Shapley variant it implements and under which axioms it is justified.","feed_headline":"Shapley's uniqueness does not pick one explanation method","feed_subtitle":"Baseline Shapley gets a real uniqueness proof; the conditional-expectation version can credit unused features.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the Shapley value and its original axioms, whose uniqueness the paper argues does not fix a single explanation method.","marker":"[16]"},{"why":"Supplies the cost-sharing uniqueness theorems for Shapley–Shubik and Aumann–Shapley that Corollaries 4.3 and 4.4 inherit.","marker":"[7]"},{"why":"Defines Aumann–Shapley values, the continuous sharing rule that Integrated Gradients generalizes.","marker":"[4]"},{"why":"Defines the conditional-expectation Shapley method and its deep-network variant, the main target of the paper's counterexamples.","marker":"[10]"},{"why":"An early application of Shapley values to conditional expectations for individual prediction, one of the variants the paper catalogues.","marker":"[21]"},{"why":"Runs Shapley on conditional expectations under a product-of-marginals distribution, another CES variant whose assumptions the paper tests.","marker":"[5]"},{"why":"Introduces Integrated Gradients, the method the paper contrasts axiomatically with BShap.","marker":"[19]"},{"why":"Introduces the baseline approach to turning features off that BShap formalizes.","marker":"[18]"},{"why":"Provides the monotonicity axiom used in the appendix to show a prior claimed axiomatization is incomplete.","marker":"[22]"}],"fun_headline_variants":["Shapley's uniqueness is meaningless; BShap brings it back","Conditional Shapley blames unused features; BShap doesn't","Baseline Shapley: the only Shapley with a true uniqueness proof","Many Shapley values, one axiomatic winner: BShap"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Shapley's uniqueness is meaningless; BShap brings it back","Conditional Shapley blames unused features; BShap doesn't","Baseline Shapley: the only Shapley with a true uniqueness proof","Many Shapley values, one axiomatic winner: BShap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000803,"raw_usage":{"total_tokens":3556,"prompt_tokens":1001,"completion_tokens":2555,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":2473}},"tokens_in":617,"tokens_out":2555,"duration_ms":17214,"temperature":1.0,"reasoning_tokens":2473,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:38:12.429273+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Shapley value and its original axioms, whose uniqueness the paper argues does not fix a single explanation method."},{"cited_title":"Three methods to share joint costs or surplus","cited_arxiv_id":null,"evidence_quote":"Supplies the cost-sharing uniqueness theorems for Shapley–Shubik and Aumann–Shapley that Corollaries 4.3 and 4.4 inherit."},{"cited_title":"J., and Shapley, L","cited_arxiv_id":null,"evidence_quote":"Defines Aumann–Shapley values, the continuous sharing rule that Integrated Gradients generalizes."},{"cited_title":"A unified approach to interpreting model predictions","cited_arxiv_id":null,"evidence_quote":"Defines the conditional-expectation Shapley method and its deep-network variant, the main target of the paper's counterexamples."},{"cited_title":"Explaining prediction models and individual predictions with feature contributions","cited_arxiv_id":null,"evidence_quote":"An early application of Shapley values to conditional expectations for individual prediction, one of the variants the paper catalogues."},{"cited_title":"Algorithmic transparency via quanti- tative input influence: Theory and experiments with learning systems","cited_arxiv_id":null,"evidence_quote":"Runs Shapley on conditional expectations under a product-of-marginals distribution, another CES variant whose assumptions the paper tests."},{"cited_title":"Axiomatic attribution for deep networks","cited_arxiv_id":null,"evidence_quote":"Introduces Integrated Gradients, the method the paper contrasts axiomatically with BShap."},{"cited_title":"Axiomatic Attribution for Multilinear Functions","cited_arxiv_id":"1102.0989","evidence_quote":"Introduces the baseline approach to turning features off that BShap formalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the monotonicity axiom used in the appendix to show a prior claimed axiomatization is incomplete."}],"review_version":1}