{"id":"3b88c56a-db20-4fc9-8f95-249b4017dfb5","arxiv_id":"2608.05031","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper organizes AI feature-attribution methods based on cooperative game theory into three 'dimensions': single features, sets of features, and feature counts. It introduces new count-based siblings of standard indices and analyzes which formal fairness properties each one obeys.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the convention-dependent distinction is disclosed and the distinctness claim is verified by direct computation.","rationale":"The reader's weakest assumption correctly identifies the convention dependence as a caveat, but it is not load-bearing: the central claim does not depend on the two indices satisfying the same named axioms, and the paper explicitly states its convention. The distinctness in Table 5 is a direct computation from the definitions, independent of how permutations are defined. The property tables are supported by proofs that appear internally consistent; the set-based results follow from the single-player results by Theorem 9; the cardinality-based Shapley coincides with Upsilon by [18, Prop. 2]; and the cardinality-based Banzhaf and Owen are derived via a transparent size decomposition in Appendix B. The paper also ships an open-source implementation, which provides an independent check. Therefore the ACCEPT verdict remains unchanged.","tokens_in":923,"tokens_out":601,"duration_ms":168386,"concrete_test":"Recompute the cardinality-based Owen and Upsilon values for the game in Table 5: N={X,Y,Z}, M_N={{X,Y},{Z}}, v(∅)=v({X})=v({Y})=v({Z})=v({X,Y})=0, and v({X,Z})=v({Y,Z})=v(N)=6, using the definitions in Appendix B.2 and Definition 10. If the Owen vector is (0,3,3) and the Upsilon/Shapley vector is (0,4,2), the distinctness claim is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The most plausible concern is that the claimed distinctness of the new cardinality-based Owen value from the Upsilon value, and its satisfaction of absolute anonymity, depend on the paper's convention that a permutation renames the a priori block structure along with the players (Section 4.1, note after Theorem 19). However, this is explicitly disclosed, and the distinctness itself is verified by the numerical example in Table 5, which does not depend on the permutation convention. The paper's convention is internally consistent: since the a priori block structure is part of the game specification for Owen-type indices, renaming players should rename the blocks. Adopting the alternative convention would change the meaning of absolute anonymity for structured games, but would not change the values shown in Table 5 or the central organizational claim. I checked the proofs of the property tables (Theorems 11-23) and found no mathematical errors.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper organizes power indices used in explainable AI into three attribution dimensions: single-player, set-based, and cardinality-based. For each dimension it reviews the corresponding indices (counterfactual, Shapley, Banzhaf, Owen, and their set-based analogues), introduces cardinality-based versions of Shapley, Banzhaf, and Owen, and analyzes which formal principles these indices satisfy. The central claims are that the cardinality-based Shapley value coincides with the Upsilon-value, that the new cardinality-based Owen value is distinct from the Upsilon-value while satisfying the same classical cardinality principles under the paper's permutation convention, and that the set-based indices inherit exactly the principles of their single-player counterparts (Theorem 9). The paper is framed as a systematic guide for practitioners, with property tables, worked examples, and an open-source reference implementation.","tokens_in":37924,"tokens_out":21640,"duration_ms":230831,"significance":"The paper's value is primarily organizational and referential rather than theorem-driven. It collects known and new power indices into a coherent three-dimensional framework and gives the XAI community a practical map for choosing an attribution method according to the intended question and the desired axiomatic guarantees. The appendix derivations are internally consistent, and the worked examples reproduce the reported values, including the distinctness of the cardinality-based Owen value in Table 5. The paper also ships an open-source implementation with documentation and tests, which materially strengthens its usefulness as a guide. Several results that look circular at first glance, such as the quantitative counterfactuality of the counterfactual index, are explicitly labeled by the authors as sanity checks, and the convention-dependence of the cardinality-based Owen distinction is disclosed rather than hidden. If the property tables and the new cardinality-based indices are taken as the contribution, the paper meets the standard for a careful and honest reference contribution.","major_comments":[{"comment":"The distinctness of the cardinality-based Owen value from the Upsilon-value under the same named axioms depends on the convention that a permutation renames the a priori block structure along with the players. The authors disclose this dependence, and the numerical distinctness in Table 5 is computed directly and does not depend on the permutation convention, so I do not regard this as an error. However, the sentence \"which would make the two indices coincide\" is too quick: under the alternative convention in [18], the absolute-anonymity axiom for structured games is not literally the same axiom, so the characterization theorem from [18] would not automatically force equality with the cardinality-based Owen value. I recommend adding a sentence that states explicitly what happens under the alternative convention, so that readers see the distinction is not merely a relabeling artifact but a substantive consequence of how structured games are treated.","section":"Section 5.3 / Section 4.1"}],"minor_comments":[{"comment":"The notation \"φ2(v)=0.167\" refers to the single-player Shapley value of feature x2, but Definition 10 reserves φ_c for the cardinality-based Shapley value, which coincides with the Upsilon-value and would equal -2/3 for c=2 in this example. Please disambiguate, for instance by writing φ_{x_2}(v) for the single-player value.","section":"Section 5.3, motivating example"},{"comment":"The \"conservation\" requirement that total power over all players equal total power over all cardinalities is a design choice that determines the resulting cardinality-based indices. The paper states the requirement, but it should be explicitly flagged in the main text as a modeling assumption rather than implied to be the unique way to cardinalize, since an alternative such as averaging over players would lead to different definitions.","section":"Appendix B.1.2, Eq. (20)"},{"comment":"The column header \"Υ Shapley\" can be read as two separate columns rather than one combined column for the Upsilon/Shapley index. Please typeset it unambiguously, for example as \"Υ / Shapley\" or \"Υ, Shapley\".","section":"Table 3, column header"},{"comment":"The value function of a cooperative game should be displayed as v: 2^N → ℝ; the current rendering of the domain is garbled in the submitted text and should be corrected.","section":"Section 2, paragraph \"Game\""},{"comment":"The list of permissible and impermissible orderings uses strikethrough notation that is hard to parse in the submitted PDF. A table with two columns, or separate lists of permissible and impermissible orders, would make the explanation much clearer.","section":"Appendix B.2.5, ordering example"},{"comment":"Because the proof of Theorem 1 is immediate from the definition of the counterfactual index, consider moving the \"sanity check\" sentence before the theorem statement and shortening the proof to a single line; this is purely a presentation issue.","section":"Section 5.1, Theorem 1"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid survey-and-guide contribution with original derivations, and it should be of interest to readers working at the intersection of cooperative game theory and explainable AI. The main risk is that readers familiar with Torra's Upsilon-value literature will object to the permutation convention used to distinguish the cardinality-based Owen value; the authors have preempted this by disclosing the convention, and the numerical examples are independent of it. I would suggest the editor ensure the requested clarification about the alternative convention is added, but I do not see a need for further technical review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a solid, honest survey-plus-derivation paper. The genuinely new pieces are the cardinality-based Banzhaf and Owen values (Definitions 11 and 12, Appendix B) and the axiom-satisfaction tables around them. The three-dimension framing—single-player, set-based, cardinality-based—is useful even if the set-based cases are just relabelings of the single-player ones, and the authors say so plainly. The proofs I checked in Appendix A and B are internally consistent, and the worked examples match the formulas. There is also a reference implementation with tests, which deserves credit. What the paper does well is scope. It does not oversell its contribution: it explicitly says the set-based indices are relabelings, it flags which principles are tailored to match a particular index, and it confines the algorithmic side to future work. The tables are the kind of thing practitioners will actually consult when choosing an attribution method. The worked examples, especially the fraud-detection one, do a good job of showing that the choice of dimension changes the question rather than just the numbers. The soft spots are real but not load-bearing. The counterfactuality principles are defined so that the counterfactual index satisfies them by construction; the paper acknowledges this, so it is a limitation of the principle design, not a hidden circularity. The more interesting caveat is the distinctness of the cardinality-based Owen value from the Upsilon value. It depends on the convention that a permutation renames the a priori block structure along with the players. The authors disclose this, and the numerical example in Table 5 does not depend on the permutation convention, so the distinctness claim holds in the sense they define. If a reader adopts the other convention, the two indices collapse into the same named axioms, but the paper is explicit about its own convention. I would call that a minor concern for the target audience, not a flaw. The paper is best for XAI researchers who want to know which power index answers which question, and for game theorists who want a compact reference on cardinality-based variants. It deserves a serious referee: the proofs are checkable, the new indices are well-defined, and the limitations are stated. I would send it to review and expect it to be accepted after light-to-moderate revision, mainly to tighten the discussion of the convention sensitivity and to avoid overemphasizing the tailored principles.","headline":"A careful, honest catalog of power indices for XAI, with two genuinely new cardinality-based indices and axiom tables that check out; the soft spots are disclosed rather than hidden.","tokens_in":696,"tokens_out":700,"would_cite":true,"duration_ms":24440,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A12","91A06"],"pacs":[],"model":"deepseek-v4-flash","headline":"Power indices for explainable AI organize into three attribution dimensions: single players, sets, and cardinalities.","keywords":["power indices","cooperative game theory","explainable AI","feature attribution","attribution granularity","Shapley value","Owen value","Upsilon value"],"falsifier":"Compute the cardinality-based Owen value under the alternative convention where a permutation acts on the game alone and does not rename the a priori block structure, and check whether the resulting index satisfies the same axioms used to characterize Upsilon values in [18]. If it does, the distinctness claimed in Table 5 depends entirely on the renaming convention. The explicit example in the appendix, with $N = \\{X, Y, Z\\}$, block structure $\\{\\{X, Y\\}, \\{Z\\}\\}$, and $v(\\{X, Z\\}) = v(\\{Y, Z\\}) = v(N) = 6$ while all other coalitions have value $0$, already exhibits a difference between the cardinality-based Owen and Shapley values and can be recomputed under each convention to settle which convention produces it.","tokens_in":37516,"feed_emoji":"🧩","tokens_out":6553,"duration_ms":69995,"temperature":0.7,"pith_summary":"The paper organizes game-theoretic power indices used in explainable AI along three attribution dimensions: single players, sets of players, and coalition cardinalities. It derives new cardinality-based versions of the Shapley, Banzhaf, and Owen values and proves which classical and non-classical principles each satisfies. The central finding is that moving to cardinality-based attribution removes player-identity information while preserving meaningful distinctions between indices: the cardinality-based Shapley value coincides with the Upsilon value, while the cardinality-based Banzhaf and Owen values remain distinct. The paper thereby gives practitioners a principle-driven map for choosing an index based on what attribution question they are asking.","feed_headline":"Three attribution dimensions organize AI power indices","feed_subtitle":"Cardinality-based Shapley, Banzhaf, and Owen values drop player identity while keeping index-level distinctions.","key_machinery":"The central mechanism is a size decomposition of marginal contributions. For Shapley and Banzhaf, the single-player sums are rewritten grouped by coalition size, summed over all players, and stripped of player identities, leaving differences $m(s)$ between the average value of size-$s$ coalitions and the average value of size-$(s-1)$ coalitions; Shapley weights cancel completely, while Banzhaf weights do not. For the Owen value, the same two-stage decomposition fixes the outer coalition $R$ and averages inner-block steps to form terms $m_{k,R}(t)$, which are then re-bundled by the resulting coalition size $c = |Q_R| + t$. This decomposition generates the new indices and powers the axiom proofs, because no Owen term is lost or double-counted when it is reassigned to a cardinality.","core_discovery":"On its own terms, the paper establishes that every power index can be classified by the player dimension it attributes to, and that the same indices behave predictably across single-player and set-based settings (Theorem 9), while the cardinality-based setting requires new principles such as absolute anonymity and dummy cardinality and reveals new distinctions. Specifically, it shows that the cardinality-based Shapley value equals the Upsilon value, that the cardinality-based Banzhaf value is the Upsilon value reweighted by a binomial factor, and that the cardinality-based Owen value, derived by re-bundling ordinary Owen terms according to the size of the coalition formed, is distinct from the Upsilon value because a permutation of players renames the a priori block structure along with the players. It also proves that the cardinality-based Banzhaf value violates efficiency and dummy cardinality while the cardinality-based Owen value satisfies both.","pith_inferences":["If the distinctness of the cardinality-based Owen value is accepted, it suggests a family of cardinality-based indices parameterized by how the block structure is treated under permutations, analogous to how ordinary Owen values generalize Shapley values; the paper does not develop this family.","The size-decomposition technique could be applied to other single-player indices, such as the Shapley-Shubik index or weighted variants of Banzhaf, to produce further cardinality-based indices whose axioms would need to be re-examined; the paper does not do this.","Because the cardinality-based counterfactual index is the only cardinality-based index satisfying counterfactuality, one could test whether a task is better served by a 'how many features' question than a 'which features' question by checking whether cardinality payouts are stable under small perturbations of the value function; the paper does not propose this test.","The failure of the success principle for every cardinality-based index implies that no cardinality-based index alone can guarantee a non-zero explanation for every non-trivial game, suggesting practical pipelines might need a fallback; this implication is left implicit in the paper."],"forward_implications":["Practitioners can use the axiom tables to decide whether single-player, set-based, or cardinality-based attribution fits their explanation task, and which index satisfies the principles they care about.","Since Theorem 9 shows single-player and set-based indices satisfy the same principles, grouping features into sets is formally a relabelling that does not change guarantees, though it changes the question being answered.","The coincidence of the cardinality-based Shapley value with the Upsilon value means the Upsilon value, originally introduced for monotone games, is the natural cardinality-based analogue of the Shapley value for general games.","The cardinality-based Banzhaf value violates efficiency and dummy cardinality, showing that the reversal of weighting behaviour observed at the single-player level persists at the cardinality level.","The distinct cardinality-based Owen value provides a way to respect a priori feature groups while attributing power to coalition sizes rather than to individual features."],"supporting_citations":[{"why":"Defines the Shapley value and its axiomatic characterization, the baseline single-player index that the paper generalizes to the cardinality setting.","marker":"[15]"},{"why":"Defines the Banzhaf index, the equal-weight single-player index whose cardinality-based version is derived and analyzed.","marker":"[2]"},{"why":"Introduces the Owen value with a priori unions, which the paper generalizes to the cardinality-based Owen value.","marker":"[12]"},{"why":"Introduces the Upsilon value via maximal chains and provides the characterization whose axioms the cardinality-based Owen value is contrasted against.","marker":"[18]"},{"why":"Supplies the classical proofs that the Banzhaf index satisfies null player, symmetry, and weak anonymity, which the paper invokes.","marker":"[4]"},{"why":"Provides the success principle that the paper adapts as a non-classical principle for power indices.","marker":"[1]"},{"why":"Provides the quantitative counterfactuality principle that the paper reinterprets for the cardinality setting.","marker":"[6]"}],"fun_headline_variants":["Power indices for AI: a three-dimension guide","Cardinality-based power indices: a fresh lens for AI attribution","AI explanations: choosing the right power index made easy","From Shapley to Owen: new distinctions in AI power indices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claimed distinctness of the cardinality-based Owen value from the Upsilon value rests on the convention that renaming players also renames the a priori block structure; if a permutation acts on the game alone, as in the characterization used for Upsilon values, the two indices satisfy the same named axioms and the distinction collapses.","fun_headline_variants_meta":{"raw":{"variants":["Power indices for AI: a three-dimension guide","Cardinality-based power indices: a fresh lens for AI attribution","AI explanations: choosing the right power index made easy","From Shapley to Owen: new distinctions in AI power indices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000414,"raw_usage":{"total_tokens":2120,"prompt_tokens":903,"completion_tokens":1217,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":1148}},"tokens_in":519,"tokens_out":1217,"duration_ms":14226,"temperature":1.0,"reasoning_tokens":1148,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:00:45.919510+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the cardinality-based Owen value under the alternative convention where a permutation acts on the game alone and does not rename the a priori block structure, and check whether the resulting index satisfies the same axioms used to characterize Upsilon values in [18]. If it does, the distinctness claimed in Table 5 depends entirely on the renaming convention. The explicit example in the appendix, with $N = \\{X, Y, Z\\}$, block structure $\\{\\{X, Y\\}, \\{Z\\}\\}$, and $v(\\{X, Z\\}) = v(\\{Y, Z\\}) = v(N) = 6$ while all other coalitions have value $0$, already exhibits a difference between the cardinality-based Owen and Shapley values and can be recomputed under each convention to settle which convention produces it.","supporting_citations":[{"cited_title":"A value for n-person games, in: Kuhn, H.W., Tucker, A.W","cited_arxiv_id":null,"evidence_quote":"Defines the Shapley value and its axiomatic characterization, the baseline single-player index that the paper generalizes to the cardinality setting."},{"cited_title":"Weighted voting doesn’t work: A mathematical analysis","cited_arxiv_id":null,"evidence_quote":"Defines the Banzhaf index, the equal-weight single-player index whose cardinality-based version is derived and analyzed."},{"cited_title":"Values of games with a priori unions, in: Henn, R., Moeschlin, O","cited_arxiv_id":null,"evidence_quote":"Introduces the Owen value with a priori unions, which the paper generalizes to the cardinality-based Owen value."},{"cited_title":"outer game","cited_arxiv_id":null,"evidence_quote":"Introduces the Upsilon value via maximal chains and provides the characterization whose axioms the cardinality-based Owen value is contrasted against."},{"cited_title":"Mathematical properties of the banzhaf power index","cited_arxiv_id":null,"evidence_quote":"Supplies the classical proofs that the Banzhaf index satisfies null player, symmetry, and weak anonymity, which the paper invokes."},{"cited_title":"Contributionfunctionsforquantitativebipolarargumentationgraphs:Aprinciple- based analysis","cited_arxiv_id":null,"evidence_quote":"Provides the quantitative counterfactuality principle that the paper reinterprets for the cardinality setting."}],"review_version":1}