Dimensions of Power: A Systematic Guide to Power Indices for Explainable AI

Arunavo Ganguly; Christopher Bl\"ocker; Filip Naudot; Timotheus Kampik; Vicen\c{c} Torra

REVIEW 1 major objections 6 minor 18 references

Dimensions of Power: A Systematic Guide to Power Indices for Explainable AI

T0 review · 1 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Power indices for explainable AI organize into three attribution dimensions: single players, sets, and cardinalities.

desk verdict 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. read the letter →

arxiv 2608.05031 v1 pith:5GOSGUAP submitted 2026-08-05 cs.GT

Filip Naudot , Arunavo Ganguly , Timotheus Kampik , Vicenç Torra , Christopher Blöcker This is my paper · ORCID

classification cs.GT MSC 91A1291A06

keywords powerindicescooperativegametheoryexplainableAIfeatureattributiongranularityShapleyvalueOwenUpsilon

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

Referee Report

1 major / 6 minor

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.

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 (1)

[Section 5.3 / Section 4.1] 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.

minor comments (6)

[Section 5.3, motivating example] 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.
[Appendix B.1.2, Eq. (20)] 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.
[Table 3, column header] 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 2, paragraph "Game"] 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.
[Appendix B.2.5, ordering example] 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 5.1, Theorem 1] 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.

Circularity Check

2 steps flagged · score 4.0 of 10

The counterfactual-index axiom-satisfaction claims reduce to definitions, but the central taxonomy, cardinality derivations, and distinctness results are self-contained; no load-bearing self-citation found.

self definitional [Section 4.2, Principle 4.7 and Section 5.1, Theorem 1]
"Principle 4.7 (Quantitative Counterfactuality): A power index μ satisfies the quantitative counterfactuality principle iff for every i∈N, it holds that μ_i(v)=v(N)-v(N\{i}). ... Theorem 1 proof: the definition of counterfactual power index γ_i(v)=v(N)-v(N\{i}) coincides with that of quantitative counterfactuality in Principle 4.7."

The principle is defined by exactly the equation that defines the counterfactual index (Definition 1). Therefore Theorem 1, claiming that the counterfactual index satisfies quantitative counterfactuality, is true by construction: it asserts that an object satisfies a property that is literally its own defining formula. The paper itself calls this 'mostly a sanity check.' This is a transparent, acknowledged self-definitional step rather than a substantive derivation, and it does not support the paper's main organizational or distinctness claims.
self definitional [Section 5.3, Principle 5.6 and Theorem 22]
"Principle 5.6 (Cardinality-based Quantitative Counterfactuality): μ_i(v)=v(N)-(1/(|N| choose i)) Σ_{|S|=i} v(S). ... Theorem 22: the task is trivial since we have that ... coincides with the requirement of quantitative counter-factuality."

The cardinality-based counterfactual index (Definition 9) and the cardinality-based quantitative counterfactuality principle (Principle 5.6) are the same formula. Consequently, Theorem 22's satisfaction result is definitional: the index satisfies the principle because the principle was written as the index's defining expression. The paper explicitly notes the 'considerable overlap' between the two, so this is not hidden, but it is still a reduction of a claimed property satisfaction to an identity by construction.

full rationale

The two flagged steps are the only places where a claimed result reduces by definition: Principle 4.7 is Definition 1 restated as a principle, and Principle 5.6 is Definition 9 restated as a principle. Both are explicitly acknowledged by the authors as sanity checks, and neither is load-bearing for the paper's core contribution, which is the three-dimensional taxonomy and the derivation of cardinality-based versions of Shapley, Banzhaf, and Owen values. Those derivations in Appendix B are self-contained algebraic size decompositions, not fits or predictions. The identification of cardinality-based Shapley with the Upsilon-value is imported from Torra [18]; although the cited author is a coauthor, this is a checkable mathematical theorem with a proof in prior published work, and the present paper also provides its own constructive derivation of the cardinality Shapley formula, so the citation is independent support rather than a circular load-bearing step. The claimed distinctness of the cardinality-based Owen value from Upsilon is supported by the concrete computation in Table 5, which does not depend on the permutation convention; the convention that a permutation renames the a priori block structure is openly stated and internally consistent. No fitted parameters are renamed as predictions, no uniqueness theorem is invoked to rule out alternatives, and no external benchmark is replaced by a self-citation. The circularity present is therefore partial, localized to definitional axiom-satisfaction checks, and not characteristic of the paper's main derivation chain.

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

The paper contains no fitted numerical parameters; all quantities are defined from the value function v. The main things the central claim rests on are standard cooperative game theory, cited prior theorems, and two paper-specific modeling conventions: the conservation rule in the cardinality derivation and the block-renaming convention for permutations. The two new cardinality-based indices are mathematical constructs without external empirical handles.

assumptions (5)

standard math Cooperative game in characteristic function form: finite player set N, v: 2^N to R, v(empty set)=0.
Used throughout Section 2 as the formal setting; standard in cooperative game theory.
domain assumption Feature attribution can be modeled as a cooperative game: features are players, and v(S) is the model output when exactly the features in S are present.
The entire XAI framing relies on this mapping, acknowledged in Sections 1 and 7; the authors also cite [8,9] for the caveat that the characteristic function must be chosen appropriately.
standard math Background characterizations: Shapley (efficiency, symmetry, null player, additivity), Dubey-Shapley Banzhaf properties, Owen's a priori union value properties, and Torra's Upsilon-value equivalence with cardinality differences.
The paper relies on these cited theorems for the single-player and cardinality-based Shapley/Upsilon rows in Tables 1-3 (Theorems 3, 4, 6, 11).
ad hoc to paper Conservation in cardinality-based derivation: the total power over all players must equal the total power over all cardinalities.
Appendix B.1.2 'Drop Player Identity' explicitly imposes this requirement to define cardinality-based Banzhaf and Owen values; it is a modeling choice introduced by this paper.
ad hoc to paper A permutation acts on the whole game specification, including the a priori block structure, so blocks are renamed along with players.
Introduced in Section 4.1 and used in Section 5.3; the note after Theorem 19 explains that this convention is what keeps the cardinality-based Owen distinct from the Upsilon-value under the same axiom names.

invented entities (2)

Cardinality-based Banzhaf value (Definition 11)
purpose: Provides Banzhaf-style attribute scores over coalition sizes rather than individual players.
Newly derived in this paper by size-decomposing the single-player Banzhaf value; no external falsifiable prediction is associated with it.
Cardinality-based Owen value (Definition 12)
purpose: Provides Owen-style attribute scores over coalition sizes while respecting an a priori partition structure.
Newly constructed in Appendix B.2; its properties are proven internally, but there is no independent empirical validation.

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Pith. "Pith review of Dimensions of Power: A Systematic Guide to Power Indices for Explainable AI." pith.science (2026). https://pith.science/paper/5GOSGUAP

@misc{pith2026260805031,
  author       = {Pith},
  title        = {Pith review of: Dimensions of Power: A Systematic Guide to Power Indices for Explainable AI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5GOSGUAP}},
  note         = {Machine review of arXiv:2608.05031}
}

read the original abstract

Power indices, originating in cooperative game theory, quantify each player's influence on the outcome of a given game. Originally designed to distribute profits or costs among players and to analyse the fairness of voting systems, power indices have recently gained prominence as methods for attributing outputs of AI-based systems to inputs, thus facilitating explainability. However, selecting the appropriate power index for a given explanation task is an understudied problem. To address this, we organise power indices along three attribution dimensions: single-player, set-based, and cardinality-based. For each dimension, we review the corresponding power indices, generalise existing ones where applicable, and analyse which formal principles they satisfy. We provide proofs for properties that are missing in the literature and show that moving to the cardinality-based setting removes player-identity information while preserving some index-level distinctions. Using concrete examples, we illustrate how the choice of dimension and index affects the resulting attributions in practice, and offer guidance for practitioners seeking to select a suitable power index for a given application context.

Figures

Figures reproduced from arXiv: 2608.05031 by the authors.

**Figure 1.** Illustration of the core concepts behind power indices. We depict players as game pieces, and coalitions as groups of players underlined with a bar. The grand coalition contains all players. In cooperative games, the value function 𝑣 assigns values 𝑣 (𝑆) to coalitions 𝑆, which we denote with a number under the coalition. In the example game shown here, each player alone “creates” a value of 1; the green player’s pre… view at source ↗

**Figure 2.** Coalition weights for the Shapley value and the Banzhaf index with 8 players. The plot shows coalition sizes 0 to 7 because marginal contributions are evaluated with respect to coalitions that exclude the player under consideration. costs based on each player’s contribution. Power indices based on marginal contributions can differ in two main ways: (i) they may differ in how the value function 𝑣 is interpreted or re… view at source ↗

**Figure 3.** The counterfactual and Banzhaf indices do not satisfy efficiency. Depending on the definition 𝑣, the solid line denotes the constant value 𝑣({𝑥1 , 𝑥2 , 𝑥3 }); the dotted line is the cumulative power index score calculated via Equation (2). In particular, the arbitrary index 𝜇 is replaced by 𝛾 and 𝛽, respectively. Shapley We turn our attention to the well-known Shapley power index. The classical principles for the Sh… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Ranking shift between the Shapley value and the Owen value on a disease prediction example. Different power indices applied to the same model may induce different rankings of feature importance [14]. To illustrate this, consider a small disease-classification model wit…

**Figure 5.** Figure 5: Comparison of attribution schemes for the fraud-detection example. One might wonder whether the same effect could be achieved more simply by treating the grouped signals as a single composite player and applying a standard set-based value. However, doing so answers a d…

**Figure 6.** Figure 6: Υ-value attribution for the fraud example. The Υ-value highlights the coalition size at which the model’s prediction becomes most valuable. 7. Discussion This paper set out to make the topic of power indices easier to navigate. There are many power indices with differe…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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work page 2017