Asymmetric Shapley Values

• Asymmetric Shapley values are allocation rules in cooperative games that relax full anonymity by incorporating weighted orderings, causal constraints, and structural hierarchies.
• They employ methodologies like nonuniform permutation weights, quasi-values, and restrictions from coalition feasibility or graph structures to adjust marginal contributions.
• Applications include model explainability, data valuation, and fair division, with efficient computation achieved by reducing admissible orderings via causal or structural constraints.

Asymmetric Shapley values are Shapley-type allocation rules in which the classical requirement of full player anonymity is relaxed, or, in a broader usage, settings in which asymmetry is carried by the underlying game so that the ordinary Shapley value already yields unequal outcomes. In the classical finite transferable-utility setting, the Shapley value is the unique value satisfying linearity, efficiency, the null-player property, and symmetry, with

$$\phi_i(v)=\sum_{S\subseteq \Omega\setminus\{i\}} \frac{|S|!(n-|S|-1)!}{n!}\bigl(v(S\cup\{i\})-v(S)\bigr).$$

The modern literature separates several non-equivalent mechanisms of asymmetry: nonuniform weights on permutations, invariance only under a subgroup $G\le S_n$, restrictions coming from coalition feasibility or graph structure, and endogenous asymmetry from heterogeneous characteristic functions while the Shapley operator itself remains standard (Kubena et al., 2012, Frye et al., 2019, Bei et al., 15 Jun 2026).

1. Weighted permutations and precedence-constrained attribution

The most direct formalization replaces the uniform average over permutations by a probability distribution $w$ on permutations. In the explainability formulation, the asymmetric Shapley value is

$$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$

Ordinary Shapley values are recovered when $w(\pi)=1/n!$ for all $\pi$. The central use case is the incorporation of causal, temporal, or precedence information by setting $w(\pi)=0$ on inadmissible orderings and averaging only over admissible ones (Frye et al., 2019).

In model-agnostic explainability, this construction is coupled with the on-manifold value function

$$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$

so that missing features are completed conditionally on observed ones rather than off-manifold. Under distal-causal weighting, only permutations consistent with ancestor-before-descendant constraints are allowed; a proximate variant reverses that preference when the explanatory target is what ancestors add after descendants are already accounted for. The same formalism supports fairness-oriented orderings, strict temporal orderings in sequence models, and feature-selection analyses in which one feature block is forced to precede another (Frye et al., 2019).

A computationally focused causal-ASV formulation specializes $w$ to the uniform distribution over topological orderings of a causal DAG $C$: $G\le S_n$0 The resulting attribution averages marginal contributions only over topological sorts. This turns causal precedence into the admissible coalition-building orders themselves and makes the asymmetry explicit at the level of the permutation measure rather than the characteristic function (Companeetz et al., 23 Jun 2026).

2. Weakening symmetry axiomatically: quasi-values and non-symmetric linear values

A second line of work formalizes asymmetry by weakening the symmetry axiom rather than by starting from a permutation distribution. A quasi-value is a value satisfying linearity, the null-player property, and efficiency, but not full symmetry. For a subgroup $G\le S_n$1, a $G\le S_n$2-symmetric quasi-value satisfies

$G\le S_n$3

If $G\le S_n$4, one recovers the ordinary Shapley value; if $G\le S_n$5, every quasi-value is $G\le S_n$6-symmetric (Kubena et al., 2012).

This perspective yields a precise classification of when weakened symmetry still determines a unique value. Writing $G\le S_n$7 and $G\le S_n$8, the affine space $G\le S_n$9 of $w$0-symmetric quasi-values has dimension

$w$1

Uniqueness holds if and only if $w$2 acts supertransitively on $w$3. The paper identifies exactly three such cases: $w$4, $w$5 for $w$6, and the exotic embedding $w$7. Outside these cases, asymmetry survives as a positive-dimensional family of admissible values (Kubena et al., 2012).

A related non-probabilistic formulation drops symmetry and replaces it with “reasonableness.” In that framework, efficiency and linearity are preserved, and player $w$8’s payoff must lie between the minimum and maximum of $w$9 over coalitions not containing $$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$0. The resulting allocation matrices satisfy the pairing condition

$$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$1

together with sign and row-sum constraints implying that each player’s payoff is a convex combination of that player’s marginal contributions. The extreme points are the “special allocations,” namely the marginal vectors associated with single permutations, and every reasonable efficient allocation is a convex combination of them (Clark et al., 2018).

Taken together, these results show that asymmetric Shapley values are not a single object but a family of linear allocation rules obtained by restricting who counts as exchangeable with whom, or by retaining only marginal-consistency constraints while abandoning anonymity.

3. Structural asymmetry from digraphs, feasible-coalition systems, and hierarchies

A third tradition does not begin with weights on permutations or weakened axioms, but with structural restrictions on coalition formation. In digraph games, a directed graph $$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$2 induces a dominance or subordination relation, and only permutations consistent with that relation are admissible. If $$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$3 is the set of consistent permutations, the Shapley value of a digraph game is defined by

$$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$4

Here asymmetry arises because the digraph rules out some player orderings and allows others. For cyclic digraphs and cardinality-based games $$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$5, the admissible permutations collapse to $$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$6 cyclic reverse orders and the resulting value is

$$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$7

showing that an asymmetrically defined rule can still yield a symmetric outcome on highly symmetric games (Khatri, 2017).

Restricted-coalition extensions push the asymmetry deeper into the domain of the game. For a coalition system $$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$8, the paper on game representations defines MM-games

$$\phi_v^{(w)}(i) = \sum_{\pi\in\Pi} w(\pi)\, \Big[ v(\{j:\pi(j)\le \pi(i)\}) - v(\{j:\pi(j)< \pi(i)\}) \Big].$$9

and shows that if $w(\pi)=1/n!$0 has full span, every $w(\pi)=1/n!$1 decomposes uniquely as $w(\pi)=1/n!$2. The generalized value is then

$w(\pi)=1/n!$3

This is not a weighted-permutation Shapley value; the asymmetry is induced by coalition feasibility, hierarchies, and semi-algebra structure (Dubey, 2024).

An even more general structural formulation replaces the coalition lattice by a weighted directed acyclic multigraph. If $w(\pi)=1/n!$4 is the generalized Möbius transform on the graph and $w(\pi)=1/n!$5 is the number of directed paths from root $w(\pi)=1/n!$6 to vertex $w(\pi)=1/n!$7, the generalized Shapley value on an unweighted DAMG is

$w(\pi)=1/n!$8

and in the weighted case

$w(\pi)=1/n!$9

Here asymmetry is generated by graph position, path multiplicity, edge weights, and root strengths. The corresponding axioms are no longer classical symmetry and null-player, but weak elements and flat hierarchy (Forré et al., 7 Oct 2025).

4. Endogenous asymmetry with the ordinary Shapley operator

A recurring misconception is that asymmetry always requires modifying the Shapley rule itself. Several papers make the opposite point: the standard Shapley operator can already generate unequal, contribution-sensitive allocations once the characteristic function is asymmetric.

In contribution-based fair division, the induced cooperative game is built from optimal welfare under heterogeneous productive valuations. For a coalition $\pi$0 and resource vector $\pi$1,

$\pi$2

and the Shapley benchmark is the ordinary expected marginal contribution

$\pi$3

The paper explicitly states that it does not introduce a formally modified asymmetric Shapley value; rather, it applies the ordinary Shapley value to a fundamentally asymmetric resource-allocation environment. The resulting entitlements are unequal because the coalition welfare function itself is heterogeneous. On the algorithmic side, the paper studies how well such entitlements can be implemented without transfers and proves tight bounds: $\pi$4 for general concave valuations, $\pi$5 for capped concave valuations with bounded aggregate demand $\pi$6, and $\pi$7 for linear valuations, with exact polynomial-time computation in the linear case (Bei et al., 15 Jun 2026).

A different structural asymmetry appears in the odd/even decomposition of a cooperative game under set complementation. For $\pi$8,

$\pi$9

and the paper proves

$w(\pi)=0$0

This is not player asymmetry in the usual sense. The asymmetry lies in the set function’s anti-symmetric component under $w(\pi)=0$1. Paired sampling is then interpreted as orthogonalizing regression so that the irrelevant even component is filtered out, and OddSHAP exploits this by regressing only on odd-order Fourier basis functions (Fumagalli et al., 1 Feb 2026).

These works broaden the phrase “asymmetric Shapley values” beyond modified permutation measures. They show that asymmetry may reside in the induced game, the welfare environment, or the symmetry class of the set function, while the Shapley formula itself remains unchanged.

5. Computation and application domains

The causal-ASV literature has developed several concrete computational gains from asymmetry. A central theorem shows that for any model family $w(\pi)=0$2, ASV is polynomial-time computable under Naive Bayes distributions if and only if ordinary Shapley values are polynomial-time computable under product distributions. The proof exploits the fact that the Naive Bayes root must appear first in every topological ordering, collapsing the causal average to a product-distribution Shapley problem after conditioning on the root. For rooted directed trees, the paper introduces equivalence classes of topological orderings defined by identical predecessor sets of the explained feature and proves exact ASV computation in time polynomial in the graph size and the number of equivalence classes. Empirically, the Child network has $w(\pi)=0$3 topological orderings but only about $w(\pi)=0$4 equivalence classes on average over features, and $w(\pi)=0$5 sampled topological orders yield an average relative error around $w(\pi)=0$6 (Companeetz et al., 23 Jun 2026).

Data valuation imports the same asymmetry into sample-space settings. One formulation uses a weight system $w(\pi)=0$7, where $w(\pi)=0$8 is an ordered partition of the dataset and $w(\pi)=0$9 is a positive weight vector. Under the intra-class uniform weight system, admissible permutations are exactly the order-preserving permutations, uniform within classes, and the value for a point in class $$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$0 becomes an average marginal contribution conditional on all lower classes already being present. The paper proves class-wise efficiency,

$$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$1

and gives an exact KNN-based algorithm with complexity $$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$2 (Zheng et al., 2024).

A closely related formulation, Asymmetric Data Shapley, writes $$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$3 and averages only over

$$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$4

For $$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$5, the subset form is

$$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$6

with $$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$7. The paper proves group efficiency,

$$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$8

derives a Monte Carlo estimator with sample size

$$v_{f_y(x)}(S)=E_{p(x' \mid x_S)} \big[ f_y(x_S \sqcup x'_{\bar S}) \big],$$9

and gives an exact $w$0 KNN-ADS procedure per test point. The main applications are synthetic and augmented data valuation, federated learning, and multi-stage LLM fine-tuning (Zheng et al., 17 Nov 2025).

Across these computational strands, asymmetry is not merely a normative modification. It changes the combinatorics of admissible orderings, sometimes dramatically reducing the effective state space.

6. Limitations, controversies, and adjacent generalizations

The main controversy concerns causal or precedence-weighted ASV as a tool for root-cause analysis. A theoretical critique shows that relaxing symmetry can produce counter-intuitive attributions when nonlinear or non-additive interactions are present. Using a variance-reduction lens, the expected contribution under a permutation is

$w$1

so positive interaction terms can cause later or proximate variables to receive inflated attribution even under a distal ordering intended to prioritize root causes. The paper provides examples based on pairwise independence, quadratic nonlinearity, and multiplicative effects, and identifies generalized additive models as a restricted class in which many of these pathologies disappear (Kelen et al., 2023).

A second boundary concerns what should count as an asymmetric Shapley value at all. The contribution-based fair-division paper explicitly states that it does not study weighted Shapley values, asymmetric permutation measures, or alternative axiomatizations of the Shapley operator; it studies the standard symmetric Shapley value on an asymmetric welfare game (Bei et al., 15 Jun 2026). The odd/even decomposition paper likewise is not about weighted players or biased arrival orders; it is about the fact that the Shapley operator depends only on the complement-antisymmetric component of the set function (Fumagalli et al., 1 Feb 2026). In both cases, the asymmetry is real, but it is not asymmetry of the operator in the narrow cooperative-game-theoretic sense.

Several adjacent generalizations are therefore best read as structurally suggestive rather than as direct asymmetric-value theories. The Hodge-theoretic extension of Shapley axioms remains symmetric under relabeling and allocates a component game $w$2 to each player and coalition; the paper itself notes that weighted edges, nonuniform Markov chains, or modified symmetry axioms would be the natural route to asymmetry, but does not develop them (Lim, 2021). Likewise, the geometric theory of efficient symmetric linear values studies the space $w$3 of efficient, symmetric, linear value maps, showing that every such value has a unique stratified decomposition by coalition size; this is a theory of symmetric nonclassical values, not of player-nonanonymous Shapley values (Feys, 15 May 2026).

Accordingly, “asymmetric Shapley values” now denotes a cluster of related ideas rather than a single canonically defined object. In the narrowest sense, it refers to nonuniform permutation averages and weakened symmetry axioms. In a broader but established sense, it includes Shapley-type values induced by precedence constraints, graph structure, coalition feasibility, and heterogeneous welfare games. The technical distinction between these meanings is central, because it determines whether asymmetry is being introduced into the rule, into the admissible coalition-order geometry, or into the game on which the standard Shapley rule is evaluated.