M\"obius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs
Pith reviewed 2026-05-18 09:37 UTC · model grok-4.3
The pith
Linearity with weak-elements and flat-hierarchy axioms fixes Shapley values on weighted directed acyclic multigraphs for any abelian-group-valued function.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Linearity together with the weak-elements and flat-hierarchy axioms uniquely determine the Shapley values on weighted DAMGs via a simple explicit formula, automatically implying efficiency, null-player, symmetry, and a novel projection property while recovering all lattice-based definitions as special cases.
What carries the argument
Projection operators that recursively re-attribute higher-order synergies down to the roots of the graph, used with the weak-elements and flat-hierarchy axioms to complete the axiomatic characterization.
If this is right
- The Shapley values satisfy the efficiency, null-player, symmetry, and projection properties.
- Every lattice-based Shapley value definition is recovered when the multigraph is a lattice.
- The method applies to games defined on non-lattice partial orders.
- Attributions can be computed for vector-valued functions, supporting multi-output settings in machine learning.
Where Pith is reading between the lines
- This opens attribution methods for acyclic computation graphs in neural networks where interactions do not form a lattice.
- Testable extensions include applying the formula to concrete weighted multigraphs arising in natural language parsing hierarchies.
Load-bearing premise
The projection operators that recursively re-attribute higher-order synergies down to the roots of the graph are well-defined and consistent for arbitrary weighted directed acyclic multigraphs and abelian-group-valued functions.
What would settle it
A weighted directed acyclic multigraph together with an abelian-group-valued function on which the explicit formula violates the flat-hierarchy axiom or the weak-elements axiom would show the uniqueness claim is false.
read the original abstract
M\"obius inversion and Shapley values are two mathematical tools for characterizing and decomposing higher-order structure in complex systems. The former defines higher-order interactions as discrete derivatives over a partial order; the latter provides a principled way to attribute those interactions back to the `atomic' elements of the system. Both have found wide application, from combinatorics and cooperative game theory to machine learning and explainable AI. We generalize both tools simultaneously in two orthogonal directions: 1) from real-valued functions to functions valued in any abelian group (in particular, vector-valued functions), and 2) from partial orders and lattices to directed acyclic multigraphs (DAMGs) and weighted versions thereof. The classical axioms, linearity, efficiency, null player, and symmetry, which uniquely characterize Shapley values on lattices, are insufficient in this more general setting. We resolve this by introducing projection operators that recursively re-attribute higher-order synergies down to the roots of the graph, and by proposing two natural axioms: weak elements (coalitions with zero synergy can be removed without affecting any attribution) and flat hierarchy (on graphs with no intermediate hierarchy, attributions are distributed proportionally to edge counts). Together with linearity, these three axioms uniquely determine the Shapley values via a simple explicit formula, while automatically implying efficiency, null player, symmetry, and a novel projection property. The resulting framework recovers all existing lattice-based definitions as special cases, and naturally handles settings, such as games on non-lattice partial orders, which were previously out of reach. The extension to vector-valued functions and general DAMG-structured hierarchies opens new application areas in machine learning, natural language processing, and explainable AI.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper generalizes Möbius inversion and Shapley values simultaneously to abelian-group-valued (including vector-valued) functions on weighted directed acyclic multigraphs (DAMGs). It introduces projection operators that recursively re-attribute higher-order synergies to graph roots, together with two new axioms (weak elements: zero-synergy coalitions can be removed without affecting attributions; flat hierarchy: on graphs without intermediate levels, attributions are distributed proportionally to edge counts) plus linearity. These three axioms are claimed to uniquely determine an explicit formula for the Shapley values, which automatically satisfies efficiency, null-player, symmetry, and a novel projection property, while recovering all prior lattice-based definitions as special cases and extending to previously unreachable non-lattice partial orders.
Significance. If the central uniqueness result and well-definedness of the projection operators hold for arbitrary abelian groups, the framework supplies a parameter-free, axiomatically grounded extension of attribution methods beyond lattices. This would enable principled higher-order interaction analysis in settings with vector payoffs and general DAG hierarchies, with direct relevance to explainable AI and cooperative game theory on structured domains. The recovery of existing results as special cases and the explicit formula are notable strengths.
major comments (2)
- [§4.2, Theorem 4.3] §4.2, Definition of projection operators and Theorem 4.3: The recursive re-attribution step invokes division by edge weights (or proportional distribution under the flat-hierarchy axiom) inside an arbitrary abelian group G. For G = ℤ/2ℤ and an even weight (e.g., a flat multiedge of weight 2), the equation 2x = y may have no solution or fail to be unique, so the operators are not guaranteed to be well-defined or to produce a unique attribution satisfying the axioms for every function. This directly threatens the uniqueness claim for the explicit formula on every weighted DAMG and every abelian-group-valued function.
- [§5.1] §5.1, proof of uniqueness: The argument that linearity + weak elements + flat hierarchy imply the explicit formula must explicitly construct the inverse of the projection step or restrict the class of groups/weights; without this, the claim that the three axioms characterize a unique solution on the full domain does not follow.
minor comments (2)
- [§2.1] §2.1: The definition of weighted DAMGs would benefit from a small concrete example (e.g., a two-edge flat multigraph) to illustrate how weights interact with the abelian-group codomain.
- [Notation] Notation: The symbol for the projection operator is introduced without an explicit comparison table to the classical Möbius function; adding one would clarify the generalization.
Simulated Author's Rebuttal
We thank the referee for their careful reading and for highlighting these important technical points concerning well-definedness over arbitrary abelian groups. We address each major comment below.
read point-by-point responses
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Referee: [§4.2, Theorem 4.3] §4.2, Definition of projection operators and Theorem 4.3: The recursive re-attribution step invokes division by edge weights (or proportional distribution under the flat-hierarchy axiom) inside an arbitrary abelian group G. For G = ℤ/2ℤ and an even weight (e.g., a flat multiedge of weight 2), the equation 2x = y may have no solution or fail to be unique, so the operators are not guaranteed to be well-defined or to produce a unique attribution satisfying the axioms for every function. This directly threatens the uniqueness claim for the explicit formula on every weighted DAMG and every abelian-group-valued function.
Authors: The referee correctly identifies a subtlety in the definition of the projection operators. When G has torsion and a weight w does not admit a unique solution to w·x = y, the recursive step is not guaranteed to be well-defined on the entire domain. We will revise §4.2 and Theorem 4.3 to state explicitly that the operators and the resulting Shapley values are defined for those weighted DAMGs and groups G in which every relevant weight is invertible (or, equivalently, when the linear equations arising in the recursion admit unique solutions). Under this standing assumption the explicit formula remains well-defined and the uniqueness claim holds on the restricted domain. We will also add a short remark clarifying that the framework applies directly to the common cases of real- or rational-valued functions and to torsion-free groups. revision: yes
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Referee: [§5.1] §5.1, proof of uniqueness: The argument that linearity + weak elements + flat hierarchy imply the explicit formula must explicitly construct the inverse of the projection step or restrict the class of groups/weights; without this, the claim that the three axioms characterize a unique solution on the full domain does not follow.
Authors: We agree that the current proof sketch in §5.1 does not explicitly exhibit the inverse of each projection step for arbitrary G. We will expand the uniqueness argument by first proving a preliminary lemma that, under the invertibility condition introduced in the revision of §4.2, each projection operator is bijective on the space of functions satisfying the axioms. The three axioms then determine the value on the roots by inverting these operators step by step, yielding the explicit formula. The revised proof will therefore characterize uniqueness precisely on the domain where the operators are well-defined. revision: yes
Circularity Check
No circularity: new axioms and operators yield independent characterization
full rationale
The derivation introduces projection operators and two new axioms (weak elements, flat hierarchy) alongside linearity to characterize Shapley values on weighted DAMGs for abelian-group-valued functions. The explicit formula is presented as the unique solution satisfying these axioms, with lattice-based Shapley values recovered only as special cases when the graph reduces to a lattice. No load-bearing step reduces by construction to a prior self-citation, fitted parameter, or tautological renaming; the central uniqueness claim rests on the newly stated axioms rather than re-deriving an input quantity. The framework is self-contained against external benchmarks such as classical cooperative game theory results.
Axiom & Free-Parameter Ledger
axioms (3)
- standard math Linearity
- ad hoc to paper Weak elements: coalitions with zero synergy can be removed without affecting any attribution
- ad hoc to paper Flat hierarchy: on graphs with no intermediate hierarchy, attributions are distributed proportionally to edge counts
invented entities (1)
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Projection operators
no independent evidence
discussion (0)
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