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M\"obius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs
Patrick Forr\'e, Abel Jansma
https://arxiv.org/abs/2510.05786

Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs
We generalize the concept of Möbius inversion and Shapley values to directed acyclic multigraphs and weighted versions thereof. We further allow value functions (games) and thus their Möbius transforms (synergy function) and Shapley values to have values in any abelian group that is a module over a ring that contains the graph weights, e.g. vector-valued functions. To achieve this and overcome the obstruction that the classical axioms (linearity, efficiency, null player, symmetry) are not str…