Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs

Shapley values, a tool used in explainable AI to attribute model predictions to input features, can now be applied to more complex systems: The paper extends Shapley values beyond their traditional use on simple hierarchical structures to work with weighted directed acyclic multigraphs, potentially enabling fairer and more accurate explanations of AI systems with intricate dependencies between components.

New mathematical axioms are needed to ensure fair attribution in complex systems: The classical axioms that uniquely define Shapley values (linearity, efficiency, null player, and symmetry) are insufficient for complex structures. The authors propose two new axioms—"weak elements" and "flat hierarchy"—that better capture fairness principles when attributing credit or responsibility across interconnected components.

The framework handles vector-valued functions, not just single numbers: By generalizing from real-valued to vector-valued functions, this work enables attribution analysis in systems where outcomes have multiple dimensions (e.g., different types of model errors, multiple objectives, or multi-stakeholder impacts), which is more realistic for responsible AI applications.

Higher-order interactions can now be properly decomposed and understood: The combination of Möbius inversion with generalized Shapley values allows researchers to identify and attribute "synergies"—interactions between multiple components that produce effects beyond what individual components alone would suggest—which is crucial for understanding complex AI system behavior.