Möbius 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.

翻译：墨比乌斯反演和沙普利值是刻画与分解复杂系统中高阶结构的两种数学工具。前者将高阶相互作用定义为偏序集上的离散导数；后者则提供了一种将此类相互作用归因于系统"原子"成分的原则性方法。两者已被广泛应用于从组合数学与合作博弈论到机器学习与可解释人工智能等多个领域。我们在两个正交方向上同时对这两种工具进行推广：1) 从实值函数推广到任意阿贝尔群取值函数（特别是向量值函数）；2) 从偏序集和格推广到有向无环多重图及其加权版本。在格上唯一表征沙普利值的经典公理——线性性、效率、零玩家和对称性——在此更一般设定下是不够的。我们通过引入投影算子（递归地将高阶协同归因到图的根节点）并提出两个自然公理——弱元素（零协同的联盟可被移除而不影响任何归因）和平坦层级（在无中间层级的图上，归因按边数比例分配）——来解决此问题。结合线性性，这三个公理通过简洁的显式公式唯一确定沙普利值，同时自动蕴含效率、零玩家、对称性以及一个新颖的投影性质。该框架将所有已有的基于格的定义作为特例恢复，并自然地处理此前无法触及的设定（如非格偏序集上的博弈）。对向量值函数和一般DAMG结构层级的拓展，为机器学习、自然语言处理和可解释人工智能开辟了新的应用领域。