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 DAMGs opens new application areas in machine learning, natural language processing, and explainable artificial intelligence.

翻译：Möbius反演和Shapley值是刻画与分解复杂系统中高阶结构的两种数学工具。前者通过偏序集上的离散导数定义高阶交互作用；后者则提供了一种将交互作用归因于系统原子元素的原则性方法。两者在组合数学、合作博弈论、机器学习及可解释人工智能等领域均有广泛应用。本文在两个正交方向上同时推广了这两种工具：1.从实值函数推广到任意阿贝尔群取值函数（特别地，向量值函数）；2.从偏序集与格推广到有向无环多重图（DAMG）及其加权版本。经典的公理体系——线性、效率、空局中人、对称性——在格上唯一刻画了Shapley值，但在本文更一般的设定下不再充分。我们通过引入投影算子将高阶协同效应递归地重新归因到图的根节点，并提出两个自然公理来解决这一问题：弱元素（零协同效应的联盟可被移除而不影响任何归因）与扁平层级（无中间层级的图中，归因按边数比例分配）。结合线性公理，这三个公理通过简洁的显式公式唯一确定了Shapley值，并自动蕴含效率、空局中人、对称性以及一个新的投影性质。本文构建的框架将所有基于格的现有定义作为特例囊括其中，并自然处理了此前无法触及的设定（如非格偏序集上的博弈）。对向量值函数与一般DAMG的拓展为机器学习、自然语言处理及可解释人工智能开辟了新的应用领域。