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 \emph{directed acyclic multigraphs} (DAMGs) and weighted versions thereof. The classical axioms, linearity, efficiency, null player, and symmetry, uniquely characterize Shapley values on lattices but are insufficient in this more general setting. We resolve this by introducing \emph{projection operators} that recursively re-attribute higher-order synergies down to the roots of the graph, and by proposing two natural axioms: \emph{weak elements} (coalitions with zero synergy can be removed without affecting any attribution) and \emph{flat hierarchy} (on graphs with no intermediate hierarchy, attributions are distributed proportionally to edge counts). Together with linearity, these axioms uniquely determine the Shapley values via a simple explicit formula, while automatically implying efficiency, null player, symmetry, and a novel \emph{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 DAMGs opens new application areas in machine learning, natural language processing, and explainable AI.

翻译：Möbius反演与Shapley值是描述和分解复杂系统中高阶结构的两种数学工具。前者将高阶交互定义为偏序集上的离散导数；后者提供了一种将交互归因于系统"原子"元素的原则性方法。两者已在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领域得到广泛应用。我们沿两个正交方向同时推广了这两种工具：(1)从实值函数推广到取值于任意阿贝尔群的函数（特别是向量值函数），(2)从偏序集与格推广到有向无环多重图(DAMG)及其加权版本。经典公理（线性性、有效性、零玩家与对称性）在格上唯一刻画了Shapley值，但在更一般的设定中并不充分。我们通过引入递归地将高阶协同归因至图根节点的投影算子，并提出两个自然公理：弱元素（零协同的联盟可删去而不影响任何归因）与扁平层级（无中间层级的图上，归因按边数比例分配），解决了这一问题。结合线性性，这些公理通过简洁的显式公式唯一确定了Shapley值，同时自动蕴含有效性、零玩家、对称性及新颖的投影性质。所得框架将现有关基于格的定义作为特例纳入，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）。向量值函数与DAMG的拓展为机器学习、自然语言处理与可解释人工智能开辟了新的应用领域。