This paper establishes a complete theoretical foundation for the Hodge-theoretic extension of the Shapley value introduced by Stern and Tettenhorst (2019). We show that a set of five axioms--efficiency, linearity, symmetry, a modified null-player condition, and an independency principle--uniquely characterize this value across all coalitions, not just the grand coalition. In parallel, we derive a probabilistic representation interpreting each player's value as the expected cumulative marginal contribution along a random walk on the coalition graph. These dual axiomatic and probabilistic results unify fairness and stochastic interpretation, positioning the Hodge-theoretic value as a canonical generalization of Shapley's framework.

翻译：本文为Stern与Tettenhorst（2019）提出的Shapley值Hodge理论扩展建立了完整的理论基础。我们证明一组五条公理——有效性、线性性、对称性、修正零参与者条件以及独立性原理——在所有联盟（而不仅限于大联盟）上唯一确定了该值。同时，我们推导出概率表示，将每个参与者的值解释为联盟图上随机游走路径的期望累积边际贡献。这两组公理化与概率论结果统一了公平性与随机解释，确立了Hodge理论值作为Shapley框架的典范推广。