Lloyd Shapley's cooperative value allocation theory is a central concept in game theory that is widely used in various fields to allocate resources, assess individual contributions, and determine fairness. The Shapley value formula and his four axioms that characterize it form the foundation of the theory. Shapley value can be assigned only when all cooperative game players are assumed to eventually form the grand coalition. The purpose of this paper is to extend Shapley's theory to cover value allocation at every partial coalition state. To achieve this, we first extend Shapley axioms into a new set of five axioms that can characterize value allocation at every partial coalition state, where the allocation at the grand coalition coincides with the Shapley value. Second, we present a stochastic path integral formula, where each path now represents a general coalition process. This can be viewed as an extension of the Shapley formula. We apply these concepts to provide a dynamic interpretation and extension of the value allocation schemes of Shapley, Nash, Kohlberg and Neyman. This generalization is made possible by taking into account Hodge calculus, stochastic processes, and path integration of edge flows on graphs. We recognize that such generalization is not limited to the coalition game graph. As a result, we define Hodge allocation, a general allocation scheme that can be applied to any cooperative multigraph and yield allocation values at any cooperative stage.

翻译：Lloyd Shapley的合作价值分配理论是博弈论中的核心概念，广泛应用于资源分配、个体贡献评估及公平性判定等众多领域。Shapley值公式及其四个公理构成了该理论的基础。Shapley值仅在假设所有合作博弈参与者最终形成大联盟时方可分配。本文旨在将Shapley理论拓展至覆盖每一部分联盟状态下的价值分配。为此，我们首先将Shapley公理拓展为一组新的五个公理，这些公理能够刻画每一部分联盟状态下的价值分配，其中大联盟处的分配与Shapley值一致。其次，我们提出一个随机路径积分公式，其中每条路径表示一个通用联盟过程。这可视为Shapley公式的扩展。我们应用这些概念对Shapley、Nash、Kohlberg与Neyman的价值分配方案提供动态解释与拓展。这一推广通过引入Hodge微积分、随机过程及图上边流的路径积分得以实现。我们认识到此类推广并不局限于联盟博弈图。因此，我们定义了Hodge分配——一种可应用于任意合作多重图并在任一合作阶段产出分配值的通用分配方案。