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：劳埃德·沙普利的合作价值分配理论是博弈论的核心概念，广泛应用于资源分配、个体贡献评估与公平性判定等领域。沙普利值公式及其刻画该值的四个公理构成了该理论的基础。然而，沙普利值仅在所有合作博弈参与者最终形成大联盟的假设下才能被分配。本文旨在将沙普利理论推广至每个局部联盟状态下的价值分配。为实现这一目标，我们首先将沙普利公理扩展为一组新的五个公理，这些公理能够刻画每个局部联盟状态下的价值分配，其中大联盟下的分配与沙普利值一致。其次，我们提出一种随机路径积分公式，其中每条路径现代表一个一般性的联盟过程，可视为沙普利公式的推广。我们将这些概念应用于沙普利、纳什、科尔伯格和奈曼的价值分配方案，提供其动态解释与扩展。这一推广得益于在图上的Hodge微积分、随机过程及边流路径积分的引入。我们认识到该推广并不局限于合作博弈图。由此，我们定义了Hodge分配——一种可应用于任意合作多重图、并在任意合作阶段产生分配值的通用分配方案。