Lloyd Shapley's cooperative value allocation theory stands as a central concept in game theory, extensively utilized across various domains to distribute resources, evaluate individual contributions, and ensure fairness. The Shapley value formula and his four axioms that characterize it form the foundation of the theory. Traditionally, the Shapley value is assigned under the assumption that all players in a cooperative game will ultimately form the grand coalition. In this paper, we reinterpret the Shapley value as an expectation of a certain stochastic path integral, with each path representing a general coalition formation process. As a result, the value allocation is naturally extended to all partial coalition states. In addition, we provide a set of five properties that extend the Shapley axioms and characterize the stochastic path integral. Finally, by integrating Hodge calculus, stochastic processes, and path integration of edge flows on graphs, we expand the cooperative value allocation theory beyond the standard coalition game structure to encompass a broader range of cooperative network configurations.

翻译：劳埃德·夏普利的合作价值分配理论是博弈论中的核心概念，被广泛应用于资源分配、个体贡献评估及公平性保障等领域。夏普利值公式及其刻画的四个公理构成了该理论的基础。传统上，夏普利值的分配基于一个假设，即合作博弈中的所有参与者最终会形成大联盟。本文重新将夏普利值解释为某种随机路径积分的期望，其中每条路径代表一个一般的联盟形成过程。由此，价值分配自然扩展到所有部分联盟状态。此外，我们提出了五条性质，这些性质扩展了夏普利公理并刻画了随机路径积分。最后，通过融合霍奇演算、随机过程以及图上边流的路径积分，我们将合作价值分配理论从标准联盟博弈结构拓展至更广泛的合作网络构型。