  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.
