Lloyd Shapley’s cooperative value allocation theory is a central concept in game theory that is widely applied in various fields to assess individual contributions and allocate resources. The Shapley value formula and his four defining axioms form the foundation of the theory.

We interpret the Shapley value as an expectation of a certain stochastic path integral, with each path representing a general coalition process. As a result, the value allocation is naturally extended to all partial coalition states. Furthermore, the new allocation scheme can be readily generalized by path-integrating various edge flows, which we refer to as the f-Shapley value. Finally, by employing Hodge theory on graphs, we show how to compute the stochastic path integral via the graph Poisson equation.