Hodge allocation for cooperative rewards: a generalization of Shapley's cooperative value allocation theory via Hodge theory on graphs

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.