Stochastic integral representation of solutions to Hodge theoretic Poisson's equations on graphs, and cooperative value allocation of Shapley and Nash

The fundamental connection between stochastic differential equations (SDEs) and partial differential equations (PDEs) has found numerous applications in diverse fields. We explore a similar link between stochastic calculus and combinatorial PDEs on graphs with Hodge structure, by showing that the solution to the Hodge-theoretic Poisson's equation on graphs allows for a stochastic integral representation driven by a canonical time-reversible Markov chain. When the underlying graph has a hypercube structure, we further show that the solution to the Poisson's equation can be fully characterized by five properties, which can be thought of as a completion of the Lloyd Shapley's four axioms