Hodge Theoretic Reward Allocation for Generalized Cooperative Games on Graphs

We define cooperative games on general graphs and generalize Lloyd S. Shapley's celebrated allocation formula for those in terms of stochastic path integral driven by the associated Markov chain each graph. then show that value operator, one player defined integral, coincides with player's component game which is solution to least squares (or Poisson's) equation, light combinatorial Hodge decomposition weighted graphs. Several motivational examples applications are also presented.