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