  This paper generalizes L.S. Shapley's celebrated value allocation theory on
coalition games by discovering and applying a fundamental connection between
stochastic path integration driven by canonical time-reversible Markov chains
and Hodge-theoretic discrete Poisson's equations on general weighted graphs.
  More precisely, we begin by defining cooperative games on general graphs and
generalize Shapley's value allocation formula for those games in terms of
stochastic path integral driven by the associated canonical Markov chain. We
then show the value allocation operator, one for each player defined by the
path integral, turns out to be the solution to the Poisson's equation defined
via the combinatorial Hodge decomposition on general weighted graphs.
  Several motivational examples and applications are presented, in particular,
a section is devoted to reinterpret and extend Nash's and Kohlberg and Neyman's
solution concept for cooperative games. This and other examples, e.g. on
revenue management, suggest that our general framework does not have to be
restricted to cooperative games setup, but may apply to broader range of
problems arising in economics, finance and other social and physical sciences.
