Hodge theoretic reward allocation for generalized cooperative games on graphs

Abstract: 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.