2021/07/22 by Tongseok Lim, Lim, Tongseok
Decision Sciences · Economics, Econometrics and Finance · #05C57 #68R01 #91A12 #Computer Science and Game Theory (cs.GT) #Economic theories and models #FOS: Computer and information sciences #FOS: Economics and business #FOS: Mathematics #Game Theory and Applications #Game Theory and Voting Systems #Mathematical Finance (q-fin.MF) #Probability (math.PR) #Theoretical Economics (econ.TH)
paper · pdf · doi:10.48550/arxiv.2107.10510
openalex publication_date 2021/07/22 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
This paper generalizes L.S. Shapley's celebrated value allocation theory on\ncoalition games by discovering and applying a fundamental connection between\nstochastic path integration driven by canonical time-reversible Markov chains\nand Hodge-theoretic discrete Poisson's equations on general weighted graphs.\n More precisely, we begin by defining cooperative games on general graphs and\ngeneralize Shapley's value allocation formula for those games in terms of\nstochastic path integral driven by the associated canonical Markov chain. We\nthen show the value allocation operator, one for each player defined by the\npath integral, turns out to be the solution to the Poisson's equation defined\nvia the combinatorial Hodge decomposition on general weighted graphs.\n Several motivational examples and applications are presented, in particular,\na section is devoted to reinterpret and extend Nash's and Kohlberg and Neyman's\nsolution concept for cooperative games. This and other examples, e.g. on\nrevenue management, suggest that our general framework does not have to be\nrestricted to cooperative games setup, but may apply to broader range of\nproblems arising in economics, finance and other social and physical sciences.\n