  Lloyd S. Shapley \cite{Shapley1953a, Shapley1953} introduced a set of axioms
in 1953, now called the {\em Shapley axioms}, and showed that the axioms
characterize a natural allocation among the players who are in grand coalition
of a {\em cooperative game}. Recently, \citet{StTe2019} showed that a
cooperative game can be decomposed into a sum of {\em component games}, one for
each player, whose value at the grand coalition coincides with the {\em Shapley
value}. The component games are defined by the solutions to the naturally
defined system of least squares linear equations via the framework of the {\em
Hodge decomposition} on the hypercube graph.
  In this paper we propose a new set of axioms which characterizes the
component games. Furthermore, we realize them through an intriguing stochastic
path integral driven by a canonical Markov chain. The integrals are natural
representation for the expected total contribution made by the players for each
coalition, and hence can be viewed as their fair share. This allows us to
interpret the component game values for each coalition also as a valid measure
of fair allocation among the players in the coalition. Our axioms may be viewed
as a completion of Shapley axioms in view of this characterization of the
Hodge-theoretic component games, and moreover, the stochastic path integral
representation of the component games may be viewed as an extension of the {\em
Shapley formula}.

Optimization and Control;Computer Science and Game Theory;Theoretical Economics