A Hodge theoretic extension of Shapley axioms

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 given by Stern and Tettenhorst, thereby suggesting that the component values for every coalition state may also serve for a valid measure of fair allocation among the players in each coalition. Our axioms may be seen as a completion of Shapley's in view of this characterization of the Hodge-theoretic component games. In addition, we provide a path integral representation of the component games which may be seen as an extension of the {\em Shapley formula}.