Upstream responsibility games – the non-tree case

In this paper the problem of sharing the cost of emission in supply chains is considered. (Gopalakrishnan et al, 2017) focus on allocation problems that can be described by rooted trees, called cost-tree problems, and on the induced transferable utility cooperative games, called upstream responsibility games. This paper generalizes the formal notion of upstream responsibility games to a non-tree model, and provides two (primal and dual) characterizations of the class of these games. Axiomatizations of the Shapley value under both characterizations are also provided. This is a followup paper of Radványi (2018); Pintér and Radványi (2019)