Axiomatizations of the Shapley Value for Upstream Responsibility Games

In this paper the problem of sharing the cost of emission in supply chains are considered. We 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 (Gopalakrishnan et al., 2017). The formal notion of upstream responsibility games is introduced, and the characterization of the class of these games is provided. The Shapley value (Shapley, 1953) is probably the most popular value for transferable utility cooperative games. In Radvanyi (2018b) we showed that Shapley's (Shapley, 1953) and Young's (Young, 1985) axiomatizations of the Shapley value are valid on the class of upstream responsibility games. According to Gopalakrishnan et al. (2017) we introduce some pollution related properties and we relate them to the TU games terminologies.