Two new classes of methods to share the cost of cleaning up a polluted river

Consider a group of agents located along a polluted river where every agent must pay a certain cost for cleaning up the polluted river. Following the model of Ni and Wang (2007), we propose weaker versions of some of their axioms, and introduce two classes of cost sharing methods. We first show that the upstream equal sharing method (for short, UES method) is characterized by relaxing independence of upstream costs and no blind cost appearing in Ni and Wang (2007). After that, we propose the classes of equal upstream responsibility methods (for short, EUR methods) and weighted upstream sharing methods (for short, WUS methods), which generalizes the local responsibility sharing method (for short, LRS method) and the UES method. We provide two axiomatizations of the class of EUR methods by replacing upstream symmetry for the UES method with weak upstream symmetry. Meanwhile, we also provide two axiomatizations of the class of WUS methods by introducing two other weak versions of upstream symmetry. Finally, we define a pollution cost-sharing game, and show that each of the EUR methods and WUS methods is obtained as a Harsanyi solution for these pollution cost-sharing games. Moreover, we also show that the average of the LRS method and UES method, referred to as the compromise method, coincides with the Shapley value of this game.