Collusion in the polluted river problem

We study the effects of collusion in the polluted river problem. In this model, agents along a river network have to share the costs of cleaning the entire river. We define a collusion neutrality property which requires that, if all agents upstream of a given point agree to collude and act as one, then the cost to be paid by this new entity equals the sum of the costs originally allocated to the colluding agents. This property is related to the Unlimited Territorial Integrity doctrine used to settle international disputes over water streams flowing through several territories. We introduce a new cost sharing method, called the Upstream Recursive Equal Sharing (URES) method, and axiomatically characterize it using our new collusion neutrality axiom. We also provide a new characterization for the well-known Local Responsibility Sharing (LRS) method that uses this axiom and which differs from the URES method only in one property that explains how costs are shared at the source of the river. Moreover, we derive and characterize a family of collusion neutral methods where the different methods are distinguished by a parametrized clean source property that balances between Unlimited Territorial Integrity and Absolute Territorial Sovereignty and that contains the LRS method as an extreme and the URES method as an intermediate method. Finally, we introduce and characterize the other extreme method in this class, which we call the limit recursive sharing method. All members of the family can be computed by means of a recursive algorithm.