Sharing a Polluted River under Waste Flow Control

When the cleaning up of a polluted transboundary river requires the cooperation of several agents (countries, regions, firms or cities) located along it, a challenging issue is how should the pollutant-cleaning costs be shared among them. An important factor ignored by literature so far concerns the ability for wastewater treatment of the river itself depending on both sediment types and ecological units (hydrophyte filter beds, aerobic digesters) in order to control waste flow from upstream to downstream. First, we introduce and implement a new cost sharing method for polluted river problems under waste ow control, called the Downstream Compensation method, which combines the two well-known conflicting theories in international river disputes, namely the Absolute Territorial Sovereignty and the Unlimitted Territorial Integrity. When the river does not have any wastewater treatment ability, the Downstream Compensation method coincides with the Downstream Equal Sharing method. At the other extreme case of full wastewater treatment within the river, the Downstream Compensation method corresponds to the Local Responsibility Sharing method. Second, we show that the Downstream Compensation method is obtained as the Shapley value of appropriately defined cooperative games with transferable utility. Finally, we prove that these games satisfy the concavity property, meaning that the proposed cost allocation scheme belongs to the core.