Sharing a Polluted River Network

Abstract

number of agents (e.g., firms, villages, municipalities, or countries) are connected to a river network. Some agents are located upstream and some downstream. Upstream agents pollute the river network. To clean up the polluted river network, costs are incurred, and they are shared among the agents. In allocating these costs, the upstream-downstream relationship between agents has to be taken into account because pollution from upstream agents affects downstream agents' costs. In general, it is difficult to measure the costs an upstream agent imposes on downstream agents. Moreover, the property rights over flowing water are not well-defined. We model the problem as a cost sharing problem on a tree network with a special node (called lake). We propose three different cost sharing methods for the problem. They are the Local Responsibility Sharing (LRS), the Upstream Equal Sharing, and the Downstream Equal Sharing, respectively. The first two methods generalize Ni and Wang (Sharing a polluted river. Games Econ. Behav., in Press, 2006), while the third generalizes Littlechild and Owen's Shapley value solution for the well-known airport landing fee problem (Littlechild, S. and Owen, G., 1973. A Simple Expression for the Shapley Value in a Special Case. Management Sci. 20, 370-372). We provide axiomatic characterizations for the three methods. We also provide a game-theoretic analysis. We introduce three different games for the problem: the stand-alone game, the Upstream-oriented game and the Downstream-oriented game. We show that the above three methods coincide with the Shapley values of these three games. We also show that they are in the core of the corresponding games.