The pipeline externalities problem

We consider a set of users who are located along a pipeline with a single source. These users consume a good that is extracted from the source and flows downstream, with diminishing marginal returns for each user. In addition, flows along each edge in the pipeline create negative externalities, which are nondecreasing as a function of flow. The users cooperate toward obtaining group welfare maximization. In both the continuous and discrete cases, we obtain the group optimal solutions, and we then use cooperative game theory to determine how best to allocate the damages, using optimistic and pessimistic formulations for the characteristic function. Using core stability as our guiding principle, we provide a set of stable allocations that apportions the damages at a location among the set of downstream users, notably an average damage allocation and a marginal damage allocation. Given that the joint optimization forces agents to reduce (unequally) their consumption, we also examine the Shapley value of the optimistic game, also in the core, that allows to compensate agents who have sacrificed their consumption for the benefit of the group. Finally, we show that our pipeline externalities model generalizes some well-known problems from the literature, including the river sharing problem of Ambec and Sprumont 2002 and the joint production problem of Moulin and Shenker 1992.