Allocating the costs of cleaning a river; estimating responsibilities versus incentive compatibility

We model a river as a segment divided into subsegments, each occupied by one region, from upstream to downstream. The waste is transferred from one region to the next at some rate t. Given that t may be unknown, Alcalde-Unzu et al. (2015) proposed the UR method to allocate the costs of cleaning a river in which each region pays the responsibility that it would have if t was its expected value. We prove in this paper that this allocation di ers from the expected responsibility of each region and propose and characterize a new solution, the EUR, that assigns to each region precisely its expected responsibility. We show that although this new solution improves the estimation of responsibilities, there is a trade-o in terms of incentive compatibility: meanwhile with the EUR solution it is possible that a region can reduce the cost allocated to it by discharging more waste, this could not happen with the UR. Moreover, the UR solution is, between the ones that do not have this problem in a general family, the solution that minimizes the bias with respect to the EUR.