The Equal Share Proportional Solution for the River Sharing Problem

Abstract

This paper considers the river sharing problem first studied in Ambec, S. and Sprumont, Y. [2002] Sharing a River, J. Econ. Theory 107, 453–462. We use the Equal Share Proportional Solution (ESPS) for the permit sharing problem introduced in Suh, S. and Wang, Y. [2023] The equal share proportional solution in a permit sharing problem, Soc. Choice Welf. 60, 477–501 to define a solution, also called the ESPS, for the river sharing problem. We first show that a river sharing problem can be divided into a list of subproblems, each of which can be considered as a permit sharing problem (Decomposition Lemma). Then, we apply the ESPS solution to each of the subproblems. The ESPS for the river sharing problem is the aggregation of the ESPS for all the subproblems. We also compare the ESPS with the well-known Downstream Incremental Distribution solution (DID) by Ambec, S. and Sprumont, Y. [2002] Sharing a River, J. Econ. Theory 107, 453–462. We show that for a dummy agent whose optimal consumption coincides with his initial endowment, the agent obtains his stand-alone benefit in the ESPS. In contrast, the DID solution may assign welfare levels to dummy agents that are higher than their stand-alone benefits. On the other hand, the ESPS violates the aspiration upper bounds.