An extension of the Myerson-Satterthwaite theorem with an application to the river sharing problem

This paper revisits the Myerson-Satterthwaite theorem. We consider a bilateral trading problem in which one seller and one buyer have single-peaked preferences on a divisible good. The seller's peak demand and the buyer's peak demand are their private information. We show that for a large family of these bilateral trading problems, there exist trading mechanisms that are incentive-compatible, individually rational, budget balanced, and ex post efficient. We provide a necessary and sufficient condition on the initial distribution of the good between the seller and the buyer under which these trading mechanisms exist. As an application, we consider a new type of river sharing problems: the river sharing problem with incomplete information. In particular, we consider a river sharing problem between two agents located along a river, in which both agents are satiable at certain amounts of water consumption and these satiation amounts are their private information. We show that the efficient trading of water between the two agents can be achieved if and only if the upstream agent's endowment of water exceeds the downstream agent's endowment by a sufficient amount.