We consider the problem of sharing water among agents located along a river, who have quasi-linear preferences over water and money, as introduced by Ambec and Sprumont (2002). Given an efficient distribution of river water, where water can be sent from upstream agents to downstream agents but not the other way around, the question is what should be the monetary transfers that downstream agents have to pay to upstream agents as compensation for the upstream agents to abstain from water consumption. Under the quasi-linear utility functions, an efficient water allocation and a transfer scheme determine a welfare distribution.

Under more general benefit functions, van den Brink, Estevez-Fernandez, van der Laan and Moes (2014) consider three basic axioms for welfare distribution and, additionally, add an independence axiom with respect to benefit functions of upstream, respectively, downstream agents. Both independence axioms yield a unique welfare distribution. In this paper we investigate the impact of similar independence axioms but with respect to water inflows. Surprisingly, together with the three basic axioms, these do not characterize a welfare distribution. Moreover, independence of upstream inflows turns out to be incompatible with the three basic axioms, while independence of downstream inflows yields mutliple solutions. We weaken one of the basic axioms to get compatibility with upstream independence, and then strengthen it to get uniqueness with downstream independence.