The river sharing problem: a survey (Q2852566)

This article gives a summary of the main solutions offered in the literature of game theory to the problem of sharing a river, from an axiomatic point of view or through the use of market mechanisms. Existing contributions from the theory of noncooperative games are not considered here.NEWLINENEWLINEIn a problem of sharing a river, optimal allocations of water among riparians are studied and also the subsequent distribution of welfare achieved as a result thereof. This is a problem that arises in real life and it is the origin of laws that are sometimes difficult to apply.NEWLINENEWLINEUnder certain natural assumptions on the problem, \textit{S. Ambec} and \textit{Y. Sprumont} in [J. Econ. Theory 107, No. 2, 453--462 (2002; Zbl 1033.91503)] proved that there exists a unique optimal consumption plan and they studied its properties. The optimal consumption leads to an optimal social welfare that takes into account the benefit to each agent originated by the amount of water consumed. From the optimal consumption, they defined a cooperative game with transferable utility that is convex and it is the starting point to find a fair distribution of the optimal social welfare.NEWLINENEWLINEA first solution is to choose vectors of the core of the game that guarantee to each group of the agents absolute right over the water flowing in its territory. Ambec and Sprumont also proved that certain marginal vectors, which are core elements, also respect a second principle: that the quantity and quality of water available for an agent cannot be altered by another. Other authors also studied other properties of different marginal vectors.NEWLINENEWLINEIn the case of cooperative games for allocating a river with multiple springs, a relevant solution with natural properties also happens to be an extreme point of the core. In addition, we can obtain a dual result for the case of a river with a delta, as a river with multiple springs is shaped like a sink-tree, a river with a delta is shaped like a rooted-tree, and a sink-tree is obtained from a rooted-tree by changing the orientation of each edge.NEWLINENEWLINEFor the case with several springs, it is also studied the principle that the surplus generated by the mixture of two disjoint coalitions is divided between them in proportion to the sum of certain weights of these two coalitions. In this case, it is also selected as a satisfactory solution a single element of the core of the game.NEWLINENEWLINEAnother interesting situation is the one in which the benefits of water consumption are defined from single-peaked profit functions. In this case, each agent will never consume more than their point of satiation and it can be generated, in a coalition, externalities in a certain sense. Again, in this context it is selected as the only desirable solution verifying desirable properties a marginal vector of a certain game.NEWLINENEWLINEAn alternative approach uses a market mechanism. It considers the social interaction between river users and it achieves a Pareto optimal distribution of welfare.NEWLINENEWLINEWith respect to the extensions, there have been considered problems with multiple sources, a delta and islands and the case in which each agent represents a community of users located along the river. Moreover, it is also developed an approach from models of bankruptcy by assuming that each agent along the river has a claim to the river flow and that the sum of downstream claims exceeds the sum of downstream endowments at each location.NEWLINENEWLINEFinally, some authors have considered that at times there are different sources that cause pollution at each location along the river, and this fact reduces water quality. Since we can estimate the cost of preventing pollution at each location, now it emerges the problem of how to allocate the total cost. In this new context, different rules have been characterized axiomatically, some of which are closely related to the well-known Shapley value.