Fair river sharing is a kind of a fair division problem in which the waters of a river has to be divided among countries located along the river. It differs from other fair division problems in that the resource to be divided—the water—flows in one direction—from upstream countries to downstream countries. To attain any desired division, it may be required to limit the consumption of upstream countries, but this may require to give these countries some monetary compensation.
In addition to sharing river water, which is an economic good, it is often required to share river pollution (or the cost of cleaning it), which is an economic bad.
River sharing in practice There are 148 rivers in the world flowing through two countries, 30 through three, nine through four and 13 through five or more. Some notable examples are: The Jordan River, whose sources run from upstream Lebanon and Syria to downstream Israel and Jordan. The attempts of Syria to divert the Jordan River, starting in 1965, are cited as one of the reasons for the Six-Day War. Later, in 1994, the Israel–Jordan peace treaty determined a sharing of the waters between Israel and Jordan, by which Jordan receives water per year.See Jordan River#Importance as a water source The Nile, running from upstream Ethiopia through Sudan to downstream Egypt. There is a long history of disputes over the Nile agreements of 1929 and 1959. The Ganges, running from upstream India to downstream Bangladesh. There was controversy over the operation of the Farakka Barrage. Between Mexico and the United States, there was controversy over the desalination facility in the Morelos Dam. The Mekong runs from China's Yunnan Province to Myanmar, Laos, Thailand, Cambodia, and Vietnam. In 1995, Laos, Thailand, Cambodia, and Vietnam established the Mekong River Commission to assist in the management and coordinated use of the Mekong's resources. In 1996 China and Myanmar became "dialogue partners" of the MRC and the six countries now work together within a cooperative framework.
Property rights In the international law, there are several conflicting views on the property rights to the river waters. The theory of absolute territorial sovereignty (ATS) states that a country has absolute property rights over any river basin in its territory. So any country may consume some or all of the waters that enter its area, without leaving any water to downstream countries. The theory of unlimited territorial integrity (UTI) states that a country shares the property rights to all the waters from the origin of the river down to its territory. So, a country may not consume all the waters in its territory, since this hurts the right of downstream countries. The theory of territorial integration of all basin states (TIBS) states that a country shares the property rights to all the waters of the river. So each country is entitled to an equal share of the river waters, regardless of its geographic location.
Efficient water allocation Kilgour and Dinar were the first to suggest a theoretical model for efficient water sharing.
The model The countries are numbered according to their location, so that country 1 is the most upstream, then 2, etc. The river picks up volume along its course: before each location , an amount of water enters the river. So, country 1 gets water, country 2 gets plus the water unconsumed by country 1, and so on. Each country has a benefit function that describes its utility from each amount of water. This function is increasing but strictly concave function, since the countries have diminishing returns. We can define for each country its marginal benefit function , which describes the price it is willing to pay for an additional unit of water given its current consumption; it is positive but strictly decreasing. Money can be transferred between countries. Countries have quasilinear utility, so a country who consumes water and receives money has utility . A consumption plan is a vector of water allocations and side-payments . The important aspect of the river sharing setting is that water only flows downstream. Therefore, the total consumption at each location must be at most the total amount of water that enters this location: . Additionally, the sum of the side-payments must be at most 0, so that the divider does not have to subsidize the division.
The situation without cooperation Without cooperation, each country maximizes its individual utility. So if a country is an insatiable agent (its benefit function is always increasing), it will consume all the water that enters its region. This may be inefficient. For example, suppose there are two countries with the following benefit functions:
The inflow is . Without cooperation, country 1 will consume 2 units and country 2 will have 0 units: . Then, the benefits will be . This is not Pareto efficient: it is possible to allocate 1 unit of water to each country: , and transfer e.g. units of money from country 2 to country 1. Then, the utilities will be which are better for both countries.
The efficient allocation Because preferences are quasi-linear, an allocation is Pareto-efficient if-and-only-if it maximizes the sum of all agents' benefits and wastes no money. Under the assumption that benefit functions are strictly concave, there is a unique optimal allocation. It structure is simple. Intuitively, the optimal allocation should equalize the marginal benefits of all countries (as in the above example). However, this may be impossible because of the structure of the river: the upstream countries do not have access to downstream waters. For example, in the above two-country example, if the inflow is , then it is not possible to equalize the marginal benefits, and the optimal allocation is to let each country consume its own water: .
Therefore, in the optimal allocation, the marginal benefits are weakly decreasing. The countries are divided to consecutive groups, from upstream to downstream. In each group, the marginal benefit is the same, and between groups, the marginal benefit is decreasing.
The possibility of calculating an optimal allocation allows much more flexibility in water-sharing agreements. Instead of agreeing in advance on fixed water quantities, it is possible to adjust the quantities to the actual amount of water that flows through the river each year. The utility of such flexible agreements has been demonstrated by simulations based on historical of the Ganges flow. The social welfare when using the flexible agreement is always higher than when using the optimal fixed agreement, but the increase is especially significant in times of drought, when the flow is below the average.
Stable monetary transfers Calculating the efficient water allocation is only the first step in solving a river-sharing problem. The second step is calculating monetary transfers that will incentivize countries to cooperate with the efficient allocation. What monetary transfer vector should be chosen? Ambec and Sprumont study this question using axioms from cooperative game theory.
Cooperation when countries are non-satiable According to the ATS doctrine, each country has full rights to the water in its region. Therefore, the monetary payments should guarantee to each country at least the utility-level that it could attain on its own. With non-satiable countries, this level is at least . Moreover, we should guarantee to each coalition of countries, at least the utility-level that they could attain by the optimal allocation among the countries in the coalition. This implies a lower bound on the utility of each coalition, called the core lower bound.
According to the UTI doctrine, each country has rights to all water in its region and upstream. These rights are not compatible since their sum is above the total amount of water. However, these rights define an upper bound - the largest utility that a country can hope for. This is the utility it could get alone, if there were no other countries upstream: . Moreover, the aspiration level of each coalition of countries is the highest utility-level it could attain in the absence of the other countries. This implies an upper bound on the utility of each coalition, called the aspiration upper bound.
There is at most one welfare-distribution that satisfies both the core-lower-bound and the aspiration-upper-bound: it is the downstream incremental distribution. The welfare of each country should be the stand-alone value of the coalition minus the stand-alone value of the coalition .
When the benefit functions of all countries are non-satiable, the downstream-incremental-distribution indeed satisfies both the core-lower-bounds and the aspiration-upper-bounds. Hence, this allocation scheme can be seen as a reasonable compromise between the doctrines of ATS and UTI.
Cooperation when countries are satiable When the benefit functions are satiable, new coalitional considerations come into play. These are best illustrated by an example.
Suppose there are three countries. Countries 1 and 3 are in a coalition. Country 1 wants to sell water to country 3 in order to increase their group welfare. If country 2 is non-satiable, then 1 cannot leave water to 3, since it will be entirely consumed by 2 along the way. So 1 must consume all its water. In contrast, if country 2 is satiable (and this fact is common knowledge), then it may be worthwhile for 1 to leave some water to 3, even if some of it will be consumed by 2. This increases the welfare of the coalition, but also the welfare of 2. Thus, cooperation is helpful not only for the cooperating countries, but also for the non-cooperating countries!
With satiable countries, each coalition has two different core-lower-bounds: The non-cooperative core-lower-bound is the value that the coalition can guarantee to itself based on its own water sources, when the other countries do not cooperate. The cooperative core-lower-bound is the value that the coalition can guarantee to itself based on its own water sources, when the other countries cooperate. As illustrated above, the cooperative core-lower-bound is higher than the non-cooperative core-lower-bound.
The non-cooperative-core is non-empty. Moreover, the downstream-incremental-distribution is the unique solution that satisfies both the non-cooperative-core-lower-bounds and the aspiration-upper-bound.
However, the cooperative-core may be empty: there might be no allocation that satisfies the cooperative-core-lower-bound. Intuitively, it is harder to attain stable agreements, since middle countries might "free-ride" agreements by downstream and upstream countries.
Sharing a polluted river A river carries not only water but also pollutants coming from agricultural, biological and industrial waste. River pollution is a negative externality: when an upstream country pollutes a river, this creates external cleaning costs for downstream countries. This externality may result in over-pollution by the upstream countries. Theoretically, by the Coase theorem, we could expect the countries to negotiate and achieve a deal in which polluting countries will agree to reduce the level of pollution for an appropriate monetary compensation. However, in practice this does not always happen.
Empirical evidence and case-studies Evidence from various international rivers shows that, at water quality monitoring stations immediately upstream of international borders, the pollution levels are more than 40% higher than the average levels at control stations. This may imply that countries do not cooperate for pollution reduction, and the reason for this may be the unclearness in property rights.
See and and for other empirical studies.