The results of different allocation schemes for the regional gains are discussed and, to assess the stability of the allocation solutions, the Shapley-Shubik (Shapley and Shubik, 1954) _power index_ is applied: The power index compares the gains of a particular player to the gains of the coalition. A more harmonized power distribution is more likely to yield a stable coalition. The coefficient of variation of the power distribution _S_ α = σα α is a measure of stability for each allocation, where σα is the variance and α is the mean value of α . The allocation resulting from economic considerations only is the most unstable; the game theoretic allocations are far more stable, with the GSVMP being the most stable and the _Shapley value_ being the least stable.

Just and Netanyahu (1998) work focuses on the difficulties in forming _grand coalitions_ , including all the riparian counties involved in the trans-boundary water basin’s management. Their observation of the multilateral agreements stock is that treaties including all riparian countries are an exception rather than a rule (World Bank, 1993).

They examine some guidelines for water apportionment, considering some international treaties, and underline some obstacles to trans-boundary water resource cooperation (asymmetric information, scientific gaps and technical uncertainties, enforcement limitations, sovereignty, conflicting national and international interests, asymmetric country characteristics, upstream-downstream considerations). Using the literature on _coalition formation_ they discuss difficulties in forming water agreements in water basins with multiple (three or more) riparian states.

Individual countries form partial coalitions rather than a grand coalition if the game is not _superadditive_ . Aumann and Dreze (1974) specify these sources of _non-superadditivity_ : inefficiency of the grand coalition, moral hazard problems (for example a country cannot perfectly monitor the clean-up effort invested by polluting riparian countries), and free rider problems.

In contrast to Aumann and Dreze (1974), Owen (1977) and Hart and Kruz (1983), the authors assume that society operates efficiently and that a grand coalition will eventually be formed. The formation of smaller coalitions is considered as a strategic step to bargaining for a larger share of the pie when the grand coalition is formed.

The authors then introduce a hypothetical example of a game in which coalition formation takes place in an environment with economic externalities. First, the example shows how the choice of partial coalitions rather than grand coalitions may be optimal (both when the transaction costs of coordination are considered and when water conservation encounters decreasing returns to scale). Second, the example shows that once the coalition is formed, optimal behavior may allow to have different environmental responsibilities or to receive different allocation opportunities, even when countries are identical, with inequities compensated by side payments.

A similar finding is proposed in another paper on Lake Ontario by Becker and Easter (1999). They use game theory to assess the potential for cooperative management. They demonstrate that a noncooperative solution is not the inevitable outcome for the use of an open access water resource. Stable cooperative coalitions are possible when a subset of users decides to cooperate. In their examples, once a minimum number of users decides to cooperate, the remaining noncooperators find that it is in their best interests to cooperate.

A _Prisoners’ Dilemma_ game is introduced in this paper, to model the case of reciprocal externalities with full cooperation, non-cooperation and partial cooperation. The paper assumes σ to be the portion of users signing an agreement, thus σ _N_ is the number of users forming a coalition, while (1 - σ ) _N_ is the number of 37