The results under the different assumptions are presented and compared: the authors conclude that an increase of 17% could occur to the regional income though cooperation with side payments; on the contrary the region can get an increase of 8% if cooperation does not involve side-payments.

On the other hand some difficulties may arise from the side payment policy (see introduction), while the model without side payments seems to be more reasonable: its assumptions are simple and easy to understand, and for this case study it is computationally simple.

If compared with most CGT application literature Aadland and Koplin follow a mirror-like approach in their two following papers (1998, 2004). Their analysis deals with some real-world cost-sharing rules for irrigation (they consider the case study of 25 irrigation ditches in Montana) and they try to see these rules under a game-theoretical point of view, providing in the first paper an ex-post axiomatic characterization. The mathematical tool does not prevail over reality: it is at its complete disposal. The problem in the use of axiomatic approaches and of the deriving solution concepts is that a reasonable axiom can be too restrictive or even cause unreasonable outputs for certain real situations. ‘Fairness’ also, on the authors’ opinion, has to be intended as long-term acceptance by people judgment. In their second paper Aadland and Koplin, while reexamining the same case study ask to themselves if there is a correspondence between some environmental aspects and the choice of a certain allocation rule by the farmers; they try to answer with the aid of an econometric model.

In his paper Dinar (2001) presents a cooperative bargaining model with NTU, and uses the Nash solution because “The allocation is feasible, fulfills Individual Rationality, is Pareto optimal, symmetric, non related with nonrelevant alternatives and to a specific presentation of the VNM utility function.” The peculiarity of Dinar’s model is that it allows an optimization of the capacity of the joint facility simultaneously with the cost allocation procedure.

The article by Norde et al. (2003) deals with another branch of CGT— _fixed tree games_ . In some irrigation systems it can be helpful to model the situation as a _tree_ where the farmers are located on one (or more in the case of FRP games) vertex, and it is possible to construct a corresponding cooperative game. The paper is almost theoretical but it is presented to give the reader an idea of a different CGT tool, which can be employed in modeling some irrigation water supply networks.

A work by Aadland and Koplin (1998) examines cost-sharing arrangements in a sample of twenty-five irrigation ditches in south-central Montana. These arrangements—variations of serial and average cost sharing—often date back to the corresponding ditch’s construction at the turn of the century. The authors present an empirical and axiomatic analysis of the allocation mechanisms employed by the ranchers in the region, using a list of static and dynamic equity principles as was revealed by direct phone surveys. The analysis shows that the observed cost-sharing arrangements can be characterized as unique solutions to equity-constrained welfare maximization.

The cost-sharing rules dominating the case study, the _serial/Shapley_ and _average cost-share_ rules, have been kept surprisingly stable, suggesting that they are really perceived as ‘fair’ by the players. This motivates the authors’ efforts to construct characterizations of the sampled mechanisms based exclusively on principles of equity.

While collecting the data, the authors directly made phone calls to one or more irrigators along each ditch, asking about (a) details regarding the cost-sharing arrangements currently in use, and (b) general feeling about what can be considered to be fair methods of sharing irrigation costs.

Two classes of sharing rules dominated: _average cost sharing_ (egalitarian) and _serial cost sharing_ . The average cost-sharing rule simply takes the total cost of ditch maintenance and divides it equally amongst all users. The serial cost-share rule divides the cost of maintaining each ditch segment equally among all agents that require the use of this segment to meet their irrigation needs. As in the _‘airport games’_ of Littlechild and Owen (1973), the allocation offered by the _serial rule_ is identical to that prescribed by the _Shapley value_ of an appropriately defined cooperative game. Per acre and per water-share weighted 14