1 Introduction

The water resources authorities are facing the challenge of ensuring the access to sufficient water resources for increasing populations and markets, while conserving healthy water ecosystems. One common approach to solve this issue is to transfer the surplus water from some basins to those with shortages, called Inter-basin Water Transfer. Optimal allocation of water from a common pool resource is usually modelled using the cooperative game theory. There are plenty of studies presenting applications of game theory in water resources management. Recent researches including Xuesen et al. (2009), Niksokhan et al. (2009), Mahjouri and Ardestani (2010), Sadegh et al. (2010), Madani (2010), Mahjouri and Ardestani (2011), Sadegh and Kerachian (2011), Nikoo et al. (2012) and Abed-Elmdoust and Kerachian (2012) have considered different game-theoretic methodologies for water resources management.

The analysis of the forming coalitions and reallocating the benefits in classic game theory is usually carried out under the assumption that the players in the game are economically rational. Inter-basin water transfer is affected by ideological-political considerations that may affect potential arrangements in the coalition formation. The literature review associated with considering political aspects of water resources is mostly restricted to some case studies. Naff and Matson (1984), Frey and Naff (1985), Dinar and Wolf (1994), and Kucukmehmetoglu (2009a, b) are among them.

In the present paper, a new economic-political methodology is proposed for the operation of inter-basin water transfer systems. The model framework proposed in this paper is explained in Section 2. The initial water allocations to players are determined in Section 3. In Section 4, how to include the political consideration in water transfer projects is explained. In Sections 5 and 6, real fuzzy cooperative games with political considerations and the effectiveness of the proposed methodology are discussed.

2 Model Framework

A flowchart is presented in Fig. 1 to provide understanding of how the proposed methodology would be carried out. As depicted in this figure, at first, basic data and information relating to the physical and hydrological characteristics of the study area are collected and used as methodology inputs.

Fig. 1
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A flowchart of the proposed methodology for inter-basin water allocation

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The purpose of this methodology is attaining optimal and efficient water allocation policies in inter-basin water transfer systems. The next major steps are determining decision-makers and stakeholders and also the objective functions and their uncertainties. Determining political criteria, including, issue position, players’ saliences and powers and evaluating the political possibility of forming coalitions based on MPAS are the main steps of the proposed methodology. In this paper, water users would participate in fuzzy coalitions to increase their profits. In this step, the characteristic function of the coalitions would also be fuzzy to consider the existing uncertainties in payoffs that players would receive by participating in different fuzzy coalitions. In this paper, the Shapley function proposed in Abed-Elmdoust and Kerachian (2012) is modified based on political considerations to reallocate the total net benefit of a coalition to its members.

In this methodology, the main objective is to maximize the total fuzzy net benefit of the system and then distribute the gained fuzzy benefit among the players in a way that they have both economic and political incentives to form coalitions. Therefore, the players, who participate in a coalition, would have fuzzy side payments. In the following sections, some of the main components of the flowchart will be discussed in more details.

3 Inclusion of Political Considerations in Water Allocation

The initial water allocation model proposed in Abed-Elmdoust and Kerachian (2012) is applied here to determine the decision variables of the initial monthly allocated water to the players. In this paper, ideological and political considerations are involved in the analysis of forming coalitions and reallocating the benefits to members of a coalition. These considerations are generally ignored in the modelling frameworks due to the complexity of calculations.

This section challenges to extend appropriate analyses that will include political consideration in the modelling framework of inter-basin water allocation. Among these important analyses would be the comprehensive definition of some terms presenting players’ political attitudes, namely issue positions, power, and salience.

In order to consider the political possibility of forming a coalition, the PAS described by Coplin and O’leary (1976) is modified and used. This PAS incorporates the modifications for hydro-politics presented by Frey and Naff (1985), as well as some extra modifications for political inter-basin water transfer project presented in this paper. In this modified PAS (MPAS), all players’ political attitudes (Issue, Power, and Salience) are ranked for each possible coalition. Three mentioned political attitudes are evaluated based on the following definitions:

  • Issue position: One of the elements in the PAS that expresses how strong the participant is for or against each of the coalitions. Values used for quantifying this element are in the range [−α1; +α1]. Frey and Naff (1985) used the scored range of [−3; +3] for evaluating Issue position for hydro-politics projects.

  • Salience: The other element in the PAS is the salience which is the importance or the rank each participant assigns to a certain coalition. Values used in Frey and Naff analysis (1985) are in the range [1,α 2], α 2 ≥ 1. The same range would be considered here.

  • Power: Power of a player over a coalition is defined as “the ability of each party to accomplish or prevent the occurrence of each coalition” (Dinar and Wolf 1994). Values used are in the range [1,α 3], α 3 ≥ 1.

By multiplying the values of these three criteria and adding those up for a specific coalition, the extent of agreement or disagreement of any player about any coalition formation is obtained. Finally, using the formula \( \xi (s)={A \left/ {{\left( {A+B+C} \right)}} \right.} \) in PAS proposed by Coplin and O’Leary (1976) absolute level of political risk of the formation of coalition s is achieved. In the above formula, A is the total sum of scores of players who are sympathetic to the coalition. B is the total sum of scores of players who are against the formation of the coalition and C is the total sum of scores of players who are indifferent about the coalition formation. ξ(s) shows the relative survival ratio of scenario s. In the present paper, scenarios are in fact coalitions which the players have three mentioned political attitudes i.e. issue position, salience, and power for or against participating in them. Therefore, by incorporating these political considerations, the possibility of forming a coalition will be corrected. The next subsection explains how to calculate these three criteria more precisely.

3.1 Evaluation of the Political Attitudes: Issue Position, Salience and Power

In order to obtain the political possibility of forming a coalition, the political criteria of issue position and salience would easily be determined for each of the players using a survey. Proposed in this paper, decision makers’ powers in inter-basin water transfer policy making that directly influence the political probability of formation of cooperative coalitions are estimated considering their economic powers. Decision makers’ economic powers are evaluated based on their net benefit coefficient (i.e. the higher net benefit coefficient, the higher decision maker’s power).

4 Real fuzzy Games with Political Considerations

When the water users receive their initial shares of water, they would participate in cooperative fuzzy coalitions with fuzzy characteristic functions (real fuzzy cooperative game) to increase their own fuzzy benefits as well as the total fuzzy profit of the system. Participation in fuzzy cooperative coalitions leads to achievement of less water for one or more water users, and greater amounts of water for others. For equity, it is necessary to reallocate the earned fuzzy profits among water users (players or participants in real fuzzy cooperative game), so that the harms to water users receiving less water than the amount of their water rights, be compensated.

Reallocation of fuzzy profits among the players is carried out by means of side payments, based on a modified version of generalized Hukuhara-Shapely function with Choquet integral form, which was introduced in Abed-Elmdoust and Kerachian (2012). The novel modified function of this paper is called Generalized Shapley Value with Modified Probabilities (GSVMP) and is able to consider political probability of coalition formation. The function is called generalized because it incorporates the uncertainties about the participation rates the users would choose for entering any coalition and the payoff they would receive from entering any coalition (this special characteristic was considered and explained in detail in Abed-Elmdoust and Kerachian (2012)). It is also called modified because it incorporates the political considerations in deciding if a coalition is possible to be formed, besides incorporating the economic criteria.

Given C∈G F (I), where, I is the set of players, G F (I) denotes the subset of players participating in real fuzzy game. let \( L(C)=\left\{ {C(i)|C(i)>0,\quad \left. {i\in I} \right)} \right. \), where, C(i) is the participation rate of player i in fuzzy coalition C and let l(C) be the cardinality of L(C). We denote the elements of L(C) in increasing order as \( {h_1}<{h_2}<\ldots <{h_{l(C) }} \). When the users participate in more than one coalition, redistributing fuzzy profits for fuzzy coalition will be achieved using the Choquet integral form game as follows:

$$ {g_i}\left( {\widetilde{\omega }} \right)(D)=\sum\limits_{l=1}^{l(D) } {{f_i}\left( \omega \right)\left( {{{{\left[ D \right]}}_{{{h_l}}}}} \right)} .\left( {{h_l}-{h_{l-1 }}} \right), $$
(1)
$$ {f_i}\left( \omega \right)(S)=\left\{ {\begin{array}{*{20}c} {\sum\limits_{{T\in P\left( {S\backslash \left\{ i \right\}} \right)}} {\xi (T).\beta \left( {|T|;|S|} \right).\left[ {\omega \left( {T\cup \left\{ i \right\}{-_H}\omega (T)} \right.} \right]\quad if\quad i\in S} } \hfill \\ {\quad \quad \quad\;\;0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad otherwise,} \hfill \\ \end{array}} \right. $$
(2)

where, \( \beta \left( {|T|;|S|} \right)=|T|!.\left( {|S|-|T|-1} \right)!/|S|!,|S|\mathrm{and}|T| \) are respectively the number of players in crisp coalitions S and T, \( {{\left[ D \right]}_{{{h_l}}}} \) is a subset of players whose participation rates in coalition D are more than h l , ξ(T) is the political probability of forming coalitions, ω(T) is the fuzzy profit of forming crisp coalition T with fuzzy characteristic function, \( \left[ {\omega \left( {T\cup \left\{ i \right\}} \right.{-_H}\omega (T)} \right] \) is the fuzzy profit which player i would add to profit of coalition T and is called the fuzzy profit margin of player i participating in coalition T. This fuzzy profit will be calculated by Hukuhara-difference between two fuzzy numbers (Banks and Jacobs 1970). In fact, the function based on Hukuhara-difference is also called the Shapely-Hukuhara function. For more information about the Hukuhara-difference between two fuzzy numbers and Shapely-Hukuhara function, the reader is referred to Abed-Elmdoust and Kerachian (2012). In Eq. 2, uncertainty in the payoff of the formed coalition is calculated by the Shapley-Hukuhara function. It also incorporates the possibilities of forming coalitions based on political and economic criteria.

5 Case Study

In order to evaluate the performance of the proposed methodology, a water transfer project from the great Karoon River to Rafsanjan plain located in Loot desert in Iran, is studied as a case study. The purpose of this project is to transfer water from Solegan reservoir, which is planned to be constructed on one of the tributaries of the great Karoon River, to Rafsanjan plain. The water users of the donor basin include modern agro-industrial (player 1), old agro-industrial (player 2), and Khuzestan local agricultural (player 3) sectors and the water user of the receiving basin is Rasanjan agricultural sector (player 4). Table 1 presents the monthly demands of the players in the study area which should be supplied by the Solegan reservoir. The main characteristics of the inter-basin water transfer system, the main statistical characteristics of water resources in water donor and receiving basins are illustrated in more details in Mahjouri and Ardestani (2010).

Table 1 The monthly demands (in million cubic meters) of the main players in the study area which should be supplied by the Solegan reservoir (Mahjouri and Ardestani 2010)
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6 Results and Discussion

6.1 Decision Makers’ Powers in the Study Area

In this paper, estimation of the players’ powers in forming coalitions among different water users in donor and receiving basins is carried out based on their net benefit coefficients which are retrieved from Mahjouri and Ardestani (2011). The players’ powers are presented in Table 2.

Table 2 The power of each player in the study area considering their Net benefit coefficient
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6.2 MPAS for the Inter-Basin Water Allocation

To reflect the real-world regional hydro-politics, all players which have both an interest (salience and position) in the issue of inter-basin water transfer, and enough power to force their interests should be included.

The salience of each scenario (coalition) for every player is estimated based on his satisfaction with that coalition comparing to other coalitions. In fact, a player would like to participate in coalitions in which he will receive more water share. The calculation of the saliences which the fourth player assigns to coalitions he can be a member of is presented in Table 3 for instance. Moreover, the position parameter of each player against each coalition is considered based on his attitude about the geographical distance of him and other players participating in that coalition. For instance, the players in the donor basin tend to cooperate with each other more than with the receiving basin. To summarize the political considerations of each scenario, the MPAS for the inter-basin water transfer project is presented in Table 4. As mentioned in Section 3, the political probabilities of forming coalitions, are calculated based on the formula \( \xi (s)={A \left/ {{\left( {A+B+C} \right)}} \right.} \).

Table 3 The calculation of the salience which the fourth player assigns to each coalition
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Table 4 A modified political accounting system for the inter-basin water transfer project in the study area
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6.3 Real Fuzzy Games Based on MPAS Results

The initial water allocations to the four players, which were presented in Abed-Elmdoust and Kerachian (2012), are considered as the players’ initial water rights in this paper. In the previous section, we examined the political probability that any of the coalitions may or may not arise at the initial steps of an inter-basin water transfer project. In this section, the GSVMP function presented in Section 5 is used. The lower and upper bounds of profit shares in the planning horizon which are the sum of the players’ profit shares participating in different coalitions are shown in Figs. 2 and 3 respectively for players 1 and 2 and for players 3 and 4.

Fig. 2
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The lower and upper bounds of profit shares for players 1 and 2 during the planning horizon

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Fig. 3
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The lower and upper bounds of profit shares for players 3 and 4 during the planning horizon

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7 Summary and Conclusion

Optimal allocation of water from a common pool resource is usually modelled using the cooperative game theory. Based on previous literature, economic efficiency is not an enough incentive for cooperation, particularly when it comes to water resource. In this paper, a new game-theoretic methodology, which incorporates ideological and political considerations in the decision-making process, was developed to adjust the probabilities of forming different coalitions in the game, and equitably reallocate the fuzzy profits. The players formed cooperative coalitions for maximizing their total net benefits. Unlike previous studies, we also included a political analysis intended for addressing related issues other than only economic considerations. The inclusion of such analysis is so important in the case of an inter-basin water allocation because of the political nature of water transfer.

The economic-political methodology was applied to a large scale inter-basin water allocation project in which the water donor and receiving basins struggle with water scarcity. The results show how including political considerations in the study may provide a more satisfactory solution comparing to the just cost-effective water allocations. More comprehensive structure for determining the political criteria of power, issue position and players’ saliences can be studied in future researches. A fuzzy core-based version of the applied game theory-based model can also be studied in future studies.