# A Study on Profit Distribution for Water Rights Cooperation Taking into Account Dominant Power of Transfer Sectors

Authors: Jiayin Tan; Dakui Jiang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, we explore the issue of profit distribution for water resources collaborative alliances that are composed of one single water rights transfer sector and several water resources requirement sectors. Considering the dominant position of the water rights transfer sector in water resources cooperation, we propose a novel 3-step approach on profit distribution for coalitions by applying game theory for interactions that include coalition structures and permission structures. We examine the effectiveness of this approach by a case study of water rights cooperation between the agriculture sector and agroindustrial sectors of the Karoon river basin in Iran. The results show that this approach allows players who have veto rights to obtain more payoffs from the coalition’s profit distribution in contrast to using the Owen value. In other words, the distribution results of the 3-step approach reflect the advantage held by the water rights transfer sector based on its veto power. Further, our proposed 3-step approach takes the permission structure among a priori unions into consideration when distributing the profits of the water resources coalition while the P-Owen value considers only the permission structure among players.


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## Body

## 1. Introduction

Water is becoming a rare resource in the 21st century. In many regions of a specific country, such as China, the volume of water initially allocated to industries by water allocation agreements could barely meet their needs. Hence, these sectors always obtain water resources through cooperation [1] with one another in the form of water rights transfers. For instance, agricultural sectors may cooperate with industrial sectors by transferring the right to use water saved through frugal irrigation and conservation [2]. The receiving industrial sectors will utilize the water for production, and then the income from industrial production is distributed among the cooperating partners according to their agreements [3–5]. Under these arrangements, the agricultural partners who serve as providers are termed “water rights transfer sectors,” while the receiving sectors are “water requirement sectors.”In some regions, due to severe water shortages, multiple sectors that have water requirements are waiting in line to arrange water rights transfer cooperation with the agricultural sectors. When the demand for water resources exceeds the supply, the water rights transfer sectors have an advantage over neighboring water requirement sectors. Although some of the water requirement sectors have opportunities to obtain water rights from other providers in nearby regions, generally these additional sources do not have a significant impact on the dominant position of the main water transfer sectors.It is important to understand the dominant position and functions of water rights transfer sectors when arranging cooperative agreements.(1) In order to obtain the best payoff from water resources cooperation, usually water rights transfer sectors will want to transfer water rights to water requirement sectors that have the highest benefit coefficients [3]. In addition, water transfer sectors may have preferences that should be taken into account. For example, a water transfer sector might believe that it can gain more utility by cooperating with water requirement industries that are already its partners in other projects. Similarly, a water rights transfer sector would prefer to work with water requirement sectors that have less sewerage. Some water rights transfer sectors even put their preferences above the water use benefit coefficient of the water requirement sectors.(2) When establishing a cooperative arrangement, the water rights transfer sectors have veto power [6]. Each water rights transfer sector has the right to veto water requirement sectors’ utilization of water saved by the transfer sectors’ conservation measures. In other words, without the permission and ongoing approval of the water rights transfer sector, water requirement sectors cannot acquire and/or exercise rights to the conserved water gained through cooperative agreement.When the water rights transfer sectors have a dominant position as described above, it is worth considering the best ways for assigning the profits that arise from a coalition between the water rights transfer sectors and water requirement sectors. Since water rights transfer sectors have veto power over water requirement sectors, they can be regarded as water requirement sectors’ superiors in a cooperative’s hierarchy. Moreover, there are hierarchical structures among a priori unions:(1)
When considering coalition partners, water rights transfer sectors will turn first to water requirement sectors that fit their preferences and build a priori union [7] with them(2)
After transferring water rights to the water requirement sectors within the a priori union, the water transfer sectors will transfer water rights to other water requirement sectors(3)
Water rights transfer process described in 2 can also be seen as a process of transferring the remaining water rights of the a priori union to water requirement sectors that do not fit water transfer sector’s preference. It means that the a priori union has veto power over these water requirement sectors. By regarding each water requirement sector that does not fit water transfer sector’s preference as a special a priori union, then a hierarchy will exist among all a priori unions, where the a priori unions that contain the water transfer sectors can be regarded as the superiors who have veto power over other kinds of a priori unions.Therefore, the cooperative games among water rights transfer sectors and water requirement sectors discussed in this paper combine the theory of cooperative games with coalition structures [7] (a priori unions) and games with conjunctive permission structures [8]. For profit sharing issues under these types of cooperative games, the existing relevant research can be classified into three categories.(  1) Research That Takes into Account Only Coalition Structures. Owen explored cooperative games that have a coalition structure and proposed the concept of the Owen value to allocate the profit of coalitions under these circumstances [7]. Vázquez-Brage axiomatized the Owen value from different angles [9], and, on this basis, Albizuri generalized coalition structures to introduce the general idea of coalition configurations and improved the Owen value [10, 11]. Sun H. X. proposed a coalitional core in games with a coalition structure and proved that the Owen value belongs to the coalition core while the game is a strong convex game [12].(  2) Research That Takes into Account Only the Hierarchy among Players. Faigle and Kern explored cooperative games with a hierarchy structure. They developed the restricted Shapley value based on the lattice structure theory and applied it to the profit distribution issue under situations in which there is a hierarchy among the players [13]. On this basis, Michel Grabisch proposed the restricted core and bounded core in games with a hierarchy structure [14, 15]. Furthermore, Gilles researched cooperative games that have a permission structure, i.e., games in which there is a hierarchy structure among players and superiors have veto rights over subordinates [16]. Then Gilles, Van den Brink, Gallardo, and other researchers went on to discuss the profit distribution problem under these types of cooperative games [17–20].(  3) Research That Takes into Account Both Coalition Structures and Hierarchy Structures. Sun H.X. discussed profit allocation issues in cooperative games that have a coalition structure [21]. In addition, she took into consideration the hierarchy structure among different a priori unions and developed the restricted Owen value. Wang L.M. explored the profit allocation issue in cooperative games that have a permission structure and proposed a new Owen value (P-Owen value) to solve this kind of problem [6].Of the three types of studies described above, the first category considered only coalition structures, and the second category considered only hierarchical structures. Although the third category considered both kinds of structures, this area of research still had limitations. Sun took account of the hierarchies that can exist among different a priori unions, but not the hierarchies among players within a priori union [21]. Wang explored the permission structures within a priori union [1], but did not consider permission structures among different a priori unions. Consequently, we cannot apply the results of any of these studies to the profit distribution issues that arise when water rights transfer sectors are dominant over water requirement sectors.Based on the theory of cooperative games with coalition structures and games with permission structures, in the remainder of this paper we explore the profit distribution for water resources collaborative alliances when the water rights transfer sectors dominate the water requirement sectors. Since solutions to the profit distribution issue correlated with the coalition structures and the payoff function of the coalitions, Section2 of this paper examines fully the cooperative preference of water rights transfer sectors and depicts the coalition structure based on cooperative games with coalition structures. And in Section 2, we constructed the payoff function for the coalitions. After fully considering the water transfer sector’s veto rights, in Section 3 we proposed a new 3-step profit distribution method based on cooperative games with a permission structure. Based on the coalition structure and the payoff function obtained in the first step, this solution can be applied to profit sharing for a water rights transfer cooperative alliance. Then, in Section 4, we tested the effectiveness and applicability of the 3-step method by applying it to a case study of water rights cooperation between the agriculture sector and agroindustrial sectors of the Karoon river basin in Iran [3–5].


## 2. Model Formulation

In this paper, we focus on the water rights cooperation problem with one sector conserving and supplying water and many sectors needing water (i.e., in a single-transfer and multidemander scenario) and try to find a feasible revenue distribution solution for water rights cooperation taking into account the dominant position (such as cooperative preference and veto power in water rights cooperation) of the transfer sector.In this section, we initially depict the coalition structure based on cooperative games with coalition structures and the cooperative preference and veto power of water rights transfer sector; then we construct the payoff function for the coalitions.


### 2.1. Coalition Structures

We consider that the water rights transfer sector has preferences for selecting collaboration partners. It is believed that it can earn more profit (utility) by collaborating with water requirement sectors known to fit its preferences. In such cases, they will provide high priority to working with water requirement industries with whom they have a previous history of successful cooperation.Let the transfer sector be denoted as player 0. Let the water requirement sector in the same region waiting to cooperate with player 0 be denoted as playeri, so that i∈N=1,2,…,n. Further, let the players who satisfy player 0’s preferences be represented as S, S⊂N. Player 0 preferentially cooperates with players in S and forms a priori union C1={0∪S}.A priori unionC1 establishes a common coalition with other water requirement sectors. The coalition structure of the common coalition can be denoted as Ω={C1,C2,…,Cm}, where Ω is a partition of the set E={0∪N} and C2,…,Cm correspond to every water requirement sector in the set N∖S, and each water requirement sector (Ct,t=2,3,…,m) in the set N∖S can be regarded as a special a priori union; in other words, each a priori union in {C2,…,Cm} includes one player. We use M={1,2,…,m} to denote the subscript sets of a priori unions C1,C2,…,Cm, and Cg∩Cr=∅,∀g,r∈M,g≠r.


### 2.2. Permission Structure among and within Coalitions

In the cooperation for water rights transfers, the transfer sector has veto power over the water requirement sectors’ use of water resources gained through the transfer. Thus, there exists a permission structure between the transfer sector and requirement sector within a priori unionC1. Furthermore, a priori union C1 represents the transfer sector when part of its water rights has already been transferred to the water requirement sectors that fit its preferences. Consequently, there are permission structures among C1 and the other a priori unions.The hierarchical organization of the permission structure within the a priori unionC1 can be described as follows. The water rights transfer sector designated as player 0 is the superior, and the other players in C1 are all subordinate sectors. The permission structure governing the players’ set E can be expressed as the mapping H1:E→2E. H1(0) represent the direct subordinates of player 0, i.e., all the water rights demanders in C1. H^1-1(i) represent the superiors of player i,i∈N={1,2,…,n} in H1, H^1-1i={0} [6, 16].Meanwhile, the hierarchy of the permission structure among a priori unions can be described as follows. A priori unionC1 is the superior, and the other a priori unions (sectors who need water rights) are all subordinates. The permission structure over the set Ω (which is the collection of a priori unions) can be expressed as the mapping H2:Ω→2Ω. H2(C1) represent the direct subordinates of C1, i.e., all Ct(t∈M,t≠1). H^2-1(Ct) represent the superiors of Ct in H2, H^2-1Ct={C1} [6].The permission structure within a priori unionC1can also be represented as a directed graph, where player 0 is the graph vertices, while the graph edges (0,j) denote j∈H(0) [6, 17] (as 0→1,0→2 in Figure 1). Similarly, the permission structure among a priori unions can also be structured as a directed graph, where the a priori union C1 is the graph vertices, while the graph edges (C1,Ct), Ct,t=2,3,…,m denote Ct∈H(C1) (as C1→C2,C1→C3 in Figure 1).Figure 1
Directed graph of permission structure.As discussed above, when the power advantage of the transfer sector is evident, the cooperative games among the main bodies involved in water rights transfer can be represented as a triple(E,v,ΩH1H2), in which v∈GE, GE represents all the cooperative strategies over the players’ set E, and v∅=0;ΩH1H2∈CH1H2E, CH1H2E is all the coalition structures over E, which has a permission structure among and within the a priori unions [17].Moreover, coalitionU is an autonomous coalition if U⊆E, H^1-1U⊆U, and for any coalition U⊆E,σH1U is the sovereign part of U, if σH1U=r∈U∣H^1-1r⊆U,r=0,1,2,…,n. Coalition T⊆Ω, which was formed by a priori unions, is an autonomous coalition, and, for any T⊆E, σH2T is the sovereign part of T, if σH2T=g∈T∣H^2-1g⊆T,g∈M [6, 16].


### 2.3. Payoff Functions for Coalitions

The payoff functions for the coalitions are related to the coalitions’ structures. Based on the analysis in Section2.1, if we know the volume of water resources saved by player 0’s water-saving and conservation in irrigating aspect and also know the water deficit of each of the other players i(i∈N), then the water rights transfer in coalition E={0∪N} can be conducted in 2 stages.(1) First, player 0 transfers the water saved through conservation efforts to playerii∈C1 based on the “priority rule” [3, 4]. In other words, player 0’s water rights transfer is given to the player with the highest benefit coefficient of water use, until that player reaches its capacity. (The capacity of a player equals the player’s water deficit plus the player’s initial water resources volume.) Then the remaining water is given to the player with the second highest benefit coefficient of water use, until that player reaches its capacity as well. This process is repeated until the total volume of water resources saved by player 0 has been transferred completely or the capacity of each player i∈C1 has been reached.(2) Second, if each playeri’s (i∈C1) capacity has been reached and player 0 still has a water surplus, then the water surplus is given to other players within E∖C1 based on the “priority rule” as well. This process is similar to the water transfer procedure in stage (1). If player 0 still has water surplus after transferring water to the players in E∖C1 and satisfying their capacity, then player 0 utilizes the surplus water resources as it sees fit; i.e., player 0 may transfer the surplus water to water requirement sectors in other regions or utilize the surplus water for irrigation.Based on the above analysis, the players inC1, except player 0, can be sorted in nonincreasing order by their water use benefit coefficient, and each player obtains a sequence number j, j=1,2,…,C1-2,C1-1. Then the payoff function of C1 can expressed as follows:(1)vC1=A0b1-F0A0<d1A0b1+α-1X1+A0b1-F0A0=d1⋮∑j=1l-1djbj+α-1bjXj+dj+A0-∑j=1l-1djbl-F0∑j=1l-1dj<A0<∑i=1ldj∑j=1l-1dj+α-1Xj+djbj+A0-∑j=1l-1dj+α-1Xl+dlbl-F0∑j=1l-1dj<A0=∑j=1ldj⋮∑l=1C1-1djbj+α-1bjXj+dj-F0∑l=1C1-1dj≤A0whereA0 is the volume of water resources that player 0 saved by conserving irrigation water. F0 is the water conservation reform cost of player 0; Xj is the initial water resources volume of playerj; dj is the water deficit of player j, and bj is player j’s benefit coefficient of water use. Player 0 gives first priority to creating a coalition with the players who fit its preferences, because player 0 believes that it can gain more utility from this kind of coalition (a priori union). Let α denote the utility coefficient of a priori union C1. Then (α-1)bjXj+dj is the value added in C1’s utility, which is contributed by player j when the water demand Xj+dj of player j has been fulfilled completely.All players inE∖C1 can be sorted in nonincreasing order of their water use benefit coefficients, and each player gets a sequence number q,q=1,2,…,E∖C1-2,E∖C1-1. Let A0L denote the water resources volume of player 0 after player 0 transfers water rights to players within C1. Xq denotes the initial water resources volume of player q, dq denotes the water deficit of player q, and bq denotes player q’s benefit coefficient of water use. Then the payoff function of coalition E is(2)vE=A0Lb1+vC1A0L≤d1⋮∑q=1l-1dqbq+Xq+dqbq+A0L-∑q=1l-1dqbl+vC1∑q=1l-1dq<A0L≤∑q=1ldq⋮∑l=1E∖C1-1dqbq+Xq+dqbq+A0L-∑l=1-1dqb0+vC1∑l=1E∖C1-1dq≤A0LIfC1 is an empty set, player 0 cooperates with all other players and forms coalition Ea. The payoff function of Ea is(3)vEa=A0b1-F0A0≤d1⋮∑q=1l-1dqbq+Xq+dqbq+A0-∑q=1l-1dqbl-F0∑q=1l-1dq<A0≤∑q=1ldq⋮∑l=1Ea-1dqbq+Xq+dqbq+A0-∑l=1-1dqb0-F0∑l=1Ea-1dq≤A0If player 0 creates a coalition with some of the other players (water requirement sectors), those players who join the coalition can be sorted in nonincreasing order of the benefit coefficients of their water use. Then we calculate payoffs of the coalition based on (1) or (2).Circumstances of the interaction may change. If playerq, who does not fit player 0’s preferences, takes on water conservation reform, for example, by installing its own water-reuse system, and then utilizes the saved water for production, the payoff function of player q is as shown in the following:(4)vq=0Xq<Xq+dq1+wq1+wsqXq-Xq+dq1+wq1+wsqbq+βq-1Xq+dqbq-FqXq≥Xq+dq1+wq1+wsqwhereFq is the water conservation reform cost of player q and wq is the original water-reuse rate of player q. wsq is the new water-reuse rate of player q after implementation of water conservation reform. When player q practices water conservation reform, its water demand can be fully satisfied with greater probability than by cooperating with player 0. If the initial water resources volume of player q after water conservation reform can completely meet its water demand Xq+dq, then player q can earn more utility by proceeding this way than by cooperating with player 0.Letβq denote the utility coefficient of player q. Then βq-1bqXq+dq is the value added to player q’s utility. If C1 is nonempty, each player outside a priori union C1 can be regarded as a priori union, which can be represented as Ct,t=2,3,…,m. The payoff function of Ct,t=2,3,…,m is vCt=v{q}.Similarly, if playerj(j∈C1) takes on water conservation reform and utilizes the saved water for agroindustrial production, the payoff function of player j is as follows:(5)vj=0Xj<Xj+dj1+wj1+wsjXj-Xj+dj1+wj1+wsjbj+βj-1Xj+djbj-FjXj≥Xj+dj1+wj1+wsjIf all players except player 0 establish coalitions, revenue from these coalitions is equal to the sum of the payoffs of all water requirement sectors.If player 0 utilizes water resources saved by water-saving irrigation for agricultural production,b0 denotes player 0’s benefit coefficient of water use; then the payoff function of player 0 is shown below:(6)v0=b0A0-F0


## 2.1. Coalition Structures

We consider that the water rights transfer sector has preferences for selecting collaboration partners. It is believed that it can earn more profit (utility) by collaborating with water requirement sectors known to fit its preferences. In such cases, they will provide high priority to working with water requirement industries with whom they have a previous history of successful cooperation.Let the transfer sector be denoted as player 0. Let the water requirement sector in the same region waiting to cooperate with player 0 be denoted as playeri, so that i∈N=1,2,…,n. Further, let the players who satisfy player 0’s preferences be represented as S, S⊂N. Player 0 preferentially cooperates with players in S and forms a priori union C1={0∪S}.A priori unionC1 establishes a common coalition with other water requirement sectors. The coalition structure of the common coalition can be denoted as Ω={C1,C2,…,Cm}, where Ω is a partition of the set E={0∪N} and C2,…,Cm correspond to every water requirement sector in the set N∖S, and each water requirement sector (Ct,t=2,3,…,m) in the set N∖S can be regarded as a special a priori union; in other words, each a priori union in {C2,…,Cm} includes one player. We use M={1,2,…,m} to denote the subscript sets of a priori unions C1,C2,…,Cm, and Cg∩Cr=∅,∀g,r∈M,g≠r.


## 2.2. Permission Structure among and within Coalitions

In the cooperation for water rights transfers, the transfer sector has veto power over the water requirement sectors’ use of water resources gained through the transfer. Thus, there exists a permission structure between the transfer sector and requirement sector within a priori unionC1. Furthermore, a priori union C1 represents the transfer sector when part of its water rights has already been transferred to the water requirement sectors that fit its preferences. Consequently, there are permission structures among C1 and the other a priori unions.The hierarchical organization of the permission structure within the a priori unionC1 can be described as follows. The water rights transfer sector designated as player 0 is the superior, and the other players in C1 are all subordinate sectors. The permission structure governing the players’ set E can be expressed as the mapping H1:E→2E. H1(0) represent the direct subordinates of player 0, i.e., all the water rights demanders in C1. H^1-1(i) represent the superiors of player i,i∈N={1,2,…,n} in H1, H^1-1i={0} [6, 16].Meanwhile, the hierarchy of the permission structure among a priori unions can be described as follows. A priori unionC1 is the superior, and the other a priori unions (sectors who need water rights) are all subordinates. The permission structure over the set Ω (which is the collection of a priori unions) can be expressed as the mapping H2:Ω→2Ω. H2(C1) represent the direct subordinates of C1, i.e., all Ct(t∈M,t≠1). H^2-1(Ct) represent the superiors of Ct in H2, H^2-1Ct={C1} [6].The permission structure within a priori unionC1can also be represented as a directed graph, where player 0 is the graph vertices, while the graph edges (0,j) denote j∈H(0) [6, 17] (as 0→1,0→2 in Figure 1). Similarly, the permission structure among a priori unions can also be structured as a directed graph, where the a priori union C1 is the graph vertices, while the graph edges (C1,Ct), Ct,t=2,3,…,m denote Ct∈H(C1) (as C1→C2,C1→C3 in Figure 1).Figure 1
Directed graph of permission structure.As discussed above, when the power advantage of the transfer sector is evident, the cooperative games among the main bodies involved in water rights transfer can be represented as a triple(E,v,ΩH1H2), in which v∈GE, GE represents all the cooperative strategies over the players’ set E, and v∅=0;ΩH1H2∈CH1H2E, CH1H2E is all the coalition structures over E, which has a permission structure among and within the a priori unions [17].Moreover, coalitionU is an autonomous coalition if U⊆E, H^1-1U⊆U, and for any coalition U⊆E,σH1U is the sovereign part of U, if σH1U=r∈U∣H^1-1r⊆U,r=0,1,2,…,n. Coalition T⊆Ω, which was formed by a priori unions, is an autonomous coalition, and, for any T⊆E, σH2T is the sovereign part of T, if σH2T=g∈T∣H^2-1g⊆T,g∈M [6, 16].


## 2.3. Payoff Functions for Coalitions

The payoff functions for the coalitions are related to the coalitions’ structures. Based on the analysis in Section2.1, if we know the volume of water resources saved by player 0’s water-saving and conservation in irrigating aspect and also know the water deficit of each of the other players i(i∈N), then the water rights transfer in coalition E={0∪N} can be conducted in 2 stages.(1) First, player 0 transfers the water saved through conservation efforts to playerii∈C1 based on the “priority rule” [3, 4]. In other words, player 0’s water rights transfer is given to the player with the highest benefit coefficient of water use, until that player reaches its capacity. (The capacity of a player equals the player’s water deficit plus the player’s initial water resources volume.) Then the remaining water is given to the player with the second highest benefit coefficient of water use, until that player reaches its capacity as well. This process is repeated until the total volume of water resources saved by player 0 has been transferred completely or the capacity of each player i∈C1 has been reached.(2) Second, if each playeri’s (i∈C1) capacity has been reached and player 0 still has a water surplus, then the water surplus is given to other players within E∖C1 based on the “priority rule” as well. This process is similar to the water transfer procedure in stage (1). If player 0 still has water surplus after transferring water to the players in E∖C1 and satisfying their capacity, then player 0 utilizes the surplus water resources as it sees fit; i.e., player 0 may transfer the surplus water to water requirement sectors in other regions or utilize the surplus water for irrigation.Based on the above analysis, the players inC1, except player 0, can be sorted in nonincreasing order by their water use benefit coefficient, and each player obtains a sequence number j, j=1,2,…,C1-2,C1-1. Then the payoff function of C1 can expressed as follows:(1)vC1=A0b1-F0A0<d1A0b1+α-1X1+A0b1-F0A0=d1⋮∑j=1l-1djbj+α-1bjXj+dj+A0-∑j=1l-1djbl-F0∑j=1l-1dj<A0<∑i=1ldj∑j=1l-1dj+α-1Xj+djbj+A0-∑j=1l-1dj+α-1Xl+dlbl-F0∑j=1l-1dj<A0=∑j=1ldj⋮∑l=1C1-1djbj+α-1bjXj+dj-F0∑l=1C1-1dj≤A0whereA0 is the volume of water resources that player 0 saved by conserving irrigation water. F0 is the water conservation reform cost of player 0; Xj is the initial water resources volume of playerj; dj is the water deficit of player j, and bj is player j’s benefit coefficient of water use. Player 0 gives first priority to creating a coalition with the players who fit its preferences, because player 0 believes that it can gain more utility from this kind of coalition (a priori union). Let α denote the utility coefficient of a priori union C1. Then (α-1)bjXj+dj is the value added in C1’s utility, which is contributed by player j when the water demand Xj+dj of player j has been fulfilled completely.All players inE∖C1 can be sorted in nonincreasing order of their water use benefit coefficients, and each player gets a sequence number q,q=1,2,…,E∖C1-2,E∖C1-1. Let A0L denote the water resources volume of player 0 after player 0 transfers water rights to players within C1. Xq denotes the initial water resources volume of player q, dq denotes the water deficit of player q, and bq denotes player q’s benefit coefficient of water use. Then the payoff function of coalition E is(2)vE=A0Lb1+vC1A0L≤d1⋮∑q=1l-1dqbq+Xq+dqbq+A0L-∑q=1l-1dqbl+vC1∑q=1l-1dq<A0L≤∑q=1ldq⋮∑l=1E∖C1-1dqbq+Xq+dqbq+A0L-∑l=1-1dqb0+vC1∑l=1E∖C1-1dq≤A0LIfC1 is an empty set, player 0 cooperates with all other players and forms coalition Ea. The payoff function of Ea is(3)vEa=A0b1-F0A0≤d1⋮∑q=1l-1dqbq+Xq+dqbq+A0-∑q=1l-1dqbl-F0∑q=1l-1dq<A0≤∑q=1ldq⋮∑l=1Ea-1dqbq+Xq+dqbq+A0-∑l=1-1dqb0-F0∑l=1Ea-1dq≤A0If player 0 creates a coalition with some of the other players (water requirement sectors), those players who join the coalition can be sorted in nonincreasing order of the benefit coefficients of their water use. Then we calculate payoffs of the coalition based on (1) or (2).Circumstances of the interaction may change. If playerq, who does not fit player 0’s preferences, takes on water conservation reform, for example, by installing its own water-reuse system, and then utilizes the saved water for production, the payoff function of player q is as shown in the following:(4)vq=0Xq<Xq+dq1+wq1+wsqXq-Xq+dq1+wq1+wsqbq+βq-1Xq+dqbq-FqXq≥Xq+dq1+wq1+wsqwhereFq is the water conservation reform cost of player q and wq is the original water-reuse rate of player q. wsq is the new water-reuse rate of player q after implementation of water conservation reform. When player q practices water conservation reform, its water demand can be fully satisfied with greater probability than by cooperating with player 0. If the initial water resources volume of player q after water conservation reform can completely meet its water demand Xq+dq, then player q can earn more utility by proceeding this way than by cooperating with player 0.Letβq denote the utility coefficient of player q. Then βq-1bqXq+dq is the value added to player q’s utility. If C1 is nonempty, each player outside a priori union C1 can be regarded as a priori union, which can be represented as Ct,t=2,3,…,m. The payoff function of Ct,t=2,3,…,m is vCt=v{q}.Similarly, if playerj(j∈C1) takes on water conservation reform and utilizes the saved water for agroindustrial production, the payoff function of player j is as follows:(5)vj=0Xj<Xj+dj1+wj1+wsjXj-Xj+dj1+wj1+wsjbj+βj-1Xj+djbj-FjXj≥Xj+dj1+wj1+wsjIf all players except player 0 establish coalitions, revenue from these coalitions is equal to the sum of the payoffs of all water requirement sectors.If player 0 utilizes water resources saved by water-saving irrigation for agricultural production,b0 denotes player 0’s benefit coefficient of water use; then the payoff function of player 0 is shown below:(6)v0=b0A0-F0


## 3. Profit Distributive Approaches for Coalitions

### 3.1. 3-Step Profit Distributive Approach for Coalitions

When the profits of the grand-coalitionE are shared among player 0 and all players ii∈N, both coalition structures and permission structures should be considered when selecting the most suitable method for dividing those profits. The Shapley value, core, least core, and nucleolus allocation methods cannot be used in the profit distribution for coalition E, since the grand-coalition E is composed of a priori unions. Player 0 (the transfer sector) forms a priori union with the players who satisfy its preferences, and each of the other players can be regarded as a priori union.We now propose a 3-step distribution approach based on the principle of the P-Owen value [6] and Owen value [7] that distributes the profit of a coalition among different a priori unions first and then distributes within each a priori union. This approach is improved according to the specific research of this paper. The 3-step distribution approach can be described as follows.Step 1.
Define a dummy allianceDE, vDE=0 and suppose that (1) there are no permission structures between the coalition E and DE, (2) E and DE cooperate with each other, and they form coalition DE∪E, and (3) the revenue of the coalition is v{DE∪E}=vE. We distribute v{DE∪E} to E and DE using the Shapley value. DE obtains u(DE)=0, and E gets u(E)=vE.Step 2.
AllocateuE among the a priori unions Cgg∈M within E by using the Shapley value. Let K denote the alliance composed by Cgg∈M, K⊆Ω, K≠∅, where Ω denotes the coalition structure of E, or the complete set of all a priori unions within E. Taking into account the permission structures among the a priori unions, the distribution of profit for each a priori union can be determined by the following equation:(7)SHCgu=∑K⊆ΩCg∈KK-1!Ω-K!Ω!uK-uK∖Cg,Cg∈Ω
whereΩ,K represents the number of a priori unions in Ω,K.
BecauseK has the possibility of cooperating with the dummy coalition DE, the profit share of K depends on its bargaining power. Let uK denote the profit share of K when K cooperates with DE in place of E. Then,(8)uK=K∪DE-1!K∪DE-K∪DE!K∪DE!vDE∪σH2K-vDE+K-1!K∪DE-K!K∪DE!vσH2K-v∅
whereσH2(K) represents the sovereign part of K. Based on the analysis in Step 1, vDE∪σH2(K)=vσH2(K), and vDE=0. The method of calculating vσH2(K) can refer to the definition of the sovereign part of U (in Section 2.2), and the utility function of the coalitions (Section 2.3) should be considered. In addition, as K∖Cg⊂K⊆Ω, u(K∖Cg) can be calculated by (8).Step 3.
AllocateSHCg(u) among the players within Cgg∈M by using the Shapley value. Since each a priori union in {C2,…,Cm} includes one player, only the revenue distribution of a priori union C1 needs to be discussed further.
LetQ⊆C1, Q≠∅, taking into account the permission structure that exists between player 0 and the other players within C1. The profit allocation solution for C1 is equal to(9)SHrγ=∑Q⊆C1r∈QQ-1!C1-Q!C1!γQ-γQ∖r,r∈C1
In (9), Q denotes the alliance composed by the players in C1, and γQ represents the profit share of Q when Q cooperates with external coalitions (C2,…,Cm) in place of C1. Then γQ can be calculated as follows:(10)γQ=∑R⊆C1Q∈RR-1!C1-R!C1!v⋃Cg∈Ω∖C1Cg∪σH1Q-v⋃Cg∈Ω∖C1CgBased on the above 3-step procedure of profit distribution, all revenue of the grand-coalitionE can be allocated to each player. Since the proposed approach completely allocates the profit of the grand-coalition E among a priori unions on the basis of the Shapley value and then completely divides the revenue of the a priori unions to each player by using the Shapley value as well, the profit of E can be distributed in full; i.e., SH0+∑SHj=vE. This distribution satisfies the definition of validity.


### 3.2. P-Owen Value and Owen Value

This section provides the formulas for the P-Owen value and Owen value for ease of comparison between the distribution results of the 3-step approach and these two methods.The P-Owen value [6] under the permission structure H1 is defined as shown in the following:(11)OwrH1v=∑K⊆ΩCg∉K∑Q⊆Cgr∈QK!Ω-K-1!Ω!Q-1!Cg-Q!Cg!v⋃Cg∈KCg∪σH1Q-v⋃Cg∈KCg∪σH1Q∖r,r∈EThe Owen value [7] is defined as shown in the following:(12)Owrv=∑K⊆ΩCg∉K∑Q⊆Cgr∈QK!Ω-K-1!Ω!Q-1!Cg-Q!Cg!v⋃Cg∈KCg∪Q-v⋃Cg∈KCg∪Qr,r∈E


## 3.1. 3-Step Profit Distributive Approach for Coalitions

When the profits of the grand-coalitionE are shared among player 0 and all players ii∈N, both coalition structures and permission structures should be considered when selecting the most suitable method for dividing those profits. The Shapley value, core, least core, and nucleolus allocation methods cannot be used in the profit distribution for coalition E, since the grand-coalition E is composed of a priori unions. Player 0 (the transfer sector) forms a priori union with the players who satisfy its preferences, and each of the other players can be regarded as a priori union.We now propose a 3-step distribution approach based on the principle of the P-Owen value [6] and Owen value [7] that distributes the profit of a coalition among different a priori unions first and then distributes within each a priori union. This approach is improved according to the specific research of this paper. The 3-step distribution approach can be described as follows.Step 1.
Define a dummy allianceDE, vDE=0 and suppose that (1) there are no permission structures between the coalition E and DE, (2) E and DE cooperate with each other, and they form coalition DE∪E, and (3) the revenue of the coalition is v{DE∪E}=vE. We distribute v{DE∪E} to E and DE using the Shapley value. DE obtains u(DE)=0, and E gets u(E)=vE.Step 2.
AllocateuE among the a priori unions Cgg∈M within E by using the Shapley value. Let K denote the alliance composed by Cgg∈M, K⊆Ω, K≠∅, where Ω denotes the coalition structure of E, or the complete set of all a priori unions within E. Taking into account the permission structures among the a priori unions, the distribution of profit for each a priori union can be determined by the following equation:(7)SHCgu=∑K⊆ΩCg∈KK-1!Ω-K!Ω!uK-uK∖Cg,Cg∈Ω
whereΩ,K represents the number of a priori unions in Ω,K.
BecauseK has the possibility of cooperating with the dummy coalition DE, the profit share of K depends on its bargaining power. Let uK denote the profit share of K when K cooperates with DE in place of E. Then,(8)uK=K∪DE-1!K∪DE-K∪DE!K∪DE!vDE∪σH2K-vDE+K-1!K∪DE-K!K∪DE!vσH2K-v∅
whereσH2(K) represents the sovereign part of K. Based on the analysis in Step 1, vDE∪σH2(K)=vσH2(K), and vDE=0. The method of calculating vσH2(K) can refer to the definition of the sovereign part of U (in Section 2.2), and the utility function of the coalitions (Section 2.3) should be considered. In addition, as K∖Cg⊂K⊆Ω, u(K∖Cg) can be calculated by (8).Step 3.
AllocateSHCg(u) among the players within Cgg∈M by using the Shapley value. Since each a priori union in {C2,…,Cm} includes one player, only the revenue distribution of a priori union C1 needs to be discussed further.
LetQ⊆C1, Q≠∅, taking into account the permission structure that exists between player 0 and the other players within C1. The profit allocation solution for C1 is equal to(9)SHrγ=∑Q⊆C1r∈QQ-1!C1-Q!C1!γQ-γQ∖r,r∈C1
In (9), Q denotes the alliance composed by the players in C1, and γQ represents the profit share of Q when Q cooperates with external coalitions (C2,…,Cm) in place of C1. Then γQ can be calculated as follows:(10)γQ=∑R⊆C1Q∈RR-1!C1-R!C1!v⋃Cg∈Ω∖C1Cg∪σH1Q-v⋃Cg∈Ω∖C1CgBased on the above 3-step procedure of profit distribution, all revenue of the grand-coalitionE can be allocated to each player. Since the proposed approach completely allocates the profit of the grand-coalition E among a priori unions on the basis of the Shapley value and then completely divides the revenue of the a priori unions to each player by using the Shapley value as well, the profit of E can be distributed in full; i.e., SH0+∑SHj=vE. This distribution satisfies the definition of validity.


## 3.2. P-Owen Value and Owen Value

This section provides the formulas for the P-Owen value and Owen value for ease of comparison between the distribution results of the 3-step approach and these two methods.The P-Owen value [6] under the permission structure H1 is defined as shown in the following:(11)OwrH1v=∑K⊆ΩCg∉K∑Q⊆Cgr∈QK!Ω-K-1!Ω!Q-1!Cg-Q!Cg!v⋃Cg∈KCg∪σH1Q-v⋃Cg∈KCg∪σH1Q∖r,r∈EThe Owen value [7] is defined as shown in the following:(12)Owrv=∑K⊆ΩCg∉K∑Q⊆Cgr∈QK!Ω-K-1!Ω!Q-1!Cg-Q!Cg!v⋃Cg∈KCg∪Q-v⋃Cg∈KCg∪Qr,r∈E


## 4. Case Study

### 4.1. Background

The Karoon river basin in Iran has two main rivers: the Karoon and the Dez. However, the basin experiences problems meeting its own water demands during parts of the year, especially in drought years [4]. As a result, various sectors in the Karoon basin collaborate to preserve and share water resources. The Khuzestan local agricultural sector, Khuzestan modern agroindustrial sector, and the Khuzestan old agroindustrial sector join in water resources cooperation [5]. The Khuzestan local agricultural sector saves irrigation water from crop production. Both the Khuzestan modern and old agroindustrial sectors hope to acquire water rights from the agricultural sector to ensure their normal industrial production operations [3].Once the agroindustrial sectors obtain water rights, they must share any revenue gained from using the water with the agricultural sector. The agricultural sector has an evident power advantage in the water resources cooperation, because it can choose one agroindustrial sector as its priority for collaboration based on its preferences for a cooperative partner. In addition, the agricultural sector has veto rights in the cooperation, which means that the agroindustrial sectors cannot acquire or exercise rights over the conserved and shared water without the agricultural sector’s permission.Consider the 3 sectors mentioned above as players 0, 1, and 2. Since industrial collaboration between player 0 and player 1 has been more frequent, player 0 prefers to transfer its frugally conserved water to player 1 rather than to player 2. Once player 1’s capacity has been reached by water transfer from player 0, then(α1-1)b1X1+d1 is the value added to the utility of the a priori union composed of players 0 and 1, where α1=1.002.The water use benefit coefficientbr, initial water resources volume Xr, water requirement capacity CAr, and water deficit dr of the 3 players are represented in Table 1.Table 1
Water use benefit coefficient and related water resources volume of players [3].
Player b r X r C A r d r 0 0.28 134(A0) 200 1 0.33 11.6 22.5 10.9 2 0.30 17 27.5 10.5Player 0’s original water-saving rate is 40%, and that rate can increase to 70% after player 0 fundsF0 for irrigation water conservation costs. When the rate rises to 70%, the water requirement of player 0 can be reduced to 200×40%/70%=114.29, and the water volume saved after water conservation by player 0 is 134-114.29=19.714. The frugally conserved water can be utilized by player 0, or it can be transferred to player 1 or player 2. Based on (6), if player 0 utilizes the conserved water, then its payoff is v0=5.5199-F0.The original water-reuse rate of player 1 is 0%, based on (4). The volume of frugally conserved water saved by player 1 is nonnegative if and only if its water-reuse rate spikes over 94%. However, this rate is difficult to reach because it costs too much, so as a result v1=0.The original water-reuse rate of player 2 is 0%, and that rate can increase to 65% after player 2 fundsF2 for water conservation reform costs. Based on (4), v2=0.1+8.25β2-F2, since v2>0; therefore, F2-8.25β2<0.1.Moreover, based on (1)–(3) and (5), the payoffs of the a priori union {0,1}, grand-coalition 0,1,3, and other coalitions are shown as follows. If player 0 utilizes the water surplus after water rights transfer in these coalitions, then BE=0.28.Based on the computed results in Table2, v0,1≥v1+v0=5.5199-F0. Since F2-8.25β2<0.1, therefore, v0,1,2≥v1+v0,2=5.7299-F0, v0,1,2≥v0,1+v2, v0,1,2≥v1,2+v0, v0,1,2≥v1+v2+v0, and v0,2≥v2+v0.Table 2
Profits of coalitions.
{ 0,1 , 2 } { 0,1 } { 0,2 } { 1,2 } 6.3897-F0 6.2134-F0 5.7299-F0 0.1 + 8.25 β 2 - F 2In addition, the coalition structure of{0,1,2} can be expressed as Ω={C1,C2}, where C1=0,1,C2={2}.


### 4.2. Profit Distribution for Coalitions

Since player 0 has veto rights as well as preferences for collaborative cooperation, permission structures exist among the a priori unions as well as within a priori unionC1. The payoff of coalition {0,1,2} can be allocated with the proposed 3-step approach, and the distribution results can be compared with the distribution results using the Owen value and P-Owen value. The profit obtained by each a priori union in the distribution is shown in Table 3, and the profit obtained by each player is shown in Table 4.Table 3
Payoffs of a priori unions obtained in profit distribution ofv{0,1,2}.
3-step P-Owen Owen C 1 6.3015-F0 6.2515-F0 + 0.5(F2-8.25β2) 6.2515-F0 + 0.5(F2-8.25β2) C 2 0.0882 0.1382-0.5(F2-8.25β2) 0.1382-0.5(F2-8.25β2)Table 4
Payoffs of players obtained in profit distribution ofC1,C2.
3-step P-Owen Owen 0 5.9382-F0 + 0.25(F2-8.25β2) 5.9132-F0 + 0.5(F2-8.25β2) 5.9132-F0 + 0.5(F2-8.25β2) 1 0.3633-0.25(F2-8.25β2) 0.3383 0.3383 2 0.0882 0.1382-0.5(F2-8.25β2) 0.1382-0.5(F2-8.25β2)Sincev1=0, the Owen value and P-Owen value return the same distribution results.To make all players accept the distribution results ofv0,1,2, the profit share of each player should not be less than the player’s water use profit if the player were to save and utilize the water for itself, which can be denoted as vr,r=0,1,2.. F2-8.25β2 should be within [0.0118,0.076].From Tables2, 3, and 4, the following can be obtained.(  1) The Allocation Results of the Proposed 3-Step Payoff Distributive Method Can Reflect the Permission Structure among A priori Unions. Since F2-8.25β2≤0.076, when we distribute v0,1,2 using the proposed 3-step method (Table 3), C1 obtains more payoff from profit distribution than using the distribution method based on the Owen value or P-Owen value. C2(player 2) obtains less payoff than it would under the other two distribution methods. The results of the 3-step method reflect the permission structure between C1,C2. Furthermore, the results acknowledge the veto rights of player 0 over player 2 during their cooperation. However, the distribution based on the Owen value or P-Owen value cannot reflect such a conclusion.(  2) The Allocation Results of the Proposed 3-Step Payoff Distributive Method Can Reflect the Permission Structure within A priori Unions. In this case study, within a priori union C1, a permission structure exists between player 0 and player 1 because player 0 has veto power over player 1’s use of water resources saved and shared by player 0.When we distributevC1 with the 3-step method (Table 4), player 0 obtains 5.9382-F0 + 0.25(F2-8.25β2), and player 1 obtains 0.3633-0.25(F2-8.25β2). When we distribute vC1 based on the Owen value, which does not consider the permission structure within a priori union C1, player 0 obtains 5.9132-F0 + 0.5(F2-8.25β2), and player 1 obtains 0.3383, where 0.0118≤F2-8.25β2≤0.076.From the distribution results, we can observe that players 0 and 1 both obtain more allocation of profit based on the 3-step method than based on the Owen value. However,v1=0 in this case study, which causes σH11=0=v1 when we divide the profit of C1 based on the third step of the 3-step method (see (10)). Consequently, the profit obtained by player 0 under the 3-step method minus the profit obtained using the Owen value equals the profit obtained by player 1 under the 3-step method minus the profit under the Owen value, which can be represented as 0.025-0.25(F2-8.25β2) =0.025-0.25(F2-8.25β2).Ifv1≠0, for example, v1=0.04, then σH11=0≠v1, based on (9)–(10). The distribution results under the proposed 3-step method will not change, but player 0 will obtain 5.8932-F0+0.5(F2-8.25β2), and player 1 will obtain 0.4383 under the Owen value. In other words, the profit obtained by player 0 under the 3-step method minus the profit obtained using the Owen value no longer equals the profit obtained by player 1 under the 3-step method minus the profit obtained using the Owen value, which can be represented as 0.045-0.25(F2-8.25β2) > 0.005-0.25(F2-8.25β2). Thus, the allocation results under the 3-step method can reflect the dominant power of player 0.(  3) The Allocation Results under the 3-Step Method and P-Owen Value Are Different Because the Profit of C 1 Obtained under the Two Methods Is Unequal. Since 0.0118≤F2-8.25β2≤0.076, when the payoff of a priori union C1 is divided, player 0 receives 5.938235-F0+0.25(F2-8.25β2) under the 3-step method. This amount is higher than the profit obtained by player 0 under the P-Owen value method: 5.913225-F0+0.5(F2-8.25β2)). Player 1 receives 0.3633-0.25(F2-8.25β2) under the 3-step method, which is higher than the profit obtained by player 1 under the P-Owen value method: 0.3383.In fact, the last step of the 3-step payoff distribution method is the same as the second step of the method that uses the P-Owen value. Both methods take into consideration the permission structure within the a priori union. Further, they both calculate the Shapley value (see (9)) of each player in C1, based on the payoff of C1’s subsets when they cooperate and share profit with exterior coalitions (C2,…,Cm) instead of C1. However, the allocations that result from using the 3-step method are different from the allocations established by using the P-Owen value, because the 3-step method adequately considers the permission structure among a priori unions. Since C1 has veto power in the permission structure, it obtains more profit sharing from v0,1,2 than the other a priori unions.


## 4.1. Background

The Karoon river basin in Iran has two main rivers: the Karoon and the Dez. However, the basin experiences problems meeting its own water demands during parts of the year, especially in drought years [4]. As a result, various sectors in the Karoon basin collaborate to preserve and share water resources. The Khuzestan local agricultural sector, Khuzestan modern agroindustrial sector, and the Khuzestan old agroindustrial sector join in water resources cooperation [5]. The Khuzestan local agricultural sector saves irrigation water from crop production. Both the Khuzestan modern and old agroindustrial sectors hope to acquire water rights from the agricultural sector to ensure their normal industrial production operations [3].Once the agroindustrial sectors obtain water rights, they must share any revenue gained from using the water with the agricultural sector. The agricultural sector has an evident power advantage in the water resources cooperation, because it can choose one agroindustrial sector as its priority for collaboration based on its preferences for a cooperative partner. In addition, the agricultural sector has veto rights in the cooperation, which means that the agroindustrial sectors cannot acquire or exercise rights over the conserved and shared water without the agricultural sector’s permission.Consider the 3 sectors mentioned above as players 0, 1, and 2. Since industrial collaboration between player 0 and player 1 has been more frequent, player 0 prefers to transfer its frugally conserved water to player 1 rather than to player 2. Once player 1’s capacity has been reached by water transfer from player 0, then(α1-1)b1X1+d1 is the value added to the utility of the a priori union composed of players 0 and 1, where α1=1.002.The water use benefit coefficientbr, initial water resources volume Xr, water requirement capacity CAr, and water deficit dr of the 3 players are represented in Table 1.Table 1
Water use benefit coefficient and related water resources volume of players [3].
Player b r X r C A r d r 0 0.28 134(A0) 200 1 0.33 11.6 22.5 10.9 2 0.30 17 27.5 10.5Player 0’s original water-saving rate is 40%, and that rate can increase to 70% after player 0 fundsF0 for irrigation water conservation costs. When the rate rises to 70%, the water requirement of player 0 can be reduced to 200×40%/70%=114.29, and the water volume saved after water conservation by player 0 is 134-114.29=19.714. The frugally conserved water can be utilized by player 0, or it can be transferred to player 1 or player 2. Based on (6), if player 0 utilizes the conserved water, then its payoff is v0=5.5199-F0.The original water-reuse rate of player 1 is 0%, based on (4). The volume of frugally conserved water saved by player 1 is nonnegative if and only if its water-reuse rate spikes over 94%. However, this rate is difficult to reach because it costs too much, so as a result v1=0.The original water-reuse rate of player 2 is 0%, and that rate can increase to 65% after player 2 fundsF2 for water conservation reform costs. Based on (4), v2=0.1+8.25β2-F2, since v2>0; therefore, F2-8.25β2<0.1.Moreover, based on (1)–(3) and (5), the payoffs of the a priori union {0,1}, grand-coalition 0,1,3, and other coalitions are shown as follows. If player 0 utilizes the water surplus after water rights transfer in these coalitions, then BE=0.28.Based on the computed results in Table2, v0,1≥v1+v0=5.5199-F0. Since F2-8.25β2<0.1, therefore, v0,1,2≥v1+v0,2=5.7299-F0, v0,1,2≥v0,1+v2, v0,1,2≥v1,2+v0, v0,1,2≥v1+v2+v0, and v0,2≥v2+v0.Table 2
Profits of coalitions.
{ 0,1 , 2 } { 0,1 } { 0,2 } { 1,2 } 6.3897-F0 6.2134-F0 5.7299-F0 0.1 + 8.25 β 2 - F 2In addition, the coalition structure of{0,1,2} can be expressed as Ω={C1,C2}, where C1=0,1,C2={2}.


## 4.2. Profit Distribution for Coalitions

Since player 0 has veto rights as well as preferences for collaborative cooperation, permission structures exist among the a priori unions as well as within a priori unionC1. The payoff of coalition {0,1,2} can be allocated with the proposed 3-step approach, and the distribution results can be compared with the distribution results using the Owen value and P-Owen value. The profit obtained by each a priori union in the distribution is shown in Table 3, and the profit obtained by each player is shown in Table 4.Table 3
Payoffs of a priori unions obtained in profit distribution ofv{0,1,2}.
3-step P-Owen Owen C 1 6.3015-F0 6.2515-F0 + 0.5(F2-8.25β2) 6.2515-F0 + 0.5(F2-8.25β2) C 2 0.0882 0.1382-0.5(F2-8.25β2) 0.1382-0.5(F2-8.25β2)Table 4
Payoffs of players obtained in profit distribution ofC1,C2.
3-step P-Owen Owen 0 5.9382-F0 + 0.25(F2-8.25β2) 5.9132-F0 + 0.5(F2-8.25β2) 5.9132-F0 + 0.5(F2-8.25β2) 1 0.3633-0.25(F2-8.25β2) 0.3383 0.3383 2 0.0882 0.1382-0.5(F2-8.25β2) 0.1382-0.5(F2-8.25β2)Sincev1=0, the Owen value and P-Owen value return the same distribution results.To make all players accept the distribution results ofv0,1,2, the profit share of each player should not be less than the player’s water use profit if the player were to save and utilize the water for itself, which can be denoted as vr,r=0,1,2.. F2-8.25β2 should be within [0.0118,0.076].From Tables2, 3, and 4, the following can be obtained.(  1) The Allocation Results of the Proposed 3-Step Payoff Distributive Method Can Reflect the Permission Structure among A priori Unions. Since F2-8.25β2≤0.076, when we distribute v0,1,2 using the proposed 3-step method (Table 3), C1 obtains more payoff from profit distribution than using the distribution method based on the Owen value or P-Owen value. C2(player 2) obtains less payoff than it would under the other two distribution methods. The results of the 3-step method reflect the permission structure between C1,C2. Furthermore, the results acknowledge the veto rights of player 0 over player 2 during their cooperation. However, the distribution based on the Owen value or P-Owen value cannot reflect such a conclusion.(  2) The Allocation Results of the Proposed 3-Step Payoff Distributive Method Can Reflect the Permission Structure within A priori Unions. In this case study, within a priori union C1, a permission structure exists between player 0 and player 1 because player 0 has veto power over player 1’s use of water resources saved and shared by player 0.When we distributevC1 with the 3-step method (Table 4), player 0 obtains 5.9382-F0 + 0.25(F2-8.25β2), and player 1 obtains 0.3633-0.25(F2-8.25β2). When we distribute vC1 based on the Owen value, which does not consider the permission structure within a priori union C1, player 0 obtains 5.9132-F0 + 0.5(F2-8.25β2), and player 1 obtains 0.3383, where 0.0118≤F2-8.25β2≤0.076.From the distribution results, we can observe that players 0 and 1 both obtain more allocation of profit based on the 3-step method than based on the Owen value. However,v1=0 in this case study, which causes σH11=0=v1 when we divide the profit of C1 based on the third step of the 3-step method (see (10)). Consequently, the profit obtained by player 0 under the 3-step method minus the profit obtained using the Owen value equals the profit obtained by player 1 under the 3-step method minus the profit under the Owen value, which can be represented as 0.025-0.25(F2-8.25β2) =0.025-0.25(F2-8.25β2).Ifv1≠0, for example, v1=0.04, then σH11=0≠v1, based on (9)–(10). The distribution results under the proposed 3-step method will not change, but player 0 will obtain 5.8932-F0+0.5(F2-8.25β2), and player 1 will obtain 0.4383 under the Owen value. In other words, the profit obtained by player 0 under the 3-step method minus the profit obtained using the Owen value no longer equals the profit obtained by player 1 under the 3-step method minus the profit obtained using the Owen value, which can be represented as 0.045-0.25(F2-8.25β2) > 0.005-0.25(F2-8.25β2). Thus, the allocation results under the 3-step method can reflect the dominant power of player 0.(  3) The Allocation Results under the 3-Step Method and P-Owen Value Are Different Because the Profit of C 1 Obtained under the Two Methods Is Unequal. Since 0.0118≤F2-8.25β2≤0.076, when the payoff of a priori union C1 is divided, player 0 receives 5.938235-F0+0.25(F2-8.25β2) under the 3-step method. This amount is higher than the profit obtained by player 0 under the P-Owen value method: 5.913225-F0+0.5(F2-8.25β2)). Player 1 receives 0.3633-0.25(F2-8.25β2) under the 3-step method, which is higher than the profit obtained by player 1 under the P-Owen value method: 0.3383.In fact, the last step of the 3-step payoff distribution method is the same as the second step of the method that uses the P-Owen value. Both methods take into consideration the permission structure within the a priori union. Further, they both calculate the Shapley value (see (9)) of each player in C1, based on the payoff of C1’s subsets when they cooperate and share profit with exterior coalitions (C2,…,Cm) instead of C1. However, the allocations that result from using the 3-step method are different from the allocations established by using the P-Owen value, because the 3-step method adequately considers the permission structure among a priori unions. Since C1 has veto power in the permission structure, it obtains more profit sharing from v0,1,2 than the other a priori unions.


## 5. Conclusions

Based on the application of game theory for interactions that include coalition structures and permission structures, this paper explored the issue of profit distribution faced by water resources collaborative alliances when the water rights transfer sector dominates the water requirement sectors. We proposed a new 3-step method of profit distribution for the coalition: utilizing the Shapley value, this method divided all the profits of the coalition among all the a priori unions within the coalition and then allocated the total profit of each a priori union to every player within each union. In this manner, the profits of the water resources cooperative coalition could be allocated in full, demonstrating that the distributive results of the 3-step method were effective.In contrast to methods using the Owen value and P-Owen value, the results of our case study showed that the 3-step method allowed a priori unions and water rights transfer sectors who have veto rights to obtain more payoffs from the coalition’s profit distribution. In other words, it was evident that, compared with previous method, the proposed 3-step method can better reflect the power held by the water rights transfer sector based on the veto power it possessed.Furthermore, considering the character of the profit distribution issue discussed in this study, our proposed 3-step method was a modification of the approach that uses the P-Owen value. The last step of the 3-step payoff distribution method was the same as the second step of the P-Owen value method. The allocation results under the 3-step method differed from the allocation results obtained using the P-Owen value because the 3-step method adequately considered the permission structure among a priori unions. Therefore, we demonstrated that the 3-step method introduced by this study can provide effective distribution of the profits of a water resources coalition in which the water transfer sector is dominant.This paper also raises a question for future research: we note that our analysis concentrates on the cooperative coalition with only one water rights transfer sector and many water requirement sectors. In future work we hope to explore the feasible revenue distribution approach for problems of many water rights transfer sectors and many water requirement sectors.



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