Benefit Distribution Mechanism of a Cooperative Alliance for Basin Water Resources from the Perspective of Cooperative Game Theory

Abstract

At present, global water resource security is facing serious threats, and the construction of a cooperative, open, and mutually beneficial water resource community is a potential solution to the global water resource crisis and water resource hegemony. Previous studies on the formation and beneficial distribution of water resources in cooperative alliances have more often focused on the idea that participants take all of their water resources when they join a cooperative alliance (i.e., a crisp cooperative alliance), while fewer studies have focused on participants including different proportions of their water resources and joining multiple cooperative alliances (i.e., fuzzy cooperative alliances), and even fewer comparative studies concern the use of different benefit-sharing mechanisms. In this paper, in order to improve the efficiency of water use, allocate water resources more optimally, and generate higher returns for water users in a given basin, we propose the establishment of a traditional crisp and improved fuzzy cooperative alliance for water resources in the basin from the perspective of cooperative game theory; we examine the water resource allocation mechanism within the alliance based on the principle of priority; we construct a benefit allocation mechanism for the cooperative alliance based on the core, least core, weak least core, and Shapley value method; and we carry out empirical research using the example of the Tarim River Basin. Our findings are as follows: (1) A cooperative alliance based on the perspective of cooperative game theory can effectively improve overall benefits and individual benefits, and a fuzzy cooperative alliance is more effective than a crisp cooperative alliance in improving the overall water benefits of the region. (2) The participants in the fuzzy cooperative alliance can obtain more benefits than in the crisp cooperative alliance, and the benefit distribution mechanism of the cooperative alliance helps the participants to determine the object of cooperation while maintaining the sustainable existence of the alliance. (3) The different methods of benefit distribution within the cooperative alliance directly affect the overall water benefits of the region. (4) The different methods of benefit distribution directly affect the stability of the alliance, and the players in the game have heterogeneous preferences for different distribution schemes. The resource-sharing mechanism and benefit distribution mechanism of a water resource cooperative alliance have good applicability as solutions to the problem of water resource optimization and allocation in river basins, and they may provide policy references for the efficient use of water resources and optimization of water resource allocation and management in areas with a shortage of water resources, such as arid and semi-arid zones.

1. Introduction

Water resources are the lifeblood of social and economic development, serving as both basic natural resources and strategic economic resources. At present, global tension is heating up in the form of local conflicts, and global water security is under a major threat, seriously affecting global soil. The scarcity of available water resources demonstrates that the era of global water hegemony has come [1]. Many are investigating how to ensure that the limited water resources are distributed under conditions of effective regional support, to optimize the framework of “resource endowment—socio-economic environment—ecological environment” and secure the normal operation of the composite system. Achieving regional economic sustainability, social sustainability, ecological sustainability, and sustainable development of water resources has become imperative within high-quality water development. This paper focuses on the intersection of water resource management, cooperative game theory, institutional economics, and other disciplines, aiming to explore the cooperative mechanisms and distribution of benefits in water shortage areas in order to achieve the sustainable use and optimal allocation of water resources. This paper proposes a win–win cooperation idea: to build a regional (basin water system) water rights cooperation alliance; to cooperate in the form of water resource integration, with this alliance acting as a “tangible hand” to achieve the optimal allocation of water resources and to meet the water use needs of the region’s “collective rationality” and “individual rationality”; and to thereby form a sustainable regional water resource community. When participating in a cooperative game, players can receive higher income than when they are “using it alone”, which creates an incentive for “cooperating” with competitors in the game. However, due to individual rationality, the players want to obtain higher excess income within the cooperative alliance, and so they have to bargain. Cooperation aims to make a bigger overall reward for players, and how this reward is distributed is at the core of the bargaining that occurs between players. The development of a scientific and reasonable resource-sharing mechanism and benefit distribution mechanism is a necessary condition for the existence of a cooperative alliance. The most important and thorny problem in forming this alliance is that of meeting the psychological expectations of all the players in the distribution program in order to ensure the sustainable existence of the cooperative alliance and the win–win cooperation of the agents. In this paper, we will discuss the considerations that regional water units must address in joining a cooperative alliance, how they might take resources to that alliance once they have joined, whether to join a large or small alliance, whether to join more than one alliance, how the cooperative alliance should allocate resources and benefits, how to ensure the perpetuity of the alliance, and how to maximize collective and individual benefits through a cooperative alliance. These realistic problems, which need to be explored in constructing a cooperative alliance, constitute the research content of this paper.

In the state of a non-cooperative game, the players, based on individual rationality, will choose a strategy for maximizing their own interest, that is, they will choose to maximize the occupation of all the resources for their own use. If all the water units in the region act on individual rational choice in this way, it will lead to a collective irrational choice and the emergence of the “tragedy of the commons” phenomenon, and the region as a whole will end up in an irrational state. In such a scenario, the system is inefficient in the utilization of resources, accompanied by the phenomenon of serious waste of resources, resulting in a series of externality problems. In a classical cooperative game, there are only two options for the players with respect to the cooperative alliance: they either bring all their resources to join the cooperative alliance or do not participate in the cooperative alliance at all. From the perspective of the whole system of the game, the existing members of the cooperative alliance would be very happy to see players bring all their resources to the alliance, but this is not realistic. Due to the uncertainty of the economic environment and the high complexity of individual decision making, it is not enough to provide only the two strategic choices of “full participation” and “no participation at all” when players participate in a cooperative alliance. In other words, a player may need multiple options in order to participate in a cooperative alliance, e.g., to carry part of their resources to an alliance or to multiple alliances (to obtain more benefits from them). With this in mind, in 1982, Aubin first proposed the concept of a fuzzy cooperative alliance in which the players can choose to carry some of their resources to participate in the cooperation, with the degree of cooperation represented by a real number [0, 1] [2]. When the concept of the fuzzy cooperative game appeared, it immediately received great attention from scholars at home and abroad, who expanded upon and applied the fuzzy cooperative alliance, which has now become an important branch of and research hotspot in the field of cooperative game theory.

The goal of a fuzzy cooperative game is to solve the problem of the uncertain expectations of the players in the game, and the fuzzy cooperative game model is applied by introducing the concepts of mathematical tools and distribution functions into the cooperative game [3]. Many domestic and foreign scholars are interested in the fuzzy cooperative game model and carry out research in terms of model expansion, characteristic function, and benefit distribution, etc. Hugh Chen et al. describe the multiple types of Shapley value features and the methods used to calculate each one, characterizing two distinct families of approaches: model-agnostic and model-specific approximations [4]. Indra Kumar et al. defined a cooperative game that described the value of different subsets of the model’s features then calculated the resulting game’s Shapley values to attribute credit additively between the features [5]. Ian Covert et al. revisited the idea of estimating Shapley values via linear regression to understand and improve upon this approach [6]. Benedek Rozemberczki et al. gave an overview of the most important applications of the Shapley value in machine learning: feature selection, explainability, multi-agent reinforcement learning, ensemble pruning, and data valuation [7]. The most important issue for stakeholders in a cooperative alliance is how they might share the profit generated by the alliance, and the fuzzy gain distribution of a fuzzy cooperative alliance also leads to imprecision in the potential gains of the players in the game, so many scholars have proceeded to study the gain distribution schemes of fuzzy cooperative alliances [8,9,10]. Jafarzadegan et al. proposed a fuzzy variable minimization kernel as a new solution for fuzzy cooperative responses, termed Fuzzy variable least core (FVLC). FVLC contains fuzzy input variables, which can be computed as the proportion of fuzzy benefits for the people involved in the coalition [11]. The current methods for solving fuzzy cooperative coalitions are mainly divided into two categories: one is the dominant solution, such as core [12], least core [13,14], stable set [15,16], etc., and the other is the valuation solution, such as Shapley’s value method [8,17,18,19,20,21,22,23], Banzhaf values [24], Myerson’s value [25], and the $\tau$ value [26,27,28], etc. The above fuzzy cooperative games are solved from different perspectives. They all have their own ways of solving and characteristics; however, there are also relative defects in these different solving methods. Shapley, for instance, although satisfying the collective rationality, satisfies the individual rationality = less; the nucleus and core have very high requirements for solution, and frequently the result is an empty set. The nucleus exists and is unique. The process of solving becomes very cumbersome when the number of people in authority increases. There is no effective solution to the stable set, and the $\tau$ value needs to satisfy the conditions of the proposed equilibrium in order to solve the equilibrium cooperative game. This means that a covariance is required under the equivalence of the strategies. The Shapley value method apportions the benefits (costs) according to the marginal benefits (costs) of the players, and the benefit of each individual participating in the cooperative game is equal to the expected value of its marginal contribution, which reflects the degree of contribution that each player can provide after participating in the cooperative alliance. The Shapley value reflects the degree of contribution in the cooperative game, avoids the egalitarian distribution, and fits practical application scenarios. In addition to reflecting the players’ process in the cooperative game, the Shapley value method is more reasonable and fairer than the allocation based on the proportion of the value of resource inputs, the proportion of resource allocation efficiency, or the combination of both [29]. Because the Shapley value method focuses on individual marginal contribution, takes collective rationality as the main objective function, and the solution conditions are relatively loose, it has become a common method that is used in the academic community to solve the distribution of gains in cooperative games.

The fuzzy cooperative game method is widely used in the optimization of water resource allocation and revenue distribution, and scholars at home and abroad have applied the fuzzy cooperative game method to water resource management and the application of various aspects of research. Sadegh used the fuzzy cooperative game theory to optimize the allocation of water resources between basins; they also redistributed the revenue to water users using the fuzzy Shapley value method [18]. Sadegh et al. also adopted a new concept of the fuzzy cooperative game, the fuzzy variable minimization kernel (FVLC), for the solution of fuzzy variable multilateral cooperation in the strategy benefit switching algorithm, which improves the applicability and efficiency of the original fuzzy cooperative game solution [30]. Liu D et al. comprehensively considered the spatial and temporal characteristics of the water demand of transnational river basin riparian countries in terms of the geographic location, water demand, and water use efficiency of each country, and they used the fuzzy Shapley value method to redistribute the benefits to the water users [18]. Abed-Elmdoust A et al. applied two fuzzy countermeasures with fuzzy eigenfunctions, namely, the Hukuhara difference and the Choquet integral, to achieve fair and efficient water allocation to water users in trans-basin and intra-basin problems [31]. Zhang Kai et al. used fuzzy cooperative game and crisp cooperative game models to address the defects of Shapley values in benefit allocation, adding investment and water use efficiency factors. They proposed a new benefit allocation strategy for the watershed water rights cooperative alliance and applied it to the Manas River Basin [32]. Sun Dongying et al. constructed a model for the optimization of basin water resource allocation based on a fuzzy alliance cooperation game, and they used the fuzzy Shapley value method to distribute alliance benefits [33]. Tan Jiayin et al. constructed a three-stage model for the optimization of water resource allocation based on the Choquet integral form of the fuzzy cooperative game, which is considered to maximize the level of overall regional water use benefits and achieve the optimal allocation of water resources in regions with water shortages [34]. Li Fang et al. constructed a cross-border water resource cooperative game model with fuzzy payments and a cross-border water resource cooperative game model with double-fuzzy elements to explore the optimal reallocation of cross-border water resources in different fuzzy scenarios [35]. In addition, many scholars have applied the cooperative game model to transboundary water resource cooperation, such as Bernauer T et al. [36], Yuan L et al. [37], Rigi H et al. [38], Duan W et al. [39], Zhu K et al. [40], Wang L et al. [41], Nishizaki I et al. [42], and Liu X et al. [43], who analyzed the impacts of the cooperative game in the context of trans-regional and trans-national water resource cooperation from the perspectives of different countries. Such studies proved that cooperative games have very important significance and a major role in cross-border water resource cooperation. Many other scholars have studied stakeholder analyses of water resource allocation by using different game methods, such as Roozbahani R et al. [38], Hargrove W L et al. [44], Mahjouri N et al. [45], and Wang S et al. [46].

In recent years, research on fuzzy cooperative games and the optimal allocation of water resources has gradually increased, especially the application of fuzzy cooperative games to different scenarios, which has become an important branch of game theory. Research on fuzzy cooperative games in China started late, and there few studies have focused on the optimal allocation of water resources, especially through targeted research taking into account regional conditions. Water in China is extremely scarce, and most simulations do not have practical significance. Therefore, although domestic and international academic research on fuzzy cooperative games and the optimal allocation of water resources has become more mature, there are still some shortcomings in this field. Firstly, it is difficult to balance collective rationality and individual rationality in the existing method for allocating water rights. The preconditions for the existence of a cooperative alliance for water rights include whether the revenue of the cooperative alliance can maintain the sustainability of the alliance; the cooperative alliance needs to ensure collective rationality, and the water users participating in the cooperative alliance need to ensure individual rationality. Existing studies consider the individual rationality of the individuals in the cooperative alliance, and the collective rationality of the collective rationality of the individual is less often taken into account. The current fuzzy cooperative game coalition, which uses the fuzzy Shapley value method to allocate the coalition’s revenue, has the following problems. The fuzzy Shapley value of the allocation result must also be the core solution of the fuzzy cooperative game as a prerequisite, but the core of the fuzzy cooperative game may not exist, and the scheme for allocating fuzzy Shapley values may not satisfy the requirements of collective rationality. The existing revenue allocation methods of a fuzzy cooperative alliance, whether this is the Shapley value method, the nucleolus, the stable set, or the $\tau$ value, cannot simultaneously satisfy the collective rationality of the cooperative game alliance and the individual rationality of the water user. Secondly, most of the current studies on the benefits of a cooperative game alliance for water rights in basins them it from the perspective of agricultural water units or industrial water units [47] and lack an overall analysis of all the water units in the basin (as well as an analysis of the strategic choices and strategic dependence of multiple water users). Thirdly, there are few studies in the existing literature that compare the allocation schemes from various allocation methods used in crisp and fuzzy cooperative games.

This research was motivated by the critical situation of global water security and a deep understanding of the inadequacy of existing water allocation mechanisms. Through the application of co-operative game theory and empirical research, this paper aims to provide new solutions and policy references for water allocation problems. This article shows the construction of a cooperative alliance for water resource sharing based on the priority principle, and we comparatively analyze the distribution of benefits in the core, least core, weak least core, and Shapley value methods in allocating to a crisp cooperative alliance and a fuzzy cooperative alliance in order to create a reasonable scheme for the optimal allocation of water resources and satisfy the collective rationality of the cooperative alliance as well as individual rationality, thus ensuring the stability of the alliance and the maximization of the individual benefit. Taking the Tarim River Basin, the largest inland river basin in China, as an example, we analyze the optimal allocation of agricultural and industrial water units and verify the validity of the framework of the cooperative alliance, resource allocation, and benefit distribution methods proposed in this paper so as to provide some reference for subsequent scholars, policy makers, and diversified water units.

2. Modelling Payment Functions and Benefit Distribution in Water Resource Cooperative Unions

2.1. Cooperative Responses with Alliance Structures

Assuming a set $N=\left\{1,2,\dots ,n\right\}$ of players with a coalitional structure, the set constituted by all the subsets of the set N is $\mathrm{F}\left(N\right)$, and the elements in $\mathrm{F}\left(N\right)$ denote a coalition. The cooperative game with a coalitional structure can be represented as an ordered dual function $\left(N,v\right)$ containing the set N of players in the game and the characteristic function $v$ from $2^N\to R^{+}$, denoted $v\left(\mathsf{\Phi }\right)=0,R^{+}=\left\{r\in R|r\ge 0\right\}$, where $v$ denotes the number of real numbers corresponding to each cooperative coalition with a coalitional structure, denoting the revenues created by the coalition. $N$ denotes the finite (non-empty) collection of n people set; $2^N$ denotes the integration of all sub-coalitions in N.

Definition 1. 

A cooperative game $\left(N,v\right)$ is super-additive if $\left(N,v\right)$ satisfies the following conditions: $v\left(s_1+s_2\right)\ge v\left(s_1\right)+v\left(s_2\right)$, in which $S_1$ and $S_2$ are any of the sets of the game consisting of two coalitions and $s_1\cap s_2=\Phi$ denotes the whole of the super-additive cooperative game as $G^N$.

Definition 2 

[48]. We can sort the players in the set N according to the value of the rate of return $b_i$ from smallest to largest to obtain $b_1\ge b_2\ge b_3\ge \dots \ge b_n$, The entire cooperative coalition consisting of the $N$ players in the set N will be referred to as S, and the set of $S$ will be $P\left(S\right)$. The cooperative coalition after the sorting $\beta =\left\{B_1,B_2,B_3,\dots B_n\right\}$ is called the coalition structure with respect to $N$. $B_i$ is called the preferred coalition, and the entirety of the preferred coalition is denoted $C^N$; the ternary set $\left(N,v,\beta \right)$ is called the cooperative response with coalition structure, and all the cooperative responses in $N$ that have coalition structures are denoted $C{G}^N$. In $C^N,$ the resources are allocated according to the resource usage yield of all members in S in descending order; the member with the highest resource usage yield is given priority in obtaining the resources, and after the upper demand limit of the member is satisfied, the resources in the cooperative coalition are allocated to the member with the next highest resource usage yield until the upper demand limit of the member is satisfied; this continues until all the resources in the cooperative coalition are allocated. This allocation is recorded as the priority principle in the cooperative alliance game.

Definition 3 

[49]. If the cooperative game $\left(N,v,\beta \right)\in C{G}^N$ for any two coalitions $S,T\in {\Omega }_{\beta }$ is formed by the players in the game and $S\cap T,S\cup T\in {\Omega }_{\beta }$ satisfies $v\left(S\cup T\right)+v\left(S\cap T\right)\ge v\left(S\right)+v\left(T\right)$, then $\left(N,v,\beta \right)$ is a cooperative coalition of convex games.

2.2. Payment Function Construction for Traditional Crisp Coalition Cooperation Games

This study makes the assumption that the initial allocation of water rights in the basin has been completed, and that the regional water resource management authorities have allocated water resources in the basin to the water users in accordance with the principles of equity and efficiency of common wealth. After the initial allocation of water rights is completed, water users in the basin can form water resource cooperative alliances, either spontaneously or in an organized manner, which will manage themselves and redistribute water resources and revenues with the “visible hand”, This will allow the limited water resources to support increased social productivity and improve the efficiency of water resource use.

If a water user participates in a cooperative alliance with all of his/her water resources, the player is said to be participating in a crisp cooperative alliance (also known as a crisp cooperative game), i.e., a generally cooperative game with a characteristic function or a cooperative game with a transfer utility. If the water topic transports some of the water resources to participate in different cooperative coalitions, then the players are said to participate in a fuzzy cooperative coalition. The key factors for the existence of a cooperative coalition and the participation of players in the cooperative coalition are the participation rate of players (the proportion of resources they carry) and the benefits they receive after participating in the cooperative game coalition. In this work, we choose the core, minimum kernel, weak minimum kernel and Shapley value method, which are representative of the benefit distribution methods used in cooperative games to solve this problem.

After the allocation of water resources in accordance with the cooperative alliance “priority principle” in Definition 2, the amount of water resources allocated to all members of the cooperative alliance and its unit rate of return on water resources are multiplied and then summed up to obtain the overall benefits $v\left(S\right)$ of the crisp cooperative alliance $S$ [29].

$$v\left(S\right)\left\{\begin{array}{c}{A}_s{b}_1,A_s\le C_1{e}_{is}\\ \dots \\ {\displaystyle \sum _{i=1}^{l-1}}{C}_i{e}_{is}{b}_i+\left(A_s-{\displaystyle \sum _{i=1}^{l-1}}{C}_i{e}_{is}\right)b_n,{\displaystyle \sum _{i=1}^{l-1}}{C}_i{e}_{is}<A_s\le {\displaystyle \sum _{i=1}^l}{C}_i{e}_{is}\\ \dots \\ {\displaystyle \sum _{i=1}^n}{C}_i{e}_{is}{b}_i,{\displaystyle \sum _{i=1}^n}{C}_i{e}_{is}\le A_s\end{array}\right.$$

(1)

where the water use efficiency of the player i participating in the cooperative alliance $S$ when using the water resources alone is $b_i$, the upper limit of water demand when using the water resources alone is $C_i$, the player i in the crisp cooperative alliance can only choose to bring all their resources to join the alliance or not join the alliance (i.e., go it alone), and all the water resources held by the cooperative alliance $S$ are $A_s=\sum _{i=1}^n{e}_{is}{X}_i$, where $e_{is}$ is a 0–1 variable indicating whether or not player i participates in the cooperative coalition $S$. If they do, then $e_{is}=1$, and the opposite is 0. $X_i$ denotes the amount of water resources that player $i$ receives in the initial allocation of water rights.

The benefit $v\left(S\right)$ of the crisp cooperative alliance $S$ can be expressed when the total amount of water resources held by $S$ is less than the upper limit of the demand $b_1$ of the most efficient individual in the cooperative alliance when using water resources individually. The benefit $v\left(S\right)$ of the alliance $S$ is the amount of water resources of $S$ multiplied by the water use efficiency of the most efficient individual in using water resources, that is, in the cooperative alliance When the total amount of water resources held by $S$ is greater than $b_1$ and less than the sum of the maximum demand of all water-using entities, the water resources will be allocated to individual $l-1$ in accordance with the principle of priority, and when it is impossible to satisfy the maximum demand of individual $l-1$ after meeting the maximum demand of individual $l$, the remaining water resources will be allocated to all other individuals in the alliance in an even manner, and the revenue of the alliance will be the total amount of the revenue of the first $l-1$ individuals. When the total water resources held by $S$ are greater than the sum of the maximum demands of all the individuals in the coalition, the coalition revenue is the sum of the revenues generated by the maximum demands of all the individuals.

2.3. Construction of a Payment Function for Improved Fuzzy Coalitional Cooperation Games

The fuzzy cooperative game can be thought of as a more general expression of the crisp cooperative game, where the level of participation of the players extends from $\left\{0,1\right\}$ in the crisp cooperative game to $\left[0,1\right]$ in the fuzzy cooperative game. A fuzzy cooperative coalition with $n$ players can be represented as an n-dimensional variable $e^{\lambda }=\left({\lambda }_1,{\lambda }_2,{\lambda }_3,\dots ,{\lambda }_n\right),$ defined ${\left[0,1\right]}^N$, where the ith coordinate ${\lambda }_i$ is a constant $\left[0,1\right]$ denoting the participation rate of player $i$. The whole of the fuzzy cooperative coalition of $\lambda$ can be denoted $F^N$. $e^{\lambda }=\left(\mathrm{1,1},1,\dots ,1\right)$ is referred to as the maximal coalition, that is, when all the players carry all their resources to join the cooperative coalition. $e^S=\sum _{i\in S}{e}^i$ means that all the players in the set $S$ join the cooperative coalition with a participation rate of 100%, while the players outside the set $S$ do not have any cooperation with the players in $S$, that is, the participation rate is 0.

Definition 4. 

Characteristic equation of a fuzzy cooperative game possessing a set $N$ of players:

$$\tilde{c}:F^N\to {\mathbb{R}}^{+},v\left(e^{\varnothing }\right)=0,{\mathbb{R}}^{+}=\left\{r\in R|r\ge 0\right\}$$

$\tilde{c}:F^N\to {\mathbb{R}}^{+}$ denotes the mapping from $F^N$ to the space of real numbers, which denotes the revenue created by the fuzzy cooperative coalition, and the revenue of the empty set is 0. $\tilde{c}\left(s\right)$ denotes the expected revenue of the fuzzy cooperative coalition S.

Definition 5. 

The configuration set $S\left(v\right)$ of a fuzzy cooperative coalition $v\in {FG}^N$ composed of $N$ people is a collection of n-dimensional vectors that satisfy the effectiveness of the distribution of gains and individual rationality.

$$S\left(v\right)=\left\{x\in {\mathbb{R}}^n|{\sum }_{i\in N}{x}_i=v\left(e^N\right),x_i\ge v\left(e^{\lambda }\right),\forall i\in N\right\}$$

In different water systems, each water user receives different water resource returns, and the trading stage of these alliances will tend to prioritize providing water users with a high rate of return. When the water subject participates in a cooperative alliance containing the highest rate of return to the water user, it is likely they will remain in the alliance due to the relatively low rate of contribution required to obtain a relatively small amount of revenue, and the allocation of water resources will produce a high rate of return to water users while using a relatively small amount of water resources. Therefore, it makes sense for the water users in the basin to join the alliance that does not include the water users with the highest yield, so that the water users with the highest unit yield are excluded from the construction of the fuzzy cooperative alliance. Based on the characteristic function $\tilde{c}$ of the fuzzy cooperative alliance that has been obtained, it is necessary to determine which fuzzy cooperative game alliances will be formed by the players, and we must find the participation rate of the players participating in different fuzzy cooperative alliances and construct the objective function and constraints as follows:

$$\begin{gathered} \mathrm{M}\mathrm{a}\mathrm{x} v\left(S\right)=\sum _{s=1}^n\sum _{i=1}^{n-1}\phi \left(i,s\right) \\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}0\le {\lambda }_{i,s}\le 1\\ {\displaystyle \sum _s}{\lambda }_{i,s}=1,\forall i\\ t\left(s\right)={\displaystyle \sum _{i=1}^{n-1}}{\lambda }_{i,s}\ast X\left(i\right),\forall s\\ w\left(s\right)=\left\{\begin{array}{c}B\left(s\right)\ast t\left(s\right),\mathrm{if}t\left(s\right)\le C\left(s\right)\\ B\left(s\right)\ast C\left(s\right),\mathrm{if}t\left(s\right)>C\left(s\right)\end{array}\right.\\ w\left(i,s\right)=B\left(i\right)\ast {\lambda }_{i,s}\ast A\left(i\right)\\ \phi \left(i,s\right)={\displaystyle \sum _{i\in T\subseteq S}}{\displaystyle \frac{\left(\left|s\right|-1\right)!\left(\left|N\right|-\left|s\right|\right)}{\left|N\right|!}}\left[v\left({\displaystyle \sum _{i\in T}}{s}_i\ast e^i\right)-v\left({\displaystyle \sum _{i\in T\setminus i}}{s}_i\ast e^i\right)\right]\end{array}\right. \end{gathered}$$

(2)

$v\left(S\right)$ is the total benefit of the system, $\phi \left(i,s\right)$ is the benefit gained by the participation of the innkeeper i in the fuzzy coalition S, ${\lambda }_{i,s}$ is the proportion of water resources that the innkeeper $i$ takes with him/her when he/she joins the fuzzy coalition $S$ (participation rate), $t\left(s\right)$ is the amount of water resources that the fuzzy coalition $S$ owns, $X\left(i\right)$ is the amount of water resources allocated to person $i$ in the first step of the game,$w\left(s\right)$ refers to the revenue of the fuzzy cooperative alliance $S$, $w\left(i,s\right)$ is the revenue independently generated by the person $i$ in the game by using the water resources they used to participate in the alliance $S$, $B\left(s\right)$ is the coefficient of return per unit of water resources of the fuzzy cooperative coalition $S$, and $t\left(s\right)$ refers to the sub-coalition. $C\left(s\right)$ of the coalition $S$ refers to the maximum water resources that can be carried by the fuzzy cooperative coalition $S$ in the case of the highest water resource rate of return coefficient $B\left(s\right)$, that is, if the players participating in the coalition $S$ carry more water resources than $C\left(s\right)$, then the coalition’s unit rate of return coefficient will not be maintained at its original level. The unit rate of return coefficient of the fuzzy coalition exists under the specific water resource constraint that the total amount of resources taken into the coalition by the players should not exceed $C\left(s\right)$. The fuzzy Shapley equation contains both the gains obtained by a single player in the non-cooperative game when he participates in the game as an independent individual and the gains obtained by the cooperative game after joining the cooperative coalition. After obtaining the composition structure of the fuzzy cooperative game coalition and the optimal participation rates of the players in the game in different fuzzy cooperative games, the total gain of the player i in the fuzzy cooperative coalition is the sum of the gains gained from participating in all the fuzzy cooperative game coalitions.

The total benefits of a fuzzy cooperative game coalition should be the sum of the benefits generated by all the players who carry different resources into different cooperative coalitions, and the proportion of water resources carried by the players to participate in different cooperative coalitions should be determined by the benefits generated by the participation in different cooperative game coalitions. It is important to note that the water resources available for allocation in a cooperative coalition come from the different players involved in this coalition. Therefore, the total amount of water carried by the players in this coalition should not exceed the maximum amount of water resource carrying capacity.

$$\phi \left(i\right)=\sum _{s=1}^n\phi \left(i,s\right)$$

(3)

2.4. Modelling the Distribution of Benefits in Water Resource Cooperative Game Coalitions

When the water resource cooperative game coalition in the basin determines the payoff function of the cooperative coalition according to the rules in Section 2.2, the optimal resource allocation model of the water resource cooperative gaming coalition can be obtained and is described as follows. There are $n$ water-using entities in the basin, and each of them, after acquiring the initial water right, can bring all or part of their water resources to join the cooperative coalition. Moreover, no more than $2^n-1$ water resource cooperative coalitions can be formed in the basin, and the cooperative alliance allocates the internal water resources and revenues to the participating alliance members. After constructing the water resource cooperative game coalition model, it is necessary to determine the participation rate of water users in each alliance as well as the distribution of water resources and benefits obtained by water users after participating in the cooperative coalition.

2.4.1. Distribution of Benefits in a Traditional Crisp Water Resource Cooperative Alliance

The benefit distribution scheme of a cooperative alliance is one of the most important factors in the existence of the alliance and in maintaining the association of the players within the alliance. The sustainability of an alliance and the reduction of individual defections can only be ensured if there is a comparable level of benefit for those participating in the alliance and no higher benefit for those using the same water resources to participate in other alliances. A crisp water resource cooperative alliance needs to satisfy both collective rationality and individual rationality when it comes to the distribution of benefits from the alliance. Collective rationality is a prerequisite for the cooperative game, and the cooperative game alliance will give priority to water users with high water use efficiency in water resource allocation, so the conditions of collective rationality are easy to meet; individual rationality involves the comparison of the contributions and benefits of water users participating in different cooperative alliances, which is more complicated and requires the application of different benefit allocation methods for comparison and consideration. In this paper, we choose to use the core, least core, weak least core and Shapley value methods to compare and analyze collective and individual returns.

(1)

Core

For any $\left(N,v,\beta \right)\in C{G}^N$, the distribution of the coalition core can be defined as follows:

$$C\left(N,v,\beta \right)=\left\{x\in I\left(v\right)|{\sum }_{i\in S}{x}_i\ge v\left(S\right),\forall S\in {\mathsf{\Omega }}_{\beta }\right\}$$

(4)

The allocation scheme of the core can satisfy the collective rationality of the main coalition and the individual rationality of the participating members. However, the core is more demanding, requiring that the benefit that every player in the game receives in the main coalition $S$ is at least at the same level and no less than the benefit that every player in the game receives from participating in any other coalition, ensuring that every player in the game has no thoughts of leaving the main coalition $S$. At this time, the main coalition $S$ has cohesion, precisely because the core is a refinement of a set of feasible configurations, and the core of a cooperative game coalition is often an empty set. In addition, the core of a cooperative game coalition may have an infinite number of solutions, which is a defect of the core [50].

(2)

Least Core

In order to compensate for the defect of the core, the restriction of the core can be appropriately relaxed and the conditions in the inequality can be expanded in order to obtain more solutions for the distribution of benefits in the crisp cooperative game coalition, and the minimal kernel and the weakly minimal kernel are the sets of solutions obtained after the relaxation of the constraints. The minimal kernel can be interpreted as a tax $\epsilon$ levied on members who leave the crisp coalition and re-form other coalitions outside the crisp coalition, which strengthens the cohesion of the crisp coalition by punishing the “betrayers” [24]. When members of a main coalition form a new coalition after exiting the main coalition, the distributed benefits are smaller than the benefits in the main coalition, and the “betrayers” therefore lose their incentive to escape from the main coalition. The tax on cooperative coalitions outside of the main coalition is in fact a way of increasing the constraints on these coalitions, thus lowering the requirements for the distribution of the main coalition’s core and freeing the core of the crisp cooperative main coalition from the problem of empty sets. The least core method can be expressed as follows:

$$C\left(N,v,\beta \right)=\left\{x\in I\left(v\right)|{\sum }_{i\in S}{x}_i\ge v\left(S\right)-,\forall S\in {\Omega }_{\beta }\right\}$$

(5)

(3)

Weak Least Core

The weak least core is similar in principle to the least core, with the difference that the weak least core is a further restrictive relaxation of the least core approach, in which the tax levied on cooperative unions other than the main coalition is represented by sets, which are related to the distributional coefficients of these unions [24]. The weak least core for cooperative coalitions is expressed as follows:

$$C_{{\epsilon }^{\prime }}\left(N,v,\beta \right)=\left\{x\in I\left(v\right)|{\sum }_{i\in S}{x}_i\ge v\left(S\right)-\left|S\right|,\forall S\in {\Omega }_{\beta }\right\}$$

(6)

In the above equation, $\left|S\right|$ denotes the number of members in the coalition $S$, which varies according to the number of participants in the cooperative coalition.

(4)

Shapley value

For the cooperative game coalition $\left(s,v\right)$, the distributable gain for each participating player $i$ in the game can be expressed as follows:

$${\phi }_i\left(v\right)=\sum _{S\in T}{\displaystyle \frac{\left(n-\left|S\right|\right)!\left(\left|S\right|-1\right)!}{n!}}\left[v\left(S\cup \left\{i\right\}\right)-v\left(S\right)\right]$$

(7)

${\phi }_i\left(v\right)$ denotes the expectation of gains allocated by player $i$ based on the Shapley value method, $T$ denotes the set of n participants $\left\{1,2,\dots ,n\right\}$, and for any subset $S$ in $T$ that can be considered a cooperative coalition, $v\left(S\right)$ represents the gain of the coalition $S$.$\left[v\left(S\cup \left\{i\right\}\right)-v\left(S\right)\right]$ denotes the gain for the overall coalition after player $i$ participates in the cooperative coalition in the game, and ${\displaystyle \frac{\left(n-\left|S\right|\right)!\left(\left|S\right|-1\right)!}{n!}}$ is the weighting term, where $\left|S\right|$ denotes the number of individuals in the participating coalition $S$.

The Shapley value is a method of equitably distributing member contributions by calculating the marginal contribution of each member to the cooperation of all possible cooperative coalitions and summing them according to the weights thus weighted [17]. In the first step, all possible coalitions are considered, including subsets containing different members; in the second step, the marginal contribution of each member in the coalition is calculated; then, in the third step, the marginal contribution of each member is weighted and summed using a weighting factor. The calculation’s Shapley value indicates the degree of the contribution of each member to the cooperation, and a higher Shapley value means that the member has made a greater contribution to the overall cooperation. The Shapley value can equitably distribute the contribution of each member and provide a rational assessment method, which can help to avoid the potential ‘free-riding’ risk of equal distribution of benefits and instead distribute the benefits according to the marginal contribution of each member. This method can promote fairness, justice and win–win cooperation, and it can motivate members to play an active role in cooperation, ensuring the cohesion and sustainability of the cooperative alliance.

2.4.2. Distribution of Benefits from an Improved Fuzzy Cooperative Alliance for Water Resources

The participation rate of water-using subjects in the fuzzy cooperative alliance can be determined by using the rules in Section 2.3, with the objective of maximizing the benefits of all water-using subjects, and using the constraints of the individual rationality of the game and the collective rationality of the fuzzy cooperative alliance. After determining the structure of the regional water resource cooperative alliance and the participation rate of water-using subjects, the distribution of benefits from the cooperative alliance can be assessed.

(1)

Fuzzy Core

Based on the description in Section 2.3, combined with the research of Yu et al. [9,16], the fuzzy core allocation method can be formulated as follows.

$$\tilde{C}\left(s\right)=\left\{\phi \in R^{+}|{\sum }_{i\in N}{\phi }_i\left(s\right)=v\left(s\right),{\sum }_{i\in q}{\phi }_i\left(s\right)\ge v\left(q\right),\forall q\subseteq s\right\}$$

(8)

(2)

Fuzzy Least Core

Using the idea of transforming fuzzy cooperative games into crisp cooperative games, the least core and weak least core methods are extended to produce fuzzy least core and fuzzy weak least core l. The fuzzy least core imposes a uniform tax $\epsilon$ on all subsets $s$ of fuzzy cooperative coalitions, and the tax is designed in such a way that the players tend to participate in all fuzzy cooperative coalitions at the same time when they typically participate in a subset s of cooperative coalitions. The solution of the fuzzy least core can be expressed as

$$\begin{gathered} \sum _{i\in s}{\phi }_i\left(s\right)\ge v\left(q\right)-\epsilon ,\forall \mathrm{q}\subseteq s \\ \sum _{i\in s}{\phi }_i\left(s\right)=v\left(s\right),\forall s\in L\left(N\right) \end{gathered}$$

(9)

where $L\left(N\right)$ is the whole fuzzy cooperative union.

(3)

Fuzzy Weak Least Core

Fuzzy weak least core is similar to the principle of fuzzy least core; the fuzzy weak least core for each fuzzy cooperative gaming coalition to collect a uniform tax $\epsilon$ should be the coefficient of the sum of the participation rate of the players in the fuzzy cooperative coalition. The fuzzy weak minimum kernel is expressed as

$$\begin{gathered} \sum _{i\in s}{\phi }_i\left(s\right)\ge v\left(q\right)-\epsilon \ast \sum _{i\in q}{\lambda }_{i,q},\forall q\subseteq \mathrm{s} \\ \sum _{i\in s}{\phi }_i\left(s\right)=v\left(s\right),\forall s\in L\left(N\right) \end{gathered}$$

(10)

where ${\lambda }_{i,q}$ denotes the proportion of players i joining the fuzzy cooperative coalition $q$.

(4)

Fuzzy Shapley value

Yu et al. [9,16] extended the Shapley value method by proposing the fuzzy Shapley value function, which can transform a fuzzy cooperative game into a corresponding crisp cooperative game, expressed as follows:

$$\begin{gathered} w_s\left(i\right)=v\left(s\ast {\lambda }_i\right) \\ {w}_s\left(i,j\right)=v\left(s\ast {\lambda }_i+s\ast {\lambda }_i\right) \\ {w}_s\left(T\right)=v\left(\sum _{i\in T}s\ast {\lambda }_i\right) \\ {w}_s\left(N\right)=v\left(\sum _{i\in N}s\ast {\lambda }_i\right) \end{gathered}$$

(11)

where $w_s$ refers to the crisp game function corresponding to the fuzzy game $v\left(S\right)$, $w_s\left(i\right)$ refers to the revenue created by the player i carrying resources to join the coalition S, ${\lambda }_i$ refers to the participation rate of the player $i$ joining the coalition, $N$ refers to the large coalition formed by all the players, and $T$ refers to the sub-coalition in the coalition $S$. The Shapley value of innings player $i$ is calculated as follows:

$${\phi }_i\left(w_s\right)=\sum _{i\in T\subseteq N}{\displaystyle \frac{\left(\left|T\right|-1\right)!\left(\left|N\right|-\left|T\right|\right)}{\left|N\right|!}}\left[w_s\left(T\right)-w_s\left(T\setminus \left\{i\right\}\right)\right]$$

(12)

where $\left|T\right|$ and $\left|N\right|$ refer to the number of people in the game in the fuzzy coalition $T$ and the fuzzy coalition $N$, respectively, and $w_s\left(T\setminus \left\{i\right\}\right)$ refers to the gain in the coalition $T$ except for person $i$ in the game. Based on the above setting, the fuzzy Shapley value obtained by the player $i$ joining the fuzzy cooperative coalition $S$ with an ${\lambda }_i$ participation rate is calculated as follows:

$${\phi }_i\left(v\right)=\sum _{i\in T\subseteq N}{\displaystyle \frac{\left(\left|S\right|-1\right)!\left(\left|N\right|-\left|S\right|\right)}{\left|N\right|!}}\left[v\left(\sum _{i\in T}{s}_i\ast {\lambda }_i\right)-v\left(\sum _{i\in T\setminus i}{s}_i\ast {\lambda }_i\right)\right]$$

(13)

A comparison of the allocation methods of the traditional crisp cooperative game and the improved fuzzy cooperative game is shown in

Table 1. Comparison of methods for distributing gains from cooperative games.

Categories	Benefit Distribution Model	Characteristic
Traditional crisp cooperative gaming	Core	The core is able to satisfy both individual and collective rationality. Core solutions are more demanding, usually featuring an empty set or an infinite number of solutions, which is a drawback of core solutions.
	Least core	Appropriate relaxation of core conditions and additional taxes (ε) on non-main coalitions
	Weak least core	More relaxed cores than the least core; the tax (ε) on non-major coalitions is represented by the set.
	Shapley value	A method of distributing benefits based on the marginal contribution of members in a cooperative alliance promotes fairness and cooperation, incentivizes members to play an active role in the cooperation, and also ensures the cohesion and permanence of the cooperative alliance.
Improved fuzzy cooperative games	Fuzzy core	Individual rationality and collective rationality are easier to satisfy after the gamer takes some of their water resources to join a cooperative coalition (compared to the core solution).
	Fuzzy least core	Similarly, in the least core solution, after the player takes some of their water resources to join the cooperative coalition, taxes (ε) are imposed on all subsets of the non-cooperative coalition, which makes the participants inclined to participate in all fuzzy cooperative coalitions.
	Fuzzy weak least core	Consistent with the fuzzy least core idea, the taxes (ε) are converted to the set.
	Fuzzy Shapley value	After a participant takes some of their water resources to join different cooperative games, the gains are calculated based on their marginal contributions in different cooperative coalitions and subsequently summed up. The fuzzy Shapley value is used to transform the fuzzy cooperative game into the corresponding crisp cooperative game.

.

3. Case Study

3.1. Overview of the Tarim River Basin

The Tarim River Basin is located in the southern part of Xinjiang Uygur Autonomous Region, between 73°10′~94°05′ east longitude and 34°55′~43°08′ north latitude. It is the largest inland river basin in China, with a basin area of 1,020,400 km² and an area of 996,800 km² within China, which accounts for about 61.27% of the land area of the whole of Xinjiang. The Tarim River Basin consists of 144 rivers from nine water systems including the Yarkant River, Hotan River, Aksu River, and Kaidu-Kongchu River, which surround the Tarim Basin. It has a multi-year (1956–2020) average natural runoff of 39.83 billion m³, and non-duplicated groundwater of about 3.07 billion m³, meaning a total water resource of 42.9 billion m³. The total length of the Tarim River is 1321 km. At present, only the Hotan River, Yarkant River, and Aksu River are connected to the main stream by surface water. The Kongchu River pumps water from Bosten Lake to the irrigation area downstream of the Tarim River through a water conservancy station; therefore, the Tarim River Basin has formed a hydrological pattern of “four sources and one trunk”. The average multi-year (1956–2020) runoff of the four sources and one trunk is 25.673 billion m³, with 1.815 billion m³ of this being unduplicated groundwater, meaning a total water resource of 27.488 billion m³. The administrative area of the Tarim River Basin includes 42 counties in five prefectures, namely Bayin’guoleng Mongol Autonomous Prefecture, Aksu Prefecture, Kashgar Prefecture, Hotan Prefecture, and Kizilsu and Kirgiz Autonomous Prefectures, as well as 58 corps of the First, Second, Third, and Fourteenth Divisions of the Xinjiang Production and Construction Corps (XPCC). At the end of 2022, the GDP of the basin was CNY 616.670 billion, with an agricultural area of 34,106,600 acres, and the disposable income of all the residents of the basin was CNY 31,200.

The total water resource supply in the Tarim River Basin “Four Sources and One Stem” region in 2022 was 32.683 billion m³; the Kashgar and Hotan regions have a larger share of allocated water, but they have a lower per capita GDP, relatively backward economic development, a less pronounced link between resources and economic development. As a result, there is an urgent need to improve the efficiency of water resource utilization in these regions. The calculation of water demand in each region of the Tarim River Basin is based on the different categories of water use quota indicators multiplied by the volume of water-using bodies (2022), and the water-using bodies are divided into four major categories (domestic, industrial, agricultural, and ecological) in accordance with the classification standards of the China Water Resources Bulletin and the China Statistical Yearbook. The classified water demand of each area in the Tarim River Basin in 2022 is shown in

Table 2. Water demand of various regions in the Tarim River Basin in 2022 (10⁸ m³) *.

District	Agriculture	Industrial	Domestic	Artificial Ecosystem Recharge	Total Water Demand	Total Supply	Supply–Demand Gap
Aksu	124.57	1.13	2.10	10.96	138.76	86.03	52.73
Bayin guoleng Mongolian Autonomous Prefecture	51.37	1.61	1.62	24.18	78.78	40.74	38.04
Kizilsu Kyrgyz	10.88	0.14	0.47	1.11	12.60	10.33	2.27
Kashgar	124.53	0.52	3.43	9.92	138.39	99.05	39.34
Hotan	37.69	0.15	1.87	7.45	47.16	42.46	4.7
First Division of XJCC	28.70	0.13	0.37	2.00	31.21	22.50	8.71
Second Division of XJCC	11.63	0.06	0.20	10.04	21.93	10.20	11.73
Third Division of XJCC	15.55	0.11	0.26	0.99	16.91	12.92	3.99
Fourteenth Division of XJCC	3.33	0.01	0.06	0.49	3.89	2.60	1.29
Total	408.25	3.86	10.37	67.14	489.63	326.83	162.8

* The figures in the table were calculated by the authors on the basis of the relevant criteria.

, and the initial scheme for allocating water rights is determined according to the actual water allocation in the Tarim River Basin in 2022, as shown in

Table 3. Initial water rights allocation plan for Tarim River Basin in 2022 (10⁸ m³) *.

District	Agriculture	Industrial	Domestic	Artificial Ecosystem Recharge	Total
Aksu	82.84	0.96	1.47	0.76	86.03
Bayin guoleng Mongolian Autonomous Prefecture	37.6	1.16	1.17	0.81	40.74
Kizilsu Kyrgyz	9.74	0.1	0.4	0.09	10.33
Kashgar	93.15	0.19	2.5	3.21	99.05
Hotan	39.9	0.12	1.38	1.06	42.46
First Division of XJCC	20.62	0.4	0.28	1.2	22.5
Second Division of XJCC	9.18	0.14	0.19	0.69	10.2
Third Division of XJCC	11.27	0.08	0.35	1.22	12.92
Fourteenth Division of XJCC	2.2	0.01	0.07	0.32	2.6
Total	306.5	3.16	7.81	9.36	326.83

* The figures in the table were calculated by the authors based on previous literature.

.

The Tarim River Basin is located in the hinterland of the Eurasian continent, surrounded by high mountains on three sides, and has a typical arid continental climate with scant precipitation, strong evaporation, many sandstorms and dust storms, abundant sunshine, and a large difference in temperature between day and night. Its average annual precipitation is 116.8 mm, with evaporation of up to 1800–2200 mm, and the number of hours of sunshine in the basin is around 2550–3500 h per year. The limited and unstable water resources of the Tarim River Basin support the harmonious, stable, and coordinated development of the regional system of soil and water resources, the ecological environmental system, and the socio-economic system. In 1950, human beings began to organize large-scale and high-intensity development of the agricultural productive activities and constructed a number of permanent canal headworks, storage reservoirs, water transport canal systems and other water conservancy facilities, and the basin of the arable land area rapidly expanded, taking over a large amount of the original occupied forest land, grassland, watersheds, and so forth. Additionally, the surge in agricultural water use has caused the original ecological environment to use less water, resulting in the deterioration of environmental water resources, a yearly decline in groundwater levels, the intensification of salinization and desertification of the land, and the discontinuation of the flow of part of the river. The area occupied by vegetation has decreased significantly, and the contradiction between the supply of and demand for water resources is prominent. The contradiction of water and soil resources in the Tarim River Basin can be summarized as follows: agricultural water use heavily crowds out other water use; there is inefficient use of water resources and continuous deterioration of the ecological environment; soil and water development are caught in a vicious circle; and water resources have become one of the bottlenecks restricting the socio-economic development of the Tarim River Basin.

The Tarim River Basin is a mosaic of autonomous regions and the XJCC; it has complex water use patterns, diverse and heterogeneous water users, and intertwined interests, which makes it necessary to rationalize the water resource management of the Tarim River Basin and the mechanism used to distribute benefits to multiple water users. With the continuous expansion of the basin to reclaim arable land, the more intense the water conflicts, the more fragile the water-carrying capacity of the natural environment, leading to crowding, theft of water resources, and the phenomenon of water encroachment. The phenomenon of encroachment on water rights is a common occurrence. Ensuring the efficient use of water resources in the basin and at the same time satisfying the water demand of multiple water users whilst straightening out the initial allocation of water rights in the basin, trading water rights, water resource management, water resource revenue and other issues is currently the most important issue in the Tarim River Basin; we must also reduce information asymmetry between water users and the government and increase the cost of bargaining, transaction costs, the cost of access to information and the cost of the game. The Tarim River Basin represents a real problem to be solved urgently.

In this paper, we propose to adopt a cooperative gaming alliance to integrate and reorganize resources through strong and weak alliances so as to increase benefits for the system as a whole and for the participating individuals. The participants in the cooperative alliance will bring all or part of their water resources when joining the cooperative alliance, so that the alliance can redistribute water resources and benefits with its “visible hand”. The “visible hand” of the alliance will redistribute the water resources and benefits and carry out “planned distribution” within a certain range in order to seek higher collective and individual benefits.

3.2. Distribution of Resources and Benefits in the Tarim River Basin Crisp Cooperative Alliance

Guaranteeing ecological water in the river channel of the basin and preserving water circulation and good water quality is the key to maintaining the benign development of the ecological environmental system of the basin. At the same time, the water needed for human life has the highest priority because it is a fundamental guarantee of economic and social stability. Based on the experience of the countries with more perfect development of their system of water rights, ecological water and the water for life are generally given the highest priority in the allocation of rights, so in the basin, ecological water and water for life are deducted in advance of the initial allocation of rights to water. In the process of water resource development and utilization in the Tarim River Basin, the water consumption of the agricultural sector accounts for 93.78% of the total water consumption in the basin (2022), while the water consumption of industry, in domestic use and in ecological use accounts for 0.97%, 2.39% and 2.86%, respectively. After deducting ecological water use and domestic water use, we can discuss the allocation of water resources and the distribution of benefits in the crisp and fuzzy cooperative games in the context of different river basin scales.

In view of the complex river affiliations and multiple water users of the Tarim River Basin, if a cooperative game alliance is constructed in the whole basin, not only can the water conservancy facilities not meet the requirements of water resource allocation in the cooperative alliance; the composition of the fuzzy cooperative game alliance will also make the allocation of resources and the distribution of benefits more complicated. Based on the “four sources and one trunk” natural geographic zoning, which is the basis of water conservancy projects’ construction and facilitates comprehensive evaluation of computability, the cooperative alliances in the Tarim River Basin are divided four as follows: the Hetian River system alliance, Yarkand River system alliance, Aksu River system alliance and Kaikong River + trunk water system alliance. The reason for merging the Kai-Kong River system with the main stream is that the location at which the Kai-Kong River joins the Ta River is in the middle of the main stream, and the majority of agricultural and industrial water use involved is included in the main stream’s system. Meanwhile, since most of the regiments of the First Division are in the Aksu River system and only some of the regiments are in the main stream, the water-using units of the First Division are completely merged into the Aksu water system alliance in order to prevent double-counting. Since the majority of the water use in the Aksu area is concentrated in the Aksu River system, and there are only small portions of the Shaya and Kuqa counties in the main stream, the water use entities in the Aksu area are merged into the Aksu Water System Union. The specific classification, membership, and water resource yields of the cooperative alliances in the Tarim River Basin are shown in

Table 4. Tarim River Basin cooperative alliances’ classification, composition, and water yield in 2022 *.

Cooperative Alliance	Membership	Rate of Return on Water Resources (CNY/m3)	Maximum Water Resources Carrying Capacity (108 m3)	Cooperative Alliance Water Revenue (108 CNY)
Hotan River Water System Alliance	1 Hotan Agriculture	2.69	44.12	637.08
	2 Hotan Industry	678.83	0.52
	17 Fourteenth Division Agriculture	8.07	3.43
	18 Fourteenth Division Industry	687.00	0.25
Yarkant River System Alliance	3 Kashi Agriculture	4.54	111.32	1348.54
	4 Kashi Industry	1493.21	0.41
	15 Third Division Agriculture	6.61	19.47
	16 Third Division Industry	745.75	0.34
Aksu River System Alliance	5 Kizilsu Kirghiz Agriculture	2.75	11.89	2672.83
	6 Kizilsu Kirghiz Industry	776.70	0.57
	7 Aksu Agriculture	5.10	96.3
	8 Aksu Industry	647.98	2.17
	11 First Division Agriculture	8.78	33.97
	12 First Division Industry	180.45	1.24
Kai-Kong River + Main Stream System Alliance	9 Bayin Guoleng Mongol Agriculture	5.79	63.19	4182.85
	10 Bayin Guoleng Mongol Industry	717.72	5.33
	13 Second Division Agriculture	6.74	45.54
	14 Second Division Industry	224.21	0.96

* The figures in the table have been compiled by the authors based on available information.

, in which the water resource yields of the cooperative alliances are derived from the calculation method in Section 2.2, and the maximum water resource-carrying capacity of the cooperative alliances is determined by the combination of the reservoir capacity and the adjustable water volume of the region in which the water-using subjects are located.

3.2.1. Hotan River Water System Alliance

As can be seen from Table 2, the total water demand of the water-using units in the Hotan River Water System alliance is slightly larger than the allocated amount of water, and the water-using units in the Hotan River Water System alliance can moderately conserve water, combine with water regulation projects to comprehensively study and judge the conditions of water resources’ dispatch, and transfer them from the agricultural sector to the industrial sector so as to enhance the overall gains of the Hotan River Water System alliance and the gains of each region. According to the computation model of the benefits of the crisp cooperative alliance in Section 2.2, optimal results of water resource cooperation in the Hotan River System are determined, and beneficial scenarios of water use are obtained for each subset in the crisp cooperative alliance, as shown in

Table 5. Water use revenue of the Hotan River Water System crisp cooperation alliance (10⁸ CNY) *.

Alliances	{1, 2}	{1, 17}	{1, 18}	{2, 17}	{2, 18}	{17, 18}	{1, 2, 17}	{1, 2, 18}	{1, 17, 18}	{2, 17, 18}
Payoffs	442.06	114.06	261.25	366.15	89.31	186.20	465.97	613.16	285.16	535.96

* The figures in the table have been compiled by the authors on the basis of available information.

. According to the method of distributing benefits in the crisp cooperative alliance in Section 2.4.1, the redistribution of benefits to the water users of the Hotan River system in the scenario of the crisp cooperative alliance is computed and is presented in

Table 6. Distribution of proceeds from the crisp cooperative alliance for the Hotan River System (10⁸ CNY) *.

Player	Core	Least Core	Weak Least Core	Shapley Value
1	No solutions	107.33	107.33	167.42
2		350.76	81.46	261.49
17		31.13	17.75	91.94
18		147.86	430.54	116.23

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

For the core method, the following conditions need to be fulfilled: (1) the benefits received by the players in the game in a cooperative alliance must not be less than those received when they go it alone, $v\left(1\right)\ge 107.33$,$v\left(2\right)\ge 81.46$, $v\left(17\right)\ge 17.75$, $v\left(18\right)\ge 6.87$; (2) it must be ensured that the player is not in other alliances in order to receive higher returns and thus cannot defect from the largest cooperative alliance (as shown in Table 5); in the crisp cooperative alliance, no player participating in the alliance can obtain higher returns, so the returns of the player participating in the large cooperative alliance after the sum of the gains must be greater than the other crisp cooperative alliance’s total gains. The core can be obtained by solving using Lingo 11.0 software. For the least core, a surtax is set on all coalitions outside the crisp cooperative main coalition to ensure the cohesion of the crisp cooperative main coalition. After using Lingo 11.0 software to solve, the result obtained from the minimal kernel is consistent with the core solution, because the original purpose of the least core is to induce the core solution to be non-empty, and in the case that the core solution is not the empty set, the least core solution does not need to relax the constraints any further in order to obtain the solution set. After further relaxation of the constraints on the least core, the weak least core results in a higher distribution of benefits to the industrial sector of Fourteenth Division of XJCC, which is the most efficient use of water resources, amounting to 67.58% of the total benefits of the coalition. For the Shapley value method, we combined the benefit values of the subset in Table 5 and introduced them into the formula of the Shapley value method in Section 2.4.1; we used MATLAB software (R2019a) to obtain the distribution of benefits to different water users in the water system.

Table 6 shows that (1) the four revenue allocation schemes of the crisp cooperative alliance of the Hotan River system have different allocation principles and methods, and the benefits of the game when using different allocation methods are different, but the sum of the benefits is unchanged. All of them are CNY 63.708 billion, and the benefits of the game in each allocation scheme are greater than those when users ‘go it alone’. (2) The core and the least core revenue allocation schemes of the crisp cooperative alliance of the Hotan River system are the same. (3) The weak least core allocation method is the same for high water use efficiency. The kernel allocation method allocates higher benefits to water-using units with high water-use efficiency, while allocating lower benefits to units with low water-use efficiency. (4) The benefits of players in the industrial sector in the Hotan River system are smaller than those of the core, least core, weak least core allocation methods when applying the Shapley value method, while the opposite is true for the benefits of players in the agricultural sector. The Shapley value method is based on the benefits of different players and allocates gains based on the marginal contribution of different players participating in different subsets of cooperative alliances, showing that the marginal contribution of players in the agricultural sector participating in different cooperative alliances is greater than that of players in the industrial sector.

3.2.2. Yarkant River System Alliance

According to the crisp cooperative alliance revenue calculation model in Section 2.2, the optimal results of water resource cooperation in the Hotan River system are determined, and scenarios of the water use revenue of each subset in the crisp cooperative alliance are obtained, as shown in

Table 7. Water use revenue of the crisp cooperation alliance in the Yarkant River system (10⁸ CNY) *.

Alliances	{3, 4}	{3, 15}	{3, 16}	{4, 15}	{4, 16}	{15, 16}	{3, 4, 15}	{3, 4, 16}	{3, 15, 16}	{4, 15, 16}
payoffs	1003.65	485.09	645.22	686.18	485.44	327.85	1096.03	1256.17	737.60	938.22

* The figures in the table have been compiled by the authors on the basis of available information.

. In accordance with the revenue allocation method of the crisp cooperative alliance in Section 2.4.1, the water use revenue of the Yarkant River system in the case of a crisp cooperative alliance is calculated, as shown in

Table 8. Distribution of proceeds from the crisp cooperative alliance for the Yarkant River system (10⁸ CNY) *.

Player	Core	Least Core	Weak Least Core	Shapley Value
5	No solutions	26.79	26.79	169.48
6		353.78	77.67	398.94
7		518.39	422.48	518.54
8		1393.29	622.06	1119.28
11		209.62	181.04	244.93
12		170.95	1342.79	221.66

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

Table 8 demonstrates that the core solutions of the Yarkant River system crisp cooperative are empty sets, indicating that there is no solution set that can satisfy all the core constraints. After relaxing the requirements of constraints, player 3 has the same benefits using the least core and weak least core allocation methods, and the other players have rising and falling benefits when using these two allocation methods, which indicates that the allocations of least core and weak least core are consistent. When the relaxation conditions are changed, the benefits received by the different participating agents change differently, which is related to the benefits of the various subsets of the cooperative game The Shapley value method’s allocation scheme provides a more even distribution to the participating subjects, and the difference between the proceeds of player 4 (who gains the most) and player 15 (who gains the least) is only CNY 32.499 billion, while the difference between the proceeds of the weak least core allocation scheme is CNY 61.007 billion, which suggests that Shapley value method of distributing gains (after considering the value of the contribution of the players to the cooperative alliance) is more fair.

3.2.3. Aksu River System Alliance

According to the model for calculating the benefits of the crisp cooperative alliance described in Section 2.2, the benefit scenarios of each subset of water use in the crisp cooperative alliance of the Aksu River system can be calculated, as shown in

Table 9. Water use revenue of the crisp cooperation alliance in the Aksu River system (10⁸ CNY) *.

Alliances	$\left\{5,6\right\}$	$\left\{5,7\right\}$	$\left\{5,8\right\}$	$\left\{5,11\right\}$	$\left\{5,12\right\}$	$\left\{6,7\right\}$	$\left\{6,8\right\}$	$\left\{6,11\right\}$
payoffs	465.41	415.85	1425.89	249.09	245.51	812.42	643.59	612.35
Alliances	$\left\{6,12\right\}$	$\left\{7,8\right\}$	$\left\{7,11\right\}$	$\left\{7,12\right\}$	$\left\{8,11\right\}$	$\left\{8,12\right\}$	$\left\{11,12\right\}$	$\left\{5,6,7\right\}$
payoffs	460.76	1770.41	597.32	591.72	1566.44	784.06	390.40	856.53
Alliances	$\left\{5,6,8\right\}$	$\left\{5,6,11\right\}$	$\left\{5,6,12\right\}$	$\left\{5,7,8\right\}$	$\left\{5,7,11\right\}$	$\left\{5,7,12\right\}$	$\left\{5,8,11\right\}$	$\left\{5,8,12\right\}$
payoffs	1867.51	724.48	687.13	1814.52	641.44	635.84	1642.39	1647.61
Alliances	$\left\{5,11,12\right\}$	$\left\{6,7,8\right\}$	$\left\{6,7,11\right\}$	$\left\{6,7,12\right\}$	$\left\{6,8,11\right\}$	$\left\{6,8,12\right\}$	$\left\{6,11,12\right\}$	$\left\{7,8,11\right\}$
payoffs	466.35	2211.09	1038.00	1032.40	2005.65	967.58	829.61	1995.99
Alliances	$\left\{7,8,12\right\}$	$\left\{7,11,12\right\}$	$\left\{8,11,12\right\}$	$\left\{5,6,7,8\right\}$	$\left\{5,6,7,11\right\}$	$\left\{5,6,7,12\right\}$	$\left\{5,6,8,11\right\}$	$\left\{5,6,8,12\right\}$
payoffs	1990.39	817.30	1783.70	2436.67	1082.11	1076.52	2081.59	2089.23
Alliances	$\left\{5,7,11,12\right\}$	$\left\{5,6,11,12\right\}$	$\left\{5,7,8,11\right\}$	$\left\{5,7,8,12\right\}$	$\left\{5,8,11,12\right\}$	$\left\{6,7,8,11\right\}$	$\left\{6,7,8,12\right\}$	$\left\{6,7,11,12\right\}$
payoffs	861.42	905.56	2040.11	2034.51	1859.65	2436.67	2431.07	1257.98
Alliances	$\left\{6,8,11,12\right\}$	$\left\{7,8,11,12\right\}$	$\left\{5,6,7,8,11\right\}$	$\left\{5,6,7,8,12\right\}$	$\left\{5,6,7,11,12\right\}$	$\left\{5,6,8,11,12\right\}$	$\left\{5,7,8,11,12\right\}$	$\left\{6,7,8,11,12\right\}$
payoffs	2222.91	2215.98	2480.79	2475.19	1302.10	2298.85	2260.09	2656.65

* The figures in the table have been compiled by the authors on the basis of available information.

. According to the method of distributing benefits in the crisp cooperative alliance in Section 2.4.1, the benefits given to the water users of the Aksu River system in the game of the Crisp cooperative alliance can be calculated, and the results are displayed in

Table 10. Distribution of proceeds from the crisp cooperative alliance for the Aksu River system (10⁸ CNY) *.

Player	Core	Least Core	Weak Least Core	Shapley Value
3	No solutions	442.9	442.9	481.78
4		283.71	283.71	489.35
15		562.27	68.75	164.36
16		59.66	553.18	213.05

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

From Table 9 and Table 10, it can be seen that in a cooperative alliance involving six players, there are 56 subset alliances associated, and the constraints of the core are relatively harsh, resulting in the core solution being an empty set; after applying a tax to the returns of the subset of alliances, the agricultural sector receives lower returns in the distribution of returns when using the least core and weak least core method compared to the industrial sector, and the difference in earnings between the highest earning player 8 and the lowest earning player 5 for the least core is CNY 136.65 billion. The difference for the weak least core is CNY 131.6 billion. The distribution scheme from the Shapley value method is more even, with the difference between the gains from the highest gainer and the lowest gainer being only CNY 94.98 billion. The gain from the highest gainer is lower than the gain from the highest gainer in the least core and weak least core scenarios, and the gain from the lowest gainer is higher than the gain from the lowest gainer in the other two scenarios. The least core allocation scheme and the Shapley value allocation scheme allocate a lower return to the agricultural sector than to the industrial sector.

3.2.4. Kai-Kong River + Main Stream Water System Alliance

According to the crisp cooperative alliance benefit calculation model in Section 2.2, the optimal result of water resource cooperation in the Kai-Kong River + Main Stream water system is determined, and beneficial scenarios of the water use of each subset in the crisp cooperative alliance are obtained, as shown in

Table 11. Water use revenue of the Kai-Kong River + Main Stream River water system crisp cooperation alliance (10⁸ CNY) *.

Alliances	$\left\{1,2\right\}$	$\left\{1,17\right\}$	$\left\{1,18\right\}$	$\left\{2,17\right\}$	$\left\{2,18\right\}$	$\left\{17,18\right\}$	$\left\{1,2,17\right\}$	$\left\{1,2,18\right\}$	$\left\{1,17,18\right\}$	$\left\{2,17,18\right\}$
payoffs	442.06	114.06	261.25	366.15	89.31	186.20	465.97	613.16	285.16	535.96

* The figures in the table have been compiled by the authors on the basis of available information.

. In accordance with the method of allocation benefits in the crisp cooperative alliance in Section 2.4.1, the water users’ benefits of the Kai-Kong River + Main Stream water system in the crisp cooperative alliance scenario can be obtained, as indicated in

Table 12. Distribution of proceeds from the crisp cooperative alliance for the Kai-Kong River + Main Stream River system (10⁸ CNY) *.

Player	Core	Least Core	Weak Least Core	Shapley Value
1	No solutions	107.33	107.33	167.42
2		350.76	81.46	261.49
17		31.13	17.75	91.94
18		147.86	430.54	116.23

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

The results of Table 12 demonstrate that except for the weak least core method, player 10 obtained the maximum benefit in both the core and the Shapley value method allocation schemes and obtained the highest benefit in the core, which is CNY 380.254 billion; players 9 and 13, who gained fewer benefits in the core, least core, and weak least core methods, received higher gains in the Shapley value allocation scheme. Player 9’s gain in the Shapley value allocation scheme was 130.59% higher than in both the least core and weak least core allocation schemes, and player 13’s gain in the Shapley value allocation scheme was 260.19% higher than in the least core allocation gain and 663.92% higher than in the weak least core allocation gain. Among the different allocation schemes, the difference between the gains of the players with the smallest nucleus is large; the difference between the highest gainer and the lowest gainer is CNY 377.115 billion. The weak least core value is CNY 300.885 billion, and the Shapley value is CNY 282.95 billion. In contrast, the Shapley value method is the allocation scheme with a more equitable return for all water-using subjects, especially the players who make small gains from solo work. Participation in the alliance produces Shapley values of revenue distribution that are several times higher than in the solo work. As a result, these players have a stronger willingness to participate in the cooperative alliance. The weak least core, after relaxing constraints for each subset of cooperative coalitions, shows a large difference in the constraints of the coalition subsets of players compared to the least core, with player 14 experiencing a surge in gains; meanwhile, player 10 only meets their minimum requirements, creating a significant difference in the distribution of gains from the rest of the allocation scheme, which suggests that further relaxation of constraints in the allocation scheme of the weak least core affects cooperative coalitions to a large extent in the allocation of excess returns.

3.3. Allocation of Resources and Benefits from the Tarim River Basin Fuzzy Cooperative Alliance

Considering that water users in each water system in the basin can obtain more benefits in the crisp cooperative alliance after joining different subsets of the cooperative alliance, rational water users will decide to take their resources into different cooperative alliances with different participation rates in order to obtain more benefits than when only joining the crisp cooperative alliance, and the whole cooperative alliance will also obtain maximum mixed benefits. In Tarim River Basin, due to the limitations of geographic location and the water conservancy projects of different water systems, the water resource cooperative game can only be established within each water system, which makes it impossible to distribute water resources and benefits among water systems. Instead, the fuzzy cooperative alliance in Tarim River Basin takes these water systems as the research object for the construction and analysis of the fuzzy cooperative alliance.

3.3.1. Coalition for Fuzzy Cooperation in the Hotan River System

According to the formula in Section 2.3, with the objective of maximizing the total benefit of the water system, the participation rate of pluralistic water users carrying different water resources to join the subset of cooperative alliance can be determined. In this paper, MATLAB software (R2019a) is used to solve this problem, and the participation rate of the diversified subjects of Hotan River Water System participating in each cooperative alliance can be obtained. The parameters of the fuzzy cooperative alliance’s benefits and its maximum carrying capacity are shown in

Table 13. Profit parameters and maximum carrying capacity of the fuzzy cooperative alliance in the Hotan River System *.

Fuzzy Alliance	Players	Rate of Return per Unit of Water Resource (CNY/m3)	Maximum Allowable Carrying Capacity (108 m3)
$s_1$	$\left\{1,2\right\}$	11.05	41.5
$s_2$	$\left\{1,17\right\}$	2.71	40.3
$s_3$	$\left\{2,17\right\}$	157.82	1.9
$s_4$	$\left\{1,2,17\right\}$	11.04	42.8

* The figures in the table have been compiled by the authors on the basis of available information.

, and the participation rate of the players participating in different cooperative alliances is shown in

Table 14. Participation rate and allocation of water quantity in the fuzzy cooperative alliance among people in the Hotan River Water System *.

Agent	${\mathit{s}}_1:\left\{1,2\right\}$	${\mathit{s}}_2:\left\{1,17\right\}$	${\mathit{s}}_3:\left\{2,17\right\}$	${\mathit{s}}_4:\left\{1,2,17\right\}$
$1$	100%	0	0	0
$2$	0	0	0	100%
$17$	0	0	86.36%	13.64%
Allocated water (108 m3)	39.9	0.00	1.89	0.42

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

The participation rate and the allocated water volume of the Hotan River System players can be seen in Table 14, and the overall benefit of the fuzzy cooperative alliance of the Hotan River System is determined to be CNY 76.611 billion. The benefits of $s_1,s_3,s_4$ are CNY 44.090 billion, CNY 29.828 billion, and CNY 464 million, respectively. Based on Table 14 and the formulas in Section 2.4.2, the distribution of benefits to each water user in the fuzzy cooperative alliance of the Hotan River System can be obtained, and the distribution of benefits to the players using the fuzzy core, fuzzy least core, fuzzy weak least core and fuzzy Shapley value methods is determined, respectively, as shown in

Table 15. Distribution of benefits using different fuzzy cooperative allocation rules for the Hotan River System players (10⁸ CNY) *.

Agent	Fuzzy Core	Fuzzy Least Core	Fuzzy Weak Least Core	Fuzzy Shapley Value
$1$	160.37	160.37	107.33	114.85
$2$	280.53	280.53	81.46	266.67
$17$	17.75	17.75	17.75	70.82
18	300.59	300.59	552.7	306.9

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

3.3.2. Coalition for Fuzzy Cooperation in the Yarkant River System

Using the formula in Section 2.3, combined with the rate of return and the maximum allowable carrying capacity of different cooperative alliances in

Table 16. Profit parameters and maximum carrying capacity of the fuzzy cooperative alliance in the Yarkant River System *.

Fuzzy Alliance	Players	Rate of Return per Unit of Water Resource (CNY/m3)	Maximum Allowable Carrying Capacity (108 m3)
$s_5$	$\left\{3,15\right\}$	4.67	95.2
$s_6$	$\left\{3,16\right\}$	6.92	92.6
$s_7$	$\left\{15,16\right\}$	28.88	13.8
$s_8$	$\left\{3,15,16\right\}$	7.06	101.5

* The figures in the table have been compiled by the authors on the basis of available information.

, the participation rate of multiple subjects participating in the fuzzy cooperative gaming alliance of the Yarkant River System can be obtained, as shown in

Table 17. Participation rate and allocation of water quantity in the fuzzy cooperative alliance among people in the Yarkant River Water System *.

Agent	${\mathit{s}}_5:\left\{3,15\right\}$	${\mathit{s}}_6:\left\{3,16\right\}$	${\mathit{s}}_7:\left\{15,16\right\}$	${\mathit{s}}_8:\left\{3,15,16\right\}$
3	0	0	0	100%
15	0	0	100%	0
16	0	0	100%	0
Allocated water (108 m3)	0.00	0.00	11.35	93.15

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

Table 17 shows the participation rate and allocated water volume of the Yarkant River System Bureau. The overall benefit of the fuzzy cooperative alliance of the Yarkant River System is CNY 126.895 billion, and the benefits of $s_7 \mathrm{a}\mathrm{n}\mathrm{d} s_8$ are CNY 32.779 billion and CNY 65.764 billion, respectively. Based on Table 17 and the formula in Section 2.4.2, the distribution of benefits to each water user in the fuzzy cooperative alliance of the Yarkant River System is shown in

Table 18. Distribution of benefits under different fuzzy cooperative allocation rules for the Yarkant River System players (10⁸ CNY) *.

Agent	Fuzzy Core	Fuzzy Least Core	Fuzzy Weak Least Core	Fuzzy Shapley Value
3	No solutions	442.9	442.9	424.48
4		283.71	283.71	327.80
15		198.97	68.75	78.72
16		59.66	189.88	154.24

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

3.3.3. Coalition for Fuzzy Cooperation in the Aksu River System

Based on the differences in the geographic location of water-using subjects, the fuzzy cooperative alliance of the Aksu Water System can be separated into seven alliances, whose profit parameters and maximum carrying capacity are shown in

Table 19. Profit parameters and maximum carrying capacity of the fuzzy cooperative alliance in the Aksu River System *.

Fuzzy Alliance	Players	Rate of Return per Unit of Water Resource (CNY/m3)	Maximum Allowable Carrying Capacity (108 m3)
$s_9$	$\left\{5,6\right\}$	102.00	8.8
$s_{10}$	$\left\{5,8\right\}$	60.30	10.2
$s_{11}$	$\left\{7,8\right\}$	11.20	81.6
$s_{12}$	$\left\{6,7\right\}$	3.95	82.5
$s_{13}$	$\left\{11,12\right\}$	18.57	30.2
$s_{14}$	$\left\{7,12\right\}$	7.11	82.5
$s_{15}$	$\left\{8,11\right\}$	72.59	30.6

* The figures in the table have been compiled by the authors on the basis of available information.

.

The participation rate and the allocated water volume of the Aksu River System Directorate can be obtained from

Table 20. Participation rate and allocation of water quantity in the fuzzy cooperative alliance among people in the Aksu River Water System *.

Agent	${\mathit{s}}_9:\left\{5,6\right\}$	${\mathit{s}}_{10}:\left\{5,8\right\}$	${\mathit{s}}_{11}:\left\{7,8\right\}$	${\mathit{s}}_{12}:\left\{6,7\right\}$	${\mathit{s}}_{13}:\left\{11,12\right\}$	${\mathit{s}}_{14}:\left\{7,12\right\}$	${\mathit{s}}_{15}:\left\{8,11\right\}$
$5$	90.35%	9.65%	0	0	0	0	0
$6$	0	0	0	100%	0	0	0
$7$	0	0	97.34%	0	0	2.66%	0
$8$	0	0	100%	0	0	0	0
$11$	0	0	0	0	0	0	100%
$12$	0	0	0	0	100%	0	0
Allocated water (108 m3)	8.77	0.97	81.31	0.10	0.40	2.49	20.62

* The figures in the table were calculated by the authors on the basis of available information and formulae.

. The overall benefit of the fuzzy cooperative alliance of the Aksu River System is CNY 348.490 billion, and the benefits of $s_9,s_{10},s_{11},s_{12},s_{13},s_{14},s_{15}$ are CNY 89.454 billion, CNY 5.849 billion, CNY 91.067 billion, CNY 40 million, CNY 743 million, CNY 1.770 billion, and CNY 63.097 billion, respectively. Based on Table 20 and the formula in Section 2.4.2, the distribution of benefits to each water-using entity in the fuzzy cooperative alliance of the Aksu River System is shown in

Table 21. Distribution of benefits under different fuzzy cooperative allocation rules for the Aksu River System players (10⁸ CNY) *.

Agent	Fuzzy Core	Fuzzy Least Core	Fuzzy Weak Least Core	Fuzzy Shapley Value
5	26.79	26.79	26.79	250.59
6	2160.35	77.67	77.67	327.97
7	422.48	422.48	422.48	496.91
8	622.06	622.06	622.06	1976.07
11	181.04	181.04	181.04	282.66
12	72.18	2154.86	2154.86	150.70

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

3.3.4. Coalition for Fuzzy Cooperation in the Kai-Kong River + Main Stream River System

According to the formula in Section 2.3, combined with the rate of return and maximum allowable carrying capacity of different cooperative alliances in

Table 22. Profit parameters and maximum carrying capacity of the fuzzy cooperative alliance in the Kai-Kong River + Main Stream River system *.

Fuzzy Alliance	Players	Rate of Return per Unit of Water Resource (CNY/m3)	Maximum Allowable Carrying Capacity (108 m3)
$s_{16}$	$\left\{9,13\right\}$	4.67	95.2
$s_{17}$	$\left\{9,14\right\}$	6.92	92.6
$s_{18}$	$\left\{13,14\right\}$	28.88	13.8
$s_{19}$	$\left\{9,13,14\right\}$	7.06	101.5

* The figures in the table have been compiled by the authors on the basis of available information.

, the participation rate of multiple players in fuzzy cooperative gaming alliances in the Kai-Kong River + Main Stream can be obtained, as shown in

Table 23. Participation rate and allocation of water quantity in the fuzzy cooperative alliance among people in the Kai-Kong River + Main Stream River System *.

Agent	${\mathit{s}}_1:\left\{1,2\right\}$	${\mathit{s}}_2:\left\{1,17\right\}$	${\mathit{s}}_3:\left\{2,17\right\}$	${\mathit{s}}_4:\left\{1,2,17\right\}$
$9$	0	6.60%	0	93.40%
$13$	0	0	92.59%	7.41%
$14$	0	100%	0	0
Allocated water (108 m3)	0.00	2.62	8.50	35.80

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

The participation rate and the amount of water allocated to the people in the Bureau of the Kai-Kong River + Main Stream Water System can be obtained from Table 22, and the overall benefit of the Fuzzy Cooperative Alliance of the River Water System of the Kai-Kong River + Main Stream Water System is CNY 135.953 billion, and the benefits of ${s_{17},s}_{18}$ and $s_{19}$ are CNY 1.813, 24.548 and 25.275 billion, respectively. Based on Table 22 and the equations in Section 2.4.2, the distribution of the benefits of each water-using entity of the Kai-Kong River + Main Stream Water System Fuzzy Cooperative Alliance is shown in

Table 24. Distribution of benefits under different fuzzy cooperative allocation rules for the Kai-Kong River + Main Stream River System players (10⁸ CNY) *.

Agent	Fuzzy Core	Fuzzy Least Core	Fuzzy Weak Least Core	Fuzzy Shapley Value
3	No solutions	No solutions	No solutions	86.15
4				1106.78
15				121.91
16				44.69

* The figures in the table were calculated by the authors on the basis of available information and formulae.

.

3.4. Analysis of the Results

For convenience of analysis, the benefits of the nine allocation schemes of the four water system cooperative alliances in the Tarim River Basin are made into surfaces in Figure 1, Figure 2, Figure 3 and Figure 4 for each of the players in the three scenarios of separate use, crisp cooperative alliance, and fuzzy cooperative alliance, and the benefits to the players in the cooperative alliance of the Hotan River System are shown in Figure 1. Those for the Yarkand River System are shown in Figure 2, for the Aksu River System are shown in Figure 3, and for the Kai-kong River + Main Stream Water System alliance are shown in Figure 4. Due to the core scenario in the crisp cooperative alliance lacking solutions in each water system alliance, the core scheme is not displayed in Figure 1, Figure 2, Figure 3 and Figure 4. In order to make the surface more intuitive, the players in the coalition are arranged on the y-axis according to demand from small to large, and the benefits of the people in the bureaus are fitted to form a smoothed surface graph according to the different allocation schemes, in which the red ball indicates the benefits of each water-using region in the different allocation schemes.

Firstly, both the crisp cooperative gaming alliance and the fuzzy cooperative alliance can effectively improve the overall water benefits of regional water resources, and the fuzzy cooperative alliance has a more significant improvement effect than the crisp cooperative alliance. A comparison of the water resource benefits of multiple actors in the Tarim River Basin is shown in

Table 25. Comparison of water resource benefits to multiple actors in the Tarim River Basin (10⁸ CNY) *.

Rainy Season	Player	Separate Use	Core	Least Core	Weak Least Core	Shapley	Fuzzy Core	Fuzzy Least Core	Fuzzy Weak Least Core	Fuzzy Shapley
Hotan River System	17	17.75	31.13	17.75	91.94	17.75	17.75	17.75	70.82	17.75
	1	107.33	107.33	107.33	167.42	160.37	160.37	107.33	114.85	107.33
	2	81.46	350.76	81.46	261.49	280.53	280.53	81.46	266.67	81.46
	18	6.87	147.86	430.54	116.23	300.59	300.59	552.7	306.9	6.87
Yarkant River System	15	74.49	562.27	68.75	164.36	−	198.97	68.75	78.72	74.49
	16	59.66	59.66	553.18	213.05	−	59.66	189.88	154.24	59.66
	4	283.71	283.71	283.71	489.35	−	283.71	283.71	327.8	283.71
	3	422.90	442.9	442.9	481.78	−	442.9	442.9	424.48	422.90
Aksu River System	5	26.79	26.79	26.79	169.48	26.79	26.79	26.79	250.59	26.79
	6	77.67	353.78	77.67	398.94	2160.35	77.67	77.67	327.97	77.67
	12	72.18	170.95	1342.79	221.66	72.18	2154.86	2154.86	150.7	72.18
	11	181.04	209.62	181.04	244.93	181.04	181.04	181.04	282.66	181.04
	7	422.48	518.39	422.48	518.54	422.48	422.48	422.48	496.91	422.48
	8	622.06	1393.29	622.06	1119.28	622.06	622.06	622.06	1976.07	622.06
Kai-Kong River + Main Stream River System	14	31.39	31.39	3070.72	189.36	−	−	−	44.69	31.39
	13	61.87	131.22	61.87	472.64	−	−	−	121.91	61.87
	9	217.70	217.7	217.7	501.99	−	−	−	86.15	217.70
	10	832.56	3802.54	832.56	3018.86	−	−	−	1106.78	832.56

* The figures in the table were calculated by the authors on the basis of available information and formulae, “−” indicates no solution.

.

In the Tarim River Basin, when water users use water resources alone, the overall benefits of water users in the Hotan River System, the Yarkant River System, the Aksu River System, and the Kai-Kong River + Main Stream System are 21.341, 84.077, 140.222 and CNY 114.352 billion, respectively. A crisp cooperative alliance is formed when the water-using entities join the cooperative game within water resources to achieve the optimal allocation of water resources within the water system. According to Table 6, Table 8, Table 10 and Table 12, under the precondition of constructing a crisp cooperative alliance, the overall benefits of the crisp cooperative alliance of the Hotan River System, the Yarkant River System, the Aksu River System, and the Kai-Kong River + Main Stream System are CNY 63.708, 134.854, 267.283, and 418.285 billion, respectively, which are 198.52%, 57.72%, 90.61% and 265.79% higher than the benefits of separate use. A crisp cooperative alliance formed by all the players in the game can deploy and optimize the allocation of water resources within the alliance for the overall benefit of the basin in which the cooperative alliance is located.

When the water-using subject takes some of their water resources to join different cooperative alliances to form a fuzzy cooperative alliance, it can be seen from Table 15, Table 18, Table 21 and Table 22 that under the premise of constructing a fuzzy cooperative alliance, the overall benefits of the fuzzy cooperative alliance of the Hotan River system, the Yarkand River system, the Aksu River system, and the Kai-Kong River + Main Stream system are CNY 76.611, 126.895, 34.849 and 135.953 billion, which are 258.98%, 50.93%, 148.53% and 18.89% higher than the benefits of separate use, respectively, and the fuzzy cooperative alliance’s effect on the overall benefits of the region is also readily apparent. Except for the Kai-Kong River + Main Stream water system, in the Tarim River Basin, the players participate in the fuzzy cooperative alliance more obviously than in the crisp cooperative alliance in terms of benefit enhancement. It is clear from the above analysis that both crisp and fuzzy cooperative alliances can effectively improve the water resource use efficiency of the region.

Secondly, participation in crisp and fuzzy cooperative alliances enables the majority of the players in the water system alliance to obtain greater benefits than the use of water resources alone. Additionally, it helps the players to identify the partners with whom to cooperate while maintaining the alliance’s perpetual existence.

Crisp and fuzzy cooperative alliances aim to maximize the benefits of water resources in the region and use the “visible hand” within the alliance to guide the allocation of water resources to water users who use water resources more efficiently, so that the benefits of the cooperative alliance can be increased significantly. Meanwhile, water users participating in the cooperative alliance can receive more benefits from the alliance. The case study of the Tarim River Basin is a good example. After participating in the cooperative alliance, all the players in the Hotan River System, the Yarkant River System, the Aksu River System, and the Kai-Kong River + Main Stream System receive greater gains than when they use water resources alone; among them, the 14th Division Industry, the 3rd Division Agriculture, and Aksu Industry, and Bayin Guoleng Industry had the largest increase in gains, and the 14th Division Agriculture and Kizilsu Kirghiz Agriculture had the smallest increase in gains. It is worth noting that the solutions obtained from the core allocation method in the crisp alliance are all empty sets; the Third Division Industry and the Second Division Industry had a larger increase in gains from the weak least core method’s allocation of the crisp alliance, which was significantly higher compared to the increase in the other allocation schemes, and the weak least core allocation method caused the Third Division Industry and the Second Division Industry to obtain higher gains through the cooperative alliance after relaxing constraints on the core of the alliance required.

Thirdly, the different methods of allocating returns to the cooperative alliance directly affect the stability of the alliance, and there is heterogeneity in the selection preferences of the players in the game when using the different allocation schemes.

Comparing the revenue allocation methods of the four crisp cooperative games, the stability of the core is the worst, and the crisp cooperative alliances of the four water systems in the Tarim River Basin cannot produce a solution set when using the core allocation method. The reason for this is that the core’s constraints are more stringent. The least core, weak least core, and Shapley value methods can ensure the stable existence of the cooperative alliance, so it can be seen that the allocation methods of the crisp cooperative alliances will directly affect their stability and permanence. Comparing the four allocation methods of the fuzzy cooperation game, the stability of fuzzy core is the worst, and both the Yarkant River Water System alliance and Kai-Kong River + Main Stream Water System alliance are empty sets when using the fuzzy core allocation method; meanwhile, the solutions of the Kai-Kong River + Main Stream Water System alliance in the fuzzy cooperative alliance using the fuzzy core, fuzzy least core, and fuzzy weak least core allocation methods are all empty sets, which indicates that the cooperation and alliance of Kai-Kong River + Main Stream Water System in the Bureau does not create additional benefits after cooperation, resulting in the fuzzy cooperative alliance not being able to exist forever.

In the four water systems of the Tarim River Basin, subjects with lower water use efficiency tend to choose the crisp Shapley value and the fuzzy Shapley value, and subjects with higher water use efficiency are more inclined to choose the crisp least core or the fuzzy least core. Water-using subjects with lower water use efficiency can obtain greater benefits using the crisp Shapley value and fuzzy Shapley value allocation methods; for example, the agricultural sector of Hotan has the lowest water use efficiency in the Hotan River System alliance and receives CNY 16.742 billion according to the crisp Shapley value method under the conditions of a crisp cooperative alliance. It can be seen that the less efficient water users are more inclined to the crisp Shapley value method and the fuzzy Shapley value method; this is probably because the Shapley value method pays more attention to the marginal contribution rate. The less efficient water users carry more water resources into the cooperative alliance despite the inefficiency of their water resource utilization, so that water resources can be allocated preferentially to water users through the optimal allocation of the cooperative alliance. Water users with high water use efficiency can be allocated benefits through the optimized allocation method of the cooperative alliance, so that subjects and the cooperative alliance as a whole can receive higher returns. Water users with higher water use efficiency, such as industrial water users in Hotan and industrial water users in Bayin Guoleng, can obtain higher returns when utilizing the least core or fuzzy least core method.

4. Discussion

In the research presented in this paper, we have drawn three main conclusions about the mechanisms for distributing benefits of the watershed to water resource cooperative alliances. In order to understand the significance and value of these conclusions more fully, our findings are compared and discussed below with related studies by other authors.

4.1. Comparison of the Effects of Clear and Fuzzy Cooperative Alliances

Our results demonstrate that both clear and fuzzy cooperative alliances are effective in enhancing the overall water use benefits of regional water resources. This finding is in line with many existing studies that have similarly highlighted the positive role of cooperative games in water resource allocation. However, we further find that fuzzy cooperative alliances are more effective than crisp cooperative alliances in enhancing overall gains. This finding complements the existing literature by pointing out that fuzzy cooperative coalitions, due to their flexibility and adaptability, may be better suited to dealing with complex and ever-changing water resource environments and diverse demands within water allocation.

Our findings provide new perspectives and evidence for the application of cooperative game theory in water resource allocation, especially in the construction of fuzzy cooperative coalitions and the evaluation of their effects. This will help future research to explore more deeply the applicability of and differences between different cooperative alliance models in various contexts.

4.2. Impact of Cooperative Alliance on the Benefits of the Players

Our study points out that most of the players in water system coalitions who participate in crisp and fuzzy cooperative coalitions are able to obtain higher gains than when using water resources alone. This finding is consistent with the results of many cooperative game theory studies, which have emphasized that cooperation can lead to additional gains, i.e., cooperative surplus. However, our study further reveals that cooperative alliances, while sustaining their own perpetual existence, can also help the players identify suitable cooperation partners, thus enhancing the stability and sustainability of cooperation.

This finding has important implications for the methods used in water management practices. It suggests that by building a cooperative alliance and designing a rational benefit distribution mechanism, more water users can be motivated to participate in water resource cooperation to meet the challenges of water scarcity. At the same time, the formation of a cooperative alliance can also help to reduce the conflict and competition among water users and promote the harmonious allocation and sustainable use of water resources.

4.3. Impact of Cooperative Alliances’ Revenue Distribution Methods on Alliances’ Stability

Our study highlights the importance of the revenue allocation methods of cooperative alliances, pointing out that different allocation methods can directly affect the stability of the alliance. This finding echoes many studies on the distribution of benefits in cooperative games, which similarly point out the important role of benefit distribution mechanisms in cooperation. However, our study further reveals the heterogeneity of the players’ preferences for different allocation schemes, which implies that the specific needs and preferences of different players need to be considered when designing the mechanism for allocating benefits in order to ensure the fairness and effectiveness of the allocation.

This finding provides important insights for policy makers. In formulating water resource management policies, policy makers need to consider the interests and needs of different water users and design a rational benefit-sharing mechanism to incentivize cooperation and maintain the stability of the alliance. Policy makers also need to pay attention to the differences in the preferences of the parties for different allocation schemes and meet the needs of different parties through flexible and diverse allocation methods in order to promote the optimal allocation and sustainable use of water resources.

As a result, our research has produced valuable findings on the effects of clear and fuzzy cooperative alliances, the effects of cooperative alliances on benefits to players, and the effects of cooperative alliances’ methods of allocating benefits on the stability of these alliances. These findings not only enrich the applied research of cooperative game theory in the field of water resource allocation but also provide important references and lessons for the practice of water resource management.

5. Conclusions

Building a water resource community of cooperation, openness, and mutual benefit is a potential solution to the future global water crisis and water resource hegemony. This paper extends the idea of the cooperative game to the field of water resource utilization, establishes a crisp cooperative alliance and fuzzy cooperative alliance from the perspective of a cooperative game, constructs a mechanism of water resource sharing and a benefit distribution mechanism, portrays the fair allocation of water resources within the cooperative alliance based on the “Priority Principle”, and employs the core, least core, weak least core, and Shapley value methods to evaluate the benefits of such an alliance. These methods were applied to the Tarim River Basin, and the Hotan River Water System alliance, Yarkant River Water System alliance, Aksu River Water System alliance, and Kaikong River + Main Stream Water System alliance were constructed for the empirical test. Our main conclusions are as follows. Firstly, a cooperative alliance that is founded on a cooperative game can effectively increase overall benefits and individual benefits and ensure that incentives for collective rationality and individual rationality are compatible and can be effective in sustaining the existence of the alliance. Secondly, the fuzzy cooperative alliance can effectively improve overall benefits and individual benefits, which ensures the compatibility of collective rationality and individual rationality and helps players to determine the object of cooperation while maintaining the sustainable existence of the alliance. Moreover, the fuzzy cooperative alliance is more effective than the crisp cooperative alliance in enhancing the overall benefits of the region’s water use. Thirdly, the fuzzy cooperative alliance allows players to bring some water resources with them when they join multiple cooperative alliances, which is an extension of the crisp cooperative alliance and represents a higher level of cooperative gaming. Players in a fuzzy cooperative alliance can receive more benefits than in a crisp cooperative alliance, and it has been proven that fuzzy cooperative alliances are more suitable for the actual situation. Different allocation methods cause the whole alliance and the players to derive different benefits, and the cohesion and sustainability of the cooperative alliance have obvious heterogeneity.

Concerning policy recommendations, it is suggested that basin-wide cooperation alliances should be steadily promoted, and a sound mechanism for benefit distribution and sharing should be established. Basin regions should conduct in-depth consultations, communication, and exchange; explore the strategic points of the convergence of water rights trading among regions; and accelerate cooperation in various water-related fields. They should continue to innovate modes of cooperation, from small-scale cooperation at a low level to all-round cooperation at a high level, and continuously promote the formation of a water rights trading alliance to realize the efficient use of water resources. They should promote the establishment of alliances between governments in upstream and downstream regions as well as between water users. Developing regional regulations for the trade of water rights will ensure that all stakeholders within the alliance can realize self-management under a common vision. Mutual political trust among regions must be enhanced, and we must advocate that regardless of the size, strength, and wealth of regional water-using entities, they should have the same right to common development, ecological products, and equal participation in decision making. We must build a horizontal transfer payment system between regional water-using entities and innovate a mechanism for ecological benefit and compensation. We should also compensate for the interests of underdeveloped regions and regions whose actual interests have been damaged, broaden more channels for compensation of interests, build a water resource cooperation alliance (organization) within the basin, distribute resources and interests within the alliance (organization) with a tangible hand, form a win–win situation for multiple water-using subjects, and enhance the welfare level of the whole basin. Through cooperative alliance, the establishment of mechanisms for external ecological governance, sharing responsibility, and cooperative governance will be synergistically promoted in order to realize the sustainable development of an ecological environment and ecological compensation in the basin.

The resource-sharing mechanism and benefit distribution mechanism of the water resource cooperative alliance constructed and developed in this paper have good applicability to the problem of allocating water resources in river basins and can provide policy references for the efficient use and optimal allocation of water resources in areas with water shortages, such as arid and semi-arid zones in China. Win–win cooperation is the main theme of future socio-economic development, which calls for creation of a social environment characterized by solidarity, equality, balance, and universality; we must deepen multi-party co-operation and share the cake while making it bigger, aiming to achieve common prosperity for all of society. In terms of research limitations, this paper has less consideration of externalities and ecological compensation in the construction of cooperative alliances for water rights, which can be remedied in future research.