Analysis on competition strategy and regulatory strategy of international river water resources system – with the participation of co-operational agencies Open Access

In recent years, climate change has seriously affected the distribution of water resources, and the development of human society has caused a certain degree of pollution to water resources, which has further aggravated the pressure of regional water shortage, so natural water shortage has become a long-term phenomenon in many countries and regions (Wheater & Gober 2015). Water resources are not only vital to people's lives, but also play an important role in global economic development (Dolan et al. 2021). The water resources of international rivers flow through many countries, due to the shortage and economic value of water resources, there are fierce water resources competition and conflict. Regional water resources competition and conflict are not conducive to regional peace, stability and development. Therefore, it is of great practical significance to study the management of water resources in international rivers, which can not only alleviate the pressure of the supply and demand of water resources, but also solve the problem of water resources competition and conflict in international rivers.

The water resource management of international rivers has been the focus of the researcher, especially the allocation of cross-boundary water resources. The EU multi-level cross-boundary water resource governance model is considered as a more comprehensive and advanced mode worldwide, but there are some shortcomings (Baranyai 2020). Due to the hydrological geography, political system and other reasons, the EU cross-boundary water resource management system ignores the water quantity management, supervision and distribution (Baranyai 2019), and pays more attention to the water resources governance framework and policy issues. Therefore, individuals establishing a scientific cross-boundary water resources regulation mechanism and finding the solution to key scientific problems and technical bottlenecks are faced with many difficulties (Daming et al. 2016).

Game theory is a discipline that studies conflicts and cooperation, and can well explain and analyze the behavioral characteristics of each participant in water resource competition (Liu et al. 2002; Wei et al. 2009; Ha et al. 2013; Fu et al. 2016). It is a powerful tool to solve the optimization of water resources allocation in international rivers. Kucukmehmetoglu & Guldmann (2002) used a cooperative game model to solve the water resource allocation problems of the Euphrates and Tigris rivers, and have achieved good results; Wu & Whittington (2006) have also achieved remarkable results in solving the problem of water resource allocation in the Nile Basin by using cooperative game theory. Wang et al. (2008) use the cooperative game method to realize the optimal reallocation of water resources based on the allocation of initial water rights. Ansink et al. (2017) used a cooperative game model to study the stability of water users alliance in international rivers, it was found that if there are four or more agents in the international rivers, there may not be a complete water market. These studies fully show the effectiveness of cooperative game theory in solving the problem of water resource allocation in international rivers.

Some researchers use non-cooperative game model to study water resources management. Kucukmehmetoglu (2012) combines the non-cooperative game with Pareto optimal theory to propose a comprehensive method of water resources optimization allocation, and applies it to practice. Parna et al. (2015) use the method of non-cooperative game to allocate groundwater and surface water, and clarify the evolutionary behavior characteristics of non-cooperative game in the process of using groundwater and surface water. Oftadeh et al. (2017) use an ultimatum non-cooperative game model to solve the conflict of water resources in international rivers, and also ensures the economic satisfaction of land owners and farmers in the upstream. Yuan et al. (2019) use game theory-related theories to construct a new weighted water allocation model on the basis of considering the economic, social and environmental differences among water users, and applied it to the optimal allocation of water resources in the Mekong River.

Some experts consider some other factors when using game theory to study water resources management. Bashar et al. (2012) consider the preference uncertainty factors in the process of constructing a game model, and construct a new conflict decomposition model, and use it in the optimal allocation of groundwater resources. The water users in the international rivers have upstream and downstream relationships, so some researchers take the upstream and downstream relationship into account when establishing the water resource game model (Degefu et al. 2016; Xiao et al. 2016; Avarideh et al. 2017; Yang et al. 2019). In addition, in order to maximize the overall benefit of water resources in international rivers, some researchers consider not only the competition between water users, but also the cooperation among water users when establishing the water resource game model (Wang & Liu 2018, 2020).

In the actual situation, water resources exist in a dynamic form, and the quantity of changes with the process of water flowing downstream, the water resources of the river are consumed and replenished when flowing through each country. Therefore, the game of water resources needs to consider not only the relationship between upstream and downstream, but also the consumption and supply of water resources. The relationship between upstream and downstream determines the order of the game, the consumption and supply determine the variables of the game, which makes the water resources game of international rivers more complex. In addition, game theory has been mainly used to solve the allocation of water resources among water users in international rivers, the participation of a third party was seldom considered. But in reality, most international rivers have co-operational agencies, which plays a crucial role in solving the competition and conflict of water resources. Therefore, the introduction of a co-operational agency in the process of studying the game of water resources is very conducive to the integrated regulation of water resources in international rivers.

In the game of international river water resources, in order to give full play to the role of co-operational agencies and reflect the changes in water volume in the process of water resources flow, this paper establishes a game model of international river water resources competition. It provides theoretical support for the optimal allocation of water resources, and provides theoretical basis for solving the frequent conflicts of water resources in international rivers. Here we analyze the influence of the regulatory strategy on the competition behavior of the water users, and the influence of the water user competition behavior on the regulatory strategy; explore the main factors that affect the competition and the regulation, and the long-term game evolution strategy of co-operational agencies and water users.

For international rivers, a river usually flows through multiple countries, which are called water users. Suppose an international river has n water users, the set composed of all water users is ⁠, the set of agreed water withdrawal of all water users is ⁠, the set of actual water withdrawal of all water users is ⁠. Due to the differences in science and technology, the benefits of water resources among water users are different, the set of unit water resources economic benefits of all water users is ⁠. When the actual water withdrawal of the water user is greater than the agreed water withdrawal, the scene of competition appears. It is assumed that extra water resources obtained by the water users can also bring certain benefits, the set of unit extra water resources benefits of all water users is ⁠. The process of water withdrawal will incur costs, due to the different development levels of different water users, the cost of water withdrawal is different. The cost of water withdrawal for different water users is composed of fixed cost and variable cost. The set of fixed costs of all water user is ⁠, which is mainly the cost of engineering facilities. The set of variable costs of all water user is ⁠, which is mainly the cost of water withdrawal per unit of water resources.

Suppose that water users of international rivers obtain a certain amount of water resources through competition. At this time, the total amount of water resources Q is distributed among water users through game. In reality, the available water resources of each water user mainly come from the remaining water resources of all water users in the upstream and the replenishment of water resources in its own region. Assume that the set of water resources replenishment quantity of all water users is ⁠. If the water demand of other water users is not considered, the minimum amount of water resources that each water user can obtain is the water replenishment in its own administrative region. Each water user needs to make efforts to obtain the corresponding water resources, the set of effort behavior of all water users is ⁠. Each water user's effort behavior needs to pay corresponding cost, the set of effort behavior cost of all water users is ⁠, the level of effort is used to represent the effort behavior cost, then ⁠.

For international rivers, there are generally co-operational agencies. The existence of co-operational agencies can regulate the vicious competition among the water users, and play a role of regulation and regulatory in the process of competition game among the water users. The co-operational agency needs everyone to work together, assume that the daily operation funds provided by water users to the co-operational agency are ⁠. If the co-operational agency regulates the water withdrawal behavior of the water users, the cost is to be paid. When there is competition among water users, the co-operational agency will impose a fine of G on the water users. If the co-operational agency does not regulate the water withdrawal behavior of water users, when there is competition among water users, the co-operational agency needs to adjust the conflicts caused by water resources competition, the cost is to be paid. When there is no competition among water users, water users will receive a better honor, assuming that the value of honor is b. This value is usually reflected in diplomacy or economic cooperation or other valuable aspects.

According to the different players of the game, the game of the international rivers can be divided into the game among water users and the game between the water users and the co-operational agency. The purpose of introducing co-operational agency into the game of water resources in international rivers is to explore the optimal allocation strategy of water resources with a new idea, so as to alleviate the current situation of frequent conflicts of water resources and promote the long-term stable development of international rivers.

To solve this dynamic game problem, we can use the forward solving method. Starting from the first water user optimal problem, we can solve the relationship of optimal water withdrawal between the most upstream water user and the downstream water users, and then it is brought into the next water user optimal problem. This process is repeated until the optimal water withdrawal quantity of the last water user is solved. At this time, the equilibrium solution of the multi-stage dynamic game problem among the water users in international rivers can be obtained, and then the optimal water withdrawal quantity of each water user can be determined.

From the game theory, the game strategies among the water users of international rivers and between the water users and the co-operational agency is analyzed. Here we analyze the long-term evolution and stability strategy of each stakeholder in an international rivers water resources competition system, as well as the key factors that affect the stability of each stakeholder's strategy.

Equation (13) shows the equilibrium point of the competition of water users, that is, when the competition strength of water users is ⁠, the overall benefit is the greatest. By observing Equation (13), we can get the following results:

• (1)

  If ⁠, there is no competition among water users, ⁠. So ⁠, then the actual water withdrawal of water users is less than the agreed water withdrawal. In this situation, when the competition strategies of other water users remain unchanged, the competitive equilibrium point of water user i will be stable. For water users i, the competitive equilibrium point is inversely proportional to their own water withdrawal cost and directly proportional to their water source benefit. In addition, we can also find that the competitive equilibrium point of water users in this situation has nothing to do with the co-operational agencies.

• (2)

  When there is competition among water users, ⁠, then the actual water withdrawal of water users is greater than the agreed water withdrawal. In this situation, when the competition strategies of other water users remain unchanged, the competitive equilibrium point of water user i is also related to the co-operational agencies, which is inversely proportional to the regulatory strength. Similarly, the competitive equilibrium point is also related to their own water withdrawal cost and water source benefit. The difference is that the competitive equilibrium point at this time is also directly proportional to the extra water resources benefit.

Equation (14) shows the relationship between the competition of one water user i and other water users in international rivers:

• (1)

  If ⁠, then ⁠, which indicates that water user i is in a dominant position in the process of competition, and the competitive equilibrium point of water user i will increase with the increase of competition intensity of other water users. If other water users strengthen the competition, water user i will also make more efforts to obtain more water resources, until the cost paid by water user i exceeds the benefits obtained by competition.

• (2)

  If ⁠, then ⁠, which indicates that water user i is at a disadvantage in the process of competition, and the competitive equilibrium point of water user i will decrease with the increase of competition intensity of other water users. It shows that under the condition of the penalty strength and the regulatory strategy are the same, the benefits obtained by water users i in the process of competition are less than the costs they pay, that is, the revenue of water user i will decrease with the increase of the competition intensity among other users. If the water user i cannot improve its water use efficiency or reduce its water withdrawal cost, the water user i will not continue to participate in the competition.

Because ⁠, ⁠, ⁠, so if then ⁠, we can deduce ⁠; if ⁠, whether is positive or negative is affected by y. Based on the above analysis, we can get:

Proposition 4.1

• (1)

  When ⁠, for any y, is the evolutionary stability strategy of co-operational agencies.

• (2)

  When and ⁠, any regulatory strategy x is stable, and will not change over time.

• (3)

  When ⁠, if ⁠, is the evolutionary stability strategy of co-operational agencies; If ⁠, is the evolutionary stability strategy of co-operational agencies.

Proposition 4.1 (1) shows that the implementation of the regulatory strategy will cause losses to the co-operational agencies when the benefits obtained are less than the costs paid in the process of regulation, then the co-operational agencies has no incentive to supervise the water users. Proposition 4.1 (2) shows that when the cost is equal to the benefit, the co-operational agencies will keep the existing strategy unchanged, and any competition strategy of water users will be stable for co-operational agencies. Proposition 4.1 (3) shows that the competition of water users affects the regulatory strategy of the co-operational agencies under certain conditions. When the competition among water users is fierce, the co-operational agencies may benefit from the regulatory, so the regulatory measures are implemented. When the competition among water users is weak, the co-operational agencies cannot benefit from the regulatory, so the regulatory measures are not implemented. Therefore, the higher competition among water users in international rivers, the more likely the co-operational agencies will take regulatory measures. In other words, the co-operational agencies will have more incentive to take regulatory measures for highly competitive international rivers, which is why the more competitive the international rivers, the more necessary it is to establish regulatory agencies.

Because⁠, ⁠, ⁠, ⁠, so if ⁠, then ⁠, we can deduce ⁠; if ⁠, whether is positive or negative is affected by x. Based on the above analysis, we can get:

Proposition 4.2

• (1)

  When ⁠, then for any x, is the evolutionary stability strategy of water users.

• (2)

  When ⁠, and ⁠, any competition strategy y is stable, and will not change over time.

• (3)

  When ⁠, if ⁠, is the evolutionary stability strategy of water users; If ⁠, is the evolutionary stability strategy of water users.

Proposition 4.2 (1) shows that when the benefits from competition are greater than the losses, the water users may take competitive strategy to obtain more water resources. Proposition 4.2 (2) shows that when the benefits are equal to the losses, the water users will maintain the existing competition strategy unchanged, and any regulatory strategy of co-operational agencies will be stable for water users. Proposition 4.2 (3) shows that the regulatory strategy of the co-operational agencies affects the competition strategy of the water users under certain conditions. When the co-operational agencies has strong regulatory, competition will damage the benefits of water users, and the water users will not take the competitive strategy. When the co-operational agencies has weak regulatory, the water users may benefit from the competition, and the water users will take the competitive strategy. Therefore, the greater the regulatory, the weaker the competition of water resources among water users.

In order to achieve the above goal, increasing and b, or reducing and can be achieved. In addition, reducing the difference of ⁠, ⁠, and f between different water users can also alleviate the competition of water resources in international rivers. Because when the difference between the economic benefits and water withdrawal costs of water users is small, it is more likely to end the problem of water resources conflict through negotiation.

For water users, the regulatory strategy of co-operational agencies, the benefit of water resources, honor value and water withdrawal cost all have an important impact on the competition. The greater the regulatory, the less likely the competition scenario will appear, the lower the frequency of the conflict of water resources competition. The greater the benefit of water resources, the greater the possibility of competitive behavior. The higher the honor value, the lower the possibility of competitive behavior. The lower the water withdrawal cost, the lower the equilibrium point of competition. In addition, for a water user, if the incentive degree of competition of other water users increases, it will further improve the chips of competition, and will not withdraw from the competition until the income of its water resources is less than zero. For co-operational agencies, the competition strategy of water users, fines and regulatory costs all have an important impact on the regulation strategy. The more intense the competition among water users of international rivers, the stronger the driving force of the co-operational agencies to implement the regulatory strategy. The higher the fine of competition, the stronger the driving force of the co-operational agencies to implement the regulatory strategy. The higher the regulatory cost, the weaker the driving force of the co-operational agencies to implement the regulatory strategy. In addition, the amount of the daily operation funds has no impact on the regulatory strategy of co-operational agencies, that is, variable income has an impact on the regulatory strategy of co-operational agencies, and fixed income has no impact on the regulatory strategy.

Narrowing the gap of water withdrawal costs, economic benefits among water user in the international rivers can inhibit the competition among water users. Because when the difference of water withdrawal costs, economic benefit among water resources is small, the difference in water resources obtained by water users is small under the same regulatory and competitive efforts. Appropriate regulatory strategies can prevent excessive competition of international river water resources, and the more intense the competition of water resources, the more the international rivers need the regulation of co-operational agencies. In addition, among these influencing factors, honor value is one of the most important factors. Co-operational agencies can alleviate the water resource competition of international rivers by improving the honor value. For example, we can strengthen the economic cooperation of water users, promote the optimal allocation of water resources through economic complementary, so as to create a win–win situation. Reducing the cost in the process of regulatory and increasing the fine in the process of regulatory can promote co-operational agencies to supervise in international rivers. Besides, the daily operating funds of co-operational agencies have no impact on its regulatory strategy, so it is also very important for co-operational agencies to determine the appropriate operation funds. Finally, good regulatory strategy of co-operational agencies can not only maintain regional stability, but also ensure the maximization of the overall benefits of water resources in international rivers, this is the main purpose of the regulatory of international rivers by co-operational agencies, and also the reason for introducing co-operational agencies in the process of studying the game of water resources in international rivers.

This work was funded by the National Social Science of China (grant no. 19BGL181); Fundamental Research Funds for the Central Universities (grant no. B220203022); Postgraduate Research & Practice Innovation Program of Jiangsu Province (grant no. KYCX22_0685); Jiangsu Social Science of China (grant no. 21GLA001).

Writing – original draft preparation, Methodology, Editing, Data curation, Conceptualization, Software: Hao Xu; Conceptualization, Methodology, Validation, Review, Project administration, Investigation: Liuxin Chen; Funding acquisition, Formal analysis, Editing, Methodology, Data curation, Regulatory, Resources: Qiongfang Li. All authors have read and agreed to the published version of the manuscript.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.