Co-Evolutionary Mechanism of Stakeholders’ Strategies in Comprehensive Agricultural Water Price Reform: The View of Evolutionary Game Based on Prospect Theory

Abstract

In today’s world, the contradiction between water supply and demand is becoming increasingly pronounced, with a particular emphasis on the severe shortage of water for agricultural purposes. As a result, it has become imperative to promote the comprehensive reform of agricultural water pricing and increase water conservation awareness among water users. However, during the actual promotion process, the conflicting interests among stakeholders often create a behavioral game that seriously hinders the effective implementation of the agricultural water price reform. Therefore, it is crucial to address this conflict of interest and find ways to overcome it in order to ensure the successful implementation of the agricultural water pricing reform. In order to explore the dynamic evolution process of the behavior and decision making of the stakeholders in the comprehensive agricultural water price reform and the influencing factors and to propose relevant strategies to effectively promote the reform. This article constructs a three-party game model based on MA-PT theory with the government, farmers, and water supply units as the main subjects, solves the equation with the perceived benefit matrix instead of the traditional benefit matrix, and calculates the objective conditions for reaching the steady state. The simulation results show that the behavioral decision of the game subject to participate in the reform will be influenced by their perception of gains and losses, and there is a mutual influence between the three strategic choices. Therefore, the reform of agricultural water prices should consider the affordability of farmers and water supply units. The government appropriately adopts a reward system to encourage farmers and water supply units to actively participate in the reform to play a positive role and appropriate penalties for promoting the reform, the early realization of the reform goals, and the development of water-saving agriculture.

1. Introduction

Water scarcity is increasingly becoming a problem in both developed and developing countries and impeding social and economic development worldwide [1]. For a long time, there have been problems of water shortage and over-exploitation in various regions of China, and the continuous growth of water demand from economic development has caused and aggravated the water shortage in China. As a sizeable water-using industry in China, agriculture is experiencing a severe water shortage, and the efficiency of water utilization is only 30~40%, which not only restricts the development of agriculture in China but also threatens food security. The agricultural water price in China is in the process of reform, and most areas are still using water in a relatively crude manner, resulting in the phenomena of water shortage and water wastage; in other words, effectively achieving agricultural water conservation has become the primary task and decisive link in optimizing agricultural water allocation in China [2]. At present, the main problems in China’s agricultural water use are old agricultural water conservancy facilities, inadequate operation and maintenance, and poor management; inadequate water pricing mechanisms, low price levels, and agricultural water prices that do not effectively reflect the cost of water use; backward irrigation methods, equipment, technology as a whole, etc. All these problems will lead to low efficiency and waste in agricultural water use [3]. Therefore, it is essential to effectively promote the comprehensive reform of agricultural water prices to solve the problem of agricultural water use.

Throughout the development of China’s agricultural water price reform, it has experienced three stages: public welfare free water use (1949–1964), policy low price water use (1965–2005), and comprehensive agricultural water price reform (2006–present) [4]. The comprehensive reform of agricultural water prices concerns the stage of cost water supply. In 2006, China issued the Notice on Strengthening the Management of Water Prices in Agricultural Last Stage Drainage Systems, which laid the foundation for the comprehensive reform of agricultural water prices in China. From 2007 onwards, China began implementing several reform pilots to promote the reform. In 2016, the General Office of the State Council promulgated the “Opinions on Promoting Comprehensive Reform of Agricultural Water Prices”, clarifying the strategic trend of the comprehensive reform of agricultural water prices in China; the comprehensive reform of agricultural water prices was thus pushed out nationwide. The 2019 Central Government Document No. 1 points out the need to accelerate the comprehensive reform of agricultural water prices and improve the incentive mechanism for water conservation. As the pilot areas in each province explore the comprehensive reform of agricultural water prices one after another, the sustainable use and development of agricultural water resources have ushered in new opportunities and challenges [5]. So far, China’s comprehensive reform of agricultural water prices from the pilot to the national scale has entered a new era of complete promotion.

But the actual process of promotion due to the conflicts of interest between the various stakeholders and the pursuit of their interests to maximize the game constantly seriously restricts the effective implementation of the agricultural water price reform. “Stakeholder theory” originated in 1959, when Penrose introduced the concept that “human resources and human relationships together build the firm” in his book “Theories of Business Growth,” which laid the foundation for stakeholder research [6]. Stakeholder theory involves various disciplines, such as management, sociology, and ethics, and has become a critical theory and tool for business management research. Academics first focused on stakeholder analysis of enterprises, then applied stakeholder theory to public management and policy making [7]. However, as the idea became more mature, it gradually began to be applied to other social practice studies. For example, stakeholder studies in rural tourism [8], tourism land development [9], community regeneration [10], rural land use [11], and photovoltaic power plants [12] can scientifically guide their management. A stakeholder in the management sense is any interested party in the organization’s external environment who is affected by the organization’s decisions and actions. There are various views in academia on the definition of stakeholders. Still, it is generally agreed that they should be defined from two perspectives: organizations or individuals who have an interest and who influence or are influenced by the organization’s activities [13]. According to these two perspectives, this paper defines “agricultural water price stakeholders” as organizations and individuals who participate in the comprehensive agricultural water price reform and are affected by agricultural water price (or affect the cost of agricultural water supply) [14]. Through the study of agricultural water price stakeholders, we can accurately identify agricultural water price stakeholders, analyze the interest demands of each subject, determine the relationship between stakeholders, predict the impact of the behavior of each interest subject on the implementation of the policy, avoid the reform resistance caused by the conflicts of interest and game, use the linkage between the issues and their interest demands, use the interest as a driving force to promote the participation of each stakeholder in the water price reform, and play its positive role to facilitate the reform process, which can effectively improve and encourage the efficiency of the comprehensive agricultural water price reform work [15]. Therefore, to better promote the comprehensive agricultural water price reform, how to accurately model the dynamic evolutionary process of each stakeholder under different decisions is a problem worth studying. This paper can fill the gap in the current research.

Compared to prior studies, the possible contributions of this paper lie in the following points. The first point applies the MA-PT combined with evolutionary game theory to examine the game behaviors among the stakeholders in the comprehensive agricultural water price reform to analyze the strategic choices of each stakeholder. Through simulating and analyzing the strategic evolution process of each party, we reveal the interactions and conflicts of interest between different stakeholders, observe the decision-making behaviors adopted by each stakeholder with the dynamic simulation of different parameter states, and predict the impact of reform measures on the interests of each party. This analysis helps identify more reasonable behavioral decision-making strategies, ultimately aiming to achieve a win–win outcome for all parties involved in the reform process. From the micro level, the perceived value of each stakeholder is divided into a gain account and a loss account, addressing the limited rationality assumption of evolutionary game theory and considering the influence of decision makers’ irrational behaviors. This approach aligns better with the complexities of human psychology. The second point is that different from the two-party game model constructed by previous studies [4,16], this paper constructs a three-party game model involving the government, farmers, and water supply units. According to the stakeholder theory, both farmers and water supply units will affect the agricultural water price and play non-negligible roles in it. Thus, from the perspective of the mechanism that affects the price of water, we take the government, farmers, and water supply units as the main participants to analyze the behavioral decision making of all parties in order to achieve the maximization of the reform objectives and to provide practical significance and a decision-making basis for the smooth implementation of the reform.

The rest of this paper is structured as follows. The second part of this article is a review of the current research on the game analysis of stakeholders in the comprehensive agricultural water price reform; the third part discusses the MA-PT theoretical approach, analyzes the relationship between the stakeholders, and constructs a model using assumptions to derive the payoff matrix; the fourth part, i.e., the solution and analysis of the model, analyzes the dynamic trend of strategy choice of the three parties and the local stability of the model; the fifth part analyzes the factors affecting the strategy choice of each party and derives the dynamic evolution process via simulation; finally, the evolution results of each stakeholder in different states are analyzed and discussed, the results are derived, and the shortcomings and directions for further research are proposed.

2. Literature Review

The comprehensive reform of agricultural water prices is a systematic reform of farmland water conservancy promoting the proper operation of farmland water and agricultural water conservation, which revolves around reasonable water prices. At present, there have been a lot of studies on the comprehensive reform of agricultural water prices in China. Wenlai Jiang [17] argued that a comprehensive agricultural water price reform should take into account multiple factors, such as socioeconomic and political factors, rather than purely economic factors. Qiang Wang [2] evaluated the changes in effects before and after the comprehensive agricultural water price reform at five levels: water saving, economic, ecological, social, and institutional impacts. Tao Zou [5] proposed that the current comprehensive reform of agricultural water prices still has the problem of a small scope of reform pilots and uneven progress in pilot areas. Haifeng Cui [18] pointed out that due to the unique nature of the agricultural production sector, it is determined that the central position of the government in the reform must be adhered to in the process of agricultural water price adjustment. Shilong Wang [6] proposed policy recommendations to further deepen the comprehensive agricultural water price reform from four perspectives: water resource owner, water supplier, water user, and other parties. Theoretically, China has developed a comprehensive, systematic, and advanced water pricing policy. However, in practice, it has been found that its reform is uncertain, its water pricing policy is contradictory, and a clear and sustainable direction and path for water pricing development needs to be developed [19]. There is also some controversy in foreign academic circles about agricultural water pricing reform policies, such as price increases for agricultural water. Currently, the international community attaches great importance to the rational allocation of water resources using market mechanisms and economic instruments to improve water use efficiency, resolve water conflicts, and promote sustainable water resource management [20,21,22]. Sapino, F. et al. [23] were the first to use a comprehensive experiment with a mathematical programming model to assess the socioeconomic impacts of agricultural water price reform. Studies have shown that the “economic value of water” ranks first among possible price scenarios based on sustainable water management criteria [24]. In many countries, such as the United States, China, Australia, and Chile, water rights trading has been recognized as an effective means of optimizing water allocation, improving water use efficiency, and alleviating water scarcity [25,26,27,28], and many water rights trading schemes have been established worldwide [29,30]. D’Odorico, P. et al. [31] agree that the water market is the best means of efficiently allocating water resources, but the commodification of water resources is still controversial. Nasiri, Parvaneh et al. [32] argued that the adverse effects must be offset with alternative policies that improve the sustainability of water resources without causing fluctuations in the economic benefits to users. Berbela and Gomez [33] pointed out that excessive agricultural water prices can compress agricultural production efficiency, reduce farmers’ willingness to produce, and threaten food security. Economic instruments such as price increases cannot be relied on exclusively to promote agricultural water conservation. Therefore, before adjusting the water price, the reform of the construction of farmland water conservancy projects and supporting water pipeline systems should be strengthened [34], and the level of farmland water supply services should be improved [35] to gain the cooperation of farmers. Agricultural water prices depend on the affordability of agricultural water in each region [36]. A reasonable agricultural water price is conducive to the transformation of agricultural water subsidies from implicit subsidies to explicit subsidies and to improving the efficiency of agricultural water conservation, which is a prerequisite and guarantee of agricultural water price reforms [37]. Still, the incentive amount should not be too high, and when the incentive amount is higher than water charges, the relationship between water charges and water consumption is reversed [38]. Kun Cheng et al. [39] developed an interval two-stage stochastic programming model, which was applied to explain the effects of agricultural water price reforms and subsidies for water-saving technologies. But contributions to the water sector, not only to farmers, can have the dual benefit of saving water and mitigating adverse impacts [40].

Therefore, in the face of complex interests, it is essential to clarify the boundaries of the interests of each stakeholder and effectively create conditions for cooperation [41]. According to Wensheng He et al. [42], reform is a complex process involving the overlap of “stakeholders, policy implementation, and policy context,” and the comprehensive agricultural water price reform should propose relevant policies in three dimensions: integrating stakeholders’ perceptions and preferences, strengthening the effectiveness of policy implementation, and optimizing the policy environment. In contrast, domestic research on agricultural water price stakeholders is relatively limited. Xin Feng [14,15] has conducted more research related to comprehensive agricultural water price reform, especially on the stakeholder aspects of water price reform, using Mitchell’s scoring method to quantitatively rank the stakeholders of comprehensive agricultural water price reform. The results present the following ranking: government departments > farmers and their organizations > other social institutions. The paper also proposes that the core stakeholders of the reform are the central government, local governments, and farmers, which provides a lot of research for the determination of stakeholders of water price reform.

In terms of research methodology, the evolutionary game is a method for studying the interactions and decision-making processes among individuals within biological and social systems. Its fundamental principle is that in a group of players, each individual engages in repeated game activities. Given the limited rationality of human beings, finding the optimal equilibrium in the initial game is highly improbable. However, in subsequent games, each player improves their strategy through imitation and learning, aiming to identify the evolutionarily stable strategy [43]. Currently, an increasing number of scholars employ evolutionary game theory to analyze various group behaviors in society. Of course, in the context of the comprehensive agricultural water price reform, there are also scholars who combine stakeholders with game theory to study the dynamics that drive the reform. The earliest one is Guoping Wang [16], who analyzed the strategic choice between water supply units and farmers from the game perspective and proposed that agricultural water price reform should balance economic and social interests and mobilize the enthusiasm of all stakeholders. Xiangcheng Jiang et al. [4] divided the stakes involved in the comprehensive agricultural water price reform into four categories: agricultural water price influencers, agricultural water price bearers, agricultural water price implementers, and managers of farmland water conservancy facilities. They also conducted a game analysis of the influencers and bearers and put forward some suggestions and countermeasures for promoting the reform. Dangyang Di et al. [26] addressed the problem of determining the optimal water trading volume and price for each region in water rights trading using a basin dynamic differential game and a pricing game approach. In summary, previous studies have employed the game approach to analyze various issues related to the comprehensive agricultural water price reform, then the application of the evolutionary game theory to the game analysis between the stakeholders of comprehensive agricultural water price reform has certain applicability. However, most studies have focused on the decision making between two stakeholders and have utilized static game models. Yet, given the absence of completely rational conditions in real society, it becomes necessary to adopt the assumption of finite rationality as the foundation [44], and then conducting dynamic simulations to analyze the evolving behavioral strategies of the main actors is essential. Therefore, in order to address these research gaps, it is of significant importance to apply the dynamic evolutionary game model to analyze the evolutionary paths and different strategies adopted by stakeholders in comprehensive agricultural water price reform.

Based on the above research, in the international market, some countries’ water price formation mechanism can be summarized as “enterprise pricing, government regulation”, and the current stakeholders for water price reform mostly focus on the game before the government and farmers, but water supply units also play a non-negligible role in the reform, so this paper constructs a tripartite subject for the government, farmers, and water supply units. The game model of government, farmers, and water supply units is, therefore, constructed. In previous studies, the influence of the perceived value of each stakeholder’s decision-making behavior was ignored, so this paper introduces the MA-PT theory, replaces the traditional benefit matrix with the perceived benefit matrix, analyzes the dynamic trend of the evolution of each subject, and finally uses MATLAB algorithms to simulate the factors influencing the choice of the tripartite strategy, deriving the dynamic evolutions of the behaviors of multiple subjects in different states. In this way, we can propose some relevant strategies to promote comprehensive agricultural water price reform. This paper can fill the gap in the current research.

3. Research Methodology

3.1. Stakeholder Description of Comprehensive Agricultural Water Price Reform

Based on a summary and analysis of the existing research literature on the field of comprehensive agricultural water price reform [3,4,14,15], it is initially concluded that the area of comprehensive agricultural water price reform involves 15 stakeholders, namely the central government, local governments, farmers’ water associations, village collective village committees, water users, water users who have switched from agriculture to non-agriculture, county-level water conservancy management departments, grassroots water conservancy stations, irrigation district management units, water-saving irrigation enterprises, water operators, financial institutions, research institutions, public welfare institutions, and the media. County-level water management departments, grassroots water stations, irrigation district management units, water-saving irrigation enterprises, water operators, financial institutions, scientific research institutions, public welfare institutions, and the press. Many stakeholders are involved in the comprehensive agricultural water price reform, so this paper has selected 11 stakeholders based on the analysis of the main stakeholders in the comprehensive agricultural water price reform and divided them into three categories: government, farmers, and water supply units. Their classification is shown in

Table 1. Classification of agricultural water stakeholders.

Stakeholders	Institution	Responsibilities and Obligations
Government	Central Government	It is the leading reformer, with absolute authority in water pricing decisions and water ownership; responsible for introducing policies, authorizing, incentivizing reform, making investments and financial subsidies, guiding and supervising local governments to promote water pricing reform
	Local Government	It is the actual reformer, carrying out specific management work, authorizing water supply units to operate and manage or purchase their services, the genuine master of the award funds, and putting policies and opinions into practice
Farmers	Water-using farmers	It is the primary bearer of agricultural water fees, paying the required fees to obtain the economic benefits of irrigation and better water supply services
	Farmers’ Water Association	Representatives of water-using farmers, formed by the farmers themselves, take care of the relevant water-using behavior, collect water charges, and pay them to the water supply unit according to regulations
	Collective village committee	Play an essential role in coordinating between farmers and government and other sectors to promote regional development, have an endogenous motivation to participate, and encourage reform
	Water for “agricultural to non-agricultural” parties	The non-agricultural water user acquires the right to use agricultural water through purchase, so part of the agricultural water use is borne by the non-agricultural water user
Water supply units	County-level water management departments	It is the representative of the irrigation district management unit and the grassroots water conservancy station to propose the pricing scheme of the preliminary water price, to guide the management of water facilities and to maintain their regular operation
	Grass-roots water conservancy stations	Guide the construction of farmland water conservancy facilities within the jurisdiction and the management of flood and drought control water conservancy projects while collecting water charges for water conservancy projects and agricultural water charges according to law
	Irrigation district management units	Based on the accounting results of the operation and maintenance costs of irrigation backbone projects, the government and farmers’ wishes are taken into account in the preliminary determination of water pricing schemes, but without final decision-making authority, manage agricultural water use in each region and make decisions on the operation of the corresponding irrigation districts
	Water-saving irrigation enterprises	Cooperate with the government in various aspects such as engineering construction, operation, and maintenance management, intelligent testing, charging, etc. To promote reform and improve water prices while obtaining corresponding revenue and considerable profits via the output of engineering and service products
	Water conservancy operators	Provision of water supply services and management of water supply projects, and collection of water charges based on the cost of water supply [4]

.

The government is the leader, the initiator of the reform, the policy maker, and the ultimate decision maker. The government aims to bring economic, social, and ecological benefits to a more effective and smooth water price reform [45]. On the financial side, raising the price of water will compensate for the cost of operation and maintenance of water conservancy projects; on the social side, it will guarantee food security and farmers’ income and create a good atmosphere for water conservation; on the ecological side, it will protect the environmental water of the basin via water conservation, maintain the habitat of living organisms, guarantee the quality of water resources, and perform the function of ecological irrigation services. Farmers are the users of agricultural water and are the critical link in the agricultural water price reform, aiming to obtain the economic benefits of irrigation and better water supply services. Water supply units can participate in various tasks, such as the development of reform programs, construction of water conservancy projects, production of irrigation or metering and other related equipment, construction of information management platforms, undertaking maintenance of farmland water conservancy projects, and supervision and evaluation of reform as a third party [46], to maximize benefits. Each stakeholder drives and influences the other. The tripartite relationship and the relationship with agricultural water use are shown in Figure 1.

3.2. Construction of Perceived Benefit Function Based on MA-PT Theory

MA-PT stands for Mental Accounting and Prospect Theory. Firstly, mental accounting refers to the psychological process in decision-making where individuals categorize, code, value, and budget outcomes, which often leads to violations of simple economic calculations. People tend to compare choices based on the outcomes rather than following economic rules. In evolutionary game theory, only the expected returns of the participants are considered, neglecting the potential transfer of funds during the game and the impact of mental accounting. This oversimplified game model fails to capture the complexities of decision-making. Secondly, prospect theory, proposed from a psychological perspective, is widely used to explain and predict people’s decision making when faced with risks [47]. It reveals irrational behavior in human decision making. However, when stating that game participants are finite rational individuals, it overlooks the influence of the psychological perceived value and risk preferences on actual decision making. In reality, participants often adopt conservative strategies due to profit and loss aversion in order to avoid excessive losses. Therefore, incorporating MA-PT theory in the evolutionary game model allows for a better understanding of how individuals weigh gains and losses and how they perceive and handle the effects of risk and uncertainty in their game behavior. This approach provides more accurate predictions and decision-making analyses that align with the actual situation. Many scholars have already combined evolutionary game theory with mental accounting and prospect theory to analyze and predict various social issues [48,49]. Since evolutionary games involve human behavioral decision making among different groups, which inevitably includes irrational behavior, adopting the MA-PT theory to study the behavioral decision making of the government, farmers, and water supply units in the comprehensive agricultural water price reform is indeed applicable.

Although there exist various theoretical approaches, such as frame dependence theory, extended mental accounting theory, and time preference theory, which explain irrational behaviors of decision makers in economics, finance, and psychology, the MA-PT theory stands out with its unique advantage. However, the MA-PT theory is better equipped to explain the psychological factors and emotional experiences that influence decision-making behaviors by considering decision-making contexts and frames and has been extensively validated via empirical studies and practical applications. Therefore, this paper utilizes the MA-PT theory to address the overlooked influence of decision makers’ irrational behavior on their strategy choices in evolutionary games.

Kahneman and Tversky [50] introduced prospect theory in 1979 as an improvement to the expected utility theory in psychology [51]. It is considered that perceived value $V$ is measured by a value function $v(\Delta x_i)$ and a weighting function $\omega \left(p_i\right)$, which can be expressed as follows:

$$V={\displaystyle \sum _{i=1}^{n}v\left(\Delta x_i\right)\omega \left(p_i\right)}$$

(1)

where $V$ represents the perceived value of the gain of the game subject, $\Delta x_i$ is the difference between the actual gain of the game subject and the gain obtained at the reference point, the weight function $\omega \left(p_i\right)$ is of the subjective perception of the game subject on the probability of occurrence of the behavioral strategy, and $p_i$ is the objective likelihood of occurrence of the behavioral approach.

Combined with the mental account theory, the value function $v(\Delta x_i)$ is further divided into $L(x)$ and $G(x)$ (a loss account and a gain account), and both the gain account and the payment account have their own perceived reference points and perceived value functions, that is, when an individual or a group faces a gain (or loss), if the gain value (or loss value) is greater than the perceived reference point, the relative gain (or loss) of the decision maker is a positive (negative) value, that is, to obtain the relative gain (or loss) perception; if the gain value (or loss value) is less than the perceived reference point, the relative gain (or loss) of the decision maker is a negative (positive) value, that is, to obtain the close loss (or gain) perception [48]. The specific functions are as follows:

$$L\left(x\right)=\left\{\begin{array}{l}\kappa {\left(x-U_1\right)}^{\gamma },x\ge U_1\\ -{\left(U_1-x\right)}^{\delta },x<U_1\end{array}\right.$$

(2)

$$G\left(x\right)=\left\{\begin{array}{l}{\left(x-U_2\right)}^{\phi },x\ge U_2\\ -\tau {\left(U_2-x\right)}^{\zeta },x<U_2\end{array}\right.$$

(3)

$U_1$ in the loss account $L(x)$ represents the base value of the losses, $\kappa$ is the sensitivity of loss aversion, and $\gamma$ and $\delta$ are the risk appetite coefficients for losses; $U_2$ in the return account $G(x)$ represents the benchmark value of the returns, $\tau$ is the sensitivity of loss aversion, and $\phi$ and $\zeta$ are the risk appetite coefficients for returns [49].

The decision weight function associated with the psychological account theory is the decision weight value derived by the decision maker who makes objective judgments about profit and loss based on the proportion of events [48]. The weight function $\omega \left(p_i\right)$ is divided into $\omega {\left(p\right)}^{+}$ and $\omega {\left(p\right)}^{-}$ (benefit decision weight function and loss decision weight function) as follows:

$$\left\{\begin{array}{l}\omega {\left(p\right)}^{+}=\frac{p^{\mu }}{p^{\mu }+{\left(1-p^{\mu }\right)}^{\frac{1}{\mu }}}\\ \omega {\left(p\right)}^{-}=\frac{p^{\eta }}{p^{\eta }+{\left(1-p^{\eta }\right)}^{\frac{1}{\eta }}}\end{array}\right.$$

(4)

where $\omega {\left(p\right)}^{+}$ and $\omega {\left(p\right)}^{-}$ represents the decision maker’s subjective judgment of the likelihood of the event occurring, and $\omega \left(0\right)=0$, $\omega \left(1\right)=1$; $\mu$ and $\eta$ represents the decision impact factor with a magnitude that determines the degree of impact of the weights. The perceived value function and decision weight function essentially depend on the objective decision environment and are influenced by the decision maker’s risk preferences, so the values of the relevant parameters are variable and vary in different backgrounds [49].

3.3. Model Underlying Assumptions

Based on the above description of the stakeholders of the comprehensive agricultural water price reform and the responsibilities and obligations in Table 1, a multi-body game model of the government, farmers, and water supply units is constructed to seek to promote the development momentum of the agricultural water price reform by studying the evolutionary dynamic process of the three parties for the comprehensive agricultural water price reform. The following hypotheses are proposed according to the research needs:

Hypothesis 1.

The government, farmers, and water supply units are all three interest groups that are little rational actors, behaving with limited rationality in the game process and influenced by their preferences and perceived values when conducting participation in the comprehensive agricultural water price reform. The government leads the reform through investment, construction projects, subsidized funds, etc., to promote the comprehensive reform of agricultural water prices in an orderly manner, to promote the improvement of water resources utilization efficiency, and to promote the development of the economic agricultural industry [52]. The goal of farmers is to obtain better economic benefits from irrigation and better water supply services. The purpose of water supply units is to maximize the benefits.

Hypothesis 2.

(1) The government, as the leading comprehensive reform of agricultural water pricing, develops and implements water pricing reform policies related to the key subjects, assuming that its behavioral strategy consists of actively implementing policies to the incentive reform ($g_1$) and routinely implementing policies to cope with the reform ($g_2$), noted as (actively implement, routine cope). (2) When the government actively implements policies, it formulates relevant policies via surveys and research and carries out specific management work to incentivize the input costs of reforms assumed to be $C_1$. (3) The government implements incentives to give farmers with high water use efficiency a certain amount of water-saving incentives and payment subsidies to promote agricultural water saving and efficiency of the cost assumptions $\alpha B_1$, where the incentives and subsidies for the strength of the $\alpha$. (4) The cost assumptions of the government’s financial support and subsidy measures for water supply units providing quality water services to incentivize reforms and ensure efficient supply of high-quality water resources are $\beta B_2$, where the subsidy is $\beta$. (5) The indirect economic benefits, such as social and ecological benefits, gained by the government via the implementation of a series of positive reform strategies to achieve the goal of water-saving irrigation as well as to improve agricultural production capacity are assumed to be $S_1$. (6) The direct economic benefit of collecting the water fees paid by farmers while the government carries out a reasonable increase in the price of agricultural water is assumed to be $S_2$. (7) Reputation and trust gains for the government are hypothesized to be $Y_1$as a result of the government’s active implementation of policies and incentives that lead to the public’s satisfaction with the government’s current performance and the formation of perceptions of the government’s behavior in terms of goodwill and competence. (8) On the contrary, if the government routinely implements policies to cope with the reform, maintains old and outdated water supply and irrigation equipment in order to minimize the cost of the reform, and does not implement incentives such as water-saving incentives to promote the development of the reform, the cost of the reform invested by the government is assumed to be $C_2$. (9) However, the government’s coping with reform will result in the loss of indirect economic benefits from the failure of the reforms to be as effective as they should be, from the failure to improve water wastage, and from the continued reduction in agricultural production capacity, assumed to be $F_1$.

Hypothesis 3.

(1) Farmers, as the main bearers of agricultural water charges, are at a certain disadvantage in the reform and need to cooperate with the reform based on the implementation of the policy. It is assumed that their behavioral strategies include proactive cooperating with reform ($p_1$) and passively cooperating with the reform ($p_2$), noted as (active, passive). (2) Farmers actively cooperate with the reform, actively participate in the decision-making process of agricultural water pricing, adopt advanced irrigation technology and water conservation measures, and strive to improve the efficiency of water resources utilization so that the energy invested as well as the cost of active payment of water fees is assumed to be $C_3$. (3) The economic benefits gained by farmers after adopting efficient water-saving irrigation technologies, improving agricultural productivity, and restructuring production are assumed to be $S_4$. (4) On the contrary, farmers passively cooperate with the reform to maintain the original concept of water use due to the lack of awareness of water conservation caused by the waste of water resources and, thus, a large number of water use of rough irrigation to pay the cost of water charges assumed to be $C_4$. (5) When the government monitors the irrational use of water in agriculture and improper behavior, the passive cooperation of farmers with the reform and rough water use of violations of the penalties and treatments, as well as being subject to certain invisible penalties, is assumed to be $F_2$.

Hypothesis 4.

(1) The primary responsibility of water supply units is to provide reliable water supply services to ensure that farmers have timely access to qualified water for agriculture; it is assumed that their behavioral strategies include providing quality water services ($e_1$) and maintaining existing water services ($e_2$), marked as (quality water supply, maintain status quo). (2) The cost of the water supply unit to guide the construction of the water supply project and the management and care of the flood control and drought relief water conservancy project, to carry out the operation and maintenance of the water supply equipment, and to provide technical support and consulting services to the farmers in agricultural water use is assumed to be $C_5$. (3) The water supply unit responds to the reform of the water tariffs and collects the water charges of the water conservancy project and the agricultural water charges according to the law on the basis of the setting of a reasonable price of the agricultural water tariffs and obtains the direct benefit assumed to be $S_6$. (4) The reputational benefits obtained by water supply units as a result of providing high-quality water supply services so that more farmers tend to believe in the rationality and effectiveness of the policy and choose to actively cooperate with the reform at the same time is assumed to be $Y_2$. (5) On the contrary, when the water supply units are willing to maintain only the existing works and equipment for water supply, the input costs after scaling down the funds for management, operation, and maintenance are assumed to be $C_6$. (6) If the water supply units do not respond to the policy and provide agricultural water resources that do not meet the relevant safety and environmental standards, resulting in agricultural production is affected by the contamination of water quality and the restriction of the source of water, and the punishment of the government’s monitoring and certain invisible penalties is assumed to be $F_3$.

Hypothesis 5.

The probability that the government will actively implement policies to incentive reform $x$, the probability of being routine cope is $1-x$; The probability that the farmer actively cooperates with the reform is $y$, the probability of a passive fit is $1-y$; The likelihood of a water supply unit providing quality water service is $z$, the likelihood of maintaining the status quo is $1-z$. And $x$, $y$, $z$takes values in the range [0, 1].

The specific meanings of the parameters in the model are shown in

Table 2. The specific meanings of the parameters.

Parameter Name	The Specific Meanings of Parameters
$C_1$	The cost of active government reform and investment in construction
$C_2$	Routine coping government reform costs
$S_1$	Indirect economic benefits gained by the government from incentive reform
$S_2$	Direct economic benefits are gained from price increases when the government is incentive reform
$F_1$	Indirect economic benefits lost due to routine coping government reform
$S_3$	Direct economic benefits are gained from price increases when the government routine coping reform
$\alpha B_1$	The government’s initiative to cooperate with the reform of the farmers to save water incentives and payment subsidies $\alpha B_1$, reward, and subsidy strength $\alpha$
$\beta B_2$	The government provides financial subsidies to water supply units that provide quality water service $\beta B_2$, and the donations are $\beta$
$Y_1$	Reputational benefits from active government implementation of relevant policies and subsidies
$C_3$	The cost of labor and payment of water bills when farmers take the initiative to cooperate with the reform
$C_4$	The cost of farmers’ passive cooperation with reform
$S_4$	Economic benefits for farmers via good irrigation and restructuring of production
$S_5$	Farmers passively reform the economic benefits gained via irrigation
$F_2$	Some invisible penalties on farmers for passively cooperating with reform
$C_5$	Cost of providing quality water services by water supply units
$C_6$	Management and operation, and maintenance costs of water supply units when maintaining existing water services
$S_6$	Revenue earned by water supply units for providing quality water services and collecting water charges
$S_7$	The water supply unit maintains the status quo water supply service to collect water charges and gains
$Y_2$	Reputational gains from water supply units providing quality water services
$F_3$	Specific invisible penalties for water supply units to maintain the status quo
$x$	The probability that the government will actively implement policies to incentive reform
$y$	Probability of farmers actively cooperating with the reform
$z$	Probability of water supply units providing quality water services

.

Based on the above assumptions and the descriptions of the parameters, a three-party game structure tree is constructed, as shown in Figure 2. It visually describes the strategy choice when the game is played by the government, farmers, and water supply units. The first row represents two strategic choices for the government, which can choose between $x$, an active implementation of policies to the incentive reform, and $1-x$, a routine implementation of policies in coping with the reform. The second row represents the strategic choice of the farmers, i.e., $y$ and $1-y$ denote the farmers’ intention to actively cooperate with the reform and passively cooperate with the reform, respectively. The third row represents the strategic choices of the water provider, which are $z$, providing quality water services, and $1-z$, maintaining existing water services; The third row represents the strategy choices of the water supply unit, which are to provide quality water service and keep the existing water service. The fourth row represents the final benefits of the eight options after all three parties of the game, i.e., the government, the farmers, and the water supply unit, have achieved their strategy choices. The lines represent the strategies that each can choose.

3.4. Stakeholder Benefit Matrix Construction

Based on the above basic assumptions, the traditional payoff matrix of the evolutionary game model is constructed, as shown in

Table 3. Traditional payoff matrix of tripartite evolutionary game.

			Farmers $\mathit{y}$	$\mathbf{Farmers} 1-\mathit{y}$
Government $x$	Water supply units $z$	Government revenue	$S_1+S_2+Y_1-C_1-\alpha B_1-\beta B_2$	$S_1+S_2+Y_1-\beta B_2$
		Farmers’ revenue	$S_4+\alpha B_1-C_3$	$S_5-C_4-F_2$
		Water supply unit revenue	$Y_2+S_6+\beta B_2-C_5$	$Y_2+S_6+\beta B_2-C_5$
	$\mathrm{Water} \mathrm{supply} \mathrm{units} 1-z$	Government revenue	$S_1+S_2+Y_1-C_1-\alpha B_1$	$S_1+S_2+Y_1-C_1$
		Farmers’ revenue	$S_4+\alpha B_1-C_3$	$S_5-C_4-F_2$
		Water supply unit revenue	$S_7-C_6-F_3$	$S_7-C_6-F_3$
Government $1-x$	Water supply units $z$	Government revenue	$S_3-F_1-C_2$	$S_3-F_1-C_2$
		Farmers’ revenue	$S_4-C_3$	$S_5-C_4-F_2$
		Water supply unit revenue	$Y_2+S_6-C_5$	$Y_2+S_6-C_5$
	$\mathrm{Water} \mathrm{supply} \mathrm{units} 1-z$	Government revenue	$S_3-F_1-C_2$	$S_3-F_1-C_2$
		Farmers’ revenue	$S_4-C_3$	$S_5-C_4-F_2$
		Water supply unit revenue	$S_7-C_6-F_3$	$S_7-C_6-F_3$

.

Combining traditional gains with MA-PT theory builds the perceived gain matrix of the three-party game, as shown in

Table 4. Perceived payoff matrix of tripartite evolutionary game.

Three-Way Game Strategy		Government Revenue	Farmers’ Revenue	Water Supply Unit Revenue
①	$\left(x,y,z\right)$	$G\left(S_1+S_2+Y_1\right)-L\left(C_1+\alpha B_1+\beta B_2\right)$	$G\left(S_4+\alpha B_1\right)-L\left(C_3\right)$	$G\left(Y_2+S_6+\beta B_2\right)-L\left(C_5\right)$
②	$\left(x,y,1-z\right)$	$G\left(S_1+S_2+Y_1\right)-L\left(C_1+\alpha B_1\right)$	$G\left(S_4+\alpha B_1\right)-L\left(C_3\right)$	$G\left(S_7\right)-L\left(C_6+F_3\right)$
③	$\left(x,1-y,z\right)$	$G\left(S_1+S_2+Y_1\right)-L\left(\beta B_2\right)$	$G\left(S_5\right)-L\left(C_4+F_2\right)$	$G\left(Y_2+S_6+\beta B_2\right)-L\left(C_5\right)$
④	$\left(x,1-y,1-z\right)$	$G\left(S_1+S_2+Y_1\right)-L\left(C_1\right)$	$G\left(S_5\right)-L\left(C_4+F_2\right)$	$G\left(S_7\right)-L\left(C_6+F_3\right)$
⑤	$\left(1-x,y,z\right)$	$G\left(S_3\right)-L\left(F_1+C_2\right)$	$G\left(S_4\right)-L\left(C_3\right)$	$G\left(Y_2+S_6\right)-L\left(C_5\right)$
⑥	$\left(1-x,y,1-z\right)$	$G\left(S_3\right)-L\left(F_1+C_2\right)$	$G\left(S_4\right)-L\left(C_3\right)$	$G\left(S_7\right)-L\left(C_6+F_3\right)$
⑦	$\left(1-x,1-y,z\right)$	$G\left(S_3\right)-L\left(F_1+C_2\right)$	$G\left(S_5\right)-L\left(C_4+F_2\right)$	$G\left(Y_2+S_6\right)-L\left(C_5\right)$
⑧	$\left(1-x,1-y,1-z\right)$	$G\left(S_3\right)-L\left(F_1+C_2\right)$	$G\left(S_5\right)-L\left(C_4+F_2\right)$	$G\left(S_7\right)-L\left(C_6+F_3\right)$

.

4. Model Solving and Analysis

4.1. Perceived Benefit Function of Each Stakeholder Based on MA-PT

In evolutionary games, the strategic choices of the subjects are usually judged by the calculation of expected returns. The mathematical meaning of expectation is the weighted average of probability, then the calculation of expected return is the product of probability and return. If there are four game strategies corresponding to the government’s active implementation of policy incentives for reform as described above, $\left(x,y,z\right)$, $\left(x,y,1-z\right)$, $\left(x,1-y,z\right)$, $\left(x,1-y,1-z\right)$, and the government’s perceived benefits corresponding to the combination of these four strategies are $G\left(S_1+S_2+Y_1\right)-L\left(C_1+\alpha B_1+\beta B_2\right)$, $G\left(S_1+S_2+Y_1\right)-L\left(C_1+\alpha B_1\right)$, $G\left(S_1+S_2+Y_1\right)-L\left(\beta B_2\right)$, and $G\left(S_1+S_2+Y_1\right)-L\left(C_1\right)$, respectively, then we obtain the following:

Perceived benefits under the government’s active implementation strategy = ${\displaystyle \sum _{i=n}^n(}$ probability of farmer’s behavior $\times$ probability of water supply unit’s behavior $\times$ government’s perceived benefits under this combination of strategies).

Similarly, the perceived benefits of routine government coping are calculated in the same way. $V{g}_1$ and $V{g}_2$ denote the perceived value function of the government’s active implementation of policies that incentivize reform and the perceived value function of routine cope, respectively. $\overline{V}g$ denotes the perceived value function of the government input reform. $V{p}_1$ and $V{p}_2$ represent the perceived value functions of farmers’ active and passive cooperation with the reform, respectively. $\overline{V}p$ represents the perceived value function of the farmer’s cooperation with the reform. $V{e}_1$ and $V{e}_2$ denote the perceived value functions of water supply units to provide quality water supply services and to maintain existing water supply services, respectively. $\overline{V}e$ denotes the perceived value function of the water supply unit in providing water supply services.

Of course, the strategy selection of the main players of the game is not completed after one selection. It is the optimal strategy of the main players to constantly repeat the game by learning and adjusting to finally derive the solution. The perceptual gain function is used to judge the strategy choice of each subject, and the dynamic description of each subject in the process of strategy adjustment needs to be calculated to analyze the replicated dynamic equations to arrive at the final equilibrium solution. The basic form of the replication dynamic equation in the evolutionary game is expressed as follows:

$$f\left(x\right)=\frac{dx}{dt}=x_i\left[f\left(s_i,x\right)-f\left(x,x\right)\right]$$

(5)

where $s_i$ is the strategy choice of the evolving subject, $x_i$ is the probability that the game subject chooses strategy $s_i$, $f\left(s_i,x\right)$ is the expected return when the game subject chooses strategy $s_i$, and $f\left(x,x\right)$ is the average expected return of the whole group. In the following, let $f\left(x\right),f\left(y\right),f\left(z\right)$ denote the replicated dynamic equations in the game model for the government, farmers, and water supply units, respectively.

Based on the above benefit matrix, the perceived value function of each stakeholder group based on MA-PT and the dynamic replication equation can be derived as follows.

(1) The government’s perceived value function

The perceived value function of active government policy implementation to incentive reform is expressed as follows:

$$\begin{array}{l}V{g}_1=\omega \left(y\right)\omega \left(z\right)\left[G\left(S_1+S_2+Y_1\right)-L\left(C_1+\alpha B_1+\beta B_2\right)\right]+\omega \left(1-y\right)\omega \left(z\right)\left[G\left(S_1+S_2+Y_1\right)-L\left(\beta B_2\right)\right]\\ +\omega \left(y\right)\omega \left(1-z\right)\left[G\left(S_1+S_2+Y_1\right)-L\left(C_1+\alpha B_1\right)\right]+\omega \left(1-y\right)\omega \left(1-z\right)\left[G\left(S_1+S_2+Y_1\right)-L\left(C_1\right)\right]\end{array}$$

(6)

The perceived value function of routine cope is expressed as follows:

$$\begin{array}{l}V{g}_2=\omega \left(y\right)\omega \left(z\right)\left[G\left(S_3\right)-L\left(F_1+C_2\right)\right]+\omega \left(1-y\right)\omega \left(z\right)\left[G\left(S_3\right)-L\left(F_1+C_2\right)\right]\\ +\omega \left(y\right)\omega \left(1-z\right)\left[G\left(S_3\right)-L\left(F_1+C_2\right)\right]+\omega \left(1-y\right)\omega \left(1-z\right)\left[G\left(S_3\right)-L\left(F_1+C_2\right)\right]\end{array}$$

(7)

The perceived value function for government to carry out the reform is expressed as follows:

$$\overline{V}g=xV{g}_1+\left(1-x\right)V{g}_2$$

(8)

The replicated dynamic differential equations for government to carry out reform is expressed as follows:

$$\begin{array}{l}f\left(x\right)=\frac{dx}{dt}=x\left(V{g}_1-\overline{V}g\right)=x\left(1-x\right)(V{g}_1-V{g}_2)=x\left(1-x\right)\left[L\right.\left(C_2-C_1+F_1\right)\\ +G\left(S_1+S_2+Y_1-S_3\right)-\omega \left(y\right)L\left(\alpha B_1\right)-\omega \left(z\right)L\left(\beta B_2-C_1\right)-\omega \left(y\right)\omega \left(z\right)L\left.\left(C_1\right)\right]\end{array}$$

(9)

(2) Farmers’ perceived value function

The perceived value function of farmers’ active cooperation with reform is expressed as follows:

$$\begin{array}{l}V{p}_1=\omega \left(x\right)\omega \left(z\right)\left[G\left(S_4+\alpha B_1\right)-L\left(C_3\right)\right]+\omega \left(x\right)\omega \left(1-z\right)\left[G\left(S_4+\alpha B_1\right)-L\left(C_3\right)\right]\\ +\omega \left(1-x\right)\omega \left(z\right)\left[G\left(S_4\right)-L\left(C_3\right)\right]+\omega \left(1-x\right)\omega \left(1-z\right)\left[G\left(S_4\right)-L\left(C_3\right)\right]\end{array}$$

(10)

The perceived value function of farmers’ passive cooperation with reform is expressed as follows:

$$\begin{array}{l}V{p}_2=\omega \left(x\right)\omega \left(z\right)\left[G\left(S_5\right)-L\left(C_4+F_2\right)\right]+\omega \left(1-x\right)\omega \left(z\right)\left[G\left(S_5\right)-L\left(C_4+F_2\right)\right]\\ +\omega \left(x\right)\omega \left(1-z\right)\left[G\left(S_5\right)-L\left(C_4+F_2\right)\right]+\omega \left(1-x\right)\omega \left(1-z\right)\left[G\left(S_5\right)-L\left(C_4+F_2\right)\right]\end{array}$$

(11)

The perceived value function of farmers’ cooperation with reform is expressed as follows:

$$\overline{V}p=yV{p}_1+\left(1-y\right)V{p}_2$$

(12)

The replicated dynamic differential equation for farm household cooperation with reform is expressed as follows:

$$\begin{array}{l}f\left(y\right)=\frac{dy}{dt}=y\left(V{p}_1-\overline{V}p\right)=y\left(1-y\right)(V{p}_1-V{p}_2)\\ =y\left(1-y\right)\left[L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)+\omega \left(x\right)G\left(\alpha B_1\right)\right]\end{array}$$

(13)

(3) Perceived value function of the water supply unit

The perceived value function of water supply units to provide quality water services is expressed as follows:

$$\begin{array}{l}V{e}_1=\omega \left(x\right)\omega \left(y\right)\left[G\left(Y_2+S_6+\beta B_2\right)-L\left(C_5\right)\right]+\omega \left(x\right)\omega \left(1-y\right)\left[G\left(Y_2+S_6+\beta B_2\right)-L\left(C_5\right)\right]\\ +\omega \left(1-x\right)\omega \left(y\right)\left[G\left(Y_2+S_6\right)-L\left(C_5\right)\right]+\omega \left(1-x\right)\omega \left(1-y\right)\left[G\left(Y_2+S_6\right)-L\left(C_5\right)\right]\end{array}$$

(14)

The perceived value function of water supply units to maintain existing water services is expressed as follows:

$$\begin{array}{l}V{e}_2=\omega \left(x\right)\omega \left(y\right)\left[G\left(S_7\right)-L\left(C_6+F_3\right)\right]+\omega \left(1-x\right)\omega \left(y\right)\left[G\left(S_7\right)-L\left(C_6+F_3\right)\right]\\ +\omega \left(x\right)\omega \left(1-y\right)\left[G\left(S_7\right)-L\left(C_6+F_3\right)\right]+\omega \left(1-x\right)\omega \left(1-y\right)\left[G\left(S_7\right)-L\left(C_6+F_3\right)\right]\end{array}$$

(15)

The perceived value function of water supply units providing water supply services is expressed as follows:

$$\overline{V}e=zV{e}_1+\left(1-z\right)V{e}_2$$

(16)

The replication of dynamic differential equations for water supply units providing water supply services is expressed as follows:

$$\begin{array}{l}f\left(z\right)=\frac{dz}{dt}=z\left(V{e}_1-\overline{V}e\right)=z\left(1-z\right)(V{e}_1-V{e}_2)\\ =z\left(1-z\right)\left[L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)+\omega \left(x\right)G\left(\beta B_2\right)\right]\end{array}$$

(17)

4.2. Dynamic Trend Analysis

From the above analysis, the dynamic replication equations of each game subject are derived, and to clarify the strategic choices of the three parties, the following dynamic trends of the government, farmers, and water supply units are analyzed separately.

(1) The replicated dynamic differential equation for the government to carry out the reform is shown in Equation (8).

1) $L\left(C_2-C_1+F_1\right)+G\left(S_1+S_2+Y_1-S_3\right)-\omega \left(y\right)L\left(\alpha B_1\right)-\omega \left(z\right)L\left(\beta B_2-C_1\right)-\omega \left(y\right)\omega \left(z\right)L\left(C_1\right)=0$, i.e.,

$$\omega \left(y\right)=\frac{L\left(C_2-C_1+F_1\right)+G\left(S_1+S_2+Y_1-S_3\right)-\omega \left(z\right)L\left(\beta B_2-C_1\right)}{L\left(\alpha B_1\right)+\omega \left(z\right)L\left(C_1\right)}$$

(18)

At $f\left(x\right)=0$, the system tends to a steady state regardless of the value of $x$.

2) When $\omega (y)\ne \frac{L(C_2-C_1+F_1)+G(S_1+S_2+Y_1-S_3)-\omega (z)L(\beta B_2-C_1)}{L(\alpha B_1)+\omega (z)L(C_1)}$, to $f\left(x\right)=0$, either $x=0$ or $x=1$ is required since the system is stable when the derivative of $f\left(x\right)$ at the point of stability is needed to be negative, then the derivative of $f\left(x\right)$ is found as follows:

$$\begin{array}{l}{f}^{\prime }\left(x\right)=\frac{df(x)}{dx}=\left(1-2x\right)\left[L\right.\left(C_2-C_1+F_1\right)+G\left(S_1+S_2+Y_1-S_3\right)\\ -\omega \left(y\right)L\left(\alpha B_1\right)-\omega \left(z\right)L\left(\beta B_2-C_1\right)-\omega \left(y\right)\omega \left(z\right)L\left.\left(C_1\right)\right]\end{array}$$

(19)

Let $u=L\left(C_2-C_1+F_1\right)+G\left(S_1+S_2+Y_1-S_3\right)-\omega \left(y\right)L\left(\alpha B_1\right)-\omega \left(z\right)L\left(\beta B_2-C_1\right)-\omega \left(y\right)\omega \left(z\right)L\left(C_1\right)$

i. When $u>0$, $f^{\prime }(1)<0$, $f^{\prime }(0)>0$, then $x=1$ is the stable point.

ii. When $u<0$, $f^{\prime }(1)>0$, $f^{\prime }(0)<0$, then $x=0$ is the stable point.

Let $\Omega =\left\{{\rm M}\left(x,y,z\right)|0\le x\le 1,0\le y\le 1,0\le z\le 1\right\}$, plane $Q_1$ be expressed as follows:

$$\omega \left(y\right)=\frac{L\left(C_2-C_1+F_1\right)+G\left(S_1+S_2+Y_1-S_3\right)-\omega \left(z\right)L\left(\beta B_2-C_1\right)}{L\left(\alpha B_1\right)+\omega \left(z\right)L\left(C_1\right)}$$

The dynamic trend of the government’s behavioral decision and the stable evolution strategy is shown in Figure 3. The space $\Omega$ is divided into $Q_{11}$ and $Q_{12}$ by the plane $Q_1$. When the initial state of the government is in the space $Q_{11}$, $x=1$ is the equilibrium solution, indicating that the government’s behavioral decision tends to actively implement the policy incentive reform; conversely, when it is in the space $Q_{12}$, the government chooses to manage to routine cope.

(2) The replicated dynamic differential equation for the farmer’s cooperation with the reform is shown in Equation (12).

1) When $L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)+\omega \left(x\right)G\left(\alpha B_1\right)=0$, i.e.,

$$\omega \left(x\right)=\frac{L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)}{-G\left(\alpha B_1\right)}$$

(20)

At this point $f\left(y\right)=0$, the system tends to a steady state regardless of the value of $y$.

2) When $\omega (x)\ne \frac{L(C_4-C_3+F_2)+G(S_4-S_5)}{-G(\alpha B_1)}$, either $y=0$ or $y=1$ is required if $f\left(y\right)=0$ since the system is stable when the derivative of $f\left(y\right)$ at the point of stability is required to be negative, then the derivative of $f\left(y\right)$ is found as follows:

$$f^{\prime }\left(y\right)=\frac{df(y)}{dy}=\left(1-2y\right)\left[L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)+\omega \left(x\right)G\left(\alpha B_1\right)\right]$$

(21)

Let $v=L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)+\omega \left(x\right)G\left(\alpha B_1\right)$

i. When $v>0$, $f^{\prime }(1)<0$, $f^{\prime }(0)>0$, then $y=1$ is the stable point.

ii. When $v<0$, $f^{\prime }(1)>0$, $f^{\prime }(0)<0$, then $y=0$ is the stable point.

Let $\Omega =\left\{{\rm M}\left(x,y,z\right)|0\le x\le 1,0\le y\le 1,0\le z\le 1\right\}$, plane $Q_2$ be expressed as follows:

$$\omega \left(x\right)=\frac{L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)}{-G\left(\alpha B_1\right)}$$

The dynamic trend of farmers’ behavioral decisions and the stable evolution strategy is shown in Figure 4. The space $\Omega$ is divided into $Q_{21}$ and $Q_{22}$ by the plane $Q_2$. When farmers are in the space $Q_{21}$, $y=1$ is the equilibrium solution and farmers’ behavioral decision tends to actively cooperate with the reform. On the contrary, they choose to passively cooperate with the reform.

(3) The replicated dynamic differential equation for water supply units providing water supply services is shown in Equation (16).

(1) When $L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)+\omega \left(x\right)G\left(\beta B_2\right)=0$, i.e.,

$$\omega \left(x\right)=\frac{L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)}{-G\left(\beta B_2\right)}$$

(22)

At this point $f\left(z\right)=0$, the system tends to a steady state regardless of the value of $z$.

(2) When $\omega (x)\ne \frac{L(C_6-C_5+F_3)+G(S_6-S_7+Y_2)}{-G(\beta B_2)}$, to $f\left(z\right)=0$, either $z=0$ or $z=1$ is required since the system is stable when the derivative of $f\left(z\right)$ at the point of stability is required to be negative, then the derivative of $f\left(z\right)$ is derived as follows:

$$f^{\prime }\left(z\right)=\frac{df(z)}{dz}=\left(1-2z\right)\left[L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)+\omega \left(x\right)G\left(\beta B_2\right)\right]$$

(23)

Let $w=L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)+\omega \left(x\right)G\left(\beta B_2\right)$

i. When $w>0$, $f^{\prime }(1)<0$, $f^{\prime }(0)>0$, then $z=1$ is the stable point.

ii. When $w<0$, $f^{\prime }(1)>0$, $f^{\prime }(0)<0$, then $z=0$ is the stable point.

Let $\Omega =\left\{{\rm M}\left(x,y,z\right)|0\le x\le 1,0\le y\le 1,0\le z\le 1\right\}$, plane $Q_3$ be expressed as follows:

$$\omega \left(x\right)=\frac{L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)}{-G\left(\beta B_2\right)}$$

The dynamic trend of the behavioral decision of the water supply unit and the stable evolution strategy is shown in Figure 5, where the space $\Omega$ is divided into $Q_{31}$ and $Q_{32}$ by the plane $Q_3$. When the initial state of the water supply unit is in space $Q_{31}$, $z=1$ is the equilibrium solution, indicating that the behavioral decision of the water supply unit tends to provide quality water service; conversely, when it is in space $Q_{32}$, the choice tends to maintain the existing water service.

According to the above evolutionary process, in the tripartite evolutionary game model consisting of the government, farmers, and water supply units, the strategy choice of each subject will not only be influenced by factors such as reform costs, invisible penalties, and incentives but also by the strategy choice of the other two parties, which will eventually present different evolutionary game results.

4.3. Local Stability Analysis

The fixed point of the replication dynamics is considered the point at which the equation satisfies the best solution, and this fixed point describes the evolution [53]. According to the above, Formulas (8), (12), and (16) are combined to obtain a system of dynamic replication equations, which are solved to describe the dynamic change characteristics of the evolutionary game model.

$$\left\{\begin{array}{l}f\left(x\right)=x\left(1-x\right)\left[L\left(C_2-C_1+F_1\right)\right.+G\left(S_1+S_2-S_3+Y_1\right)-\omega \left(z\right)L\left(\beta B_2\right)\\ \left.-\omega \left(y\right)L\left(\alpha B_1\right)+\omega \left(z\right)L\left(C_1\right)-\omega \left(y\right)\omega \left(z\right)L\left(C_1\right)\right]\\ f\left(y\right)=y\left(1-y\right)\left[L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)+\omega \left(x\right)G\left(\alpha B_1\right)\right]\\ f\left(z\right)=z\left(1-z\right)\left[L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)+\omega \left(x\right)G\left(\beta B_2\right)\right]\end{array}\right.$$

(24)

Let $f\left(x\right)=f(y)=f(z)=0$ in this system of equations, we can find the 12 local equilibrium points of the government, farmers, and water supply units, of which 8 are local equilibrium stability points, respectively: $E_1(0,0,0)$, $E_2(1,0,0)$, $E_3(0,1,0)$, $E_4(0,0,1)$, $E_5(1,1,0)$, $E_6(1,0,1)$, $E_7(0,1,1)$, and $E_8(1,1,1)$. According to the conclusion proposed by Ritzberger and Weibull [54], for the three-party game strategy of the government, farmers, and water supply units, only the stability of these eight equilibrium points need to be analyzed, and the rest are non-asymptotic steady states. According to the method proposed by Friedman, the strength of the above eight equilibria can be obtained by analyzing the Jacobian matrix [55]. The corresponding Jacobian matrices are obtained by taking partial derivatives of $f\left(x\right)$, $f\left(y\right)$, and $f\left(z\right)$, respectively, as follows:

$$J=\left(\begin{array}{ccc}\frac{\partial F\left(x\right)}{\partial x}& \frac{\partial F\left(x\right)}{\partial y}& \frac{\partial F\left(x\right)}{\partial z}\\ \frac{\partial F\left(y\right)}{\partial x}& \frac{\partial F\left(y\right)}{\partial y}& \frac{\partial F\left(y\right)}{\partial z}\\ \frac{\partial F\left(z\right)}{\partial x}& \frac{\partial F\left(z\right)}{\partial y}& \frac{\partial F\left(z\right)}{\partial z}\end{array}\right)$$

(25)

The eight local equilibrium stabilization points are substituted into the matrix, and the eigenvalues of each equilibrium point are calculated as follows in

Table 5. The eigenvalues of the Jacobian.

Balancing Point	$\mathbf{Eigenvalue} {\mathit{\lambda }}_1$	$\mathbf{Eigenvalue} {\mathit{\lambda }}_2$	$\mathbf{Eigenvalue} {\mathit{\lambda }}_3$
$E_1\left(0,0,0\right)$	$L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)$	$L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)$	$L\left(C_2-C_1+F_1\right)+G\left(S_1+S_2-S_3+Y_1\right)$
$E_2\left(1,0,0\right)$	$L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5+\alpha B_1\right)$	$L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2+\beta B_2\right)$	$-\left[L\left(C_2-C_1+F_1\right)+G\left(S_1+S_2-S_3+Y_1\right)\right]$
$E_3\left(0,1,0\right)$	$-\left[L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)\right]$	$L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)$	$L\left(C_2-C_1-\alpha B_1+F_1\right)+G\left(S_1+S_2-S_3+Y_1\right)$
$E_4\left(0,0,1\right)$	$L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)$	$-\left[L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)\right]$	$L\left(C_2-C_1+F_1\right)+G\left(S_1+S_2-S_3+Y_1\right)$
$E_5\left(1,1,0\right)$	$-\left[L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5+\alpha B_1\right)\right]$	$L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2+\beta B_2\right)$	$-\left[L\left(C_2-C_1-\alpha B_1+F_1\right)+G\left(S_1+S_2-S_3+Y_1\right)\right]$
$E_6\left(1,0,1\right)$	$L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5+\alpha B_1\right)$	$-\left[L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2+\beta B_2\right)\right]$	$-\left[L\left(C_2-\beta B_2+F_1\right)+G\left(S_1+S_2-S_3+Y_1\right)\right]$
$E_7\left(0,1,1\right)$	$-\left[L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)\right]$	$-\left[L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)\right]$	$L\left(C_2-C_1+F_1-\alpha B_1-\beta B_2\right)+G\left(S_1+S_2-S_3+Y_1\right)$
$E_8\left(1,1,1\right)$	$-\left[L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5+\alpha B_1\right)\right]$	$-\left[L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2+\beta B_2\right)\right]$	$-\left[L\left(C_2-C_1+F_1-\alpha B_1-\beta B_2\right)+G\left(S_1+S_2-S_3+Y_1\right)\right]$

. Then, they can be assigned and discussed analytically according to different situations.

According to the principles of evolutionary games, dynamic games usually stop only when they are stable [56]. Using Lyapunov’s method, the asymptotic stability point is obtained when all the eigenvalues of the Jacobi matrix have negative genuine parts, the instability point is obtained when at least one of the eigenvalues of the matrix has a positive fundamental interest, the equilibrium point is a critical point when all the eigenvalues of the matrix have negative genuine interests except for the eigenvalue with a genuine interest of zero, and it cannot be determined by the sign of the eigenvalue to determine its stability [57].

Assumptions: $\begin{array}{l}L\left(C_2-C_1+F_1-\alpha B_1-\beta B_2\right)+G\left(S_1+S_2-S_3+Y_1\right)>0\\ L\left(-C_3+C_4+F_2\right)+G\left(S_4-S_5+\alpha B_1\right)>0\\ L\left(-C_5+C_6+F_3\right)+G\left(S_6-S_7+Y_2+\beta B_2\right)>0\end{array}$. The results show that the government, farmers, and water supply units are all actively involved in the comprehensive agricultural water price reform, and the perceived benefits are greater than the perceived benefits when the government has water-saving incentives and payment subsidies for farmers, as well as financial subsidies for water supply units, than when they are not actively involved. The evolutionary game stabilization strategy is analyzed in the four scenarios below.

Scenario 1: When $L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)>0$, $L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)>0$, there are no government incentives and contributions for farmers to save water and no financial subsidies from the government for water suppliers, the perceived benefits of both farmers and water suppliers to cooperating and promoting comprehensive agricultural water price reform are more significant than the perceived benefits of no effort.

Scenario 2: When $L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)>0$, $L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)<0$, this indicates that there is no government incentive and payment subsidy for farmers and no government subsidy for water supply units, the perceived benefits of farmers’ active cooperation with the comprehensive agricultural water price reform are more significant than the perceived benefits of passive cooperation with the reform, while the perceived benefits of water supply units in providing quality water supply services are less than the perceived benefits of maintaining the existing water supply services due to the lack of financial subsidies.

Scenario 3: When $L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)<0$, $L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)>0$, without government incentives and subsidies for farmers to save water and water supply units without government financial subsidies, the perceived benefits of water supply units providing quality water services are more significant than the perceived benefits of maintaining the status quo, while the perceived benefits of farmers actively cooperating with the reform are less than the perceived benefits of passively cooperating with the reform because they cannot afford the increase in agricultural water price after the reform without government support.

Scenario 4: When $L\left(C_4-C_3+F_2\right)+G\left(S_4-S_5\right)<0$, $L\left(C_6-C_5+F_3\right)+G\left(S_6-S_7+Y_2\right)<0$, when the government does not provide water-saving incentives and financial subsidies to farmers and water supply units, the perceived benefits when both farmers and water supply units make efforts to cooperate and promote comprehensive agricultural water price reform are smaller than the perceived benefits of not making efforts.

According to

Table 6. Stability analysis of equilibrium points in different cases.

Balancing Point	Scenario 1				Scenario 2				Scenario 3				Scenario 4
	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	Stability	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	Stability	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	Stability	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	Stability
$E_1\left(0,0,0\right)$	+	+	+	Saddle Point	+	-	+	Unstable	-	+	+	Unstable	-	-	+	Unstable
$E_2\left(1,0,0\right)$	+	+	-	Unstable	+	+	-	Unstable	+	+	-	Unstable	+	+	-	Unstable
$E_3\left(0,1,0\right)$	-	+	+	Unstable	-	-	+	Unstable	+	+	+	Saddle Point	+	-	+	Unstable
$E_4\left(0,0,1\right)$	+	-	+	Unstable	+	+	+	Saddle Point	-	-	+	Unstable	-	+	+	Unstable
$E_5\left(1,1,0\right)$	-	+	-	Unstable	-	+	-	Unstable	-	+	-	Unstable	-	+	-	Unstable
$E_6\left(1,0,1\right)$	+	-	-	Unstable	+	-	-	Unstable	+	-	-	Unstable	+	-	-	Unstable
$E_7\left(0,1,1\right)$	-	-	+	Unstable	-	+	+	Unstable	+	-	+	Unstable	+	+	+	Saddle Point
$E_8\left(1,1,1\right)$	-	-	-	ESS	-	-	-	ESS	-	-	-	ESS	-	-	-	ESS

, only $E_8(1,1,1)$ is the equilibrium stabilization point of the evolutionary game in all cases and the corresponding evolutionary game strategy combination is (actively implement, active, and quality water supply).

5. Case Study

According to the 2019 annual agricultural water price comprehensive reform pilot implementation project program in Minquan County, Henan Province, the total investment in the reform pilot project is CNY 1,992,600, including CNY 910,000 of central funds, CNY 880,000 of provincial funds, and CNY 202,600 of county-level financial and association income or donation funds; after the completion of the water price reform, the water fee income of the irrigation district is guaranteed, which solves the expenses of project maintenance and management, and regular operation improves the project support, improve the channel’s water diversion and distribution capacity and water supply guarantee rate, improve agricultural production conditions, increase and improve the effective irrigation area, save labor, improve the total agricultural production capacity, and indirectly obtain economic, social, and ecological benefits.

From the current stage of China’s comprehensive agricultural water price reform, it is clear that the purpose of the evolutionary game analysis is to promote the reform through research, so the MATLAB R2018b software will be used to simulate a numerical analysis of the three-party evolutionary game, how to achieve the strategic situation of the government actively implementing policies to incentivize the reform, farmers actively cooperating with the reform and water supply units providing quality water supply services, and how to promote China’s agricultural water price. The comprehensive reform of agricultural water prices in China is accelerated, and the corresponding changes in the reform when certain influencing factors are changed are analyzed. According to the above analysis of the various scenarios, the results are reached at $E_8(1,1,1)$ to stabilize the state, so the following based on the assumptions and the conditions of scenario 4, the various parameters are assigned values. Combined with the above project program, reference to the relevant literature [53,58] in the initial assignment, and on this basis of consulting the experts of the Henan Provincial Water Resources Department for correction, we ensure the reasonableness of the parameters and finally arrive at the following results of the assignment: $C_1=8$, $C_2=5$, $S_1=3$, $S_2=8$, $S_3=7$, $S_4=7$, $B_1=3$, $B_2=3$, $C_3=6$, $C_4=3$, $S_5=5$, $S_6=5$, $C_5=7$, $C_6=3$, $S_7=5$, $Y_1=2$, $Y_2=1$, $F_1=4$, $F_2=2$, $F_3=2$, $\alpha =0.5$, and $\beta =0.5$. Their assigned values are all calculated perceived values. The value of the risk appetite parameter for MA-PT is $\gamma =0.98$, $\delta =0.98$, $\phi =0.88$, $\zeta =0.88$, $\kappa =2$, and $\tau =2$. The reference values for loss accounts and loss accounts are $U_1=U_2=1$, and the decision influence coefficient of the weight function are set as $\mu =0.61$, $\eta =0.69$ [49].

5.1. The Influence of Game Subjects’ Willingness to Participate in Evolutionary Behavior

Keeping the other states constant, only the probability that the three parties choose to make an effort to participate in the reform is changed to simulate its evolution. As shown in Figure 6, as the probability of the initial willingness increases, the government’s strategy is to implement policies to incentivize the reform actively, the farmers’ strategy is to cooperate with the reform actively, and the water supply units are also actively involved in the reform and choose to provide quality water supply services. The findings in Table 5 confirm that all three parties in the game reach a steady state in the combination of strategies (actively implement, active, and quality water supply) in multiple scenarios.

With other parameters unchanged, when the initial willingness of the three parties is $x=y=z=0.5$, as shown in Figure 7a, the strategy diagram of the three parties of the game evolves, its evolutionary trend is to strive to participate in the reform, the government first chooses to carry out the active implementation of policies to incentivize the reform, the farmers follow and also choose to cooperate with the reform actively, the water supply units eventually select the strategy of providing quality water supply services, but tends to stabilize more slowly. When the initial willingness of water supply units is increased, i.e., the willingness of water supply units to provide quality water supply services is increased, as shown in Figure 7b, so that $x=y=0.5$ and $z=0.8$, the initial desire of water supply units is more prominent, and the speed of water supply units choosing to provide quality water supply services and stabilization is accelerated. The time is shortened because the willingness to participate in the game parties affects each other. It can be observed that the change in the desire of water supply units to participate and the willingness to participate interact with each other. It can be observed that the difference in the willingness to partake of the water suppliers will accelerate the speed of stabilization of the government, and finally, all three parties converge to the evolutionary stabilization strategy.

5.2. The Influence of Reference Values on the Evolutionary Behavior of the Subject

As shown in Figure 8, keeping the other states unchanged, we change the reference values $U_1$ and $U_2$ of the loss and gain accounts of the game parties, respectively, and observe the strategy choices of the three parties through evolution, where the initial willingness of the three parties is $x=y=z=0.2$.

$U_1$ is the reference value of the loss account, which is assumed to take a range of values [0, 4], and it can be observed from Figure 8a that as $U_1$ increases, the three parties are more likely to choose to actively participate in the reform, indicating that the strategic choices of the game parties show risk preferences. $U_2$ is the reference value of the payoff account and also assumed that the same range of [0, 4], as shown in Figure 8b, with the increase of $U_2$, the strategy choice of the three parties changes. When $U_2$ increases to 3, firstly, the water supply unit will change from choosing to provide quality water supply service to maintaining the existing water supply service; when $U_2=4$, the strategy of the three parties will reach a stable state at (0, 1, 0), i.e., the government will choose to routine cope, the water supply unit will decide to maintain the existing water supply service, and only the farmers choose to actively cooperate with the reform, indicating that the strategy choice of the game parties shows risk aversion. It suggests that the game parties will be influenced by the MA-PT theory when making strategy choices, i.e., people’s risk preferences are not consistent when facing gains and losses, and they show risk aversion when facing increases and risk propensity when facing losses.

5.3. Impact of Input Cost Changes on Evolutionary Behavior

The cost of construction work invested by the government in carrying out the reform is objective data, so the following evolutionary analysis is performed on the input costs of farmers and water supply units. Other conditions are unchanged, the input costs of farmers and water supply units are changed, evolution is performed, and changes in their strategic choices are observed where the initial willingness of the three parties is $x=y=z=0.2$.

When farmers’ input costs are $C_3=2$, $C_3=4$, $C_3=6$, $C_3=8$, and $C_3=10$ respectively, their strategy choice changes, as shown in Figure 9a. When $C_3=2$, $C_3=4$, $C_3=6$, and $C_3=8$, all three parties choose to participate in the reform actively, but as the cost increases to $C_3=10$, the farmers’ strategic choice becomes passive to cooperate with the reform. As shown in the figure, when $C_3=2$, farmers do not hesitate to choose to cooperate with the reform actively because, at this time, the input cost is low, and the return is significant; thus, farmers will more easily choose to actively cooperate. When $C_3=4$, farmers begin to hesitate, although the speed of stabilization will be reduced, but the direction of choice is still active with the reform. When $C_3=6$, that is, the reference point of the above parameters is taken, at this time, compared with the first two lines, this time, the speed has also slowed down. But when $C_3=8$, because the input cost is too large, the farmers’ choice is more indecisive, showing a tendency to passively cooperate with the reform first, and then actively cooperate with the reform and stabilize. When $C_3=10$, the farmers’ input cost has exceeded the farmers’ estimated range, and the cost of this input greatly exceeds the income obtained, so the farmers no longer choose to actively cooperate with the reform, as their strategy choice is passive. Instead, they decide to passively cooperate with the reform.

For the water supply units, their input costs are $C_5=1$, $C_5=3$, $C_5=5$, $C_5=7$, and $C_5=9$, and their strategy choices change, as shown in Figure 9b. When $C_5=1$, $C_5=3$, $C_5=5$, and $C_5=7$, the same three parties will choose the incentive reform, actively cooperate, and provide quality water supply services, and as the input costs of the water supply units increase to $C_5=9$, the water supply units no longer want to provide quality water supply services. When $C_5=1$, $C_5=3$, and $C_5=5$, the input cost is small, and the water supply unit is happy to choose to provide quality water supply services and obtain more revenue. When $C_5=7$, because the input cost increases, the water supply unit will hesitate to provide quality water supply services, showing a tendency to maintain the status quo, but later choose to provide quality water supply and reach a stable state. When the cost increases to $C_5=9$, the water supply unit no longer wants to provide quality water supply, chooses to maintain the status quo, and decides to maintain the status quo.

5.4. Impact of Changes in Government Incentives on Evolutionary Behavior

The evolution of the strategic choices of the government, farmers, and water supply units over time when the values of government incentives $\alpha$ and $\beta$ are changed to $\alpha =0, \beta =0$, $\alpha =0.5, \beta =0.5$, $\alpha =1, \beta =1$, $\alpha =1.5, \beta =1.5$, and $\alpha =2, \beta =2$, respectively, with other parameters unchanged, as shown in Figure 10. According to the above parameter assumptions, changing the value of $\alpha$ mainly affects the choice of farmers, and changing the value of $\beta$ mainly affects the choice of water supply units. Still, since the strategic decisions of the three parties of the game have a mutual influence on each other, the strategic decisions of the three parties are observed by changing the values of $\alpha$ and $\beta$ simultaneously. The initial willingness of the three parties is $x=y=z=0.5$.

The change in the government’s choice is shown in Figure 10a. When $\alpha =0, \beta =0$, $\alpha =0.5, \beta =0.5$, and $\alpha =1, \beta =1$, all converge to 1, that is, the choice of active implementation of policies to the incentive reform, and with the reduction in the intensity, the speed and time of the tendency to stabilize is accelerated because the government is the implementer who carries out the reward, the less the government input, the easier to choose the incentive reform; when $\alpha =1.5, \beta =1.5$, the change in the government’s strategy choice shows a tendency to first favor the incentive reform and then opted for routine coping but, with time, shows a tendency to choose the incentive reform. When $\alpha =2, \beta =2$, the government will have a tendency to choose the incentive reform but will eventually choose the routine coping reform because the reward and subsidy are too large to support the rest of the reform.

The strategy choice of farmers due to the change in the incentive system is shown in Figure 10b. Farmers are the target of water-saving incentives and contribution subsidies, so when the above conditions are changed, their strategy choices are all active with the reform, and when there is no government incentive, although slow, they will eventually choose to cooperate with the reform actively and, with the increase of the motivation, will be more quickly to stabilize the state.

The change in strategy choice of water supply units is shown in Figure 10c. The choice changes of water supply units are more complicated than those of the government and farmers. Although water supply units are also subject to financial subsidies from the government, the choice changes are different from those of farmers. When $\alpha =0, \beta =0$, water supply units may choose to maintain their existing water service because they cannot make ends meet and do not want to manage and maintain the water supply equipment. When $\alpha =0.5, \beta =0.5$ and $\alpha =1, \beta =1$, water supply units will both choose to provide quality water service, but the former is slower than the latter, and the choice is not as strong. When $\alpha =1.5, \beta =1.5$ and $\alpha =2, \beta =2$, there is a tendency to converge to 1 and then to 0 over time. One part of the reason is that the strategic choices of the government and water supply units will influence each other, and the selection of the government in this condition is routine coping reform, which will affect the selection of water supply units to maintain the existing water supply services; another part of the reason is the increase in government awards, in which water supply units may fail to provide earmarked funds, and other situations may occur.

5.5. Impact of the Degree of Change in Water Prices on Evolutionary Behavior

The purpose of the comprehensive agricultural water price reform is to increase the price of water within the affordability of farmers, to make water resources more efficient, and to promote the development of the economic and agricultural industry. Therefore, the impact of the degree of change in water price on the evolution is analyzed to enhance the effect of water price on the three parties. The government and water supply units are the supply side of agricultural water prices, and with the increase in water prices, the government and water supply units will have more income; farmers are the bearers of agricultural water prices, so the water fee paid by farmers will increase. The water price of the 2019 comprehensive reform of agricultural water price in Minquan County is tentatively set at 0.19 yuan/m³, assuming that A is the value of agricultural water price, after assigning values so that $A=0.05$, $A=0.1$, $A=0.2$, $A=0.4$, and $A=0.6$, the government, farmers, and water supply units with the evolutionary process over time is shown in Figure 11a–c, respectively, where the initial willingness of the three parties is $x=y=z=0.5$.

When $A=0.05$, the government and water supply units will tend to favor routine coping reform and maintain the status quo, respectively, and farmers will actively participate in the reform; as $A$ increases to 0.1, the government will change from the routine coping reform to the incentive reform. When $A=0.2$, that is, increased to close to the tentative value of water price, the water supply units will change from maintaining the status quo to quality water supply, and all three parties will actively participate in the reform with the evolution of time. When $A=0.4$, farmers’ willingness to reform will be affected, and they will slowly join in the reform, while the government and water supply units will actively participate in the reform because of the benefits gained. When $A$ increases to 0.6, farmers cannot accept the increase in water price and passively cooperate with the reform.

5.6. The Influence of Invisible Punishment on the Evolution of the Subject’s Behavior

Holding the other parameters constant, the invisible penalties mentioned here are mainly the reform pressure exerted by the government on farmers and water supply units. The government will impose fines on farmers who waste water and have no awareness of water conservation and penalties on teams that do not supply water according to the regulations, etc. However, the losses that farmers and water supply units will suffer when they choose to passively reform and maintain their existing water supply services, i.e., the invisible penalties go beyond this. The hidden penalty parameters of the two parties are changed to $F_1=0, F_2=0$, $F_1=1, F_2=1$, $F_1=2, F_2=2$, $F_1=3, F_2=3$, and $F_1=4, F_2=4$ to observe the changes in their strategic choices where the initial willingness of the three parties is $x=y=z=0.5$.

The changes in strategy choices of farmers and water supply units are shown in Figure 12a,b, respectively. First, farmers will be more willing to cooperate with the reform voluntarily as the invisible penalty increases. When $F_1=0, F_2=0$ and $F_1=1, F_2=1$, the water supply unit will relax the management of water supply because of the government’s little regulation and then maintain the existing water supply service; after that, as the government increases its penalty, it chooses to actively participate in the reform to provide farmers with quality water supply service and tends to stabilize at one more rapidly as the penalty increases. After analysis, the government can appropriately increase the penalties for farmers and water supply units in the reform, thus prompting both parties to actively participate in the reform and promote the orderly implementation of the comprehensive agricultural water price reform.

5.7. Discussion

After the above-simulated evolution of the factors affecting the behavioral decision of each interest group, the following will optimize and simulate the growth of each parameter. Combined with the above simulation analysis, the parameters are changed to the following: $C_1=10$, $C_2=7$, $S_1=3$, $S_2=8$, $S_3=7$, $S_4=7$, $B_1=3$, $B_2=3$, $C_3=2$, $C_4=5$, $S_5=5$, $S_6=5$, $C_5=5$, $C_6=5$, $S_7=5$, $Y_1=2$, $Y_2=1$, $F_1=4$, $F_2=4$, $F_3=2$, $\alpha =1$, $\beta =1$. The reference values of the loss account and the loss account are $U_1=U_2=0$, and the simulation results are shown in Figure 13. Figure 13a shows the results of 50 three-way evolutions, which converge to $E_8(1,1,1)$ faster and reach a steady state compared to before parameter optimization. As shown in Figure 13b, when the initial willingness of the three parties is $x=y=z=0.1$, the government actively implements policies to the incentive reform, farmers actively participate, and water supply units provide quality water services and reach a steady state.

In summary, the comprehensive agricultural water price reform is a process involving multi-stakeholder coordination, and it is crucial to clarify the direction of the reform and regulate it from there. According to the model analysis and numerical simulation results, the government, farmers, and water supply units will eventually choose to participate in the reform actively, and efforts to achieve the reform are the common goal of all parties, while determining how to accelerate the reform is the goal of this paper’s research.

From the numerical simulation results, the strategy choice of the water supply unit is highly related to the government’s decision, and the two are closely related. Of course, there is also a certain connection between farmers and the government and water supply units, so the strategic choices of the three parties are influenced by each other. In this game, to essentially balance the interests of the various stakeholders as a way to encourage water price reform [59], the government should actively implement policies to the incentive reform, farmers should take the initiative to cooperate with the reform, and water supply units should also provide quality water supply services when a stable state can be achieved. In the evolutionary game, the three-party strategy choice is influenced by a variety of factors, and the choice of one party will have some influence on the strategy choice of the other two parties [49]. Therefore, by changing the influence factors, the countermeasures to accelerate the reform can be studied from an evolutionary perspective.

From the above simulation analysis of various influencing factors, the following results were obtained: (1) The willingness to participate in the reform has a direct impact on the convergence rate of each game subject to a steady state. Increasing the willingness to participate accelerates the convergence rate while altering the willingness to participate of one subject also influences the convergence rate of other subjects [53]. Moreover, the strategic choices of the government, farmers, and water supply units mutually influence each other. (2) Perceptions of gains and losses significantly influence the decision-making process of each game subject. They exhibit risk aversion in the face of gains and risk preference in the face of losses when selecting strategies [49]. Reducing the loss reference value as well as providing the gain reference value allows the parties to actively engage in the reform and accelerates the rate of convergence to a steady state. (3) The formulation of agricultural water prices is crucial in the water price reform. Setting low prices hinders the efficient utilization of water resources [60] and diminishes the enthusiasm of the government and water supply units to actively participate in the reform. Conversely, excessively high water prices make it difficult for farmers to afford water and may lead to reluctant participation in the reform. (4) Appropriate government water conservation incentives and financial subsidies for farmers and water supply units can reduce the input costs of farmers and water supply units for the reform, and a clear compensation mechanism for agricultural water can incentive farmers to save more water [61], thus regulating the behaviors of farmers in water use and guaranteeing the income of farmers [62]. At the same time, appropriate penalties can be imposed on farmers who passively cooperate with the reform and on water supply units that fail to provide quality water supply services in accordance with the provisions of the law [63] to enable farmers and water supply units to actively and quickly participate in the reform.

6. Conclusions

Currently, regions in China that have not implemented water price reforms still have not changed their crude water use, and there are phenomena such as unsound management mechanisms and chaotic water use management. Deepening the comprehensive agricultural water price reform is a significant breakthrough in solving the difficult agricultural water use situation in China. The essence of the reform is to coordinate the contradictions and conflicts among various stakeholders. Therefore, this paper introduces the MA-PT theory based on the traditional evolutionary game theory, applies the perceived value function to optimize the conventional benefit matrix, and uses the MATLAB algorithm to evolve the influence of game subjects on behavioral decision making under different degrees of change in six-point factors: willingness to participate, reference value, input cost, incentive, subsidy, degree of water price change and invisible penalty. We also obtain the following conclusions:

(1) The comprehensive agricultural water price reform necessitates the collective participation of all stakeholders to drive the reform forward. Numerical simulations indicate that an increase in the willingness to participate by one party, whether it be the government, farmers, or water supply units, can promote the convergence rate of the other two parties to a steady state. This sparks the enthusiasm of other stakeholders to engage in the reform. In particular, the strategic choices of the water supply units have a significant impact on the government’s decision making. The government can motivate water supply units to provide high-quality water services, thereby mobilizing the willingness of farmers to actively cooperate with the reform. In other words, when the government actively implements policies that incentivize reform and water supply units deliver high-quality services, farmers become more inclined to participate actively in the reform. Hence, comprehensive agricultural water price reform is not solely the government’s responsibility; it requires the participation of farmers and water supply units. Furthermore, the strategic choices of the government, farmers, and water supply units interact with one another. Only when all three parties actively engage in the reform and their strategic choices reach a stable state (i.e., actively implement, active, and quality water supply) can the effective implementation of comprehensive agricultural water price reform and the rapid development of water-saving agriculture be effectively promoted.

(2) The psychological behavior of each stakeholder in the comprehensive agricultural water tariff reform, specifically their reference values for gains and losses, plays a crucial role in influencing their strategy choices. The integration of evolutionary game theory with mental accounting and prospect theory is equally applicable to the government, farmers, and water supply units involved in the water tariff reform. Because the generation of the game is actually between the subjects of human behavior, which is different and lead to the subjects to choosing their own behavioral strategy, we will inevitably assess the actual benefits of the emotional costs of inputs, as subjects will be affected by their own perception of the psychological value of the gains and losses. In particular, in the face of the gains, they will demonstrate risk aversion, while in the face of the losses, they will demonstrate risk preference. According to the results of evolutionary simulations, reducing the reference value of losses and increasing the reference value of gains for each stakeholder can encourage active participation in the reform from parties and expedite the rate of convergence to a stable state.

(3) The main focus of comprehensive agricultural water price reform is the agricultural water price, and the increase in water price should be implemented within a reasonable range. As the supply group of agricultural water, the government and water supply units want to raise the price of water, while as the use group of agricultural water, farmers want to lower the price of water, so the price increase in the reform should take into account affordability for farmers. The purpose of raising water prices is to make farmers gradually become aware of water conservation, reduce the current phenomenon of sloppy agricultural water use in some areas, and thus alleviate the shortage of water resources worldwide. According to the results of this study, if the price of water is too high, the cost will be too much for farmers to cooperate with the reform, which will weaken the enthusiasm of farmers to cooperate with the reform actively and lead to groups of farmers choosing to cooperate with the reform passively. If the price of water is too low, the meaning of water price reform will not be well displayed; farmers and water supply units will still maintain the current methods of using water and may even increase the waste of water resources.

(4) In the comprehensive reform of agricultural water prices, government incentives and penalties play a very important role. The government needs to implement relevant policies to promote the reform, mainly through water-saving incentives and financial subsidies to reduce the pressure of reform on farmers and water supply units. In the reform, the cost of managing and maintaining water supply projects and equipment is too significant for water supply units to preserve their existing water supply services, so the government needs to give certain financial subsidies to help water supply units carry out the reform. The water-saving incentives for farmers can mobilize their enthusiasm for water saving and reduce the waste of water resources. However, it is necessary to grasp the strength of the incentive system, albeit not too much, to avoid the occurrence of undesirable phenomena. At the same time, it is also necessary to appropriately punish farmers who cooperate with the reform passively and water supply units that do not provide quality water supply services as required. Appropriate punishment can encourage farmers and water supply units to actively participate in the reform and speed up the reform process, realize the comprehensive reform of agricultural water prices as soon as possible, promote the development of water-saving agriculture, and alleviate the shortage of water resources in China.

In terms of theoretical contributions, according to the stakeholder theory, the reform requires not only the efforts of the government but also farmers and water supply units who can influence the reform process and play significant roles in it. Different from the two-party game model constructed in previous studies, this paper constructs a three-party game model of the government, farmers, and water supply units to analyze the effective factors affecting the behavioral strategies of each stakeholder. According to the simulation results, it can be seen that the dynamic evolution of the game system and the strategy choices of the three parties are related to a variety of factors, such as the initial willingness of each party, the psychological reference value of payments and benefits, the government’s incentive subsidies and supervision and punishment of farmers and water supply units, the losses borne by negatively promoting the work, and the degree of increase or decrease in the water price. Especially through the analysis, we can observe the different impacts of the water price and the strengths of the reward and punishment system on the three parties and find that the strategic choices of each subject have similar mutual influences on each other. Therefore, from the perspective of the mechanism affecting water prices, it is meaningful to analyze the behavioral decisions of the three parties, with the government, farmers, and water supply units as the main participants for the study of water price reforms.

In summary, the purpose of this research paper is to propose several strategies for the comprehensive agricultural water price reform in China to speed up the reform process and help improve the current form of agricultural water use in China. However, there are some shortcomings in this study. Firstly, this study was conducted under ideal conditions, and various problems may arise in practice; furthermore, this study is not perfect in classifying the stakeholders of the comprehensive agricultural water price reform and should go further to study the stakeholders included in the three interest groups of the government, farmers, and water supply units. These two aspects will be the focus of the following study.