Evolutionary game analysis of water-saving behavior of energy enterprises and food producers from the perspective of water-energy-food nexus

China's economy has rapidly risen to become the world's second-largest in the last 40 years of reform and opening up. The demand for water, energy, and food is also increasing. Since 2010, China has become the largest energy consumer in the world. China's total energy consumption has increased from 3.6 billion tons of standard coal in 2010–5 billion tons of standard coal in 2020, nearly 40% (Yuan & Luo 2019). Since 1986, Chinese grain output has steadily ranked first in the world. In 2019, the total grain output of China was 664 million tons, which far exceeded that of other countries and accounted for about 24.4% of the total grain output of the world. However, water resources, energy, and food are mutually supplied and inseparable. Food and energy production depends on the safe supply of water resources, and agriculture and energy are also the two major sectors consuming water (Mahjabin et al. 2020). In 2017, the total water consumption of China reached 604.3 billion m³, where the water consumption from agriculture and energy accounted for more than 70% of the total water consumption (MWR 2017). As food and energy production increases year by year, China's consumption of water resources also increases (Liu et al. 2013). However, with climate change, the severe drought in China has caused a shortage of water resources. The shortage of water resources has become an important factor restricting the safe production of energy and food (Marvin et al. 2013; Gasbarro et al. 2016; Vogel et al. 2019).

With the severe shortage of water resources, the competition for water use between the energy and food systems has gradually become the norm (Damerau et al. 2016; D'Odorico et al. 2018; Hua et al. 2020; Qin 2021). Suppose energy enterprises and food producers fail to transfer from the ‘competition of water-consuming’ to the ‘cooperation of water-saving.’ In that case, the optimal utilization of water resources cannot be fully realized, and the high-quality development of China will be affected. Therefore, guiding energy enterprises and food producers to save water is significant for ensuring water-energy-food security and sustainable water use management, while it is the research of this paper. In recent years, the Chinese government has paid more and more attention to the conservation and utilization of water resources. In April 2019, the National Development and Reform Commission and the Ministry of Water Resources jointly issued the ‘National Water Saving Action plan’ in China.

The water-saving behavior of energy enterprises and food producers is significant for alleviating the shortage of water resources. The researchers proposed that the water for food production can be reduced by popularizing water-saving facilities (Tian et al. 2017; Cetin & Kara 2019; Nouri et al. 2019) and cultivating the drought-resistant crops (Zhang et al. 2021). The technological progress of water-saving (Ge et al. 2020; Aili et al. 2021) and wastewater recycling (Pan et al. 2012) can reduce the water for energy production. Furthermore, the development and improvement of the water-energy nexus are essential for several industries and enterprises (Panagopoulos 2021a, 2021b). Although these studies have made significant contributions to alleviating the shortage of water resources, most of them focus on the internal water-saving behavior of the independent water user. They do not comprehensively consider the competition of water use and water-saving behavior among multiple water users from the holistically external perspective.

The Bonn Conference, held in Germany in 2011, firstly combined water, energy, and food as a ‘Nexus’ (water-energy-food Nexus) (Bazilian et al. 2011; Dubreuil et al. 2013; Zhang & Vesselinov 2017). With the increasingly severe problem of climate change and the development of economic globalization, the nexus of water resources, energy, and food are getting tighter and tighter. Research from the perspective of the water-energy-food nexus has become a hot spot. Scholars have carried out many types of research about the coupling and coordination of water-energy-food (Rasul & Sharma 2016; Albrecht et al. 2018; Pahl-Wostl 2019), the relationship between the water-energy-food system and economic growth (Xu et al. 2022), the water-energy-food-land nexus (Lazaro et al. 2021; Susnik et al. 2021), the water-energy-food-climate nexus (Zhou et al. 2022) and so on. The above studies have contributed to the in-depth understanding of the water-energy-food nexus. However, most of these studies focus on the macro-level research and lack of research on the sustainable water use management of the micro subjects with behavioral consciousness such as energy enterprises and food producers.

Societies worldwide have long struggled with the planning and managing of water resources amid growing populations and resource use, competition among users, and, more recently, widespread ecosystem degradation and climate change (Chartzoulakis & Bertaki 2015). Scholars have carried out a series of studies on the necessity (Rosa et al. 2020), practice (Qu et al. 2013; Ahmad et al. 2016), and effectiveness (Long et al. 2020) of sustainable water management. These studies effectively promote the progress of researches on sustainable water use management and provide an essential reference for the research of sustainable water management in this paper.

From the perspective of the water-energy-food nexus, this paper focuses on the effect of sustainable water use management of local governments on the energy enterprises and food producers. Because of the publicity and complexity of water resources, the water-saving action is no longer the individual's responsibility. The joint promotion of local governments, energy enterprises, and food producers is necessary. Therefore, promoting water-saving in energy and food production is a technical problem and a real problem of complex stakeholders’ different interest demands and behavior conflicts. Evolutionary game theory is widely used to reveal the interests and behaviors of multiple stakeholders and has achieved fruitful results (Wei et al. 2010; Kahil et al. 2016; Chen et al. 2020). As a result, this paper examines the water-saving behavior of energy enterprises and food producers using an evolutionary game model and the critical roles of local governments in guiding the transformation of food producers’ and energy enterprises’ behaviors from the ‘competition of water-consuming’ to the ‘cooperation of water-saving’ over time.

To demonstrate the importance of this work, the innovations and contributions of this paper are summarized as follows: (1) This paper innovatively studies the behavioral choice among multiple sectors with high water consumption from the holistically external perspective of the water-energy-food nexus. (2) This paper innovatively focuses the research of the ‘water-energy-food nexus’ on the sustainable water use management of the micro-subject energy enterprises and food producers. Their behavioral choices are further analyzed when facing the water use competition. (3) This paper provides the theoretical support for local governments of arid areas to achieve sustainable water use management for high water-consuming energy enterprises and food producers. (4) Putting forward the theory of water-saving development stage according to the theory of economic development stage of Rostow (Rostow 1959).

The following are the primary objectives and contents of this paper: (1) developing an evolutionary game model to understand better the behavioral patterns of local governments, energy enterprises, and food producers; (2) discussing and examining changes in the behavioral choices of energy enterprises and food producers at various stages of economic development; (3) analyzing the process of energy enterprises and food producers from ‘water-consuming competition’ to ‘water-saving cooperation’ and demonstrating the critical role of local government in this process.

Energy and food production has always been the large consumers of water resources, which account for 70% of the total water consumption of China in 2019 (MWR 2019). The shortage of water resources caused by the extreme climate is severe in China, and the water consumption of energy affects the security of grain irrigation. Because of the publicity and complexity of water resources, the water-saving action is no longer the individual's responsibility. The joint promotion of local governments, energy enterprises, and food producers is necessary. The three stakeholders in the water-saving action are all bounded rational individuals: local governments, energy enterprises, and food producers. In this paper, we use local governments to represent the current management departments of water-energy-food, which mainly include the water conservancy departments, the agricultural departments, the energy departments, the environment departments, and the river, lake, and sea management departments. Coal, oil refining, coal-fired power generation, and other energy firms with high water use are among the energy enterprises studied in this article. Food producers are the individuals engaging in grain production, agricultural production enterprises, and farmers.

To explain the model more clearly, this paper makes the following assumptions according to the actual situation:

Hypothesis 1: Local governments, energy enterprises, and food producers constitute a stably complete system. The three stakeholders are all individuals with learning abilities and have the right to choose their behavior plans. Local governments have two strategies: guiding and not guiding, and energy enterprises have two strategies: saving water and not saving water. Food producers also have two strategies: saving water and not saving water. At the time t, the probabilities of correspondingly strategic choices are expressed by x, y, and z (x, y, z ∈ [0,1]).

Hypothesis 2: The cost of guidance of the local governments is C_g. Whether the local governments guide or not, when the energy enterprises and food producers choose the water-saving strategy simultaneously, the regional water resources are fully utilized, and the benefit obtained by the local governments is S_g. As long as energy enterprises or food producers do not choose the water-saving strategy, the value of water resources in the region cannot be maximized. The loss of political achievements is caused by reducing food production, and the substandard energy production within the jurisdiction is W_g.

Hypothesis 3: The benefit of the energy enterprises producing according to the existing technology is S_e, the increased benefit after water-saving is ΔS_e, the cost that energy enterprises pay for water-saving technologies is C_e. If food producers save water, the extra benefit of energy enterprises not saving water is πS_e. In the long term, energy enterprises that do not save water will further aggravate the shortage of water resources, and the losses of energy enterprises and food producers caused by the water shortage are δW_e and δW_f, respectively. The guidance of local governments to energy enterprises includes two means of water-saving reward and resource tax, the executive strengths of the two means are α and β, respectively. The local governments’ reward for the energy enterprises saving water is αJ. The costs of energy enterprises for local governments levying resource tax is βT.

Hypothesis 4: The benefit of food producers producing according to the current technology is S_f, the increased benefit after choosing a water-saving strategy is ΔS_f, and the cost that food producers pay for water-saving measures is C_f; If energy enterprises save water, the extra benefits of food producers not saving water is πS_f. In the long term, food producers that do not save water will further aggravate the shortage of water resources, and the losses of energy enterprises and food producers caused by the water shortage are ηW_e and ηW_f, respectively. The local governments’ reward for food producers saving water is γJ.

According to the above assumptions about the water-saving behavior of local governments, energy enterprises, and food producers, we construct the profit matrix of game behavior of players as shown in Tables 1 and 2.

In the evolutionary game, to finally obtain a stable strategy, players need to constantly adjust the probability values x, y, and z according to the above profit matrix after many games (Lu et al. 2021). The process of the dynamic adjustment strategy is the dynamic process of copying. When the dynamic copying equation equals 0, the tripartite strategy evolves to the locally balanced state. Then the equilibrium point of the evolution process of the game players is obtained (Zhao & Liu 2019).

1. According to the game profit matrix, the expected benefit of local governments choosing the strategy of guidance is

   

   (1)

Therefore, the average benefit of local governments is

• 2. According to the game profit matrix, the expected benefit of food producers choosing a water-saving strategy is

  

  (3)

Therefore, the average benefit of food producers is

• 3. The expected benefit of energy enterprises choosing a water-saving strategy can be calculated as

  

  (5)

Therefore, the average benefit of energy enterprises is

According to the Lyapunov theory, the equilibrium point is asymptotically stable when all eigenvalues of the Jacobian matrix λ < 0; the equilibrium point is not stable when there are positive or zero eigenvalues Jacobian matrix.

From Equation (10), the eigenvalues of the above Jacobian matrix J₁ are ⁠, ⁠, ⁠. If ⁠, ⁠, and are all less than 0 (i.e., ⁠, ⁠, and ⁠), E₁(0,0,0) is an asymptotically stable point. The stability of the equilibrium point of other pure strategies can be obtained similarly. The stability analysis of equilibrium points is shown in Table 3.

According to the above analysis, in the tripartite copying dynamic system of local governments, energy enterprises, and food producers, the equilibrium of the evolutionary game of behavioral strategies is affected by many factors. For energy enterprises and food producers, their strategic choice is not only affected by governmental regulation but also by their benefit. The evolutionary stability analysis of this paper is from the following two aspects. The first one is that the water-saving cost of energy enterprises and food producers is higher than the benefit. The second one is that the water-saving cost of energy enterprises and food producers is less than the benefit. Combining with the theory of economic development stage of Rostow (Rostow 1959), this paper puts forward the theory of the water-saving development stage. It discusses game players’ behavior changes in the extensive expansion period, the water-saving starting period, the water-saving development period, and the mature water-saving period, respectively (Figure 1).

The MATLAB simulation tool examines the asymptotically stable evolution trajectory of the game model under changing restrictions to depict better the dynamic evolutionary process of strategic decisions of local governments, energy enterprises, and food producers. The initial point of evolution is set to x = 0.5, y = 0.5, z = 0.5. The vertical axis represents the strategies chosen by the local government (x), the strategies chosen by the food producer (y), the strategies chosen by the energy enterprise (z), and the horizontal axis represents the time (t).

When the water-saving costs of energy enterprises and food producers are higher than the benefits, and ⁠. The asymptotic stability of the tripartite evolutionary game system of local governments, energy enterprises, and food producers is shown in Table 4.

From Table 4, when the water-saving costs of energy enterprises and food producers are higher than the benefits, their strategic choice depends on whether the local governments guide or not and the intensity of guidance. The following are three situations for further discussion.

Situation (1): When ⁠, the guiding cost is higher than the water resources tax, the local governments will choose the strategy of not guiding. The energy enterprises and food producers who have opportunistic behavior will choose the strategy of not saving water. The asymptotic stable point E₁(0,0,0) is the only evolutionary stable point. According to the behavioral strategy, local governments’ purpose in the early stages of development is to foster economic development, industrialization, and urbanization. Therefore, the local governments pay less attention to the intensive utilization of water resources. During this period, agricultural development is still in a highly labor-intensive stage. Water-saving behavior is more expensive for food producers due to the high cost and low popularity of agricultural mechanization. At the same time, the shortage of energy makes accelerating production a top priority. The high water consumption in energy production and the weak awareness of water-saving both make the waste of water resources seriously. And the relatively low level of scientific and technological development makes water-saving technological innovation difficult and leads to the high cost of energy enterprises. Driven by loose regulation and interests, the bounded rational energy enterprises and food producers choose not to save water, resulting in the severe waste of water resources and a potential water shortage crisis. This is a relatively extensive expansion period, and the degree of intensification is low. There was ‘competition of water-consuming’ between energy enterprises and food producers at this stage.

During the period of extensive expansion, the initial parameter shall meet the conditions: ⁠, and ⁠. Based on the above the conditions, the initial values of the parameters are set as: the cost of guidance of the local governments C_g = 0.8, the local governments’ reward for food producers saving water γJ = 0.3, the increased benefit of food producers after choosing a water-saving strategy ΔS_f = 0.2, the losses of food producers caused by the water shortage ηW_f = 0.3, the cost that food producers pay for water-saving measures C_f = 0.8, the extra benefits of food producers not saving water πS_f = 0.4, the local governments’ reward for the energy enterprises saving water αJ = 0.3, the costs of resource tax of local governments βT = 0.3, the increased benefit of energy enterprises after saving water ΔS_e = 0.2, the losses of energy enterprises caused by the water shortage δW_e = 0.3, the cost that energy enterprises pay for water-saving technologies C_e = 0.8, the extra benefit of energy enterprises not saving water πS_e = 0.4. According to the above parameters, the tripartite evolutionary stability strategies of local governments, energy enterprises, and food producers under the benchmark scenario are shown in Figure 2(a). Over time, the three players of the game finally stabilize at the combined strategy that local governments do not guide, and energy enterprises and food producers both do not save water. Changing the incentives of local governments for water-saving behavior. When γJ = αJ = 0.6, the influence of the change of reward policy on the evolutionary stability strategy is shown in Figure 2(b). Compared to the benchmark scenario, although local governments enhance the reward for water-saving behavior, energy enterprises and food producers still do not adopt the water-saving strategy. Changing the tax of water resources, when βT is 0.6, the influence of the change of water resources tax on the evolutionary stability strategy is shown in Figure 2(c). Compared with the benchmark situation, although an increase in the water resources tax does not affect the strategic choices of the three players in the game, the greater the intensity of the resource tax, the longer the evolutionary time for energy enterprises to reach a stable strategy of not saving water. In the extensive expansion period, local governments are stable in the non-guidance strategy due to the lack of attention to water-saving problems. Energy enterprises and food producers are stable in the non-water-saving strategy since the water-saving strategy cost is higher than the benefit.

Situation (2): When local governments add the reward of water-saving behavior, ⁠, ⁠, ⁠. The total cost of energy enterprises and food producers is still significantly higher than the total benefit of water-saving, which indicates that the guidance of the local governments is not enough. Therefore, the energy enterprises and food producers will still adopt the strategy of non-water-saving. The asymptotically stable point E₄(1,0,0) is the only evolutionary stable point. Compared with Situation (1), after a period of economic development, local governments began to pay attention to the issue of saving water and utilized the water-saving reward and the water resources tax for encouraging large water consumers such as energy enterprises and food producers to save water. During this period, the finance of local governments has some initial accumulation. However, due to the weak financial foundation and many projects requiring support, the guidance of local governments for the intensive use of water resources is not enough. At this time, agriculture is still in the stage of labor-intensive production, the mechanized cost of large-scale water-saving is still very high, and the popularization rate is low. Therefore, food producers still choose the non-water-saving strategy with lower costs despite the temptation of a water-saving reward. Even though local governments will reward water-saving behavior and impose a water resources tax on non-water-saving behavior, energy enterprises will continue to refuse to save water under non-compulsory local government supervision because water-saving technology is still in its infancy and the cost is high. This is the initial period of water-saving, and the degree of water-saving is still low. At this stage, there was a ‘competition of water-consuming’ between energy enterprises and food producers under the water-saving policy of the government.

In the initial period of water-saving, the initial parameter shall meet the conditions: ⁠, ⁠, ⁠, ⁠, and ⁠. Based on the above the conditions, the initial values of the parameters are set as: the cost of guidance of the local governments C_g = 0.2, the local governments’ reward for food producers saving water γJ = 0.3, the increased benefit of food producers after choosing a water-saving strategy ΔS_f = 0.2, the losses of food producers caused by the water shortage ηW_f = 0.3, the cost that food producers pay for water-saving measures C_f = 1.5, the extra benefits of food producers not saving water πS_f = 0.4, the local governments’ reward for the energy enterprises saving water αJ = 0.3, the costs of energy enterprises for local governments levying resource tax βT = 0.3, the increased benefit of energy enterprises after saving water ΔS_e = 0.2, the losses of energy enterprises caused by the water shortage δW_e = 0.3, the cost that energy enterprises pay for water-saving technologies C_e = 1.8, the extra benefit of energy enterprises not saving water πS_e = 0.4. According to the above parameters, the tripartite evolutionary stability strategies of local governments, energy enterprises, and food producers under the benchmark scenario are shown in Figure 3(a). The local governments transfer from the ‘initial strategy of not guiding’ to the ‘strategy of guiding,’ and the energy enterprises and food producers still choose to adopt the strategy of not saving water since the high water-saving cost and insufficient guidance. As shown in Figure 3(a), with time, the three players of the game finally stabilize at the stage of local governments guiding, and both energy enterprises and food producers do not save water. Changing the incentives of the local governments for the water-saving behavior. When γJ = αJ = 0.6, the influence of the change of reward policy on the evolutionary stability strategy is shown in Figure 3(b). Compared with the benchmark situation, although the local governments increase the reward for the water-saving behavior, energy enterprises and food producers still stabilize at the strategy of not saving water since the water-saving cost is too high. Changing the tax of water resources, when βT is 0.6, the influence of the change of water resources tax on the evolutionary stability strategy is shown in Figure 3(c). Compared with the benchmark situation, although an increase in resource tax does not affect the strategic choices of the three parties in the game, the greater the intensity of the water resources tax, the longer the evolution time for energy enterprises to reach a stable strategy of not conserving water. In summary, in the initial period of water-saving, local governments began to pay attention to intensive water resources. They will guide the water-saving strategies of large water consumers. However, energy enterprises and food producers will stabilize at the strategies of not saving water since the water-saving cost is higher than the benefit.

Situation (3): When and ⁠, the local governments strengthen the guidance so that the total cost of saving water for food producers is less than the total benefit of saving water. Thus, food producers choose the strategy of saving water. The asymptotically stable point E₅(1,1,0) is the only evolutionary stable point. During this period, local governments continue to pay attention to the issue of saving water and utilize the water-saving reward and the water resources tax for encouraging large water consumers such as energy enterprises and food producers to save water. Compared with Situation (2), the improvement of agricultural water-saving technology leads to a decrease in the water-saving cost. The total water-saving benefit is greater than the water-saving cost of food producers. As a rational ‘economic man,’ food producers will voluntarily choose the water-saving strategy. However, similar to Situation (2), even though local governments will reward water-saving behavior and levy a water resources tax for non-water-saving behavior, energy enterprises will continue to choose not to save water under non-compulsory local government supervision because water-saving technology is complex and expensive. This is the developing period of water-saving, and the level of water-saving is still modest. At this point, energy enterprises and food producers are transitioning from ‘water-consuming competition’ to ‘water-saving cooperation.’

In the developing period of water-saving, the initial parameter shall meet the conditions: ⁠, ⁠, and ⁠. Based on the above the conditions, the initial values of the parameters are set as: the cost of guidance of the local governments C_g = 0.1, the local governments’ reward for food producers saving water γJ = 0.3, the increased benefit of food producers after choosing a water-saving strategy ΔS_f = 0.2, the losses of food producers caused by the water shortage ηW_f = 0.3, the cost that food producers pay for water-saving measures C_f = 0.5, the extra benefits of food producers not saving water πS_f = 0.4, the local governments’ reward for the energy enterprises saving water αJ = 0.3, the costs of energy enterprises for local governments levying resource tax βT = 0.8, the increased benefit of energy enterprises after water-saving ΔS_e = 0.2, the losses of energy enterprises caused by the water shortage δW_e = 0.3, the cost that energy enterprises pay for water-saving technologies C_e = 2, the extra benefit of energy enterprises saving water water πS_e = 0.4. According to the above parameters, the tripartite evolutionary stability strategies of local governments, energy enterprises, and food producers under the benchmark scenario are shown in Figure 4(a). The local governments are stable in the guidance strategy. The water-saving benefit of food producers is greater than the water-saving cost. Food producers gradually transferred from the strategy of not saving water to the strategy of saving water and finally stabilized at the strategy of saving water. Due to the bottleneck of water-saving technology and the high cost, energy enterprises still adopt the strategy of not saving water. The three players of the game finally stabilize in the strategies local governments choose to guide, food producers save water, and energy enterprises do not save water. Changing the incentives of local governments for water-saving behavior when γJ = αJ = 0.6, the influence of the change of reward policy on the evolutionary stability strategy is shown in Figure 4(b). Compared with the benchmark situation, even though food producers continue to use the water-saving strategy, the greater the intensity of the water-saving reward, the quicker the time it takes for food producers to attain a stable water-saving strategy. Changing the tax of water resources, when βT is 1.5, the influence of the change of the water resources tax on the evolutionary stability strategy is shown in Figure 4(c). Although an increase in the water resources tax does not affect the strategic choices of the three players in the game, the greater the intensity of the water resources tax, the longer it takes for energy enterprises to establish a stable strategy of not saving water. Therefore, in the developing period of water-saving, the reward strategy of local governments promotes food producers to stabilize in the water-saving strategy. The water resources tax of local governments has a reverse coercion effect on the water-saving strategy of energy enterprises in the short term. In the long run, the strategic choice of energy enterprises depends on multiple factors such as the regulation of local governments and the water-saving cost.

When the water-saving costs of energy enterprises and food producers are less than the benefit, and ⁠. The asymptotic stability of the tripartite evolutionary game system of local governments, energy enterprises, and food producers is shown in Table 5.

From Table 5, when the water-saving cost is less than the water-saving benefit, energy enterprises and food producers will eventually choose the strategy of saving water after long-term evolution, whether the local government guides it. At this time, the asymptotic stable point E₇(0,1,1) is the only evolutionary stable point, which indicates that compared with the reward and tax of local governments, the profit of water-saving technology influences the strategic choice of energy enterprises and food producers more directly and significantly. During this period, the awareness of intensive water resources of the whole society is robust, the water-saving technology is mature, and the water-saving cost is low. As the bounded rational ‘economical people,’ energy enterprises and food producers will also spontaneously choose water-saving strategies. The serious water-wasting and the potential water shortage crisis will be gradually alleviated. This is the mature period of water-saving, and the degree of water-saving is high. At this point, energy companies and food producers have realized the change from ‘water-consuming competition’ to ‘water-saving cooperation ‘ and optimized water resource usage.

In the mature period of water-saving, the initial parameter shall meet the conditions: and ⁠. Based on the above the conditions, the initial values of the parameters are set as: the cost of guidance of the local governments C_g = 0.1, the local governments’ reward for food producers saving water γJ = 0.3, the increased benefit of food producers after choosing a water-saving strategy ΔS_f = 0.5, the losses of food producers caused by the water shortage ηW_f = 0.2, the cost that food producers pay for water-saving measures C_f = 0.1, the extra benefits of food producers not saving water πS_f = 0.2, the local governments’ reward for the energy enterprises saving water αJ = 0.3, the costs of energy enterprises for local governments levying resource tax βT = 0.3, the increased benefit of energy enterprises after saving water ΔS_e = 0.5, the losses of energy enterprises caused by the water shortage δW_e = 0.2, the cost that energy enterprises pay for water-saving technologies C_e = 0.1, the extra benefit of energy enterprises not saving water πS_e = 0.2. According to the above parameters, the tripartite evolutionary stability strategies of local governments, energy enterprises, and food producers under the benchmark scenario are shown in Figure 5(a). As people become more conscious of the importance of conserving water and science and technology advance, the cost of conserving water is less than the benefit of conserving water. Energy enterprises and food producers gradually transferred to the water-saving strategy and finally stabilized in the water-saving strategy. The large water consumers spontaneously save water, and the local governments also gradually stabilize from the initial guiding strategy to the strategy of not guiding. From Figure 5(a), with time, the three players of the game finally stabilized in the strategy that the local governments do not guide, and the energy enterprises and food producers both save water. Changing the reward of local governments for the water-saving behavior, when γJ = αJ = 0.6, the influence of the change of reward policy on the evolutionary stability strategy is shown in Figure 5(b). It can be seen that after the increase in the reward of local governments for water-saving, energy enterprises and food producers are still stable in the water-saving strategy, and there is no difference from the benchmark situation. Changing the tax of water resources, when βT is 0.6, the influence of the water resources tax on the evolutionary stability strategy is shown in Figure 5(c). Compared with the benchmark situation, although a change in the water resources tax does not affect the strategic choices of the three players in the game, the greater the intensity of the water resources tax, the shorter the evolutionary time for energy enterprises to establish a stable water-saving strategy. Therefore, in the mature period of water-saving, energy enterprises and food producers will adopt the water-saving strategy.

According to the theory of economic development stage and the analysis and discussion of the simulations in the previous section, the main contradictions faced by different stages of economic development are different. Therefore, we further summarize and interpret the results as follows:

1. The stability strategy of System (I) is not immutable, and the stability strategy of the game system is different with different stages of economic development. In the extensive expansion period of water-saving, energy enterprises and food producers are stable in the non-water-saving strategy. On the contrary, in the mature period of water-saving, energy enterprises and food producers will adopt the water-saving strategy.

2. During the progress from the period of extensive expansion to the mature period of water-saving, large water consumers’ water-saving behavior requires the government's guidance. The study of this paper shows that when the local government levies the tax on water resources, the time for energy enterprises to choose the strategy of not saving water becomes more extended, and the time of choosing the strategy of saving water becomes shorter. By adopting a series of water-saving policies, the local governments promote the water-saving behavior of energy enterprises and food producers.

3. The choice of water-saving behavior of energy enterprises and food producers is influenced by the guiding policies of local governments and depends on the water-saving cost. During the period of extensive expansion, the initial period of water-saving, and the developing period of water-saving, with the strengthening guidance and the reduction of water-saving costs, energy enterprises and food producers gradually transfer from the strategy of not saving water to the strategy of saving water together.

4. In the mature period of water-saving, energy enterprises and food producers have realized the transformation from the ‘competition of water-consuming’ to the ‘cooperation of water-saving,’ and the utilization of water resources has been maximized.

The shortage of water resources caused by extreme climate change has hugely exacerbated the water resource competition between energy enterprises and food producers. Therefore, from the perspective of the water-energy-food nexus, this paper introduces the game theory, puts forward the theory of the water-saving development stage according to the theory of economic development stage of Rostow (Rostow 1959), constructs the evolutionary game model composed of local governments, energy enterprises, and food producers, and further analyzes the strategic choices of the three players in different development stages. This work could apply to countries and regions with serious water use competition between food and energy production, such as Northwest China, Ethiopia, etc. Future research should focus on analyzing the influence of specific policies issued by local governments on the water-saving behavior of energy enterprises and food producers.

The study in this paper would provide a basis for the behavioral choices of energy enterprises and food producers and provide a reference for the political decision-making of the local governments. According to the above results and conclusions, the political recommendations are given as follows:

1. At different stages of economic development, the government should adopt different guidance strategies for water-saving. Local governments must guide the water-saving behavior through ‘one policy at a time.’ In the period of extensive expansion, local governments should actively promote the intensive water resources while devoting the economic development to prepare for the early entry into the society of intensive resources and the realization of regionally sustainable development. In the initial period of water-saving and the developing period of water-saving, the government should design the guiding policy of ‘rewarding the good and punishing the bad’ and use the two-way incentive effect of special funds.

2. To reduce the water-saving cost of large water consumers, the governments should support the innovation of water-saving technology. Local governments should encourage high-tech innovation, including constructing facilities to save water and promote drought-resistant grain varieties. As one of the major water consumers, the energy industry should be included in the annually technological transformation and upgrading plan of the Ministry of Industry and Information Technology. The financial support for technological transformation projects should be strengthened through industrial funds, bond financing, and special taxes on resources.

3. To improve the water-saving concept of large water consumers, the governments should strengthen the publicity of intensive water resources. Local governments should adopt the suggestions of experts and scholars in many aspects, especially economists’ suggestions about the economic development stage and sustainable development. Moreover, local governments should also strengthen the guidance of intensive water resources and improve the water-saving consciousness of large water consumers. Thus, a society of intensive resources will be constructed as soon as possible, and sustainable development will be realized.

4. The government should raise the levying standard of resources tax to reverse coerce the water-saving behavior of large water consumers. Keeping the water resources tax in the local area can make up for the cost of guidance of local governments. On the one hand, the government should dynamically increase resource taxes to achieve the double dividend of user payment and resource tax reform; on the other hand, the government should implement the resource tax unique fund system and ensure that all resource tax revenue is used as special funds for intensive use of local natural resources and environmental protection.

We would like to acknowledge the support of the Postgraduate Research Innovation Program of Jiangsu Province (no. KYCX21_0444), the Fundamental Research Funds for the Central Universities, the Major Project of National Social Science Foundation of China (Grant No. 19ZDA084), and the Fundamental Research Funds for the Central Universities (no. B200207010). We appreciate the editor and anonymous reviewers for their beneficial comments and suggestions that improve the quality of this paper.

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

The authors declare there is no conflict.