Game Analysis and Simulation Study of Hydropower Development Interests

Abstract

In China, hydropower development is an important strategic initiative for the structural reform of energy supply and the development of poor areas. With technology at a mature stage, hydropower development is now constrained by relocation issues. Building a mechanism to balance the interests of all the stakeholders is the key to realizing the shared benefits of hydropower development. This paper takes reservoir-displaced people, hydropower developers, and government departments as the research objects, combines the relationship between them, and constructs two evolutionary game models: “reservoir-displaced people versus hydropower developer” and “reservoir-displaced people versus hydropower developer versus government department.” We then analyze strategy selection and evolution for reservoir-displaced people and hydropower developers, as well as the changes in the evolutionarily stable strategies of both players under the government’s macro-control and determine the boundary conditions for stable behavior of each player. Based on the results of the evolutionary analysis, the initial willingness and parameters of the players can be input into a simulation, and the evolutionary paths under different scenarios are calculated in MATLAB to further verify and analyze the evolutionary results. According to the boundary conditions of the different strategies they adopt, the equilibrium point of benefits for the three players can be calculated, which provides a new research framework for the formulation of land acquisition compensation policy, and also provides an idea and scientific basis for the macro regulation of the distribution of the benefits of hydropower development.

1. Introduction

Hydropower is a renewable and clean energy source with mature technology, low operating costs, high efficiency, and flexible regulation, and is one of the best sources of energy to achieve carbon neutrality goals [1,2]. In recent years, the total installed hydropower capacity in the world has continued to increase, but its development process has encountered many obstacles [3]. Its development inevitably expropriates large areas of farmland, resulting in the involuntary relocation of a large number of people. Low compensation standards and insufficient consideration of livelihood issues for reservoir-displaced people (RDP) led to social instability and thus hampered hydropower development [4,5]. In more recent projects, resettlement policies and management systems have improved, the proportion of investment in migrant compensation has increased, and satisfaction among RDP has risen. However, if relocation requirements are too costly, developers will be unable to justify hydropower projects. Determining how to balance the various interests is the key to the sustainable promotion of hydropower development [6,7].

Land acquisition compensation and resettlement investment are mainly measured based on the utility value of resources to RDP, and according to the value transfer theory, RDP benefit little [8]. In addition to the loss of physical amenities, such as land, houses, attached buildings, and scattered trees, RDP face intangible losses through the non-applicability of survival skills, destruction of social relationship networks, changes in living habits, and new psychological pressures [9,10]. Although their life circumstances can gradually return to or exceed their original level through early compensation, subsidies, and later support, resources also have time value, and scholars generally believe that RDP are denied a fair share of the fruits of hydropower development [11,12]. If disputed compensation standards are not reasonably resolved for a long period, the original economic problems can easily become social problems and even protest movements [13]. On the surface, the conflict is due to the unmet demands of RDP and insufficient protection of their rights and interests. In essence, it is the result of a complex game among RDP, hydropower developers, and government departments, and an imbalance between competing interests.

In 2018, the National Development and Reform Commission proposed a plan to establish a sound long-term mechanism that would let RDP, government departments, and enterprises share the benefits of hydropower development. The core of benefit sharing lies in benefit distribution [14]. Research on the benefit-sharing game of hydropower development can be broadly divided into two approaches. One is to use the Shapley value method for cooperative game analysis and adjust the benefit shares based on the marginal contribution of the participants. For example, Chen S uses the Shapley value method to propose a compensation amount correction algorithm [15]. The other is to compare and analyze the decision choices of different participants to find the Nash equilibrium solution, i.e., the optimal strategy for all participants. For example, Han Z constructs a “hawk-dove game” model to analyze the interests and strategic choices of the government versus RDP [16]. The results so far have laid a good research foundation but ignore the dynamic adjustment of relationships between the different decision-makers in hydropower development. This paper uses an evolutionary game model to replicate the conflict of interests and the cooperation between rational decision-makers, calculate strategy choices and their evolution under different interest distributions, and provide a research framework for macro regulation of the distribution of benefits from hydropower development.

2. Theoretical Basis

2.1. Research Status

Hydropower development involves many stakeholders [17]. Zhao Y uses a “Smart Pig Game” model to analyze the distribution of interests between the central government and local governments in hydropower development [18]. Jia L analyzes the game of claims between engineering contractors and owners [19]. Sheng J introduces an “Eagle-Dove Model” to analyze the economic conflicts between RDP and government departments [20]. Zeng Y proposes a multiplayer game model for the government, contractors, and local people [21]. Ren L studies the land redistribution problem of resettlement-area residents and submerged-area migrants through an “Eagle Dove Model” [22]. Xu J points out that the stakeholders in hydropower development include the central government, local governments, displaced people, people already living in the resettlement areas, consultants, engineering construction units, NGOs, and others. [23]. For a more focused-interest game analysis, the scope of stakeholder research needs to be defined.

Following Mitchell et al.’s generalization, Shi G proposes that stakeholders have three characteristics: They have invested certain resources in the project; they can be actively invested in resources or passively invested in resources; and they have taken certain risks [24]. From the above definition, we define the stakeholders in hydropower development as RDP, hydropower developers, and government departments, and we use this as the scope of research to conduct a game analysis (

Table 1. Stakeholder characteristics of hydropower development.

Stakeholders	Characteristic	Behavior	Committed Resources	Main Risks	Expected Return
Hydropower developers	Proactive	Investment, construction	Financial resources, human resources	Financial	Project economic benefits
Government departments	Proactive	Administration	Administrative resources	Political	Tax, social benefits
Displaced persons	Passive	Relocation	Land resources, socio-economic relations, etc.	Poverty	Compensation, subsidies, and post-completion support

).

2.2. Game Relationship Analysis

2.2.1. The Game between RDP and Hydropower Developers

For RDP, the game has the following characteristics. Due to the urgency of water conservancy projects and the reshaping of the environment, the RDP do not have a choice about whether to relocate. If they cooperate with the relocation and resettlement, they receive the corresponding compensation. RDP who complete their relocation on time may receive further incentives, while those who choose not to cooperate will need to invest more time and effort in negotiating with the hydropower developer. If their demands are met, they may receive additional compensation (which is an additional cost to the developer). Furthermore, if the completion of relocation is delayed through RDP non-cooperation, the benefits generated by the project will be postponed, and the developer will bear additional losses.

2.2.2. The Game between RDP and Government Departments

Government departments should properly protect and restrain RDP [25]. To alleviate conflicts over land acquisition and achieve some measure of social justice, government departments tend to protect vulnerable RDP when formulating policies, meet reasonable demands from the RDP, and try to maintain a positive public image. Government departments cannot directly punish RDP who do not cooperate with the relocation, but if RDP take an overly aggressive approach, they may meet resistance from the government—for example, protesters may be detained. Such struggles raise the costs on both sides.

2.2.3. The Game between Government Departments and Hydropower Developers

The relationship between hydropower developers and government departments is one of win–win cooperation. The developer invests capital to develop local resources and pays taxes that promote local economic development. Government departments assist in promoting resettlement and protecting the legitimate rights and interests of RDP; and to maintain their own reputation, government departments fight for the maximum rights and interests of RDP (within the acceptable range of the developer) and ensure the stable development of the local area. The government can guide the behavior of hydropower developers by rewarding or punishing them, and indirectly protect the rights and interests of RDP.

2.3. Game Evolution Theory

An evolutionary game is a dynamic framework for analyzing the learning and strategy adjustment process of players, and the object of study is a dynamically changing group [26]. The players are not fully rational, but they can modify and improve their strategies through trial and error, learning and imitation, and eventually maximize their own interests [27]. Replication dynamics and evolutionarily stable strategy (ESS) are the core concepts of evolutionary game theory [28].

The replication dynamics equation is the change in the number of people who adopt a strategy in unit time dt:

$$F\left(x\right)=\frac{d{x}_k}{dt}=x_k\left(U_{x_k}-U_x\right)$$

(1)

$$U_x=\frac{\sum x_k\times U_{x_k}}{\sum x_k}$$

(2)

where $x_k$ is the proportion of people who choose strategy $k$ over time; $U_{x_k}$ is the individual expected return of choosing strategy $k$; $U_x$ is the average expected return of the group choosing different strategies.

An evolutionarily stable strategy is one that, if it is adopted by a majority of the individuals in the population, any small variation in behavior will not lead to larger changes, and over time, eventually all individuals will adopt that strategy [29]. Such a strategy satisfies both equilibrium and stability conditions [30].

3. Analysis of the Evolutionary Game between RDP and Hydropower Developers

3.1. Basic Assumptions of the Model

To make the evolutionary game model more objective, we define the parameters in terms of benefits, costs, rewards, and penalties, taking into account the real economic environment, and make the following assumptions:

(1)

Both players are finite rational groups, with imperfect symmetry of information; the game is random and uncertain; and the goal is to maximize benefits.

(2)

Each participant has only two strategies to choose from. The set of strategies for RDP is {cooperate, do not cooperate}; and for hydropower developers it is {compromise, do not compromise}.

(3)

Assumptions about RDP. The compensation received by RDP,$R_1$, includes compensation for the expropriation of requisitioned land, compensation for seedlings, compensation for forest trees, compensation for houses and appurtenances, relocation subsidy, transitional living subsidy, land acquisition and infrastructure construction for new sites, compensation for scattered trees, and so on. Their cost when cooperating with the relocation is $C_1$, and when not cooperating is $C_2.$ If the hydropower developer chooses a compromise strategy, the RDP who cooperate receive a reward of$W_1.$ If the demands of RDP who do not cooperate are satisfied, the RDP receive additional compensation, $F_1$. If not cooperating, the RDP will need to invest more time and effort. Therefore, $R_1>C_2>C_1$.

(4)

Assumptions about hydropower developers. The benefits to the developer are $R_2$. The cost under the compromise strategy is $C_3$. The cost under the no-compromise strategy is$C_4$. Under a no-compromise strategy, a hydropower developer needs to invest more human resources to deal with RDP disputes. Therefore, $R_2>C_4>C_3$. If the RDP issue is not handled and the project is delayed, the developer bears additional losses, $P$.

(5)

If the probability that the RDP cooperate with relocation is $x$, then their probability of not cooperating is $1-x$. If the probability that the developer compromises is $y$, then their probability of adopting a no-compromise strategy is $1-y$.

The relevant parameters of the game model of RDP and hydropower developers and their meanings are detailed in

Table 2. Relevant parameters and their meanings in the game model.

Parameter	Meaning
$R_1$	Compensation received by RDP in the project
$C_1$	Costs when RDP cooperate with relocation
$C_2$	Costs when RDP do not cooperate with relocation
$W_1$	Rewards for RDP who cooperate under a compromise strategy
$F_1$	Additional compensation given by hydroelectric developer after compromise when reservoir migrants did not cooperate
$R_2$	Hydropower developer’s gain from the project
$C_3$	Cost for hydropower developer under compromise strategy
$C_4$	Cost for hydropower developer under a no-compromise strategy
$P$	Additional losses for hydropower developer due to project delays due to RDP conflicts
$x$	Probability of RDP choosing to cooperate
$y$	Probability of hydropower developer choosing a compromise strategy

.

3.2. Model Construction

Using these assumptions and related parameters, an evolutionary game model of “RDP versus hydropower developer” is established, and the corresponding benefit matrix is obtained (

Table 3. Matrix of benefits for RDP and the hydropower developer.

Developer	$\mathbf{Compromise} \left(\mathit{y}\right)$	$\mathbf{Not} \mathbf{Compromise} \left(1-\mathit{y}\right)$
RDP
$\mathrm{Cooperate} \left(x\right)$	$R_1-C_1+W_1; R_2-C_3-W_1$	$R_1-C_1; R_2-C_4$
$\mathrm{Not} \mathrm{cooperate} \left(1-x\right)$	$R_1-C_2+F_1; R_2-C_3-F_1$	$R_1-C_2; R_2-C_4-P$

).

Recall that the expected benefits for RDP of cooperation and non-cooperation are $U_{x1}$ and $U_{x2}$, and the average benefit is $U_x$. Then

$$U_{x_1}=y\times \left(R_1-C_1+W_1\right)+\left(1-y\right)\times \left(R_1-C_1\right)$$

(3)

$$\begin{gathered} U_{x_2}=y\times \left(R_1-C_2+F_1\right)+\left(1-y\right)\times \left(R_1-C_2\right) \\ {U}_x=x\times U_{x_1}+\left(1-x\right)\times U_{x_2} \end{gathered}$$

From the first equation, we obtain

$$F\left(x\right)=x\left(1-x\right)\left[y\left(W_1-F_1+C_2-C_1\right)+\left(1-y\right)\left(C_2-C_1\right)\right]$$

Similarly, we can obtain

$$F\left(y\right)=y\left(1-y\right)\left[x\left(C_4-C_3-W_1\right)+\left(1-x\right)\left(C_4-C_3+P-F_1\right)\right]$$

3.3. Stability Analysis of Equilibrium Points

(1)

Equilibrium Conditions

Let $F\left(x\right)=\frac{d{x}_k}{dt}=0$, and $F\left(y\right)=\frac{d{y}_k}{dt}=0$. Then five points of evolutionary equilibrium of this game system can be obtained: ${\mathrm{S}}_1\left(0, 0\right), {\mathrm{S}}_2\left(0, 1\right), {\mathrm{S}}_3\left(1, 0\right), {\mathrm{S}}_4\left(1, 1\right), {\mathrm{and} \mathrm{S}}_5 \left(x^{*}, y^{*}\right)$, where

$$x^{*}=\frac{C_3-C_4+F_1-P}{F_1-P-W_1}$$

$$y^{*}=\frac{C_1-C_2}{W_1-F_1}$$

(2)

Stability Conditions

Friedman proposed the Jacobi matrix local analysis: A stable point in system evolution is an equilibrium point that satisfies two conditions [25]:

$$\mathrm{tr}\left(J\right)=\frac{\partial F\left(x\right)}{\partial x}+\frac{\partial F\left(y\right)}{\partial y}<0$$

(4)

$$\det \left(J\right)=\frac{\partial F\left(x\right)}{\partial x}\times \frac{\partial F\left(y\right)}{\partial y}-\frac{\partial F\left(x\right)}{\partial y}\times \frac{\partial F\left(y\right)}{\partial x}>0$$

(5)

This gives us five equilibrium points corresponding to $\mathrm{tr}\left(J\right)$ and $\det \left(J\right)$—see

Table 4. Judgment basis for local stability of equilibrium points.

Balancing Point	$\mathit{t}\mathit{r} \left(\mathit{J}\right)$	$\mathit{d}\mathit{e}\mathit{t} \left(\mathit{J}\right)$
${\mathrm{S}}_1\left(0,0\right)$	$C_2-C_1+C_4-C_3+P-F_1$	$\left(C_2-C_1\right)\left(C_4-C_3+P-F_1\right)$
${\mathrm{S}}_2\left(0,1\right)$	$C_2-C_1-C_4+C_3+W_1-P$	$-\left(C_2-C_1+W_1-F_1\right)\left(C_4-C_3+P-F_1\right)$
${\mathrm{S}}_3\left(1,0\right)$	$C_1-C_2+C_4-C_3-W_1$	$-\left(C_2-C_1\right)\left(C_4-C_3-W_1\right)$
${\mathrm{S}}_4\left(1,1\right)$	$C_1-C_2+C_3-C_4+F_1$	$\left(C_2-C_1+W_1-F_1\right)\left(C_4-C_3-W_1\right)$
${\mathrm{S}}_5\left(x^{*},y^{*}\right)$	0	-

.

Equilibrium point ${\mathrm{S}}_5\left(x^{*},y^{*}\right)$ has $\mathrm{tr}\left(J\right)=0$, which does not satisfy $\mathrm{tr}\left(J\right)<0$, so ${\mathrm{S}}_5$ is not a stable point of system evolution, and only the local stability of the other four equilibrium points needs to be judged. Since ${\mathrm{C}}_2>{\mathrm{C}}_1$, it is only necessary to judge the positive and negative of $C_2-C_1+W_1-F_1$, $C_4-C_3+P-F_1$, and $C_4-C_3-W_1$. See

Table 5. Stability determination of the equilibrium point of the evolutionary game model.

	Binding Conditions	Balancing Point	$\mathit{t}\mathit{r} \left(\mathit{J}\right)$	$\mathit{d}\mathit{e}\mathit{t} \left(\mathit{J}\right)$	Results
Condition 1	$C_2-C_1+W_1-F_1>0$ $C_4-C_3+P-F_1>0$ $C_4-C_3-W_1>0$	${\mathrm{S}}_1$ ${\mathrm{S}}_2$ ${\mathrm{S}}_3$ ${\mathrm{S}}_4$	+ N N −	+ − − +	Unstable Saddle point Saddle point Stability (ESS)
Condition 2	$C_2-C_1+W_1-F_1>0$ $C_4-C_3+P-F_1>0$ $C_4-C_3-W_1<0$	${\mathrm{S}}_1$ ${\mathrm{S}}_2$ ${\mathrm{S}}_3$ ${\mathrm{S}}_4$	+ N − N	+ − + −	Unstable Saddle point Stability (ESS) Saddle point
Condition 3	$C_2-C_1+W_1-F_1>0$ $C_4-C_3+P-F_1<0$ $C_4-C_3-W_1>0$	${\mathrm{S}}_1$ ${\mathrm{S}}_2$ ${\mathrm{S}}_3$ ${\mathrm{S}}_4$	N + N −	− + − +	Saddle point Unstable Saddle point Stability (ESS)
Condition 4	$C_2-C_1+W_1-F_1<0$ $C_4-C_3+P-F_1>0$ $C_4-C_3-W_1>0$	${\mathrm{S}}_1$ ${\mathrm{S}}_2$ ${\mathrm{S}}_3$ ${\mathrm{S}}_4$	+ − N N	+ + − −	Unstable Stability (ESS) Saddle point Saddle point
Condition 5	$C_2-C_1+W_1-F_1>0$ $C_4-C_3+P-F_1<0$ $C_4-C_3-W_1<0$	${\mathrm{S}}_1$ ${\mathrm{S}}_2$ ${\mathrm{S}}_3$ ${\mathrm{S}}_4$	N + − N	− + + −	Saddle point Unstable Stability (ESS) Saddle point
Condition 6	$C_2-C_1+W_1-F_1<0$ $C_4-C_3+P-F_1>0$ $C_4-C_3-W_1<0$	${\mathrm{S}}_1$ ${\mathrm{S}}_2$ ${\mathrm{S}}_3$ ${\mathrm{S}}_4$	+ − − +	+ + + +	Unstable Stable Stability (ESS) Unstable
Condition 7	$C_2-C_1+W_1-F_1<0$ $C_4-C_3+P-F_1<0$ $C_4-C_3-W_1>0$	${\mathrm{S}}_1$ ${\mathrm{S}}_2$ ${\mathrm{S}}_3$ ${\mathrm{S}}_4$	N N N N	− − − −	Saddle point Saddle point Saddle point Saddle point
Condition 8	$C_2-C_1+W_1-F_1<0$ $C_4-C_3+P-F_1<0$ $C_4-C_3-W_1<0$	${\mathrm{S}}_1$ ${\mathrm{S}}_2$ ${\mathrm{S}}_3$ ${\mathrm{S}}_4$	N N − +	− − + +	Saddle point Saddle point Stability (ESS) Unstable

Remarks: “+” means greater than 0, “−” means less than 0, N means uncertain.

for details.

The following scenarios can be inferred from Table 5.

Scenario 1. When $C_4<C_3+W_1$, the developer loses more if they choose the compromise strategy, so they will tend not to compromise. In this case, the RDP cannot obtain additional incentives or compensation no matter what strategy they choose, so they choose cooperation, which costs less for them. The evolutionarily stable points of the system all tend to ${\mathrm{S}}_3\left(1,0\right)$, as in conditions 2, 5, 6, and 8.

Scenario 2. When $C_4>C_3+W_1$, the developer loses more if they choose the no-compromise strategy, so they will tend to compromise. For the RDP, $F_1-C_2<W_1-C_1$, so they will cooperate with the relocation, and the evolutionarily stable points of the system all tend to ${\mathrm{S}}_4\left(1,1\right)$, as in conditions 1 and 3.

Scenario 3. When $C_4>C_3+W_1$ and $C_4+P>C_3+F_1$, no matter what strategy the RDP choose, the developer benefits more from a compromise strategy. If $F_1-C_2>W_1-C_1$, the RDP gain more by not cooperating than they would by cooperating, and the evolutionarily stable points of the system all tend to ${\mathrm{S}}_2\left(0,1\right)$, as in condition 4.

Scenario 4. When $C_4>C_3+W_1$, $C_4+P<C_3+F_1$ and $F_1-C_2>W_1-C_1$, the strategies of the RDP and the developers affect each other, and they also depend on the initial strategy choice of both, so there is no point of system stability.

3.4. Numerical Simulation

Simulation can be used to visualize the data and is a complement to the theoretical study of the model. We perform a numerical simulation of strategy selection and evolution in the game model by replicating the dynamic equations. $W_1$, $F_1$, and $P$ are the main parameters for adjusting the benefit distribution. These are assigned according to the four scenarios described in the previous section (

Table 6. Parameter assignment for MATLAB simulation of RDP vs. hydropower developer.

Scenario	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$	${\mathit{W}}_1$	${\mathit{F}}_1$	$\mathit{P}$
1	0.7	0.9	0.4	0.6	0.3	0.2	0.3
2	0.7	0.9	0.4	0.6	0.1	0.2	0.3
3	0.7	0.9	0.4	0.6	0.1	0.4	0.3
4	0.7	0.9	0.4	0.6	0.1	0.4	0.1

).

Let x take the values x = 0.1/0.3/0.5/0.7/0.9 and y = 0.5. Figure 1, Figure 2, Figure 3 and Figure 4 show the results of the evolution simulation for the RDP and the developer in four different scenarios.

Taking together the calculations of the previous section and the simulations of this section, we find the following.

(1)

The simulation results are consistent with the theoretical calculations, with corresponding evolutionary results obtained when different evolutionary conditions are satisfied. By adjusting parameters such as the rewards and punishments from hydropower developers to RDP and the losses to hydropower developers when conflicts break out, the gains of both sides can be adjusted so that the strategies adopted by both sides gradually evolve in the preferred direction.

(2)

When there is an evolutionarily stable strategy, the initial willingness of the players to participate does not affect the final game outcome, but it will change the evolutionary duration. As seen in Figure 1, variations in RDP willingness to cooperate all eventually lead to full cooperation, but that evolution takes longer when the initial willingness is low.

(3)

Since the strategies of the players mutually interact, they may be perturbed away from the stable strategy and then return, as shown in Figure 2. The developer is more likely to compromise when the RDP do not cooperate, and more likely not to compromise (to reduce expenses) when the RDP do cooperate. So, when RDP’s willingness to cooperate starts out low, the developer is likely to compromise, but as the probability of RDP cooperation increases, the developer will become less likely to compromise.

(4)

When the game does not have an evolutionarily stable strategy, its outcome is influenced by the initial willingness of both. As shown in Figure 4, the RDP can gain more benefits by not cooperating with relocation when the developer adopts a compromise strategy, and by cooperating when the developer chooses not to compromise. Meanwhile, the developer will benefit more from compromising if the RDP do not cooperate, and from not compromising if the RDP do cooperate, so a cycle develops: Compromise/no cooperation/no compromise/cooperation/compromise with amplitude and frequency influenced by the initial willingness to cooperate.

4. Evolutionary Game Analysis among RDP, Hydropower Developers, and Government Departments

4.1. Basic Assumptions of the Model

We assume that government departments have just two strategies to choose from: “active regulation” and “loose regulation”. Active supervision means that the government rewards or punishes RDP and hydropower developers by formulating policies and regulations, thereby macro-controlling players’ strategic choices. Loose regulation means that the government relies on market mechanisms for governance. The sets of strategies for hydropower developers and RDP remain unchanged, as detailed in Section 2.

The cost to the government of choosing active regulation is $C_5$. The government department imposes fines $F_2$ on hydropower developers who adopt an uncompromising strategy, and rewards $W_2$ for developers who adopt a compromise strategy. If the regulation is successful, the government department receives social benefits $R_3$. The cost to the government is negligible when the department chooses loose regulation, but if the hydropower developer adopts an uncompromising strategy and the RDP do not cooperate with the relocation, the government’s credibility decreases, resulting in a loss, $S$. Since there are only two choices, if the probability of active regulation is $z$, then the probability of loose regulation is $1-z$.

4.2. Model Construction

With the foregoing assumptions and relevant parameters, the evolutionary game model of the three parties, “RDP versus hydropower developer versus government department” is established, and the corresponding benefit matrix is obtained (

Table 7. Benefit matrix for the three-player game.

Strategy Choice	RDP Cooperate		RDP Do Not Cooperate
	Developer Compromises	Developer Does Not Compromise	Developer Compromises	Developer Does Not Compromise
Gov’t uses active regulation	$R_1-C_1+W_1$	$R_1-C_1$	$R_1-C_2+F_1$	$R_1-C_2$
	$R_2-C_3-W_1+W_2$	$R_2-C_4-F_2$	$R_2-C_3-F_1+W_2$	$R_2-C_4-P-F_2$
	$R_3-C_5-W_2$	$F_2-C_5$	$-W_2-C_5$	$F_2-C_5$
Gov’t uses loose regulation	$R_1-C_1+W_1$	$R_1-C_1$	$R_1-C_2+F_1$	$R_1-C_2$
	$R_2-C_3-W_1$	$R_2-C_4$	$R_2-C_3-F_1$	$R_2-C_4-P$
	$R_3-S$	$-S$	$-S$	$-S$

).

Recall that the expected benefits to the RDP of cooperative and non-cooperative relocation are $K_{x1}$ and $K_{x2}$, and the average benefit is $K_x$. Then

$$\begin{gathered} K_{x1}=yz\left(R_1-C_1+W_1\right)+y\left(1-z\right)\left(R_1-C_1+W_1\right)+\left(1-y\right)z\left(R_1-C_1\right)+\left(1-y\right)\left(1-z\right)\left(R_1-C_1\right) \\ {K}_{x2}=yz\left(R_1-C_2+F_1\right)+y\left(1-z\right)\left(R_1-C_2+F_1\right)+\left(1-y\right)z\left(R_1-C_2\right)+\left(1-y\right)\left(1-z\right)\left(R_1-C_2\right) \end{gathered}$$

From the first equation, we obtain

$$F\left(x\right)=x\left(1-x\right)\left[yz\left(C_1-C_2\right)+y\left(W_1-F_1\right)+C_2-C_1\right]$$

Similarly, we can obtain

$$\begin{gathered} F\left(y\right)=y\left(1-y\right)\left[z\left(F_2+W_2\right)+x\left(F_1-W_1-P\right)+C_4-C_3+P-F_1\right] \\ F\left(z\right)=z\left(1-z\right)\left[-y\left(F_2+W_2\right)+S+F_2-C_5\right] \end{gathered}$$

4.3. Stability Analysis of Equilibrium Points

(1)

Equilibrium Conditions

Let $F\left(x\right)=\frac{dx}{dt}=0$, $F\left(y\right)=\frac{dy}{dt}=0$, and $F\left(z\right)=\frac{dz}{dt}=0$. This gives us nine evolutionary equilibrium points: $E_1\left(0,0,0\right), E_2\left(0,0,1\right), E_3\left(0,1,0\right), E_4\left(1,0,0\right), E_5\left(0,1,1\right), E_6\left(1,0,1\right), E_7\left(1,1,0\right), E_8\left(1,1,1\right), \mathrm{and} E_9\left(x^{*},y^{*},z^{*}\right)$, where

$$\begin{gathered} x^{*}=\frac{\left(W_1-F_1\right)\left(W_2+F_2\right)}{\left(C_2-C_1\right)\left(W_1+P-F_1\right)}+\frac{{\left(W_2+F_2\right)}^2}{\left(S+F_2-C_5\right)\left(W_1+P-F_1\right)}+\frac{C_4-C_3+P-F_1}{W_1+P-F_1} \\ {y}^{*}=\frac{S+F_2-C_5}{W_2+F_2} \\ {z}^{*}=\frac{W_1-F_1}{C_2-C_1}+\frac{W_2+F_2}{S+F_2-C_5} \end{gathered}$$

(2)

Stability Conditions

The eigenvalue is the extraction of matrix information, which represents the change in different participants’ strategies over time. When the three eigenvalues are all less than 0, the final evolution result is unique.

The stability of the equilibrium point is judged according to the Lyapunov stability theory, as follows [28]:

• If all the characteristic roots λ of the Jacobi matrix are less than 0, the equilibrium point is an evolutionarily stable strategy (ESS).
• If all the characteristic roots λ of the Jacobi matrix are greater than 0, the equilibrium point is unstable.
• If some of the characteristic roots λ of the Jacobian matrix are greater than 0 and some are less than 0, the equilibrium point is a saddle point.

Since $E_9\left(x^{*},y^{*},z^{*}\right)$ has eigenvalues greater than 0, it is not a stable point for system evolution. The eigenvalues for the other eight points are given in

Table 8. System equilibrium points and their characteristic values.

Balancing Point	$\mathbf{Eigenvalue} {\mathit{\lambda }}_1$	$\mathbf{Eigenvalue} {\mathit{\lambda }}_2$	$\mathbf{Eigenvalue} {\mathit{\lambda }}_3$
$E_1\left(0,0,0\right)$	$C_2-C_1$	$C_4-C_3+P-F_1$	$S+F_2-C_5$
$E_2\left(0,0,1\right)$	$C_2-C_1$	$F_2+W_2+C_4-C_3+P-F_1$	$-S-F_2+C_5$
$E_3\left(0,1,0\right)$	$W_1-F_1+C_2-C_1$	$-C_4+C_3-P+F_1$	$-W_2+S-C_5$
$E_4\left(1,0,0\right)$	$-C_2+C_1$	$-W_1+C_4-C_3$	$S+F_2-C_5$
$E_5\left(0,1,1\right)$	$W_1-F_1$	$-F_2-W_2-C_4+C_3-P+F_1$	$W_2-S+C_5$
$E_6\left(1,0,1\right)$	$-C_2+C_1$	$F_2+W_2-W_1+C_4-C_3$	$-S-F_2+C_5$
$E_7\left(1,1,0\right)$	$-W_1+F_1-C_2+C_1$	$W_1-C_4+C_3$	$-W_2+S-C_5$
$E_8\left(1,1,1\right)$	$-W_1+F_1$	$-F_2-W_2+W_1-C_4+C_3$	$W_2-S+C_5$

.

With $C_2>C_1$, $E_1 \mathrm{and} E_2$ have eigenvalues greater than 0, so they are not stable points of system evolution.

Scenario 5. When $S<C_5+W_2$, the government will lose more prestige under active regulation, so it will favor loose regulation. When $C_3+F_1<C_4+P$, the developer will incur higher costs if they do not compromise, so they will tend to compromise. When $C_2-F_1<C_1-W_1$, the RDP will incur higher costs if they cooperate, so they will tend not to cooperate. When these conditions are satisfied, the three eigenvalues are less than 0, so $E_3\left(0,1,0\right)$ is an evolutionarily stable strategy.

Scenario 6. When $S<C_5-F_2$, the government will lose more prestige under active regulation, so it will favor loose regulation. When $C_4<C_3+W_1$, the developer will incur higher costs if they compromise, so they will tend not to compromise. However, in this case for the DRP, resisting relocation costs more $(C_1<C_2)$, so they will tend to cooperate. When these conditions are met, the three eigenvalues are less than 0, so $E_4\left(1,0,0\right)$ is an evolutionarily stable strategy.

Scenario 7. When $S>C_5+W_2$, the government will lose more prestige under loose regulation, so it will favor active regulation. When $C_3+F_1-W_2<C_4+P+F_2$, the developer will incur higher costs if they do not compromise, so they will tend to compromise. When $W_1<F_1$, the incentives the RDP receive for cooperating with the relocation are smaller than the additional compensation they receive when they resist, so they will tend to resist. When these conditions are satisfied, the three eigenvalues are less than 0, so $E_5\left(0,1,1\right)$ is an evolutionarily stable strategy.

Scenario 8. When $S>C_5-F_2$, the government will lose more prestige under loose regulation, so it will favor active regulation. When $C_4+F_2<C_3+W_1-W_2$, the developer will incur higher costs if they compromise, so they will tend not to compromise. However, in this case, for the DRP, resisting relocation costs more $(C_1<C_2)$, so they will tend to cooperate. When these conditions are satisfied, the three eigenvalues are less than 0, so $E_6\left(1,0,1\right)$ is an evolutionarily stable strategy.

Scenario 9. When $S<C_5+W_2$, the government will lose more prestige under active regulation, so it will favor loose regulation. When $C_4>C_3+W_1$, the developer will incur higher costs if they do not compromise, so they will tend to compromise. When$C_2-F_1>C_1-W_1$, the RDP will incur higher costs if they do not cooperate with the relocation, so they will tend to cooperate. When these conditions are satisfied, the three eigenvalues are less than 0, so $E_7\left(1,1,0\right)$ is an evolutionarily stable strategy.

Scenario 10. When $S>C_5+W_2$, the government will lose more prestige under loose regulation, so it will favor active regulation. When $C_4+F_2>C_3+W_1-W_2$, the developer will incur higher costs if they do not compromise, so they will tend to compromise. When$W_1>F_1$, the incentives the RDP receive for cooperating with the relocation are greater than the additional compensation they receive when they resist, so they will tend to cooperate. When these conditions are satisfied, the three eigenvalues are less than 0, so $E_8\left(1,1,1\right)$ is an evolutionarily stable strategy.

4.4. Numerical Simulation

For a more intuitive presentation of the evolutionary path and results of the three-player game, we conduct numerical simulations of the six scenarios laid out in the previous subsection, with parameters set as shown in

Table 9. Parameter assignment for MATLAB simulation of RDP vs. hydropower developer vs. government department.

Scenario	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$	${\mathit{C}}_5$	$\mathit{S}$	${\mathit{F}}_1$	${\mathit{F}}_2$	${\mathit{W}}_1$	${\mathit{W}}_2$	$\mathit{P}$
5	0.7	0.9	0.4	0.6	0.5	0.7	0.4	0.3	0.1	0.3	0.3
6	0.7	0.9	0.4	0.6	0.5	0.3	0.1	0.1	0.3	0.1	0.3
7	0.7	0.9	0.4	0.6	0.5	0.7	0.2	0.1	0.1	0.1	0.3
8	0.7	0.9	0.4	0.6	0.5	0.7	0.1	0.1	0.5	0.1	0.3
9	0.7	0.9	0.4	0.6	0.5	0.7	0.1	0.1	0.1	0.3	0.3
10	0.7	0.9	0.4	0.6	0.5	0.7	0.1	0.1	0.2	0.1	0.3

.

Assuming an initial willingness to cooperate of x = 0.5, y = 0.5, and z = 0.5, the simulations evolve as shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10.

Taking together the calculations from the previous section and the simulations from this section, we find the following.

(1)

The simulation results are consistent with the theoretical calculations, with corresponding evolutionary results obtained when different evolutionary conditions are satisfied. The strategy finally chosen by each player is closely related to its cost—the punishments and rewards it faces. Varying the parameter values leads to obvious differences in the results of the three-player game: Regardless of the initial values of x, y, and z, the direction of evolution of each player’s behavior can be changed by means of rewards and punishments to promote the convergence of the system to the ideal state.

(2)

When the government adopts a loose regulatory strategy, the strategy choice between the RDP and the hydropower developer operates according to a market mechanism, and the evolutionary results are consistent with the “RDP versus hydropower developer” model, as detailed in the previous section.

(3)

When the government department adopts an active regulatory strategy, it can regulate the hydropower development environment by changing the rewards and penalties for hydropower developers and the confrontation costs for uncooperative RDP. The theoretical calculations in the previous section indicate that the evolutionary conditions of all six of these scenarios are related to the rewards and punishments meted out by the government, and the evolutionary direction of the system can be effectively controlled under this macro-regulation. The regulation should start from its evolutionary stable condition. For example, the evolutionary stability condition of $E_4\left(1,0,0\right)$ is $S<C_5-F_2$, $C_4<C_3+W_1$, $C_1<C_2$. In this case, hydropower developers will tend to choose the no-compromise strategy, even if the government increases the incentives given to the developer $\left( W_2\right)$.

5. Conclusions and Recommendations

5.1. Conclusions

In this paper, a dynamic evolutionary game model is introduced to quantitatively analyze the distribution of benefits of hydropower development, which can directly present the dynamic adjustment mechanism between the game participants. According to the final decision-making goal, the corresponding benefit distribution scheme can be obtained, which provides a research framework for rationally distributing the benefits of hydropower development.

Simplifying the roles, relationships, and operation mechanisms from the perspective of core stakeholders, we constructed two models, “RDP versus hydropower developer” and “RDP versus hydropower developer versus government department”. The evolutionary processes of different player strategies were calculated according to replication dynamics and evolutionarily stable strategies. According to the replicator dynamic equation and evolutionarily stable strategy theory, the evolutionary conditions for different strategies of all players are calculated. MATLAB is used for numerical simulation, and the simulation results are compared with the calculation results to further verify the correctness of the model.

In the calculation process, the distribution of benefits of hydropower development mainly revolves around four aspects—benefits, costs, rewards, and penalties—and changing these parameters changes the game outcome significantly. Regardless of the initial willingness of the players to cooperate, the evolution of their behavior can be changed by means of rewards and penalties to promote the convergence of the system in the desired direction.

When certain conditions are met—that is, $S>C_5+W_2$, $C_4+F_2>C_3+W_1-W_2$, and $W_1>F_1$—the system’s evolutionarily stable point tends toward $E_8\left(1,1,1\right)$. In this case, under the macro-control of government departments, both hydropower developers and RDP are willing to adopt cooperative strategies, which is the most favorable stable state for the sharing of benefits from hydropower development. Under this evolutionary path, the initial willingness of RDP to cooperate can be improved, and convergence can be accelerated, by adjusting the benefit distribution.

5.2. Recommendations

Based on this research, we offer the following recommendations. Related recommendations are shown in Figure 11.

5.2.1. Strengthen Public Policy to Foster the Active Participation of RDP

RDP are often in a passive position with respect to hydropower projects, with little understanding of policies related to resettlement, fearful of losses, and lacking any effective means of responding to unreasonable treatment in the relocation process. The government should strengthen its outreach policy, meet the reasonable demands of the RDP, guide them toward active participation, encourage them to cooperate, and speed up the convergence of system evolution.

5.2.2. Adjust Hydropower Development Revenue within Reasonable Limits to Achieve a Win–Win Situation

Delays due to people’s resistance to relocation reduce the overall benefits of projects and seriously hamper hydropower development. Benefit distribution can be adjusted by means of incentives, penalties, and so on. Consideration of the additional losses caused by conflicts between hydropower developers and RDP can inform the calculation of incentives for RDP to cooperate, which can maximize the overall benefits of the project and achieve a win–win situation.

5.2.3. Strengthen the Government’s Close Supervision to Protect the Rights and Interests of All Parties

Our game-theoretic studies suggest that government macro-regulation is closely tied to the strategic choices of hydropower developers and RDP and can change the direction of the evolution of stakeholders’ behavior. Regulatory means such as rewards and penalties can help the system converge in the desired direction. In addition to rewards and punishments for hydropower developers, government departments should also improve in-kind compensation methods, combine RDP demands, and set reasonable compensation standards. At the same time, they should increase the costs to RDP of not cooperating, so as to encourage both developers and RDP to resolve their conflicts through negotiation.