Green Development of Traditional Villages: Stakeholder Game Perspectives Under Reward and Punishment Policies

Abstract

Government, companies, and residents are the most important stakeholders in the sustainable development of traditional villages. Balancing the participants' interests is a challenge for the green development of traditional villages. By constructing a tripartite evolutionary game model and introducing reward and punishment policies under the stakeholder theory, this study examined changes in traditional village conservation strategies regarding the timing of the regulation in the development process. Based on the reform data of Wan Liushu Village and government support policies, the proposed theoretical model was employed using numerical simulations, and the influence of critical parameters on stakeholders' decisions was discussed. The results show that (1) four stable points exist in the theoretical model, but only the (0,1,0) strategy point is consistent with traditional village conservation and development interests. At this point, the government adopts weak regulation, the company adopts green development, and the residents choose not to participate in the strategy. (2) Only when the government incentives exceed the companies' illegal benefits. The companies will choose the conservation and development strategy for stability. (3) The government has to ensure that penalties are more substantial and significant than the cost of regulation to avoid the (0,0,0) strategy point. At the (0,0,0) point, the government adopts weak regulation, the company adopts a no green development strategy, and the residents adopt a non-participation strategy. (4) Under the government's limited regulatory strategy, residents can only be motivated if their incentives exceed participation costs. This study can provide a reference for the green development of traditional villages.

Appendices

Appendix

Appendix A

$U_{{{\text{ij}}}}$ represents the payoff of the stakeholder $i$ when the strategy $j$ is chosen, where $i = x,y,z$, representing the government, companies, and residents. $j = 1,2$ represents the first strategy and the second strategy of each stakeholder being chosen. For example: $U(y_{1} )$ and $U(y_{2} )$ represent the payoff of the companies in the green developed and non-green developed, respectively.

The expected benefits under the government's strong and weak regulatory policies as follows:

$$ U(x_{1} ) = yz(E_{1} + B_{1} - C_{1} - M_{1} - B_{4} ) + (1 - y)z(E_{1} + F_{1} - C_{1} - M_{1} ) + y(1 - z)(E_{1} + B_{1} - C_{1} - M_{1} - B_{4} ) + (1 - z)(1 - y)(E_{1} + F_{1} - C_{1} - M_{1} ) = E_{1} - C_{1} - M_{1} + {\text{y(}}B_{1} - B_{4} ) + (1 - {\text{y}})F_{1} $$

(A.1)

$$ U(x_{2} ) = yz(E_{1} + B_{1} - M_{1} - B_{4} ) + (1 - y)z(E_{1} + (1 - a)F_{1} - M_{1} ) + y(1 - z)(E_{1} + B_{1} - M_{1} - B_{4} ) + (1 - y)(1 - z)(E_{1} - M_{1} ) = E_{1} - M_{1} + y(B_{1} - B_{4} ) + (1 - y)(1 - a)zF_{1} \\ $$

(A.2)

So, the replicator dynamic equations of the local governments that utilize the “with strong regulatory policies” strategy can be expressed as follows:

$$ F(x) = \frac{dx}{{dt}} = x(1 - x)(U(x_{1} ) - U(x_{2} )) = x(1 - x)[(1 - z + za)(1 - y)F_{1} - C{}_{1}] $$

(A.3)

According to (A.3), the derivative function can be obtained as follows:

$$ F^{\prime}(x) = (1 - 2x)[(1 - z + za)(1 - y)F_{1} - C_{1} ] $$

(A.4)

Proof of inference about government strategy choice:

According to (A.4)

$$ \begin{gathered} \left\{ \begin{gathered} f(y) = (1 - {\text{z}} + az)(1 - y)F_{{_{1} }} - C_{1} \hfill \\ \frac{\partial f(y)}{{\partial y}} = (z - 1 - az)F_{{_{1} }} \hfill \\ \end{gathered} \right. \hfill \\ \hfill \\ \end{gathered} $$

(A.5)

It can be found that $(z - 1 - az)$ is always less than 0, so (A.5) must always be less than 0, (A.5) is a decreasing function. When $y_{{}} = y_{1} = 1 - \frac{{C_{1} }}{{(1 - z + az)F_{1} }}$, we can calculate $(z - 1 - az)F_{{_{1} }} = 0$, $f(y{}_{1}) = 0$, $F(0) = 0$,$F^{\prime}(0) = 0$. When $y > y_{1}$, we can calculate $f(y) < 0$, $F(x = 0) = 0$,$F^{\prime}(x = 0) < 0$. When $y_{{}} < y_{1}$, we can calculate $f(y) > 0$, $F(x = 1) = 0$,$F^{\prime}(x = 0) < 0$. Therefore, $x = 0,1$ are stable points that meet the definition of stable point judgment.

The expected benefits under the companies as follows:

$$ U(y_{1} ) = xz(E_{2} + M_{1} - C_{2} + B_{4} ) + x(1 - z)(E_{2} + M_{1} - C_{2} + B_{4} ) + (1 - x)z(E_{2} + M_{1} - C_{2} + B_{4} ) + (1 - x)(1 - z)(E_{2} + M_{1} - C_{2} + B_{4} ) = E_{2} + M_{1} - C_{2} + B_{4} $$

(A.6)

$$ U(y_{1} ) = xz(M_{1} + B_{2} - C_{2} - F_{1} - E_{2} ) + x(1 - z)(M_{1} + B_{2} - C_{2} - F_{1} - E_{2} ) + (1 - x)z(M_{1} + B_{2} - C_{2} - E_{2} - F_{1} ) + (1 - x)(1 - z)(E_{2} + B_{2} + M_{1} - C_{2} ) = M_{1} + B_{2} + (1 + 2xz - 2x - 2z)E_{2} - C_{2} - (x + z + xz)F_{1} $$

(A.7)

So, the replicator dynamic equations of the companies that utilize the “with green development” strategy can be expressed as follows:

$$ F(y) = \frac{dy}{{dt}} = y(1 - y)(U(y_{1} ) - U(y_{2} )) = y(1 - y)[(x + z + xz)F_{1} - (2xz - 2z - 2x)E_{2} + B_{4} - B_{2} ] $$

(A.8)

According to (A.3), the derivative function can be obtained as follows:

$$ \begin{aligned} F^{\prime}(y) &= (1 - 2y)[(x + z + xz)F_{1} - (2xz - 2z - 2x)E_{2}\\ &\quad + B_{4} - B_{2} ]\end{aligned} $$

(A.9)

Proof of inference about government strategy choice:

According to (A.11), let

$$ \left\{ \begin{gathered} f(x) = (x + z - xz)F_{1} - (2xz - 2z - 2x)E_{2} + B_{4} - B_{2} \hfill \\ \frac{\partial f(x)}{{\partial x}} = (1 - z)F_{1} - (2z - 2)E_{2} \hfill \\ \end{gathered} \right. $$

(A.10)

(A.10) is always greater than 0, so that $f(x)$ is an increasing function. When $x = x_{1} = \frac{{B_{2} - B_{4} - z(F_{1} + 2E_{2} )}}{{(1 - z)F_{1} - (2z - 2)E_{2} }}$, can calculate $f(x) = 0$, $F(y) = 0$,$F^{\prime}(y) = 0$. When $x > x_{1}$, we can calculate $f(x) > 0$,$F(y = 1) = 0$. When $x < x_{1}$, we can calculate $f(x) < 0$, $F(y = 0) = 0$,$F^{\prime}(y = 0) < 0$. Similar to the government's judgment, $y = 0,1$ are stable points.

The expected benefits under the companies as follows:

$$ \begin{aligned} U(z_{1} ) & = xy(E_{3} + B_{3} - C_{3} ) + x(1 - y)( - C_{3} ) + (1 - x)y\\ &\quad\times (E_{3} + B_{3} - C_{3} ) + (1 - x)(1 - y)(aF_{1} - C_{3} ) \\ &= y(E_{3} + B_{3} ) - C_{3} + (1 + xy - x - y)aF_{1} \\ \end{aligned} $$

(A.11)

$$ \begin{aligned} U(z_{2} ) & = xy(E_{3} + B_{3} ) + (1 - x)(1 - y)(E_{3} + B_{3} ) \\ &= y(E_{3} + B_{3} ) \\ \end{aligned} $$

(A.12)

So, the replicator dynamic equations of the companies that utilize the “with participation” strategy can be expressed as follows:

$$ F(z) = \frac{dz}{{dt}} = z(1 - z)(U(z_{1} ) - U(z_{2} )) = z(1 - z)[(1 + xy - x - y)aF_{1} - C_{3} ] $$

(A.13)

According to (A.13), the derivative function can be obtained as follows:

$$ F^{\prime}(z) = (1 - 2z)[(1 + xy - x - y)aF_{1} - C_{3} ] $$

(A.14)

Proof of inference about residents' strategy choice:

According to (A.14), let

$$ \left\{ \begin{gathered} f_{2} (y) = (1 + xy - x - y)aF_{1} - C \hfill \\ \frac{{\partial f_{2} (y)}}{\partial y} = (x - 1)aF_{1} \hfill \\ \end{gathered} \right. $$

(A.15)

(A.15) is always less than 0, so $f_{2} (y)$ is a decreasing function. When $y = y_{2} = \frac{{C_{3} - aF_{1} (1 - x)}}{{aF_{1} (x - 1)}}$, can calculate $y = y_{2} = \frac{{C_{3} - aF_{1} (1 - x)}}{{aF_{1} (x - 1)}}$, $f_{2} (y) = 0$, $F(z) = 0$,$F^{\prime}(z) = 0$. When $y < y_{2}$, we can calculate $f_{2} (y) > 0$, $F(z = 1) = 0$,$F^{\prime}(z = 1) < 0$. When $y > y_{2}$, we can calculate $f_{2} (y) < 0$, $F(z = 0) = 0$,$F^{\prime}(z = 0) < 0$. Therefore, ${\text{z}} = 0,1$ are stable points, which meet the definition of stable point judgment.

Appendix B

Three-way evolutionary game Jacobi matrix derivation as follows:

$$ J = \left[ {\begin{array}{*{20}c} {{{\partial F(x)} \mathord{\left/ {\vphantom {{\partial F(x)} {\partial x}}} \right. \kern-0pt} {\partial x}}} &\quad {{{\partial F(x)} \mathord{\left/ {\vphantom {{\partial F(x)} {\partial y}}} \right. \kern-0pt} {\partial y}}} &\quad {{{\partial F(x)} \mathord{\left/ {\vphantom {{\partial F(x)} {\partial z}}} \right. \kern-0pt} {\partial z}}} \\ {{{\partial F(y)} \mathord{\left/ {\vphantom {{\partial F(y)} {\partial x}}} \right. \kern-0pt} {\partial x}}} &\quad {{{\partial F(y)} \mathord{\left/ {\vphantom {{\partial F(y)} {\partial y}}} \right. \kern-0pt} {\partial y}}} &\quad {{{\partial F(y)} \mathord{\left/ {\vphantom {{\partial F(y)} {\partial z}}} \right. \kern-0pt} {\partial z}}} \\ {{{\partial F(z)} \mathord{\left/ {\vphantom {{\partial F(z)} {\partial x}}} \right. \kern-0pt} {\partial x}}} &\quad {{{\partial F(z)} \mathord{\left/ {\vphantom {{\partial F(z)} {\partial y}}} \right. \kern-0pt} {\partial y}}} &\quad {{{\partial F(z)} \mathord{\left/ {\vphantom {{\partial F(z)} {\partial z}}} \right. \kern-0pt} {\partial z}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\lambda_{1} } & 0 & 0 \\ 0 & {\lambda_{2} } & 0 \\ 0 & 0 & {\lambda_{3} } \\ \end{array} } \right] $$

(B.1)

where $\lambda_{1} = (2z - 1)C_{3} + (a - ax - ay - 2az + axy + 2axz + 2ayz - 2axyz)F_{1}$,

$\lambda_{2} = (2x - 1)C_{1} + (1 - 2x - y - z + az + 2xy + 2xz + yz - 2axz - ayz - 2xyz + 2axyz)F_{1}$,

$\lambda_{3} = (1 - 2y)B_{4} + (2{\text{y}} - 1)B_{2} + (4xy + 2xz + 4yz - 2x - 2z - 4xyz)E_{2} + (x + z - xy - 2yz + xyz)F_{1}$.

The set of individual strategy points is brought into (B.1) and calculated to obtain the eigenvalues of each point.