*6.2. Asymptotic Stability Analysis of Evolutionary Game*

In the process of energy-saving renovations of existing rural residential buildings, there is serious information asymmetry between the government, energy-saving service enterprises and rural residents in terms of energy-saving renovation information. The three parties in the game will choose the strategies that maximize their own interests in the process of trial and error. When all three parties reach a stable state, all the participants in the game achieve Nash equilibrium through the process of trial and error. According to the stability principle of differential equations, the duplicated dynamic equation of the tripartite subject strategy is simultaneously established. By setting *F*(*x*) = *F*(*y*) = *F*(*z*) = 0, we find that there are eight pure strategy equilibrium points in the equation, namely (0,0,0); (1,0,0); (0,1,0); (0,0,1); (1,1,0); (1,0,1); (0,1,1); (1,1,1).

In view of the asymptotic stability of the equilibrium points, this study uses the Lyapunov discriminant method (indirect method) to judge, list the Jacobian matrix (as in Formula (13)), and discuss the stability of the above equilibrium points. Firstly, the Jacobian matrix is calculated as in Formula (14):

$$
\begin{bmatrix}
\frac{dx/dt}{dx} & \frac{dx/dt}{dy} & \frac{dx/dt}{dz} \\
\frac{dy/dt}{dx} & \frac{dy/dt}{dy} & \frac{dy/dt}{dz} \\
\frac{dz/dt}{dx} & \frac{dz/dt}{dy} & \frac{dz/dt}{dz}
\end{bmatrix}
\tag{13}
$$

$$\begin{bmatrix} (1-2x)[y(2R\_1 - zR\_2 - S\_1 - P\_1) \\ -z(S\_2 + P\_2) + P\_1 + P\_2 - C\_1 \end{bmatrix} \qquad \begin{aligned} &\mathbf{x}(1-x)(zR\_1 - zR\_2 - S\_1 - P\_1) \\ &(1-x)(zR\_1 - zR\_2 - S\_1 - P\_1) \\ &(1-2y)[z(E\_1 + C\_3 + C\_4 - C\_2) \\ &+x\mathsf{S}\_1 + x\mathsf{P}\_1 - C\_4] \end{aligned} \qquad \begin{aligned} &\mathbf{x}(1-x)(yR\_1 - yR\_2 - S\_2 - P\_2) \\ &y(1-y)(E\_1 + C\_3 + C\_4 - C\_2) \\ &(1-z)(E\_1 + C\_3 + C\_4) + y(E\_3 - C\_3) \\ &(+\mathbf{C}\_3 - \mathbf{C}\_5) - \mathbf{C}\_6 \end{bmatrix} \end{aligned} \tag{14}$$

According to Lyapunov's stability theorem, when the characteristic roots of the Jacobian matrix are all negative, the equilibrium point is a stable node. By substituting the equilibrium points into the Jacobian matrix, the eigenvalue corresponding to each equilibrium point can be obtained, as shown in Table 3.


**Table 3.** Eigenvalues corresponding to pure strategy equilibrium points.

It can be seen from Table 3 that there are three situations in which evolutionary stability can be achieved to meet the eigenvalue requirements of the Lyapunov discriminant method (indirect method).

Scenario 1: If the external conditions remain unchanged, only equilibrium point H1 (0,0,0) can meet the requirements of Liapunov's discriminant method (indirect method) for the eigenvalue, and other equilibrium points cannot form evolutionary stability, that is, the equilibrium point H1 (0,0,0) (laissez-faire, no energy-saving service, no energy-saving renovations) is an evolutionary stability strategy. The phase diagram is shown in Figure 1.

**Figure 1.** Phase diagram of equalization point H1 (0,0,0).

Scenario 2: if the external conditions change, i.e., *R*<sup>1</sup> < *R*<sup>2</sup> + *S*<sup>1</sup> + *S*<sup>2</sup> + *C*1, then equilibrium points H1 (0,0,0) and H4 (0,1,1) can achieve evolutionary stability, that is, H1 (0,0,0) (laissez-faire, not providing energy-saving services, refusing energy-saving renovation) and H4 (0,1,1) (laissez-faire, providing energy-saving services, fulfilling energysaving renovations) are evolutionary stable strategies. The phase diagram is shown in Figure 2.

**Figure 2.** Phase diagram of equilibrium point H4 (0,1,1) when *R*<sup>1</sup> < *R*<sup>2</sup> + *S*<sup>1</sup> + *S*<sup>2</sup> + *C*1.

Scenario 3: if the external conditions change, i.e., *R*<sup>1</sup> < *R*<sup>2</sup> + *S*<sup>1</sup> + *S*<sup>2</sup> + *C*1, then equilibrium points H1 (0,0,0) and H8 (1,1,1) can achieve evolutionary stability, that is, H1 (0,0,0) (laissez-faire, not providing energy-saving services, refusing energy-saving renovations) and H8 (1,1,1) (supervision and encouragement, providing energy-saving services, fulfilling energy-saving renovations) are evolutionary stable strategies. The phase diagram is shown in Figure 3.

**Figure 3.** Phase diagram of equalization point H8 (1,1,1) when *R*<sup>1</sup> > *R*<sup>2</sup> + *S*<sup>1</sup> + *S*<sup>2</sup> + *C*1.

However, in the case of a laissez-faire approach by the government, the energy-saving service enterprises choose to provide energy-saving renovation services by themselves, and this provides an ideal state for rural residents to perform energy-saving renovations by themselves, without considering the renovation costs and the impact on their daily lives. According to the evolutionary game hypothesis, the government, energy-saving service enterprises and the rural residents are all bounded and rational. Therefore, under the premise of bounded rationality, all players in the three-party game hope to maximize their interests. Without the support of incentive policies and the guidance of relevant energy-saving renovation policies, low enthusiasm is displayed by energy-saving service enterprises and rural residents in the existing rural residential buildings to actively carry out energy-saving renovations, as is also confirmed in the actual investigation. Therefore, during the energy-saving renovations of existing rural residential buildings, the equilibrium points for realizing the tripartite evolutionary and stable strategy among the government, energy-saving service enterprises and rural residents are mainly H1 (0,0,0) (laissez-faire, no energy-saving service, no energy-saving renovation] and H8 (1,1,1) [supervision and encouragement, providing energy-saving service, fulfilling energy-saving renovations). This study also focuses on these two situations.

#### **7. Numerical Simulation Analysis**

To further verify the accuracy of the model and more intuitively show the results that the government, energy-saving service enterprises and rural residents in the existing rural residential buildings achieved, as well as the evolutionary stability under different constraints and strategies, this study uses MATLAB2020A to analyze the evolution of equilibrium points H1 (0,0,0) and H8 (1,1,1) from the perspective of cost and benefit, subsidies and penalties, combined with the replication of dynamic equations and assumptions.
