*7.1. When the Benefits of Energy-Saving Renovations Are Greater than the Cost, the Change in the Initial Proportion of the Three Parties in the Game Will Affect the Evolution Results*

Suppose x0, y0 and z0 represent the initial proportions of the government's choice to implement supervision and incentive strategies, energy-saving service enterprises' choice to provide energy-saving service strategies and rural residents' choice to implement energysaving transformation strategies, respectively. The initial test time is 0, the evolution end time is 5, and the initial states are (0.1, 0.2, 0.3), (0.4, 0.5, 0.6), (0.7, 0.8, 0.9). *F*<sup>1</sup> = 40, *F*<sup>2</sup> = 10, *S*<sup>1</sup> = 30, *S*<sup>2</sup> = 5, *R*<sup>1</sup> = 300, *R*<sup>2</sup> = 180, *E*<sup>1</sup> = 120, *E*<sup>2</sup> = 50, *C*<sup>1</sup> = 60, *C*<sup>2</sup> = 40, *C*<sup>3</sup> = 15, *C*<sup>4</sup> = 8, *C*<sup>5</sup> = 25, *C*<sup>6</sup> = 5. The stability of the equilibrium point and system evolution results are shown in Figure 4.

It can be found that when the profits of energy-saving renovations are greater than the cost of energy-saving renovations, no matter which initial state the three parties chose, the three parties choose the cooperative strategy H8 (1,1,1) (supervision and encouragement, providing energy-saving services and performing energy-saving renovations), but the time taken to reach a stable and balanced state differs. Figure 4 shows that when the government, the energy-saving service enterprises and the rural residents choose supervision and encouragement, providing energy-saving services and performing energy-saving transformations with a strategy ratio of 0.1, 0.2 and 0.3, the system reaches an evolutionary stable equilibrium state at around t = 3.5. When the initial ratio increases by 0.3 or 0.6, that is, 0.4, 0.5, 0.6 or 0.7, 0.8, 0.9, the system reaches an evolutionary stable equilibrium state at about t = 2.5 or t = 1.8. This shows that, when the government chooses to implement supervision and incentive strategies for energy-saving renovations of existing rural residential buildings, energy-saving service enterprises choose to provide energy-saving service strategies, and rural residents choose to improve the proportion of energy-saving renovation, this helps to shorten the system evolution and achieve a stable and balanced state, which also helps to stimulate participants' enthusiasm for energy-saving renovations of existing rural residential buildings, and promote the development of energy-saving renovations of existing rural residential buildings.

**Figure 4.** System evolution results under different initial intentions when the benefits of energysaving renovations outweigh the costs. (**a**) The time at which the system evolves to a stable state. (**b**) Evolution rate.

#### *7.2. When the Cost of Energy Savings Is Greater than the Income, the Change in the Initial Proportion of the Three Parties in the Game Will Affect the Evolution Result*

It is assumed that the initial test time is 0, the evolution end time is 5, the initial states are (0.1, 0.2, 0.3), (0.2, 0.3, 0.4), (0.4, 0.6, 0.8), and the parameters are *F*<sup>1</sup> = 20, *F*<sup>2</sup> = 10, *S*<sup>1</sup> = 15, *S*<sup>2</sup> = 5, *R*<sup>1</sup> = 100, *R*<sup>2</sup> = 60, *E*<sup>1</sup> = 80, *E*<sup>2</sup> = 30, *C*<sup>1</sup> = 160, *C*<sup>2</sup> = 100, *C*<sup>3</sup> = 40, *C*<sup>4</sup> = 10, *C*<sup>5</sup> = 30, *C*<sup>6</sup> = 8. The stability of the equilibrium point and system evolution results is shown in Figure 5. Through the simulation results, it can be found that when the cost of energy-saving renovation is greater than the profits, the choice strategies of the tripartite government, energy-saving service enterprises and rural residents will not change due to the change in the initial state ratio; they all choose the behavior strategy H1 (0,0,0), in which energy-saving service enterprises do not provide energy-saving renovation services, and rural residents do not perform energy-saving renovations. The reason for this is that, as the number of rational people is limited, the ultimate goal of the energy-saving service enterprises and the rural residents participating in the energy-saving renovations of existing rural residential buildings is to obtain relevant benefits. Therefore, when the cost of energy-saving renovation is greater than the profits, because the renovations are unprofitable, energy-saving service enterprises and rural residents naturally choose to revoke their participation, and the ultimate behavior strategy of the three parties in the game is uncooperative.
