**6. Equilibrium Analysis of Tripartite Evolutionary Game**

*6.1. Expectation Function Construction*

(1) The expected return of the government's choice of regulatory incentives:

$$\begin{array}{l} P\_{11} = yz(R\_1 - S\_1 - S\_2 - \mathbb{C}\_1) + y(1 - z)(F\_2 - S\_1 - L\_2 - \mathbb{C}\_1) + (1 - y)z(F\_1 - S\_2 - L\_1 - \mathbb{C}\_1) + (1 - y)(1 - y)(F\_2 - S\_2 - L\_1 - \mathbb{C}\_1) \\\ (1 - y)(1 - z)(F\_1 + F\_2 - L\_1 - L\_2 - \mathbb{C}\_1) \\\ = yzR\_1 + y(L\_1 - S\_1 - F\_1) + z(L\_2 - S\_2 - F\_2) + (F\_1 + F\_2 - L\_1 - L\_2 - \mathbb{C}\_1) \end{array} \tag{1}$$

The expected return of the government's laissez-faire:

$$P\_{12} = yzR\_2 = y(1-z)(-L\_2) + (1-y)z(-L\_1) + (1-y)(1-z)(-L\_1 - L\_2) = yzR\_2 + yL\_1 + zL\_2 - L\_1 - L\_2 \tag{2}$$

Average expected revenue of government:

$$\begin{aligned} P\_1 &= xP\_{11} + (1-x)P\_{12} = xyzR\_1 - xyzR\_2 - xyS\_1 - xyF\_1 - xzS\_2 - xzF\_2 + xF\_1 + xF\_2 \\ &- xC\_1 + yzR\_2 + zL\_2 + yL\_1 - L\_1 - L\_2 \end{aligned} \tag{3}$$

Replication dynamic equation of government's choice of regulatory incentives:

$$F(\mathbf{x}) = \frac{d\mathbf{x}}{dt} = \mathbf{x}(P\_{11} - P\_1) = \mathbf{x}(1 - \mathbf{x})[y(zR\_1 - zR\_2 - S\_1 - F\_1) - z(S\_2 + F\_2) + F\_1 + F\_2 - C\_1] \tag{4}$$

(2) The expected benefits of the energy-saving service enterprises choosing to provide energy-saving services:

$$\begin{cases} P\_{21} = x\mathbf{z}(E\_1 + \mathbf{S}\_1 - \mathbf{C}\_2) + x(1 - z)(\mathbf{S}\_1 - \mathbf{C}\_4) + (1 - x)z(E\_1 - \mathbf{C}\_2) + (1 - x)(1 - z)(1 - z)(-\mathbf{C}\_4) \\ \mathbf{x} = x\mathbf{S}\_1 + zE\_1 - z\mathbf{C}\_2 - \mathbf{C}\_4 + z\mathbf{C}\_4 \end{cases} \tag{5}$$

Expected benefits of the energy-saving service enterprises choosing not to provide energy-saving services:

$$P\_{22} = xz(-F\_1 - C\_3) + x(1 - z)(-F\_1) + (1 - x)z(-C\_3) + (1 - x)(1 - z) \times 0 = -xF\_1 - zC\_3 \tag{6}$$

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Average expected income of the energy-saving service enterprises:

$$P\_2 = yP\_{21} + (1 - y)P\_{22} = xyS\_1 + yzE\_1 + yzC\_4 - yzC\_2 - yC\_4 - xF\_1 - zC\_3 + xyF\_1 + yzC\_3 \tag{7}$$

Replication dynamic equation of the energy-saving service enterprises choosing to provide energy-saving services:

$$F(y) = \frac{dy}{dt} = y(P\_{21} - P\_2) = y(1 - y)[z(E\_1 + \mathbb{C}\_3 + \mathbb{C}\_4 - \mathbb{C}\_2) + x(\mathbb{S}\_1 + F\_1) - \mathbb{C}\_4] \tag{8}$$

(3) The expected income of rural residents who choose to perform energysaving transformations:

$$\begin{aligned} P\_{31} &= \mathbf{x}y(\mathbf{E}\_2 + \mathbf{S}\_2 - \mathbf{C}\_5) + \mathbf{x}(1-y)(\mathbf{S}\_2 - \mathbf{C}\_6) + (1-\mathbf{x})y(\mathbf{E}\_2 - \mathbf{C}\_5) + (1-\mathbf{x})(1-y)(1-y)(-\mathbf{C}\_6) \\ &= \mathbf{x}\mathbf{S}\_2 + y\mathbf{E}\_2 + y\mathbf{C}\_6 - y\mathbf{C}\_5 - \mathbf{C}\_6 \end{aligned} \tag{9}$$

The expected benefits of rural residents choosing to refuse energy-saving renovations:

$$P\_{32} = xy(-F\_2) + x(1-y)(-F\_2) + (1-x)y \times 0 + (1-x)(1-y) \times 0 = -xF\_2 \tag{10}$$

Average expected income of rural residents:

$$P\_3 = zP\_{31} + (1 - z)P\_{32} = xz(S\_2 + F\_2) + zy(E\_2 + C\_6 - C\_5) - xF\_2 - zC\_6 \tag{11}$$

Replication dynamic equation of rural residents choosing to perform energysaving transformations:

$$F(z) = \frac{dz}{dt} = z(P\_{31} - P\_3) = z(1 - z)[x(S\_2 + F\_2) + y(E\_2 + \mathbb{C}\_6 - \mathbb{C}\_5) - \mathbb{C}\_6] \tag{12}$$
