*3.2. Payoff Matrix and Replicator Dynamics Equations*

Table 2 gives the payoff matrix of the government agencies, waste recycles and waste producers, which is defined based on the principles shown in Figure 1 and each element of the Payoff Matrix are shown in Equation (1).

**Figure 1.** The three-party game tree of government agencies, waste recyclers and waste producers.



⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *G* − *Cg* − *SS* − *Sj* + *S*<sup>1</sup> + *S*<sup>2</sup> (1 − *λ*)*R* − (*C* − *ηC*1) + *Sj λR* − *ηC*<sup>1</sup> + *SS G* + *G*<sup>1</sup> + *F*<sup>1</sup> + *F*<sup>2</sup> − *Cg* − *Eg* − *Sj* + *S*<sup>1</sup> *Pj* + *Sj* − Δ*Cj* −*C*<sup>0</sup> − *F*<sup>2</sup> *G* + *G*<sup>1</sup> + *F*1 + *F*<sup>2</sup> − *Cg* − *Eg* − *SS* + *S*<sup>2</sup> *Pj* − *F*<sup>1</sup> −*C*<sup>0</sup> + *Ss* − *ηC*<sup>1</sup> *G* + *C*<sup>1</sup> + *F*<sup>1</sup> + *F*<sup>2</sup> − *Cg* − *Eg Pj* −*C*<sup>0</sup> − *F*<sup>2</sup> *S*<sup>1</sup> + *S*<sup>2</sup> (1 − *λ*)*R* − (*C* − *ηC*1) *λR* − *ηC*<sup>1</sup> −*Eg* + *S*<sup>1</sup> *Pj* − Δ*Cj* −*C*<sup>0</sup> −*Eg* + *S*<sup>2</sup> *Pj* −*C*<sup>0</sup> − *ηC*<sup>1</sup> −*Eg Pj* −*C*<sup>0</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1)

Let *N*<sup>11</sup> and *N*<sup>12</sup> denote the expected utility when government agency conducts positive supervision and negative supervision, respectively, and their average is represented by *N*¯ 1.

$$\begin{aligned} N\_{11} &= yz \left( G - \mathbb{C}\_{\mathcal{S}} - \mathbb{S}\_{\mathcal{S}} - \mathbb{S}\_{\mathcal{I}} + \mathbb{S}\_{\mathcal{I}} + \mathbb{S}\_{\mathcal{I}} \right) + y(1 - z) \left( G + G\_1 + F\_1 + F\_2 - \mathbb{C}\_{\mathcal{S}} - \mathbb{E}\_{\mathcal{S}} - \mathbb{S}\_{\mathcal{I}} + \mathbb{S}\_{\mathcal{I}} \right) \\ &+ (1 - y)z \left( G + G\_1 + F\_1 + F\_2 - \mathbb{C}\_{\mathcal{S}} - \mathbb{E}\_{\mathcal{S}} - \mathbb{S}\_{\mathcal{S}} + \mathbb{S}\_{\mathcal{I}} \right) + (1 - y)(1 - z) \\ &\left( G + G\_1 + F\_1 + F\_2 - \mathbb{C}\_{\mathcal{S}} - \mathbb{E}\_{\mathcal{S}} \right) \end{aligned} \tag{2}$$

$$\begin{split} \mathsf{N}\_{\mathsf{I}2} &= yz(\mathsf{S}\_{1} + \mathsf{S}\_{2}) + y(1 - z)\left(-E\_{\mathsf{g}} + S\_{1}\right) + (1 - y)z\left(-E\_{\mathsf{g}} + S\_{2}\right) + (1 - y)(1 - z)\left(-E\_{\mathsf{g}}\right) \\ &= yz\left(\mathsf{E}\_{\mathsf{S}}\right) - y\mathsf{S}\_{1} + zS\_{2} - E\_{\mathsf{g}} \end{split} \tag{3}$$

$$
\hat{N}\_1 = \mathfrak{x} \times N\_{11} + (1 - \mathfrak{x}) \times N\_{12} \tag{4}
$$

Then the replicator dynamic formula of government agency conducting positive supervision is given, as shown in Equation (5):

$$\begin{split} F(\mathbf{x}) &= \frac{d\mathbf{x}}{dt} = \mathbf{x}(\mathbf{N\_{l1}} - \mathbf{N\_{l}}) = \mathbf{x}(\mathbf{1} - \mathbf{x})(\mathbf{N\_{l1}} - \mathbf{N\_{l2}}) \\ &= \mathbf{x}(\mathbf{1} - \mathbf{x}) \left[ y\mathbf{z}(-\mathbf{G\_{l}} - \mathbf{F\_{l}} - \mathbf{F\_{l}}) - y\mathbf{S\_{j}} - z\mathbf{S\_{s}} + \left( \mathbf{G + G\_{l} + F\_{l} + F\_{2} - G\_{\mathbf{g}} \right) \right] \end{split} \tag{5}$$

Similarly, let *N*<sup>21</sup> and *N*<sup>22</sup> represent that waste recycler enterprise selects to implement and not implement construction waste recycling, respectively. And *N*¯ <sup>2</sup> denotes the average revenues.

$$\begin{aligned} N\_{21} &= \text{xz}[(1-\lambda)R - (\mathbb{C} - \eta \mathbb{C}\_1)] + \text{x}(1-z)\{P\_{\hat{\jmath}} + \mathbb{S}\_{\hat{\jmath}} - \Delta \mathbb{C}\_{\hat{\jmath}}\} \\ &+ (1-\mathsf{x})z[(1-\lambda)R - (\mathbb{C} - \eta \mathbb{C}\_1)] + (1-\mathsf{x})(1-z)\{P\_{\hat{\jmath}} - \Delta \mathbb{C}\_{\hat{\jmath}}\} \\ &= -\mathsf{x}z\mathbb{S}\_{\hat{\jmath}} + \mathsf{x}\mathbb{S}\_{\hat{\jmath}} + z\left[(1-\lambda)R - (\mathbb{C} - \eta \mathbb{C}\_1) - P\_{\hat{\jmath}} + \Delta \mathbb{C}\_{\hat{\jmath}}\right] + \{P\_{\hat{\jmath}} - \Delta \mathbb{C}\_{\hat{\jmath}}\} \end{aligned} \tag{6}$$

$$N\_{22} = xz(P\_{\dot{f}} - F\_1) + x(1 - z)P\_{\dot{f}} + (1 - x)zP\_{\dot{f}} + (1 - x)(1 - z)P\_{\dot{f}} = P\_{\dot{f}} - xzF\_1 \tag{7}$$

$$N\_2 = y \ast N\_{21} + (1 - y) \ast N\_{22} \tag{8}$$

Then, according to Equations (6) and (7), the replicator dynamic equation of waste producers conducting construction waste recycling strategy is given as follows:

$$\begin{split} F(y) = \frac{dy}{dt} &= y(\mathcal{N}\_{21} - \bar{\mathcal{N}}\_{2}) = y(1-y)(\mathcal{N}\_{21} - \mathcal{N}\_{22}) \\ &= y(1-y)\left\{-\text{xz}\mathcal{S}\_{\bar{\mathcal{I}}} + \text{x}\mathcal{S}\_{\bar{\mathcal{I}}} + z\left[(1-\lambda)\mathcal{R} - (\mathcal{C} - \eta\mathcal{C}\_{1}) - P\_{\bar{\mathcal{I}}} + \Lambda\mathcal{C}\_{\bar{\mathcal{I}}}\right] + \text{xz}F\_{\bar{\mathcal{I}}} - \Lambda\mathcal{C}\_{\bar{\mathcal{I}}}\right\} \end{split} \tag{9}$$

Finally, let *N*<sup>31</sup> and *N*<sup>32</sup> denote the expected utility that the waste producer chooses to implement and not implement the waste recycling and their average is represented by *N*¯ 3, which are formulated as follows:

$$\begin{split} N\_{31} &= \left. \text{xy} (\lambda R - \eta \mathbb{C}\_1 + \mathbb{S}\_s) + \text{x} (1 - y) (-\mathbb{C}\_0 + \mathbb{S}\_s - \eta \mathbb{C}\_1) \\ &+ (1 - \mathbb{x}) y (\lambda R - \eta \mathbb{C}\_1) + (1 - \mathbb{x}) (1 - y) (-\mathbb{C}\_0 - \eta \mathbb{C}\_1) \\ &= \left. x \mathbb{S}\_s + y (\lambda R + \mathbb{C}\_0) - \mathbb{C}\_0 - \eta \mathbb{C}\_1 \end{split} \tag{10}$$

$$\begin{split} \mathbf{N}\_{32} &= \mathbf{x}y(-\mathsf{C}\_{0} - \mathsf{F}\_{2}) + \mathbf{x}(1-y)(-\mathsf{C}\_{0} - \mathsf{F}\_{2}) - (1-\mathsf{x})y\mathsf{C}\_{0} - (1-\mathsf{x})(1-y)\mathsf{C}\_{0} \\ &= -\mathsf{x}F\_{2} - \mathsf{C}\_{0} \end{split} \tag{11}$$

$$N\_3 = z \times N\_{31} + (1 - z) \times N\_{32} \tag{12}$$

Then, the replicator dynamic formula of waste recycler conducting construction waste recycling strategy is defined as follows:

$$\begin{split} F(z) &= \frac{dz}{dt} = z(\mathcal{N}\_{\text{31}} - \mathcal{N}\_{\text{3}}) = z(1 - z)(\mathcal{N}\_{\text{31}} - \mathcal{N}\_{\text{32}}) \\ &= z(1 - z)[\varkappa(\mathcal{S}\_{\text{s}} + \mathcal{F}\_{\text{2}}) + y(\lambda R + \mathcal{C}\_{\text{0}}) - \mathcal{C}\_{\text{0}} - \eta \mathcal{C}\_{\text{1}}] \end{split} \tag{13}$$

#### *3.3. Replicator Dynamics Analysis*

According to Equations (5), (9) and (13), the multi-agent dynamic replication formula of construction waste recycling system is achieved, i.e.,

$$\begin{cases} F(\mathbf{x}) = \mathbf{x}(1-\mathbf{x}) \left[ yz(-G\_1 - F\_1 - F\_2) - yS\_j - zS\_s + \left( G + G\_1 + F\_1 + F\_2 - \mathcal{C}\_{\mathbf{y}} \right) \right] \\\ F(y) = y(1-y) \left\{ -xzS\_j + xS\_j + z\left[ (1-\lambda)R - (\mathbb{C} - \eta \mathcal{C}\_1) - P\_j + \Lambda \mathcal{C}\_j \right] + \mathbf{x}zF\_1 - \Lambda \mathcal{C}\_j \right\} \\\ F(z) = z(1-z) \left[ \mathbf{x}(S\_s + F\_2) + y(\lambda R + \mathcal{C}\_0) - \mathbb{C}\_0 - \eta \mathcal{C}\_1 \right] \end{cases} \tag{14}$$

Let ⎧ ⎨ ⎩ *F*(*x*) = 0 *F*(*y*) = 0 *F*(*z*) = 0 , 8 corresponding strategy solutions for the construction waste recy-

cling system can be achieved, i.e., *A*1(0, 0, 0), *A*2(0, 0, 1), *A*3(0, 1, 0), *A*4(0, 1, 1), *A*5(1, 0, 0), *A*6(1, 0, 1), *A*7(1, 1, 0), and *A*8(1, 1, 1). Additionally, there also exists a mixed strategy solution *O*((*x*∗, *y*∗, *z*∗)), which satisfies Equation (15)

$$\begin{cases} F(\mathbf{x}^\*) = y^\* z^\* \left( -G\_1 - F\_1 - F\_2 \right) - y^\* S\_{\bar{f}} - z^\* S\_{\bar{s}} + \left( G + G\_1 + F\_1 + F\_2 - C\_{\mathfrak{F}} \right) = 0\\ F(y^\*) = -\mathbf{x}^\* z^\* S\_{\bar{f}} + \mathbf{x}^\* S\_{\bar{f}} + z^\* \left[ (1 - \lambda) \mathbf{R} - (\mathbb{C} - \eta \mathbb{C}\_1) - P\_{\bar{f}} + \Lambda \mathbb{C}\_{\bar{f}} \right] + \mathbf{x}^\* z^\* F\_1 - \Lambda \mathbf{C}\_{\bar{f}} = 0\\ F(\mathbf{z}^\*) = \mathbf{x}^\* (\mathbb{S}\_{\bar{s}} + F\_2) + y^\* \left( \lambda \mathbf{R} + \mathbb{C}\_0 \right) - \mathbf{C}\_0 - \eta \mathbb{C}\_1 = 0 \end{cases} \tag{15}$$

Therefore, the following equations can be achieved

$$\mathbf{x}^\* = \frac{\mathbf{C}\_0 + \eta \mathbf{C}\_1}{S\_s + F\_2} \tag{16}$$

$$y^\* = \frac{\mathbb{C}\_0 + \eta \mathbb{C}\_1}{\lambda R + \mathbb{C}\_0} \tag{17}$$

$$z^\* = \frac{(S\_\sf s + F\_2)\Delta C\_j - (C\_0 + \eta C\_1)S\_j}{(C\_0 - \eta C\_1)(F\_1 - S\_j) + (S\_\sf s + F\_2)[(1 - \lambda)R - (C - \eta C\_1) - P\_j + \Delta C\_j]} \tag{18}$$

where 0 < *x*∗ < 1, 0 < *y*∗ < 1 and 0 < *z*∗ < 1.

In addition, it is obvious that 1 − *x*, 1 − *y*, and 1 − *z* are non-negative, so they will not influence the results of the evolution analysis. Next, the replicator dynamic formulas of government agencies, waste recyclers, and waste producers can be rewritten as:

$$\begin{cases} F(\mathbf{x}) = d\mathbf{x}/dt = \mathbf{x} \left[ yz(-G\_1 - F\_1 - F\_2) - yS\_j - zS\_s + \left( G + G\_1 + F\_1 + F\_2 - C\_{\mathbf{g}} \right) \right] \\\ F(y) = dy/dt = y \left\{ -\mathbf{x}zS\_j + \mathbf{x}S\_j + z \left[ (1-\lambda)\mathbf{R} - \left( \mathbf{C} - \eta \mathbf{C}\_1 \right) - P\_j + \Delta \mathbf{C}\_j \right] + \mathbf{x}zF\_1 - \Delta \mathbf{C}\_j \right\} \\\ F(z) = dz/dt = z[\mathbf{x}(S\_s + F\_2) + y(\lambda R + \mathbf{C}\_0) - \mathbf{C}\_0 - \eta \mathbf{C}\_1] \end{cases} \tag{19}$$
