*4.2. Equilibrium Solutions Analysis*

It is known that Equation (20) is *Ito*ˆ-type stochastic differential formula, therefore, at initial time *x*(0) = 0, *y*(0) = 0, and *z*(0) = 0, respectively. Then according to Equation (20), the following equations are given:

$$\begin{cases} dx(t) = \left[ yz\left(-G\_1 - F\_1 - F\_2\right) - yS\_\uparrow - zS\_\uparrow + \left(G + G\_1 + F\_1 + F\_2 - C\_\S\right) \right] \cdot 0 + \delta x(t) d\omega(t) \\ dy(t) = \left\{-zzS\_\uparrow + xS\_\uparrow + z\left[ (1-\lambda)R - (C - \eta\mathbb{C}\_1) - P\_\uparrow + \Delta\mathbb{C}\_\uparrow \right] + xzF\_1 - \Delta\mathbb{C}\_\uparrow \right\} \cdot 0 + \delta y(t) d\omega(t) \\ dz(t) = \left[ x(S\_\uparrow + F\_2) + y(\lambda R + \mathbb{C}\_0) - \mathbb{C}\_0 - \eta\mathbb{C}\_1 \right] \cdot 0 + \delta z(t) d\omega(t) \end{cases} \tag{21}$$

Based on Equation (21), it can be seen that *dω*(*t*)|*t*=<sup>0</sup> = *ω* (*t*)*dt*||*t*=<sup>0</sup> = 0, and there at least have zero solution, which indicates the construction waste recycling system will stay in this state without the interference of external white noise. To this end, zero solution is the best in this situation.

However, the construction recycling system will always be disturbed by the internal and external environment, which influences system stability. Therefore, the system stability under random noise circumstances must be considered and analyzed.

Given stochastic differential equation [16]

$$\begin{cases} \,d\mathbf{x}(t) = f(t, \mathbf{x}(t))dt + \mathcal{g}(t, \mathbf{x}(t))d\omega(t) \\ \qquad \mathbf{x}(t\_0) = \mathbf{x}\_0 \end{cases} \tag{22}$$

It is assumed that there has a function *V*(*t*, *x*) for which there exist positive constant *σ*1, *σ*2, such that

$$
\sigma\_1 |\mathbf{x}|^p \le V(t, \mathbf{x}) \le \sigma\_2 |\mathbf{x}|^p, t \ge 0 \tag{23}
$$

Then, two kinds of specific scenarios are analyzed concerning system stability.

**I.** If a positive constant *α* is existing, making *LV*(*t*, *x*) ≤ −*αV*(*t*, *x*), *t* ≥ 0, the null solution of Equation (22) is therefore globally exponentially stable in p-th mean. Then, *E*|*x*(*t*, *x*0)| *<sup>p</sup>* < *<sup>σ</sup>*<sup>2</sup> *σ*1 |*x*0| *p e*−*α<sup>t</sup>* , *t* ≥ 0.

**II.** When a positive constant *α* is existing, making *LV*(*t*, *x*) ≥ *αV*(*t*, *x*), *t* ≥ 0. In this case, the null solution of Equation (22) is not exponentially stable in p-th mean. Then, *E*|*x*(*t*, *x*0)| *<sup>p</sup>* <sup>≥</sup> *<sup>σ</sup>*<sup>2</sup> *σ*1 |*x*0| *p e*−*α<sup>t</sup>* , *t* ≥ 0.

To this end, for the Equation (19), let *V*(*t*, *x*) = *x*(*t*), *V*(*t*, *y*) = *y*(*t*), and *V*(*t*, *z*) = *z*(*t*), where *x*, *y*, *z* ∈ [0, 1]. In particular, when *σ*<sup>1</sup> = *σ*<sup>2</sup> = 1, *p* = 1, and *α* = 1, the following equations can be attained:

$$\begin{aligned} LV(t, \mathbf{x}) &= f(t, \mathbf{x}) = \mathbf{x} \left[ yz(-\mathbf{G}\_1 - \mathbf{F}\_1 - \mathbf{F}\_2) - y\mathbf{S}\_{\bar{\jmath}} - z\mathbf{S}\_{\bar{\jmath}} + \left( \mathbf{G} + \mathbf{G}\_1 + \mathbf{F}\_1 + \mathbf{F}\_2 - \mathbf{C}\_{\mathcal{S}} \right) \right] \\ LV(t, y) &= f(t, y) = y \left\{ -\mathbf{x}z\mathbf{S}\_{\bar{\jmath}} + \mathbf{x}\mathbf{S}\_{\bar{\jmath}} + z \left[ (1 - \lambda)\mathbf{R} - (\mathbb{C} - \eta\mathbf{C}\_1) - \mathbf{P}\_{\bar{\jmath}} + \Delta\mathbf{C}\_{\bar{\jmath}} \right] + \mathbf{x}z\mathbf{F}\_1 - \Delta\mathbf{C}\_{\bar{\jmath}} \right\} \\ LV(t, z) &= f(t, z) = z \left[ \mathbf{x}(\mathbf{S}\_{\bar{\imath}} + \mathbf{F}\_2) + y(\lambda\mathbf{R} + \mathbf{C}\_0) - \mathbf{C}\_0 - \eta\mathbf{C}\_1 \right] \end{aligned} \tag{24}$$

So, if the conditions

$$\begin{cases} yz\left[yz\left(-G\_1 - F\_1 - F\_2\right) - yS\_{\bar{j}} - zS\_s + \left(G + G\_1 + F\_1 + F\_2 - C\_{\mathcal{G}}\right)\right] \le -\infty\\ y\left\{-xzS\_{\bar{j}} + xS\_{\bar{j}} + z\left[(1-\lambda)R - (C - \eta\mathbb{C}\_1) - P\_{\bar{j}} + \Delta\mathbb{C}\_{\bar{j}}\right] + xzF\_1 - \Delta\mathbb{C}\_{\bar{j}}\right\} \le -y\\ f(t, z) = z\left[\mathbf{x}(S\_s + F\_2) + y(\lambda R + C\_0) - \mathbf{C}\_0 - \eta\mathbb{C}\_1\right] \le -z \end{cases} \tag{25}$$

are satisfied, the null solutions of Equation (19) are globally exponentially stable in *p*-th mean, respectively.

#### *4.3. Taylor Expansion of Evolution Equation*

It is known that there is no clear solution for a nonlinear *Ito*ˆ stochastic differential formula. To this end, the random Taylor expansion for *Ito*ˆ equation is conducted and the numerical approximations are used to solve it.

For a existing stochastic differential equation, i.e., Equation (26)

$$dx(t) = f(t, x(t))dt + g(t, x(t))d\omega(t)\tag{26}$$

where *t* ∈ [*t*0, *T*], *x*(*t*0) = *x*0, *x*<sup>0</sup> ∈ *R*, and *ω*(*t*) is the standard winner process. Assume that, when *h* = (*T* − *t*0)/*N*, *tn* = *t*<sup>0</sup> + *nh*, the equation of random Taylor expansion is given in Equation (26)

$$\mathbf{x}(t\_{n+1}) = \mathbf{x}(t\_n) + I\_0 f(\mathbf{x}(t\_n)) + I\_1 \mathbf{g}(\mathbf{x}(t\_n)) + I\_{11} L^1 \mathbf{g}(\mathbf{x}(t\_n)) + I\_{00} L^0 f(\mathbf{x}(t\_n)) + \mathbb{R} \tag{27}$$

where *L*<sup>0</sup> = *f*(*x*) *<sup>∂</sup> <sup>∂</sup><sup>x</sup>* <sup>+</sup> <sup>1</sup> <sup>2</sup> *<sup>g</sup>*2(*x*) *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> , *<sup>L</sup>*<sup>1</sup> <sup>=</sup> *<sup>g</sup>*(*x*) *<sup>∂</sup> <sup>∂</sup><sup>x</sup>* , *<sup>I</sup>*<sup>0</sup> <sup>=</sup> *<sup>h</sup>*, *<sup>I</sup>*<sup>1</sup> <sup>=</sup> <sup>Δ</sup>*ωn*, *<sup>I</sup>*<sup>00</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> *<sup>h</sup>*2, *I*<sup>11</sup> = <sup>1</sup> 2 (Δ*ωn*) <sup>2</sup> <sup>−</sup> *<sup>h</sup>* , and *R* is the remainder of the Taylor expansion.

Therefore, Equation (27) can be rewritten as follows

$$\begin{split} \mathbf{x}(t\_{n+1}) &= \mathbf{x}(t\_n) + h f(\mathbf{x}(t\_n)) + \Delta \omega\_n \mathbf{g}(\mathbf{x}(t\_n)) + \frac{1}{2} \Big[ (\Delta \omega\_n)^2 - h \Big] \mathbf{g}(\mathbf{x}(t\_n)) \mathbf{g}'(\mathbf{x}(t\_n)) \\ &+ \frac{1}{2} h^2 \Big[ f(\mathbf{x}(t\_n)) f'(\mathbf{x}(t\_n)) + \frac{1}{2} \mathbf{g}^2(\mathbf{x}(t\_n)) f''(\mathbf{x}(t\_n)) \Big] + R \end{split} \tag{28}$$

To this end, the Milstein approach is used to solve the approximation problem. The Taylor expansions are further conducted for government agencies, waste recyclers, and waste producers, which leads to

$$\begin{split} \mathbf{x}(t\_{n+1}) &= \mathbf{x}(t\_n) + h\Big(\mathbf{y}(t\_n)\mathbf{z}(t\_n)(-\mathbf{G}\_1 - \mathbf{F}\_1 - \mathbf{F}\_2) - \mathbf{y}(t\_n)\mathbf{S}\_j - \mathbf{z}(t\_n)\mathbf{S}\_s + \\ &+ \left(\mathbf{G} + \mathbf{G}\_1 + \mathbf{F}\_1 + \mathbf{F}\_2 - \mathbf{C}\_\mathbf{g}\right)\mathbf{x}(t\_n) + \frac{1}{2}\Big((\Delta\omega\_n)^2 - h\Big)\sigma^2\mathbf{x}(t\_n) + \frac{1}{2}h^2\Big(y(t\_{n+1})\mathbf{z}(t\_{n+1}) \\ &- (\mathbf{G}\_1 - \mathbf{F}\_1 - \mathbf{F}\_2) - y(t\_{n+1})\mathbf{S}\_j - \mathbf{z}(t\_{n+1})\mathbf{S}\_5 + \left(\mathbf{G} + \mathbf{G}\_1 + \mathbf{F}\_1 + \mathbf{F}\_2 - \mathbf{C}\_\mathbf{g}\right)\Big)^2 \mathbf{x}(t\_n) + \Delta\omega\_{\mathbb{R}}\sigma\mathbf{x}(t\_n) + R\_1 \end{split} \tag{29}$$

$$\begin{aligned} y(t\_{n+1}) &= y(t\_n) + h\left(-\mathbf{x}(t\_n)z(t\_n)S\_j + \mathbf{x}(t\_n)S\_j + z(t\_n)\left[(1-\lambda)R - (\mathbb{C} - \eta\mathbb{C}\_1) - P\_j + \Delta\mathbb{C}\_j\right]\right) \\ &+ \mathbf{x}(t\_n)z(t\_n)F\_1 - \Delta\mathbb{C}\_j\big)y(t\_n) + \frac{1}{2}\big[(\Delta\omega\_n)^2 - h\big|\sigma^2 y(t\_n) + \frac{1}{2}h^2\Big(-\mathbf{x}(t\_n)z(t\_n)S\_j + \mathbf{x}(t\_n)S\_j \\ &+ z(t\_n)\left[(1-\lambda)R - (\mathbb{C} - \eta\mathbb{C}\_1) - P\_j + \Delta\mathbb{C}\_j\right] + \mathbf{x}(t\_n)z(t\_n)F\_1 - \Delta\mathbb{C}\_j\Big)^2 y(t\_n) + \Delta\omega\_n\sigma y(t\_n) + R\_2 \\ &z(t\_{n+1}) = z(t\_n) + h\Big(\mathbf{x}(t\_n)(S\_i + F\_2) + y(t\_n)(\lambda R + \mathbb{C}\_0) - \mathbb{C}\_0 - \eta\mathbb{C}\_1\Big)z(t\_n) + \frac{1}{2}h^2\Big(\mathbf{x}(t\_n)(S\_i + F\_2) + \Delta\mathbb{C}\_j\Big)^2 y(t\_n) \\ &+ F\_2\big) + y(t\_n)(\lambda R + \mathbb{C}\_0) - \mathbb{C}\_0 - \eta\mathbb{C}\_1\Big)^2 z(t\_n) + \Delta\omega\_n\sigma z(t\_n) + \frac{1}{2}\Big((\Delta\omega\_n)^2 - h\Big)\sigma^2 z(t\_n) + R\_3 \end{aligned} \tag{31}$$

#### **5. Numerical Simulations**

To the best of our knowledge, it is hard to achieve the precise solution of nonlinear *Ito*ˆ differential formula. To this end, numerical simulation is applied to simulate the trajectory of three-party dynamic evolution. Especially, in this study, a three participants stochastic evolutionary game framework is proposed for construction waste recycling by analyzing the effect principle of sorting cost of construction waste, construction waste producer disposal cost when recyclers and producers do not implement construction waste recycling, effort level when waste producer implement construction waste recycling, and Gaussian white noise on the three-party evolutionary trajectory. In addition, the stability and convergence rate of the evolutionary trajectory is also analyzed. In the beginning, the following two different cases are considered: (1) for the numerical study of sorting costs and effort level, let both waste recyclers and producers implement waste recycling under positive government supervision. In this case, *x* = 0.5, which means the government agency conducts the positive supervision. In particular, the government agency does not favor any one of the positive and negative strategies at the game start, and the same with the waste producers and waste recyclers. To this end, the initial points are defined as *x*<sup>0</sup> = *y*<sup>0</sup> = *z*<sup>0</sup> = 0.5. (2) In contrast, for the disposal costs study, let both waste recyclers and producers do not want to implement waste recycling, while government agency tends to negative supervision strategies at the beginning. Here *x* = 0.4. While the waste producers and recyclers do not implement construction waste recycling. Therefore the initial points are defined as *x* = 0.4, *y*<sup>0</sup> = 0.2, *z*<sup>0</sup> = 0.3.

### *5.1. Sorting Cost of Construction Waste*

Sorting cost is an essential factor when conducting construction waste recycling. Therefore, it is necessary for the waste producers and recyclers to take this factor into consideration. Figure 3 shows the results of the evolutionary trajectory of government agencies, waste recyclers, and waste producers, respectively. From Figure 3a, it is observed that with the increase of waste sorting cost, government agency always keeps the state under positive supervision. From the perspective of stability of evolution system and convergence rate, *C*<sup>1</sup> = 5 is the first one to reach the equilibrium point, while *C*<sup>1</sup> = 2 tends to reach the stable point. However, when the value of sorting cost (i.e., *C*<sup>1</sup> = 19) belongs to some ranges that lead to *LLE* > 0, the evolutionary trajectory of the three parties shows a very instability characteristic. Meanwhile, it can be seen from Figure 3b,c, the waste producers and recyclers also can implement waste recycling with the increasing sorting costs. However, when sorting cost *C*<sup>1</sup> = 19, the trajectories show strong instability.

Furthermore, the analysis of sorting cost between the waste recyclers and producers without external interference (*δ* = 0) is conducted. In particular, from Figure 4, it can be seen that when *C*<sup>1</sup> = 2 and *C*<sup>1</sup> = 5, the trajectory of recyclers and producers presents a fast convergence to implement construction waste recycling, and there exist Nash equilibrium. This means that if waste producers and recyclers bear fewer sorting costs, it will promote its enthusiasm to implement construction waste recycling. This is because the more sorting is, the more complex the dynamic system will show. Furthermore, with the increase of

the sorting cost, the probabilities of waste producers and recyclers choosing to implement construction recycling will reduce.

(**c**)

**Figure 3.** Multi-agent dynamic evolutionary trajectories under different construction waste sorting costs. (**a**) The probability when government agency conducting positive supervision. (**b**) The probability when waste recyclers implement construction waste recycling. (**c**) The probability when waste producers implement construction waste recycling. When *Ss* = 10, *Sj* = 10, *F*<sup>1</sup> = 15, *F*<sup>2</sup> = 20, *G* = 30, *G*<sup>1</sup> = 15, *Eg* = 8, *Cg* = 5, *Pj* = 15, Δ*Cj* = 8, *C* = 30, *λ* = 0.2, *R* = 45, *η* = 0.3, *C*<sup>0</sup> = 11, *δ*<sup>1</sup> = *δ*<sup>2</sup> = *δ*<sup>3</sup> = 0.1.

**Figure 4.** Evolution of waste recycles and waste producers.

Therefore, the waste producers and recyclers should find a suitable sorting cost that enhances their enthusiasm for construction waste recycling under the positive supervision of government agencies.

#### *5.2. Disposal Costs from Waste Producers*

An essential assumption in the construction waste recycling management system is that if both waste producers and recyclers do not implement waste recycling. The producers should pay the fee for waste landfills. To this end, how disposal cost affects the waste recycling system is further studied. Figure 5 shows the numerical simulation results. From the perspective of the government agency, the suitable *C*<sup>0</sup> leads the evolutionary trajectory to quickly converge to the stability points. However, unsuitable value brings disturbance to the dynamic system, which makes the construction recycling system is easily affected by external factors. In contrast, it can be seen that waste recyclers and producers are prone to not implement waste recycling when the value of *C*<sup>0</sup> results in *LLE* > 0. This means although an unreasonable value of *C*<sup>0</sup> can speed the trajectory evolution, the system is easily influenced by the external environment. In addition, the government agency can quickly reach the equilibrium point with a reasonable *C*<sup>0</sup> and choose positive supervision. Waste recyclers and producers aim to not conduct recycling construction waste in this situation.

#### *5.3. Effects of Effort Level of Waste Producers*

The effort level represents how waste producers implement waste recycling. Generally, the smaller *η* is, the rougher the waste producers dispose of the construction waste. In contrast, the larger *η* represents the waste producers dispose of the construction waste finer. Therefore, how effort level affects the dynamic system is also considered. To this end, the effect of effort level is discussed. Figure 6 gives the results. From Figure 6a, it is observed that reasonable and higher value of effort level make the government agency reach the equilibrium faster and more stable under positive supervision. In contrast, a lower reasonable value of *η* also reduces the convergence time of the system, which even brings disturbance to the system. And when selecting the unreasonable value of *η*, the system will be more easily affected by the external environments. In this case, the waste recyclers select to implement waste recycling under positive supervision from the government agency. A larger and reasonable value of effort level will make a faster and more stable system. This means waste recyclers can quickly reach the balance and a lower value of effort level will bring disturbance for the system. In contrast, waste producers reach the balance under the reasonable effort level value. In addition, the dynamic shows more vulnerable characteristics under the unreasonable effort level.

In the construction waste recycling system, the more effort from waste producers to recycle the construction waste is, the more enthusiasm for government agencies implementing positive supervision is and the more enthusiasm for waste recyclers implementing waste recycling is. This undoubtedly brings great benefits for the construction waste recycling system. Therefore, the waste producers need to try their best to recycle the construction waste generated by themselves, which will promote the activities of government agencies and waste recyclers.

**Figure 5.** Multi-agent dynamic evolutionary trajectories under different disposal costs. (**a**) The probability when government agency conducting positive supervision. (**b**) The probability when waste recyclers implement construction waste recycling. (**c**) The probability when waste producers implement construction waste recycling. When *Ss* = 10, *Sj* = 10, *F*<sup>1</sup> = 15, *F*<sup>2</sup> = 20, *G* = 30, *G*<sup>1</sup> = 15, *Eg* = 8, *Cg* = 5, *Pj* = 15, Δ*Cj* = 8, *C* = 30, *λ* = 0.2, *R* = 45, *η* = 0.3, *C*<sup>1</sup> = 5, *δ*<sup>1</sup> = *δ*<sup>2</sup> = *δ*<sup>3</sup> = 0.1.

**Figure 6.** Multi-agent dynamic evolutionary trajectories under different effort level. (**a**) The probability when government agency conducting positive supervision. (**b**) The probability when waste recyclers implement construction waste recycling. (**c**) The probability when waste producers implement construction waste recycling. When *Ss* = 10, *Sj* = 10, *F*<sup>1</sup> = 15, *F*<sup>2</sup> = 20, *G* = 30, *G*<sup>1</sup> = 15, *Eg* = 8, *Cg* = 5, *Pj* = 15, Δ*Cj* = 8, *C* = 30, *λ* = 0.2, *R* = 45, *C*<sup>0</sup> = 6, *C*<sup>1</sup> = 5, *δ*<sup>1</sup> = *δ*<sup>2</sup> = *δ*<sup>3</sup> = 0.1.
