*4.1. Multi-Agent Stochastic Evolutionary Game Framework*

To the best of our knowledge, there exist high uncertainty in the game among the government agencies, waste recyclers, and waste producers because the complexity of the external environment. To this end, the different participants will have different strategic selections because of their profits. In particular, there always exists random noise in the replicator dynamics formula, leading to bad performance for the deterministic evolutionary game framework, since the existing uncertainty around different participants. Therefore, it is necessary to take random noise into account in the tripartite game model. To further improve the previous deterministic game model, in this study, the replicator dynamic formula is combined with Gaussian white noise, which results in the multi-agent stochastic evolutionary game framework, as follows:

$$\begin{cases} dx(t) = \left[yz(-\mathcal{G}\_1 - \mathcal{F}\_1 - \mathcal{F}\_2) - y\mathcal{S}\_\uparrow - z\mathcal{S}\_\uparrow + \left(\mathcal{G} + \mathcal{G}\_1 + \mathcal{F}\_1 + \mathcal{F}\_2 - \mathcal{C}\_\downarrow\right)\right]x(t)dt + \delta x(t)d\omega(t) \\\ dy(t) = \left[-xz\mathcal{S}\_\uparrow + x\mathcal{S}\_\uparrow + z\left[(1-\lambda)\mathcal{R} - (\mathcal{C} - \eta\mathcal{C}\_1) - P\_\uparrow + \Delta\mathcal{C}\_\uparrow\right] + xzF\_\uparrow - \Delta\mathcal{C}\_\uparrow\right]y(t)dt + \delta y(t)d\omega(t) \\\ dz(t) = \left[x(\mathcal{S}\_\uparrow + F\_2) + y(\lambda R + \mathcal{C}\_\uparrow) - \mathcal{C}\_\mathcal{O} - \eta\mathcal{C}\_\downarrow\right]z(t)dt + \delta z(t)d\omega(t) \end{cases} \tag{20}$$

where *ω*(*t*) is Brownian movement. *dω*(*t*) denotes Gaussian white noise, where *t* > 0 should stratify and *h* is time step, *h* > 0. Δ*ω*(*t*) = *ω*(*t* + *h*) − *ω*(*t*) and it can be represented as normal distribution *<sup>N</sup>*(0, <sup>√</sup>*h*), and *<sup>δ</sup>* denotes noise intensity.

To this end, the Equation (20) denotes one-dimensional multi-agent stochastic differential formula, which also describes the tripartite evolutionary replicator dynamics equation of government agency, waste recyclers, and waste producers under random noise, respectively.
