Construction waste is an issue that has attracted increasing worldwide attention recently. With the rapid development of socioeconomic and urbanization in China, the building industry has emerged as a pillar of the national economy. In particular, a large number of raw materials are used and massive construction waste is generated along a gradient of increasing urbanization, resulting in environmental pollution and scarcity of nature resource ^[1][2]. According to a study published by the Chinese Academy of Engineering, construction waste increased by 15.4% per year from 1990 to 2000, and by 16.2% per year from 2000 to 2013 ^[1][3]. Because of limited technology, e.g., a lack of professional construction waste recycling enterprises, and a lack of unified technical standards, China’s construction waste resource rate is less than 10%, which is far below the developed countries ^[1][3]. Furthermore, to the best of researchers' knowledge, the traditional disposal methods of construction waste in many countries in China are landfill and open-air stacking, which not only cause secondary pollution to soil, groundwater, rivers, and air, but also continuously occupy valuable land resources. To that end, representatives from the Chongqing Technology Evaluation and Transfer Service Center of the Chongqing Academy of Science and Technology suggested that the government should do everything possible to supervise construction waste recycling and ensure that it meets the requirements of construction sustainability development ^[2][4].

At present, construction waste recycling has been proven to be the most effective method of managing construction trash. In the meantime, many existing works ^{[5][6][7][8][9]} have already investigated its positive social, environmental, and sustainable influences and pointed out that many factors, such as positive government agency supervision, or waste recyclers implement waste recycling, can influence construction waste recycling. Huang et al. ^[7] pointed out that government takes a decisive role in directing and promoting construction waste recycling in China. Furthermore, Bakshan et al. ^[10] used Bayesian network analysis to investigate the causal behavioral determinants of practice improvement in construction waste management, and they concluded that proper supervision is critical in construction waste recycling systems. Lately, Fu et al. ^[11] further investigated the influence of the government’s supervision for waste recycling enterprises. Tam et al. ^[12] emphasized that the government’s incentives can encourage construction waste producers and waste recyclers to actively recycle construction waste. However, these studies almost discussed construction waste recycling from the standpoint of an interview and questionnaire survey, and there are no existing studies that focus on how different factors influence the behavior between government and recycling enterprises.

To address the above mentioned issues, Ma et al. ^[1] introduced a dynamic evolutionary game theory into the construction management system and investigated the effects of government incentive policies on the evolution process. The experiments show that: (1) government subsidies for waste enterprises are critical for construction waste recycling; (2) government subsidies for waste recyclers are not always necessary since the behavior of waste recyclers is influenced by the waste producers. Furthermore, increasing the landfill cost will encourage cooperation when the government does not provide a subsidy. In contrast, Long et al. ^[13] investigated the evolutionary game theory between construction waste producers and construction waste recyclers in the context of the government’s reward-penalty mechanism. However, it focuses primarily on the dynamic evolution process between different enterprises with and without government incentives, ignoring how the government influences the behavior of construction waste producers and waste recyclers during the evolution process. To this end, Su ^[2] stated that recycling construction waste is extremely beneficial for reducing environmental pollution and conserving resources, and the three-party evolutionary game theory is investigated, which included government agencies, construction waste producers, and construction waste recyclers. In particular, it was discovered that the government plays different roles during different construction waste recycling periods. Du et al. ^[14] presented a theoretical evolutionary game theory framework to analyze the behavior of governments, construction contractors, and the public. It first investigated the impact of various factors on stakeholders’ decision-making and discovered that incentives and penalties can reduce the illegal dumping of construction waste. To that end, this entry mainly investigated what is the best choice for penalties and incentives selection.

2. Three-Party Evolutionary Game Framework

The government agencies, waste recyclers, and waste producers are members of the construction waste recycling system. In this system, the government agency aims to increase the proportion of implementing construction waste recycling to realize and promote construction sustainability development. As for waste recyclers and producers, they try to maximize their interests. It is worth noting that if waste producers do not implement construction waste recycling, the construction waste will increase, which will further pollute the environment and lead to higher environmental management costs. Therefore, strategies from waste recyclers and producers play an essential role for the environment and eco-system, the more these two enterprises adopt waste recycling, the less pollution led by construction waste. Following Ref. [2], this work first introduces a more precise multi-agent evolutionary model by introducing environmental benefits and penalties for waste recyclers and producers, respectively. In particular, it is assumed that government is more prone to support waste recyclers than waste producers. Then the evolution behavior of three participants is analyzed during the procedure of construction waste recycling through the evolutionary game framework. The three-party game tree of government agencies, waste recyclers, and waste producers is shown in Figure 1.

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

Based on Figure 1, the following replicator dynamics equation is given:

$$F(x)=x(1-x)\left[y z\left(-G_{1}-F_1-F_2\right)-y S_{j}-z S_{s}+\left(G+G_{1}+F_1+F_2-C_{g}\right)\right]$$

$$F(y)=y(1-y)\left\{-x z S_j+x S_{j}+z\left[(1-\lambda) R-\left(C-\eta C_{1}\right)-P_{j}+\Delta C_{j}\right]+x z F_1-\Delta C_{j}\right\} $$

$$F(z)=z(1-z)\left[x\left(S_{s}+F_2\right)+y\left(\lambda R+C_{0}\right)-C_{0}-\eta C_{1}\right] $$

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:

$$F(x)=dx / dt = x\left[y z\left(-G_{1}-F_1-F_2\right)-y S_{j}-z S_{s}+\left(G+G_{1}+F_1+F_2-C_{g}\right)\right]$$

$$F(y)=dy / dt = y\left\{-x z S_j+x S_{j}+z\left[(1-\lambda) R-\left(C-\eta C_{1}\right)-P_{j}+\Delta C_{j}\right]+x z F_1-\Delta C_{j}\right\} $$

$$F(z)=dz / dt = z\left[x\left(S_{s}+F_2\right)+y\left(\lambda R+C_{0}\right)-C_{0}-\eta C_{1}\right] $$

3. Stochastic Evolutionary Game Framework

To the best of researchers' knowledge, there exist high uncertainty in the game among the government agencies, waste recyclers, and waste producers because of 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 entry, the replicator dynamic formula is combined with Gaussian white noise, which results in the multi-agent stochastic evolutionary game framework, as follows:

$$dx(t)= \left[y z\left(-G_{1}-F_1-F_2\right)-y S_{j}-z S_{s}+\left(G+G_{1}+F_1+F_2-C_{g}\right)\right]x(t)dt +\delta x(t)d\omega(t) $$

$$dy(t) = \left\{-x z S_j+x S_{j}+z\left[(1-\lambda) R-\left(C-\eta C_{1}\right)-P_{j}+\Delta C_{j}\right]+x z F_1-\Delta C_{j}\right\}y(t)dt+\delta y(t)d\omega(t) $$

$$dz(t) = \left[x\left(S_{s}+F_2\right)+y\left(\lambda R+C_{0}\right)-C_{0}-\eta C_{1}\right]z(t)dt+\delta z(t)d\omega(t) $$

4. The Effect of Noise Intensity

Figure 2 shows the results that how noise intensity affects the trajectory of the evolutionary game model. It can be observed that the uncertainty will bring random disturbance into the evolution process and then affect the evolution process. In addition, it also can be seen that the higher the noise intensity is, the more fluctuation exists in the evolutionary trajectories. This means the uncertainty can affect the strategy choice of the government agencies, waste recyclers, and waste producers.

Figure 2. Multi-agent dynamic evolutionary trajectories under different noise intensities

5. Conclusions

Facilitating the implementation of construction waste recycling is the primary basis to realize construction sustainability and it has a great practical significance for the quality improvement of construction waste recycling. In this entry, the three-party stochastic evolutionary game framework is proposed for construction waste recycling, making the payoff matrix and combining the Gaussian white noise with the replicator dynamic formula. Then the random Taylor expansion is used to solve the numerical approximation, and finally, the numerical simulations are conducted to study the dynamic evolution between the government agencies, waste recyclers, and waste producers. The main conclusions are as follows: (1) Smaller sorting costs make, the group strategy more stable and effective. (2) Larger disposal costs make waste producers do not implement construction waste recycling. (3) The more waste producers put into disposing of the construction waste enthusiasm the waste recyclers recycle construction waste. (4) Based on the comparative analysis of Gaussian white noise intensity, the effect of uncertainty external environments brings the random disturbance into the evolution trajectory of different participants, which leads to fluctuation of a smooth curve. To evade strategy fluctuation for different participants, it is necessary to let government agencies actively guide the waste producers and waste recyclers.

In a brief, this entry investigated the tripartite stochastic evolutionary game model for construction waste recycling policies analysis, filling the multi-agent stochastic game study of construction waste recycling and offering a practical basis for different agencies to implement construction waste recycling.