**1. Introduction**

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 our 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 [5–9] have already investigated its positive social, environmental, and sustainable influences and

**Citation:** Sun, Y.; Gu, Z. Implementation of Construction

Waste Recycling under Construction Sustainability Incentives: A Multi-Agent Stochastic Evolutionary Game Approach. *Sustainability* **2022**, *14*, 3702. https://doi.org/10.3390/ su14063702

Academic Editors: Carlos Morón Fernández and Daniel Ferrández Vega

Received: 7 February 2022 Accepted: 16 March 2022 Published: 21 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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' decisionmaking and discovered that incentives and penalties can reduce the illegal dumping of construction waste. To that end, this paper mainly investigated what is the best choice for penalties and incentives selection.

Many significant efforts have been made to use evolutionary game theory to investigate the impact of various factors on construction waste recycling, e.g., construction sustainability incentives, positive/negative government supervision, etc. However, in these existing works [1,2,13,14], the evolutionary game process analysis for construction waste recycling is based on a deterministic model that ignores the effect of external uncertainty. It is well understood that various random factors play an important role in decision-making between each participant during the evolution process, which should be taken into account in terms of construction waste recycling [15,16]. The purpose of this paper is to build a three-party stochastic game framework that can answer the following corresponding questions. (1) How should the three-party payoff matrix and replicator dynamic formula for the construction recycling system be defined? (2) Is there an equilibrium solution in the random replicator dynamic differential formula when Gaussian white noise is introduced? If so, what kinds of boundary conditions must be met?

To address the aforementioned issues, a three-party stochastic game framework is proposed for construction waste recycling based on bounded rationality theory, in which the payoff matrix is first constructed and then the replicator dynamic equation is formalized. In particular, the Lyapunov exponent diagram is employed to investigate the nonlinear dynamic characteristics of replicator dynamic equations based on the Benettin

method, and then Gaussian white noise is introduced into the *Ito*ˆ equation. The numerical approximations are then solved using the Taylor expansion method. Finally, a numerical simulation is run to demonstrate the dynamic evolutionary trajectory. In conclusion, the following contributions have been made:

