# Promoting Sponge City Construction through Rainwater Trading: An Evolutionary Game Theory-Based Analysis

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^{5}

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## Abstract

**:**

## 1. Introduction

^{3}of rainwater resources stored by Hunan High-tech Property Co., Ltd. in Changsha, Hunan Province, China at the price of 0.7 RMB/m

^{3}. After rainwater treatment, Hunan Yuchuang Environmental Protection Engineering Co., Ltd. transferred the rainwater resources to Changsha High-tech Zone Municipal Sanitation Co., Ltd. in Changsha, Hunan Province, China at the price of 3.85 RMB/m

^{3}(20% lower than the local water price) for landscaping and road cleaning, instead of using high-quality tap water. This was the country’s first rainwater trading. The transaction of the right to use rainwater resources in a sponge city creates a new model of intensive use of rainwater resources and the marketization of ecological value.

## 2. Evolutionary Game Theory and Key Stakeholders

#### 2.1. Applicability of the EGT

#### 2.2. Analysis of Interactions of Key Stakeholders

**Municipal enterprises (MEs):**Although the MEs in various regions are state-owned, they all depend on their revenue. MEs generally need a large amount of water, but the price of municipal water is generally higher than that of tap water. The price of urban water supply in China is classified. According to the nature of use, urban water can be divided into five categories: residential, industrial, administrative, business service and special water. The prices of all types of water are determined by the local people’s government and the administrative department of water supply in considering the actual situation. However, municipal water belongs to the category of non-residential water in all provinces and cities, the price of which is higher than that of tap water. Taking 10 cities in Jilin Province as an example (Appendix A), municipal water in each city is approximately 10–15% higher than the price of tap water. MEs are more willing to actively look for alternatives. Both government departments and MEs are willing to motivate EPEs through subsidies and other incentive measures to attract more EPEs to recycle rainwater, to promote the increase in rainwater trading volume (RWTV). However, subsidies cannot be blindly increased, which will lead to increased costs for MEs. Therefore, the government can consider whether some incentive policies are useful or to increase others.

**Environmental protection enterprises (EPEs):**Economic benefits are the ultimate pursuit of any enterprise [38]. Companies are free to decide whether to recycle and trade rainwater, which is not currently mandated by the government, even though rainwater recycling plays a crucial role in promoting SCCSs. Although some EPEs have begun trading rainwater, many are still dubious, fearing that the additional expense of treating stormwater may not be justified by the absence of long-term execution of the agreement. Therefore, many EPEs are willing to cooperate with MEs in the case of incentive policies. However, when incentive policies are insufficient, they will choose to cooperate with enterprises with lower water quality requirements.

#### 2.3. Evolutionary Game Modeling between EPEs and MEs

#### 2.3.1. Model Assumption

**Hypothesis**

**1**

**(H1):**

**Hypothesis**

**2**

**(H2):**

_{E}, tax T is paid to the government, and revenue R

_{E}is obtained. If MEs choose the negative strategy, EPEs cooperate with non-MEs and adopt a series of preferential policies to attract additional sales revenue E. If EPEs cooperate with MEs, additional costs will be incurred (recreational, ornamental landscape environmental water, green pouring of roads, and other high water quality requirements) $\mathsf{\delta}$ C

_{E}($\mathsf{\delta}$ > 0), in this case, the increased revenue is $\mathsf{\gamma}$ R

_{E}($\mathsf{\gamma}$ > 0), and financial subsidy S is received.

**Hypothesis**

**3**

**(H3):**

_{M}, and the gain is R

_{M}. If MEs choose the positive strategy, the additional cost is $\mathsf{\theta}$ C

_{M}($\mathsf{\theta}$ > 0), the increased-revenue is $\mathsf{\omega}$ R

_{M}($\mathsf{\omega}$ > 0), attracting more EPEs to join and increasing RWT revenue N. If EPEs choose to cooperate with non-MEs, MEs adopt positive strategies to introduce a series of preferential subsidy policies and win the recognition of superior government departments, which will lead to subsidy income $\mathsf{\mu}$T ($\mathsf{\mu}$ > 0).

#### 2.3.2. Parameters and Income Matrix

#### 2.3.3. Stakeholder Replication Dynamic Equation

_{11}represent the expected payoff of the EPEs if they cooperate with Mes, and let U

_{12}represent the expected payoff of the EPEs if they cooperate with non-Mes. U

_{1}represents the average expected payoff of the EPEs. U

_{11}, U

_{12}, and U

_{1}can be expressed as:

_{21}represents the expected payoff of the MEs if they choose positive strategies, and U

_{22}represents the expected payoff of the MEs if they choose negative strategies. U

_{2}represents the average expected payoff of the MEs. U

_{21}, U

_{22}, and U

_{2}can be expressed as:

#### 2.3.4. Equilibrium Point and Stability Analysis

- (i).
- Equilibrium point

_{1}(0, 0), E

_{2}(0, 1), E

_{3}(1, 0), and E

_{4}(1, 1) and a mixed strategy equilibrium point may exist in in system I: E

_{5}(x*, y*), where x* = $\frac{\mathsf{\theta}{\mathrm{C}}_{\mathrm{M}}-\mathsf{\mu}\mathrm{T}}{\mathrm{N}-\mathsf{\mu}\mathrm{T}}$ and y* = $\frac{\mathrm{E}+\mathsf{\delta}{\mathrm{C}}_{\mathrm{E}}-\mathsf{\gamma}{\mathrm{R}}_{\mathrm{E}}}{\mathrm{S}+\mathrm{T}+\mathrm{E}}$.

- (ii).
- Stability analysis of equilibrium point

_{1}(0, 0), E

_{2}(0, 1), E

_{4}(1, 1), E

_{3}(1, 0), and E

_{5}(x*, y*). For an unstable initial situation with a specific value of (x*, y*), the evolutionary trend shows a convergence from point E

_{5}(x*, y*) to E

_{1}(0, 0) or E

_{4}(1, 1). The trend is expressed by the arrows in Figure 3. The arrow represents the two evolutionary results of EPEs and MEs after a long-term game in the RWT. E

_{4}(1, 1) is an ideal evolutionary game stable strategy. It means that EPEs choose to cooperate with MEs for more profit, and MEs choose positive strategies to gain social reputation or recognition from superior departments. In this case, the two sides jointly promoted the construction and maintenance of sponge cities.

#### 2.4. ESS Analysis between EPEs and MEs

_{5}is not the ESS. Based on the above equations, Table 2 displays $\mathrm{D}\mathrm{e}\mathrm{t}\mathrm{J}$ and $\mathrm{T}\mathrm{r}\mathrm{J}$ for each equilibrium point.

_{1}(0, 0), E

_{2}(0, 1), E

_{3}(1, 0), and E

_{4}(1, 1). In scenarios 1–16, the first and second columns represent $\mathrm{D}\mathrm{e}\mathrm{t}\mathrm{J}$ and $\mathrm{T}\mathrm{r}\mathrm{J}$, respectively, while the third column represents the states at the four equilibrium points.

_{4}(willing to cooperate) to E

_{1}(unwilling to cooperate); when a < 0, the improvement in MEs reputation is less than the increased cost of choosing a positive strategy after EPEs choose to cooperate with non-MEs; when b < 0, the net profit value of EPEs cooperating with MEs is less than the increase in sales value attracted by cooperation with non-MEs; when c < 0, EPEs cooperate with MEs, and after MEs adopt the positive strategy, the benefit of attracting more EPEs to join is less than the cost of MEs choosing the positive strategy; when d < 0, it means that when EPEs choose to cooperate with the non-MEs, and the tax paid is less than the net cost of EPEs choosing to cooperate with MEs. To summarize, if EPEs gain more profit from cooperating with non-MEs than from cooperating with MEs and pay less taxes than the net cost of cooperating with MEs, then EPEs’ strategies will change from “Cooperating with MEs” to “Cooperating with Non-MEs”. However, when the income of MEs is less than the increased cost of the MEs’ positive strategies, the MEs’ strategies will change from “positive strategy” to “negative strategy”. The evolutionary game system converges to E

_{1}(0, 0). Conversely, in scenario 16, the initial strategy equilibrium point would converge at E

_{4}(1, 1). This is the expected outcome of the eventual evolution of the two parties.

_{1}(0, 0); In scenario 5, the ultimate point of equilibrium would converge at E

_{1}(1, 0); in scenario 9, the ultimate point of equilibrium would converge at E

_{1}(0, 1). To summarize, we concluded that if only one party gains when changing the strategy, the two parties will not reach cooperation in the long-term evolution process, and thus the ultimate point of equilibrium would converge at (0, 0); if the external benefits (subsidies, etc.) directly obtained by the EPEs and MEs are greater than the benefits generated by the cooperation between the two parties, no matter whether the other party chooses cooperation, the evolutional game path of the party with benefits tends to be 1.

_{2}(0, 1), suggesting that regardless of the strategy chosen by the EPEs, the benefits of the MEs will not be affected, so the result of the final evolutionary game of the MEs tends to 1; similarly, in scenario 6, the ultimate point of equilibrium would converge at E

_{3}(1, 0), and the final evolutionary game result of EPEs also tends to be 1.

_{4}(1, 1), and in scenario 14, the ultimate point of equilibrium would converge at E

_{3}(1, 0), suggesting that the strategy choice of EPEs in this evolutionary game tends toward cooperation with MEs, If the MEs choose the active strategy, the evolutionary game system converges to E

_{4}(1, 1), conversely, it converges toward E

_{3}(1, 0). Similarly, in scenario 12, the ultimate point of equilibrium would converge at E

_{4}(1, 1); in scenario 15, the ultimate point of equilibrium would converge at E

_{2}(0, 1), suggesting that the strategy choice of MEs in evolutionary game tends toward cooperation with MEs. If the MEs choose the active strategy, the evolutionary game system converges to E

_{4}(1, 1), suggesting that the MEs choose an active strategy. If the EPEs choose to cooperate with the MEs, the ultimate point of equilibrium would converge at E

_{4}(1, 1), otherwise the ultimate point of equilibrium would converge at E

_{2}(0, 1).

_{1}(0, 0) and E

_{4}(1, 1) are the two equilibrium points; in scenario 13, E

_{1}(1, 1) is an ideal evolutionary game stable strategy. The choice of system evolution strategy depends on the area of quadrangle ${\mathrm{S}}_{{\mathrm{E}}_{1}{\mathrm{E}}_{2}{\mathrm{E}}_{5}{\mathrm{E}}_{3}}$ and ${\mathrm{S}}_{{\mathrm{E}}_{4}{\mathrm{E}}_{2}{\mathrm{E}}_{5}{\mathrm{E}}_{3}}$; if ${\mathrm{S}}_{{\mathrm{E}}_{1}{\mathrm{E}}_{2}{\mathrm{E}}_{5}{\mathrm{E}}_{3}}>{\mathrm{S}}_{{\mathrm{E}}_{4}{\mathrm{E}}_{2}{\mathrm{E}}_{5}{\mathrm{E}}_{3}}$, the likelihood is higher that the ultimate point of equilibrium would converge at (0,0); if ${\mathrm{S}}_{{\mathrm{E}}_{1}{\mathrm{E}}_{2}{\mathrm{E}}_{5}{\mathrm{E}}_{3}}<{\mathrm{S}}_{{\mathrm{E}}_{4}{\mathrm{E}}_{2}{\mathrm{E}}_{5}{\mathrm{E}}_{3}}$, the likelihood is higher that the ultimate point of equilibrium would converge at (1, 1); if ${\mathrm{S}}_{{\mathrm{E}}_{1}{\mathrm{E}}_{2}{\mathrm{E}}_{5}{\mathrm{E}}_{3}}={\mathrm{S}}_{{\mathrm{E}}_{4}{\mathrm{E}}_{2}{\mathrm{E}}_{5}{\mathrm{E}}_{3}}$, the final equilibrium point has the same probability of converging at two points. Therefore, the factors influencing the final strategy of the evolutionary game are the parameters that determine the area of ${\mathrm{S}}_{{\mathrm{E}}_{4}{\mathrm{E}}_{2}{\mathrm{E}}_{5}{\mathrm{E}}_{3}}$.

## 3. Results and Discussion

#### 3.1. Related Data

#### 3.2. Simulation and Parameter Analysis

#### 3.2.1. Impact of S on Evolutionary Game

#### 3.2.2. Impact of T on Evolutionary Game

#### 3.2.3. Impact of E on Evolutionary Game

#### 3.2.4. Impact of N on Evolutionary Game

## 4. Conclusions and Policy Implications

- (1)
- EGT should be used for analyzing the strategy selection of EPEs and MEs to improve the implementation of RWT. The incentive policy influences the behavior and strategic choices of players. Therefore, an ESS between the EPEs and MEs with “Cooperate with MEs” and “Positive strategy” can be realized.
- (2)
- Sufficient financial subsidies, the reduction of additional sales, the increase in taxes, and the participation of more EPEs can accelerate the realization of ESS between EPEs and MEs. Therefore, relevant incentive measures must be formulated to promote a stronger interactive relationship between players on both sides and to develop the rainwater trading market, as well as the normalization and standardization of sponge cities.
- (3)
- The incentive policies should be considered not only from the perspective of enterprises and focus on economic means, but also from the perspective of the government to formulate relevant industry standards and regulations. Constraints and supervision by governments are equally important, which maximize the interests for all parties.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Parameters | |

C_{E} | cost of rainwater treatment |

R_{E} | return from sales to non-MEs |

C_{M} | cost of negative strategy |

R_{M} | return of negative strategy |

S | subsidies |

T | pay taxes to government departments |

E | additional sales revenue |

N | incremental benefits |

μ | MEs’ profit percentage |

γ | the percentage of increased revenue from EPEs cooperation with non-MEs |

δ | the percentage of increased cost of EPEs cooperation with non-MEs |

θ | the percentage of cost increase after MEs choose an active strategy |

ω | the percentage of revenue increase after MEs choose an active strategy |

Variables | |

x | the probability of EPEs cooperate with MEs |

y | the probability of MEs choosing to positive strategies |

Acronyms | |

EPE(s) | environmental protection enterprise(s) |

ME(s) | municipal enterprise(s) |

SCCS(s) | sponge city construction strategy (ies) |

ESS | evolutionarily stable strategy |

EGT | evolutionary game theory |

RWT | rainwater trading |

RWTV | rainwater trading volume |

RHUS(s) | rainwater harvesting and utilization system(s) |

## Appendix A. Municipal and Residential Water Prices in Jilin Province

District | Residential Water Price (yuan/m^{3}) | Municipal Water Price (yuan/m^{3}) |

Changchun | 4.55 | 4.85 |

Jilin | 2.6 | 4.75 |

Yanbian Korean Autonomous Prefecture | 3.2 | 6.4 |

Songyuan | 4.85 | 5.05 |

Siping | 3.7 | 5.8 |

Tonghua | 4.75 | 7.1 |

Baicheng | 3.65 | 6.4 |

Baishan | 3.45 | 5.4 |

Liaoyuan | 3.5 | 4.11 |

Note: Source: http://www.jl.gov.cn/ (accessed on 5 February 2023). |

## References

- Chan, F.K.S.; Griffiths, J.A.; Higgitt, D.; Xu, S.; Zhu, F.; Tang, Y.T.; Xu, Y.; Thorne, C.R. “Sponge city” in China—A breakthrough of planning and flood risk management in the urban context. Land Use Policy
**2018**, 76, 772–778. [Google Scholar] [CrossRef] - Valenca, R.; Le, H.; Zu, Y.; Dittrich, T.M.; Tsang, D.C.W.; Datta, R.; Sarkar, D.; Mohanty, S.K. Nitrate removal uncertainty in stormwater control measures: Is the design or climate a culprit? Water Res.
**2021**, 190, 116781. Available online: https://www.sciencedirect.com/science/article/pii/S0043135420313142 (accessed on 8 September 2022). [CrossRef] [PubMed] - Wang, Y.; Meng, F.; Liu, H.; Zhang, C.; Fu, G. Assessing catchment scale flood resilience of urban areas using a grid cell based metric. Water Res.
**2019**, 163, 114852. Available online: https://www.sciencedirect.com/science/article/pii/S0043135419306189 (accessed on 8 September 2022). [CrossRef] [PubMed] - Liu, H.; Jia, Y.; Niu, C. “Sponge city” concept helps solve China’s urban water problems. Environ. Earth Sci.
**2017**, 76, 1–5. [Google Scholar] [CrossRef] - Ding, L.; Ren, X.; Gu, R.; Che, Y. Implementation of the “sponge city” development plan in China: An evaluation of public willingness to pay for the life-cycle maintenance of its facilities. Cities
**2019**, 93, 13–30. [Google Scholar] [CrossRef] - Gao, J.; Li, J.; Li, Y.; Xia, J.; Lv, P. A distribution optimization method of typical lid facilities for sponge city construction. Ecohydrol. Hydrobiol.
**2021**, 21, 13–22. [Google Scholar] [CrossRef] - Yang, M.; Chen, D.; Shi, L.; He, J.; Liu, L.; Shang, X. How do construct a sponge city that can improve residents’ satisfaction? Evidence from a suburb of Huizhou city, China. Ecol. Indic.
**2022**, 142, 109238. [Google Scholar] [CrossRef] - Zhou, H.; Li, H.; Zhao, X.; Ding, Y. Emergy ecological model for sponge cities: A case study of China. J. Clean. Prod.
**2021**, 296, 126530. Available online: https://www.sciencedirect.com/science/article/pii/S0959652621007502 (accessed on 18 September 2022). [CrossRef] - Li, H.; Ding, L.; Ren, M.; Li, C.; Wang, H. Sponge city construction in China: A survey of the challenges and opportunities. Water
**2017**, 9, 594. [Google Scholar] [CrossRef] [Green Version] - Jia, H.; Wang, Z.; Zhen, X.; Clar, M.; Yu, S.L. China’s sponge city construction: A discussion on technical approaches. Front. Environ. Sci. Eng.
**2017**, 11, 1–11. [Google Scholar] [CrossRef] - Wang, H.; Mei, C.; Liu, J.; Shao, W. A new strategy for integrated urban water management in China: Sponge city. Sci. China Technol. Sci.
**2018**, 61, 317–329. [Google Scholar] [CrossRef] - Zhao, H.; Zhai, X.; Guo, L.; Yang, Y.; Li, J.; Ren, C.; Wang, K.; Liu, X.; Zhan, R.; Wang, K. Comparing protected cucumber and field cucumber production systems in China based on emergy analysis. J. Clean. Prod.
**2019**, 236, 117648. [Google Scholar] [CrossRef] - Liu, Y.; Li, G.; Zeng, P.; Zhang, X.; Tian, T.; Feng, H.; Che, Y. Challenge of rainwater harvesting in shanghai, China: A public psychological perspective. J. Environ. Manag.
**2022**, 318, 115584. Available online: https://www.ncbi.nlm.nih.gov/pubmed/35753131 (accessed on 1 September 2022). [CrossRef] [PubMed] - Gross, A.; Shmueli, O.; Ronen, Z.; Raveh, E. Recycled vertical flow constructed wetland (rvfcw)—A novel method of recycling greywater for irrigation in small communities and households. Chemosphere
**2007**, 66, 916–923. Available online: https://www.ncbi.nlm.nih.gov/pubmed/16844197 (accessed on 2 September 2022). [CrossRef] - Ding, A.; Wang, J.; Lin, D.; Zeng, R.; Yu, S.; Gan, Z.; Ren, N.; Li, G.; Liang, H. Effects of gac layer on the performance of gravity-driven membrane filtration (gdm) system for rainwater recycling. Chemosphere
**2018**, 191, 253–261. Available online: https://www.ncbi.nlm.nih.gov/pubmed/29035797 (accessed on 2 October 2022). [CrossRef] - Su, D.; Zhang, Q.H.; Ngo, H.H.; Dzakpasu, M.; Guo, W.S.; Wang, X.C. Development of a water cycle management approach to sponge city construction in Xi’an, China. Sci. Total Environ.
**2019**, 685, 490–496. Available online: https://www.ncbi.nlm.nih.gov/pubmed/31176234 (accessed on 19 October 2022). [CrossRef] - Schuetze, T. Rainwater harvesting and management—Policy and regulations in Germany. Water Supply
**2013**, 13, 376–385. [Google Scholar] [CrossRef] - Brown, R.R.; Keath, N.; Wong, T.H. Urban water management in cities: Historical, current and future regimes. Water Sci. Technol.
**2009**, 59, 847–855. Available online: https://www.ncbi.nlm.nih.gov/pubmed/19273883 (accessed on 4 October 2022). [CrossRef] [PubMed] - Ashley, R.; Lundy, L.; Ward, S.; Shaffer, P.; Walker, L.; Morgan, C.; Saul, A.; Wong, T.; Moore, S. Water-sensitive urban design: Opportunities for the UK. Proc. Inst. Civ. Eng. Munic. Eng.
**2013**, 166, 65–76. [Google Scholar] [CrossRef] - Amos, C.C.; Rahman, A.; Gathenya, J.M. Economic analysis of rainwater harvesting systems comparing developing and developed countries: A case study of Australia and Kenya. J. Clean. Prod.
**2018**, 172, 196–207. [Google Scholar] [CrossRef] - Ahiablame, L.M.; Engel, B.A.; Chaubey, I. Effectiveness of low impact development practices: Literature review and suggestions for future research. Water Air Soil Pollut.
**2012**, 223, 4253–4273. [Google Scholar] [CrossRef] - Qiao, X.-J.; Liao, K.-H.; Randrup, T.B. Sustainable stormwater management: A qualitative case study of the sponge cities initiative in China. Sustain. Cities Soc.
**2020**, 53, 101963. [Google Scholar] [CrossRef] - Ji, S.-F.; Zhao, D.; Luo, R.-J. Evolutionary game analysis on local governments and manufacturers’ behavioral strategies: Impact of phasing out subsidies for new energy vehicles. Energy
**2019**, 189, 116064. [Google Scholar] [CrossRef] - Dou, Y.; Sun, X.; Ji, A.; Wang, Y.; Xue, X. Development strategy for prefabricated construction projects: A tripartite evolutionary game based on prospect theory. Eng. Constr. Archit. Manag.
**2021**, 28, 51–71. [Google Scholar] [CrossRef] - Coninx, K.; Deconinck, G.; Holvoet, T. Who gets my flex? An evolutionary game theory analysis of flexibility market dynamics. Appl. Energy
**2018**, 218, 104–113. [Google Scholar] [CrossRef] - Eissa, R.; Eid, M.S.; Elbeltagi, E. Current applications of game theory in construction engineering and management research: A social network analysis approach. J. Constr. Eng. Manag.
**2021**, 147, 04021066. [Google Scholar] [CrossRef] - Chen, Z.; Wang, T. Photovoltaic subsidy withdrawal: An evolutionary game analysis of the impact on Chinese stakeholders’ strategic choices. Sol. Energy
**2022**, 241, 302–314. [Google Scholar] [CrossRef] - Tuyls, K.; Parsons, S. What evolutionary game theory tells us about multiagent learning. Artif. Intell.
**2007**, 171, 406–416. [Google Scholar] [CrossRef] [Green Version] - Zhang, L.; Chen, L.; Wu, Z.; Zhang, S.; Song, H. Investigating young consumers’ purchasing intention of green housing in China. Sustainability
**2018**, 10, 1044. [Google Scholar] [CrossRef] [Green Version] - Zhao, H.; Liu, X.; Wang, Y. Evolutionary game analysis of opportunistic behavior of sponge city ppp projects: A perceived value perspective. Sci. Rep.
**2022**, 12, 8798. Available online: https://www.ncbi.nlm.nih.gov/pubmed/35614166 (accessed on 7 October 2022). [CrossRef] - Chen, Y.; Chen, H. The collective strategies of key stakeholders in sponge city construction: A tripartite game analysis of governments, developers, and consumers. Water
**2020**, 12, 1087. [Google Scholar] [CrossRef] [Green Version] - Lv, J.; Lin, M.; Zhou, W.; Xu, M. How ppp renegotiation behaviors evolve with traffic changes: Evolutionary game approach. J. Constr. Eng. Manag.
**2021**, 147, 04021032. [Google Scholar] [CrossRef] - Jiang, S.; Wei, X.; Jia, J.; Ma, G. Toward sustaining the development of green residential buildings in China: A tripartite evolutionary game analysis. Build. Environ.
**2022**, 223, 109466. [Google Scholar] [CrossRef] - Fawcett, T.W.; Hamblin, S.; Giraldeau, L.A. Exposing the behavioral gambit: The evolution of learning and decision rules. Behav. Ecol.
**2012**, 24, 2–11. [Google Scholar] [CrossRef] - Wang, J.; Qin, Y.; Zhou, J. Incentive policies for prefabrication implementation of real estate enterprises: An evolutionary game theory-based analysis. Energy Policy
**2021**, 156, 112434. [Google Scholar] [CrossRef] - Antoci, A.; Borghesi, S.; Sodini, M. Water resource use and competition in an evolutionary model. Water Resour. Manag.
**2016**, 31, 2523–2543. [Google Scholar] [CrossRef] - Li, F.; Pan, B.; Wu, Y.; Shan, L. Application of game model for stakeholder management in construction of ecological corridors: A case study on Yangtze river basin in China. Habitat Int.
**2017**, 63, 113–121. [Google Scholar] [CrossRef] - Luo, L.-Z.; Mao, C.; Shen, L.-Y.; Li, Z.-D. Risk factors affecting practitioners’ attitudes toward the implementation of an industrialized building system. Eng. Constr. Archit. Manag.
**2015**, 22, 622–643. [Google Scholar] [CrossRef] - Sun, X.; Wang, W.; Pang, J.; Liu, X.; Zhang, M. Study on the evolutionary game of central government and local governments under central environmental supervision system. J. Clean. Prod.
**2021**, 296, 126574. [Google Scholar] [CrossRef] - Fisher-Vanden, K.; Olmstead, S. Moving pollution trading from air to water: Potential, problems, and prognosis. J. Econ. Perspect.
**2013**, 27, 147–172. [Google Scholar] [CrossRef] [Green Version] - Fan, K.; Hui, E.C.M. Evolutionary game theory analysis for understanding the decision-making mechanisms of governments and developers on green building incentives. Build. Environ.
**2020**, 179, 106972. [Google Scholar] [CrossRef] - Bai, Y.; Song, S.; Jiao, J.; Yang, R. The impacts of government r&d subsidies on green innovation: Evidence from Chinese energy-intensive firms. J. Clean. Prod.
**2019**, 233, 819–829. [Google Scholar] [CrossRef] - Chen, Y.; Zhu, D.; Zhou, L. A game theory analysis of promoting the spongy city construction at the building and community scale. Habitat Int.
**2019**, 86, 91–100. [Google Scholar] [CrossRef] - Quentin Grafton, R.; Horne, J.; Wheeler, S.A. On the marketisation of water: Evidence from the Murray-darling basin, Australia. Water Resour. Manag.
**2015**, 30, 913–926. [Google Scholar] [CrossRef] - Deng, J.; Li, C.; Wang, L.; Yu, S.; Zhang, X.; Wang, Z. The impact of water scarcity on Chinese inter-provincial virtual water trade. Sustain. Prod. Consum.
**2021**, 28, 1699–1707. [Google Scholar] [CrossRef] - Du, M.; Huang, C.; Chen, Z. Evaluating the water-saving and wastewater-reducing effects of water rights trading pilots: Evidence from a quasi-natural experiment. J. Environ. Manag.
**2022**, 319, 115706. Available online: https://www.ncbi.nlm.nih.gov/pubmed/35834845 (accessed on 6 October 2022). [CrossRef]

Municipal Enterprises | Environmental Protection Enterprises | |
---|---|---|

Cooperate with ME (x) | Cooperate with Non-ME (1 − x) | |

Positive (y) | $(\left(1+\mathsf{\gamma}\right){\mathrm{R}}_{\mathrm{E}}-\left(1+\mathsf{\delta}\right){\mathrm{C}}_{\mathrm{E}}+\mathrm{S}$$,\text{}\left(1+\mathsf{\omega}\right){\mathrm{R}}_{\mathrm{M}}-\left(1+\mathsf{\theta}\right){\mathrm{C}}_{\mathrm{M}}+\mathrm{N}$) | $(\left(1+\mathsf{\gamma}\right){\mathrm{R}}_{\mathrm{E}}-\left(1+\mathsf{\delta}\right){\mathrm{C}}_{\mathrm{E}}$$,\text{}\left(1+\mathsf{\omega}\right){\mathrm{R}}_{\mathrm{M}}-{\mathrm{C}}_{\mathrm{M}}$) |

Negative (1 − y) | $({\mathrm{R}}_{\mathrm{E}}-{\mathrm{C}}_{\mathrm{E}}-\mathrm{T}$$,\text{}{\mathrm{R}}_{\mathrm{M}}-\left(1+\mathsf{\theta}\right){\mathrm{C}}_{\mathrm{M}}+\mathsf{\mu}\mathrm{T}$) | $({\mathrm{R}}_{\mathrm{E}}-{\mathrm{C}}_{\mathrm{E}}+\mathrm{E}$$,\text{}{\mathrm{R}}_{\mathrm{M}}-{\mathrm{C}}_{\mathrm{M}}$) |

Equilibrium | Eigenvalue | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ |
---|---|---|---|

${\mathrm{E}}_{1}\left(0,0\right)$ | ${\mathsf{\lambda}}_{11}=\mathrm{a}$, ${\mathsf{\lambda}}_{12}=\mathrm{b}$ | ${\mathsf{\lambda}}_{11}+{\mathsf{\lambda}}_{12}$ | ${\mathsf{\lambda}}_{11}\cdot {\mathsf{\lambda}}_{12}$ |

${\mathrm{E}}_{2}\left(0,1\right)$ | ${\mathsf{\lambda}}_{21}=-\mathrm{a}$, ${\mathsf{\lambda}}_{22}=\mathrm{d}$ | ${\mathsf{\lambda}}_{21}+{\mathsf{\lambda}}_{22}$ | ${\mathsf{\lambda}}_{21}\cdot {\mathsf{\lambda}}_{22}$ |

${\mathrm{E}}_{3}\left(1,0\right)$ | ${\mathsf{\lambda}}_{31}=\mathrm{c}$, ${\mathsf{\lambda}}_{32}=-\mathrm{b}$ | ${\mathsf{\lambda}}_{31}+{\mathsf{\lambda}}_{32}$ | ${\mathsf{\lambda}}_{31}\cdot {\mathsf{\lambda}}_{32}$ |

${\mathrm{E}}_{4}\left(1,1\right)$ | ${\mathsf{\lambda}}_{41}=-\mathrm{c}$, ${\mathsf{\lambda}}_{42}=-\mathrm{d}$ | ${\mathsf{\lambda}}_{41}+{\mathsf{\lambda}}_{42}$ | ${\mathsf{\lambda}}_{41}\cdot {\mathsf{\lambda}}_{42}$ |

Scenario | a | b | c | d |
---|---|---|---|---|

Scenario1 | − | − | − | − |

Scenario2 | − | − | − | + |

Scenario3 | − | − | + | − |

Scenario4 | − | − | + | + |

Scenario5 | − | + | − | − |

Scenario6 | − | + | − | + |

Scenario7 | − | + | + | − |

Scenario8 | + | − | − | + |

Scenario9 | + | − | − | − |

Scenario10 | − | + | + | + |

Scenario11 | + | − | + | − |

Scenario12 | + | − | + | + |

Scenario13 | + | + | − | − |

Scenario14 | + | + | − | + |

Scenario15 | + | + | + | − |

Scenario16 | + | + | + | + |

Equilibrium | Scenario1 | Scenario2 | Scenario3 | Scenario4 | ||||||||

$\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | |

E_{1}(0, 0) | − | + | ESS | − | + | ESS | − | + | ESS | − | + | ESS |

E_{2}(0, 1) | ? | − | saddle | + | + | unstable | ? | − | saddle | + | + | unstable |

E_{3}(1, 0) | ? | − | saddle | ? | − | saddle | + | + | unstable | + | + | unstable |

E_{4}(1, 1) | + | + | unstable | ? | − | saddle | ? | − | saddle | − | + | ESS |

Equilibrium | Scenario5 | Scenario6 | Scenario7 | Scenario8 | ||||||||

$\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | |

E_{1}(0, 0) | ? | − | saddle | ? | − | saddle | ? | − | saddle | ? | − | saddle |

E_{2}(0, 1) | ? | − | saddle | + | + | unstable | ? | − | saddle | ? | − | saddle |

E_{3}(1, 0) | − | + | ESS | − | + | ESS | ? | − | saddle | ? | − | saddle |

E_{4}(1, 1) | + | + | unstable | ? | − | saddle | ? | − | saddle | ? | − | saddle |

Equilibrium | Scenario9 | Scenario10 | Scenario11 | Scenario12 | ||||||||

$\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | |

E_{1}(0, 0) | ? | − | saddle | ? | − | saddle | ? | − | saddle | ? | − | saddle |

E_{2}(0, 1) | − | + | ESS | + | + | unstable | − | + | ESS | ? | − | saddle |

E_{3}(1, 0) | ? | − | saddle | ? | − | saddle | + | + | unstable | + | + | unstable |

E_{4}(1, 1) | + | + | unstable | − | + | ESS | ? | − | saddle | − | + | ESS |

Equilibrium | Scenario13 | Scenario14 | Scenario15 | Scenario16 | ||||||||

$\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | $\mathbf{T}\mathbf{r}\mathbf{J}$ | $\mathbf{D}\mathbf{e}\mathbf{t}\mathbf{J}$ | state | |

E_{1}(0, 0) | + | + | unstable | + | + | unstable | + | + | unstable | + | + | unstable |

E_{2}(0, 1) | − | + | ESS | ? | − | saddle | − | + | ESS | − | saddle | |

E_{3}(1, 0) | − | + | ESS | − | + | ESS | ? | − | saddle | − | saddle | |

E_{4}(1, 1) | + | + | unstable | ? | − | saddle | ? | − | saddle | − | + | ESS |

Parameter | Initial Value | Parameter | Initial Value |
---|---|---|---|

x | 0.5 | ${\mathrm{C}}_{\mathrm{E}}$ | 3000 |

y | 0.5 | ${\mathrm{C}}_{\mathrm{M}}$ | 17,600 |

E | 3000 | $\mathsf{\gamma}$ | 0.2 |

S | 1200 | $\mathsf{\delta}$ | 0.2 |

T | 800 | $\mathsf{\theta}$ | 0.2 |

N | 7000 | $\mathsf{\omega}$ | 0.2 |

R_{E} | 10,000 | $\mathsf{\mu}$ | 0.2 |

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## Share and Cite

**MDPI and ACS Style**

Shi, C.; Miao, X.; Xu, T.; Gao, W.; Liu, G.; Li, S.; Lin, Y.; Wei, X.; Liu, H.
Promoting Sponge City Construction through Rainwater Trading: An Evolutionary Game Theory-Based Analysis. *Water* **2023**, *15*, 771.
https://doi.org/10.3390/w15040771

**AMA Style**

Shi C, Miao X, Xu T, Gao W, Liu G, Li S, Lin Y, Wei X, Liu H.
Promoting Sponge City Construction through Rainwater Trading: An Evolutionary Game Theory-Based Analysis. *Water*. 2023; 15(4):771.
https://doi.org/10.3390/w15040771

**Chicago/Turabian Style**

Shi, Chunyan, Xinyue Miao, Tongyu Xu, Weijun Gao, Gen Liu, Siwen Li, Yingzi Lin, Xindong Wei, and Hui Liu.
2023. "Promoting Sponge City Construction through Rainwater Trading: An Evolutionary Game Theory-Based Analysis" *Water* 15, no. 4: 771.
https://doi.org/10.3390/w15040771