Introduction

The United Nations defines sustainable development as “development that meets the needs of the present without compromising the ability of future generations to meet their own needs” in the Brundtland Report (United Nations, 2015). Since then, sustainable development has gradually become a global consensus. The 2030 Agenda for Sustainable Development established 17 immediate development goals and one is emphasized: “to build resilient infrastructure, promote inclusive and sustainable industrialization, and foster innovation” (United Nations, 2015). However, inadequate investment in infrastructure is a significant barrier to sustainable development in most developing countries within the Asia-Pacific region (United Nations, 2017). Various modes have been adopted to alleviate financial pressure to encourage private sector to invest in infrastructure projects, such as PPP (Public-Private Partnership) mode (Hodge and Greve, 2007). Nevertheless, private investors often aim to maximize their profits (Kumaraswamy and Zhang, 2001; Liu et al., 2018) and thus may sacrifice the environmental sustainability to pursue maximum profits (Keers and van Fenema, 2018). Infrastructure projects, especially waste-to-energy (WTE) incineration projects (Xu et al. 2015), water environment treatment projects (Xue and Wang, 2020; Xiong et al., 2022), and high-speed railway projects (Kang and Cao, 2022) can have a significant impact on the surrounding environment and often lead to strong reactions from nearby residents. Consequently, supervision over the environmental sustainability of infrastructure projects has become increasingly critical. Infrastructure projects often involve an extremely broad geographical scope—for instance, a comprehensive treatment project for river water may span multiple provinces. This poses significant challenges for government regulation. Due to information asymmetry in such projects, it is difficult for governments to implement round-the-clock and comprehensive monitoring, making the supervisory role of whistleblowers critically important.

The United Nations Convention against Corruption defines a whistleblower as an individual who, in good faith and with reasonable grounds, reports facts to the appropriate authorities concerning offenses established by the Convention (Peng, 2019). Internally, whistleblowers can report alleged wrongdoing to leaders within the organization, such as supervisors or higher-level officials (Ferrell et al., 2008). Externally, whistleblowers can expose information to the media, government agencies, law enforcement, and other relevant organizations. The reported wrongdoing can include violations of company policies or rules, laws, regulations, threats to the public interest or national security, fraud, and corruption (Near and Miceli, 1985). The Watergate Incident in the United States in 1972 led to the passage of the Whistleblower Protection Act by the US Congress in 1989. Relevant laws have been enacted in many other countries, including New Zealand, Canada, Britain, Australia, Ireland, South Africa, Japan, South Korea, Belgium, the Netherlands, Hungary, Israel, and Slovenia, to safeguard and promote whistleblowing. In 2019, China’s State Council issued the Guiding Opinions on Strengthening and Standardizing Interim and Ex-post Supervision, emphasizing the importance of public supervision, the establishment of a whistleblower system, and the protection and rewards for individuals reporting serious violations of laws, regulations, and major risks. Many local governments have formulated corresponding policies to motivate informants to report illegal pollution violations to the authorities. With the rapid development of the Internet and social media in recent years, whistleblowers can now more conveniently relay their reports to governments and the public through these new media channels. Severe environmental issues often attract significant attention from both authorities and citizens, sometimes even leading to public protests. In response, governments will typically investigate these matters and implement appropriate measures to prevent further environmental harm. Therefore, whistleblowers are essential in environmental protection (Iwasaki, 2024)

Prior studies directly related to whistleblowers are quite limited. These studies have primarily concentrated on business management and the laws of various countries (Jeong, 2015; Leclerc, 2023), with little research dedicated explicitly to whistleblowers in the environmental field. Research on infrastructure projects has mainly centered on public supervision without addressing whistleblowers. Many scholars have emphasized the importance of public participation in infrastructure projects and analyzed the impact of public opposition (He et al., 2021; Krajangsri and Pongpeng, 2017). Several scholars even pointed out that public participation is a critical success factor for infrastructure projects (Krajangsri and Pongpeng, 2017; He et al., 2021). In addition, some scholars incorporated public participation as an indicator within risk assessment frameworks for such projects (Xu et al., 2015; Liu et al., 2018). Whistleblowers are the first to detect and report irregularities with substantial evidence, making it known to the public and the government. The interests and risks faced by whistleblowers are more complex than those faced by the general public. Whistleblowers serve as a bridge among the private sector, the government and the public. Therefore, this paper is an important supplement to public supervision of infrastructure. Infrastructure projects often have long timelines and cover extensive geographical areas, making it difficult for the government to conduct comprehensive supervision alone. The oversight provided by whistleblowers can significantly enhance regulatory efficiency. Similarly, Iwasaki (2024) emphasized that whistleblowers serve as an essential complement to law enforcement in environmental protection.

Previous studies on environmental impacts of infrastructure projects have considered the influence of public participation, but there remain significant knowledge gaps in the following aspects. (1) Although whistleblower reports enable the public to identify project issues, no studies have yet examined the critical role of whistleblowers in addressing environmental concerns within infrastructure projects. (2) There is a lack of systematic analysis regarding the interrelationships between different factors influencing whistleblowers’ decision-making processes. (3) Existing literature fails to provide dynamic analysis and predictive modeling of whistleblower behavior. Rational whistleblowers may progressively adapt their choices over time through observational learning to identify optimal decision-making strategies. To our knowledge, this study makes the first academic exploration of whistleblower behavior in infrastructure projects, thereby addressing this critical research gap. This paper unveils the dynamic evolution of environmental sustainability in infrastructure projects from the perspective of the rationality of whistleblower by building a novel tripartite game model, which can predict the strategic evolution of environmental sustainability among the government, the private sector, and the whistleblower. It can provide evidence for the role of whistleblowers and assist the government in predicting the evolving environmental sustainability in infrastructure projects. The paper delves into the impact of main factors on the strategy adoption of the government, the private sector, and the rational whistleblower. It will aid the government in taking measures to monitor and regulate the behaviors of various stakeholders, enhancing the governance efficiency of infrastructure projects. Last, the paper conducts an in-depth sensitivity analysis of whistleblowers’ willingness to participate and their role in infrastructure projects. Based on these research findings, relevant management implications and policy recommendations are proposed to enhance the government’s governance efficiency in the environmental sustainability of infrastructure projects.

The paper is arranged as follows. Section “Literature review” presents literature review regarding theoretical background and methodology. Section “Methodology” constructs the tripartite game model and solves the model among the government, the private sector, and the whistleblower. Section “Numerical simulation analysis” analyzes the impact of important factors on strategy evolution in infrastructure projects by conducting a numerical simulation. Sections “Discussion”–”Conclusions and management insights” summarize this paper and puts forward policy recommendations.

Literature review

Environmental sustainability of infrastructure projects

As the global awareness of environmental protection increases, the importance of environmentally friendly and sustainable development in infrastructure projects is highly valued. Song et al. (2013) stated that errors in inappropriate project settings could worsen the surrounding environment of WTE projects, eliciting public opposition. Krajangsri and Pongpeng (2017) used structural equation modeling (SEM) to validate the impact of sustainable infrastructure assessments on the success of construction projects. Wu et al. (2018) contended that the annual generation of a significant quantity of municipal solid waste (MSW) posed a substantial impediment to substantially impedes China’s sustainable development. He et al. (2021) proposed that achieving health, safety, and environmental (HSE) goals in megaprojects should be valued highly because of increasing relevant issues.

Environmental sustainability is a crucial aspect of infrastructure projects. Many scholars stressed that environmental risk is a key factor in infrastructure projects. Xu et al. (2015) summarized the key risk factors of waste-to-energy PPP projects as insufficient waste supply, disposal of waste without proper licensing, environmental risks, payment issues, and the absence of necessary supporting infrastructure. Liu et al. (2018) investigated 35 WTE incineration plants in China and identified the most critical risks as public opposition, environmental pollution, government decision-making, and a defective legal and regulatory framework. Cui et al. (2020) posited that public opposition typically arises from the local residents’ risk perception of the harmful environmental and health impact.

Other scholars engaged in how to achieve the sustainability of infrastructure projects. Koo et al. (2009) developed a sustainability assessment model for underground infrastructure projects, and 10 qualitative indicators under the environmental aspect were included in the model. Chan et al. (2022) identified the driving factors for sustainable infrastructure development from technological, organizational, environmental, and financial aspects, evaluated their correlation, and ranked them based on their relative importance. The need for sustainability in infrastructure development is to address its negative environmental impact and harness the fantastic opportunity to foster sustainable socio-economic development through infrastructure projects. Koppa et al. (2023) emphasized the importance of continuously monitoring, evaluating, and reporting on the progress of infrastructure sustainability goals to enhance sustainable outcomes while minimizing negative environmental impacts.

In summary, practice and research have widely emphasized environmental sustainability and the potential for public opposition due to environmental issues. However, there remains a significant research gap regarding whistleblowers in infrastructure projects. Unlike the general public, whistleblowers, as the first to report irregularities with substantial evidence face more complex situations. Studies on whistleblowers related to environmental protection or infrastructure projects are very scarce and still need to be further expanded in the future.

Whistleblowing on environmental issues

Previous studies on whistleblowers have mainly focused on legal systems (Jeong, 2015; Leclerc, 2023), business management (Near and Miceli, 1985), and corruption issues (Kim and Min-Woo, 2009). Research on environmental whistleblowers is rare, and the main research on environmental whistleblowers is summarized as follows.

Kim and Min-Woo (2009) stated that many actions in society harm public interests, such as the production and distribution of harmful food and environmental pollution, and that it is crucial to establish and implement a whistleblower protection system for such actions. They suggested enacting a streamlined whistleblower protection law specifically for the private sector. Jeong (2015) stated that although the National Assembly of Korea has enacted the “Act on the Protection of Public Interest Whistleblowers” (2011) to comprehensively protect public health and safety, the environment, etc. Jeong strived to eliminate the ambiguity and uncertainty in the interpretation of the core concept of the act to facilitate the implementation of the act. Based on the latest revision of the French whistleblower law, following the transposition of Directive 2019/1937 on October 23, 2019, Leclerc (2023) explored whether this unified system adequately addressed the specific nature of whistleblowing related to public health and environmental risks.

Some scholars studied the effect of environmental whistleblower. Yang and Yang (2019) proposed whistleblowers can improve the air pollution control and that increasing rewards are effective strategies for motivating and enhancing whistleblowers’ supervisory efforts. Guo (2024) analyzed the crucial role played by Rachel Carson, the whistleblower, in studying the effects of dichlorodiphenyltrichloroethane (DDT) in the history of environmental protection. Iwasaki (2024) stressed that whistleblowers are an essential complement to law enforcement in environmental protection, and the law should not only protect whistleblowers but also actively reward their contributions. The public online complaint platforms provide a convenient way for whistleblowers to contact relevant government departments to make reports. To explore the impact of public report sentiment on environmental pollution, Li et al. (2024) gathered public report texts from the “Environmental Pollution Complaint Platform” between 2013 and 2020. By conducting sentiment analysis, they identified the emotions embedded within these complaints. Subsequently, they employed a double logarithmic model to empirically demonstrate that public report sentiment exerted a direct inhibitory influence on air pollution. Similarly, Leng et al. (2022) found that introducing an environmental report platform improved the air quality in Leshan City, enhanced public recognition of grassroots governments, and increased their life satisfaction.

Prior studies on whistleblowers in environmental protection mainly focus on the importance of whistleblowers for environmental protection and the necessity of rewarding them. However, there is a lack of research on the dynamic evolution of whistleblowers’ behavior in the game process within infrastructure projects. The private sector has an incentive to damage the environment in pursuit of high profits (Keers and van Fenema, 2018). Such behavior often changes dynamically as the private sector engages in a dynamic game with the government and whistleblowers, thereby evading government regulation. Therefore, it is essential to encourage whistleblower supervision and study the behavioral evolution of the government, private sector, and whistleblowers in the dynamic game within infrastructure projects.

Evolutionary game theory

Existing studies have proven that game theory is a powerful tool for addressing strategic interactions (Shan and Yang, 2019; Zhao and Bai, 2021) and behavioral decision-making among multiple participants (Du et al., 2020). Players in evolutionary game theory have bounded rationality (Taylor and Jonker, 1978; Schmidt, 2004; Yin and Zhao, 2024), a more realistic hypothesis. Evolutionary game theory draws on the essence of Darwinian species competition, aiming to analyze the evolution of strategy adoption (Weibull, 1995). Due to bounded rationality, players in evolutionary game theory may not immediately choose the optimal strategy. In this context, players can adapt their strategies (Gao et al., 2019). Furthermore, players in evolutionary games possess the capacity for continuous learning, enabling them to improve their strategies over time gradually (Fawcett et al., 2012; Wang et al., 2020; Ma and Zhang, 2020).

The evolutionary game theory has been applied in studies related to infrastructure projects. Shi et al. (2018) developed an evolutionary game to analyze the cooperative tendency and the evolution between multi-suppliers adopting prefabrication in major construction projects. Chen et al. (2019) studied the strategic interaction between governments and developers to promote sponge city construction at the building and community scale based on an evolutionary game theory. Xing et al. (2020) built an evolutionary game model on the governments’ and investors’ renegotiation strategies and studied the model’s evolutionary stability strategy. Xue and Wang (2020) constructed an evolutionary game model which considered a performance-based stepped payment to analyze the dynamic changes in the cooperative behaviors of stakeholders involved in China’s river water environment comprehensive treatment PPP projects. Song et al. (2021) developed an evolutionary game theory to analyze the main factors affecting cooperation behaviors in user-pay PPP projects. Chen and Chen (2023) established a tripartite evolutionary game model between the collusion body, government, and the public to study the evolution of collusion in mega projects. They found that the evolutionarily stable strategies differed under different cost and benefit conditions. Zhu et al. (2024) constructed an evolutionary game model considering the opportunistic behavior of the participants to analyze the cooperative strategies in low-carbon PPP projects. Xue et al. (2024) built an evolutionary game model to study the evolution of safety regulation strategies with different stakeholders in major infrastructure projects.

A comprehensive review of the literature from Sections “Environmental sustainability of infrastructure projects”, “Whistleblowing on environmental issues”, and “Evolutionary game theory” reveals that previous research primarily focuses on public participation in infrastructure projects and the role of whistleblowers in environmental issues. However, significant research gaps exist. One notable gap is the absence of studies examining whistleblower reports on environmental damage within infrastructure projects. Such reports are crucial as they enable the public and government to identify issues within projects and play a vital role in environmental supervision. Another critical gap is the lack of analysis regarding the dynamic characteristics of whistleblower behavior, which is influenced by a combination of policy, environmental, and individual factors. Existing literature fails to explore how these multifaceted influences collectively shape the dynamic evolution of whistleblowing behaviors. Infrastructure projects usually last for many years, and all parties may change their selected strategies over a long period (Robinson and Scott, 2009). This paper aims to address these research gaps by developing an evolutionary game model involving whistleblowers, the private sector, and the government. The model simulates the dynamic evolutionary process of strategic interactions within infrastructure projects, focusing on investigating the influencing factors and dynamic changes in whistleblowers’ decision-making through numerical simulations. Based on these analyses, the paper provides policy recommendations to enhance environmental governance in infrastructure development.

Methodology

Problem description

This paper will establish a tripartite evolutionary game model involving the private sector, the government, and the whistleblower to simulate the dynamic process of the tripartite decision-making on environmental sustainability in infrastructure projects. In this model, all stakeholders are bounded rational and cannot accurately know the strategic choices that other parties will make in advance. Therefore, they can’t make an optimal strategy at the beginning of the game. But over time, all parties can learn and adjust their decisions to improve their returns. To facilitate the study, the following hypotheses are made about the relevant strategies and other factors to be considered. In infrastructure projects, the private sector can adopt either a sustainable or an unsustainable strategy. When the private sector adopts the sustainable strategy, environmental harms in infrastructure projects, such as wastewater, waste gas, and waste residue, will be properly treated. Consequently, the project’s environmental impact is better than the unsustainable strategy’s. Similarly, the government can choose between two strategies: strong environmental sustainability promotion and weak environmental sustainability promotion. Whistleblowers are often driven by a strong sense of justice or a desire to protect the interests of the public or their own interests related to environmental pollution. However, whistleblowing may influence the whistleblower’s benefit, so the whistleblower needs to decide whether to supervise the infrastructure projects. The factors that a rational whistleblower perceives are modeled in the following hypotheses.

Model construction

The evolutionary game model focuses on the dynamic evolution of strategy adoption among different parties, measured by the changing trends in the proportion of different strategies adopted. The government’s strategy set consists of (strong promotion of sustainability, weak promotion of sustainability), with adoption probabilities of (x, 1-x). The private sector’s strategy set consists of (sustainable strategy, unsustainable strategy), with adoption probabilities of (y, 1-y). Likewise, the whistleblower’s strategy set comprises (supervision, non-supervision), with adoption probabilities of (z, 1-z).

Governments need to decide whether to take strong measures to promote environmental sustainability. Cost is an important factor influencing the strategy evolution of government in infrastructure projects (Chen and Chen, 2023). A strong environmental sustainability promotion strategy will generate a cost (\({C}_{11}\)) and a return (\({R}_{11}\)), while weak promotion strategies will generate a lower cost (\({C}_{12}\)) and a lower return (\({R}_{12}\)). If the government can obtain additional compensation for choosing a strong environmental sustainability promotion, the compensation amount is affected by the environmental effects achieved by the private sector in the infrastructure project. The additional compensations for sustainable and unsustainable strategies are \({R}_{13}\) and \({R}_{14}\), respectively. It is obvious that \({R}_{13} > {R}_{14}\). When environmental problems that have not been discovered by the government are exposed by the whistleblower, the government will suffer losses (\({L}_{11}\)), including reputational damage and punishment from higher authorities.

As for the private sector, choosing a sustainable strategy will generate a normal cost \({C}_{21}\). Cost is very important for the private sector’s strategies (Li et al., 2016; Shi and Zhang, 2022). The private sector choosing sustainable strategy will receive a normal return (\({R}_{21}\)) with a probability of \({P}_{21}\), or receive a higher return (\({R}_{21}+{R}_{22}\)) due to excellent environmental effects with a probability of \(1-{P}_{21}\), where \({R}_{22}\) denotes an additional reward. If the unsustainable strategy is adopted, there will be a lower cost (\({C}_{22}\)) and a lower return (\({R}_{23}\)). It is evident that \({C}_{22} < {C}_{21}\) and \({R}_{23} < {R}_{21}\). The probability of the government verifying a severe environmental problem is denoted as \({P}_{22}\). In such a case, the private sector may incur losses (\({L}_{21}\)), which include reputation damage, penalties, etc. (Cao et al., 2016).

Whistleblowers’ rationality mainly involves a careful evaluation of the potential outcomes of supervising infrastructure projects. Rational whistleblowers consider the impact that whistleblowing may have on their own lives, careers, and personal relationships. They weigh the benefits of reporting wrongdoing against the potential negative effects, such as retaliation, ostracization, or even prosecution. These factors determine whether a rational whistleblower supervises the project. For a whistleblower, supervising the project will incur a cost (\({C}_{31}\)), which includes expenses for evidence and other expenses. The private sector’s unsustainable strategy imposes losses (\({L}_{31}\)) on the whistleblower, including physical health damage and medical expenses. After discovering environmental pollution, the whistleblower decides whether to expose the incident, and the exposure probability is \({P}_{31}\). Under strong and weak government promotion strategies, the whistleblower’s losses caused by exposure are, respectively, \({L}_{32}\) and \({L}_{33}\), mainly including retaliation and other losses caused by information leakage. Laboratory experiments have proved that rewards can encourage whistleblowing (Butler et al., 2019; Mechtenberg et al., 2020). If the whistleblower exposes environmental pollution that the government has not discovered and it is then investigated and confirmed to be true by the government, the whistleblower can obtain an additional reward. Under strong and weak government promotion strategies, the rewards are, respectively, \({R}_{31}\) and \({R}_{32}\), including rewards for whistleblowing behavior and the return that the government stops further pollution and treats pollution. The government verifies the exposed pollution incidents with a probability of \({P}_{32}\). After supervising the environmental problems in the project under the government’s strong sustainability promotion, the whistleblower can receive an indirect return (\({R}_{33}\)), including reputation, opportunities, etc.

\({U}_{11}\) denotes the expected return of government choosing strong promotion and can be computed using Eq. (1).

$${U}_{11}=y({R}_{11}+{R}_{13})+(1-y)[{R}_{12}+{R}_{11}-z{P}_{31}{P}_{32}{L}_{11}(1-{P}_{22})]-{C}_{11}$$
(1)

\({U}_{12}\) denotes the expected return of government choosing weak promotion and can be computed using Eq. (2).

$${U}_{12}=(1-y)({R}_{12}-z{P}_{31}{P}_{32}{L}_{11})+y{R}_{11}-{C}_{12}$$
(2)

The average expected return of government can be computed using Eq. (3).

$$\overline{{U}_{1}}={U}_{11}x+{U}_{12}(1-x)$$
(3)

Based on Eqs. (1)–(3), the replicator dynamic equation for the government is expressed as Eq. (4).

$${F}_{1}(x)=\frac{dx}{dt}=x(1-x)[(1-y)({R}_{13}+z{L}_{11}{P}_{31}{P}_{32}{P}_{22})+y{R}_{14}+{C}_{11}-{C}_{12}]$$
(4)

\({U}_{21}\) denotes the expected return of the private sector choosing a sustainable strategy. \({U}_{21}\) can be obtained from Eq. (5).

$${U}_{21}=x{R}_{22}(1-{P}_{21})+{R}_{21}-{C}_{21}$$
(5)

\({U}_{22}\) denotes the expected return of the private sector choosing the unsustainable strategy, which can be obtained from Eq. (6).

$${U}_{22}={R}_{23}-{P}_{22}{L}_{21}x-z{L}_{21}{P}_{31}{P}_{32}(1-x{P}_{22})-{C}_{22}$$
(6)

The average expected return of the private sector can be computed using Eq. (7).

$$\overline{{U}_{2}}={U}_{21}y+{U}_{22}(1-y)$$
(7)

Based on Eqs. (5)–(7), the replicator dynamic equation for the private sector is expressed by Eq. (8) as follows:

$$\begin{array}{l}{F}_{2}(y)=\frac{dy}{dt}=y(1-y)\left[x{R}_{22}(1-{P}_{21})+x{P}_{22}{L}_{21}+z{L}_{21}{P}_{31}{P}_{32}(1-{P}_{22}x)\right.\\\qquad\quad\left.+{R}_{21}-{C}_{21}-{R}_{23}+{C}_{22}\right]\end{array}$$
(8)

\({U}_{31}\) denotes the expected return of whistleblower choosing supervision strategy. \({U}_{31}\) can be obtained from Eq. (9).

$$\begin{array}{l}{U}_{31}=-{C}_{31}-{L}_{31}(1-y){C}_{31}-{L}_{31}(1-y)(1-x{P}_{22})(1-{P}_{31}{P}_{32})\\\qquad\quad+x{R}_{31}{P}_{31}{P}_{32}(1-{P}_{22})(1-y)\\\qquad\quad \left.+(1-x)(1-y){P}_{31}{P}_{32}{R}_{32}\right]\\\qquad\quad+{P}_{31}(1-y)(-x{L}_{32}-{L}_{33}+x{L}_{33})+x{R}_{33}\end{array}$$
(9)

\({U}_{32}\) denotes the expected return of the whistleblower choosing the non-supervision strategy. \({U}_{32}\) can be obtained from Eq. (10).

$${U}_{32}=-x{L}_{31}(1-y)(1-{P}_{22})-{L}_{31}(1-x)(1-y)$$
(10)

The average expected return of the whistleblower can be computed using Eq. (11).

$$\overline{{U}_{3}}=y{U}_{31}+(1-y){U}_{32}$$
(11)

The replicator dynamic equation of the whistleblower supervising the project is expressed by Eq. (12), which determines the rate of change in the proportion of the whistleblower supervising the project.

$$\begin{array}{lll}{F}_{3}(z)=\frac{dz}{dt}=z(1-z)\left[{P}_{31}{P}_{32}{L}_{31}(1-{P}_{22}x)(1-y)\right.\\\qquad\qquad\qquad+x{R}_{31}{P}_{31}{P}_{32}(1-{P}_{22})(1-y)\\\qquad\qquad\qquad+{R}_{32}{P}_{31}{P}_{32}(1-x)(1-y)\\\qquad\qquad\qquad +{P}_{31}(1-y)(x{L}_{33}-x{L}_{32}-{L}_{33})\\\qquad\qquad\qquad\left.+x{R}_{33}-{C}_{31}\right]\end{array}$$
(12)

A three-dimensional dynamic system is constructed by incorporating the aforementioned calculation process. The dynamic system is represented by Eq. (13). To help readers understand the model, Table 1 lists all the parameters contained and their meanings.

$$\left\{\begin{array}{l}{F}_{1}(x)=x(1-x)[(1-y)({R}_{13}+z{L}_{11}{P}_{31}{P}_{32}{P}_{22})+y{R}_{14}+{C}_{11}-{C}_{12}]\\ {F}_{2}(y)=y(1-y)\left[x{R}_{22}(1-{P}_{21})+x{P}_{22}{L}_{21}+z{L}_{21}{P}_{31}{P}_{32}(1-{P}_{22}x)\right.\\\quad\quad\quad+\left.{R}_{21}-{C}_{21}-{R}_{23}+{C}_{22}\right]\\ {F}_{3}(z)=z(1-z)\left[{P}_{31}{P}_{32}{L}_{31}(1-{P}_{22}x)(1-y)+x{R}_{31}{P}_{31}{P}_{32}(1-{P}_{22})(1-y)\right.\\\quad\quad\quad+{R}_{32}{P}_{31}{P}_{32}(1-x)(1-y)\\\quad\quad\quad\left.+{P}_{31}(1-y)(x{L}_{33}-x{L}_{32}-{L}_{33})+x{R}_{33}-{C}_{31}\right]\end{array}\right.$$
(13)
Table 1 Parameters and their descriptions.
Full size table

Strategy stability analysis of different parties

When \({F}_{1}(x)\) is set equal to 0, it can be obtained either \(x=0\), \(x=1\), or \(y=\frac{-{C}_{11}+{C}_{12}-{R}_{13}-z{P}_{31}{P}_{32}{L}_{11}{P}_{22}}{{R}_{14}-{R}_{13}-z{P}_{31}{P}_{32}{L}_{11}{P}_{22}}\). \({y}^{\ast }\) denotes \(\frac{-{C}_{11}+{C}_{12}-{R}_{13}-z{P}_{31}{P}_{32}{L}_{11}{P}_{22}}{{R}_{14}-{R}_{13}-z{P}_{31}{P}_{32}{L}_{11}{P}_{22}}\). When \({F}_{1}(x)\equiv 0\), indicating that any value (\(0\le x\le 1\)) is at a stable state. The ESS is located on the shaded surface \(y={y}^{\ast }\) in Fig. 1a. It means if the probability of the private sector adopting strong promotion is equal to \({y}^{\ast }\), \(x\) (the probability of the government adopting strong promotion) will be stable.

Fig. 1: Strategy evolutionary diagram of government, private sector, and whistleblower.
figure 1

Note: The x-axis, y-axis, and z-axis correspond sequentially to the proportion of strong promotion strategy, sustainable strategy, and supervision strategy adopted by the government, the private sector, and the whistleblower. The shaded surfaces represent the positions of ESS when certain conditions are met as mentioned above, and the arrows represent the evolution direction of the whistleblower’s strategy. a depicts the strategy evolutionary diagram of the government. b depicts the strategy evolutionary diagram of the private sector. c depicts the strategy evolutionary diagram of the whistleblower.

Full size image

If \(y\ne {y}^{\ast }\), the stable point must satisfy the condition \({{F}_{1}}^{{\prime} }(x)=\frac{d{F}_{1}(x)}{dx} < 0\). When \(0 < {y}^{\ast } < y < 1\), \({{F}_{1}}^{{\prime} }(0) > 0\) and \({{F}_{1}}^{{\prime} }(1) < 0\), and therefore \(x=1\) is the ESS. It means that strong sustainability promotion will be adopted more frequently in the project. The evolution phase diagram is shown in region I of Fig. 1a. When \(0 < y < {y}^{\ast } < 1\), \({{F}_{1}}^{{\prime} }(0) < 0\) and \({{F}_{1}}^{{\prime} }(1) > 0\). In this case, \(x=0\) is the ESS and the government will tend to choose weak sustainability promotion in the long run. The evolution phase diagram is shown in region II of Fig. 1a.

When \({F}_{2}(y)\) is set equal to 0, we can obtain \(z=\frac{{C}_{21}-{R}_{21}-x{R}_{22}(1-{P}_{21})+{R}_{23}-{C}_{22}-x{L}_{21}{P}_{22}}{{P}_{31}{P}_{32}{L}_{21}(1-{P}_{22}x)}\), \(y=0\) or \(y=1\). \({z}^{\ast }\) denotes \(\frac{{C}_{21}-{R}_{21}-x{R}_{22}(1-{P}_{21})+{R}_{23}-{C}_{22}-x{L}_{21}{P}_{22}}{{P}_{31}{P}_{32}{L}_{21}(1-{P}_{22}x)}\). Let \(z={z}^{\ast }\), \({F}_{2}(y)\equiv 0\), indicating that any value of \(y\) (\(0\le y\le 1\)) is ESS located on the shaded surface in Fig. 1b. If the probability of whistleblower supervising the project is equal to \({z}^{\ast }\), \(y\) (the probability of private sector adopting sustainable strategy) will be in a stable state. When \(0 < {z}^{\ast } < z < 1\), \({{F}_{2}}^{{\prime} }(0)={\frac{d{F}_{2}(y)}{dy}|}_{y=0} > 0\) and \({{F}_{2}}^{{\prime} }(1) < 0\). In this case, \(y=1\) is the ESS, indicating that the strategy of the private sector will evolve towards the sustainable strategy in infrastructure projects. The evolution phase diagram is shown in the region I of Fig. 1b. When \(0 < z < {z}^{\ast } < 1\), \({{F}_{2}}^{{\prime} }(0) < 0\) and \({{F}_{2}}^{{\prime} }(1) > 0\). \(y=0\) is the ESS. The strategy of the private sector will evolve towards the unsustainable strategy. The strategy evolutionary diagram of the private sector is shown in region II of Fig. 1b.

Let \({F}_{3}(z)\) = 0, we can obtain \(x=\frac{-{C}_{31}+{P}_{31}{P}_{32}(1-y)({L}_{31}+{R}_{32})-{P}_{31}{L}_{33}(1-y)}{{P}_{31}{P}_{32}(1-y)[{L}_{31}{P}_{22}-{R}_{31}(1-{P}_{22})+{R}_{32}]+{P}_{31}(1-y)({L}_{32}-{L}_{33})-{R}_{33}}\), \(z=0\) or \(z=1\). \({x}^{\ast }\) denotes \(\frac{-{C}_{31}+{P}_{31}{P}_{32}(1-y)({L}_{31}+{R}_{32})-{P}_{31}{L}_{33}(1-y)}{{P}_{31}{P}_{32}(1-y)[{L}_{31}{P}_{22}-{R}_{31}(1-{P}_{22})+{R}_{32}]+{P}_{31}(1-y)({L}_{32}-{L}_{33})-{R}_{33}}\). If \(x={x}^{\ast }\), \({F}_{3}(z)\equiv 0\), meaning that \(x\) (\(0\le x\le 1\)) is in a stable state. The ESS is located on the shaded surface in Fig. 1c. When \(x > {x}^{\ast }\), \({{F}_{3}}^{{\prime} }(0)={\frac{d{F}_{3}(z)}{dz}|}_{z=0} < 0\) and \({{F}_{3}}^{{\prime} }(1) > 0\), so \(z=0\) is the ESS in this case. Non-supervision is the evolutionary stable strategy for the whistleblower in the project. The evolution phase diagram is shown in region I of Fig. 1c. When \(x < {x}^{\ast }\),\({{F}_{3}}^{{\prime} }(0) > 0\) and \({{F}_{3}}^{{\prime} }(1) < 0\). It means that \(z=1\) is the ESS. The strategy of the whistleblower will evolve towards supervising the project. The evolution phase diagram is shown in region I of Fig. 1c.

Model solution

After solving the replicator dynamic equations, it can be found that there are 13 local equilibrium points (LEPs) in this dynamic system, including 8 pure strategy equilibrium points and 5 mixed strategy equilibrium points. Friedman put forward that a stable point must be strictly at the Nash equilibrium of pure strategy (Friedman, 1998), and cannot be a Nash equilibrium of mixed strategy point (Wainwright and Hsu, 1989). Therefore, we mainly analyze 8 pure strategy equilibrium points (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0) and (1,1,1). According to Friedman’s theory (Friedman, 1991), the Jacobian matrix (as shown in Eq.(14)) can be used to solve the evolutionary equilibrium stability,

$$J=\left(\begin{array}{ccc}\frac{\partial {F}_{1}(x)}{\partial x} & \frac{\partial {F}_{1}(x)}{\partial y} & \frac{\partial {F}_{1}(x)}{\partial z}\\ \frac{\partial {F}_{2}(y)}{\partial x} & \frac{\partial {F}_{2}(y)}{\partial y} & \frac{\partial {F}_{2}(y)}{\partial z}\\ \frac{\partial {F}_{3}(z)}{\partial x} & \frac{\partial {F}_{3}(z)}{\partial y} & \frac{\partial {F}_{3}(z)}{\partial z}\end{array}\right)=\left(\begin{array}{ccc}{F}_{11} & {F}_{12} & {F}_{13}\\ {F}_{21} & {F}_{22} & {F}_{23}\\ {F}_{31} & {F}_{32} & {F}_{33}\end{array}\right)$$
(14)

where

$${F}_{11}=(1-2x)[{C}_{12}-{C}_{11}+y{R}_{13}+(1-y){R}_{14}+z(1-y){P}_{31}{P}_{32}{L}_{11}{P}_{22}]$$
(15)
$${F}_{12}=x(1-x)({R}_{13}-{R}_{14}-z{P}_{31}{P}_{32}{L}_{11}{P}_{22})$$
(16)
$${F}_{13}=x(1-x)(1-y){P}_{31}{P}_{32}{L}_{11}{P}_{22}$$
(17)
$${F}_{21}=y(1-y)[{R}_{22}(1-{P}_{21})+{L}_{21}{P}_{22}-z{P}_{31}{P}_{31}{L}_{21}{P}_{22}]$$
(18)
$$\begin{array}{l}{F}_{22}=(1-2y)\left[-{C}_{21}+{R}_{21}+x{R}_{22}(1-{P}_{21})-{R}_{23}\right.\\\quad\quad\quad\left.+{C}_{22}+x{L}_{21}{P}_{22}+z{P}_{31}{P}_{31}{L}_{21}(1-{P}_{22}x)\right]\end{array}$$
(19)
$${F}_{23}=y{P}_{31}{P}_{31}{L}_{21}(1-y)(1-{P}_{22}x)$$
(20)
$$\begin{array}{l}{F}_{31}=z(1-z)\left[-{P}_{22}{L}_{31}{P}_{31}{P}_{32}(1-y)+{P}_{31}{P}_{32}{R}_{31}(1-{P}_{22})\right.\\\qquad\quad(1-y)-(1-y){P}_{31}{P}_{32}{R}_{32}\\\qquad\quad\left.+{P}_{31}(1-y)({L}_{32}-{L}_{33})+{R}_{33}\right]\end{array}$$
(21)
$$\begin{array}{l}{F}_{32}=-z(1-z)\left[{P}_{31}{P}_{32}{L}_{31}(1-{P}_{22}x)\right.\\\quad\quad\quad+{P}_{31}{P}_{32}{R}_{31}x(1-{P}_{22})\\\quad\quad\quad+(1-x){P}_{31}{P}_{32}{R}_{32}\\\quad\quad\quad\left.+{P}_{31}(x{L}_{32}+{L}_{33}-x{L}_{33})\right]\end{array}$$
(22)
$$\begin{array}{l}{F}_{33}=(1-2z)\left[-C{}_{31}+{L}_{31}{P}_{31}{P}_{32}(1-y)(1-{P}_{22}x)\right.\\\qquad\quad+{P}_{31}{P}_{32}{R}_{31}x(1-{P}_{22})(1-y)\\\qquad\quad+(1-x)(1-y){P}_{31}{P}_{32}{R}_{32}\\\qquad\quad\left.+{P}_{31}(1-y)(x{L}_{32}+{L}_{33}-x{L}_{33})+x{R}_{33}\right].\end{array}$$
(23)

At the local equilibrium point (0, 0, 0),

$${J}_{1}=\left(\begin{array}{ccc}{C}_{12}-{C}_{11}+{R}_{14} & 0 & 0\\ 0 & -{C}_{21}+{R}_{21}-{R}_{23}+{C}_{22} & 0\\ 0 & 0 & -{C}_{31}+{L}_{31}{P}_{31}{P}_{32}+{P}_{31}{P}_{32}{R}_{32}+{P}_{31}{L}_{33}\end{array}\right).$$
(24)

At the local equilibrium point (0, 0, 1),

$${J}_{2}=\left(\begin{array}{ccc}{C}_{12}-{C}_{11}+{R}_{14}+{P}_{31}{P}_{32}{L}_{11}{P}_{22} & 0 & 0\\ 0 & -{C}_{21}+{R}_{21}{-}_{23}+{C}_{22}+{P}_{31}{P}_{22}{L}_{21} & 0\\ 0 & 0 & {C}_{31}-{L}_{31}{P}_{31}{P}_{32}-{P}_{1}{P}_{2}{R}_{32}-{P}_{31}{L}_{33}\end{array}\right).$$
(25)

At the local equilibrium point (0, 1, 0),

$${J}_{3}=\left(\begin{array}{ccc}{C}_{12}-{C}_{11}+{R}_{13} & 0 & 0\\ 0 & {C}_{21}-{R}_{21}+{R}_{23}-{C}_{22} & 0\\ 0 & 0 & -{C}_{31}\end{array}\right).$$
(26)

At the local equilibrium point (0, 1, 1),

$${J}_{4}=\left(\begin{array}{ccc}{C}_{12}-{C}_{11}+{R}_{13} & 0 & 0\\ 0 & {C}_{21}-{R}_{21}+{R}_{23}-{C}_{22}-{P}_{31}{P}_{32}{L}_{21} & 0\\ 0 & 0 & {C}_{31}\end{array}\right).$$
(27)

At the local equilibrium point (1, 0, 0),

$${J}_{5}=\left(\begin{array}{ccc}-{C}_{12}+{C}_{11}-{R}_{14} & 0 & 0\\ 0 & -{C}_{21}+{R}_{21}+{R}_{22}(1-{P}_{21})-{R}_{23}+{C}_{22}+{L}_{21}{P}_{22} & 0\\ 0 & 0 & -{C}_{31}+{L}_{31}{P}_{31}{P}_{32}(1-{P}_{22})+{P}_{31}{P}_{32}{R}_{31}(1-{P}_{22})+{P}_{31}{L}_{32}+{R}_{33}\end{array}\right).$$
(28)

At the local equilibrium point (1, 0, 1),

$${J}_{6}=\left(\begin{array}{ccc}-{C}_{12}+{C}_{11}-{R}_{14}-{P}_{31}{P}_{32}{L}_{11}{P}_{22} & 0 & 0\\ 0 & -{C}_{21}+{R}_{21}+{R}_{22}(1-{P}_{21})-{R}_{23}+{C}_{22}+{L}_{21}{P}_{22}+{P}_{31}{P}_{32}{L}_{21}(1-{P}_{22}) & 0\\ 0 & 0 & {C}_{31}-{L}_{31}{P}_{31}{P}_{32}(1-{P}_{22})-{P}_{31}{P}_{32}{R}_{31}(1-{P}_{22})-{P}_{31}{L}_{32}-{R}_{33}\end{array}\right).$$
(29)

At the local equilibrium point (1, 1, 0),

$${J}_{7}=\left(\begin{array}{ccc}-{C}_{12}+{C}_{11}-{R}_{13} & 0 & 0\\ 0 & {C}_{21}-{R}_{21}-{R}_{22}(1-{P}_{21})+{R}_{23}-{C}_{22}-{L}_{21}{P}_{22} & 0\\ 0 & 0 & -{C}_{31}+{R}_{33}\end{array}\right).$$
(30)

At the local equilibrium point (1, 1, 1),

$${J}_{8}=\left(\begin{array}{ccc}-{C}_{12}+{C}_{11}-{R}_{13} & 0 & 0\\ 0 & {C}_{21}-{R}_{21}-{R}_{22}(1-{P}_{21})+{R}_{23}-{C}_{22}-{L}_{21}{P}_{22}-{P}_{31}{P}_{32}{L}_{21}(1-{P}_{22}) & 0\\ 0 & 0 & {C}_{31}-{R}_{33}\end{array}\right).$$
(31)

Then, the eigenvalues of the Jacobian matrix for each LEP of pure strategy can be obtained from the above result. According to Lyapunov’s stability conditions (Lyapunov, 1992), the LEP, of which all eigenvalues are smaller than 0, is an asymptotically stable point. We can analyze the evolutionary stability of different strategies. The eigenvalues of 8 pure strategy equilibrium points are listed in Table 2.

Table 2 Eigenvalues of each LEP.
Full size table

As can be seen from Table 2, ESS points change under different conditions. By dividing different scenarios based on ESS points, there are 5 scenarios in total. The scenario in which different projects are situated may differ significantly. By inputting the data of their respective projects into the following expression, both the government and the private sector can ascertain the scenario of the project, thereby enabling predictions of the evolutionary trends in behavior for the government, private sector, and whistleblowers.

Scenario 1: When \({C}_{12}-{C}_{11}+{R}_{14} < 0\), \(-{C}_{21}+{R}_{21}-{R}_{23}+{C}_{22} < 0\), \(-{C}_{31}+{L}_{31}{P}_{31}{P}_{32}+{P}_{31}{P}_{32}{R}_{32}+{P}_{31}{L}_{33} < 0\), \(-{C}_{12}+{C}_{11}-{R}_{13} < 0\), \({C}_{21}-{R}_{21}-{R}_{22}(1-{P}_{21})+{R}_{23}-{C}_{22}-{L}_{21}{P}_{22} < 0\) and \(-{C}_{31}+{R}_{33} < 0\), there are two ESS points (0, 0, 0) and (1, 1, 0), namely (weak promotion, unsustainable strategy, non-supervision) and (strong promotion, sustainable strategy, non-supervision). Whether the strategies of the three parties ultimately evolve towards (0, 0, 0) or (1, 1, 0) depends on the initial state (\({x}_{0}\), \({y}_{0}\), \({z}_{0}\)). When \({x}_{0}\) or \({y}_{0}\) is small, the system evolves towards (0, 0, 0). As \({x}_{0}\) and \({y}_{0}\) increase, the system gradually evolves towards (1, 1, 1).

Scenario 2: When conditions \({C}_{12}-{C}_{11}+{R}_{14} < 0\), \(-{C}_{21}+{R}_{21}-{R}_{23}+{C}_{22} < 0\) and \(-{C}_{31}+{L}_{31}{P}_{31}{P}_{32}+{P}_{31}{P}_{32}{R}_{32}+{P}_{31}{L}_{33} < 0\) are met simultaneously, and neither \(-{C}_{12}+{C}_{11}-{R}_{13} < 0\) nor \({C}_{21}-{R}_{21}-{R}_{22}(1-{P}_{21})+{R}_{23}-{C}_{22}-{L}_{21}{P}_{22} < 0\) is met, there is only one ESS point (0,0,0), namely (weak promotion, unsustainable strategy, non-supervision). In the long run, it is the worst scenario for both the government and the public, and the environmental quality will suffer significantly.

Scenario 3: When \({C}_{12}-{C}_{11}+{R}_{13} < 0\), \(-{C}_{21}+{R}_{21}-{R}_{23}+{C}_{22} > 0\), there is only one ESS point (0, 1, 0), namely (weak promotion, sustainable strategy, non-supervision). This scenario requires the government to pay extremely high fees or reward to the private sector, which is difficult to afford in reality given the current tight fiscal budgets.

Scenario 4: When conditions \(-{C}_{12}+{C}_{11}-{R}_{13} < 0\), \({C}_{21}-{R}_{21}-{R}_{22}(1-{P}_{21})+{R}_{23}-{C}_{22}-{L}_{21}{P}_{22} < 0\) and \(-{C}_{31}+{R}_{33} < 0\) are met simultaneously, and conditions \({C}_{12}-{C}_{11}+{R}_{14} < 0\), \(-{C}_{21}+{R}_{21}-{R}_{23}+{C}_{22} < 0\) and \(-{C}_{31}+{L}_{31}{P}_{31}{P}_{32}+{P}_{31}{P}_{32}{R}_{32}+{P}_{31}{L}_{33} < 0\) are not met at the same time, there is only one ESS point (1, 1, 0), namely (strong promotion, sustainable strategy, non-supervision). This scenario requires the government to identify environmental issues in projects with a high degree of accuracy and keep the regulatory costs within an acceptable range. Therefore, it is challenging to meet these conditions in practice.

Scenario 5: When conditions \(-{C}_{12}+{C}_{11}-{R}_{13} < 0\), \({C}_{21}-{R}_{21}-{R}_{22}(1-{P}_{21})+{R}_{23}-{C}_{22}-{L}_{21}{P}_{22}-{P}_{31}{P}_{32}{L}_{21}(1-{P}_{22}) < 0\) and \(-{C}_{31}+{R}_{33} > 0\) are met simultaneously while conditions \({C}_{12}-{C}_{11}+{R}_{14} < 0\), \(-{C}_{21}+{R}_{21}-{R}_{23}+{C}_{22} < 0\) and \(-{C}_{31}+{L}_{31}{P}_{31}{P}_{32}+{P}_{31}{P}_{32}{R}_{32}+{P}_{31}{L}_{33} < 0\) are not met at the same time, there is only one ESS point (1, 1, 1), namely (strong promotion, sustainable strategy, supervision).

Numerical simulation analysis

In this section, MATLAB R2023a is utilized to conduct a numerical simulation. By conducting sensitivity analysis through parameter changes and the dynamic evolution of the resulting tripartite strategies, it can identify factors that have a significant impact on environmental sustainability and analyze their influencing mechanisms. The values of parameters combine online data, expert interviews and reasonable calculations.

Impact of \({R}_{13}\) and \({C}_{11}\) on the government and the private sector

Set \({R}_{14}\) = 200, \({C}_{12}\) = 240, \({C}_{11}\) = 460, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({L}_{21}\) = 280, \({P}_{22}\) = 0.6, \({C}_{21}\) = 6180, \({P}_{21}\) = 0.7, \({R}_{21}\) = 6700, \({R}_{22}\) = 60, \({P}_{31}\) = 0.8, \({P}_{32}\) = 0.6, \({L}_{11}\) = 150, \({C}_{31}\) = 8, \({L}_{31}\) = 10, \({R}_{31}\) = 4, \({R}_{32}\) = 1, \({R}_{33}\) = 10, \({L}_{32}\) = 1, \({L}_{33}\) = 3 and initial point (0.5, 0.5). Figure 2a depicts the evolutionary trajectories for the value of \({R}_{13}\) set at 210, 230, and 250, respectively. \({R}_{13}\) denotes the reward granted to the government when the private sector adopts the sustainable strategy and the government implements strong promotion. The strategy of the government evolves towards the weak environmental sustainability promotion strategy when \({R}_{13}\)=210. The strategy of the government evolves towards the strong environmental sustainability promotion strategy when \({R}_{13}\) increases to 230 and 250. The finding means that the government’s motivation to promote environmental sustainability in the infrastructure project is enhanced when it can achieve higher rewards associated with the project’s environmental quality.

Fig. 2: Influence of R13 and C11.
figure 2

a depicts the evolutionary trajectories for different values of R13. b depicts the evolutionary trajectories for different values of C11. The x-axis, y-axis, and z-axis correspond sequentially to the proportions of strong promotion strategy, sustainable strategy, and supervision strategy adopted by the government, the private sector, and the whistleblower.

Full size image

Set \({R}_{14}\) = 200, \({R}_{13}\) = 230, \({C}_{12}\) = 240, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({L}_{21}\) = 280, \({P}_{22}\) = 0.6, \({C}_{21}\) = 6180, \({P}_{21}\) = 0.7, \({R}_{21}\) = 6700, \({R}_{22}\) = 60, \({P}_{31}\) = 0.8, \({P}_{32}\) = 0.6, \({L}_{11}\) = 150, \({C}_{31}\) = 8, \({L}_{31}\) = 10, \({R}_{31}\) = 4, \({R}_{32}\) = 1, \({R}_{33}\) = 10, \({L}_{32}\) = 1, \({L}_{33}\) = 3 and initial point (0.5, 0.5). Figure 2b depicts the evolutionary trajectories for the value of \({C}_{11}\) set as 440, 460 and 480, respectively. The strategy of the government evolves towards strong environmental sustainability promotion when \({C}_{11}\) = 440 and 460. It evolves towards weak environmental sustainability promotion when \({C}_{11}\) increases to 480. \({C}_{11}\) denotes the cost incurred by the government when it adopts the strong promotion of environmental sustainability in infrastructure projects. As the cost associated with strong promotion of environmental sustainability rises sharply, the likelihood of the government maintaining a strong commitment to environmental sustainability in these projects decreases.

Impact of \({P}_{21}\) and \({L}_{21}\) on the private sector

\({R}_{14}\) = 180, \({R}_{13}\) = 220, \({C}_{12}\) = 240, \({C}_{11}\) = 450, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({L}_{21}\) = 270, \({P}_{22}\) = 0.5, \({C}_{21}\) = 6180, \({R}_{21}\) = 6700, \({R}_{22}\) = 50, \({P}_{31}\) = 0.5, \({P}_{32}\) = 0.5, \({L}_{11}\) = 150, \({C}_{31}\) = 8, \({L}_{31}\) = 10, \({R}_{31}\) = 4, \({R}_{32}\) = 1, \({R}_{33}\) = 5, \({L}_{32}\) = 1, \({L}_{33}\) = 5 and initial point (0.5, 0.5). Figure 3a depicts the evolutionary trajectories for the value of \({P}_{21}\) set at 0.3, 0.6, and 0.9, respectively. \({P}_{21}\) is the probability that the private sector will receive a normal return without additional reward when choosing the sustainable strategy. The strategy of the private sector evolves towards the sustainable strategy when \({P}_{21}\)=0.3 and 0.6, and towards the unsustainable strategy when \({P}_{21}\) increases to 0.9. It means that when the private sector faces difficulties in achieving the high-level performance required for incentives or rewards, their willingness to adopt the sustainable strategy within the infrastructure project tends to decline.

Fig. 3: Influence of P21 and L21.
figure 3

a depicts the evolutionary trajectories for different values of P21. b depicts the evolutionary trajectories for different values of L21. The x-axis, y-axis, and z-axis correspond sequentially to the proportions of strong promotion strategy, sustainable strategy, and supervision strategy adopted by the government, the private sector, and the whistleblower.

Full size image

Set \({R}_{14}\) = 190, \({R}_{13}\) = 220, \({C}_{12}\) = 240, \({C}_{11}\) = 450, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({P}_{22}\) = 0.5, \({C}_{21}\) = 6180, \({P}_{21}\) = 0.7, \({R}_{21}\) = 6700, \({R}_{22}\) = 50, \({P}_{31}\) = 0.5, \({P}_{32}\) = 0.5, \({L}_{11}\) = 150, \({C}_{31}\) = 8, \({L}_{31}\) = 10, \({R}_{31}\) = 4, \({R}_{32}\) = 1, \({R}_{33}\) = 5, \({L}_{32}\) = 1, \({L}_{33}\) = 5 and initial point (0.5, 0.5). Figure 3b depicts the evolutionary trajectories for the value of \({L}_{21}\) set at 230, 250 and 270, respectively. \({L}_{21}\) is the total loss that the private sector possibly faces after the government confirms the environmental problems in the infrastructure project. Figure 3b shows that the private sector’s strategy evolves towards the unsustainable strategy when \({L}_{21}\)=230 and 250, and towards the sustainable strategy when \({L}_{21}\) increases to 270. This shift towards the sustainable strategy proves the necessity for the government to implement stringent punishments, thereby ensuring that the private sector increasingly adopts environmentally friendly measures.

Impact of \({L}_{31}\), \({R}_{31}\) and \({C}_{31}\) on whistleblower and private sector

The rationality of the whistleblowers enables them to identify the potential risks, benefits, and associated costs that may result from engaging in whistleblowing. Rational thinking allows whistleblowers to weigh the harm of staying silent against the potential benefits of blowing the whistle.

Set \({R}_{14}\) = 200, \({R}_{13}\) = 230, \({C}_{12}\) = 240, \({C}_{11}\) = 460, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({L}_{21}\) = 300, \({P}_{22}\) = 0.5, \({C}_{21}\) = 6180, \({P}_{21}\) = 0.7, \({R}_{21}\) = 6600, \({R}_{22}\) = 60, \({P}_{31}\) = 0.7, \({P}_{32}\) = 0.7, \({L}_{11}\) = 130, \({C}_{31}\) = 8, \({R}_{31}\) = 10, \({R}_{32}\) = 2, \({R}_{33}\) = 8, \({L}_{32}\) = 1, \({L}_{33}\) = 4 and initial point (0.5, 0.5). Figure 4a depicts the evolutionary trajectories for the value of \({L}_{31}\) set at 10, 30, and 50, respectively. \({L}_{31}\) denotes the losses incurred by whistleblowers as a result of the private sector adopting the unsustainable strategy. In infrastructure projects, if the unsustainable strategy is implemented, it can lead to severe environmental degradation. When the whistleblower perceives these pollution-related issues as posing a significant threat to health and well-being (including air pollution, water pollution, and soil pollution), the losses become significant. Such losses can range from physical health problems to psychological stress and even financial losses due to the need for medical treatment or relocation. Figure 4a shows that the ESS changes from (0,0,0) to (1, 1, 1). This indicates that when losses are higher, the whistleblower is more inclined to supervise the project. Consequently, relevant event may be reported to the government or the public, drawing their attention to the environmental issues caused by the project. The government needs to act to prevent further deterioration of the environment, resulting in various measures being implemented to penalize the private sector.

Fig. 4: Influence of L31, R31, and C31.
figure 4

a–c, respectively, depict the evolutionary trajectories for different values of L31, R31, or C31. a depicts the evolutionary trajectories for different values of L31. b depicts the evolutionary trajectories for different values of R31. c depicts the evolutionary trajectories for different values of C31. The x-axis, y-axis, and z-axis correspond sequentially to the proportions of strong promotion strategy, sustainable strategy, and supervision strategy adopted by the government, the private sector, and the whistleblower, respectively.

Full size image

Set \({R}_{14}\) = 200, \({R}_{13}\) = 230, \({C}_{12}\) = 240, \({C}_{11}\) = 460, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({L}_{21}\) = 300, \({P}_{22}\) = 0.5, \({C}_{21}\) = 6180, \({P}_{21}\) = 0.7, \({R}_{21}\) = 6600, \({R}_{22}\) = 60, \({P}_{31}\) = 0.7, \({P}_{32}\) = 0.7, \({L}_{11}\) = 130, \({C}_{31}\) = 8, \({L}_{31}\) = 12, \({R}_{32}\) = 2, \({R}_{33}\) = 8, \({L}_{32}\) = 1, \({L}_{33}\) = 3 and initial point (0.5, 0.5). Figure 4b depicts the evolutionary trajectories for the value of \({R}_{31}\) set at 10, 20 and 30, respectively. \({R}_{31}\) is defined as the reward granted to the whistleblower for reporting a significant environmental issue discovered within the project, provided that the government verifies the authenticity of the reported concern under strong sustainability promotion. It can be seen from Fig. 4b that the strategy of whistleblower evolves toward non-supervision when \({R}_{31}\)=10 and 20, and towards supervision strategy when \({R}_{31}\) increases to 30. When whistleblowers are offered higher incentives for supervising projects, they are more inclined to actively engage in supervising infrastructure projects. It proves that reward can be an effective means to motivate whistleblowers to participate (Mechtenberg et al. 2020).

Set \({R}_{14}\) = 200, \({R}_{13}\) = 230, \({C}_{12}\) = 240, \({C}_{11}\) = 460, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({L}_{21}\) = 300, \({P}_{22}\) = 0.5, \({C}_{21}\) = 6180, \({P}_{21}\) = 0.7, \({R}_{21}\) = 6600, \({R}_{22}\) = 60, \({P}_{31}\) = 0.7, \({P}_{32}\) = 0.7, \({L}_{11}\) = 130, \({C}_{31}\) = 8, \({L}_{31}\) = 12, \({R}_{31}\) = 10, \({R}_{32}\) = 2, \({R}_{33}\) = 8, \({L}_{32}\) = 1, \({L}_{33}\) = 3 and initial point (0.5, 0.5). Figure 4c depicts the evolutionary trajectories for the value of \({C}_{31}\) set at 5, 7, and 9, respectively. The cost of supervision strategy, denoted as \({C}_{31}\), incurred by whistleblower encompasses expenses related to the evidence, including environmental testing and other associated costs. The ESS evolves towards (strong promotion, sustainable strategy, supervision) when \({C}_{31}\)=5, and towards (weak promotion, unsustainable strategy, non-supervision) when \({C}_{31}\) increases to 7 and 9. When the cost of supervision is high, it will be more difficult for the whistleblower to supervise the infrastructure project. This can prevent the whistleblower from obtaining sufficient evidence and channels to report environmental pollution issues.

Impact of \({L}_{32}\), \({P}_{32}\), and \({R}_{33}\) on whistleblower and private sector

Set \({R}_{14}\) = 210, \({R}_{13}\) = 230, \({C}_{12}\) = 240, \({C}_{11}\) = 460, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({L}_{21}\) = 300, \({P}_{22}\) = 0.6, \({C}_{21}\) = 6180, \({P}_{21}\) = 0.7, \({R}_{21}\) = 6600, \({R}_{22}\) = 60, \({P}_{31}\) = 0.2, \({P}_{32}\) = 0.2, \({L}_{11}\) = 150, \({C}_{31}\) = 8, \({L}_{31}\) = 10, \({R}_{31}\) = 4, \({R}_{32}\) = 1, \({R}_{33}\) = 10, \({L}_{33}\) = 5 and initial point (0.5, 0.5). Figure 5a depicts the evolutionary strategy results of the whistleblower and the private sector when \({L}_{32}\) is respectively set as 0, 3 and 6. \({L}_{32}\) denotes the potential losses that the whistleblower may face after exposing environmental events under the government’s strong sustainability promotion. The whistleblower’s report often has a significant impact on the reputation of the private sector, potentially leading to interested parties of the project being investigated and penalized by superior government agencies. The whistleblower may face retaliation as a result, and may even bear other losses resulting from the leakage of personal information. Figure 5a depicts the ESS changes from (strong promotion, sustainable strategy, supervision) to (weak promotion, unsustainable strategy, non-supervision) because of the increase in \({L}_{32}\). This demonstrates that when the whistleblower faces high risks and losses, the enthusiasm for supervision may be dampened, which can result in a higher frequency of the private sector employing unsustainable strategies in the projects.

Fig. 5: Influence of L32, P32 and R33.
figure 5

a depicts the evolutionary trajectories for different values of L32. b depicts the evolutionary trajectories for different values of P32. c depicts the evolutionary trajectories for different values of R33. The x-axis, y-axis, and z-axis correspond sequentially to the proportions of strong promotion strategy, sustainable strategy, and supervision strategy adopted by the government, the private sector, and the whistleblower, respectively.

Full size image

Set \({R}_{14}\) = 210, \({R}_{13}\) = 230, \({C}_{12}\) = 240, \({C}_{11}\) = 460, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({L}_{21}\) = 300, \({P}_{22}\) = 0.6, \({C}_{21}\) = 6180, \({P}_{21}\) = 0.7, \({R}_{21}\) = 6600, \({R}_{22}\) = 60, \({P}_{31}\) = 0.2, \({L}_{11}\) = 150, \({C}_{31}\) = 8, \({L}_{31}\) = 10, \({R}_{31}\) = 4, \({R}_{32}\) = 1, \({R}_{33}\) = 10, \({L}_{32}\) = 1, \({L}_{33}\) = 5 and initial point (0.5, 0.5). Figure 5b depicts the evolutionary trajectories for the value of \({P}_{32}\) set at 0.2, 0.5 and 0.8, respectively. \({P}_{32}\) denotes the probability that a pollution problem brought to light by the whistleblower will be verified as true and subsequently tackled by the government, following a thorough investigation. Upon encountering media reports on environmental issues on social platforms, authorities must decide how to address the issues. Figure 5b demonstrates that the possibility of the whistleblower supervising the infrastructure project grows due to an increase in \({P}_{32}\). Similarly, Fig. 5b illustrates that the possibility of the private investor adopting the sustainable strategy becomes larger in the infrastructure project due to an increase in \({P}_{32}\). The effectiveness of the government’s response to whistleblower reports significantly impacts the strategic evolution of both whistleblowers and the private sector.

Set \({R}_{14}\) = 220, \({R}_{13}\) = 230, \({C}_{12}\) = 245, \({C}_{11}\) = 460, \({R}_{23}\) = 6500, \({C}_{22}\) = 5900, \({L}_{21}\) = 300, \({P}_{22}\) = 0.5, \({C}_{21}\) = 6180, \({P}_{21}\) = 0.5, \({R}_{21}\) = 6600, \({R}_{22}\) = 60, \({P}_{31}\) = 0.2, \({P}_{32}\) = 0.8, \({L}_{11}\) = 150, \({C}_{31}\) = 8, \({L}_{31}\) = 10, \({R}_{31}\) = 4, \({R}_{32}\) = 1, \({L}_{32}\) = 1, \({L}_{33}\) = 5 and initial point (0.5, 0.5). Figure 5c depicts the evolutionary trajectories for the value of \({R}_{33}\) set at 0, 5 and 10, respectively. \({R}_{33}\) denotes the indirect income of the whistleblower resulting from the supervision of the infrastructure project. When \({R}_{33}\) increases from 0 to 10, the evolution result of whistleblower changes from non-supervision to supervision strategy, and the evolution result of private sector changes from unsustainable strategy to sustainable strategy. This demonstrates that the indirect income of the whistleblower can also motivate the willingness to supervise. This underscores the importance of indirect returns and emphasizes that society can significantly incentivize whistleblowers through such means. In situations where there is a strong societal concern for the environment, the whistleblower has the potential to obtain significant indirect returns from supervising the project.

Discussion

As proved in Section “Numerical simulation analysis”, under different conditions, strategies of the government, the private sector, and the whistleblower evolve towards different ESS points. The government should play a leading role in promoting sustainable development (Yuan et al., 2019). The government may adopt a weak promotion strategy due to insufficient funds or human resources. It is found here that reward and cost are important for the government to choose the optimal strategy (Chen and Chen, 2023). It is recommended to adopt cost-effective new digital technologies, which can improve supervision efficiency without causing a significant increase in costs. Meanwhile, it is essential to ensure that the adoption of these innovative technologies complies with the relevant regulations on confidentiality and patents (Balsa-Barreiro et al., 2023). Efficient government supervision is a crucial factor in determining the success of infrastructure projects (Cao et al., 2015). In order to protect the public interest, the government should develop appropriate regulation mechanisms to regulate different parties’ behavior in the infrastructure project (Shi and Zhang, 2022).

The private sector may adopt opportunism to pursue high profits, hurting project quality (Cao et al., 2015). If the private sector chooses an unsustainable strategy in infrastructure projects, the environmental quality of the project will be poor. Except for the return and cost (Xiong and Zhang, 2016), the study finds that penalty, reputation (Cao et al., 2016), and the strategy of the government are important influencing factors for the private sector’s strategy evolution. Our study finds that the uncertainty of return and the role of whistleblower also significantly affect the private sector’s strategy evolution. When the sustainable strategy can achieve good returns with a high possibility and the losses of the unsustainability strategy are high, the private sector will be inclined to adopt the sustainable strategy. On the contrary, if the performance standard is difficult to achieve, stricter supervision is necessary (Shi and Zhang, 2022).

The role of whistleblowers in supervising infrastructure projects is examined in Section “Numerical simulation analysis”. Bounded rationality allows whistleblowers to gradually learn what strategies are optimal for themselves, thereby adjusting their strategies. When whistleblowers perceive greater losses from environmental damage incidents, their motivation to monitor projects intensifies. Rewards (Iwasaki, 2024; Yang and Yang, 2019) and indirect incomes can provide whistleblowers with gains from both government and society, thereby enhancing their willingness to report environmental problems. Efficient verification of whistleblower reports by the government ensures timely resolution of environmental issues, boosting public enthusiasm for reporting. Conversely, high supervision costs and the losses arising from reporting (Jeong, 2015) diminish whistleblowers’ willingness to report. These challenges indicate difficulties in evidence collection, lack of proper reporting channels, and potential high risks of retaliation, all undermining whistleblowers’ inclination to monitor and report environmental issues. The whistleblower’s exposure often causes strong opposition from the public, which is an essential event in the infrastructure projects (Zhang and Tariq, 2020; Xiong et al., 2022). For example, a public protestation in constructing the Hong Kong-Zhuhai-Macao Bridge (Xiong et al., 2017).

Whistleblowers play an essential role in the supervision of environmental activities (Iwasaki, 2024). To encourage whistleblowing, the government must create conditions that are accessible, friendly, and safe, including the implementation of online report platforms and the utilization of new media (Leng et al., 2022, Li et al., 2024). Enacting relevant policies to encourage whistleblowers and protect their legitimate rights and interests (Jeong, 2015) will be an effective measure for the government to enhance whistleblowers’ willingness to supervise infrastructure projects. Meanwhile, the government should promptly investigate and verify the evidence reported by whistleblowers. It is necessary to establish a blacklist of whistleblowers within the government. Whistleblowers who maliciously spread rumors should be added to the blacklist after verification by relevant government departments.

Conclusions and management insights

The paper builds a novel evolutionary game model to explore the dynamic evolution of environmental sustainability strategy among the government, the private sector, and the rational whistleblower. The whistleblower’s supervision mechanism in the model is highly consistent with the reality and policies issued by the Chinese government. A numerical simulation is then conducted to analyze the factors influencing supervision willingness and the role of the whistleblower in infrastructure projects.

The theoretical contributions of this paper are as follows. First, it fills the research gap concerning whistleblowers in the infrastructure sector and complements existing studies on environmental sustainability and public participation in infrastructure projects. While some scholars have emphasized the critical role of public participation in infrastructure projects (Li et al., 2016; Wang et al., 2020; Chen and Chen, 2023; Wu et al., 2019; Shen and Zhou, 2022), and public opposition often significantly influences infrastructure project decision-making (Shen and Zhou, 2022), whistleblowers serve as a bridge connecting the public, government, and private sectors. The evidence they report can assist the public and government uncover previously undetected issues. However, previous studies have not examined whistleblowers in infrastructure projects. Second, this paper comprehensively considers the interplay of multiple factors influencing whistleblowers’ decision-making processes and the dynamic nature of their choices. The role of whistleblowers fundamentally differs from that of the general public, with associated risks and benefits being highly complex. Rational whistleblowers must weigh various factors when deciding whether to report misconduct. The model developed in this study not only incorporates influencing factors such as reward (Iwasaki, 2024; Yang and Yang, 2019), supervision costs (Leng et al., 2022; Li et al., 2024), losses (Jeong, 2015), discovery probability, government efficiency, and indirect income but also enables in-depth analysis of relationships between these factors. This allows prediction of behavioral trends among whistleblowers and their resultant impacts on projects.

Based on the research findings, the following management implications and policy recommendations are proposed.

First, given the government’s pivotal role in promoting sustainable development (Yuan et al., 2019), it is recommended that a performance tracking system for government agencies and officials be established. This system should record key indicators such as the number of green projects promoted, environmental complaint resolution rates, and compliance with sustainability standards. Departments and officials making outstanding contributions should be provided with material rewards or spiritual incentives.

Second, governments should create a tiered reward mechanism for whistleblowers. Rewards could scale with the severity of issues reported and the environmental/social benefits achieved post-investigation. Relevant laws must be established to mandate anonymous reporting channels and digital identity encryption systems. Penalties for privacy breaches, retaliatory actions, or harassment against whistleblowers or their families should be severely escalated, including criminal liability and corporate blacklisting.

Third, public awareness campaigns should be enhanced to educate citizens on the critical role of environmental protection and whistleblowers in sustainable development. Integrate environmental citizenship education into school curricula to cultivate respect for whistleblowers. Additionally, legal safeguards should be introduced to ensure whistleblowers face no employment discrimination and provide job placement assistance to them.

Fourth, the government should pay close attention to the progress of new technology research and development and the distinctive features of different emerging technologies (Balsa-Barreiro et al., 2023), while increasing R&D investment in this field. This strategic approach not only fosters national innovation capacity but also enhances the environmental sustainability of infrastructure projects.

The limitations of this study are as follows. Firstly, due to words constraints, only a sensitivity analysis of the impact of changes in certain key parameters on the evolution of strategic choices could be conducted. The analysis of other parameters’ influences necessitates simulation by readers or project managers based on their specific needs. Secondly, different projects exhibit distinct characteristics, and determining the specific scenario a project falls into requires inputting project data into the model, thereby predicting the evolutionary trends of environmental sustainability. Future research plans to employ a multi-case comparative approach to deeply analyze the characteristics of parameter values across diverse projects, particularly focusing on the risks whistleblowers face in different projects.