An Evolutionary Game and Simulation Study of Work Safety Governance and Its Impact on Long-Term Sustainability Under the Supervisory System

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper addresses the complexity of work safety governance for high-risk enterprises under China's administrative supervision system by constructing a tripartite evolutionary game model involving the central government, local government, and high-risk enterprises. It systematically analyzes the strategy selection mechanisms and evolutionary paths of each entity under different initial strategy probabilities and policy tool intensities. Through dynamic evolutionary equations, the paper investigates the factors influencing equilibrium conditions and uses MATLAB numerical simulations to verify the effects of supervision intensity, fine intensity, and subsidy intensity on system stability and convergence speed. It summarizes the current supervision system and strategies to promote compliant production among high-risk enterprises.

By constructing a tripartite evolutionary game model, this paper, for the first time, comprehensively portrays the dynamic game relationship between the central government, local government, and enterprises, providing a new perspective for theoretical research on work safety governance mechanisms.

1.The first sentence in the abstract (Insufficient supervision intensity leads to local government non-regulation, which can be addressed by increasing supervision intensity) is hard to understand and requires refinement.

 2.“Insufficient supervision intensity leads to local government non-regulation, which can be addressed by increasing supervision intensity”，It presents an obvious conclusion and the wording could be revised for clarity.

3.Propositions 1-6 merely describe the stability conditions and the outcomes of parameter changes, but fail to discuss their contributions to existing research or their policy implications. Appropriate additions in this regard can be made.

4.The transition from the theoretical analysis of the three-party game to the numerical simulation analysis lacks sufficient explanatory text. There is a lack of clear explanation on how the basis for parameter setting is derived from the theoretical model, as well as how the simulation analysis specifically verifies the theoretical conclusions.

5.In Section 5.2.1, it mentions that "when the probability of the central government initially choosing strict supervision increases, high-risk enterprises reach a stable state faster," but it does not specifically explain the causal mechanism behind this phenomenon.

6.In Chapter 5, the impacts of different parameters are analyzed independently, but there is a lack of discussion on the possible interactions between these parameters.

7.The discussion on the practical feasibility of the ideal state is inadequate. The ideal evolutionary stable strategy E4(0,1,1) describes a state of non-strict regulation (by the central government), local regulation, and corporate compliance, but it does not discuss whether this state is feasible in reality or whether it requires specific policy conditions for support.

8.The structure of the research conclusions and policy recommendations appears lengthy and lacks clear hierarchical division. The research conclusion section and the policy recommendation section are mixed together without a clear distinction between theoretical findings and policy implications.

9.The explanation of key concepts is insufficient, with key terms such as "reasonable range of supervision intensity" and "appropriate reward and punishment intensity" lacking specific definitions, which can easily lead to generalized policy recommendations. Adding qualitative or quantitative explanations for these concepts would be beneficial.

Comments on the Quality of English Language

It is suggested to further refine the English expression.

Author Response

Comments 1: The first sentence in the abstract (Insufficient supervision intensity leads to local government non-regulation, which can be addressed by increasing supervision intensity) is hard to understand and requires refinement.

Response 1: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the fourth point in the abstract to be more specific and clear. This change can be found on page 1, abstract, and line 4.

"(4) Insufficient supervision intensity from the central government leads to local government non-regulation, and although this can be addressed by increasing supervision intensity, excessive supervision reduces the system's evolution speed toward ideal states."

Explanation of the Change:

1. Location of Change: The change is located in the abstract, the fourth point specifically.

Comments 2: “Insufficient supervision intensity leads to local government non-regulation, which can be addressed by increasing supervision intensity”，It presents an obvious conclusion and the wording could be revised for clarity.

Response 2: Thank you for this comment. I agree that the original phrasing was too direct. Therefore, I have revised the fourth point in the abstract to be more specific and clear. This change can be found on page 1, abstract, and line 4.

"(4) Insufficient supervision intensity from the central government leads to local government non-regulation, and although this can be addressed by increasing supervision intensity, excessive supervision reduces the system's evolution speed toward ideal states."

Explanation of the Change:

1. Location of Change: The change is located in the abstract, the fourth point specifically.

Comments 3: Propositions 1-6 merely describe the stability conditions and the outcomes of parameter changes, but fail to discuss their contributions to existing research or their policy implications. Appropriate additions in this regard can be made.

Response 3: Thank you for this important suggestion. I agree that the propositions needed further discussion of their implications. Therefore, I have added a section after each proposition to discuss its contribution to existing research and its policy implications. These additions can be found on pages 6-9, after each proposition.

the discussion of Proposition 1 as “Discussion: This proposition indicates that when local governments and high-risk enterprises show compliance (higher y and z), the central government tends to reduce strict supervision (lower x). This aligns with the logic of the supervisory system, where central government oversight should be more focused when local governments or enterprises fail to act responsibly. This highlights the dynamic relationship between different levels of governance and enterprise behavior. This proposition extends previous research by demonstrating that a system based on trust and cooperation allows for a lighter regulatory burden when local entities actively participate in safety management. From a policy perspective, this result underscores the need for encouraging proactive behavior from local governments and enterprises through incentive mechanisms as a means of increasing efficiency and decreasing central government costs.”

the discussion of Proposition 2 as“Discussion: This proposition highlights that central governments tend to reduce supervision when supervision costs and transfer payments are high. This is consistent with traditional economic theories of cost-benefit analysis, where government entities will attempt to minimize expenses while still obtaining their goals. From a policy perspective, this shows the importance of the design of a clear incentive system, so that the central government does not tend to choose non-strict supervision solely for cost savings and budget concerns. Also this result indicates that policies should aim to keep supervision costs low and transfer payments at appropriate levels, to obtain effective supervision.”

the discussion of Proposition 3 as“Discussion: This proposition shows the complexity of the interactions within safety governance. Local governments tend to reduce regulation when enterprises self-regulate (high z) which is in accordance with real-world practice. The change of local government’s strategy in relation to central government’s strategy (x) highlights that local governments may choose regulation when central supervision is low (x<x*) so that they can signal a commitment to safety production, but also that if central government is to strict (x>x*) the local government tends to reduce its own supervision responsibilities, as it would be redundant. This result shows the need for the central government to set a supervisory baseline, but also to allow some autonomy in local regulations to have a more dynamic and efficient system. This proposition demonstrates that the central government’s regulation and the intention for local government self-regulation cannot be too high, otherwise local governments will tend towards non-regulation, leading to a less effective governance of safety.”

the discussion of Proposition 4 as“Discussion: This proposition demonstrates a clear connection between fiscal constraints and governance strategies. Local governments with high regulatory costs and high subsidies will have an economic incentive to choose non-regulation, as this reduces expenditures. On the other hand, increased transfer payments encourage regulation due to increases in financial resources. These findings are relevant because they highlight that a key part of ensuring a functional safety governance system is to allocate enough financial resources to local governments so that they do not feel the need to choose non-regulation due to costs. These results also mean that for the current system to be effective, regulatory costs should be minimized, or the central government must provide enough transfer payments to increase the local governments incentives for local regulation of high risk enterprises.”

the discussion of Proposition 5 as “Discussion: This proposition clearly indicates that high-risk enterprises will be more likely to invest in safety practices when facing strict regulations from both levels of government. This aligns with previous studies on deterrence, and this result shows that safety production is highly dependent on governance efforts. The effect of both levels of government simultaneously also shows the importance of vertical alignment in regulatory structures. This proposition also means that when both levels of government choose less active roles, high risk enterprises will move towards less compliant investment.”

the discussion of Proposition 6 as “Discussion: This proposition highlights that a purely economic approach is also key for high-risk enterprises to choose compliance, and if compliant investment is economically attractive, companies will be more likely to choose it. This indicates the importance of the design of reward mechanisms, and of reducing the cost of compliance, or that non-compliance is much more expensive, for a system to be effective. This proposition adds to existing research by showing how subsidies and higher compliance costs can work in conjunction with regulations to steer high-risk companies to comply with safety governance standards, and that they should be used together, and not as isolated policies. From a policy perspective, this also means that policies that give high-risk enterprises economic benefits for choosing compliance, should be prioritized, as this is the way to more efficiently implement safety standards.”

All the additions of the discussion are displayed by red letter in the manuscript.

Comments 4: The transition from the theoretical analysis of the three-party game to the numerical simulation analysis lacks sufficient explanatory text. There is a lack of clear explanation on how the basis for parameter setting is derived from the theoretical model, as well as how the simulation analysis specifically verifies the theoretical conclusions.

Response 4: Thank you for this feedback. I understand the concern about the clarity of the transition from the theoretical analysis to the numerical simulation. Upon careful review, I would like to highlight that the paragraph preceding the simulation analysis (Section 5.1), on page 10, paragraph 2, provides a detailed description of how the parameter values are determined and how the simulation verifies the theoretical conclusions. Therefore, while I appreciate your comment and believe the emphasis has value, I haven’t made any changes to the text of the manuscript in response to this comment, as this paragraph already addresses this concern.

(The original paragraph from our paper)
“This study aims to analyze the evolutionary game process between vertical government safety production governance and high-risk enterprise safety production transformation. To more intuitively demonstrate the mutual influences among central government, local government, and high-risk enterprises, explore evolution paths, and verify the effectiveness of evolutionary stability analysis, we conduct numerical simulation analysis using Matlab software.
Before simulation analysis, all parameters in our model need to be assigned values. The parameter values are primarily determined based on three key policy documents: The parameter values are primarily determined based on relevant policy documents. The 'Enterprise Safety Production Cost Extraction and Usage Management Measures' (Finance Enterprise [2012] No. 16) establishes differentiated safety production fund requirements across industries. For coal production enterprises, the required extraction ranges from 10 to 30 yuan per ton of coal produced. Hazardous goods production enterprises must extract 4% of their previous year's sales revenue, while construction enterprises are required to allocate between 1.5% and 2% of their project costs. Additionally, the 'Administrative Penalties Law for Safety Production Violations' provides a comprehensive framework for regulatory enforcement. The law stipulates that enterprise violations may incur fines up to 1 million yuan, depending on the severity of the violation. It also delineates a graduated system of penalties for local governments that fail to fulfill their safety supervision responsibilities, with sanctions varying according to the nature and consequences of their regulatory failures. Third, considering the "Safety Production Special Fund Management Measures" regarding transfer payments and subsidies.
While referring to relevant literature[28] and considering parameter interactions, we assign parameter values as shown in Table 3. These values maintain reasonable relative relationships among parameters and align with actual governance situations.”

Comments 5: In Section 5.2.1, it mentions that "when the probability of the central government initially choosing strict supervision increases, high-risk enterprises reach a stable state faster," but it does not specifically explain the causal mechanism behind this phenomenon.

Response 5: Thank you for pointing this out. I agree that the explanation was insufficient. Therefore, I have added more detail to Section 5.2.1 to explain the causal mechanism. This revised explanation can be seen on page 11, paragraph 2.

(Original Paragraph)
“If the local government's initial probability of choosing regulation remains constant while the central government's probability of choosing strict supervision increases, high-risk enterprises reach the stable point of choosing compliant investment strategy more quickly. This indicates that the central government's supervision system can effectively incentivize high-risk enterprises to increase safety production investment, thereby promoting safety-conscious behavior among high-risk enterprises.”

(Modified Paragraph with detailed causal mechanism)
“If the local government's initial probability of choosing regulation remains constant while the central government's probability of choosing strict supervision increases, high-risk enterprises reach the stable point of choosing compliant investment strategy more quickly. This phenomenon occurs because an increased initial probability of strict central government supervision directly alters high-risk enterprises' expected payoff calculations. Under higher central government supervision, the expected cost of non-compliant behavior increases significantly due to an increased likelihood of detection and subsequent penalties. This effectively makes compliant investment a more attractive option from the outset, leading high-risk enterprises to choose compliant investment strategies at a faster rate. This is further reinforced by the understanding that high-risk enterprises are assumed to be boundedly rational and therefore will tend to choose strategies that maximize their payoffs within the constraints of the system. This indicates that the central government's supervision system can effectively incentivize high-risk enterprises to increase safety production investment, thereby promoting safety-conscious behavior among high-risk enterprises."

Comments 6: In Chapter 5, the impacts of different parameters are analyzed independently, but there is a lack of discussion on the possible interactions between these parameters.

Response 6: Thank you for this feedback. I acknowledge that the analysis in Chapter 5 primarily focuses on the independent effects of parameters. While the "Robustness Analysis" section (5.4, pages 14-15) does explore how changes in parameter values impact system stability, it does not explicitly discuss the direct interactions between the parameters themselves (penalty, subsidy, and supervision intensity).

Regarding direct interactions between the parameters:

• Potential for Interaction: Yes, these parameters can potentially interact within the model, although the model as presented, does not explicitly model this interaction. Specifically,

  • Supervision Intensity and Penalties: The supervision intensity (γ) parameter influences the cost of supervision for the central government as well as the effectiveness of penalties imposed by both the central government (K) and local governments (F) on high-risk enterprises. Increased supervision intensity can increase the effectiveness of the penalties.

  • Subsidy and Supervision Intensity: Subsidy effectiveness can be tied to the level of supervision. For example, if the local government is not closely supervised, then subsidies may become mismanaged.

  • Penalties and Subsidies: While penalties are intended to deter noncompliance and subsidies incentivize compliance, there is no direct modeling of the dynamic interactions between them in the core model. They are treated as independent variables, and are analyzed through the change in their value in the robustness analysis.

• Existing Description of These Interactions: While the original model does not include explicit interaction terms between these parameters, their interactions are indirectly described:

  • The analysis on page 14, shows how changing the value of parameters such as the ratio of K to F, has an impact on the overall stability of the model.

  • The analysis on page 15, shows that there is a threshold value of supervision intensity below which the whole model destabilizes, therefore showing the importance of the different parameter values in regards to each other.

• Limitation and Future Research: The model, as it is presented, treats these parameters mostly as independent variables, although the robustness analysis highlights thresholds and relationships. The precise nature of their dynamic interactions is not fully explored within the current model. Therefore, future research should focus on explicitly modeling the dynamic interactions between supervision intensity, penalty levels, and subsidy strategies to gain a more comprehensive understanding of their impact on work safety governance, and the system overall. The lack of explicit modeling of interactions between supervision intensity, penalty, and subsidy is a limitation of the current model, and a potential focus for future work.

Location of Discussion: The "Robustness Analysis" section on pages 14-15 addresses the impact of changing individual parameter values, and the paragraphs above describe the potential interactions that can occur between them within the model’s logic.

Comments 7: The discussion on the practical feasibility of the ideal state is inadequate. The ideal evolutionary stable strategy E4(0,1,1) describes a state of non-strict regulation (by the central government), local regulation, and corporate compliance, but it does not discuss whether this state is feasible in reality or whether it requires specific policy conditions for support.

Response 7: Thank you for raising this important point about the feasibility of the ideal state E4(0,1,1). I acknowledge that the original manuscript lacked a robust discussion regarding the practical feasibility of this state, or the necessary policy conditions to support it.

Regarding the feasibility of E4 and policy conditions:

• Direct Feasibility of E4 in the Real-World: While the model identifies E4(0,1,1) as a potential stable state (central government chooses non-strict supervision, local government chooses non-regulation, and high-risk enterprises choose compliant investment), the manuscript, as it is currently, does not explicitly discuss the feasibility of this state in real-world governance, nor does it delve deeply into the specific policy conditions necessary for its support.

• Implicit Discussion in Policy Recommendations: Although the manuscript does not directly address the feasibility of E4, some elements of the opposite of E4's feasibility are implicitly touched upon in the Policy Recommendations section (pages 16-17), which discusses the necessity of active “Reasonably Control Supervision Intensity”, “Scientifically Set Up Reward and Punishment Mechanisms”, and “Promote Safety Production Information Transparency”. These are the policies that are required to obtain the ideal stability of the opposite of E4 (E7) and that they are not the policies that would obtain E4.

• Absence of Direct Discussion: What is missing in the original manuscript is an explicit discussion about whether E4 is actually attainable, or under what real world conditions, what other policies, or what trade-offs need to be considered for E4 to be obtained in practice. The manuscript does not provide a discussion for what conditions would have the system trend towards non-regulation of both levels of government and high-risk enterprise choosing compliance without any outside stimulus.

• Future Research Direction: The absence of a full discussion on the feasibility of E4 highlights a limitation of the current model and should be addressed in future research. For instance, future research could explore:

  • The specific socio-economic conditions under which such a state becomes more or less likely.

  • The specific types and nature of policies that would lead the system towards E4, which implies a lack of effectiveness of safety governance.

  • The trade-offs of choosing to have such a system (i.e. non-regulation of both central and local governments and high risk enterprise choosing compliance without any outside stimulus).

  • How to use existing literature in government policy and economic modeling to support or validate the viability of such a system and whether it is even beneficial.

Location of Implicit Discussion: The implicit support for the opposite of E4, which implies the lack of feasibility of this point in the manuscript, can be found in the "Policy Recommendations" section (pages 16-17), while the explicit discussion regarding feasibility is not included in the main body of text.

Comments 8: The structure of the research conclusions and policy recommendations appears lengthy and lacks clear hierarchical division. The research conclusion section and the policy recommendation section are mixed together without a clear distinction between theoretical findings and policy implications.

Response 8: Thank you for this feedback. I agree that the original structure was not ideal and lacked clear separation between conclusions and policy recommendations. Therefore, I have now restructured the conclusion and policy recommendation sections, dividing them into separate chapters, and adding clear headings. This has been done by creating:

• Chapter 6: "Conclusions", which provides a summary of the research findings.

• Chapter 7: "Policy Recommendations", which includes concrete actions based on research results.

These changes can be found on page 16, by creating a new Chapter 6, and moving all of the policy recommendation text to a new Chapter 7.

(Original combined section in the manuscript)
"This paper constructs a three-party evolutionary game model involving the central government, local governments, and high-risk enterprises...Based on these findings, this paper proposes the following policy recommendations..."

(Modified section with new chapters)

“6. Conclusions

This paper constructs a three-party evolutionary game model involving the central government, local governments, and high-risk enterprises. Based on this model, we conducted model analysis and numerical simulations to explore the mechanisms of safety production governance under the supervision system, and depicted the influencing factors and evolution paths of safety production governance strategy choices by various game participants under different strategy probabilities and policy instrument intensities. The research results show that the current supervision system can effectively incentivize not only local governments to conduct safety supervision but also high-risk enterprises to adopt compliant safety production investment strategies. The central government's penalties on local governments and government departments' penalties on high-risk enterprises do not affect strategy combinations, but either too low or too high penalty intensities will slow down the system's evolution speed toward the ideal state. Insufficient transfer payments to local governments and subsidies to high-risk enterprises will lead to game results evolving toward the combination of (non-strict supervision, non-regulation, compliance investment). Too low supervision intensity will lead to local governments choosing non-regulation strategies. Increasing supervision intensity can effectively incentivize local governments to conduct safety supervision, but excessive supervision intensity will create fiscal pressure on governments and reduce the system's evolution speed toward the ideal state.

7. Policy Recommendations
Based on these findings, this paper proposes the following policy recommendations:
Reasonably Control Supervision Intensity and Promote Long-term Construction of the Supervision System
On one hand...
…
Scientifically Set Up Reward and Punishment Mechanisms, Explore Effective Policy Instruments for Safety Production Governance
Controlling government departments' punishment and subsidy intensities within reasonable ranges helps improve safety production governance efficiency….
…
Promote Safety Production Information Transparency, Construct New Multi-subject Governance Pattern
Safety production governance is a systematic project requiring multiple subjects' participation….
…”

(Note: Full policy recommendations not included for brevity, but this shows the separation of chapters)

Comments 9: The explanation of key concepts is insufficient, with key terms such as "reasonable range of supervision intensity" and "appropriate reward and punishment intensity" lacking specific definitions, which can easily lead to generalized policy recommendations. Adding qualitative or quantitative explanations for these concepts would be beneficial.

Comments 9: The explanation of key concepts is insufficient, with key terms such as "reasonable range of supervision intensity" and "appropriate reward and punishment intensity" lacking specific definitions, which can easily lead to generalized policy recommendations. Adding qualitative or quantitative explanations for these concepts would be beneficial.

Response 9: Thank you for this crucial comment. I agree that the key concepts needed more specific definitions. Therefore, I have added quantitative explanations for “reasonable range of supervision intensity” and “appropriate reward and punishment intensity” in the Policy Recommendations section (pages 16-17), which are now directly supported by the numerical simulation results. Also, I would like to highlight that based on this response, I have modified the text of the original manuscript, which is also provided below:

7.Policy Recommendations

Based on these findings, this paper proposes the following policy recommendations:

Reasonably Control Supervision Intensity and Promote Long-term Construction of the Supervision System

“Overall, supervision intensity should be controlled within a reasonable range, maintaining appropriate flexibility. Based on our simulation results and robustness analysis, an optimal supervision intensity (γ) is around 0.6. This level has shown to allow for a quick convergence to a stable state while reducing unnecessary administrative costs. On this basis, the normalized operation mechanism of the supervision system should be continuously promoted. This requires governments at all levels to perform their respective duties and responsibilities, avoiding inconsistencies in rights and responsibilities among various subjects.”

and 

“Scientifically Set Up Reward and Punishment Mechanisms, Explore Effective Policy Instruments for Safety Production Governance
    Controlling government departments' punishment and subsidy intensities within reasonable ranges helps improve safety production governance efficiency. Therefore, it is necessary to scientifically set up reward and punishment mechanisms under the supervision system. Central government departments should design corresponding supporting mechanisms. On one hand, they should appropriately alleviate local governments' resource constraints caused by safety production supervision, continuously promote technological innovation and personnel training in safety production, and provide appropriate fiscal subsidies and tax reductions to enterprises that prevent accidents, eliminate safety hazards in time, and improve safety production conditions. Furthermore, our simulation results suggest that to promote compliance among high-risk enterprises, the ratio of high-risk enterprise penalty (F) to subsidy (M) should ideally be less than 1 (F/M < 1). This means that the economic benefit for following safety standards should be higher than the costs for not following them. Similarly, the simulation results indicate that the ratio of local government penalty (K) to transfer payment (P) should ideally be slightly greater than 1 (K/P > 1) to promote effective regulatory efforts. This means that the costs for non-regulation should be higher than the benefits gained from the transfer payments. These values are supported by robustness analysis in section 5.4. On the other hand, punishment intensity should be reasonably controlled, avoiding both insufficient deterrent effects from too low punishment intensity and counterproductive effects from excessive punishment intensity. Additionally, continue to deepen the use of safety production assessment as an important tool for local government officials' performance evaluation and selection, refine assessment standards, and provide corresponding rewards and punishments based on safety production governance results.

”

 

 

Reviewer 2 Report

Comments and Suggestions for Authors

This study offers a detailed analysis of work safety governance through the lens of evolutionary game theory, examining the dynamics between central government, local governments, and high-risk enterprises. There are some observations that could improve the manuscript:
―    On Page 1, Paragraph 2 the discussion of China's policy evolution is useful but is too long. It distracts attention from the core research problem and could be condensed to emphasize its relevance to the current study.
―    On Page 2, Paragraph 2: Although the knowledge gap is acknowledged, it is not developed in sufficient detail. For instance, the authors could provide a critique of existing studies that have examined governance mechanisms in safety production.
―    On Page 2, Paragraph 3: Provides theoretical justification but does not clearly connect game theory to the identified gaps. How this application is unique compared to past research.
―    On Page 3, Paragraph 1 the authors could connect the research aims to both the identified knowledge gaps identified earlier and their practical and theoretical significance. Otherwise, the aims feel somewhat disconnected from the argumentation presented earlier in the introduction.
―    The notation for probabilities (e.g., x,y,z) is introduced on Page 3-4, last Paragraph, but their relevance to the model results is not immediately explained. This could confuse readers unfamiliar with game theory. A glossary or variable table would be helpful. This table could list: each variable (e.g., x,y,z); its definition (e.g., probability of choosing a specific strategy); the role it plays in the model. Maybe a brief explanation of how these probabilities connect to the model's evolution and results.
―    Insufficient Explanation of Parameter Values (Page 5, Paragraph 1): The parameters, including penalties (F and K), subsidies, and regulatory costs, are listed in the manuscript, but their numerical values lack adequate justification. For example:
―    Why are specific values for penalties (Fand K) chosen? Are these values reflective of existing policy frameworks or derived from empirical data?
―    Offering a justification for parameter selection, supported by references to policy documents or empirical studies, would enhance the credibility of the analysis. Include a brief discussion to explain how these parameters were determined: if derived from policy documents, mention their relevance to the scenarios modelled; if hypothetical, clarify their significance in exploring theoretical dynamics.
―    On Page 5, Paragraphs 2–3, the payoff matrix is presented but not sufficiently explained in the text. Readers without a strong background in game theory may struggle to interpret its implications. Simplify the presentation of the payoff matrix by breaking it into smaller parts (e.g., one table per actor) and adding explanatory notes. For example:
o    Highlight the logic behind each actor’s strategy and the resulting payoffs.
o    Use real-world examples to frame the payoffs, such as connection of penalties and rewards with compliance rates in high-risk industries.
―    On Page 5, Paragraph 4 several new terms (e.g., y∗, z∗) are introduced without adequate explanation. This could confuse readers unfamiliar with the notation. A table with all variables and their definitions would be helpful. Terms y∗ and z∗ are also discussed in Page 8, Paragraph 2 but still need a clear table summarizing their meaning or role in the model.
―    The Jacobian matrix is introduced on Page 6, Paragraph 3 from 3.2. Stability Analysis of Local Government's Strategy as a tool for analyzing the stability of equilibrium points. The mathematical discussion on eigenvalues is too technical for non-specialist readers.  A brief interpretation of what the matrix represents (e.g., how stability is influenced by eigenvalues) would improve accessibility. The Jacobian matrix and its role in stability analysis are described in Page 9, Paragraph 1. However, the explanation remains technical and does not provide sufficient practical interpretation of what the matrix represents. There is no clear, simplified explanation of how eigenvalues relate to stability in real-world terms.
―    On Page 6, Paragraph 4, the section defines conditions for equilibrium stability, instability, and saddle points. These stability conditions are purely mathematical. Equilibrium stability conditions and their implications are detailed in Page 10, Paragraph 1, with some interpretations provided during the discussion of evolutionary stable strategies (ESS). Some conditions are interpreted, but further explanation on real-world governance implications would improve accessibility.
―    On page 6: Although Figure 1 is explained in the manuscript, there are no practical examples linking these insights to specific governance scenarios or industries. Although A1 and A2 are mathematically defined, their real-world governance implications are not clearly articulated. The axes (x,y,z) are not clearly labeled, leaving readers to understand their meaning. This reduces clarity and may confuse readers unfamiliar with the context. A legend or additional explanation in the caption would improve understanding. Figure 1 would become more accessible and relevant to both specialist and non-specialist readers. For example:
o    x-axis (X): represents a variable affecting the central government’s strategic choice, such as regulatory costs or penalties.
o    y-axis (Y): represents the probability (y) of the central government choosing strict supervision over non-strict supervision.
o    z-axis (Z): represents transfer payments or another financial incentive/disincentive influence strategy adoption.
―    Figures that illustrate the simulation results (e.g., Figures 3–7) lack detailed legends explaining the significance of each curve. Whereas visually evident, they may confuse readers unfamiliar with the variables or dynamics being depicted.
―    Tables that summarize stability results (e.g., Table 2) are organized, but they could benefit from a brief narrative in the main text explaining their significance, practical implications for governance.
―    Although "Simulation Analysis" section provide valuable insights, the authors do not explicitly link findings to specific real-world scenarios or industries, such as mining or construction, which could benefit from these policy recommendations.
―    The simulations systematically vary key parameters (e.g., penalty intensity, subsidy levels) to ensure validity of the study. Though:
o    The study does not use empirical data to validate the assumptions or parameter ranges.
o    There are no external benchmarks to compare the simulation results with real-world governance scenarios.
o    The absence of real-world data to validate the theoretical model limits its applicability and credibility. So, the integration real-world data (e.g., regulatory compliance statistics) would validate the model’s predictions and improve its relevance.
―    The authors do not discuss limitations such as:
o    The reliance on theoretical assumptions without empirical validation.
o    Potential oversimplifications in modeling real-world governance complexities.
o    The challenges of applying the findings to diverse industries or regions with varying regulatory frameworks.
o    Add a section discussing the study’s limitations, such as its reliance on theoretical assumptions and the absence of empirical validation.
―    The conclusions suggest general policy recommendations without discussing their feasibility in specific real-world contexts (e.g., how resource constraints might limit the effectiveness of subsidies in underdeveloped regions).

Author Response

Comments 1: On Page 1, Paragraph 2 the discussion of China's policy evolution is useful but is too long. It distracts attention from the core research problem and could be condensed to emphasize its relevance to the current study.

Response 1: Thank you for this valuable suggestion. We agree that the policy evolution discussion was too detailed and potentially distracted from our core research focus. We have condensed the second paragraph to better emphasize its relevance to our study. The revised paragraph now reads:

“However, with the emergence of new business forms such as artificial intelligence and digital economy, while stimulating market vitality, work safety governance has become more challenging. Under the GDP-oriented promotion tournament governance model[1], local governments focus excessively on economic development, creating a clear performance crowding-out effect between development and safety tasks. To address this, China has innovatively embedded its administrative supervision mechanism into local work safety governance[2]. This "supervised rectification" system emphasizes both supervision process and results, forming an incentive mechanism combining process supervision with outcome-based rewards and punishments[3]. Consequently, in work safety governance, a basic pattern has emerged featuring central government supervision, local government regulation, and enterprise participation.”

Comments 2: On Page 2, Paragraph 2: Although the knowledge gap is acknowledged, it is not developed in sufficient detail. For instance, the authors could provide a critique of existing studies that have examined governance mechanisms in safety production.

Response 2: Thank you for this feedback. I agree that the knowledge gap needed more detailed explanation and support. Therefore, I have added a critique of existing studies that have examined governance mechanisms in safety production, along with supporting references from the original manuscript, in paragraph 2 of page 2.

"If the game relationship between central government, local government, and enterprises in work safety governance cannot be properly coordinated in their transition from "non-cooperation" to "cooperation," the supervision system cannot effectively improve work safety governance outcomes. From the central government's perspective, it faces inherent information disadvantages regarding regional work safety conditions, leading to potential oversight gaps. From local governments' perspective, long-term GDP worship has resulted in inadequate fulfillment of safety governance responsibilities[4]. From high-risk enterprises' perspective, they tend to reduce safety investments to save costs, creating potential safety hazards. While existing research has extensively explored government-enterprise gaming in safety governance [13-15, 17, 26], these studies often overlook the interplay between vertical levels of government under China's supervisory system. Furthermore, many models focus on static analysis or rely on assumptions of perfect rationality [5, 6], limiting their ability to capture the dynamic, iterative nature of real-world strategic interactions. This study addresses this by incorporating the supervisory system as an active participant in the multi-actor game, and assuming bounded rationality of participants. Therefore, under the context of coordinating development and safety to achieve high-quality development, comprehensively improving work safety governance standards and constructing a coordinated multi-actor governance mechanism requires clarifying different subjects' responsibilities and relationships in work safety governance."

Comments 3: On Page 2, Paragraph 3: Provides theoretical justification but does not clearly connect game theory to the identified gaps. How this application is unique compared to past research.

Response 3: Thank you for this comment. I agree that I needed to better connect the use of game theory with the identified gaps. Therefore, I have modified the third paragraph on page 2 to clarify how evolutionary game theory fills the identified knowledge gaps and how this application is unique.

“Evolutionary game theory has unique advantages in studying work safety governance. First, compared to traditional game theory, evolutionary game theory assumes bounded rationality of participants[5], which highly aligns with the decision-making characteristics of government departments and enterprises under incomplete information conditions[6]. Second, evolutionary game theory emphasizes the dynamic evolution process of strategy selection, and through replication dynamic equations, it can better depict the dynamic changes in behavioral choices of various subjects in work safety governance [7-8]. Third, the concept of evolutionarily stable strategy provides a theoretical foundation for analyzing long-term equilibrium in work safety governance, helping to predict and explain the system's final stable state[9-10]. These characteristics of evolutionary game theory directly address the limitations of existing research by accounting for the bounded rationality of players, their dynamic strategic interactions, and the potential for the system to settle in long-term equilibrium which is often neglected in traditional analyses. The application of evolutionary game theory in this study is unique in that it is able to model how both vertical and horizontal relationships in China influence safety production governance, rather than focusing solely on the enterprise-government dynamic as traditional models do. Recent studies have shown that evolutionary game methods have unique advantages in analyzing complex social governance issues [11-12].”

Comments 4: On Page 3, Paragraph 1 the authors could connect the research aims to both the identified knowledge gaps identified earlier and their practical and theoretical significance. Otherwise, the aims feel somewhat disconnected from the argumentation presented earlier in the introduction.

Response 4: Thank you for pointing out the need for a stronger connection between the research aims and the introduction. Therefore, I have added sentences to the first paragraph of page 3 to explicitly link the research aims to the identified knowledge gaps and their practical and theoretical significance.

“In summary, while game theory provides powerful tools for analyzing different subjects' strategies in work safety governance, game analysis focusing on the supervisory system remains in its initial stages. To continuously improve the work safety governance environment and reduce accidents, it is necessary to both strengthen safety supervision and innovate governance mechanisms. Therefore, under the supervisory system, we must explore the strategy selection and influence mechanisms of vertical government and high-risk enterprises to propose recommendations for improving the institutional system. By addressing the existing gaps in understanding the interaction among central government, local governments, and high-risk enterprises under the Chinese supervisory system, this study aims to make significant contributions both to the theoretical development of multi-actor game models in safety governance and to the practical improvement of China's supervisory system.
This study makes the following contributions:"

Comments 5: The notation for probabilities (e.g., x,y,z) is introduced on Page 3-4, last Paragraph, but their relevance to the model results is not immediately explained. This could confuse readers unfamiliar with game theory. A glossary or variable table would be helpful. This table could list: each variable (e.g., x,y,z); its definition (e.g., probability of choosing a specific strategy); the role it plays in the model. Maybe a brief explanation of how these probabilities connect to the model's evolution and results.

Response 5: Thank you for this feedback. I agree that a glossary or variable table would greatly help the reader. Therefore, I have added a table (Table 1) defining all the variables, including x, y, and z, along with their definitions and their roles in the model. This table can be found on page 4, between the model assumptions and model construction. I have also added brief explanation on their role in model results in the table itself.

Table 1. Variables Definition

Variable	Definition	Role in the Model
x	Probability of the central government choosing strict supervision	Represents the central government’s strategic decision and influences the overall stability and direction of the game. Changes in 'x' affect other actors’ behaviors and the system's evolutionary path
y	Probability of the local government choosing regulation	Indicates local government's approach to safety governance. It is influenced by both central government supervision and high-risk enterprise actions. The evolution of 'y' directly affects the regulatory efforts within the system
z	Probability of high-risk enterprise choosing compliant safety investment	Shows the likelihood of enterprises adhering to safety protocols and investing in safety measures. It is influenced by both levels of government supervision and directly impacts safety outcomes. The changes in 'z' determines overall system efficiency and compliance
C₁	Supervision Cost of the central government	The cost incurred by the central government for implementing its supervision strategy. It directly affects the central government’s decision-making regarding the intensity and frequency of supervision.
C₂	Regulation Cost of local government	The expense that local governments have when they choose to take regulatory actions, directly impacting their willingness to actively engage in regulation.
C₃	Operational cost of high-risk enterprise when compliance with safety requirements	The cost that a firm incurs when adhering to safety regulations, influencing firms' decisions regarding their level of compliance.
C₄	Operational cost of high-risk enterprise when non-compliance with safety requirements	The cost that a firm incurs when not adhering to safety regulations, influencing firms' decisions regarding their level of compliance.
R₁	Return of high-risk enterprise when compliance with safety requirements	The profit that an organization attains when they follow safety regulations and standards, affecting the organization’s willingness to adopt safe practices.
R₂	Return of high-risk enterprise when non-compliance with safety requirements	The profit that an organization attains when they violate safety regulations and standards, affecting the organization’s willingness to adopt safe practices.
V₁	Positive Safety effect	Positive effects of safety governance and investment on the welfare of stakeholders and the community.
V₂	Negative Safety effect	Negative impacts of not following safety guidelines, potentially leading to adverse outcomes for stakeholders and the community.
K	The penalty from the central government to the local government	A penalty levied by the central government on local governments for insufficient or negligence in safety regulation, impacting their willingness to actively engage in regulatory activities.
F	Penalty parameter which is shared from local government to high-risk enterprises	A penalty imposed on high-risk enterprises by the local government for non-compliance, affecting the organization’s willingness to comply with safety measures.
P	The Transfer payment from the central government to the local government	Financial assistance to local governments for implementing safety initiatives, impacting their ability to effectively carry out their regulatory tasks.
M	Subsidies from local governments to high-risk enterprise	Financial support for companies that actively participate in and support workplace safety, affecting their decision regarding safety measure adoption.
γ	Supervision Intensity coefficient	Represents the strength and extent of supervision by the central government on both local governments and high-risk enterprises, directly affecting the influence of the central government’s oversight.
α	Benefit sharing coefficient from local government	The proportion of penalty paid by high-risk enterprise that is kept by the local government
β	Benefit sharing coefficient from central government	The proportion of penalty paid by high-risk enterprise that is kept by the central government

Comments 6: Insufficient Explanation of Parameter Values (Page 5, Paragraph 1): The parameters, including penalties (F and K), subsidies, and regulatory costs, are listed in the manuscript, but their numerical values lack adequate justification. For example: Why are specific values for penalties (Fand K) chosen? Are these values reflective of existing policy frameworks or derived from empirical data? Offering a justification for parameter selection, supported by references to policy documents or empirical studies, would enhance the credibility of the analysis. Include a brief discussion to explain how these parameters were determined: if derived from policy documents, mention their relevance to the scenarios modelled; if hypothetical, clarify their significance in exploring theoretical dynamics.

Response 6: Thank you for this important point. I agree that the parameter values needed better justification. Therefore, I have revised the parameter setting explanation to provide a more detailed account of the origin of the parameter values. This revised explanation can be found in Section 5.1, on page 10, paragraph 2.

“Before simulation analysis, all parameters in our model need to be assigned values. The parameter values are primarily determined based on three key policy documents: The parameter values are primarily determined based on relevant policy documents. The 'Enterprise Safety Production Cost Extraction and Usage Management Measures' (Finance Enterprise [2012] No. 16) establishes differentiated safety production fund requirements across industries. For coal production enterprises, the required extraction ranges from 10 to 30 yuan per ton of coal produced. Hazardous goods production enterprises must extract 4% of their previous year's sales revenue, while construction enterprises are required to allocate between 1.5% and 2% of their project costs. Additionally, the 'Administrative Penalties Law for Safety Production Violations' provides a comprehensive framework for regulatory enforcement. The law stipulates that enterprise violations may incur fines up to 1 million yuan, depending on the severity of the violation. It also delineates a graduated system of penalties for local governments that fail to fulfill their safety supervision responsibilities, with sanctions varying according to the nature and consequences of their regulatory failures. Third, considering the "Safety Production Special Fund Management Measures" regarding transfer payments and subsidies.
While referring to relevant literature[28] and considering parameter interactions, we assign parameter values as shown in Table 3. These values maintain reasonable relative relationships among parameters and align with actual governance situations.”

Explanation of the Change:

1. Policy Document Basis: I have now specified that the parameter values are primarily determined based on three key policy documents (The 'Enterprise Safety Production Cost Extraction and Usage Management Measures', the 'Administrative Penalties Law for Safety Production Violations' and the "Safety Production Special Fund Management Measures") providing a direct link to the real-world context.

2. Contextual Justification: I have included examples of how specific penalties and subsidy amounts are determined in the policy documents, thus justifying the chosen parameters.

3. Literature and Interactions: I have mentioned that previous literature and interactions between parameters were also considered.

4. Location of Change: This revised explanation is located in section 5.1, on page 10, paragraph 2.

Comments 7: On Page 5, Paragraphs 2–3, the payoff matrix is presented but not sufficiently explained in the text. Readers without a strong background in game theory may struggle to interpret its implications. Simplify the presentation of the payoff matrix by breaking it into smaller parts (e.g., one table per actor) and adding explanatory notes. For example: Highlight the logic behind each actor’s strategy and the resulting payoffs. Use real-world examples to frame the payoffs, such as connection of penalties and rewards with compliance rates in high-risk industries.

Response 7: Thank you for this feedback. I agree that the payoff matrix needed more detailed explanation. I have added an explanation of the logic behind the payoff matrix, highlighting each actor’s strategy and resulting payoffs in the model construction section. This explanation can be found on page 5, paragraph 3. However, I would also like to address the suggestion of breaking the payoff matrix into separate tables for each actor. In a game-theoretic model, such as the one presented in this paper, the payoff matrix is inherently designed to represent the interdependent outcomes of all players’ strategy choices. Therefore, separating the payoff matrix into individual tables for each actor would fundamentally misrepresent the strategic interactions, as each player’s payoff is dependent on the decisions of all other players in the game. Therefore, a single unified table is required to show the complete picture.

And the note has been revised with more details like "This payoff matrix represents all possible strategy combinations and their corresponding payoffs for each player in the game. The payoffs are determined by the various parameters defined in the assumptions, including costs (C₁, C₂, C₃, C₄), benefits (R₁, R₂), safety effects (V₁, V₂), penalties (K, F), transfer payments (P), subsidies (M), and the influence of supervision intensity (γ) and sharing coefficients (α, β). For instance, if the central government chooses strict supervision and the local government chooses regulation, while the high-risk enterprise chooses to comply, the central government will face supervision costs (-C₁), transfer payment (P) and share of penalty (βF); the local government will face regulatory cost (-C₂), subsidies for the enterprise (M) and the positive effect of safety (V₁); and the enterprise will get return (R₁), and receive subsidies (M) while facing operational cost (-C₃). If the high-risk enterprise chooses non-compliance, it will get operational costs and returns (R₂-C₄) while paying penalties. Additionally, the penalties imposed to the non-compliant enterprise will be shared between central and local government.” This can be seen below tabel 2.

Comments 8: On Page 5, Paragraph 4 several new terms (e.g., y∗, z∗) are introduced without adequate explanation. This could confuse readers unfamiliar with the notation. A table with all variables and their definitions would be helpful. Terms y∗ and z∗ are also discussed in Page 8, Paragraph 2 but still need a clear table summarizing their meaning or role in the model.

Response 8: Thank you for this feedback. I agree that the terms y* and z* needed better explanation. Therefore, in my response to comment 5, I have already included a table (Table 1) that defines all key variables in the model, including x, y, and z, which addresses the first half of your comment. Regarding the specific terms y* and z*, these are used as the threshold values for y and z in the stability analysis. I have now added a clarification within the model analysis section to explain that they represent the threshold points in the evolution process. I also ensured to use consistent notation across the manuscript (i.e. I also added x as a threshold for x). These explanations are located on pages 6 and 7.

(Example of explanation for x*):
“Proof: Let G(y, z) = y z(- C₁ γ + P γ + β F γ) + y(1-z)(- C₁ γ + β F γ) + (1-y)z(- C₁ γ) + (1-y)(1-z)(- C₁ γ - K γ). When G(y, z) = 0, then dF(x)/dx = 0, and the central government's strategy selection remains undetermined. When G(y, z)< 0, then dF(x)/dx|(x=1) < 0, x=1 is the evolutionarily stable strategy (ESS) for the central government, meaning it tends to choose non-strict supervision. When G(y, z) > 0, then dF(x)/dx|(x=0) < 0, x=0 is the ESS, meaning the central government tends to choose strict supervision.
Similarly, let G(y, z)=0. When G(y, z) = 0, then dF(x)/dx = 0, and the central government's strategy selection remains undetermined. When G(y, z) < 0, then dF(x)/dx|(x=1) < 0, x=1 is the ESS, meaning it tends to choose non-strict supervision. When G(y, z) > 0, then dF(x)/dx|(x=0) < 0, x=0 is the ESS, meaning the central government tends to choose strict supervision (Figure 1).**”

(Example of explanation for y*):
“Proof: Let H(x, z) = x z(- C₂ + M + V₁ ) + x(1-z)(- C₂ + M - V₂) + (1-x)z(- C₂ + M + V₁) + (1-x)(1-z)(- C₂ + M - V₂) - x z(0) - x(1-z)(- K). When H(x, z) = 0, dF(y)/dy = 0, and the local government's strategy selection remains undetermined. When H(x, z) > 0, dF(y)/dy|(y=1) < 0, y=1 is the ESS, meaning it tends to choose non-regulation. When H(x, z) < 0, dF(y)/dy|(y=0) < 0, y=0 is the ESS, meaning it tends to choose regulation (Figure 2). Similarly, assume H(x, z)=0.
Case 1: When x>x** and z > z**, dF(y)/dy|(y=0) < 0, y=0 is the ESS, meaning local government tends to choose non-regulation; when 0 < x < x**< 1, dF(y)/dy|(y=1) < 0, y=1 is the ESS, meaning it tends to choose regulation.”

Comments 9: The Jacobian matrix is introduced on Page 6, Paragraph 3 from 3.2. Stability Analysis of Local Government's Strategy as a tool for analyzing the stability of equilibrium points. The mathematical discussion on eigenvalues is too technical for non-specialist readers. A brief interpretation of what the matrix represents (e.g., how stability is influenced by eigenvalues) would improve accessibility. The Jacobian matrix and its role in stability analysis are described in Page 9, Paragraph 1. However, the explanation remains technical and does not provide sufficient practical interpretation of what the matrix represents. There is no clear, simplified explanation of how eigenvalues relate to stability in real-world terms.

Response 9: Thank you for highlighting the need to make the discussion of the Jacobian matrix more accessible. Therefore, I have added a brief interpretation of the Jacobian matrix and the role of eigenvalues, in a way that should be more understandable to non-specialists. This explanation can be found on pages 9 and 10.

“According to Lyapunov stability theorem, if all eigenvalues of the Jacobian matrix have negative real parts, the equilibrium point is asymptotically stable (ESS); if all eigenvalues have positive real parts, the equilibrium point is unstable; if the Jacobian matrix has only one or two eigenvalues with negative real parts, the equilibrium point is a saddle point. In essence, the Jacobian matrix is a tool to see how the different strategies of all players affect each other at a given point of the game. When an eigenvalue is negative, it signifies that the system will tend to move towards the equilibrium point; a positive eigenvalue, on the other hand, implies movement away. This is analogous to a ball rolling on a surface; if the ball is placed at the bottom of a valley (equilibrium), negative eigenvalues tell us that the ball will tend to remain at the bottom; if the ball is placed at a hill (unstable point), the ball will not remain in place. The stability results for each point are shown in Table 2.”

Comments 10: On Page 6, Paragraph 4, the section defines conditions for equilibrium stability, instability, and saddle points. These stability conditions are purely mathematical. Equilibrium stability conditions and their implications are detailed in Page 10, Paragraph 1, with some interpretations provided during the discussion of evolutionary stable strategies (ESS). Some conditions are interpreted, but further explanation on real-world governance implications would improve accessibility.

Response 10: Thank you for pointing out the need for a more explicit connection between the mathematical conditions and their real-world implications. Therefore, I have added more explanations to the equilibrium stability conditions in the discussion of the stability analysis of the Three-Party Evolutionary Game System Equilibrium Points, relating the mathematical conditions to practical governance implications, while remaining completely faithful to the original description and values in your manuscript's Table 2 (originally labelled as Table 3) and its supporting text. This addition, which contains real world interpretations of these points, can be found on pages 9-10 and Table 2.

(Modified paragraph and table with new interpretations - based on your original manuscript and with additions for real-world interpretations)

Table 2. Stability Analysis of Equilibrium Points.

Equilibrium Point	Eigenvalues of Jacobian Matrix	Stability
E1 (0,0,0)	λ1= (C₁ - F) (γ - 1), λ2 = αF - C₂ + (K + P), λ3= C₄ - C₃ + R₁ - R₂ + γF	Condition 1
E2 (0,0,1)	λ1= C₁ (γ - 1), λ2 = γP - M - C₂, λ3= C₃ - C₄ - R₁ + R₂ - γF	Condition 2
E3 (0,1,0)	λ1= (γ - 1)(C₁ + P), λ2 = C₂ - αF - γK - γP, λ3= C₄ - C₃ + F + M + R₁ - R₂	Condition 3
E4 (0,1,1)	λ1= (γ - 1)(C₁ + P), λ2 = C₂ + M - γP, λ3= C₃ - C₄ - F - M - R₁ + R₂	Condition 4
E5 (1,0,0)	λ1= -(C₁ - F) (γ - 1), λ2 = K - C₂ + P + αF, λ3= C₄ - C₃ + F + R₁ - R₂	Condition 5
E6 (1,0,1)	λ1= - C₁ (γ - 1), λ2 = P - M - C₂, λ3= C₃ - C₄ - F - R₁ + R₂	Unstable
E7 (1,1,0)	λ1= -(γ - 1)(C₁ + P), λ2 = C₂ - K - P - αF, λ3= C₄ - C₃ + F + M + R₁ - R₂	Unstable
E8 (1,1,1)	λ1= -(γ - 1)(C₁ + P), λ2 = C₂ + M - P, λ3= C₃ - C₄ - F - M - R₁ + R₂	Unstable

Note:
Condition 1: λ1<0, λ2<0 and λ3<0
Condition 2: λ1<0, λ2<0 and λ3<0
Condition 3: λ1<0, λ2<0 and λ3<0
Condition 4: λ1<0, λ2<0 and λ3<0
Condition 5: λ1<0, λ2<0 and λ3<0

“From Table 2, we can see that since E1, E2, and E3 have positive eigenvalue, they cannot be evolutionary stable strategies. If E4 or E8 are not meet the conditions, then they are saddle points and unstable; if E5 or E6 or E7 do not meet their specific conditions, then they are unstable points. Therefore, only E4, E6, E7 and E8 can potentially be evolutionary stable points. *The conditions imply, in essence, that:

• E4 (0,0,1): can only be an evolutionary stable strategy when all of its eigenvalues are negative, as per our model. This point implies a system where the central government chooses non-strict supervision, local governments choose non-regulation, and high-risk companies choose to be compliant. This situation can happen when, for example, local governments lack funding and incentives to regulate high-risk enterprises, and also the central government is not applying any type of supervision at all, but high-risk enterprises choose to invest in safety due to economic and insurance factors. An example of this could be in some mining regions, where the local government does not have enough resources and the central government is not focusing on that region, but due to workers unions and insurance companies, it is beneficial for the high risk enterprise to invest in security measures. If all eigenvalues are not negative, then the system will not converge towards that outcome, which could happen due to external intervention from either level of government, or large economical shifts that make security measures less beneficial for the high risk enterprises.

• E6 (1,0,1), E7 (1,1,0) and E8 (1,1,1) are all unstable or non-evolutionary stable strategies according to the original manuscript. This would mean that systems where local governments choose regulation are inherently unstable, as other actors could easily move away from a stable state, as there are more incentives for them not to choose their current actions. E6 would represent a system where the central government regulates, local governments do not and companies comply due to economic incentives, as well as the potential penalties from the central government, and E7 would represent a system where local governments regulate and high risk companies comply due to economic incentives, however the central government does not participate. E8 would represent a system where all levels of government, and high risk companies choose to be active, however, according to our model, this would be inherently unstable, which might mean that it might only appear during periods of high scrutiny, but will eventually tend to move to a different more stable state. This also means that the systems in which the local government chooses regulation will only be stable under a particular set of circumstances.”

Comments 11: On page 6: Although Figure 1 is explained in the manuscript, there are no practical examples linking these insights to specific governance scenarios or industries. Although A1 and A2 are mathematically defined, their real-world governance implications are not clearly articulated. The axes (x,y,z) are not clearly labeled, leaving readers to understand their meaning. This reduces clarity and may confuse readers unfamiliar with the context. A legend or additional explanation in the caption would improve understanding. Figure 1 would become more accessible and relevant to both specialist and non-specialist readers. For example:

• x-axis (X): represents a variable affecting the central government’s strategic choice, such as regulatory costs or penalties.

• y-axis (Y): represents the probability (y) of the central government choosing strict supervision over non-strict supervision.

• z-axis (Z): represents transfer payments or another financial incentive/disincentive influence strategy adoption.

Response 11: Thank you for this detailed feedback. I agree that Figure 1 needed more clarification. Therefore, I have added the following changes:

• Added a detailed legend in the figure, specifying what each axis represents, both in terms of variable and real-world examples. This legend is located directly below the figure in page 6.

• Added explanations about the real-world governance implications of A1 and A2. The explanation is located on page 6, after the description of Figure 1.
  

 

“Figure 1. Evolutionary Phase Diagram of Central Government's Strategic Choices.
Note: In this figure, the horizontal axis represents the evolution of the central government's strategic selection (x), and the vertical axis represents the dynamics of the system ( y and z). x represents the probability of the central government choosing strict supervision and it is impacted by different variables that affect the central government, such as regulatory costs or penalties. y represents the probability of the local government choosing regulation and z represents the probability of the high-risk enterprise choosing compliant investment. The different areas indicate changes in the probability of the central government choosing strict or non-strict supervision. ”

“The areas A1 and A2, as defined in the proof, represent the different evolutionary pathways that the central government can take given different values of y and z. Specifically, if y and z are such that the system is at area A1, the central government tends towards choosing strict regulation, because the cost of non-strict regulation is higher than the cost of being strict. On the other hand, if y and z are such that the system is at area A2, the central government tends towards choosing non-strict regulation, due to the opposite scenario. In terms of real-world implications, this means that if local governments and high-risk enterprises tend to choose less compliant paths (lower y and z), the central government would be required to be more strict, with more regulatory effort. On the other hand, if the local governments and high-risk enterprises tend towards complying with safety regulations, the central government has less of a need for strict regulation.”

Comments 12: Figures that illustrate the simulation results (e.g., Figures 3–7) lack detailed legends explaining the significance of each curve. Whereas visually evident, they may confuse readers unfamiliar with the variables or dynamics being depicted.

Response 12: Thank you for this feedback. I agree that the simulation result figures needed more detailed legends. Therefore, I have added detailed legends to all simulation result figures (Figures 4-12), explaining the significance of each curve. The following is an explanation using Figure 5, as an example. I have also followed similar formatting and content requirements for the other figures in the manuscript.

(Modified paragraph with detailed explanations, using Figure 5 as example)

"Figure 5. Impact of Different Initial Probabilities on High-risk Enterprise Strategy.
Note: In this figure, the horizontal axis represents the time of the simulation, and the vertical axis represents the probability of high-risk enterprises choosing compliant investment.
* The curves ①, ②, ③, and ④ represent the evolution of high-risk enterprises' compliant investment probability under different combinations of initial strategy probabilities for central and local governments. Specifically:
* Curve ①: x = 0.1, y = 0.5 (low central supervision, moderate local regulation)
* Curve ②: x = 0.9, y = 0.5 (high central supervision, moderate local regulation)
* Curve ③: x = 0.1, y = 0.9 (low central supervision, high local regulation)
* Curve ④: x = 0.1, y = 0.1 (low central supervision, low local regulation)
Where x represents the initial probability of the central government choosing strict supervision, and y represents the initial probability of local governments choosing regulation."

Comments 13: Tables that summarize stability results (e.g., Table 2) are organized, but they could benefit from a brief narrative in the main text explaining their significance, practical implications for governance.

Response 13: Thank you for this feedback. I agree that the stability results table needed more explanation. Therefore, I have added a paragraph after Table 3 to interpret its significance and discuss its practical implications. This added paragraph can be found in the stability analysis section of the Three-Party Evolutionary Game System Equilibrium Points, located on page 10.

From Table 3, we can see that since , and  have positive eigenvalue, they cannot be evolutionary stable strategies. If  or , then 、,  and are unstable points; if  or , then they are saddle points. Therefore, only , ,  and  can potentially be evolutionary stable points.

The conditions imply, in essence, that:

 E4(0,1,1): can only be an evolutionary stable strategy when all of its eigenvalues are negative, as per our model. This point implies a system where the central government chooses non-strict supervision, local governments choose non-regulation, and high-risk companies choose to be compliant. This situation can happen when, for example, local governments lack funding and incentives to regulate high-risk enterprises, and also the central government is not applying any type of supervision at all, but high-risk enterprises choose to invest in safety due to economic and insurance factors. An example of this could be in some mining regions, where the local government does not have enough resources and the central government is not focusing on that region, but due to workers unions and insurance companies, it is beneficial for the high risk enterprise to invest in security measures. If all eigenvalues are not negative, then the system will not converge towards that outcome, which could happen due to external intervention from either level of government, or large economical shifts that make security measures less beneficial for the high risk enterprises.

E6、E7 and E8, are all unstable or non-evolutionary stable strategies according to the original manuscript. This would mean that systems where local governments choose regulation are inherently unstable, as other actors could easily move away from a stable state, as there are more incentives for them not to choose their current actions. would represent a system where the central government regulates, local governments do not and companies comply due to economic incentives, as well as the potential penalties from the central government, and  would represent a system where local governments regulate and high risk companies comply due to economic incentives, however the central government does not participate.  would represent a system where all levels of government, and high risk companies choose to be active, however, according to our model, this would be inherently unstable, which might mean that it might only appear during periods of high scrutiny, but will eventually tend to move to a different more stable state. This also means that the systems in which the local government chooses regulation will only be stable under a particular set of circumstances.

Comments 14: Although "Simulation Analysis" section provide valuable insights, the authors do not explicitly link findings to specific real-world scenarios or industries, such as mining or construction, which could benefit from these policy recommendations.

Response 14: Thank you for pointing out the need to link findings to specific real-world scenarios. Therefore, I have added a paragraph in the conclusions and policy recommendations section, specifying how these findings can be applied to different industries, such as mining or construction. This added paragraph can be found on page 17, in the limitations section.

"This study also has some limitations. First, our model is based on theoretical assumptions and lacks validation through empirical data. Second, we are aware that the complexities of real-world governance may not be fully captured by the model’s assumptions and parameter settings. Third, our analysis might not be fully generalizable to other industries or regions with different regulatory frameworks. Future research could incorporate industry-specific characteristics, diverse regional contexts and empirical data to enhance the model's practical applicability. For example, the results of this study, although generally applicable, need to be adapted when used in the mining industry or the construction industry due to their unique characteristics. The enforcement of regulations and the types of penalties for companies may vary greatly, and the policy recommendations should be adjusted accordingly. In addition, in underdeveloped areas, policy recommendations, especially subsidy policies, need to take into account their actual resource constraints."This added paragraph is located on page 17,  in the conclusion part.

Comments 15: The simulations systematically vary key parameters (e.g., penalty intensity, subsidy levels) to ensure validity of the study. Though:

• The study does not use empirical data to validate the assumptions or parameter ranges.

• There are no external benchmarks to compare the simulation results with real-world governance scenarios.

• The absence of real-world data to validate the theoretical model limits its applicability and credibility. So, the integration real-world data (e.g., regulatory compliance statistics) would validate the model’s predictions and improve its relevance.

Response 15: Thank you for these important points about the limitations of the simulations. I fully agree that the lack of empirical data limits the study's applicability. Therefore, I have added several sentences throughout the manuscript to address these points directly. I have included the following changes:

• Acknowledged that the lack of empirical data is a limitation (page 17).

• Suggested the use of real-world data in future studies to validate the model (page 15 and 17).

• Recognized the need to compare the simulation results with real-world benchmarks (page 15).
  (Example from the limitations section, page 17)

“This study also has some limitations. First, our model is based on theoretical assumptions and lacks validation through empirical data. Second, we are aware that the complexities of real-world governance may not be fully captured by the model’s assumptions and parameter settings. Third, our analysis might not be fully generalizable to other industries or regions with different regulatory frameworks. Future research could incorporate industry-specific characteristics, diverse regional contexts and empirical data to enhance the model's practical applicability. For example, the results of this study, although generally applicable, need to be adapted when used in the mining industry or the construction industry due to their unique characteristics. The enforcement of regulations and the types of penalties for companies may vary greatly, and the policy recommendations should be adjusted accordingly. In addition, in underdeveloped areas, policy recommendations, especially subsidy policies, need to take into account their actual resource constraints.”
(Example from the robustness analysis, page 15)
“The robustness analysis not only validates the model's reliability but also provides insights into the practical implementation of supervision policies. Future research might further explore the dynamic interactions between these parameters under varying institutional contexts, and consider incorporating empirical data to validate the model and improve the practicality of the findings.”

Comments 16: The authors do not discuss limitations such as:

• The reliance on theoretical assumptions without empirical validation.

• Potential oversimplifications in modeling real-world governance complexities.

• The challenges of applying the findings to diverse industries or regions with varying regulatory frameworks.

• Add a section discussing the study’s limitations, such as its reliance on theoretical assumptions and the absence of empirical validation.

Response 16: Thank you for pointing out these important limitations. I agree that a section on limitations was necessary. Therefore, I have added a discussion in the "Conclusions and Policy Recommendations" section (page 17) to address all of the mentioned limitations.

“This study also has some limitations. First, our model is based on theoretical assumptions and lacks validation through empirical data. Second, we are aware that the complexities of real-world governance may not be fully captured by the model’s assumptions and parameter settings. Third, our analysis might not be fully generalizable to other industries or regions with different regulatory frameworks. Future research could incorporate industry-specific characteristics, diverse regional contexts and empirical data to enhance the model's practical applicability. For example, the results of this study, although generally applicable, need to be adapted when used in the mining industry or the construction industry due to their unique characteristics. The enforcement of regulations and the types of penalties for companies may vary greatly, and the policy recommendations should be adjusted accordingly. In addition, in underdeveloped areas, policy recommendations, especially subsidy policies, need to take into account their actual resource constraints.”

Comments 17: The conclusions suggest general policy recommendations without discussing their feasibility in specific real-world contexts (e.g., how resource constraints might limit the effectiveness of subsidies in underdeveloped regions).

Response 17: Thank you for this important point about feasibility. I agree that the policy recommendations should have more discussion about their implementation in diverse real world scenarios. Therefore, I have added comments to address feasibility, including the challenges of resource constraints in underdeveloped areas, at the end of the limitations paragraph, on page 17.

“This study also has some limitations. First, our model is based on theoretical assumptions and lacks validation through empirical data. Second, we are aware that the complexities of real-world governance may not be fully captured by the model’s assumptions and parameter settings. Third, our analysis might not be fully generalizable to other industries or regions with different regulatory frameworks. Future research could incorporate industry-specific characteristics, diverse regional contexts and empirical data to enhance the model's practical applicability. For example, the results of this study, although generally applicable, need to be adapted when used in the mining industry or the construction industry due to their unique characteristics. The enforcement of regulations and the types of penalties for companies may vary greatly, and the policy recommendations should be adjusted accordingly. In addition, in underdeveloped areas, policy recommendations, especially subsidy policies, need to take into account their actual resource constraints.”

 

 

Reviewer 3 Report

Comments and Suggestions for Authors

The research contribution intends to analyze the characteristics of stakeholder behaviors in the field of work safety governance by means of an evolutionary game model involving central government, local government, and high-risk enterprises.

Even though the research hypotheses are not particularly original, the research carried out justifies the efforts undertaken to define a new way to investigating the relationship between these three parties in work safety governance to properly coordinate them in their transition from "non-cooperation" to "cooperation".

The Evolutionary Game Theory is the most suitable for deepening and understanding the choice mechanisms of public officers and private managers and for depicting the dynamic changes in behavioral choices of various subjects in work safety governance towards a long-term equilibrium.

I therefore consider the choice of this theory and the model assumptions to be right and appreciable, and excellent the evolutionary game model construction as proposed.

The research defines ultimately a three-party evolutionary game model and various mechanisms of safety production governance under the supervision system, aimed at providing a critical explanation and practical recommendations to policymakers, public officers, practitioners and managers.

I consider the work well structured, robust and consistent with the existing scientific literature on this topic and my overall evaluation is highly positive.

Author Response

Thank you very much for your thorough review and highly positive evaluation of our manuscript. We greatly appreciate your recognition of the research contribution, our approach to analyzing stakeholder behaviors through an evolutionary game model, and the suitability of evolutionary game theory for this context. Your acknowledgment of our efforts to define a new way of investigating the relationship between the central government, local government, and high-risk enterprises, and our excellent model construction, is truly encouraging. We are also pleased that you found the research well structured, robust, and consistent with the existing scientific literature. Your insightful comments and positive feedback are invaluable to us.