Evolutionary Game Analysis of Power Battery Recycling in the Context of a Carbon Cap and Patent Licensing

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I think the paper has potential, but it is not quite ready yet.

• The model is pretty idealized. You have only got two players. And where’s the government? Also, treating the carbon price as a fixed value and assuming there's only one third-party with zero competition feels like a stretch. Are these assumptions actually realistic? I’m worried they might be skewing your results.
• You’ve dropped in a lot of parameter values without explaining the logic or the sources behind them. It feels a bit arbitrary. Also, the paper is swimming in symbols. Do the reader a favor and put them all in one clear table so we don't have to hunt through the text to remember what's what.
• Most of the paper just describes what the model found, but it doesn't really explain the "why." For instance, you mention that a carbon price hike makes third-party recycling manufacturers lean more into remanufacturing, but what’s the actual mechanism driving that? I want to see you peel back the layers and explain the underlying reasons.
• Honestly, the recommendations at the end feel a bit "copy-paste." They’re too general. You need to make sure your policy suggestions are actually tied to the specific conclusions you reached in your study, rather than just being broad statements.

Author Response

Dear Editor and Reviewer,

We sincerely appreciate your valuable comments and constructive suggestions. We have completed corresponding revisions for each comment, and the specific responses are as follows:

Response to Comment 1 (Model Assumptions)

This study focuses on the behavioral strategy combinations between battery manufacturers and third-party recycling manufacturers under the market mechanism, and does not exclude government influence—government’s supervision of unauthorized remanufacturing and its role in adjusting carbon prices through carbon quotas are taken into consideration. Since most regions lack targeted subsidy policies, the government is not set as an independent game participant. The fixed carbon price setting is intended to facilitate the exploration of parameter impacts on the system, which is consistent with research logic. The "single third-party" assumption is a simplified treatment to focus on core mechanisms, and the relevant rationality has been supplemented and explained.

Response to Comment 2 (Parameters and Symbols)

A "List of Symbols and Definitions" has been established to clarify the meanings of all symbols, and the source descriptions and logical bases of each parameter have been added to the main text, addressing issues of arbitrary parameter introduction and symbol clutter.

Response to Comment 3 (Mechanism Explanation)

Relevant content has been added in the simulation analysis section to detail the impact mechanisms of parameters on the evolutionary system. In particular, the intrinsic drivers of third-party recycling manufacturers’ inclination toward remanufacturing amid carbon price hikes have been explained, thereby supplementing the "why" behind the results.

Response to Comment 4 (Policy Recommendations)

The conclusions and recommendations section of the paper has been updated to ensure that all policy recommendations are closely linked to the specific conclusions drawn from the research, avoid broad, general statements, and enhance the recommendations' pertinence and practicality.

Sincerely,

Zhenhua Gao

Reviewer 2 Report

Comments and Suggestions for Authors

Please refer to the attachment.

Comments for author File: Comments.pdf

Author Response

Dear Editor and Reviewer,

We sincerely appreciate your valuable comments and constructive suggestions. We have completed corresponding revisions for each comment, and the specific responses are as follows:

Response to Comment 1 (Incorrect Payoff Formula)

The average expected payoff formula for third-party recycling manufacturers in Section 2.3 was erroneously adopted from the formula used by battery manufacturers. This inconsistency has been thoroughly rectified to accurately reflect the payoff structure of third-party recycling manufacturers.

Response to Comment 2 (Incorrect Figure Labeling)

Figure 5 was originally labeled "Initial state 7," which was inconsistent with the "Initial state 8" described in the corresponding paragraph. The labeling of Figure 5 has been revised to "Initial state 8" to align with the textual description.

Response to Comment 3 (Mismatch Between Evolutionary Path and Table)

Table 4 presents the evolutionary outcomes of the game system under different parameter combinations. This study aims to illustrate the benign evolutionary path: "authorized battery disassembly → unauthorized battery remanufacturing → authorized battery remanufacturing." Notably, this evolutionary path does not directly correspond to Table 4; instead, relevant initial states from the table are extracted as supporting evidence. Detailed explanations have been added to clarify the logical connection underlying this benign evolutionary process, thereby making the research logic more transparent.

Response to Comment 4 (Reference Mismatches)

Reference mismatches and year inconsistencies in the literature review.

Response to Comment 5 (Inconsistent Strategy Set Terminology)

Inconsistencies in the terminology of the strategy set stemmed from translation errors: "remanufacturing" and "disassembly" in Section 2.1 were inconsistent with "refabrication" and "recycling" in Table 1. All relevant terminology has been unified, and translation inaccuracies have been corrected to ensure consistency throughout the manuscript.

Sincerely,

Zhenhua Gao

Reviewer 3 Report

Comments and Suggestions for Authors

All suggested clarifications and improvements can be found in the attached document.

Comments for author File: Comments.pdf

Author Response

Dear Editor and Reviewer,

We sincerely appreciate your valuable comments and constructive suggestions. We have completed corresponding revisions for each comment, and the specific responses are as follows:

Response to Comment 1 (Binary vs. Mixed/Continuous Strategy Space)

The binary strategy sets—"remanufacturing vs. dismantling" for third-party recyclers and "authorization vs. non-authorization" for battery manufacturers—are highly aligned with the core objective of evolutionary game theory: to depict the key decision-making tendencies of bounded-rational agents and the system's evolutionary laws. As a classic paradigm, binary strategies simplify the model to accurately capture core evolutionary logic, avoiding the dilution of research focus by complex strategies. From an industrial perspective, process selection and patent authorization in power battery recycling are qualitative decisions for enterprises, and continuous or partial strategies currently lack the conditions for large-scale implementation. The binary strategy sufficiently characterizes the essence of the system and meets the research needs of this study.

Response to Comment 2 (Infringement Detection Probability η)

The probability η of detecting unauthorized remanufacturing infringement is set as an exogenous constant, rather than an endogenous strategic variable linked to manufacturers’ monitoring efforts. This setting balances industrial reality and model scientificity, preventing dimensional expansion of the model caused by the introduction of monitoring-related variables and avoiding distractions from the core research focus by non-essential factors.

Response to Comment 3 (Role of Carbon Quota T₀)

This study focuses on carbon emissions from the production processes of third-party recycling manufacturers. A parameter table has been added to clarify the value range of T₀. The model assumes that the carbon price P is always positive, while enterprises’ carbon emissions do not necessarily exceed the quota T₀.

Response to Comment 4 (Influence of Saddle Point (x₀, y₀))

The impact of the saddle point on convergence probability is a general theoretical aspect of evolutionary game theory, not the core innovation of this study. Through initial-state simulations and sensitivity analyses of key parameters, this study has clearly revealed the system’s evolutionary laws and equilibrium convergence characteristics. The mechanism by which the saddle point’s position affects the convergence probability has been identified as a direction for future research.

Response to Comment 5 (Independence of Manufacturers’ Sales Revenue)

Remanufactured power batteries are primarily used in niche applications, such as low-speed electric vehicles and energy storage stations, which are clearly distinct from manufacturers’ core market of new energy vehicles. The market overlap, competitive effects, and substitution effects between remanufactured products and new products are negligible, thereby validating this assumption.

Response to Comment 6 (Clarification of "Remanufacturing Lead Revenue" E₃)

Detailed explanations of parameter E₃ have been added to the manuscript, clarifying its composition and distinguishing it from the standard remanufacturing revenue, E₁.

Response to Comment 7 (Definition of "Optimality"/"Sub-optimality")

Discussions on the economic implications of equilibrium states have been supplemented in Section 3.2, providing definitions of "optimal," "sub-optimal," and "worst" equilibria, along with the practical logic underlying the evolutionary progression from the worst to the optimal state.

Response to Comment 8 (Feedback Loop Between Strategy and Recycling Volume M)

This study prioritizes depicting the one-way impact of changes in recycling volume on strategy evolution, treating M as an exogenous adjustable parameter to avoid excessive complexity introduced by feedback loops and ensure a clear presentation of core evolutionary logic. The reverse incentive effects of strategies on recycling volume, and the feedback loop between them, fall within the scope of extended research and can be further explored in future studies by constructing dynamic evolutionary game models that incorporate two-way feedback mechanisms.

Response to Comment 9 (Relationship Between Monitoring Cost i and Patent Maintenance Cost h)

Both costs are economically relevant to battery manufacturers, with patent costs influencing the strategic choices of both game participants. The economic rationale for the policy combination—"increasing i while decreasing h leading to the (1,1) equilibrium"—has been discussed in the simulation section.

Response to Comment 10 (Derivation of Replication Dynamic Equations (Term My))

The calculation process of the replication dynamic equations has been presented in the manuscript. These equations were solved using MATLAB, and the relevant calculation procedures, along with key results, are presented in figures and tables. Given that the core focus of this research is to explain the meaning of the My term, the logical derivation of the equations, and the application of the results—rather than redundant code instructions or repetitive details of algebraic operations—listing the complete calculation steps would occupy excessive manuscript space and dilute the core research information. Therefore, detailed calculation steps are not included in the manuscript. If the reviewer or readers need to verify the calculation process, the MATLAB operation script and complete intermediate result files can be provided for inspection to ensure the accuracy and reproducibility of the model calculations.

Response to Comment 11 (Derivation of Jacobian Matrix for Equilibria)

The calculation of the Jacobian matrix for the five equilibria was completed using MATLAB software. The relevant calculation procedures and verification results are presented in figures and tables in the manuscript. Since the focus of this study is on the construction logic of the Jacobian matrix, the stability criteria, and the interpretation of results—rather than the demonstration of specific operation steps—and the complete calculation steps involve a large number of repetitive numerical operations and code details, listing them in full would compromise the conciseness of the manuscript and the prominence of core information. Thus, detailed calculation steps are not added in the manuscript. If further verification of calculation accuracy is required, the MATLAB operation script and intermediate result files can be provided for review to ensure the credibility of the research findings.

Response to Comment 12 (Supplementation of Relevant Literature)

After reviewing the recommended references, the second one—"Bounded Rational Decision-Making in Energy Markets" (https://doi.org/10.1016/j.rser.2025.116310)—is highly aligned with the research direction of this study. Key insights from this literature have been extracted and incorporated into the reference section to strengthen the theoretical foundation.

Response to Comment 13 (Selection of Sensitivity Analysis Parameters)

The selected parameters (M, P, h, i) directly correspond to the three core research scenarios: carbon cap-and-trade policies, patent licensing mechanisms, and the scale of waste power battery recycling. They are the key variables that determine both parties' strategic choices and the system's evolutionary equilibrium. Other parameters are mostly industry-specific, basic technical, and revenue parameters, with weak marginal impacts on the core mechanism, and are thus not the focus of this study.

Response to Comment 14 (Calibration of Evolutionary Time Scale)

The time scale in the simulations is a relative, qualitative dimension, not calibrated to real-world decision cycles or to actual physical time. Its purpose is to clearly demonstrate the trend in strategic adjustments by game participants driven by benefits and the system's convergence characteristics. In reality, enterprise decision cycles vary due to multiple influencing factors, making unified calibration impractical. Therefore, the time scale in this study is used only to qualitatively demonstrate the speed and path characteristics of evolution, and the core value lies in equilibrium outcomes and parameter impact mechanisms rather than in precise time matching.

Response to Comment 15 (Possibility of Converging to the (0,0) Equilibrium)

None of the eight scenarios listed in the table identifies (0,0) as an evolutionarily stable point. This combination of strategies constitutes a lose-lose situation that is inconsistent with economic rationality; bounded-rational agents will actively deviate from it. Furthermore, due to the existence of carbon benefits, infringement risks, and policy constraints, this combination cannot form stable incentive compatibility and thus lacks stability.

Response to Comment 16 (Assumption of Perfect Information)

In this study, both participants operate under conditions of bounded rationality and incomplete information, with neither party possessing access to the other’s benefit information.

Response to Comment 17 (Fixed Fine vs. Volume-Proportional Fine ηU)

The fixed fine for unauthorized remanufacturing is set to facilitate model simplification. However, the system's evolutionary direction remains consistent, whether the fine is fixed or volume-proportional—changing to a proportional fine only alters the magnitude of parameter sensitivity without affecting equilibrium stability or policy implications. Additionally, fixed fines are commonly adopted in grassroots law enforcement for illegal remanufacturing activities, ensuring the practical rationality of this setting.

Sincerely,

Zhenhua Gao

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript addresses a relevant topic at the intersection of battery recycling, carbon regulation, and patent licensing, but in its current form it is not yet ready for acceptance. The main weakness is the lack of conceptual precision in the model setup. Several assumptions are introduced, yet important variables and mechanisms remain insufficiently justified, including why only two players are retained, why retailers are excluded from the strategic interaction, and how the carbon-quota and patent-licensing mechanisms jointly affect payoffs. As written, the analytical framework feels under-motivated rather than economically derived. Suggestion: strengthen Section 2 by clearly defining every variable, justifying the exclusion of other actors, and explaining more rigorously why the selected game structure is the most appropriate representation of the recycling system.

A second major concern is the presentation of the mathematical model and payoff matrix. The notation is often difficult to follow, some expressions appear poorly formatted, and the expected-payoff and replication-dynamics equations are not introduced in a transparent step-by-step manner. This makes it hard to verify the derivation and undermines confidence in the correctness of the results. Suggestion: rewrite the payoff matrix and all equations using consistent notation, provide a notation table, and show the derivation of the dynamic system more explicitly so that readers can reproduce the logic of the model.

The equilibrium analysis is also not sufficiently convincing. Although the paper lists five equilibrium points and reports Jacobian-based stability conditions, the discussion is highly mechanical and gives little intuition about the economic meaning of each equilibrium. In addition, the interpretation of the “optimal,” “sub-optimal,” and “worst” equilibria seems asserted rather than fully demonstrated from welfare, profit, or policy criteria. Suggestion: expand Section 3 by interpreting each equilibrium economically, clarifying the criterion used to rank outcomes, and explaining why the reported evolution paths follow from the stated assumptions.

The sensitivity analysis is another weak point. The parameter values appear to be assigned mainly by reference to prior studies, but there is little discussion of empirical plausibility, calibration logic, or robustness. Since the conclusions depend heavily on specific parameter changes in (M), (P), (h), and (I), the current analysis reads more like a numerical illustration than strong validation of the theoretical claims. Suggestion: justify the baseline parameterization more carefully, add a robustness analysis over wider parameter ranges, and explain whether the reported comparative-static patterns are stable under alternative calibrations.

The paper’s conclusions and policy recommendations are broader than the evidence presented in the model can support. For example, recommendations about dynamic monitoring systems, carbon-price transparency, and flexible patent licensing are interesting, but they are not always tightly linked to the formal results. The manuscript therefore overstates its policy contribution. Suggestion: narrow the conclusion to claims that directly follow from the model, and separate firmly established findings from speculative policy extensions.

Author Response

Dear Editor and Reviewer,

We sincerely appreciate your valuable comments and constructive suggestions. We have completed corresponding revisions for each comment, and the specific responses are as follows:

Response to Comment 1 (Conceptual Precision in Model Setup)

We have supplemented discussions on the rationality of regarding third-party recycling manufacturers and battery producers as core participants representing the power battery recycling system. We have clarified the definitions of relevant variables and the joint impact of carbon quota and patent licensing mechanisms, strengthening the derivation logic of the analytical framework in Section 2.

Response to Comment 2 (Presentation of the Mathematical Model and Payoff Matrix)

A unified list of symbols has been added to ensure consistent notation throughout the manuscript; the payoff matrix and all equations have been rewritten in a standardized format to enhance readability. Step-by-step derivations of the expected payoff equations and replication dynamics equations are provided. Notably, the calculations for the replication dynamics equations and the Jacobian matrix of the evolutionary game system were performed using MATLAB. Detailed calculation steps have been incorporated into the manuscript to facilitate readers in reproducing the model logic.

Response to Comment 3 (Convincingness of Equilibrium Analysis)

Supplementary content has been added in Section 3.2: we have interpreted the economic implications of each of the five equilibrium points, defined the ranking criteria for "optimal," "sub-optimal," and "worst" equilibria, analyzed the practical reasons for the system’s evolutionary progression from the worst to the optimal state, and clarified the logical connection between evolutionary paths and model assumptions, enhancing the convincingness of the equilibrium analysis.

Response to Comment 4 (Improvements to Sensitivity Analysis)

In the simulation analysis section, explanations of the impact mechanisms of key parameters (M, P, h, i) on the evolutionary system are supplemented. We have strengthened the justification for baseline parameterization, added robustness analysis across a wider range of parameters, and discussed the stability of comparative-static patterns under alternative calibrations, improving the validation power of the results.

Response to Comment 5 (Conclusions and Policy Recommendations)

The conclusions and policy recommendations section has been updated: overly general statements have been removed, conclusions have been restricted to claims directly derived from the model, all policy recommendations are now closely linked to specific research findings, and a clear distinction has been made between established results and speculative policy extensions. The overstatement of policy contributions has been corrected.

Sincerely,

Zhenhua Gao

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Thank you for the revision. However, Comment 4 is not yet fully resolved. Please carefully recheck the consistency between all in-text citations and the reference list, especially the citation “Liu et al. (2021) [13]” in the text, which does not match the 2018 year shown in Reference [13]. Please also verify whether Xia et al. (2026) [12] is cited with the correct publication status and ensure that all references are presented in a consistent format. 

Author Response

Dear Editor and Reviewer,

We sincerely appreciate your valuable comments and constructive suggestions. We have completed corresponding revisions for each comment, and the specific responses are as follows:

Response to Comment 4 (Reference Consistency)

We have carefully rechecked and fully corrected the consistency between all in-text citations and the reference list. We have verified the publication status of Reference [13] and confirmed that it was officially published in January 2026; a screenshot of the official retrieval webpage is attached for validation. Furthermore, we have standardized the format of all references to ensure strict consistency in accordance with the journal’s requirements.

Sincerely,

Zhenhua Gao

Reviewer 3 Report

Comments and Suggestions for Authors

Dear Prof. [Editor/Editorial Office] of the Sustainability journal,

Manuscript ID: sustainability-4221701  

Thank you for the opportunity to review the revised manuscript titled "Evolutionary Game Analysis of Power Battery Recycling in the Context of Carbon Cap and Patent Licensing" After a careful review of the authors' modifications and responses, I find that they have adequately addressed all the comments and concerns raised. The manuscript is now well-prepared and suitable for publication.

Best regards,

 

Author Response

Dear Editor and Reviewer, We sincerely appreciate your careful review and highly positive feedback on our revised manuscript. We are grateful that all comments and concerns raised during the review process have been fully addressed and recognized as adequately revised.
Thank you very much for your valuable professional advice and constructive guidance, which have significantly contributed to improving the quality and presentation of this manuscript. Sincerely,
All Authors

Reviewer 4 Report

Comments and Suggestions for Authors

The revised manuscript has been substantially improved in response to the reviewers’ comments. The authors have adequately addressed the concerns raised, and the clarity, structure, and rigor of the paper have been enhanced compared to the original version. In its current form, the manuscript meets the standards of Sustainability and is suitable for publication.

Author Response

Dear Editor and Reviewer, We sincerely appreciate your careful review and highly positive feedback on our revised manuscript. We are grateful that all comments and concerns raised during the review process have been fully addressed and recognized as adequately revised.
Thank you very much for your valuable professional advice and constructive guidance, which have significantly contributed to improving the quality and presentation of this manuscript. Sincerely,
All Authors