Evolutionary game analysis of stakeholder privacy management in the AIGC model

Lv, Yali; Yang, Jian; Sun, Xiaoning; Wu, Huafei Article Evolutionary game analysis of stakeholder privacy management in the AIGC model Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: Lv, Yali; Yang, Jian; Sun, Xiaoning; Wu, Huafei (2025) : Evolutionary game analysis of stakeholder privacy management in the AIGC model, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 14, pp. 1-14, https://doi.org/10.1016/j.orp.2025.100327 This Version is available at: https://hdl.handle.net/10419/325804 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Contents lists available at ScienceDirect Operations Research Perspectives journal homepage: www.elsevier.com/locate/orp Evolutionary game analysis of stakeholder privacy management in the AIGC model Yali Lv, Jian Yang ∗, Xiaoning Sun, Huafei Wu School of Information, Shanxi University of Finance and Economics, Taiyuan, 030006, Shanxi, China ARTICLE INFO Keywords: AIGC Data privacy Evolutionary game Replicator dynamic equations ABSTRACT The technological development powered by Artificial Intelligence Generated Content (AIGC) models, exemplified by Generative Pre-trained Transformer 4 (GPT-4) and Bidirectional Encoder Representations from Transformers (BERT), has completely transformed machine language processing and fostered substantial technological advancements. However, their extensive deployment has amplified concerns regarding data privacy risks, which are attributed not only to technological vulnerabilities but also to the intricate conflicts of interest among model providers, application service providers, and privacy regulators. To tackle this challenge, this research develops a tripartite evolutionary game model that examines the strategic interactions and dynamic relationships among large language model providers, application service providers, and privacy regulatory agencies. By employing replicator dynamic equations and Jacobian matrices, the research investigates the stability of strategic equilibria and simulates optimal adjustment paths across diverse policy scenarios. Drawing on the research findings, this paper offers practical recommendations to strengthen data privacy protection in large language models, delivering a solid theoretical foundation for policymakers and industry practitioners. 1. Introduction Large language models as a significant breakthrough in artificial intelligence (AI) technology, are reshaping service modes across multiple domains. Artificial Intelligence Generated Content (AIGC) models represented by Generative Pre-trained Transformer 4 (GPT-4) and Bidirectional Encoder Representations from Transformers (BERT) [1] demonstrate exceptional capabilities in semantic understanding [2] and content generation [3], particularly in highly specialized fields such as medical diagnostic assistance [4] and financial analysis, significantly enhancing service efficiency and decision support capabilities through accurate comprehension of professional texts and contextual relationships [5]. These models not only handle routine language tasks but also deeply understand domain-specific requirements, bringing innovative solutions to various industries. With the extensive application of large language models in sensitive fields such as therapy [6], education [7], and healthcare [8– 10], their data security risks have become increasingly prominent. Research indicates these risks manifest in three aspects: first, potential leakage of users’ personal information due to model memory mechanisms [11]; second, malicious attacks against models, including inference reconstruction [12] and exploitation of personalized configuration vulnerabilities [13]; and third, data leakage risks at the technical interface level [14]. These multi-dimensional security challenges require ∗Corresponding author. E-mail address: [email protected] (J. Yang). not only technical protection measures but also the establishment of comprehensive regulatory frameworks and industry standards. Privacy protection for large language models is a complex systems engineering challenge involving collaboration among multiple stakeholders [15]. This system encompasses various entities including data providers, technology developers, service users, and regulatory agencies, with complex interactions and trade-offs among them [16]. Each participant’s decisions and behaviors affect the overall effectiveness of privacy protection: from data providers’ privacy awareness to the security design of technical solutions to the formulation and implementation of regulatory policies, all require coordination and optimization within a unified framework. This study focuses on these complex systemic characteristics, attempting to construct an analytical model that reflects the interaction mechanisms among all parties. This study adopts evolutionary game theory (EGT) as its theoretical framework, leveraging its unique advantages in analyzing multiagent dynamic decision processes [17–19]. By constructing a tripartite evolutionary game model, this study dynamically tracks the strategy evolution processes among large language model providers, application service providers, and regulatory agencies, while identifying the Nash equilibrium—a stable state where no party can improve its payoff by unilaterally changing its strategy [20]. The model examines several key https://doi.org/10.1016/j.orp.2025.100327 Received 13 September 2024; Received in revised form 13 January 2025; Accepted 5 February 2025 Operations Research Perspectives 14 (2025) 100327 Available online 13 February 2025 2214-7160/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). Y. Lv et al. variables, including the economic benefits of all parties, violation costs, regulatory efficiency, and the social reputation impacts. This analytical framework not only uncovers the optimal strategies for different parties under varying conditions but also predicts the long-term evolutionary trends of the system, thereby providing robust theoretical support for developing privacy protection policies that balance efficiency and security. This study establishes a novel tripartite evolutionary game model to analyze the interaction mechanisms among large language model providers, application service providers, and privacy regulatory agencies, demonstrating significant value in both theoretical contributions and practical implications: (a) From a theoretical perspective, the model thoroughly examines key factors such as provider profit growth rates, penalty amounts, social reputation of regulatory agencies, and regulatory costs, offering a novel analytical perspective for large-scale artificial intelligence (AI) privacy governance. (b) From a practical standpoint, the research findings provide actionable policy recommendations for balancing compliance requirements and stakeholder incentives. However, the study is limited by its simplified assumptions in the evolutionary game framework and reliance on simulation data, which may restrict the applicability of its findings to real-world scenarios. Further empirical research is required to enhance its generalizability and practical impact. 2. Related work 2.1. Large language models Large language models, epitomized by ChatGPT, have showcased their astonishing emergent capabilities, further propelling technological breakthroughs in the direction of General Artificial Intelligence (AGI). In this revolutionary shift within the AI paradigm, the academic, industrial, and research communities are actively exploring and studying the potential of large language models, embarking on a series of experiments and applied research. Vaswani, A et al. [21] introduced the self-attention mechanism in their proposed Transformer architecture, a significant innovation that enhanced the capabilities of NLP and laid the foundation for the training of large language models. Ouyang [22] built upon the GPT-3 architecture, incorporating instruction based learning and reinforcement learning from human feedback to guide the model’s training utilizing fine tuning and policy alignment and successfully developed InstructGPT and ChatGPT. Zeng et al. [23] have trained the PanGualpha large-scale autoregressive language model, which shows remarkable capabilities in various scenarios based on massive high-quality Chinese industry data within the MindSpore framework. In the research and development of large language models, studies on data privacy protection and corresponding AI governance are of paramount importance. Xu et al. [24] reviewed key privacy-preserving machine learning (PPML) techniques, such as differential privacy, homomorphic encryption, and secure multi-party computation. They identified significant challenges, including high computational costs, scalability limitations, and trade-offs between privacy and utility, while proposing strategies to integrate these techniques into large-scale AI systems. Li et al. [25] conducted a comprehensive analysis of privacy attacks and defense strategies for large language models, categorizing the types of attacks based on the capabilities of potential attackers and revealing critical vulnerabilities within LLMs. Previous studies have demonstrated that attackers can extract or reconstruct precise training samples from LLMs, potentially leading to the leakage of personal identity information. To mitigate this risk, Rehnia et al. [26] proposed a novel framework known as EW-Tune. This framework employs advanced gradient perturbation techniques to safeguard a limited number of samples while introducing minimal noise. Kandpal N et al. [27] discovered that reducing duplicate data in the training sets of large language models significantly decreases the likelihood of privacy breaches when handling sensitive information, thereby enhancing the overall security of the models regarding data privacy. Recent advancements in privacy-preserving computation technologies have further expanded this field. For instance, Xu et al. [28] proposed a federated learning framework that incorporates privacy-preserving data pricing, ensuring sensitive data remains protected while enabling equitable valuation across stakeholders. This approach underscores the potential of integrating advanced privacy-preserving mechanisms into large language model ecosystems to balance data utility and privacy in multi-stakeholder scenarios. Rajagopal M et al. [29] introduced a conceptual framework for AI governance in public administration, combining regulatory theory and ethical principles. The framework emphasizes transparency, accountability, and stakeholder collaboration while addressing challenges such as bias, privacy concerns, and the complexity of AI systems. 2.2. Game theory Game theory has become a widely used tool [30,31]. Scholars in academia have analyzed the game behavior of various stakeholders in different scenarios, providing solid theoretical support for logical decision-making processes within these fields. Zhang et al. [32] proposed a game-theoretic framework for privacy-preserving federated learning, referred to as the Federated Learning Privacy Game. This framework considers the strategic interactions between defenders and attackers, accounting for computational costs, model utility, and privacy leakage risks. By addressing incomplete information scenarios, the study provides a structured approach to balancing privacy protection and performance in federated learning environments. Shah H et al. [33] reviewed the applications of game theory models in privacy protection, cybersecurity, intrusion detection, and resource optimization. Xu et al. [34] transformed the privacy issues arising from data collection, anonymization, and release into a game problem. Within this framework, they explored the interactive behaviors amongst data providers, collectors, and users, utilizing a game model based on k-anonymity to propose a general method for finding Nash equilibria. ‘‘Free-riding’’ is a common behavior studied in game theory, often observed among stakeholders in supply chains where some participants benefit from shared resources or cooperative efforts without contributing proportionally to the associated costs. For example, Ju et al. [35] highlighted that in the adoption of blockchain technology within shipping supply chains, certain stakeholders may strategically avoid investing in the technology while still reaping its benefits. Sagduyu Y E et al. [36] introduces a game-theoretic framework to analyze free-riding behavior in federated learning (FL) over wireless networks. The study highlights how selfish clients, seeking to avoid computational and communication costs, engage in free-riding by not participating in model updates while still benefiting from the global model. This behavior adversely impacts the accuracy of the global model and reduces overall system utility. By formulating a non-cooperative game, the research derives Nash equilibrium strategies for free-riding probabilities and quantifies the trade-offs between participation costs and global accuracy. The results emphasize the need for incentive mechanisms to mitigate free-riding and enhance FL’s resilience while preserving privacy through decentralized data sharing. In addition to evolutionary game theory (EGT), several foundational approaches have been explored in the context of privacy protection within multi-stakeholder AI governance. Static game theory is effective for analyzing single-shot interactions with fully rational participants; however, it lacks the capacity to model the long-term evolution of strategies, rendering it unsuitable for capturing the iterative adjustments observed in dynamic multi-party scenarios like privacy investments and regulatory actions [37,38]. Agent-based modeling offers Operations Research Perspectives 14 (2025) 100327 2 Y. Lv et al. detailed simulations of micro-level interactions and accounts for stakeholder heterogeneity, making it valuable for studying decentralized systems and emergent behaviors; yet, it lacks the theoretical generalizability and analytical precision of equilibrium-based methods like EGT, which are more adept at deriving global strategic insights. System dynamics excels at analyzing macro-level trends and modeling feedback loops and long-term system behaviors but struggles to represent the nuanced and strategic interactions among individual stakeholders essential in privacy protection and investment decisions [39,40]. In contrast, EGT addresses these limitations by modeling bounded rationality and the dynamic evolution of strategies over time, effectively capturing the interplay of competition and cooperation among stakeholders under conditions of uncertainty, thereby making it well-suited for analyzing privacy governance in AI ecosystems. Building upon the strengths of EGT, this study explores its application to the domain of user privacy protection in the context of large language models, where dynamic and iterative interactions among stakeholders are particularly prominent. To achieve this, a tripartite evolutionary game model is developed, offering an in-depth analysis of privacy protection strategies and their evolution among key participants. Unlike traditional game theory, which assumes fully rational behavior, this study adopts the more realistic framework of bounded rationality, enabling a nuanced understanding of strategic decisionmaking under uncertainty. Through numerical simulations, the study further investigates how vested interests drive strategic adjustments during ongoing interactions, providing valuable insights into the challenges and opportunities of privacy governance in large language model ecosystems. 3. Basic assumptions and model construction 3.1. Model assumptions To ensure the realism and validity of the constructed model, this study grounds its assumptions in empirical evidence and industry practices observed in the AI supply chain. Specifically, the roles of large language model providers (Participant 1), application service providers (Participant 2), and privacy regulatory authorities (Participant 3) are consistent with the stakeholder interactions described in existing literature. For instance, [41] highlights the complex dynamics among stakeholders in the large language model supply chain, including the need for coordinated investment in privacy and security measures. Similarly, [42] emphasizes the bounded rationality of stakeholders and the importance of balancing costs, benefits, and regulatory pressures. In this context, the strategic choices of these participants are modeled as evolving over time and stabilizing at optimal strategies, as summarized in Table 1. Assumption 1. There are three key game participants in the data privacy game and investment decision-making. Firstly, large language model providers (S) possess the technology frameworks of deep learning and machine learning, large language model interfaces, as well as related storage and computational capabilities. They choose to invest in data privacy protection with a probability of 𝑥, and choose not to with a probability of 1 −𝑥. Secondly, application service providers (H) utilize the basic technology offered by large language model providers to provide sector-specific software solutions, support, and maintenance. They choose to invest in data privacy protection with a probability of 𝑦, and not to with a probability of 1 −𝑦. The third entity is privacy regulatory authorities (G). Their responsibilities include issuing privacy protection guidelines and conducting technical audits and certifications. They also have the power to investigate and penalize non-compliant behaviors, with the assumption that the probability of them enforcing strict regulation is 𝑧, and lax regulation is 1 −𝑧.𝑥,𝑦, and 𝑧are all defined within the interval [0,1]. Assumption 2. From the perspective of large language model providers, when neither they nor the application service providers invest in privacy protection, their profit is 𝑃𝑆. However, under the supervision of regulatory authorities, large language model providers will face a fine of 𝐹. Even if large language model providers choose to invest in privacy protection (at a cost of 𝐶𝑆), they cannot ensure that the privacy protection measures are 100% effective, and there is a risk of failure. This means they may still face fines of 𝐹due to privacy breaches. Nevertheless, investing in privacy protection can generally be expected to increase their profit to (1 +𝛼0)𝑃𝑆, where 𝛼0is the profit growth rate when large language model providers invest in privacy protection alone. Additionally, under strict regulation by regulatory authorities, large language model providers who invest in privacy protection may receive an additional reward subsidy of 𝐹. When large language model providers do not invest while application service providers do, they can free-ride and obtain an additional benefit of 𝜉𝑆. However, due to not investing in privacy protection themselves, there is a higher risk of privacy breaches, which may lead to fines of 𝐹. Assumption 3. From the perspective of application service providers, when neither they nor the large language model providers invest in privacy protection, their profit is 𝑃𝐻. However, under strict regulation, they will face a fine of 𝐹. Even if application service providers choose to invest in privacy protection (at a cost of 𝐶𝐻), the privacy protection measures may not be completely effective, and there is a risk of failure, leading to the possibility that they may still face fines of 𝐹due to privacy breaches. Nevertheless, investing in privacy protection can generally be expected to increase their profit to (1 +𝛽0)𝑃𝐻, where 𝛽0 is the profit growth rate when application service providers invest in privacy protection alone. Furthermore, under strict regulation, application service providers who invest in privacy protection may receive an additional reward subsidy of 𝐹. When application service providers do not invest while large language model providers do, they can free-ride and obtain an additional benefit of 𝜉𝐻. However, due to not investing in privacy protection themselves, there is a higher risk of privacy breaches, which may lead to fines of 𝐹. Assumption 4. The motivations for regulatory authorities to enforce strict regulations arise from fiscal, reputational, and ethical considerations. Strict regulation incurs a cost of 𝐶𝐺. Non-investing providers are penalized with a fine 𝐹, while investing providers receive an equivalent subsidy of 𝐹. Strict regulation enhances the social reputation of regulatory authorities, with an increase of 𝑅0. In contrast, lax regulation results in no reputation gain, and ineffective regulation leads to a reputation loss of 𝐿. When all providers choose to invest, regulatory authorities achieve the maximum reputation gain, 𝑅1, where 𝑅1> 𝑅0. In scenarios where regulatory authorities act as government entities, their decisions are not solely driven by fiscal or reputational factors. Ethical responsibilities, such as protecting public interests and ensuring privacy standards, may also influence their choices. While the current model simplifies this complexity by focusing primarily on reputational considerations, future extensions could incorporate additional factors (e.g., ethical responsibilities or public expectations) into the decisionmaking framework to better reflect the multifaceted motivations of government regulators. Assumption 5. In industrial practice, due to constraints of funding, technical challenges, and difficulties in the evaluation process, regulatory authorities are unable to comprehensively monitor strategies like privacy investment or free-riding. Therefore, the actual fines may be lower than the reputation gains due to regulation, i.e., 2𝐹 < 𝑅0. Assumption 6. When both large language model providers and application service providers choose to invest in privacy protection, both parties achieve a win–win situation. At this time, the profit growth rate Operations Research Perspectives 14 (2025) 100327 3 Y. Lv et al. Table 1 Main symbols used in the paper. Symbol Description Unit 𝑥The probability that large language model providers (S) invest in data privacy protection, with 𝑥∈ [0,1] – 𝑦The probability that application service providers (H) invest in data privacy protection, with 𝑦∈ [0,1] – 𝑧The probability that privacy regulatory authorities (G) enforce strict regulation, with 𝑧∈ [0,1] – 𝑃𝑆The profit of large language model providers when not investing in privacy protection Monetary Unit (e.g., $) 𝑃𝐻The profit of application service providers when not investing in privacy protection Monetary Unit (e.g., $) 𝐶𝑆The investment cost of privacy protection for large language model providers Monetary Unit (e.g., $) 𝐶𝐻The investment cost of privacy protection for application service providers Monetary Unit (e.g., $) 𝐶𝐺The fiscal expenditure produced by regulatory authorities for enforcing strict regulation Monetary Unit (e.g., $) 𝐹The fine imposed by regulatory authorities for non-compliance Monetary Unit (e.g., $) 𝛼0The profit growth rate of large language model providers when investing alone in privacy protection Fraction (e.g., 0.1) 𝛼1The profit growth rate when both large language model providers and application service providers invest, 𝛼1> 𝛼0>0 Fraction (e.g., 0.1) 𝛽0The profit growth rate of application service providers when investing alone in privacy protection Fraction (e.g., 0.1) 𝛽1The profit growth rate when both application service providers and large language model providers invest, 𝛽1> 𝛽0>0 Fraction (e.g., 0.1) 𝜉𝑆The additional benefit that large language model providers obtain from free-riding on the investments of application service providers Monetary Unit (e.g., $) 𝜉𝐻The additional benefit that application service providers obtain from free-riding on the investments of large language model providers Monetary Unit (e.g., $) 𝑅0The social reputation gained by regulatory authorities from enforcing strict regulation – 𝑅1The more significant social reputation gained when both providers invest, 𝑅1> 𝑅0– 𝐿The reputation loss faced by regulatory authorities due to lax regulation – Fig. 1. Tripartite Evolutionary Game Decision Tree. of large language model providers increases to 𝛼1,i.e., 𝛼1> 𝛼0>0; the profit growth rate of application service providers increases to 𝛽1, i.e., 𝛽1> 𝛽0>0. Although privacy protection measures may still fail, the collaborative investment of both parties can reduce the risk of failure, enhance the overall effectiveness of privacy protection, and thereby reduce the likelihood of fines and reputational losses. To delve deeper into strategic interactions in large language model data privacy protection, we have constructed a decision tree for evolutionary game involving three participants. As shown in Fig. 1, this decision tree illustrates the potential strategies and evolutionary paths of large language model providers, application service providers, and privacy regulatory authorities in the data privacy game. Each game participant has two choices at every decision node; these decisions unfold throughout the decision tree, eventually forming eight end-states representing the evolutionary results of the tripartite bodies under different strategy combinations. This model highlights the uncertainty of strategic choices and the complexity of strategy evolution, providing a theoretical framework for understanding and predicting the behavior patterns of each participant in data privacy protection. 3.2. Game participant relations Fig. 2depicts a tripartite evolutionary game model for data privacy protection, involving large language model providers, application service providers, and privacy regulatory authorities as principal players. Fig. 2. Tripartite Evolutionary Game Framework. In this framework, users and hackers play pivotal roles. Users, as data generators, supply information to both service provider categories. Conversely, hackers aim to breach these systems to pilfer critical user data, directly jeopardizing system data privacy. Large language model providers possess key technologies and computational resources, which application service providers leverage to offer tailored solutions. Their investment decisions in privacy protection are mutually dependent: if one invests and the other does not, the non-investor may indirectly benefit from the investor’s commitment. However, mutual investment in privacy protection enables both to achieve a symbiotic gain, enhancing their profitability. Privacy regulatory authorities play a critical role in this game by influencing the investment behavior of service providers through the formulation and enforcement of privacy protection policies. The strictness of their regulatory strategy is directly linked to the net benefits and investment motivations of service providers: too strict regulation may lead to fines for non-compliant providers, while reward mechanisms may encourage providers to comply with regulations. The effectiveness of the regulatory institutions not only affects the economic benefits of Operations Research Perspectives 14 (2025) 100327 4 Y. Lv et al. Table 2 The profit and loss matrix of the three game participants. LLM provider App provider Privacy authority Strict regulation(z) Lax regulation(1-z) Invest(x) Invest(y) (1 +𝛼1)𝑃𝑆−𝐶𝑆(1 +𝛼1)𝑃𝑆−𝐶𝑆 (1 +𝛽1)𝑃𝐻−𝐶𝐻(1 +𝛽1)𝑃𝐻−𝐶𝐻 𝑅1−𝐶𝐺𝑅1 Not Invest(1-y) (1 +𝛼0)𝑃𝑆−𝐶𝑆+𝐹(1 +𝛼0)𝑃𝑆−𝐶𝑆 𝜉𝐻−𝐹 𝜉𝐻 𝑅0−𝐶𝐺−𝐿 Not Invest(1-x) Invest(y) 𝜉𝑆−𝐹 𝜉𝑆 (1 +𝛽0)𝑃𝐻−𝐶𝐻+𝐹(1 +𝛽0)𝑃𝐻−𝐶𝐻 𝑅0−𝐶𝐺−𝐿 Not Invest(1-y) 𝑃𝑆−𝐹 𝑃𝑆 𝑃𝐻−𝐹 𝑃𝐻 2𝐹−𝐶𝐺−𝐿 the service providers but is also related to their own social credibility and authority. Effective regulation cannot only improve their reputation among the public but can also enhance social welfare; conversely, it may lead to damage to their reputation. 3.3. Model establishment After synthesizing the assumptions and analyses proposed in Sections 3.1 and 3.2, we have constructed a detailed payoff matrix to quantitatively describe the interactions and expected payoffs of the game entities — large language model providers, application service providers, and privacy regulatory authorities — under different strategy combinations. Detailed information is outlined in Table 2. 3.3.1. Replicator dynamics equation and phase diagram for large language model providers Based on Table 2, it is known that large language model providers face two strategic choices: to invest or not invest in privacy protection. When they choose the former, the expected payoff is 𝐸𝑆1; for the latter, it is 𝐸𝑆2. We define the specific calculation formulas for 𝐸𝑆1 and 𝐸𝑆2as follows: If choosing to invest in privacy protection, the expected payoff 𝐸𝑆1 is calculated via the formula: 𝐸𝑆1=𝑦𝑧[(1 +𝛼1)𝑃𝑆−𝐶𝑆] +𝑦(1 −𝑧)[(1 +𝛼1)𝑃𝑆−𝐶𝑆] + (1 −𝑦)𝑧[(1 +𝛼0)𝑃𝑆−𝐶𝑆+𝐹] + (1 −𝑦)(1 −𝑧)[(1 +𝛼0)𝑃𝑆−𝐶𝑆] =𝑃𝑆−𝐶𝑆+𝑃𝑆𝛼0+𝐹 𝑧−𝑃𝑆𝛼0𝑦+𝑃𝑆𝛼1𝑦−𝐹 𝑦 𝑧 (1) If choosing not to invest in privacy protection, the expected payoff 𝐸𝑆2is calculated via the formula: 𝐸𝑆2=𝑦𝑧[𝜉𝑆−𝐹] +𝑦(1 −𝑧)𝜉𝑆 + (1 −𝑦)𝑧[𝑃𝑆−𝐹] + (1 −𝑦)(1 −𝑧)𝑃𝑆 =𝑃𝑆−𝐹 𝑧−𝑃𝑆𝑦+𝜉𝑆𝑦 (2) Subsequently, the average expected payoff 𝐸𝑆for large model providers can be represented by the formula: 𝐸𝑆=𝑥𝐸𝑆1+ (1 −𝑥)𝐸𝑆2(3) To delve into the pathways and equilibrium points of strategy evolution for the tripartite game participants, we solve the replicator dynamics equation for large model providers: 𝐹(𝑥) =d𝑥 d𝑡=𝑥(𝐸𝑆1−𝐸𝑆) =𝑥(𝑥− 1) (𝐶𝑆−𝑃𝑆𝛼0− 2𝐹 𝑧−𝑃𝑆𝑦 +𝜉𝑆𝑦+𝑃𝑆𝛼0𝑦−𝑃𝑆𝛼1𝑦+𝐹 𝑦𝑧) (4) Designating 𝑦0=𝑃𝑆𝛼0−𝐶𝑆+2𝐹 𝑧 𝜉𝑆−𝑃𝑆+𝑃𝑆𝛼0−𝑃𝑆𝛼1+𝐹 𝑧and calculating the partial derivative of the replicator dynamics equation 𝐹(𝑥)with respect to variable 𝑥, we obtain: 𝑑 𝐹(𝑥) 𝑑 𝑥=(2𝑥− 1)(𝐶𝑆−𝑃𝑆𝛼0− 2𝐹 𝑧−𝑃𝑆𝑦+𝜉𝑆𝑦 +𝑃𝑆𝛼0𝑦−𝑃𝑆𝛼1𝑦+𝐹 𝑦𝑧) =(2𝑥− 1)[(𝜉𝑆−𝑃𝑆+𝑃𝑆𝛼0−𝑃𝑆𝛼1+𝐹 𝑧)𝑦 −(−𝐶𝑆+𝑃𝑆𝛼0+ 2𝐹 𝑧)] (5) If 𝑦=𝑦0, we can obtain 𝐹(𝑥) = 0, where regardless of the value of 𝑥, the strategic choice of large language model providers is in a stable state. If 𝑦 < 𝑦0, we can derive that 𝑑 𝐹(𝑥) 𝑑 𝑥||||𝑥=0 >0and 𝑑 𝐹(𝑥) 𝑑 𝑥||||𝑥=1 <0, at which point 𝑥= 1is an equilibrium point. When the probability of application service providers choosing to ‘‘invest in privacy protection’’ is lower than a certain threshold, large language model providers will choose the ‘‘invest in privacy protection’’ strategy. If 𝑦 > 𝑦0, we can deduce that 𝑑 𝐹(𝑥) 𝑑 𝑥||||𝑥=0 <0and 𝑑 𝐹(𝑥) 𝑑 𝑥||||𝑥=1 >0, at which point 𝑥= 0is an equilibrium point. When the probability of application service providers choosing to ‘‘invest in privacy protection’’ exceeds a certain threshold, large language model providers will opt for the ‘‘not invest in privacy protection’’ strategy. According to the above analysis, the large language model providers’ replication dynamic phase diagram can be obtained, as shown in Fig. 3. 3.3.2. Application service provider’s replicator dynamics equation and phase diagram For application service providers, their decision-making strategies can be divided into ‘‘investing in privacy protection’’ and ‘‘not investing in privacy protection’’. When choosing to ‘‘invest in privacy protection’’, the expected payoff is defined as 𝐸𝐻1; when choosing ‘‘not investing in privacy protection’’, the expected payoff is defined as 𝐸𝐻2. The specific formulas are: Operations Research Perspectives 14 (2025) 100327 5 Y. Lv et al. Fig. 3. Replication dynamic phase diagram of large language model providers: (a) 𝑦=𝑦0; (b) 𝑦 < 𝑦0; (c) 𝑦 > 𝑦0. If choosing to invest in privacy protection, the expected payoff 𝐸𝐻1 is calculated as: 𝐸𝐻1=𝑥𝑧[(1 +𝛽1)𝑃𝐻−𝐶𝐻] +𝑥(1 −𝑧)[(1 +𝛽1)𝑃𝐻−𝐶𝐻] + (1 −𝑥)𝑧[(1 +𝛽0)𝑃𝐻−𝐶𝐻+𝐹] + (1 −𝑥)(1 −𝑧)[(1 +𝛽0)𝑃𝐻−𝐶𝐻] =𝑃𝐻−𝐶𝐻+𝑃𝐻𝛽0+𝐹 𝑧 −𝑃𝐻𝛽0𝑥+𝑃𝐻𝛽1𝑥−𝐹 𝑥𝑧 (6) If not choosing to invest in privacy protection, the expected payoff 𝐸𝐻2is calculated as: 𝐸𝐻2=𝑥𝑧[𝜉𝐻−𝐹] +𝑥(1 −𝑧)𝜉𝐻 + (1 −𝑥)𝑧(𝑃𝐻−𝐹) + (1 −𝑥)(1 −𝑧)𝑃𝐻 =𝑃𝐻−𝐹 𝑧−𝑃𝐻𝑥+𝑥 𝜉𝐻 (7) The average expected payoff 𝐸𝐻for application service providers can be expressed by the following formula: 𝐸𝐻=𝑦𝐸𝐻1+ (1 −𝑦)𝐸𝐻2(8) The replicator dynamics equation for application service providers is: 𝐹(𝑦) =d𝑦 d𝑡 =𝑦(𝐸𝐻1−𝐸𝐻) =𝑦(𝑦− 1) (𝐶𝐻−𝑃𝐻𝛽0− 2𝐹 𝑧−𝑃𝐻𝑥+𝑥𝜉𝐻 +𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥+𝐹 𝑥𝑧) (9) Setting 𝑧0to make the growth rate neutral: 𝑧0=𝐶𝐻−𝑃𝐻𝛽0−𝑃𝐻𝑥+𝑥𝜉𝐻+𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥 2𝐹−𝐹 𝑥, calculating the partial derivative of the replicator dynamics equation 𝐹(𝑦)with respect to the variable 𝑦, Fig. 4. Replication dynamic phase diagram of application service providers: (a) 𝑧=𝑧0; (b) 𝑧 < 𝑧0; (c) 𝑧 > 𝑧0. we get: 𝑑 𝐹(𝑦) 𝑑 𝑦=(2𝑦− 1)(𝐶𝐻−𝑃𝐻𝛽0− 2𝐹 𝑧−𝑃𝐻𝑥+𝑥𝜉𝐻 +𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥+𝐹 𝑥𝑧) =(2𝑦− 1)[(𝐹 𝑥− 2𝐹)𝑧+𝐶𝐻−𝑃𝐻𝛽0 −𝑃𝐻𝑥+𝑥𝜉𝐻+𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥] (10) If 𝑧=𝑧0, we have 𝐹(𝑦) = 0, so no matter the value of 𝑦, the strategic choice of the application service provider is in a stable state. If 𝑧 < 𝑧0,𝑑 𝐹(𝑦) 𝑑 𝑦||||𝑦=0 <0and 𝑑 𝐹(𝑦) 𝑑 𝑦||||𝑦=1 >0, thus at 𝑦= 0there is an equilibrium point. When the probability of the regulatory body choosing a ‘‘strict regulation’’ strategy is below a specific threshold, the application service provider will choose the ‘‘not investing in privacy protection’’ strategy. If 𝑧 > 𝑧0,𝑑 𝐹(𝑦) 𝑑 𝑦||||𝑦=0 >0and 𝑑 𝐹(𝑦) 𝑑 𝑦||||𝑦=1 <0, thus at 𝑦= 1there is an equilibrium point. When the probability of the regulatory body choosing a ‘‘strict regulation’’ strategy exceeds a certain threshold, the application service provider will choose the ‘‘investing in privacy protection’’ strategy. According to the above analysis, the application service providers’ replication dynamic phase diagram can be obtained, as shown in Fig. 4. 3.3.3. Regulatory authority’s replicator dynamics equation and phase diagram For privacy regulatory authorities, when implementing the strategy of ‘‘strict regulation’’, the expected payoff is defined as 𝐸𝐺1; while implementing ‘‘lax regulation’’, the expected payoff is 𝐸𝐺2. The formulas are: 𝐸𝐺1=𝑥𝑦 [𝑅1−𝐶𝐺] +𝑥(1 −𝑦)[𝑅0−𝐶𝐺] + (1 −𝑥)𝑦[𝑅0−𝐶𝐺] + (1 −𝑥)(1 −𝑦)[2𝐹−𝐶𝐺] (11) Operations Research Perspectives 14 (2025) 100327 6 Y. Lv et al. 𝐸𝐺2=𝑥𝑦𝑅1 +𝑥(1 −𝑦)[−𝐿] + (1 −𝑥)𝑦[−𝐿] + (1 −𝑥)(1 −𝑦)[−𝐿] (12) To fully evaluate the impact of these strategies, we calculate the average expected benefit 𝐸𝐺for the regulatory authorities using the following formula: 𝐸𝐺=𝑧𝐸𝐺1+ (1 −𝑧)𝐸𝐺2(13) The replicator dynamics equation for the regulatory authority is: 𝐹(𝑧) =𝑑 𝑧 𝑑 𝑡=𝑧(𝐸𝐺1−𝐸𝐺)(14) Setting 𝑥0to make the growth rate neutral: 𝑥0=2𝐹−𝐶𝐺+𝐿−2𝐹 𝑦+𝑅0𝑦 2𝐹−𝑅0−2𝐹 𝑦+𝐿𝑦+2𝑅0𝑦, calculating the partial derivative of the replicator dynamics equation 𝐹(𝑧)with respect to the variable 𝑧, we obtain: 𝑑 𝐹(𝑧) 𝑑 𝑧=(2𝑧− 1)(𝐶𝐺− 2𝐹−𝐿+ 2𝐹 𝑥+ 2𝐹 𝑦 −𝑅0𝑥−𝑅0𝑦− 2𝐹 𝑥𝑦 +𝐿𝑥𝑦 + 2𝑅0𝑥𝑦) =(2𝑧− 1)[(2𝐹−𝑅0− 2𝐹 𝑦+𝐿𝑦 + 2𝑅0𝑦)𝑥 −(−𝐶𝐺+ 2𝐹+𝐿− 2𝐹 𝑦+𝑅0𝑦)] (15) If 𝑥=𝑥0,𝐹(𝑧) = 0, and no matter the value of 𝑧, the strategy choice of the regulatory authority is in a stable state. If 𝑥 < 𝑥0,𝑑 𝐹(𝑧) 𝑑 𝑧||||𝑧=0 >0and 𝑑 𝐹(𝑧) 𝑑 𝑧||||𝑧=1 <0, thus at 𝑧= 1there is an equilibrium point. When the probability of the big language model provider choosing to ‘‘invest in privacy protection’’ is below a certain threshold, the regulatory authority will choose the ‘‘strict regulation’’ strategy. If 𝑥 > 𝑥0,𝑑 𝐹(𝑧) 𝑑 𝑧||||𝑧=0 <0and 𝑑 𝐹(𝑧) 𝑑 𝑧||||𝑧=1 >0, thus at 𝑧= 0there is an equilibrium point. When the probability of the big language model provider choosing to ‘‘invest in privacy protection’’ exceeds a certain threshold, the regulatory authority will choose the ‘‘lax regulation’’ strategy. According to the above analysis, the privacy regulatory authorities’ replication dynamic phase diagram can be obtained, as shown in Fig. 5. 4. Stability analysis of the model’s equilibrium points 4.1. Jacobian matrix Through Eqs. (4),(9) and (14), we derive the state equations for the tripartite game involving large language model providers, application service providers, and privacy regulatory authorities in the context of data privacy protection: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 𝐹(𝑥) =𝑥(𝑥− 1) (𝐶𝑆−𝑃𝑆𝛼0− 2𝐹 𝑧−𝑃𝑆𝑦 +𝜉𝑆𝑦+𝑃𝑆𝛼0𝑦−𝑃𝑆𝛼1𝑦+𝐹 𝑦𝑧 ) 𝐹(𝑦) =𝑦(𝑦− 1) (𝐶𝐻−𝑃𝐻𝛽0− 2𝐹 𝑧−𝑃𝐻𝑥 +𝑥𝜉𝐻+𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥+𝐹 𝑥𝑧 ) 𝐹(𝑧) =𝑧(𝑧− 1) (𝐶𝐺− 2𝐹−𝐿+ 2𝐹 𝑥+ 2𝐹 𝑦 −𝑅0𝑥−𝑅0𝑦− 2𝐹 𝑥𝑦 +𝐿𝑥𝑦 + 2𝑅0𝑥𝑦 ) (16) In system dynamics, an analysis of the eigenvalues of the Jacobian matrix is a key step in assessing local stability. The determination of Evolutionarily Stable Strategy (ESS) often relies on a local stability analysis of the Jacobian matrix near the equilibrium point. Specifically, an equilibrium point’s ESS is considered stable only if all eigenvalues of its Jacobian matrix are negative; otherwise, the equilibrium point is considered unstable. On this theoretical basis, we first derive the Jacobian matrix based on Formula (16) and further use Lyapunov’s method to conduct a comprehensive assessment of the stability of each equilibrium point in the game system, providing a strict mathematical Fig. 5. Replication dynamic phase diagram of regulatory authority: (a) 𝑥=𝑥0; (b) 𝑥 < 𝑥0; (c) 𝑥 > 𝑥0. foundation for the system’s stability analysis. The Jacobian matrix is shown in Formula (17): 𝐽=⎡⎢⎢⎢⎣ 𝑗11 𝑗12 𝑗13 𝑗21 𝑗22 𝑗23 𝑗31 𝑗32 𝑗33 ⎤⎥⎥⎥⎦ (17) Among them, 𝑗11 = (2𝑥− 1)(𝐶𝑆−𝑃𝑆𝛼0− 2𝐹 𝑧−𝑃𝑆𝑦+𝜉𝑆𝑦+𝑃𝑆𝛼0𝑦 −𝑃𝑆𝛼1𝑦+𝐹 𝑦𝑧) 𝑗12 =𝑥(𝑥− 1)(𝜉𝑆−𝑃𝑆+𝑃𝑆𝛼0−𝑃𝑆𝛼1+𝐹 𝑧) 𝑗13 =𝐹 𝑥(𝑥− 1)(𝑦− 2) 𝑗21 =𝑦(𝑦− 1)(𝜉𝐻−𝑃𝐻+𝑃𝐻𝛽0−𝑃𝐻𝛽1+𝐹 𝑧) 𝑗22 = (2𝑦− 1)(𝐶𝐻−𝑃𝐻𝛽0− 2𝐹 𝑧−𝑃𝐻𝑥+𝑥𝜉𝐻+𝑃𝐻𝛽0𝑥 −𝑃𝐻𝛽1𝑥+𝐹 𝑥𝑧) 𝑗23 =𝐹 𝑦(𝑥− 2)(𝑦− 1) 𝑗31 =𝑧(𝑧− 1)(2𝐹−𝑅0− 2𝐹 𝑦+𝐿𝑦 + 2𝑅0𝑦) 𝑗32 =𝑧(𝑧− 1)(2𝐹−𝑅0− 2𝐹 𝑥+𝐿𝑥 + 2𝑅0𝑥) 𝑗33 = (2𝑧− 1)(𝐶𝐺− 2𝐹−𝐿+ 2𝐹 𝑥+ 2𝐹 𝑦−𝑅0𝑥−𝑅0𝑦 − 2𝐹 𝑥𝑦 +𝐿𝑥𝑦 + 2𝑅0𝑥𝑦) 4.2. Stability analysis 4.2.1. Eigenvalues at equilibrium points By inserting the 8 local equilibrium points into the Jacobian matrix and following the assumptions provided, we obtain the respective eigenvalues for each equilibrium point. The results are as shown in the Table 3. The table clearly demonstrates a significant relationship between parameters such as regulatory costs, regulatory benefits, providers’ profit growth rates, and fines/subsidies, and the ESS of the three Operations Research Perspectives 14 (2025) 100327 7 Y. Lv et al. Table 3 System equilibrium points and eigenvalues. 𝜆1𝜆2𝜆3 𝐸𝑝1(0,0,0) 𝑃𝑆𝛼0−𝐶𝑆𝑃𝐻𝛽0−𝐶𝐻2𝐹−𝐶𝐺+𝐿 𝐸𝑝2(0,0,1) 2𝐹−𝐶𝑆+𝑃𝑆𝛼02𝐹−𝐶𝐻+𝑃𝐻𝛽0𝐶𝐺− 2𝐹−𝐿 𝐸𝑝3(0,1,0) 𝐶𝐻−𝑃𝐻𝛽0𝐿−𝐶𝐺+𝑅0𝑃𝑆−𝐶𝑆−𝜉𝑆+𝑃𝑆𝛼1 𝐸𝑝4(0,1,1) 𝐶𝐺−𝐿−𝑅0𝐶𝐻− 2𝐹−𝑃𝐻𝛽0𝐹−𝐶𝑆+𝑃𝑆−𝜉𝑆+𝑃𝑆𝛼1 𝐸𝑝5(1,0,0) 𝐶𝑆−𝑃𝑆𝛼0𝐿−𝐶𝐺+𝑅0−𝜉𝐻+𝑃𝐻−𝐶𝐻+𝑃𝐻𝛽1 𝐸𝑝6(1,0,1) 𝐶𝐺−𝐿−𝑅0𝐶𝑆− 2𝐹−𝑃𝑆𝛼0−𝜉𝐻+𝐹−𝐶𝐻+𝑃𝐻+𝑃𝐻𝛽1 𝐸𝑝7(1,1,0) −𝐶𝐺𝐶𝑆−𝑃𝑆+𝜉𝑆−𝑃𝑆𝛼1𝜉𝐻+𝐶𝐻−𝑃𝐻−𝑃𝐻𝛽1 𝐸𝑝8(1,1,1) 𝐶𝐺𝐶𝑆−𝐹−𝑃𝑆+𝜉𝑆−𝑃𝑆𝛼1𝜉𝐻+𝐶𝐻−𝐹−𝑃𝐻−𝑃𝐻𝛽1 Table 4 Parameter values for different propositions. 𝑃𝑆𝑃𝐻𝐶𝑆𝐶𝐻𝐶𝐺𝐹 𝛼0𝛼1𝛽0𝛽1𝜉𝑆𝜉𝐻𝑅0𝑅1𝐿 Proposition 130 25 10 8 44 0 0.24 1.09 0.05 0.31 70 60 5 15 20 Proposition 248 37 15 10 40 0 0.19 0.25 0.51 0.59 65 55 2 12 10 Proposition 348 37 15 10 40 0 0.41 0.49 0.19 0.62 65 55 10 110 10 Proposition 448 37 15 10 40 0 0.92 0.94 0.79 0.86 65 55 2 12 10 Proposition 548 37 15 10 33 10 0.07 0.10 0.27 0.62 65 55 30 35 10 Proposition 648 37 20 15 27 6 0.45 0.57 0.07 0.17 65 55 45 55 10 Proposition 748 37 20 15 16 6 0.00 0.30 0.01 0.10 65 55 17 22 10 major game participants. For simplification of the analysis process, we assume that there is only one regulatory authority responsible for overseeing all large language model providers and application service providers. Based on this assumption, we set parameter values under different propositions and analyze three critical parameter ranges. The parameter values are shown in Table 4. 4.2.2. When 𝐶𝐺> 𝑅0+𝐿 When 𝐶𝐺> 𝑅0+𝐿, the regulatory authority will opt for a lax regulation policy regardless of whether the providers invest due to the high cost of regulation. Proposition 1. When the conditions 0< 𝛼0< 𝐶𝑆∕𝑃𝑆,𝛼0< 𝛼1< 𝜉𝑆+𝐶𝑆−𝑃𝑆 𝑃𝑆 ,0< 𝛽0< 𝐶𝐻∕𝑃𝐻, and 𝛽0< 𝛽1<𝜉𝐻+𝐶𝐻−𝑃𝐻 𝑃𝐻 are met, as shown in Fig. 6, the system tends to take the strategy combination of noninvestment and lax regulation (0, 0, 0), which constitutes an ESS. Evaluating the cost–benefit of the investment return 𝐸𝑆(1,0,0) = (1 +𝛼0)𝑃𝑆−𝐶𝑆 for large language model providers, we find 𝐸𝑆(1,0,0) to be less than the non-investment return 𝐸𝑆(0,0,0) = (1 +𝐶𝑆∕𝑃𝑆)𝑃𝑆−𝐶𝑆=𝑃𝑆. A similar cost–benefit assessment for application service providers shows that investing in data privacy protection 𝐸𝐻(0,1,0) = (1 +𝛽0)𝑃𝐻−𝐶𝐻does not exceed the straightforward return 𝐸𝐻(0,0,0) = (1 +𝐶𝐻∕𝑃𝐻)𝑃𝐻−𝐶𝐻=𝑃𝐻. In addition, the government faces a situation where regulatory costs 𝐶𝐺exceed the sum of basic fines and losses 𝑅0+𝐿, which in itself is greater than twice the fines and losses 2𝐹+𝐿. In this context, the government is more inclined to opt for lax regulation. In summary, due to limited profit margins, both large language model providers and application service providers will choose not to invest in data privacy protection, while the government opts for lax regulation. Therefore, the system will tend to evolve into a state where all parties choose not to invest and not to strictly regulate, solidifying (0,0,0) as the system’s ESS under this condition. Proposition 2. When the conditions 0< 𝛼0< 𝐶𝑆∕𝑃𝑆,𝛼0< 𝛼1< 𝜉𝑆+𝐶𝑆−𝑃𝑆 𝑃𝑆 , and 𝐶𝐻∕𝑃𝐻< 𝛽0< 𝛽1<𝜉𝐻+𝐶𝐻−𝑃𝐻 𝑃𝐻 are met, as shown in Fig. 7, the system reaches the system’s equilibrium point (0,1,0) after 50 evolutions. If 𝐶𝐺> 𝑅0+𝐿, a cost–benefit analysis of the regulatory authority yields: 𝐸𝐺(0,1,1) =𝑅0−𝐶𝐺< 𝐸𝐺(0,1,0) (18) Fig. 6. Diagram of the evolution path under Proposition 1. In this case, the regulatory authority will enforce lax regulation. Due to 𝐶𝐻∕𝑃𝐻< 𝛽0< 𝛽1<𝜉𝐻+𝐶𝐻−𝑃𝐻 𝑃𝐻 , a cost–benefit assessment of the investment return for application service providers is conducted: 𝐸𝐻(0,1,0) = (1 +𝛽0)𝑃𝐻−𝐶𝐻 >(1 +𝐶𝐻 𝑃𝐻)𝑃𝐻−𝐶𝐻 =𝑃𝐻=𝐸𝐻(0,0,1) (19) This indicates that the application service provider is inclined to undertake data privacy protection. A cost–benefit assessment of the investment return for large language model providers shows: 𝐸𝑆(1,1,0) = (1 +𝛼1)𝑃𝑆−𝐶𝑆 <(1 +𝜉𝑆+𝐶𝑆−𝑃𝑆 𝑃𝑆)𝑃𝑆−𝐶𝑆 =𝜉𝑆=𝐸𝑆(0,1,0) (20) Therefore, the large language model provider is not inclined to invest under these conditions. Proposition 3. When the conditions 𝐶𝑆∕𝑃𝑆< 𝛼0< 𝛼1<𝜉𝑆+𝐶𝑆−𝑃𝑆 𝑃𝑆 , 0< 𝛽0< 𝐶𝐻∕𝑃𝐻, and 𝛽0< 𝛽1<𝜉𝐻+𝐶𝐻−𝑃𝐻 𝑃𝐻 are met, as shown in Fig. 8, the system reaches the system’s equilibrium point (1,0,0) after 50 evolutions. Similarly, it is found that 𝐸𝐺(1,0,1) =𝑅0−𝐶𝐺< 𝐸𝐺(1,0,0), hence the regulatory authority is inclined to choose lax regulation. Based on the given parameter range, a cost–benefit assessment of the investment returns for both large language model providers and application service Operations Research Perspectives 14 (2025) 100327 8