Research on Collaborative Governance of Cross-Domain Digital Innovation Ecosystems Based on Evolutionary Game Theory

Abstract

The complexities inherent in resource management within cross-domain digital innovation ecosystems have significantly intensified, giving rise to heightened challenges in collaborative interactions among diverse stakeholders, thereby directly impacting systemic stability. Conventional governance frameworks for innovation ecosystems are inadequate in effectively managing the uncertainties and risks inherent in these environments. To address the collaborative governance dilemma and enhance governance efficiency, this paper aims to construct an effective collaborative governance mechanism for a cross-domain digital innovation ecosystem and explore the optimal strategy choices of key governance stakeholders, including the government, digital platform enterprises, and other relevant parties. This research utilizes evolutionary game theory to construct a model comprising three governing entities: the government, digital platform enterprises, and stakeholders. It investigates the evolutionary dynamics of collaborative governance strategies among these entities and the factors that influence governance. Following this, a system dynamics methodology is employed for simulation analysis. The results reveal the following: (1) As the initial intentions of the governing entities evolve, governance decisions within the system tend to stabilize, characterized by a strategic combination of proactive regulation, active cooperative governance, and engaged participation. This equilibrium governance strategy significantly fosters the stable advancement of cross-domain digital innovation ecosystems. (2) The punitive measures enacted by the government and the internal incentive structures of the system positively influence the evolution of governance decisions towards collaborative governance. (3) The cost–benefit assessment of the primary governing entity, the digital platform enterprise, demonstrates a detrimental effect on the evolution of governance decisions towards collaborative governance. These findings are vital for refining the collaborative governance frameworks of cross-domain digital innovation ecosystems and for promoting the robust and stable progression of the system.

1. Introduction

As the current wave of technological revolution advances, digital technology emerges as a pivotal production factor, significantly influencing and engaging in innovation activities [1,2]. This evolution is catalyzing the expansion and refinement of traditional innovation theories. The advent of digital technologies has initiated a transformation in production factors and redefined the value co-creation model among innovation stakeholders, thereby broadening the theoretical frameworks of existing innovation ecosystems. The continuous advancement of next-generation digital information technologies—encompassing 5G, artificial intelligence, the Internet of Things, and big data—is driving innovation entities toward digital transformation, intelligent upgrading, virtualization, and strategic diversification. Digitalization not only revolutionizes the organizational methodologies of innovation but also fosters the profound integration of digital technologies within innovation ecosystems, a process that permeates every phase and the entire lifecycle of innovation. As a result, a variety of new innovation paradigms have emerged, such as democratized innovation, intelligent innovation, and platform-based innovation. These paradigms encapsulate the intrinsic combinatorial, open, and distributed nature of the digital innovation process and its outcomes [3]. This trajectory of digital innovation further accelerates the deep integration of digital technologies with innovation ecosystems, culminating in what is termed the digital innovation ecosystem [4]. The cross-domain digital innovation ecosystem represents a distinct manifestation of the digital innovation ecosystem, highlighting the utilization of digital technologies as enabling instruments, with data serving as the central production factor, aimed at facilitating cross-domain collaboration, resource sharing and integration, and value co-creation within a complex economic framework [5,6].The cross-disciplinary digital innovation ecosystem exhibits significant characteristics such as the digitization of innovative elements, the virtualization of participating entities, and the ecological nature of interrelationships. These features increase the complexity of resource management among elements, elevate the challenges of collaborative interactions among stakeholders, and may impact the stability of the system [7].

Taking the cross-domain digital innovation ecosystem of intelligent transportation and healthcare as an example, differences in data formats, standards, and security exacerbate the “data silos” [8]. Divergences among different industry stakeholders (e.g., transportation enterprises prioritizing real-time performance, while medical institutions emphasizing privacy and security [9,10]) regarding collaboration models, responsibility, and benefit distribution significantly constrain collaborative effectiveness, directly threatening system stability. Faced with such challenges, the governance strategies of traditional innovation ecosystems exhibit clear limitations. There is an urgent need to explore new collaborative governance pathways to achieve system health, stability, and value maximization.

Collaborative governance provides a core theoretical perspective for addressing the multi-stakeholder, cross-domain governance issues in digital innovation ecosystems. This paper conceptualizes collaborative governance as follows: under conditions of bounded rationality, key governance entities engage in strategic interactions to collectively address system complexity, promote resource sharing, and foster value co-creation, forming a dynamic process. This theory is chosen because it effectively explains how diverse, heterogeneous entities, operating under differing objectives and asymmetric information, can overcome collective action dilemmas and move toward cooperative equilibrium through mechanisms such as rules, incentives, and sanctions.

This paper aims to construct an integrated analytical framework combining evolutionary game theory and system dynamics to investigate the strategic evolution and interaction mechanisms among three key stakeholders—government, digital platform firms, and stakeholders—in a cross-domain digital innovation ecosystem. The core research questions include the following: Under bounded rationality constraints, how will the collaborative governance strategies of the three stakeholders evolve over time and what equilibrium will they ultimately reach? Which key factors significantly influence the governance decisions of each stakeholder and the overall evolution of the system?

The novelty of this research is reflected in three aspects: (1) theoretically, it operationalizes collaborative governance theory within a cross-domain digital innovation ecosystem as a dynamic governance strategy game among three stakeholders; (2) methodologically, it innovatively integrates evolutionary game theory with system dynamics, overcoming the limitations of purely qualitative or two-stakeholder game models; and (3) practically, it deeply analyzes the cost–benefit structure of core enterprises and its impact on governance performance, offering new perspectives for optimizing governance mechanisms.

The subsequent structure of this paper is as follows: Section 2 reviews the relevant literature on digital innovation ecosystems, evolutionary game theory, and collaborative governance; Section 3 explores the composition of governance entities and their interaction relationships in cross-domain digital innovation ecosystems; Section 4 constructs a tripartite evolutionary game model involving government, digital platform firms, and stakeholders; Section 5 conducts a stability analysis of the model; Section 6 performs simulation analysis using system dynamics; Section 7 summarizes the research conclusions and recommendations; and Section 8 briefly discusses the limitations of this study and future research directions.

2. Literature Review

2.1. Digital Innovation Ecology

2.1.1. Digital Innovation

Currently, the popularization and application of digital technology has led to a fundamental shift in the connotation of innovation, and digital innovation, as an emerging innovation paradigm, has had a disruptive impact on innovation theory [11]. Digital innovation was first defined as the use of emerging digital avenues, tools, and strategies by firms to enhance their operational effectiveness and drive innovation outcomes [12]. Fichman R G et al. took a macro perspective and defined digital innovation as a product, process, or business that is considered novel [13]. Abrell et al. argued that digital innovation utilizes digital technologies to accelerate the completion of work tasks and facilitate the innovation process [14]. Yoo Y et al. state that digital innovation leads to the widespread penetration of digital technologies deep into the core products, services, and operations of an organization, which fundamentally transforms the essential characteristics of products and services [15]. Bharadwaj A et al. suggest that digital innovation is the process of innovation that integrates information, computation, communication, and connectivity technologies to enable new product or service delivery, the optimization of production processes, the innovation of existing organizations, and the fundamental transformation of business [16]. John Zhen et al. theorized the synergy mechanism between digital innovation and economic high from the perspective of synergy theory. They found that the degree of fit between digital innovation and economic high is a key factor in determining the economic and synergistic effects. The higher the degree of fit and integration, the more it helps to enhance and improve the synergistic effect of digital innovation and economic high in the region [17]. Based on the availability theory, Hong Jiangtao and Zhang Siyue constructed a theoretical framework with “foundation-transformation-innovation” as the core clue and conducted an in-depth study on the role of organization and digital technology on the internal mechanism of digital innovation, which clearly pointed out the key role of organization within the enterprise in the process of digital innovation [18].

2.1.2. Digital Innovation Ecosystems

Digital innovation ecosystem is a fusion of the two concepts of “digital innovation” and “innovation ecosystem” [4], which refers to an ecological organization system in which innovation subjects complement and share innovation resources across levels and fields through digital technology and achieve value co-creation through competition and cooperation [19]. Regarding the characteristics of the digital innovation ecosystem, Adner R’s study points out that the innovation elements, participating subjects and inter-subject relations of this system, exhibit specific characteristics, which are high interdependence, interrelated and influential decision-making, coordinated challenge of interests, and dynamic changes in roles [20]. First, the digitization of innovation factors is the core, in which digital resources, digital technology and digital infrastructure constitute the key elements of the digital innovation ecosystem [21]. Secondly, the virtualization of the participating subjects is remarkable, the application of digital technology makes the connection between the participating subjects more extensive and flexible, many highly heterogeneous participants become the main body of value creation, and the virtual link between the innovation subjects through the platform realizes the limit [22]. Finally, the relationship between the participating subjects presents ecological characteristics, the participants carry out activities at different levels of digital architecture, the interaction relationship is more dynamic and open, and the innovation cooperation is transformed from a linear supply chain to a multi-layer networked ecosystem [23,24]. Lin Yan and Lu Junyao believe that the digital innovation ecosystem is a complex system of competition and symbiosis between different innovation subjects and innovations in the region relying on digital technology to realize new combinations of digital and non-digital resources to create new products or services [25]. Suseno et al. believe that the digital innovation ecosystem is a complex system that creates new products and value through digital technology [26]. Within the framework of digital innovation ecosystems, platforms serve as a bridge connecting the supply and demand sides of the innovation body, which promotes the enhancement of the creativity of enterprises in the digital era. Qu Yongyi believes that platform-based enterprises are the most important new organizational form in the digital innovation ecosystem in the context of digital innovation [27].

The digital innovation ecosystem exhibits inherent complexities in factor resource management and challenges in achieving synergistic interactions among stakeholders, thereby generating significant governance obstacles. Consequently, the academic community is dedicated to developing practical governance mechanisms to effectively address these challenges. Wei Jiang and Zhao Yuhan reveal three kinds of digital innovation ecosystem governance: the massive and heterogeneous participation of virtual subjects in digital innovation ecosystem makes it difficult for innovation subjects to interact and collaborate, the dynamic mesh inter-subjective relationship makes the innovation process face more uncertainty, and the digitized innovation elements make the sharing of data resources become a focal point of governance. They also put forward a proposal to realize the coordination of multilateral relations based on the construction of digital platforms and control of innovation behaviors based on the application of digital technologies. Their study also proposes three new mechanisms of governance based on digital platform construction to achieve the coordination of multilateral relations, based on digital technology application, to achieve the control of innovation behavior, based on digital resource synergy, and to achieve incentives for innovation results, and it deconstructs the characteristics of governance subjects, explores the microscopic mechanism of digital innovation ecosystem governance, expands the relationship of system nodes to construct its unique governance of equal importance to competition and cooperation, and explores the evolution of governance mechanisms to construct the trend of evolution of governance of digital innovation ecosystem and other key scientific issues of governance of the three kinds of governance [5]. Three scholars, including Sun Yonglei and Zhu Nonjie, revealed the three core dimensions of the governance mechanism of digital innovation ecosystems, the relationship mechanism bonded by trust, the incentive mechanism centered on knowledge, and the control mechanism based on practices, and discussed in depth the management strategies of each participating subject in the system [7]. Adopting fsQCA and selecting 28 provinces and cities in China as research samples, Yang Wei and Lao Xiaoyun et al. deeply explored the impact of five governance elements, namely knowledge, knowledge diffusion, market formation, resource allocation, and legitimacy, on the resilience of regional digital innovation ecosystems. Their study reveals two types of governance elements that can contribute to high resilience, “knowledge-resource”-driven and “market-driven” elements, and also identifies two types of governance elements that can lead to low resilience: “resource-driven” and “market-driven”. At the same time, two types that may lead to low resilience are also identified: “resource-driven” and “knowledge-driven” elements [28]. Hou Erxiu and Xu Rongqi et al. pointed out that domestic research on the governance of digital innovation ecosystems mainly focuses on the digital technology, digital connectivity, and innovation synergy of participants. In addition, the platform governance mechanism, as an important factor influencing the innovation behavior of platform owners and participants and their outcomes, has also received extensive attention. They proposed that combining digital platform architecture with governance theory and exploring the interaction between the two is a frontier direction in the field [29].

2.2. Evolutionary Games

Evolutionary game theory is a game theory that combines the game theory of assuming perfectly rational game subjects and perfectly informed games with dynamic evolution [30,31]. Evolutionary stable strategies and replication dynamics constitute the two core elements of evolutionary game theory. Smith and Price proposed the concept of evolutionary stable strategies, and then Taylor and Jonker introduced the theoretical framework of replication dynamics [32]. With the continuous evolution of evolutionary game theory and the deepening of its application in many fields, many scholars have utilized the theory to study issues in the field of managerial economics. Fan et al. studied the impact of incentives on firms’ green building decision-making based on evolutionary game theory and proposed four types of interactions, which are the dominant strategy, dependent on the choices of the merchant; the merchant-dominated strategy, dependent on the choices of the merchant; and the two-party dominant strategy, independent of each other. These dominant strategies depend on each other, and these types affect the effectiveness of incentives [33]. Mahmoudi and Rasti-Barzoki constructed an evolutionary game model of the effects of subsidies and taxes on the performance of supply chain firms and found that the tax policy, which is a trade-off between benefits and objectives, has a significant effect on firm performance and behavior [34]. Wang Invention and Zhu Meijuan constructed an evolutionary game model specifically to study the value co-creation process between focal firms and cooperative firms in an innovation ecosystem. Their study reveals that key factors such as the proportion of excess benefit distribution, cooperation costs, fraud strategy gains, and penalty strength have a significant impact on the stability of value co-creation behavior in innovation ecosystems [35]. Shen et al. constructed an evolutionary game model of contractors and manufacturers in the construction industry based on the premise of behavioral economics theory, pointing out that the initial strategy, costs and the degree of technology and perception will have an impact on the system’s stability and that after a reasonable allocation of the factors, its stability will be affected. state, and the system will evolve to a stable state only after a reasonable allocation of factors, and then they put forward suggestions that are conducive to cooperation and innovation between the two parties [36]. Yu Lijing and other scholars used the evolutionary game theory to deepen the collaborative innovation relationship between logistics enterprises and manufacturing industries and revealed the necessary conditions to realize the stable state of collaborative innovation [37]. Xue Lei and other scholars revealed the conditions under which the two parties can reach the equilibrium state of cooperative innovation by constructing an evolutionary game model between shipbuilding enterprises and suppliers [38]. Zhou Yanping and other scholars constructed a model describing the evolutionary game between enterprises and research institutes based on the theory of incomplete contract and accordingly put forward policy suggestions aimed at enhancing the enthusiasm of cooperative innovation between the two parties [39]. Zhang S et al. constructed an evolutionary game model with manufacturers, and the results of the study showed that the dynamic carbon trading pricing strategy is an effective mechanism to promote the process of emission reduction and the introduction of green technology and the speculative behavior of manufacturers are related to the introduction of green technology and the speculative behavior of manufacturers [40]. positive correlation exists between the introduction and the speculative behavior penalty of manufacturers [40]. The core advantage of evolutionary game theory lies in its bounded rationality assumption aligning with the decision-making reality of governance entities; its dynamic evolutionary perspective can depict the formation and transformation processes of collaborative governance strategies; and its multi-agent interaction framework can analyze the strategic interplay and interdependencies among diverse entities. Therefore, this theory has become a robust foundation for exploring the evolutionary patterns and equilibrium states of multi-agent collaborative governance within cross-domain digital innovation ecosystems.

2.3. Collaborative Governance

The theory of collaborative governance, also known as “synergetics” or “concordance”, originated in 1973 and was first proposed by the famous German physicist Hacken in his international conference proceedings “Synergetics” [41]. According to Ansell C and Gash A, collaborative governance is a type of governance that requires direct dialog between public institutions, the public and non-governmental organizations in a collective decision-making process to reach consensus and solve public problems in society [42]. Scholars, such as John M. and Bryson, define collaborative governance as the joint achievement of tasks that a single sector cannot accomplish on its own through the sharing of resources, exchange of information, etc. [43], between two or more sectors. Hutter B M suggests that the role change from “management” to “governance” should be realized, and the interaction of multiple subjects, the integration and utilization of resources and the core functions of the organization should be focused on the three main aspects of supervision, guidance and governance [44]. Pirsoul N, based on the system perspective and conducting an in-depth examination of governance, pointed out that governance is a key component of the overall implementation system of society and is composed of multiple stakeholders through the formation of synergistic cooperation between the main actors. The behavioral characteristics of its behavior cannot be summarized by a single processing behavior but is rather the product of the interactions of multiple subjects, and this interaction stems from the inherent organizational characteristics of the governance structure itself, rather than driven by external forces [45]. Li asserts that collaborative governance is a governance structure implemented in a collective decision-making process with non-governmental stakeholders in pursuit of a common goal [46]. Ansell et al. explored the mechanisms of collaborative governance in their study, noting that the formation of collaborative governance is associated with reliance on inclusive preconditions such as incentives, interdependence, and trust. The integration of stakeholders into a collaborative governance network in an inclusive manner helps to build cooperative relationships to address and solve problems together [47]. Lai advanced pointed out that the core value of collaborative governance is to stimulate the willingness of participating subjects to collaborate and serve as a driving force to achieve collaborative goals. Constructing public value goals is crucial to enhance the intrinsic motivation of collaborative governance [48]. The subjects of collaborative governance are characterized by diversity, and the resources of each governance subject are different, thus forming a complex relationship of both competition and cooperation. Li Hanqing emphasizes that collaborative governance can cut across different subjects such as sectors and include non-governmental organizations, businesses, and citizens, and its core process involves the interaction of power and resources. Therefore, self-organized synergy and interaction between respective subjects is crucial for support [49]. Yu Jianxing and other researchers suggest that the mechanism of collaborative governance involves playing a leading role in building communication channels and platforms through institutionalized means and fostering social power in order to promote social autonomy, participation in services, and collaborative management [50]. Yang Huafeng pointed out that the dynamics and operational mechanisms of collaborative governance are composed of rules, institutions, interests, and psychological factors. He suggests that in order to advance the process of collaborative governance, it is necessary to pursue networking and the internalization of rules [51].

2.4. Literature Commentary

In summary, research on digital innovation ecosystems primarily focuses on concept definition, structural characteristics, and governance mechanisms, providing a solid theoretical foundation and value orientation for this study. Research on governance mechanisms of digital innovation ecosystems mainly relies on qualitative analysis, with relatively limited applications of quantitative research methods. Research on collaborative governance primarily focuses on the theoretical analysis of concepts, value orientations, governance subjects, and governance mechanisms, offering theoretical references for this study. Additionally, scholars typically employ evolutionary game models independently, lacking comprehensive analysis with other models, and existing studies predominantly concentrate on strategy selection among two parties, with an insufficient exploration of collaborative evolution among three or more parties. Therefore, to address these gaps, this study conducts the following work: (1) In terms of theoretical methods, based on system dynamics and evolutionary game theory, a tripartite collaborative governance model for cross-domain digital innovation ecosystems is constructed, considering the impact of parameter changes such as government punishment mechanisms and system internal incentive mechanisms. (2) In terms of research significance, this study takes digital innovation as the core context, combines actual circumstances, and conducts simulation analysis on the tripartite collaborative governance model within cross-domain digital innovation ecosystems, aiming to enhance the universality of research results.

3. Governance Actor Analysis

3.1. Governance Constituents

In the context of collaborative governance within cross-domain digital innovation ecosystems, the key participants can be delineated across six dimensions [52]. Firstly, the government assumes a vital role in governance activities, functioning as both a regulator and a guide, with its primary responsibilities encompassing the formulation of pertinent policies and the provision of administrative services. Secondly, digital platform enterprises emerge as central actors in the collaborative governance framework, facilitating the integration and sharing of technological, data, and human resources across diverse sectors while also establishing and enforcing platform regulations. Thirdly, user groups, which comprise individual consumers and business users, are integral to this ecosystem. Fourthly, traditional enterprises, representing various industries such as manufacturing, finance, and retail, contribute to the governance landscape. Fifthly, developers and entrepreneurs of digital platforms, particularly those engaged in cross-sector integration, utilize their domain-specific advantages and resources to innovate new business models and products. Lastly, social institutions, including higher education establishments, research organizations, and various non-governmental entities, play an indispensable role in fostering ecosystem governance, often acting as intermediaries in areas such as legal frameworks, media communication, talent development, and governance accountability oversight. For the purposes of this study, the latter four components will be collectively designated as stakeholders, thereby categorizing the governance entities of the digital innovation ecosystem into three distinct categories, government, digital platform enterprises, and stakeholders, as depicted in Figure 1.

3.2. The Relationship Between Governance Actors

In the management of cross-sector digital innovation ecosystems, challenges related to information asymmetry and irrational decision-making often emerge among diverse governance participants. The efficacy of governance is intricately tied to the strategies adopted by governmental bodies, digital platform companies, and relevant stakeholders. As the primary architect of governance frameworks and overseer of public interests, the government holds principal accountability for regulatory oversight. The strategies at the government’s disposal during the governance process encompass both proactive regulatory measures and permissive regulatory structures. Proactive regulation involves the government enhancing its scrutiny of digital technology applications, instituting rigorous regulations and standards for cross-sector digital innovation, and offering significant policy and financial incentives to steer digital platform companies and stakeholders towards compliant innovation. In contrast, permissive regulation reflects a more relaxed oversight stance, allowing digital platform companies and stakeholders to increase their autonomy in their developmental pursuits.

Digital platform enterprises play a pivotal role in cross-domain digital innovation ecosystems, with the primary objective of investing in innovative resources for technological research and development. They are dedicated to the deep integration of various fields, achieving the cross-domain integration of digital technologies and fostering technological collaborative innovation among enterprises within the ecosystem. In the governance process, digital platform enterprises can adopt strategies that include proactive collaborative governance and passive collaborative governance. Proactive collaborative governance reflects the active engagement of digital platform enterprises in integrating digital and technological resources across domains. For instance, in the cross-domain innovation ecosystem of smart transportation and healthcare, digital platform enterprises actively integrate various data resources, such as traffic flow data and healthcare impact data, aiming to eliminate data silos between fields [8]. This initiative seeks to achieve the complementarity and sharing of data resources across different domains, providing a richer foundation for innovative activities. Furthermore, digital platform enterprises leverage their platform advantages to establish mechanisms and environments that facilitate cross-domain communication and collaboration, effectively promoting cooperative innovation projects among stakeholders in transportation, healthcare, finance, and other sectors. Conversely, under passive collaborative governance strategies, digital platform enterprises tend to focus excessively on their own interests, demonstrating a lack of proactive attitude towards the integration of cross-domain digital and technological resources and showing low enthusiasm for cross-domain collaborative innovation projects conducted through their platforms.

Stakeholders include users, established businesses, developers, and entrepreneurs of digital platforms, along with social institutions, each possessing unique interests yet collectively striving to obtain relevant advantages within the cross-sector digital innovation ecosystem. In the governance framework, stakeholders may employ strategies that encompass both active and passive participation. Active participation signifies that stakeholders are inclined to engage collaboratively with digital platform enterprises in their innovation endeavors, proactively offering feedback on requirements, sharing data, and providing innovative resources while also adapting to the transformations instigated by digital technologies. In contrast, passive participation denotes a skeptical perspective towards digital innovation, wherein stakeholders passively accept the results of innovation with limited involvement in the innovation process.

4. Evolutionary Game Model Construction

4.1. Model Assumptions and Variable Descriptions

To construct a triadic evolutionary game model of governance entities and derive optimal evolutionary strategies for governance mechanisms, the following model assumptions are proposed:

Assumption 1.

The governance game entities within cross-domain digital innovation ecosystems primarily consist of three categories, government, digital platform enterprises, and stakeholders, all of which are boundedly rational agents. The government is primarily responsible for regulation and guidance; digital platform enterprises assume the role of governance leaders, facilitating the participation of other entities in the governance process; and stakeholders act as governance facilitators, actively contributing to collaborative governance efforts.

Assumption 2.

Strategy spaces $a=\left(a_1,a_2\right)$, $b=\left(b_1,b_2\right)$ and  $c=\left(c_1,c_2\right)$ represent the governance strategy selection spaces of three governance entities: the government, digital platform enterprises, and stakeholders. Specifically, the government’s strategy space, $a$, consists of binary choices: $a_1$ represents proactive regulation, and $a_2$ represents permissive regulation. The digital platform enterprise’s strategy space, $b$, also involves binary choices: $b_1$ signifies active collaborative governance, while $b_2$ indicates passive collaborative governance. Similarly, the stakeholders’ strategy space, $c$, includes binary choices: $c_1$ denotes active participation, and $c_2$ denotes passive participation.

Assumption 3.

The costs incurred by the government in executing its regulatory functions are denoted as $c_g$. Through effective regulation, the government can achieve a positive social image benefit represented by $r_g$. Additionally, the support and subsidies provided by the government to digital platform enterprises are indicated as $c_s$. Conversely, if the government fails to regulate effectively and does not identify violations by digital platform enterprises, it will result in a negative social image loss quantified as $p_g$.

Assumption 4.

In the process of engaging in proactive collaborative governance, the costs incurred by digital platform enterprises amount to $c_{a1}$, while their achievable normal returns are represented by $r_{a1}$. Additionally, the returns from cross-sector collaborative innovation with stakeholders are denoted as $r_u$, and the distribution ratio of the collaborative innovation returns received by the digital platform enterprises is $\lambda$. Conversely, during instances of reactive collaborative governance, the costs faced by these enterprises are $c_{a2}$, and the potential excess returns they can obtain are $r_{a2}$. Should the digital platform enterprises engage in non-compliant behavior during this process and be detected by the government, they will face the corresponding penalty, $p_a$.

Assumption 5.

In the context of active stakeholder engagement, the incurred costs can be denoted as $c_b$. Correspondingly, the achievable returns are represented as  $r_b$, while the anticipated benefits from cross-sector collaborative innovation with digital platform enterprises are $(1-\lambda )r_u$. However, there exists a risk of losses, denoted as $c_l$, arising from the negative cooperative governance of digital platform enterprises. Furthermore, if the government detects the stakeholders’ passive participation, the potential penalty may amount to $p_b$, and the opportunity cost incurred during stakeholder disengagement is represented as $c_0$. The specific model parameters and their significance are presented in

Table 1. Model parameters and their descriptions.

Parameter	Description
$c_g$	The costs associated with proactive government regulation
$r_g$	The positive social image benefits derived from effective government regulation
$c_s$	Government support and subsidies for digital platform enterprises
$p_g$	The government has failed to regulate effectively and has not recognized the social image losses caused by the misconduct of digital platform companies
$c_{a1}$	The costs associated with collaborative governance among digital platform enterprises
$r_{a1}$	The normal returns generated when digital platform enterprises actively engage in collaborative governance
$r_u$	The benefits of cross-domain collaborative innovation between digital platform enterprises and stakeholders
$\lambda$	The distribution ratio of collaborative innovation benefits obtained by digital platform enterprises
$c_{a2}$	The costs associated with the governance of digital platform enterprises during instances of passive collaboration
$r_{a2}$	Excess returns of digital platform enterprises during negative collaborative governance
$p_a$	The penalties faced by digital platform enterprises when non-compliance is detected by the government during instances of passive cooperation governance
$c_b$	The costs associated with active stakeholder engagement
$r_b$	The benefits of active stakeholder engagement
$(1-\lambda )r_u$	The benefits of cross-domain collaborative innovation between stakeholders and digital platform enterprises
$c_l$	The risk of losses resulting from the negative collaborative governance of digital platform enterprises by stakeholders
$p_b$	The penalties faced by stakeholders when their passive participation is detected by the government
$c_0$	Opportunity costs arising from stakeholder disengagement

.

4.2. The Construction of the Tripartite Payoff Matrix

The tripartite payoff matrix is the core of the tripartite evolutionary game, quantifying the payoff of each agent under different strategy combinations. The necessity of the payoff matrix in this paper lies in clarifying the payoff distribution mechanism of tripartite collaborative governance, supporting the derivation of replicator dynamic equations, and validating the economic rationality of evolutionary strategies. Based on the model assumptions, it is assumed that the probability of adopting positive regulation is $x$ and the probability of choosing lax regulation is $1-x$; the probability of digital platform enterprises adopting positive co-operative governance is $y$ and the probability of choosing negative co-operative governance is $1-y$; and the probability of stakeholders adopting active participation is $z$ and the probability of choosing negative participation is $1-z$. Based on this, the tripartite payoff matrix of digital platform enterprises and stakeholders can be obtained, as shown in

Table 2. The matrix of tripartite payoff to digital platform enterprises and stakeholders.

Digital Platform Enterprise			Stakeholder
			Active Participation (z)	Negative Participation (1 − z)
Government	Positive regulation ($x$)	Positive co-operative governance ($y$)	$r_g-c_g-c_s$	$r_g-c_g+p_b$
			$r_{a1}+\lambda r_u+c_s-c_{a1}$	$r_{a1}-c_{a1}$
			$r_b+\left(1-\lambda \right)r_u-c_b$	$-c_0-p_b$
		Negative co-operative governance ($1-y$)	$r_g-c_g+p_a$	$r_g-c_g+p_a+p_b$
			$r_{a2}-c_{a2}-p_a$	$r_{a2}-c_{a2}-p_a$
			$r_b-c_b-c_l$	$-p_b$
	Lax regulation ($1-x$)	Positive co-operative governance ($y$)	$0$	$0$
			$r_{a1}+\lambda r_u-c_{a1}$	$r_{a1}-c_{a1}$
			$r_b+\left(1-\lambda \right)r_u-c_b$	$-c_0$
		Negative co-operative governance ($1-y$)	$p_a-p_g$	$p_a-p_g$
			$r_{a2}-c_{a2}-p_a$	$r_{a2}-c_{a2}-p_a$
			$r_b-c_b-c_l$	$0$

.

4.3. Replication of Dynamic Equation Building

First, assume that the expected return from adopting an aggressive regulatory strategy is $E_{11}$, the expected return from adopting a lax regulatory strategy is $E_{12}$, the average expected return is $E_1$, and the replicated dynamic equation is $F\left(x\right)$. Then, based on the payoff matrix in Table 2, we can obtain the following:

$$\begin{array}{c}{E}_{11}=yz(r_g-c_g-c_s)+y(1-z)(r_g-c_g+p_b)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+(1-y)z(r_g-c_g+p_a)+(1-y)(1-z)(r_g-c_g+p_a+p_b)\end{array}$$

(1)

$$E_{12}=(1-y)z(p_a-p_g)+(1-y)(1-z)(p_a-p_g)$$

(2)

$$E_1=x{E}_{11}+(1-x)E_{12}$$

(3)

$$\begin{array}{c}F(x)={\displaystyle \frac{dx}{dt}}=x(E_{11}-E_1)=x(1-x)(E_{11}-E_{12})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=x(x-1)(c_g-p_b-p_g-r_g+y{p}_g+z{p}_b+yz{c}_s)\end{array}$$

(4)

Second, assume that the expected return for digital platform firms adopting positive collaborative governance is $E_{21}$, the expected return for adopting negative collaborative governance is $E_{22}$, the average expected return is $E_2$, and the replicated dynamic equation is $F\left(y\right)$. Then, based on the payoff matrix in Table 2, we can obtain the following:

$$\begin{array}{c}{E}_{21}=xz(r_{a1}+\lambda r_u+c_s-c_{a1})+x(1-z)(r_{a1}-c_{a1})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +(1-x)z(r_{a1}+\lambda r_u-c_{a1})+(1-x)(1-z)(r_{a1}-c_{a1})\end{array}$$

(5)

$$\begin{array}{c}{E}_{22}=xz(r_{a2}-c_{a2}-p_a)+x(1-z)(r_{a2}-c_{a2}-p_a)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+(1-x)z(r_{a2}-c_{a2}-p_a)+(1-x)(1-z)(r_{a2}-c_{a2}-p_a)\end{array}$$

(6)

$$E_2=y{E}_{21}+(1-y)E_{22}$$

(7)

$$\begin{array}{c}F(y)={\displaystyle \frac{dy}{dt}}=y(E_{21}-E_2)=y(1-y)(E_{21}-E_{22})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =y(1-y)(c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}+xz{c}_s+z\lambda r_u)\end{array}$$

(8)

Third, assuming that stakeholders have an expected return of $E_{31}$ for adopting active participation and $E_{32}$ for adopting passive participation and an average expected return of $E_3$, and the replicated dynamic equation is $F(z)$. Then, based on the payoff matrix in Table 2, we can obtain

$$\begin{array}{c}{E}_{31}=xy\left[r_b+\left(1-\lambda \right)r_u-c_b\right]+x(1-y)\left(r_b-c_b-c_l\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+(1-x)y\left[r_b+\left(1-\lambda \right)r_u-c_b\right]+(1-x)(1-y)\left(r_b-c_b-c_l\right)\end{array}$$

(9)

$$E_{32}=xy(-c_0-p_b)+x(1-y)(-p_b)+(1-x)y(-c_0)$$

(10)

$$E_3=z{E}_{31}+(1-z)E_{32}$$

(11)

$$\begin{array}{c}F(z)={\displaystyle \frac{dz}{dt}}=z(E_{31}-E_3)=z(1-z)(E_{31}-E_{32})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=z(1-z)(r_b-c_l-c_b+y{c}_0+y{c}_l+x{p}_b+y{r}_u-y\lambda r_u)\end{array}$$

(12)

5. Stability Analysis

5.1. Stability of Governance Decisions

The replicated dynamic equation for decision-making is derived as shown in Equation (13):

$$F^{\prime }\left(x\right)={\displaystyle \frac{dF(x)}{dx}}=\left(2x-1\right)(c_g-p_b-p_g-r_g+y{p}_g+z{p}_b+yz{c}_s)$$

(13)

According to the principles of stability in equations, the condition for the probability of government decision-making to remain stable is $F(x)=0$ and $F^{\prime }(x)<0$. Consequently, the government’s stability strategy should be evaluated based on the values of $c_g-p_b-p_g-r_g+y{p}_g+z{p}_b+yz{c}_s$. Thus, we define $G\left(y\right)=c_g-p_b-p_g-r_g+y{p}_g+z{p}_b+yz{c}_s$, and when $G(y)=0$, we obtain $y_0={\displaystyle \frac{-c_g+p_b+p_g+r_g-z{p}_b}{p_g+z{c}_s}}$. By differentiating $G(y)$, we find that ${\displaystyle \frac{\partial G\left(y\right)}{\partial y}}=p_g+z{c}_s>0$, indicating that $G(y)$ is an increasing function with respect to $y$. When $y>y_0$, $G(y)>0$, leading to $F^{\prime }(0)<0$ and $F^{\prime }(1)>0$, which implies that $x=0$ represents an evolutionarily stable strategy, signifying a relaxed regulatory approach by the government as an evolutionarily stable strategy. Similarly, when $y<y_0$, $G(y)<0$, resulting in $F^{\prime }(0)>0$ and $F^{\prime }(1)<0$, thus $x=1$ serves as an evolutionarily stable strategy, indicating an active regulatory stance by the government as an evolutionarily stable strategy. At this moment, the trajectory of the evolutionary game under active government regulation is illustrated in Figure 2.

Figure 2 illustrates that when the government’s decision-making state is situated in space $v(x_1)$, $x=0$ represents an evolutionarily stable strategy, indicating that the governance approach is stable under a regime of relaxed regulation. Conversely, when the government’s decision-making state is located in space $v(x_2)$, $x=1$ signifies an evolutionarily stable strategy, suggesting that the governance approach is stable under a regime of proactive regulation.

5.2. An Analysis of the Stability of Governance Decision-Making in Digital Platform Enterprises

We differentiate the dynamic equation of replication for digital platform enterprises, as illustrated in Equation (14).

$$F^{\prime }\left(y\right)={\displaystyle \frac{dF(y)}{dy}}=\left(1-2y\right)(c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}+xz{c}_s+z\lambda r_u)$$

(14)

According to the principles of stability in equations, the condition for the decision-making probability of digital platform enterprises to remain stable is $F\left(y\right)=0$ and $F^{\prime }\left(y\right)<0$. Consequently, the stability strategies of digital platform enterprises should be considered based on the values of $c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}+xz{c}_s+z\lambda r_u$. Thus, let $H\left(x\right)=c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}+xz{c}_s+z\lambda r_u$ be defined such that when $H\left(x\right)=0$, we obtain $x_0={\displaystyle \frac{-c_{a2}+c_{a1}-p_a-r_{a1}+r_{a2}-z\lambda r_u}{z{c}_s}}$. By differentiating $H\left(x\right)$, we find that ${\displaystyle \frac{\partial H\left(x\right)}{\partial x}}=z{c}_s>0$, indicating that $H\left(x\right)$ is an increasing function with respect to $x$. When $x>x_0$, $H\left(x\right)>0$, leading to $F^{\prime }\left(0\right)>0$ and $F^{\prime }\left(1\right)<0$. In this scenario, $y=1$ represents an evolutionarily stable strategy, signifying that active cooperative governance is the evolutionarily stable strategy for digital platform enterprises. Conversely, when $x<x_0$, $H\left(x\right)<0$, resulting in $F^{\prime }\left(0\right)<0$ and $F^{\prime }\left(1\right)>0$, where $y=0$ denotes an evolutionarily stable strategy, indicating that passive cooperative governance is the evolutionarily stable strategy for digital platform enterprises. At this juncture, the evolutionary game trajectory of collaborative governance among digital platform enterprises is illustrated in Figure 3.

Figure 3 illustrates that when the decision-making state of digital platform enterprises is situated in space $v(y_1)$, where $y=1$ represents an evolutionarily stable strategy, the governance strategy of digital platform enterprises stabilizes at active cooperative governance. Conversely, when the decision-making state is located in space $v(y_2)$, where $y=0$ denotes an evolutionarily stable strategy, the governance strategy stabilizes at passive cooperative governance.

5.3. An Analysis of the Stability of Governance Decision-Making Among Stakeholders

We derive the dynamic equation for stakeholder decision-making, as illustrated in Equation (15):

$$F^{\prime }\left(z\right)={\displaystyle \frac{dF(z)}{dz}}=\left(1-2z\right)(r_b-c_l-c_b+y{c}_0+y{c}_l+x{p}_b+y{r}_u-y\lambda r_u)$$

(15)

According to the principles of stability in equations, the condition for the probability of stakeholder decision-making to be in a stable state is $F\left(z\right)=0$ and $F^{\prime }\left(z\right)<0$. Consequently, the stable strategies of stakeholders should be considered based on the values of $r_b-c_l-c_b+y{c}_0+y{c}_l+x{p}_b+y{r}_u-y\lambda r_u$. Thus, let $L\left(x\right)=r_b-c_l-c_b+y{c}_0+y{c}_l+x{p}_b+y{r}_u-y\lambda r_u$ be defined such that when $L\left(x\right)=0$, we obtain $x_{*}={\displaystyle \frac{-r_b+c_l+c_b-y{c}_0-y{c}_l-y{r}_u+y\lambda r_u}{p_b}}$. By differentiating $L\left(x\right)$, we find that ${\displaystyle \frac{\partial L\left(x\right)}{\partial x}}=P_b>0$, indicating that $L\left(x\right)$ is an increasing function with respect to $x$. When $x>x_{*}$, $L\left(x\right)>0$, leading to $F^{\prime }\left(0\right)>0$ and $F^{\prime }\left(1\right)<0$, which implies that $z=1$ represents an evolutionarily stable strategy, signifying active stakeholder participation as the evolutionarily stable strategy. Similarly, when $x<x_{*}$, $L\left(x\right)<0$, resulting in $F^{\prime }\left(0\right)<0$ and $F^{\prime }\left(1\right)>0$; thus, $z=0$ also serves as an evolutionarily stable strategy, indicating passive stakeholder participation as the evolutionarily stable strategy. At this point, the evolutionary game trajectory of stakeholder engagement is illustrated in Figure 4.

Figure 4 illustrates that when the stakeholder decision-making state is situated in space $v(z_1)$, where $z=1$ represents an evolutionarily stable strategy, the stakeholder governance strategy stabilizes at active participation. Conversely, when the stakeholder decision-making state is located in space $v(z_2)$, where $z=0$ denotes an evolutionarily stable strategy, the stakeholder governance strategy stabilizes at passive participation.

5.4. An Analysis of the Stability of the Overall System Evolution Strategy

From the replicator dynamic equations $F\left(x\right)=0$, $F\left(y\right)=0$, and $F\left(z\right)=0$, i.e., Equations (4), (8) and (12) equal to zero, respectively, we can derive the system’s overall equilibrium points as $E_1\left(0,0,0\right)$, $E_2\left(1,0,0\right)$, $E_3\left(1,1,0\right)$, $E_4\left(0,1,0\right)$, $E_5\left(0,1,1\right)$, $E_6\left(0,0,1\right)$, $E_7\left(1,0,1\right)$, $E_8\left(1,1,1\right)$, and $E_9\left(x^{*},y^{*},z^{*}\right)$. The points $E_1$ through $E_8$ represent pure governance strategies, while $E_9\left(x^{*},y^{*},z^{*}\right)$ corresponds to a mixed strategy that satisfies the conditions $F\left(x\right)=0$, $F\left(y\right)=0$, and $F(z)=0$.

$$\left\{\begin{array}{c}{x}^{*}={\displaystyle \frac{-r_b+c_l+c_b-y{c}_0-y{c}_l-y{r}_u+y\lambda r_u}{p_b}}\\ y^{*}={\displaystyle \frac{-c_g+p_b+p_g+r_g-z{p}_b}{p_g+z{c}_s}}\\ z^{*}={\displaystyle \frac{-c_{a2}+c_{a1}-p_a-r_{a1}+r_{a2}}{x{c}_s+\lambda r_u}}\end{array}\right.$$

(16)

Based on the replicated dynamic equations $F\left(x\right)$, $F\left(y\right)$, and $F\left(z\right)$, the Jacobian matrix can be derived.

$$J=\left[\begin{array}{ccc}{\displaystyle \frac{\partial F(x)}{\partial x}}& {\displaystyle \frac{\partial F(x)}{\partial y}}& {\displaystyle \frac{\partial F(x)}{\partial z}}\\ {\displaystyle \frac{\partial F(y)}{\partial x}}& {\displaystyle \frac{\partial F(y)}{\partial y}}& {\displaystyle \frac{\partial F(y)}{\partial z}}\\ {\displaystyle \frac{\partial F(z)}{\partial x}}& {\displaystyle \frac{\partial F(z)}{\partial y}}& {\displaystyle \frac{\partial F(z)}{\partial z}}\end{array}\right]=\left[\begin{array}{ccc}{J}_{11}& J_{12}& J_{13}\\ J_{21}& J_{22}& J_{23}\\ J_{31}& J_{32}& J_{33}\end{array}\right]$$

(17)

Among them,

$$J_{11}=\left(2x-1\right)(c_g-p_b-p_g-r_g+y{p}_g+z{p}_b+yz{c}_s)$$

(18)

$$J_{12}=x(x-1)(p_g+z{c}_s)$$

(19)

$$J_{13}=x(x-1)(p_b+y{c}_s)$$

(20)

$$J_{21}=-yz{c}_s(y-1)$$

(21)

$$J_{22}=\left(1-2y\right)(c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}+xz{c}_s+z\lambda r_u)$$

(22)

$$J_{23}=-y(y-1)(x{c}_s+\lambda r_u)$$

(23)

$$J_{31}=-z(z-1)p_b$$

(24)

$$J_{32}=-z(z-1)(c_0+c_l+r_u-\lambda r_u)$$

(25)

$$J_{33}=\left(1-2z\right)(r_b-c_l-c_b+y{c}_0+y{c}_l+x{p}_b+y{r}_u-y\lambda r_u)$$

(26)

According to Lyapunov’s theorem, if all eigenvalues of the Jacobian matrix are negative, the equilibrium point exhibits asymptotic stability, referred to as an $ESS$ point; if all eigenvalues are positive, the equilibrium point lacks asymptotic stability, thus being classified as an unstable point; and if the eigenvalues of the Jacobian matrix are mixed, comprising both positive and negative values, the equilibrium point is identified as a saddle point. The eigenvalues corresponding to each equilibrium point are presented in

Table 3. Equilibrium points and eigenvalues of the system.

Equilibrium Point	$\mathbf{Eigenvalue} {\mathit{\lambda }}_1$	$\mathbf{Eigenvalue} {\mathit{\lambda }}_2$	$\mathbf{Eigenvalue} {\mathit{\lambda }}_3$
$E_1=(0,0,0)$	$p_b-c_g+p_g+r_g$	$c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}$	$r_b-c_l-c_b$
$E_2=(1,0,0)$	$c_g-p_b-p_g-r_g$	$c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}$	$p_b-c_l-c_b+r_b$
$E_3=(1,1,0)$	$c_g-p_b-p_g$	$c_{a1}-c_{a2}-p_a-r_{a1}+r_{a2}$	$c_0-c_b+p_b+r_b+r_u-\lambda r_u$
$E_4=(0,1,0)$	$p_b-c_g+r_g$	$c_{a1}-c_{a2}-p_a-r_{a1}+r_{a2}$	$c_0-c_b+r_b+r_u-\lambda r_u$
$E_5=(0,1,1)$	$r_g-c_g-c_s$	$c_{a1}-c_{a2}-p_a-r_{a1}+r_{a2}-\lambda r_u$	$c_b-c_0-r_b-r_u+\lambda r_u$
$E_6=(0,0,1)$	$p_g-c_g+r_g$	$c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}+\lambda r_u$	$c_b+c_l-r_b$
$E_7=(1,0,1)$	$c_g-p_g-r_g$	$c_{a2}-c_{a1}+c_s+p_a+r_{a1}-r_{a2}+\lambda r_u$	$c_b+c_l-r_b-p_b$
$E_8=(1,1,1)$	$c_g+c_s-r_g$	$c_{a1}-c_{a2}-c_s-p_a-r_{a1}+r_{a2}-\lambda r_u$	$c_b-c_0-p_b-r_b-r_u+\lambda r_u$
$E_9=(x^{*},y^{*},z^{*})$	nexus point

.

Scenario 1.

When $c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}<0$ and  $p_b-c_l-c_b+r_b<0$ are present, the difference between the benefits and costs of active cooperative governance chosen by digital platform enterprises is less than the difference between the benefits and costs of passive cooperative governance minus the penalties imposed by the government. Additionally, when stakeholders opt for active participation, the difference between the costs and risks of losses and benefits is greater than the government penalties associated with passive participation. $E_2=(1,0,0)$ represents a stable point, and according to Table 3, the corresponding evolutionary stable strategy is active regulation, active cooperative governance, and passive participation.

Scenario 2.

When $c_{a1}-c_{a2}-p_a-r_{a1}+r_{a2}<0$ and $c_0-c_b+p_b+r_b+r_u-\lambda r_u<0$ are present, the difference between the benefits and costs of active cooperative governance chosen by digital platform enterprises exceeds the difference between the benefits and costs of passive cooperative governance, adjusted for government penalties. Additionally, when stakeholders opt for active participation, the disparity between costs and normal returns, as well as collaborative innovation returns, surpasses the opportunity costs and government penalties associated with passive participation. In this context, as illustrated in Table 3, $E_3=(1,1,0)$ represents a stable point, with the corresponding evolutionary stable strategy being active regulation, active cooperative governance, and passive participation.

Scenario 3.

When $c_{a2}-c_{a1}+c_s+p_a+r_{a1}-r_{a2}+\lambda r_u<0$ and $c_b+c_l-r_b-p_b<0$ are present, indicating that digital platform enterprises opt for proactive collaborative governance, the difference between normal returns, collaborative innovation gains, and government subsidies us costs is less than the difference between returns and costs when opting for passive collaborative governance, adjusted for government penalties. Additionally, the penalties faced by stakeholders choosing passive participation exceed the difference between costs and loss risks versus gains when opting for active participation. According to Table 3, $E_7=(1,0,1)$ represents a system stability point, with the corresponding evolutionary strategy being proactive regulation, passive collaborative governance, and active participation.

Scenario 4.

When $c_g+c_s-r_g<0$, $c_{a1}-c_{a2}-c_s-p_a-r_{a1}+r_{a2}-\lambda r_u<0$, and $c_b-c_0-p_b-r_b-r_u+\lambda r_u<0$ are present, specifically when the positive social image resulting from proactive government regulation yields greater benefits than the combined costs of such regulation and subsidies to digital platform enterprises, and when the benefits from active cooperation in governance, collaborative innovation, and government subsidies exceed the costs, the difference surpasses that of passive innovation cooperation adjusted for government penalties. Additionally, when stakeholders opt for active governance, the cost–benefit ratio and the difference in collaborative innovation benefits are less than the opportunity costs and government penalties associated with passive participation. In this context, as indicated in Table 3, $E_8=(1,1,1)$ represents a stable point in the system, with the corresponding evolutionary strategy being (proactive regulation, active cooperative governance, and active participation). The specific results of the stability analysis are presented in

Table 4. An analysis of the stability of system equilibrium points.

Equilibrium Point	Stability Conditions	Outcome
$E_1=(0,0,0)$	Any conditions	Instability
$E_2=(1,0,0)$	$c_{a2}-c_{a1}+p_a+r_{a1}-r_{a2}<0$ $p_b-c_l-c_b+r_b<0$	ESS
$E_3=(1,1,0)$	$c_{a1}-c_{a2}-p_a-r_{a1}+r_{a2}<0$ $c_0-c_b+p_b+r_b+r_u-\lambda r_u<0$	ESS
$E_4=(0,1,0)$	Any conditions	Instability
$E_5=(0,1,1)$	Any conditions	Instability
$E_6=(0,0,1)$	Any conditions	Instability
$E_7=(1,0,1)$	$c_{a2}-c_{a1}+c_s+p_a+r_{a1}-r_{a2}+\lambda r_u<0$ $c_b+c_l-r_b-p_b<0$	ESS
$E_8=(1,1,1)$	$c_g+c_s-r_g<0$ $c_{a1}-c_{a2}-c_s-p_a-r_{a1}+r_{a2}-\lambda r_u<0$ $c_b-c_0-p_b-r_b-r_u+\lambda r_u<0$	ESS
$E_9=(x^{*},y^{*},z^{*})$	Any conditions	Saddle point

.

6. Simulation Analysis

6.1. Establishment of System Dynamics Models

To address the limitations of evolutionary game theory in analyzing system dynamic complexity and visualizing strategy evolution paths, this paper introduces system dynamics methods to construct a simulation model for the interactive governance strategies among three parties. Evolutionary game theory provides the microfoundations for the analysis, while system dynamics builds a causal feedback structure incorporating level variables, rate variables, and key parameters, transforming the replicator dynamic equations from evolutionary games into a functional dynamic system. This approach visually simulates the evolutionary trajectories of governance strategy choices among the three parties over time and efficiently conducts parameter sensitivity analyses, thereby quantitatively revealing the mechanisms by which key factors influence the system’s convergence to an equilibrium state of active regulation, active cooperation, and active participation. This integration achieves a complementary relationship between theoretical modeling and dynamic simulation.

Based on the triadic evolutionary game model of collaborative governance within cross-domain digital innovation ecosystems, this study employs VensimPLE software to develop a system dynamics model for the triadic evolutionary game. Through the application of system dynamics methodology, this research investigates the strategic choices of governance entities during the evolutionary game process and conducts simulation analysis, aiming to explore the impact of varying parameters on the governance behavior selection of these entities. The system dynamics model is illustrated in Figure 5.

The model encompasses various variable components, where the level variables include the probabilities of government, digital platform enterprises, and stakeholders opting for collaborative governance, denoted as $x$, $y$, and $z$, respectively. The rate variables reflect the rate of change in the probabilities of the three governance entities selecting collaborative governance strategies, represented as $F\left(x\right)$, $F\left(y\right)$ and $F\left(z\right)$. The mediating variables consist of the expected returns from different decision-making strategies chosen by the three governance entities, along with the differences in these expected returns. The exogenous auxiliary variables include $c_g$, $r_g$, $c_s$, $p_g$, $c_{a1}$, $r_{a1}$, $r_u$, $\lambda$, $c_{a2}$, $r_{a2}$, $p_a$, $c_b$, $r_b$, $c_l$, $p_b$ and $c_0$.

To further explore the mechanism of governance mechanisms, in terms of model parameter assignment, based on the actual situation of collaborative governance in cross-domain digital innovation ecosystems, we reference relevant studies by Wang Yuting, Yi Jiabin, and Yu Lantian [53], Wei Xiaochao, and Pan Gangmei [54], and Sajib Mandal [55], and combine them with the average collaborative governance costs of digital platform enterprises from the “Report on High-Quality Development of China’s Digital Economy (2023)” [56]. The values of the exogenous auxiliary variables in the model under the initial state are shown in

Table 5. Variable assignment cases.

Parameter	Description	Assignment	References
$c_g$	The costs associated with proactive government regulation	15	[53,54,55,56]
$r_g$	The positive social image benefits derived from effective government regulation	25
$c_s$	Government support and subsidies for digital platform enterprises	8
$p_g$	The government has failed to regulate effectively and has not recognized the social image losses caused by the misconduct of digital platform companies	8
$c_{a1}$	The costs associated with collaborative governance among digital platform enterprises	16
$r_{a1}$	The normal returns generated when digital platform enterprises actively engage in collaborative governance	19
$r_u$	The benefits of cross-domain collaborative innovation between digital platform enterprises and stakeholders	12
$\lambda$	The distribution ratio of collaborative innovation benefits obtained by digital platform enterprises	0.5
$c_{a2}$	The costs associated with the governance of digital platform enterprises during instances of passive collaboration	10
$r_{a2}$	Excess returns of digital platform enterprises during negative collaborative governance	27
$p_a$	The penalties faced by digital platform enterprises when non-compliance is detected by the government during instances of passive cooperation governance	15
$c_b$	The costs associated with active stakeholder engagement	20
$r_b$	Benefits of active stakeholder engagement	15
$c_l$	The risk of losses resulting from the negative collaborative governance of digital platform enterprises by stakeholders	8
$p_b$	The penalties faced by stakeholders when their passive participation is detected by the government	5
$c_0$	Opportunity costs arising from stakeholder disengagement	7

.

6.2. Initial Evolutionary Pathway Analysis

To investigate the scenario where the decision-making strategies of collaborative governance entities evolve over time in a cross-domain digital innovation ecosystem when there are differences in their initial willingness levels, this paper sets different initial probability values and analyzes them using the system dynamics simulation method. The results are shown in Figure 6.

The comparison of the simulation results presented in Figure 6a–c indicates that variations in initial probability settings do not significantly affect the evolutionary decision-making strategies of the government, digital platform enterprises, and stakeholders. All entities tend to converge towards a stable state characterized by proactive regulation, collaborative governance, and active participation. As observed in the results of Figure 6, the willingness of stakeholders to engage actively exhibits considerable volatility, with a noticeable increase in their willingness to participate only as the government approaches an equilibrium state. This phenomenon suggests that stakeholders’ decisions to engage in collaborative governance are significantly influenced by government policies. Furthermore, as the initial probabilities of collaborative governance entities increase, the speed at which digital platform enterprises and stakeholders reach equilibrium also accelerates. Thus, enhancing the willingness and awareness of all governance game participants towards collaborative governance plays a crucial role in fostering value co-creation within cross-sector digital innovation ecosystems and achieving long-term stability.

6.3. The Impact of Changes in Government Punitive Mechanisms

Figure 7 shows the impact of the role of the punishment mechanism on the governance decision-making strategies of digital platform enterprises and stakeholders. According to Figure 7a, it can be seen that when the digital platform enterprise’s excess benefit, $r_{a2}$, is taken as 30 and the penalty, $p_a$, is taken as 15, the digital platform enterprise starts to take the risk and evolve towards negative innovation cooperation. Meanwhile, according to Figure 7b, it can be seen that as digital platform enterprises begin to choose negative cooperative governance, stakeholders also choose negative participation, which suggests that the choice of governance strategy of digital platform enterprises is key to the stabilization of the entire cross-domain digital ecosystem. From Figure 7c,d, it can be seen that when the amount of punishment, $p_a$, is increased, the punishment mechanism can promote the cross-domain digital innovation ecosystem collaborative governance utility, which can effectively avoid fluctuations in the process of the governance game between the digital platform enterprise and stakeholders and realize the state of positive regulation, negative cooperative governance, and negative participation in the cross-domain digital innovation ecosystem.

6.4. The Impact of Digital Platform Enterprises on the Collaborative Governance of Cost and Revenue Fluctuations

To conduct an in-depth analysis of the variations in costs and benefits associated with core digital platform enterprises during the active collaborative governance process and their impact on the evolutionary game of governance entities, this study sets the cost of active collaborative governance, $c_{a1}$, at 10, 16 (initial state), 24, 25, and 26. The simulation results are illustrated in Figure 8a,b. When $c_{a1}$ is relatively low, the government, digital platform enterprises, and stakeholders gradually stabilize at a state of active regulation, active collaborative governance, and active participation. Conversely, when $c_{a1}$ is significantly high, digital platform enterprises must weigh the trade-offs between benefits and costs, resulting in a prolonged period to reach equilibrium. During this phase, stakeholders may lean towards adopting a strategy of passive participation due to the uncertainties surrounding the governance strategies of digital platform enterprises.

With other parameters unchanged, this paper makes the gains $r_{a1}$ be −5, −4, 0, 10, 14, and 19 (the initial state) and compares the evolution paths of governance strategies of digital platform enterprises and stakeholders, and the results are shown in Figure 8c,d. Comparing the initial state, it can be seen that the size of the benefits of digital platform enterprises actively cooperating in governance affects the stability of the game system of the three-party governance subjects. If the digital platform enterprise in cooperative governance achieves higher gains at the same moment it adopts the active regulation strategy, then the digital platform enterprise should adopt the active cooperative governance strategy and stakeholders should adopt the active participation strategy. If the digital platform enterprise in the cooperative governance has lower revenue or even losses, the digital platform enterprise will consider the costs and benefits of active cooperative governance and will tend to choose the strategy of negative cooperative governance, and at this time, the cross-domain digital innovation ecosystem collaborative governance tends to deteriorate and the stakeholders will take into account the losses brought about by the negative cooperative governance of the digital platform enterprise, $c_l$, and will also choose the governance strategy of negative participation. It can be seen that the high cost consumed by the digital platform enterprise’s positive cooperative governance will lead it to choose the negative cooperative governance; however, the digital platform enterprise, as the key core subject of collaborative governance, will lead to a change in the stakeholders’ governance strategy by its choice of governance strategy. Therefore, in the process of the collaborative governance of cross-domain digital innovation ecosystems, we should play a supervisory role as far as possible, appropriately give certain incentives and subsidies to the core body of digital platform enterprises, and at the same time, actively mobilize stakeholders to participate in the governance initiative.

6.5. The Impact of Changes in Internal Incentive Mechanisms

Figure 9 illustrates the impact mechanism of collaborative innovation benefits on governance decisions within digital platform enterprises and their stakeholders. In cross-domain digital innovation ecosystems, the collaborative governance process can stimulate the willingness of various governance entities to cooperate, thereby facilitating the realization of collaborative innovation benefits. This system functions to implement an incentive mechanism for the distribution of collaborative innovation benefits. As depicted in Figure 9, an increase in collaborative innovation benefits ($r_u$) accelerates the convergence of digital platform enterprises and stakeholders towards unity. This indicates that the cross-domain digital innovation ecosystem, through the implementation of collaborative governance mechanisms, can impose certain constraints on the decision-making behaviors of governance entities within the system. This phenomenon reveals that the benefits derived from collaborative innovation can effectively mitigate stakeholders’ divergence from the overall interests of the system, thereby reducing moral hazard.

Figure 10 illustrates the impact of the distribution ratio of collaborative innovation benefits on governance decisions within digital platform enterprises and their stakeholders. Core entities of cross-domain digital innovation ecosystems, specifically digital platform enterprises, incentivize stakeholder engagement in collaborative governance by sharing the benefits of collaborative innovation. As depicted in Figure 10, a higher distribution ratio of collaborative innovation benefits leads both digital platform enterprises and stakeholders to converge towards a value of 1, with stakeholders exhibiting a greater sensitivity to changes in the distribution ratio compared to digital platform enterprises. This indicates that the cross-domain digital innovation ecosystem exerts a significant influence on the governance decisions of its members through appropriate incentive mechanisms. This is evidenced by the ability of digital platform enterprises to foster active stakeholder participation in the collaborative governance of the system through the sharing of collaborative innovation benefits.

6.6. Results and Discussion

This paper employs evolutionary game theory and system dynamics to investigate the collaborative governance of cross-domain digital innovation ecosystems. The findings indicate the following:

1. The decision-making evolution of governance entities tends toward a stable equilibrium: During the evolution of the initial intentions of governance entities, systemic governance decisions aim to achieve a balanced state characterized by a strategic combination of proactive regulation, collaborative governance, and active participation. This equilibrium governance strategy significantly promotes the stable development of cross-domain digital innovation ecosystems. This finding supports the perspectives of Smith and Price, as well as Taylor and Jonker, in evolutionary game theory, which posit that boundedly rational agents tend to stabilize their strategies in dynamic environments [31,32]. Furthermore, it validates that in cross-domain digital innovation ecosystems, governance entities can ultimately reach a relatively stable collaborative governance state through continuous strategy adjustments, thereby facilitating the stable development of the system. This represents an extension of research on governance in digital innovation ecosystems within practical application contexts.

2. The positive impact of government penalties and systemic incentive mechanisms: The punitive measures established by the government, along with the incentive structures developed within the system, play a crucial role in facilitating the evolution of collaborative governance decision-making. Specifically, the imposition of higher penalty amounts and an increased distribution ratio of collaborative innovation benefits will expedite the system’s attainment of a stable collaborative governance state. The enhancement of collaborative innovation benefits will more effectively motivate the system to adopt collaborative governance strategies. This aligns with the findings of Ansell, Gash and Doberstein, which emphasize the significant role of incentive mechanisms in fostering cooperation among diverse stakeholders [42,47]. Furthermore, it elucidates the specific mechanisms through which the government’s punitive measures and systemic incentives guide the behavior of governance actors and promote collaborative governance within the context of digital innovation ecosystems. This research deepens and expands the application of collaborative governance theory in the realm of digital innovation.

3. The negative impact of governance costs and benefits on digital platform enterprises: The costs and benefits incurred by core digital platform enterprises during collaborative governance processes adversely affect the evolution of their systemic collaborative governance decisions. Specifically, as the costs of collaborative governance for digital platform enterprises increase, the tendency for these enterprises and their stakeholders to engage in passive collaborative governance and participation accelerates. This study addresses the gap in previous research regarding the influence of cost–benefit factors on governance decisions within the realm of digital innovation ecosystems, offering a new perspective for the study of governance mechanisms in these ecosystems. It underscores the importance of carefully considering the governance costs and benefits of digital platform enterprises during the governance process to mitigate any detrimental effects on systemic collaborative governance.

7. Conclusions and Recommendations

This paper is predicated on the bounded rationality of governance decision-makers and employs methods from evolutionary game theory and system dynamics in the study of collaborative governance within cross-domain digital innovation ecosystems. A tripartite evolutionary game model comprising the government, digital platform enterprises, and stakeholders is established. Through the analysis of the stability of governance strategies and simulation modeling, the following conclusions are drawn:

1. From the perspective of the initial willingness levels of a singular governance entity, regardless of how the willingness of the governance entity fluctuates, the systemic governance decisions ultimately stabilize at active regulation, collaborative governance, and active participation. Furthermore, during the interplay among the government, digital platform enterprises, and stakeholders, the governance strategies of these three parties mutually influence one another. The higher the initial probability of active governance engagement by the governance entities, the more rapidly the system reaches an equilibrium state. Therefore, enhancing the willingness and awareness of each governance participant to engage in collaborative governance plays a crucial role in fostering value co-creation within cross-sector digital innovation ecosystems and achieving long-term stability.

2. From the perspective of the government’s punitive mechanisms and the internal incentive structures within the system, there is a positive facilitative effect on the evolution of the system towards collaborative governance. Specifically, an increase in the amount of government penalties and the distribution ratio of collaborative innovation benefits will aid the system in achieving a stable state of collaborative governance more rapidly. The greater the collaborative innovation benefits, the more likely the system is to be motivated to adopt collaborative governance strategies.

3. From the perspective of cost–benefit analysis regarding the proactive collaborative governance of core digital platform enterprises, the costs associated with such governance negatively influence the evolution of the system towards collaborative governance decision-making. Specifically, as the governance costs for digital platform enterprises increase, the tendency for these enterprises and their stakeholders to engage in passive collaborative governance and participation accelerates.

Based on the aforementioned research findings and in light of the current state of collaborative governance within cross-sector digital innovation ecosystems, this paper offers the following recommendations:

1. From the Government Perspective—Firstly, the government should refine a precision-based regulatory framework: The government should formulate and dynamically update laws, regulations, and standards for data sharing, security, rights, and responsibilities that are adaptable to cross-domain scenarios. It should establish penalty mechanisms based on the nature and impact of violations, clearly define penalty rules, ensure transparent enforcement, and significantly increase the expected risk cost of passive governance by digital platform enterprises.

Secondly, the government should construct diversified incentive mechanisms: The government should set up special funds or provide tax incentives to offer quantified rewards to digital platform enterprises and core stakeholders that demonstrate significant results in data interoperability, standard co-creation, and major collaborative innovation projects.

Finally, the government should build trusted collaboration infrastructure: The government should lead or support the construction of regional or industry-specific data circulation platforms. These platforms should provide basic services such as rights confirmation, valuation, and transaction matching, systematically reducing the costs and trust barriers of collaborative innovation and promoting ecosystem-level collaboration.

2. From the Digital Platform Enterprise Perspective—Firstly, digital platform enterprises should reduce collaborative governance costs: Through technological innovation, process standardization, and economies of scale, they should meticulously manage the costs of integrating cross-domain digital innovation resources, interface development, and coordination. They should seek to utilize government subsidies to share upfront investments.

Secondly, digital platform enterprises should promote the value of collaborative governance: Enterprises should clearly calculate and communicate the long-term ecosystem value generated by active governance, including market expansion, strategic positioning, and data asset appreciation. Concurrently, they should establish dedicated governance bodies responsible for partner management, rule enforcement, and conflict mediation.

Finally, digital platform enterprises should enhance platform openness and compatibility: Enterprises should invest in building open ecosystems and user-friendly toolchains to significantly lower the entry barriers and participation costs for heterogeneous entities. Moreover, they should proactively initiate cross-domain projects to demonstrate the effectiveness of collaboration through successful case studies.

3. From the Stakeholder Perspective—Firstly, stakeholders should proactively enhance digital literacy and collaborative capabilities: They should actively participate in training organized by platforms or the government and fully utilize platform tools to reduce participation costs.

Secondly, stakeholders should actively contribute domain-specific data, scenario knowledge, or digital innovation resources while ensuring security and compliance, thereby becoming sources of value co-creation.

Finally, stakeholders should, through established governance mechanisms, actively monitor the behavior of platforms and enterprises, safeguarding their own rights and interests while fostering systemic trust.

8. Limitations and Future Directions

Firstly, while this paper constructs an evolutionary game model of triadic governance involving government, digital platform enterprises, and stakeholders, it is important to note that stakeholders encompass various types of entities that can influence the ultimate equilibrium point of the system governance combination. Therefore, future research could benefit from a more nuanced segmentation of stakeholders to develop a multi-party evolutionary game model that is more targeted and precise. Secondly, the parameter settings in the simulation analysis of this study are primarily based on assumptions and references to real-world situations, which may not fully capture all actual conditions. Consequently, future studies could optimize the parameter setting methodology by employing advanced technologies to enhance the scientific rigor and accuracy of model parameters. Finally, this paper primarily investigates the issue of system collaborative governance through theoretical frameworks and simulations, lacking an analysis of real-world cases. Thus, future research could select multiple representative cases for comparative analysis to strengthen the practical foundation of the theory and enhance the generalizability of the research findings.