In an innovation ecosystem based on cross-regional operations, self-organization is an important mechanism that drives the co-evolution of cross-regional operating enterprises, local enterprises in target areas, and target regions. From a micro perspective, cross-regional operating enterprises show the characteristics of self-emergence and self-adaptation in the process of integrating resources, technology, and knowledge across regions. By continuously acquiring new market information and absorbing external talents and technologies, the internal management methods and innovation strategies of enterprises gradually adapt to the external environment and form new synergy models. While pursuing the maximization of their own interests, micro-subjects often give rise to innovative behaviors that benefit the system as a whole in dynamic games and cooperation networks, thereby achieving efficient utilization and integration of heterogeneous resources [1]. From a meso perspective, the industrial clusters formed in the same region are like organisms of “innovation populations.” These clusters will continuously interact with local policies, industrial systems, technological atmosphere, and other environmental factors, and at the same time, they will show ups and downs and non-linear evolution due to the relationship between government and enterprises. When an enterprise achieves a first-mover advantage through cross-regional development or forms a new technological breakthrough, the entire innovation population will experience a ripple effect, promoting the reflow and reallocating resource elements within the cluster [2,3]. In this process, both win-win cooperation and joint expansion of new value chains between cross-regional enterprises and local governments of cross-entry places may occur. New industrial restructuring may be spawned due to the break-up of market competition patterns. From a macro perspective, the formation and evolution of innovation clusters result from mutual synergy among cross-regional resources, policies, and industrial environments. At the macro level, the formation and evolution of innovation clusters result from mutual synergy among cross-regional resources, policies, and industrial environments. In the ecosystem of cross-regional operations, the governments, industry associations, and various social resource networks of cross-regional regions constitute the external environment that supports the evolution of clusters. While exchanging information and energy in the macro-environment, the innovation community will continuously break the original equilibrium state and move towards a more advanced and orderly structure. Along with the fluctuation of market demand and the adjustment of policy support, non-linear effects such as competition, learning, and imitation will be generated among innovation subjects, which will provide endogenous power for the rise and fall of the system, thus accelerating the realization of new synergies and upgrading of enterprises [4,5].

In this process of self-organized evolution, the choices and strategies of each innovation subject may have an impact on the system. When a new technology or business model changes the original competition–cooperation pattern, the equilibrium state of the system is broken [6]. Subsequently, under the dominance of core technology, financial support, market potential, etc., the innovation cluster will form a new order that is more advanced and stable. It is in the rise and fall and reorganization over and over again that the innovation ecosystem operating across districts develops the robustness of self-adaptation, self-sustenance, and self-development, which continuously spawns the interactive game between advantageous cross-district firms and the government of the cross-entry place [7,8]. Through continuous self-organized evolution, innovation agents in the cross-regional context jointly promote economic growth as well as collaborative innovation, providing solid support for the overall sustainable competitiveness of the system.

The formation of cross-regional business strategies is closely related to the upgrading of the enterprise’s own industrial structure, the regional competition pattern, the regional economic environment, and the local policy orientation, and its decision-making motivation mainly comes from the integration of external resources and the need for complementarity of advantages, as well as the incentives of the local government for attracting investment and upgrading industry [9,10]. Therefore, the interaction between firms and the governments of cross-entry regions not only involves elements such as the scale of investment, policy incentives, and industrial synergies but also profoundly affects the innovation activities of the firms and the regional innovation ecosystem improvements.

Specifically, when choosing whether to conduct cross-regional operations and selecting target regions, cross-regional enterprises usually hope to obtain favorable conditions such as tax incentives, supporting resources and high-quality R&D environments, to reduce the overall risks and costs of innovation [11] and accelerate the connection with the local industrial chain and relevant public research institutions. Meanwhile, the government of the trans-boundary region expects enterprises to transfer their rich practical experience and cutting-edge technologies to inject new R&D impetus and innovation concepts into the local area and to drive the related industries and enterprises in the region to follow up through the knowledge spillover effect, to form the diffusion of innovation effect, enhance the technological innovation capability of the region or industry as a whole, further optimize the regional industrial structure, and improve the overall competitiveness and sustainable development capability. If the two sides can form an effective incentive mechanism between cross-region operations and policy support, cross-region enterprises will have more incentives to plough into the local innovation ecosystem, and the government will reap the double benefits of economic growth and regional innovation. In contrast, if the two sides fail to reach a reasonable balance, not only can the innovation potential of enterprises not be fully released, but it may also lead to resource mismatch and competitive imbalance. Based on the above analyses, a suitable mutual participation model is constructed, as shown in Figure 1.

The formation of cross-regional business strategies is closely related to the upgrading of the enterprise’s own industrial structure, the regional competition pattern, the regional economic environment, and the local policy orientation. Its decision-making motivation mainly comes from the integration of external resources and the need for complementarity of advantages, as well as the incentives of the local government for investment attraction and industrial upgrading [9,10]. Therefore, the interaction between firms and the governments of cross-entry regions not only involves elements such as the scale of investment, policy incentives, and industrial synergies but also profoundly affects the innovation activities of the firms as well as the improvement of the regional innovation ecosystem. Specifically, when choosing whether to conduct cross-regional operations and selecting target regions, cross-regional enterprises usually hope to obtain favorable conditions such as tax incentives, supporting resources and high-quality R&D environments, so as to reduce the overall risks and costs of innovation [11], and accelerate the connection with the local industrial chain and relevant public research institutions. Meanwhile, the government of the trans-boundary region expects enterprises to transfer their rich practical experience and cutting-edge technologies to inject new R&D impetus and innovation concepts into the local area and to drive the related industries and enterprises in the region to follow up through the knowledge spillover effect, so as to form the diffusion of innovation effect, enhance the technological innovation capability of the region or industry as a whole, further optimize the regional industrial structure, and improve the overall competitiveness and sustainable development capability.

If the two sides can form an effective incentive mechanism between cross-region operations and policy support, cross-region enterprises will have more incentives to plough into the local innovation ecosystem, and the government will be able to reap the double benefits of economic growth and regional innovation. In contrast, if the two sides fail to reach a reasonable balance, not only can the innovation potential of enterprises not be fully released, but it may also lead to resource mismatch and competitive imbalance. Based on the above analyses, a suitable model of mutual participation is constructed, as shown in Figure 1.

The specific assumptions made to construct the game model of cross-region enterprises and cross-entry place government follow.

Assumption 1: Participating subject 1 is the enterprise, and participating subject 2 is the local government. Both parties are limited-rational, information-limited economic players who will adjust their strategies in response to external changes [12]. Firms may choose to operate across regions and set up subsidiaries in the target region to seek external heterogeneous resources to enhance their innovation capabilities, or they may not operate across regions and concentrate their resources locally to enhance their innovation capabilities through existing resources [13]. Therefore, the firm’s strategy choice space s 1 = O p e r a t e a c r o s s t h e r e g i o n , d o n o t o p e r a t e a c r o s s t h e r e g i o n . Meanwhile, the government of the cross-entry location can choose to provide supportive policies to attract external firms to set up subsidiaries locally so as to enhance the regional economic vitality and innovation capacity, or it can choose to remain neutral and not provide additional policy incentives. Therefore, the strategy choice space of the local government s 2 = a c t i v e s u p p o r t , n o a c t i v e s u p p o r t .

Assumption 2: Firms choose with probability x to engage in cross-regional operations and with probability 1 − x not to engage cross-regional operations; the government of the place where they cross into chooses with probability y to actively support them and with probability 1 − y not to actively support them. x , y ∈ 0,1 and are both functions of time t .

Assumption 3: For enterprises, it is their choice whether or not to operate across regions. If it chooses to operate across regions, its ability to access resources and the extent to which it can do so may be affected by resource conditions in the target region, government support, and local market expansion. The basic innovation investment cost for local development is C 1 , and the innovation gain for local development is R 1 . Cross-regional operations lead to increased internal management costs, resulting in direct costs of C 2 . Cross-regional operations also incur competition costs in different locations, innovation investments in these areas, and initial investments and adaptation expenses required to enter new regions, collectively forming C 3 . Additionally, heterogeneous resources are obtained in different regions, generating innovation benefits R 2 and market expansion benefits R 3 . When the host government adopts proactive support policies, companies can also receive fiscal subsidies and tax incentives provided by the host government, which constitute policy benefits S .

Assumption 4: An inbound government that chooses active support must provide policy support such as financial subsidies and tax incentives to attract external enterprises to operate in the region and increase their investment, and the cost of this is S . In addition to direct policy support, the inbound government must strengthen the supervision of the external enterprises and coordinate the competition and cooperation between them and the local enterprises, and this generates an additional cost of C 4 . At the same time, the direct economic growth and tax benefits brought about by external firms’ cross-border operations in the local area are D . When the inbound government chooses to actively support them, the inbound government can promote collaborative innovation through coordinating the establishment of close cooperation between the external firms and the local firms or research institutes, thus generating knowledge spillovers to the local area as H , and when the inbound government does not actively support them or sets up barriers to entry, the external firms might be inclined to give up cross-border operations and concentrate their resources on their own locations. Thus, a government of a trans-entry location that does not actively support collaborative innovation may miss out on potential opportunity gains such as economic growth, employment opportunities, knowledge spillovers, and increased tax revenue that can be brought about by the external firm, which are defined as T . Although local governments must bear the additional cost of C 4 when actively supporting external enterprises, C 4 represents the initial short-term costs. The opportunity benefits brought about by the introduction of external enterprises, including long-term economic growth, increased tax revenue, creation of job opportunities, knowledge spillover, and technological innovation, are far greater than the cost of C 4 in total.

The corresponding parameters are shown in Table 1, and the payoff matrix is shown in Table 2.

Based on the above game analysis between cross-region enterprises and local government, the behavioral strategy choices are suitable for further analysis using the dynamic replication equation in evolutionary game theory. Therefore, this section constructs the dynamic replication equations of cross-district enterprises and local government to further describe the strategy evolution process of both sides of the game.

The firm’s expected return from choosing to operate across the region is U 1 , and the earnings expectation function, defined in Equations 1–13: U 1 = y R 2 + R 3 + S − C 2 − C 3 + 1 − y R 2 + R 3 − C 2 − C 3 (1) The expected return from not choosing to operate across the region is U 1 n , and the earnings expectation function is U 1 n = y R 1 − C 1 + 1 − y R 1 − C 1 (2) The average expected return for a firm adopting a mixed strategy is U ̄ 1 , and the earnings expectation function is U ̄ 1 = x U 1 + 1 − x U 1 n (3) Therefore, the dynamic replication equation for a firm undertaking a cross-regional business strategy is F x = d x d t = x U 1 − U ̄ 1 = x 1 − x U 1 − U 1 n = x 1 − x S y + C 1 + R 3 − C 2 − C 3 − R 1 + R 2 (4) The expected payoff of the local government’s choice of active support is U 2 given by the earnings expectation function: U 2 = x D + H − S − C 4 + 1 − x − C 4 (5) The expected payoff for a local government that chooses not to actively support is U 2 n , and the earnings expectation function is U 2 n = x D + 1 − x − T (6) The average expected return of a mixed strategy by a local government is U ̄ 2 , and the earnings expectation function is U ̄ 2 = y U 2 + 1 − y U 2 n (7) Therefore, the dynamic replication equation for the trans-entry local government is F y = d x d t = y U 2 − U ̄ 2 = y 1 − y U 2 − U 2 n = y 1 − y H − S − T x − C 4 + T (8) The union of the above two replicated dynamic equations leads to a two-dimensional dynamical system I reflecting the evolution of the behavior of the two parties over time. F x = d x d t = x 1 − x S y + C 1 + R 3 − C 2 − C 3 − R 1 + R 2 F y = d x d t = y 1 − y H − S − T x − C 4 + T (9) The essential properties of a system are determined by the system’s stationary state, which is often portrayed by the equilibrium point equation of the system, which consists of the points in the system where the derivatives of all state variables with respect to time are zero. When the system is in a stationary state, the state of the system no longer changes; that is, the system is in equilibrium. Therefore, the equilibrium point equation of the system is F x = d x d t = 0 F y = d x d t = 0 (10) From Equations 6–10, the points O 0,0 , A 0,1 , B 1,0 , C 1,1 are the equilibrium points of the system, and let x 0 = C 4 − T H − S − T (11) y 0 = − C 1 + C 2 + C 3 + R 1 − R 2 − R 3 S (12) Because both X O and y 0 are probabilities, they must be satisfied such that they take values between 0 and 1 when 0 < C 4 − T < H − S − T , 0 < − C 1 + C 2 + C 3 + R 1 − R 2 − R 3 < S , in order to ensure that 0 < x 0 < 1 , 0 < y 0 < 1 , but because the hypothesis analysis has already been stated as C 4 − T < 0 , and X O can be considered to lack practical significance, x 0 , y 0 is not an evolutionary stabilization point.

According to Friedman’s findings, it is known that the equilibrium point of the system is not necessarily the stable point of the system, and the local stability of the equilibrium point can be judged by the Jacobian matrix. Taking J to represent the Jacobian matrix of the two-dimensional dynamical system () in the static regime, it is obtained that J = ∂ F x ∂ x ∂ F x ∂ y ∂ F y ∂ x ∂ F y ∂ y = a 11 a 12 a 21 a 22 = 1 − 2 x S y + C 1 + R 3 − C 2 − C 3 − R 1 + R 2 x 1 − x S y y 1 − y H − S − T x 1 − 2 y H − S − T x − C 4 + T (13)

An equilibrium point is an evolutionarily stable strategy for the system only if the Jacobian matrix corresponding to the equilibrium point simultaneously satisfies the determinant D e t J > 0 and T r J < 0 . The stability analysis of the five equilibrium points obtained for the dynamical system I is shown in Table 3.

According to the calculation results in Table 3 and the judgment conditions of the evolutionary stability strategy, it can be seen that the signs of the equilibrium points 0,0 , 0,1 and 1,1 are uncertain, and it is necessary to further analyze the parameters before determining whether they are evolutionary stability points (ESSs).

In order to reflect more intuitively the influence of the changes of each decision variable on the evolution process and the results of the game between the enterprise and the government of the trans-entry place, MATLAB is used to carry out the simulation analysis. In case 1, the following values are assigned to each parameter: the basic innovation input cost of local development is 8, the direct cost of inter-regional operations is 15, the competition, innovation and market entry costs of inter-regional operations are 12, the innovation gain from accessing heterogeneous resources is 6, the market expansion gain is 9, the financial subsidies and tax incentives are 5, the government regulation and coordination cost is 4, the potential loss of gains when the government is not active is 15, the gain from external firms to local knowledge spillovers when the government is active is 8.5, and the innovation gain from local development is 10,15,20 . The potential loss of gains is 15, the gain in local knowledge spillovers from external firms is 8.5 when the government actively supports them, and the innovation gains from local development are 10,15,20 . The strategy evolution process and results for firms and cross-entry local governments are shown in Figure 2. It can be seen that firms tend to choose local development innovations when the net benefits from local development are higher than the net benefits from innovations through cross-boundary operations. In addition, as the gap between the net return on innovation and the net return on innovation from operating across districts continues to widen, the evolution of firms’ strategies toward local development accelerates significantly. This suggests that the increasing gap in net innovation gains reinforces firms’ preference for local development, prompting them to adjust their strategies faster to maximize their economic efficiency and innovation capabilities. This suggests that the return gap is a significant driver of strategy choice.

In case 2, the following values are assigned to each parameter: the basic innovation input cost of local development is 8, the direct cost of inter-regional operations is 15, the competition, innovation and market entry costs of inter-regional operations are 12, the market expansion gain is 9, the financial subsidies and tax incentives are 5, the cost of government regulation and coordination is 4, the gain that may be lost if the government does not actively support it is 15, the gain in local knowledge spillovers from external firms if the government actively supports it is 8.5, the innovation gain from local development is 10, and the innovation gain from accessing heterogeneous resources is 25,30,35 8.5 for local knowledge spillovers from external firms with active government support, 10 for innovation gains from local development, and 25,30,35 for innovation gains from access to heterogeneous resources. The strategy evolution process and results of the firms and the cross-entry local governments are shown in Figure 3. Enterprises tend to choose cross-entry operations when the net innovation benefit from cross-entry operations is higher than the net innovation benefit from local development. In addition, as the gap between the net innovation gains from cross-district operations and the net innovation gains from local development continues to widen, the evolution of firms’ strategies for choosing cross-district operations accelerates significantly.

In case 3, the following values are assigned to each parameter: the basic innovation input cost of local development is 8, the innovation gain of local development is 10, the direct cost of inter-regional development is 15, the competition, innovation, and market entry costs of inter-regional development are 12, the innovation gain from accessing heterogeneous resources is 21, the market expansion gain is 9, the fiscal subsidy and tax incentives are 5, the government regulation and coordination costs are 4, the potential loss of gains when the government is not active is 15, and the gain in local knowledge spillovers from external firms when the government is active is 10,15,20 . The strategy evolution process and the results of firms and cross-entry local governments are shown in Figure 4. Firms tend to choose to operate across regions when the net innovation benefit they obtain from doing so is higher than the net innovation benefit from local development. When the benefits that the inbound government can obtain exceed the subsidies it provides and the costs it bears to support cross-border operations, the inbound government tends to introduce supportive policies. At the same time, the evolution of the cross-border government’s strategy towards active support accelerates significantly as its benefits increase.

This paper explores the formation and dynamic evolution process of the interaction mechanism between cross-regional operating enterprises and local government based on evolutionary game theory. The study shows that the strategic choices of both parties are influenced by multiple factors such as benefit distribution, policy environment, risk cost, and cooperation expectation, and their interactive behavior shows significant dynamic adaptability and path dependence. The strategic evolution between cross-regional enterprises and local government is stable only if their respective returns are in line with their rational decision-making, and the choice of strategy is strongly influenced by each party’s own net returns. In the long run, enterprises tend to invest more in regions with high policy support and low institutional costs, while local governments balance the goals of economic growth and the preservation of public interest by adjusting the intensity of regulation and incentives. It is found that the establishment of interest interaction and government-enterprise gaming mechanisms can effectively promote the two sides from short-term gaming to stable cooperation and reduce strategic uncertainty. In addition, the optimization of the external institutional environment (e.g., regional synergy policy and cross-regional governance framework) plays a key role in promoting the realization of cooperative equilibrium. Future research can further combine case validation and multi-subject simulation to deepen the exploration of the dynamic law of the interaction mechanism in the context of differentiated regions, so as to provide theoretical support for the design of cross-region governance policies.