Tripartite evolutionary game analysis of collaborative innovation in the low-altitude economy: a case study of Shenzhen

In the Low-Altitude Economy, effective innovation is crucial for sustainable urban development, requiring governance that balances the interests of government, enterprises, and research institutions. This study develops a tripartite evolutionary game model based on the case of Shenzhen as a pioneering city in China to explore this synergy. The findings provide operational policy tools for urban planners: (1) Cooperation stability peaks when technological upgrade benefits are distributed between 0.65 and 0.7; (2) Economic value from technology commercialization drives convergence to an ideal equilibrium, particularly within a range of 20 to 600 million yuan; (3) A critical threshold of approximately 6 million yuan exists for government policy costs—exceeding this significantly reduces willingness to participate. These parameters and mechanisms are essential for effective policy design. In conclusion, the study identifies key mechanisms and conditions that promote collaborative governance in the Low-Altitude Economy based on incentive compatibility principles, making its proposed framework valuable for other cities’ low-altitude economic policies.

1 Introduction

Low-Altitude Economy, an emerging paradigm driven by technological convergence and sustainable development demands, is significantly transforming urban transportation and regional economies. However, its growth faces governance challenges: it relies not only on core technologies like aircraft manufacturing but also on public infrastructure such as airspace management and take-off/landing stations, along with necessary regulatory frameworks. This interplay of “technology—facilities—institutions” necessitates collaboration among government, enterprises, and research institutions—not just for efficiency but as a cornerstone for system success. Yet, mismatched incentives and uncertain cooperative benefits often trap stakeholders in “innovation islands,” hindering the realization of potential synergies.

Significant progress has been made in analyzing the role of unilateral policies, particularly government subsidies, in reducing Research and Development (R&D) risks and stimulating market entry (Peters et al., 2022; Grossman and Helpman, 2021; Huergo and Moreno, 2017). However, most studies adopt a static or bilateral perspective, failing to fully capture the strategic interactions and dynamic feedback among the three core actors: government, enterprises, and research institutions. This gap limits our understanding of policy effectiveness in practice. For instance, some research suggests that without coordinated intellectual property protection and fair benefit-sharing mechanisms, much of the subsidy effectiveness may be lost (Awe et al., 2021). While this assessment is rough and illustrative, it highlights an urgent need for an analytical framework that considers more factors. Consequently, current research cannot systematically address a fundamental question: Under what conditions will these key actors with bounded rationality transition from isolation to cooperation and ultimately form a stable innovation ecosystem?

To address the identified gaps, this study introduces the Triple Helix model as its core theory. This model highlights that the interrelationship and co-evolution among universities (research), industry, and government drive innovation in the knowledge economy era (Etzkowitz and Leydesdorff, 2000). Based on this framework, we develop a tripartite evolutionary game model. This approach aims to capture actors’ limited rationality, strategic learning processes, and system evolution’s path dependence, thereby moving beyond static equilibrium analysis to dynamically reveal conditions for collaboration and evolutionary pathways in governance.

Based on this, the study aims to address the following core questions: (1) What key conditions and mechanisms drive the strategic behaviors of government, enterprises, and research institutions in the low-altitude economic innovation ecosystem towards a stable and sustainable cooperation? (2) How do these conditions—particularly policy cost thresholds, benefit distribution proportions, and technology’s commercial value—quantitatively influence the system’s convergence path and equilibrium outcome?

The main contributions of this paper are twofold: (1) Theoretically, it integrates evolutionary game theory with Triple Helix innovation theory, offering a quantitative framework to analyze the dynamic complexity of multi-agent collaborative governance in the Low-Altitude Economy. (2) Practically, by calibrating the model using empirical data from Shenzhen, China, and conducting scenario simulations, it provides evidence-based decision support for policymakers. Key parameter thresholds identified—such as critical policy cost values and optimal benefit distribution ranges—offer a scientific basis for designing transferable dynamic policy toolkits like an intellectual property sharing framework to align individual rationality with sustainable development goals effectively (Tables 1–3).

Table 1

Table 1. Description of model parameters.

Table 2

Table 2. Perceived benefit matrix of the tripartite subjects.

Table 3

Table 3. Equilibrium points and eigenvalues.

2 Literature review

2.1 Low-altitude economy related literature

The Low-Altitude Economy, as a global emerging frontier, is still evolving in its definition. In international academic discussions, it is often framed within “Urban Air Mobility” (UAM) and its broader concept “Advanced Air Mobility” (AAM). NASA initially described it as “an automated air transportation system serving densely populated areas to safely and efficiently transport people and goods” (Thipphavong et al., 2018). This concept later expanded into AAM, which includes various aircraft applications—from large drones to electric vertical take-off and landing vehicles (eVTOLs)—in both urban and rural settings (Straubinger et al., 2020). In China, the Low-Altitude Economy emphasizes economic attributes, defined as a comprehensive economic form driven by diverse low-altitude flight activities. It centers on general aviation and the drone industry while promoting coordinated development across related fields (Qin, 2023).

Current research emphasizes technological and application-oriented aspects, summarized into three main areas: First, technical feasibility and core systems focus on breakthroughs in aircraft technology, including eVTOL range, battery energy density, and autonomous flight systems (Rajendran and Srinivas, 2023). It also addresses the necessary supporting infrastructure like communication, navigation, and surveillance (CNS) for safe operations (Toreini et al., 2020). Second, market potential and social acceptance involve studies predicting market size, assessing economic impact, and exploring public concerns about noise, privacy, and safety issues (Fu et al., 2022; Al Haddad et al., 2020). Third, airspace management and integration challenges highlight research on integrating high-density heterogeneous aircraft within airspace paradigms as well as traffic management rules and intelligent path planning algorithms (Vascik and Hansman, 2017; Cohen et al., 2021).

It is noteworthy that recent research has shifted from a purely technical focus to exploring governance and coordination mechanisms. For example, Ye et al. (2025) conducted an empirical analysis in Yunnan Province, China, examining how the Low-Altitude Economy promotes regional private sector development. Their findings highlight the interactive relationship between policy coordination and market activation, providing valuable regional insights for this study. However, while these studies acknowledge the need for coordination, most remain at a descriptive level without offering a theoretical model to systematically reveal the strategic interactions among three core actors: government, enterprises, and research institutions. This theoretical gap has resulted in many coordination initiatives facing implementation challenges due to misaligned incentives—this constitutes the central issue this study seeks to address.

2.2 Literature on multi-stakeholder innovation ecosystems and governance frameworks

The healthy development of the Low-Altitude Economy relies on a stable innovation ecosystem. Governance challenges, particularly in coordinating strategies among government, industry, and research institutions, require robust theoretical support. The innovation ecosystem theory posits that technological innovation arises from complex interactions among diverse entities (de Vasconcelos Gomes et al., 2018; Rabelo and Bernus, 2015). Within this framework, the Triple Helix model provides a clear analytical lens, highlighting that the collaboration and co-evolution of universities (research), industry, and government are key drivers of innovation in the knowledge economy era (Etzkowitz and Leydesdorff, 2000).

The contemporary development of this model has integrated specific policy coordination mechanisms, such as designing dynamic subsidies, establishing an intellectual property sharing framework, and implementing adaptive regulation to effectively align the incentive structures across different departments (Cai and Li, 2023). Complementarily, research on profit-sharing arrangements in R&D collaborations provides insights into the stability of synergy. Zhang and Li (2022) found that a fair and clear profit-sharing ratio is crucial for maintaining alliance stability in green technology innovation networks. Further studies have highlighted the importance of dynamically combining contractual governance with relational governance to enhance university-industry cooperation project performance (Bstieler et al., 2018). Regarding policy coordination… Platform ecosystem theory posits that the government can serve as a “platform leader” by creating common infrastructure and standardized interfaces, thereby reducing transaction costs for multi-party collaboration and fostering system-level innovation (Gawer and Cusumano, 2014).

These studies highlight a key insight: A successful policy-driven innovation ecosystem depends on well-designed benefit-sharing mechanisms and flexible policy coordination tools that align individual incentives with collective goals, ensuring the long-term stability of collaborative innovation.

In summary, existing theories have established clear connections among policy coordination, multi-agent dynamics, and sustainable innovation outcomes. This study aims to integrate the tripartite evolutionary game model into this theoretical framework, utilizing its analytical strengths to explain how the triple helix operates within the Low-Altitude Economy, thereby providing methodological justification and enhancing the research’s theoretical relevance.

2.3 Literature on environmental pressure and the resilience of the innovation ecosystem

The evolution of the innovation ecosystem is increasingly influenced by external environmental pressures, in addition to internal policy incentives. Research is focusing on how factors like climate change, stricter regulations, and extreme weather are reshaping enterprises’ innovation strategies and inter-organizational cooperation (Horbach, 2020). Studies show that companies facing higher climate risks are more inclined to develop green technologies in collaboration with research institutions when supported by public funds (Noailly and Shestalovich, 2017). Recently Li et al. (2025) confirmed that even uncertainties from policies can catalyze breakthroughs in green technologies.

Recent studies have deepened the understanding of environmental pressure as a key driver in innovation ecosystems. Chen et al. (2025) argued that strict environmental policies can steer technological change in low-altitude transportation, with their effectiveness reliant on collaboration among government, enterprises, and research institutions. Zhou and Smith (2025) highlighted green finance’s role in stabilizing multi-stakeholder innovation ecosystems under climate pressure, noting that ESG investment channels align incentives across sectors. On a macro level, Liu and Kumar (2025) emphasized governance challenges in aligning urban air mobility with the UN Sustainable Development Goals (SDGs), necessitating policy coordination beyond traditional technological incentives. Davis and Yang (2025) empirical research showed that ESG investment significantly influences enterprise innovation strategies in sustainable aviation, encouraging greater resource allocation to green technology R&D. Additionally, Wang and Li (2025) analyzed resilient innovation ecosystem construction in the Low-Altitude Economy using a systematic approach, stressing dynamic feedback and adaptive governance amid external environmental shocks.

The “Porter Hypothesis” suggests that well-designed environmental regulations can boost enterprise competitiveness through an innovation compensation effect (Popp, 2019). Ullah et al. (2025) expand on this by discussing how enterprises can convert regulatory pressure into development momentum via adaptive innovation strategies. In the transportation sector, such technology shifts driven by pressure rely heavily on cross-sectoral R&D cooperation and flexible policy frameworks (Acemoglu et al., 2021; Garcia and Smith, 2025). Zhang and Zhou (2025) further explore the dynamic feedback mechanism for creating a resilient innovation ecosystem under environmental stress using a system dynamics approach. Empirical studies indicate that in response to external pressures like environmental regulations, enterprises modify their green product innovation strategies, impacting both internal R&D investments and external collaboration models (Berchicci et al., 2021). Additionally, capital markets influence resource allocation through ESG investment channels (e.g., green bonds), prompting firms to allocate more resources to green innovations and indirectly fostering collaborative R&D aimed at mitigating environmental risks (Flammer, 2021).

By integrating these literatures, we situate our research within the broader discourse of sustainable innovation and highlight environmental pressure as a key driver of tripartite coordination, complementing policy incentives.

2.4 Evolutionary game theory

To analyze the complex micro-mechanisms of collaborative governance in the low-altitude economic innovation ecosystem, we need an approach that addresses bounded rationality, strategic interaction, and dynamic evolution. Evolutionary Game Theory (EGT) is a suitable theoretical tool for this analysis. Initially developed in biology to explain the evolution of strategic behavior within populations (Weibull, 1995), EGT has been widely utilized in social sciences to explore institutional change, cooperative evolution, and innovation policy (Friedman, 1998). It relaxes the strict assumption of “perfect rationality” found in classical game theory, allowing participants to adapt their strategies through trial and error, imitation, and learning—better reflecting real decision-making behaviors amid complexity and uncertainty (Hofbauer and Sigmund, 2003).

In innovation and governance research, EGT demonstrates unique advantages in modeling multi-agent coordination. For example, Zhang and Li (2021) developed a government-enterprise game model to illustrate how enterprises adopt green technologies under environmental regulations, highlighting the synergistic effects of subsidies and penalties. Similarly, Yang, J. (2018) applied EGT to demonstrate that sustainable collaboration in strategic emerging industries’ innovation networks can be achieved through self-enforcing agreements and reputation mechanisms, even in the absence of central authority intervention.

However, there is a significant gap in the literature when applying EGT to the emerging field of Low-Altitude Economy. Firstly, most related EGT studies focus on technical aspects, such as task allocation and cooperative control in multi-UAV systems (Gao et al., 2023). These engineering-oriented studies have little connection to governance research that emphasizes institutions and incentives. Secondly, while some studies have begun addressing innovation policies, their models fail to capture essential governance features of the Low-Altitude Economy, including high policy dependence, urgent iterative interactions between technology and regulation, and the distribution of common benefits from technological upgrades.

The mechanisms of key variables such as policy implementation costs, technology commercialization value, and the distribution of technology upgrade benefits in the tripartite collaboration of the Low-Altitude Economy have not been systematically examined or empirically calibrated within evolutionary game theory. This study aims to integrate the core actors defined in the Triple Helix model (Etzkowitz and Leydesdorff, 2000) with the analytical capabilities of evolutionary game theory to construct a tailored tripartite evolutionary game model for Low-Altitude Economy governance. We focus on both equilibrium outcomes and revealing the evolutionary path from “collaboration failure” to “collaboration stability.” Through numerical simulation, we quantitatively analyze how these key parameters influence strategic choices among stakeholders and system convergence efficiency. This research expands EGT’s application in emerging frontier industry governance and deepens collaborative governance theory in the Low-Altitude Economy from conceptual frameworks to computable models.

3 Methodology

3.1 Model assumptions

The successful implementation of technological collaborative innovation in the Low-Altitude Economy’s emerging sector relies on dynamic coordination among governments, businesses, and research organizations. Currently, the global Low-Altitude Economy is rapidly developing, with the interaction between technology advancement and policy support being vital for building an industrial ecosystem. This paper presents seven assumptions to develop an evolutionary game model.

Hypothesis 1: The first participant is the government; the second is a low-altitude business; and the third is a technological R&D organization. The government leads collaborative innovation in low-altitude flight technology by providing policy support, resource coordination, and supervision, while reducing information asymmetry through incentive mechanisms. Low-altitude businesses primarily conduct economic activities and leverage cooperative innovation for technological transformation and market competitiveness. R&D institutes serve as the main source of technology for this collaboration, lowering R&D costs and enhancing technology transfer efficiency. All three parties are bounded rational actors focused on maximizing their own interests, with strategy choices influenced by factors such as information transparency, policy incentives, and penalty coefficients.

Hypothesis 2: The government’s strategic space is (G1, G2) = (participation, non-participation). “Participation” involves the government using policy tools like tax breaks, R&D subsidies, and pilot areas to support collaborative innovation between low-altitude economic enterprises and technology R&D institutions. “Non-participation” means the government maintains minimal oversight without additional financial investment or policies. The strategic space of low-altitude economic enterprises is (K1, K2) = (collaboration, independent R&D). “Collaboration” entails sharing resources and risks with technology R&D institutions for joint research; “Independent R&D” relies on their own resources and capabilities without sharing technological achievements. The strategic space of technology R&D institutions is (N1, N2) = (collaboration, independent R&D). Here, “collaboration” refers to joint research on key technologies with low-altitude economic enterprises while sharing benefits; “Independent R&D” focuses on developing technology based solely on their own capabilities and generating revenue through technology transfer.

Hypothesis 3: When the government adopts a participation strategy, it incurs policy costs such as research subsidies and financial support, quantified as C1 (in ten thousand yuan), alongside basic regulatory costs amounting to C2 (in ten thousand yuan). Conversely, when the government refrains from engaging in collaborative innovation, it does not provide any form of policy or other support and only bears the basic regulatory costs. In this scenario, the government secures a fixed basic income of R1 (in ten thousand yuan). Upon commercialization of the technology, an additional economic value represented by H1 (in ten thousand yuan) is generated, from which the government receives a share denoted as a (dimensionless).

Hypothesis 4: Under government policy support, Low-Altitude Economy enterprises can receive a subsidy of V1 (in ten thousand yuan). For collaborative research and development, they obtain a basic income of R2 (in ten thousand yuan) while incurring a basic R&D cost of C3 (in ten thousand yuan). In independent research and development, the basic income is R3 (ten thousand yuan) with an associated cost of C4 (in ten thousand yuan). Upon commercialization, the additional economic value generated is H1 (in ten thousand yuan), from which Low-Altitude Economy enterprises can claim a proportion b (dimensionless). From the technological upgrades yielding additional income H2 (in ten thousand yuan), they can claim a proportion w (dimensionless).

Hypothesis 5: With government policy support, research and development institutions can receive financial aid of V2 (in ten thousand yuan). For collaborative R&D, they earn a basic income of R4 (in ten thousand yuan) while incurring a basic cost of C5 (in ten thousand yuan). In independent R&D, the income is R5 (in ten thousand yuan) with a cost of C6 (in ten thousand yuan). After commercialization, the technology generates additional economic value H1 (in ten thousand yuan), with research institutions receiving a share c (dimensionless). From the extra income H2 (in ten thousand yuan) due to technology upgrades, their share is 1-w (dimensionless).

Hypothesis 6: Only when the government chooses “participation” (G1), enterprises opt for “collaboration” (K1), and research institutions select “collaboration” (N1) will they jointly commercialize the technology, generating additional economic value H1 (in ten thousand yuan), distributed in proportion (a + b + c ≤ 1). When enterprises choose “collaboration” (K1) and research institutions select “collaboration” (N1), they will collaborate to upgrade the technology, producing additional economic benefits of H2 (in ten thousand yuan), which will also be distributed in a certain proportion.

Hypothesis 7: Considering “free-rider” behavior, Low-Altitude Economy enterprises and R&D institutions will sign contracts with advance payments as deposits. If one party breaches the contract, the deposit is forfeited as a penalty. Thus, the breach of contract cost is set at F (in ten thousand yuan). As shown in the following text, Table 1 is the relevant description of the model parameters.

3.2 Subject strategy

From the strategy set of the government, drone companies, and consumers, it can be seen that there are a total of 8 strategy combinations of the three parties, as shown in Figure 1.

Figure 1

Figure 1. The tripartite strategic combination relationships.

To summarize, the dynamic interactions among the government, low-altitude economic enterprises, and R&D institutions yield eight strategic combinations: I. {G1, K1, N1} = {Participation, Collaboration, Collaboration}. II. {G1, K1, N2} = {Participation, Collaboration, Independent R&D}. III. {G1, K2, N1} = {Participation, Independent R&D, Collaboration}. IV. {G1, K2, N2} = {Participation, Independent R&D, Independent R&D}. V. {G2, K1, N1} = {Non-Participation, Collaboration, Collaboration}. VI. {G2, K1, N2} = {Non-Participation, Collaboration, Independent R&D}. VII. {G2, K2, N1} = {Non-Participation, Independent R&D, Collaboration}. VIII. {G2, K2, N2} = {Non-Participation, Independent R&D, Independent R&D}. According to the main strategy, the perceived benefit matrix of the three main parties is constructed as shown in Table 2 below.

3.3 Model analysis

Assume that the expected return received by the government in selecting the participation strategy is E1, the expected benefit of opting out of the subsidy is E12, and the average return is E1, Equations (1) to (3).

$\begin{array}{l}\mathrm{E}11={\mathrm{Y}}^{\ast }{\mathrm{Z}}^{\ast }(\mathrm{Ha}1+\mathrm{R}1-\mathrm{C}1-\mathrm{C}2)+{\mathrm{Y}}^{\ast }(1-\mathrm{Z})\ast (\mathrm{R}1-\mathrm{C}1-\mathrm{C}2)+\\ {\mathrm{Z}}^{\ast }(1-\mathrm{Y})\ast (\mathrm{R}1-\mathrm{C}1-\mathrm{C}1)+(1-\mathrm{Y})\ast (1-\mathrm{Z})\ast (\mathrm{R}1-\mathrm{C}1-\mathrm{C}1)\end{array}\text{ (1)}$

$\begin{array}{l}\mathrm{E}12={\mathrm{Y}}^{\ast }{\mathrm{Z}}^{\ast }(\mathrm{R}1-\mathrm{C}2)+{\mathrm{Y}}^{\ast }(1-\mathrm{z})\ast (\mathrm{R}1-\mathrm{C}2)+\\ {\mathrm{Z}}^{\ast }(1-\mathrm{Y})\ast (\mathrm{R}1-\mathrm{C}2)+(1-\mathrm{Y})\ast (1-\mathrm{Z})\ast (\mathrm{R}1-\mathrm{C}2)\end{array}\text{ (2)}$

$\begin{array}{ll}\mathrm{E}1={\mathrm{X}}^{\ast }\mathrm{E}11+(1-\mathrm{X})\ast \mathrm{E}12& (3)\end{array}$

Consequently, the replication dynamic equation for the conduct of government decision-making is as follows Equation (4):

$\begin{array}{l}F\left(x\right)=\frac{d_x}{d_t}={\mathrm{X}}^{\ast }(\mathrm{E}11-\mathrm{E}1)={\mathrm{X}}^{\ast }(1-\mathrm{X})\ast (\mathrm{E}11-\mathrm{E}12)\\ ={\mathrm{X}}^{\ast }(\mathrm{X}-1)\ast (\mathrm{C}1-\mathrm{Ha}{1}^{\ast }{\mathrm{Y}}^{\ast }\mathrm{Z})\end{array}\text{ (4)}$

Hypothesis, because, G(z) is about z being a decreasing function.

The partial derivative of F(x) is Equation (5).

$\begin{array}{ll}F\prime \left(\mathrm{X}\right)=\frac{\mathit{dF}\left(x\right)}{\mathit{dx}}=(2^{\ast }\mathrm{X}-1)(\mathrm{C}1-\mathrm{Ha}{1}^{\ast }{\mathrm{Y}}^{\ast }\mathrm{Z})& (5)\end{array}$

The following requirements must be fulfilled in order for the government to accomplish game optimization, based on the differential equation stability theorem: $F\left(x\right)=0$ and $F\prime \left(\mathrm{X}\right)<0$.

When $z=\frac{C1}{\mathit{Ha}{1}^{\ast }Y}=z^{\ast }$, $G\left(z\right)=0$, and $d\left(F\right(x\left)\right)/\mathit{dx}=0$, so the game is in equilibrium, regardless of the value of x.

When $z\ne z^{\ast }$, Let $F\left(\mathrm{X}\right)=0$, we know that $x=0$ and $x=1$ are the two steady states of X, respectively. At this point, there are two scenarios:

① When $0<z<z^{\ast }$, $F\prime \left(0\right)<0$, $F\prime \left(1\right)>0$, In this case, $x=0$ is the evolutionary stability point, and the government’s policy choice is “not to participate”.

② When$z^{\ast }<z<1$, $F\prime \left(0\right)>0$, $F\prime \left(1\right)<0$, In this case, $x=1$ is the evolutionary stability point, and the government’s strategy choice is “participation”.

Corollary 1: Government involvement has a negative correlation with policy cost C1 and a positive correlation with Ha1. The predicted benefits of government participation exceed those of non-participation, leading to convergence at x = 1 as the government’s allocation of Ha1 from technology commercialization increases. Conversely, as shown in Figure 2, the evolutionary stabilization approach will converge to x = 0 when policy costs C1 rise and net income declines.

Figure 2

Figure 2. Phase diagram of government decision-making.

Suppose that the expected return obtained by low-altitude economic enterprises choosing collaborative strategy is E21, and the expected income obtained by choosing independent R&D is E22, and the average income is E2 Equations (6) to (8).

$\begin{array}{l}\begin{array}{l}\mathrm{E}21={\mathrm{X}}^{\ast }{\mathrm{Z}}^{\ast }(\mathrm{Hb}1+\mathrm{R}2+\mathrm{V}1+\mathrm{W}2-\mathrm{C}3)\\ +{\mathrm{X}}^{\ast }(1-\mathrm{Z})\ast (\mathrm{F}+\mathrm{R}2+\mathrm{V}1-\mathrm{C}3)\end{array}\\ \begin{array}{l}+{\mathrm{Z}}^{\ast }(1-\mathrm{X})\ast (\mathrm{R}2+\mathrm{W}2-\mathrm{C}3)\\ +(1-\mathrm{X})\ast (1-\mathrm{Z})\ast (\mathrm{F}+\mathrm{R}2-\mathrm{C}3)\end{array}\end{array}\text{ (6)}$

$\begin{array}{l}\mathrm{E}22={\mathrm{X}}^{\ast }{\mathrm{Z}}^{\ast }(-\mathrm{F}+\mathrm{R}3+\mathrm{V}1-\mathrm{C}4)+{\mathrm{X}}^{\ast }(1-\mathrm{Z})\ast (\mathrm{R}3+\mathrm{V}1-\mathrm{C}4)\\ +{\mathrm{Z}}^{\ast }(1-\mathrm{X})\ast (-\mathrm{F}+\mathrm{R}3-\mathrm{C}4)+(1-\mathrm{X})\ast (1-\mathrm{Z})\ast (\mathrm{R}3-\mathrm{C}4)\end{array}\text{ (7)}$

$\begin{array}{ll}\mathrm{E}2={\mathrm{Y}}^{\ast }\mathrm{E}21+(1-\mathrm{Y})\ast \mathrm{E}22=\mathrm{E}22+{\mathrm{Y}}^{\ast }(\mathrm{E}21-\mathrm{E}22)& (8)\end{array}$

Therefore, the replication dynamic equation for the decision-making behavior of low-altitude economic enterprises is as follows Equation (9):

$\begin{array}{l}F\left(y\right)=\frac{d_y}{d_t}={\mathrm{Y}}^{\ast }(\mathrm{E}11-\mathrm{E}1)={\mathrm{Y}}^{\ast }(1-\mathrm{Y})\ast (\mathrm{E}11-\mathrm{E}12)=\\ {\mathrm{Y}}^{\ast }(\mathrm{Y}-1)\ast (\mathrm{C}3+\mathrm{R}3-\mathrm{F}-\mathrm{R}2-\mathrm{W}{2}^{\ast }\mathrm{Z}-\mathrm{Hb}{1}^{\ast }{\mathrm{X}}^{\ast }\mathrm{Z}-\mathrm{C}4)\end{array}\text{ (9)}$

Hypothesis $J\left(x\right)=\mathrm{C}3+\mathrm{R}3-F-\mathrm{R}2-\mathrm{W}{2}^{\ast }Z-\mathrm{Hb}{1}^{\ast }{\mathrm{X}}^{\ast }\mathrm{Z}-\mathrm{C}4$, because $\partial J\left(X\right)/\partial X<0$, $J\left(x\right)$ is about x being a decreasing function.

The partial derivative of F(y) is Equation (10).

$F\prime \left(y\right)=\frac{\mathit{dF}\left(y\right)}{\mathit{dy}}=(2^{\ast }\mathrm{Y}-1)\left(\begin{array}{l}\mathrm{C}3+\mathrm{R}3-F-\mathrm{R}2-\mathrm{W}{2}^{\ast }Z\\ -\mathrm{Hb}{1}^{\ast }{\mathrm{X}}^{\ast }\mathrm{Z}-\mathrm{C}4\end{array}\right)\text{ (10)}$

The following requirements must be fulfilled in order for low-altitude economic firms to attain game optimization, based on the differential equation stability theorem: $F\left(y\right)=0$ and $F\prime \left(y\right)<0$.

(1) When $x=\frac{\mathrm{C}3+\mathrm{R}3-F-\mathrm{R}2-\mathrm{W}{2}^{\ast }\mathrm{Z}-\mathrm{C}4}{\mathrm{Hb}{1}^{\ast }\mathrm{Z}}=x^{\ast }$, $J\left(x\right)=0$, and $d\left(F\right(y)/\mathit{dy})=0$, so the game is in equilibrium, regardless of the value of y.

(2) When $x\ne x^{\ast }$, Let $F\left(y\right)=0$, we know that $y=0$ and $y=1$ are the two steady states of y, respectively. At this point, there are two scenarios:

① When $0<x<x^{\ast }$, $F\prime \left(0\right)<0$, $F\prime \left(1\right)>0$, In this case, $y=0$ is the evolutionary stability point, and the low-altitude economic enterprises’s policy choice is “independent R&D.”

② When $x^{\ast }<x<1$, $F\prime \left(0\right)>0$, $F\prime \left(1\right)<0$. In this case, $y=1$ is the evolutionary stability point, and the low-altitude economic enterprise’s strategy choice is “collaboration.”

Corollary 2: The collaborative R&D probability of enterprises positively correlates with policy subsidy V1 and the distribution ratio w of technology upgrading income, while negatively correlating with penalty F and independent R&D cost C4. When the government participates in the policy, businesses’ evolutionary stability strategy converges to X = 1 due to increased benefits from the collaborative strategy provided by subsidy V1. Additionally, collaboration with technology R&D institutions enhances technological upgrading W2, further encouraging cooperation among businesses. Conversely, a reduction in penalty F diminishes the risks associated with independent R&D choices, and lower independent R&D costs C4 lead Low-Altitude Economy enterprises’ evolutionary stabilization strategy to converge towards x = 0. See Figure 3 for details.

Figure 3

Figure 3. Phase diagram of low-altitude economic enterprises.

The technical R&D institution is expected to receive anticipated returns of E31 from collaboration, E32 from independent R&D, and an average return of E3, Equations (11). to (13).

$\begin{array}{l}\begin{array}{l}\mathrm{E}31={\mathrm{X}}^{\ast }{\mathrm{Y}}^{\ast }(\mathrm{Hc}1+\mathrm{R}4+\mathrm{V}2+\mathrm{W}1-\mathrm{C}5)\\ +{\mathrm{X}}^{\ast }(1-\mathrm{Y})\ast (\mathrm{F}+\mathrm{R}4+\mathrm{V}2-\mathrm{C}6)+\end{array}\\ \begin{array}{l}{\mathrm{Y}}^{\ast }(1-\mathrm{X})\ast (\mathrm{R}4+\mathrm{W}1-\mathrm{C}5)+(1-\mathrm{X})\\ \ast (1-\mathrm{Y})\ast (\mathrm{F}+\mathrm{R}4-\mathrm{C}6)\end{array}\end{array}\text{ (11)}$

$\begin{array}{l}\mathrm{E}32={\mathrm{X}}^{\ast }{\mathrm{Y}}^{\ast }(-\mathrm{F}+\mathrm{R}4+\mathrm{V}2-\mathrm{C}5)+{\mathrm{X}}^{\ast }(1-\mathrm{Y})\ast (\mathrm{R}4+\mathrm{V}2-\mathrm{C}5)+\\ {\mathrm{Y}}^{\ast }(1-\mathrm{X})\ast (-\mathrm{F}+\mathrm{R}4-\mathrm{C}5)+(1-\mathrm{X})\ast (1-\mathrm{Y})\ast (\mathrm{R}4-\mathrm{C}5)\end{array}\text{ (12)}$

$\begin{array}{ll}\mathrm{E}3={\mathrm{Z}}^{\ast }\mathrm{E}31+(1-\mathrm{Z})\ast \mathrm{E}32=\mathrm{E}32+Z^{\ast }(\mathrm{E}31-E32)& (13)\end{array}$

Consequently, the replication dynamic equation for technological R&D institutions’ decision-making behavior is displayed below Equation (14).

$\begin{array}{l}F\left(\mathrm{Z}\right)=\frac{d_z}{d_t}={\mathrm{Z}}^{\ast }(\mathrm{E}31-\mathrm{E}3)={\mathrm{E}}^{\ast }(1-\mathrm{Z})\ast (\mathrm{E}31-\mathrm{E}32)\\ ={\mathrm{Z}}^{\ast }(\mathrm{Z}-1)\ast (\mathrm{C}6-\mathrm{F}+(\mathrm{C}5-\mathrm{W}1-\mathrm{C}6)\ast \mathrm{Y}-\mathrm{Hc}{1}^{\ast }{\mathrm{X}}^{\ast }\mathrm{Y}-\mathrm{C}5)\end{array}\text{ (14)}$

Hypothesis $\mathrm{H}\left(\mathrm{y}\right)=\mathrm{C}6-\mathrm{F}+(\mathrm{C}5-\mathrm{W}1-\mathrm{C}6)\ast \mathrm{Y}-\mathrm{Hc}{1}^{\ast }{\mathrm{X}}^{\ast }\mathrm{Y}-\mathrm{C}5$, because $\partial H\left(\mathrm{y}\right)/\partial Y<0$, $\mathrm{H}\left(y\right)$ is about y being a decreasing function.

The partial derivative of $F\left(z\right)$ is Equation (15).

$F\prime \left(\mathrm{Z}\right)=\frac{\mathit{dF}\left(\mathrm{z}\right)}{\mathrm{dz}}=(2^{\ast }\mathrm{Z}-1)\ast \left(\begin{array}{l}\mathrm{C}6-\mathrm{F}+(\mathrm{C}5-\mathrm{W}1-\mathrm{C}6)\\ \ast \mathrm{Y}-\mathrm{Hc}{1}^{\ast }{\mathrm{X}}^{\ast }\mathrm{Y}-\mathrm{C}5\end{array}\right)\text{ (15)}$

The following requirements must be fulfilled in order for a technology R&D organization to accomplish game optimization, based on the differential equation stability theorem: $F\left(z\right)=0$ and $F\prime \left(z\right)<0$.

(1) When $y=\frac{\mathrm{C}6-\mathrm{F}-\mathrm{C}5}{\mathrm{Hc}{1}^{\ast }\mathrm{X}-\mathrm{C}5+\mathrm{W}1+\mathrm{C}6}=y^{\ast }$, $\mathrm{H}\left(y\right)=0$, and $d\left(F\right(z)/\mathit{dz})=0$, so the game is in equilibrium, regardless of the value of z.

(2) When $y\ne y^{\ast }$, Let F(z) = 0, we know that $z=0$ and $z=1$ are the two steady states of z, respectively. At this point, there are two scenarios:

① When $0<y<y^{\ast }$, $F\prime \left(0\right)<0$, $F\prime \left(1\right)>0$. In this case, $y=0$ is the evolutionary stability point, and the technology R&D institution’s policy choice is “independent R&D.”

② When $y^{\ast }<y<1$_, $F\prime \left(0\right)>0$, $F\prime \left(1\right)<0$. In this case, $y=1$ is the evolutionary stability point, and the technology R&D institution’s strategy choice is “collaboration.”

Corollary 3: The synergistic probability of R&D institutions is positively correlated with government funding support V2 and the technology commercialization revenue distribution ratio c, while negatively correlated with independent R&D costs C6. When the government adopts a participation strategy, the evolutionary stability strategy of technology R&D institutions converges to x = 1 due to increased funding support V2, which reduces collaborative innovation’s marginal cost. Additionally, advantages in technology commercialization enhance collaboration appeal. Conversely, lower independent R&D costs (C6) improve the viability of independent strategies, leading to convergence towards x = 0 for evolutionary stability. See Figure 4 for details.

Figure 4

Figure 4. Phase diagram of the technology R&D institution strategy.

3.4 Evolutionary stability analysis of tripartite agent strategy

The replication dynamic system for the tripartite agent is as follows, which can be used to examine the system’s evolutionary stability state: Equation (16)

$\{\begin{array}{c}\begin{array}{l}F\left(x\right)=X^{\ast }(X-1)\ast \\ (C1-\mathit{Ha}{1}^{\ast }{Y}^{\ast }Z)\end{array}\\ \begin{array}{l}F\left(y\right)=Y^{\ast }(Y-1)\ast (C3+R3-F-R2\\ -W{2}^{\ast }Z-\mathit{Hb}{1}^{\ast }{X}^{\ast }Z-C4)\end{array}\\ \begin{array}{l}F\left(z\right)=Z^{\ast }(Z-1)\ast (C6-F+(C5-\\ W1-C6)\ast Y-\mathit{Hc}{1}^{\ast }{X}^{\ast }Y-C5)\end{array}\end{array}\text{ (16)}$

Only the stability of eight pure strategy equilibrium points is discussed here, as Ritzberger and Weibull (1995) work suggests that many groups’ evolutionary stability strategy must be pure strategy. If $F\left(x\right)=0$, $F\left(y\right)=0$, and $F\left(z\right)=0$, points of equilibrium for pure strategy and mixed strategy equilibrium points are two examples of the several local equilibrium points that we might acquire. First, the system’s Jacobian matrix is solved in order to examine the equilibrium point’s asymptotic stability: Equations (17) and (18).

$J=\left[\begin{array}{ccc}\frac{\partial F\left(x\right)}{\partial x}& \frac{\partial F\left(x\right)}{\partial y}& \frac{\partial F\left(x\right)}{\partial z}\\ \frac{\partial F\left(y\right)}{\partial x}& \frac{\partial F\left(y\right)}{\partial y}& \frac{\partial F\left(y\right)}{\partial z}\\ \frac{\partial F\left(z\right)}{\partial x}& \frac{\partial F\left(y\right)}{\partial y}& \frac{\partial F\left(y\right)}{\partial z}\end{array}\right]=\left[\begin{array}{ccc}{J}_{11}& J_{12}& J_{13}\\ J_{21}& J_{22}& J_{23}\\ J_{31}& J_{32}& J_{33}\end{array}\right]\text{ (17)}$

$\begin{array}{l}{J}_{11}=(2^{\ast }X-1)\ast (C1-H\mathrm{a}{1}^{\ast }{Y}^{\ast }Z)\\ J_{12}=X^{\ast }(X-1)\ast (-H\mathrm{a}{1}^{\ast }Z)\\ J_{13}=X^{\ast }(X-1)\ast (-H\mathrm{a}{1}^{\ast }Y)\\ J_{21}=Y^{\ast }(Y-1)\ast (-H\mathrm{b}{1}^{\ast }Z)\\ J_{22}=(2^{\ast }Y-1)\ast (C3+R3-F-R2-\\ W{2}^{\ast }Z-\mathit{Hb}{1}^{\ast }{X}^{\ast }Z-C4)\\ J_{23}=Y^{\ast }(Y-1)\ast (-\mathit{Hb}{1}^{\ast }X-W2)\\ J_{31}=Z^{\ast }(Z-1)\ast (-H\mathrm{c}{1}^{\ast }Y)\\ J_{32}=Z^{\ast }(Z-1)\ast (C5-W1-C6-\mathit{Hc}{1}^{\ast }X)\\ J_{33}=(2^{\ast }Z-1)\ast (C6-F\\ +(C5-W1-C6)\ast Y-\mathit{Hc}{1}^{\ast }{X}^{\ast }Y-C5)\end{array}\text{ (18)}$

The equilibrium points exhibit asymptotic stability or evolutionary stability strategy (ESS) when the corresponding eigenvalues are less than 0, as indicated by Lyapunov’s condition for equilibrium point stability (Lyapunov, 1992). An unstable point occurs when the eigenvalues are positive. A saddle point exists when one or both eigenvalues are positive. Table 3 shows the characteristic values of each equilibrium state as follows:

(1) For $\mathrm{E}1(0,0,0)$to be evolutionary equilibrium, the following conditions must be met① –C1<0, ② F + R2 + C4–R3–C3 <0, that is F + R2 + C4<R3 + C3 ③ F + C5–C6 <0, is that F + C5 <C6.

(2) $\mathrm{E}2(1,0,0)$ cannot evolve equilibrium for the following reasons:

Among the three eigenvalues of the equilibrium point, ${\lambda }_1$ is C1 and C1 is the parameter, which is not 0 itself.

(3) $\mathrm{E}3(0,1,0)$ cannot evolve equilibrium for the following reasons:

Among the three eigenvalues of the equilibrium point, ${\lambda }_3$ is $F+\mathrm{W}1$, and F and W1 are parameters, which are not 0 themselves.

(4) $\mathrm{E}4(0,0,1)$ cannot evolve equilibrium for the following reasons:① –C1<0, ② F+R2+W2+C4–R3–C3 <0, that is, F+R2+W2+C4<R3+C3 ③ The formula is –F–C5+C6 <0, or C6 <F+C5

where C6 is the cost of independent R&D for technology R&D institutions and $C5$is the cost of collaborative R&D. This is not consistent with the design because the cost of independent R&D is higher than the cost of collaborative research;

(5) $\mathrm{E}5(1,1,0)$ cannot evolve equilibrium for the following reasons:

Among the three eigenvalues of the equilibrium point, ${\lambda }_1$ is C1 and C1 is the parameter, which is not 0 itself.

(6) $\mathrm{E}6(1,0,1)$ cannot evolve equilibrium for the following reasons:

Among the three eigenvalues of the equilibrium point, ${\lambda }_1$ is C1 and C1 is the parameter, which is not 0 itself.

(7) For $\mathrm{E}7(0,1,1)$ to be evolutionary equilibrium, the following conditions must be satisfied:

① $\mathrm{Ha}1-\mathrm{C}1<0$, that is $\mathrm{Ha}1<\mathrm{C}1$ ② $\mathrm{R}3+\mathrm{C}3-F-\mathrm{R}2-\mathrm{W}2-\mathrm{C}4<0$ that is $\mathrm{R}3+\mathrm{C}3<F+\mathrm{R}2+\mathrm{W}2+\mathrm{C}4$ ③$-F-W1<0$.

(8) For $\mathrm{E}8(1,1,1)$ to be evolutionary equilibrium, the following conditions must be satisfied:① C1-Ha1<0, that is C1<Ha1 ② R3+C3-F-RW-W2-Hb1-C4 <0 that is, R3+C3<F+R2-W2+Hb1+C4 ③ -F-W1-Hc1<0:

Table 4 clearly shows that the equilibrium point obtained by the previously given derivation is stable, which organizes the equilibrium point analysis in accordance with the model derivation analysis mentioned above.

Table 4

Table 4. Results of local equilibrium stability analysis.

The table shows that stability is satisfied at only three points:$\mathrm{E}1(0,0,0)$, $\mathrm{E}7(0,1,1)$, and $\mathrm{E}8(1,1,1)$.

Scenario 1: The equilibrium point E1(0,0,0) is the system’s stability point when condition ① $-\mathrm{C}1<0$_, ② $F+\mathrm{R}2+\mathrm{C}4<\mathrm{R}3+\mathrm{C}3$_, ③ $F+\mathrm{C}5<\mathrm{C}6$ is met. From condition ②, it can be seen that the net income of independent R&D is higher than that of collaborative R&D, and enterprises are more inclined to rely on their own resources for R&D and choose independent R&D. From condition③, it can be seen that when the research costs and fines of independent R&D are greater than the R&D costs of collaborative R&D, R&D institutions obtain stable income through independent R&D. At this time, all three parties choose a more conservative strategy, and the system falls into equilibrium, which is an inefficient equilibrium at that time.

Scenario 2: The equilibrium point E7(0,1,1) is the system’s stability point when condition ① $\mathrm{Ha}1<\mathrm{C}1$, ② $\mathrm{R}3+\mathrm{C}3<F+\mathrm{R}2+\mathrm{W}2+\mathrm{C}4$, ③ $-F-\mathrm{W}1<0$ is met. From condition①, it can be seen that the distribution of income $a1$ obtained by the government from technology commercialization at this time is smaller than the policy cost at the time of participation, and the government still chooses to withdraw from participating in collaborative innovation and only bears regulatory responsibility; From condition②, it can be seen that the net income of low-altitude economic enterprises in independent R&D is lower than that of collaborative R&D, and the fine has a certain influence at this time, and low-altitude economic enterprises choose collaborative R&D; Collaborative R&D will be chosen by technology R&D groups if its net revenue is greater than that of independent R&D. This is evident from condition ③, which highlights the advantages and penalties of technology upgrading under collaborative R&D. The three parties’ current decision demonstrates that, in the event that the government does not engage in collaborative innovation, technology R&D organizations and low-altitude economic enterprises will naturally form collaborative innovation due to the advantages of technological upgrading and high fines. However, the government will not benefit in the long run from this decision, and the balance will be impacted in the future.

Scenario 3: The equilibrium point E8(1,1,1) is the system’s stability point when condition ① $\mathrm{C}1<\mathrm{Ha}1$, ② $\mathrm{R}3+\mathrm{C}3<\mathrm{F}+\mathrm{R}2+\mathrm{W}2+\mathrm{Hb}1+\mathrm{C}4$, ③ $-F-\mathrm{W}1-\mathrm{Hc}1<0$ is met. The government will profit more from technological commercialization than the policy cost of engaging in collaborative innovation, as can be observed from condition ①, and the government will typically engage in collaborative innovation. From condition ②, it can be seen that the net income of low-altitude economic enterprises under collaborative innovation is greater than that of independent R&D, and low-altitude economic enterprises tend to collaborative innovation. It is evident from condition ③ that the government’s financial assistance and the advantages of technology commercialization lower the cost of independent R&D, leading technology R&D institutions to opt for joint innovation. At this time, the decision-making of the three parties is to actively participate in collaborative innovation, and the three parties form an efficient coordination mechanism, and policy incentives, penalty constraints and income distribution mechanisms jointly drive technological innovation to achieve a win-win “ideal equilibrium” for all parties.

4 Evolutionary stability strategy’s simulation analysis

Numerical simulations are performed using Matlab2023b to confirm the validity of the evolutionary stability analysis, visualize the evolutionary paths of each equilibrium point, and show how changes in each parameter affect the evolutionary stabilization strategy through the simulation diagrams. The simulation data are based on the example of Shenzhen City, China. The basis of parameter setting: firstly, referring to “Several Measures to Support the High-Quality Development of Low Altitude Economy in Shenzhen”, the scale of low altitude economy industry in Shenzhen reaches 505.95 billion yuan in 2023, so the economic value of commercialization of the technology (H1) is set to 5,000, and secondly, according to the “Regulations on Promotion of Low Altitude Economy Industry in Shenzhen Special Economic Zone”, “the defaulting party needs to return the pre-paid funds and pay a 20% penalty”. The pricing of the contract is set at 10 million RMB. Then, according to the “Shenzhen Low Altitude Economy Industry Innovation and Development Implementation Plan (2022–2025)”, it is proposed that “over 5 billion RMB will be allocated to support the development of the whole industry chain of the low altitude economy within 3 years”, and then focusing on the “special input for collaborative innovation”, the annual input of 200 million RMB will be taken. The annual input is 200 million yuan, so the policy cost (C1) is set at 200. Finally, the remaining parameters are assigned values according to the actual market situation of low altitude economy and the balance relationship between the parameters. All the specific parameters are assigned as follows:

To explore the dynamic interactions and strategic evolution among government, enterprises, and R&D institutions, this study employs an agent-based simulation method—specifically multi-agent modeling—to create a tripartite evolutionary game simulation model. This approach effectively captures the complexity and adaptability of limited rational agents, allowing each to adjust its strategy based on system states and personal benefits. The simulation is conducted in the MATLAB 2023b environment. Each stakeholder—government, Low-Altitude Economy enterprises, and technology R&D institutions—is defined as an autonomous agent. The parameter settings are derived from a thorough analysis of Shenzhen City’s Low-Altitude Economy policy documents, industry market data, and relevant academic literature to accurately reflect real market conditions. Table 5 details all core parameters’ symbols, meanings, benchmark values, and assignment bases. For example: The policy cost is informed by Shenzhen’s three-year special investment plan for the entire industrial chain. The value of technology commercialization corresponds proportionally to Shenzhen’s Low-Altitude Economy industry scale in 2023. Parameters like revenue distribution ratios are grounded in theoretical research on value creation within innovation ecosystems. For further details, please refer to Table 5. Specific parameter assignments are as follows:

$\begin{array}{l}\begin{array}{l}\mathrm{C}1=200,\mathrm{C}2=50,\mathrm{R}1=100,\mathrm{H}1=5000,a=0.3,\mathrm{V}1=5,\\ \mathrm{R}2=800,\mathrm{C}3=300,\mathrm{R}3=600,b=0.5\end{array}\\ \begin{array}{l}\mathrm{C}4=400,\mathrm{H}2=750,\mathrm{W}=0.6,\mathrm{V}2=3,\mathrm{R}4=200,\mathrm{C}5=100,\\ \mathrm{R}5=150,\mathrm{C}6=120,F=2,c=0.2\end{array}\end{array}$

Table 5

Table 5. Key parameter design table.

To ensure the robustness of the simulation results, we conducted 1,000 independent runs for each scenario, with each run comprising 50 iterations. This number has been verified as sufficient for most cases to reach a stable equilibrium or clearly show long-term strategy trends. The initial strategy selection probabilities of agents were randomly assigned within the range of [0, 1] to promote diversity in initial conditions and avoid biases from specific starting values.

4.1 Technological upgrading, the income distribution ratio w

This study sets the initial technical allocation ratio at 0.6. To systematically explore its impact on game results, five comparison groups were established: w = 0.1, 0.2, 0.7, 0.8, and 0.9, using the control variable method for simulation analysis. Figure 5 illustrates how changes in parameter w affect system evolution: with w = 0.1, low-altitude economic enterprises converge to a coordinated state slowly; as w increases to 0.7, this process accelerates significantly but then slows down again. In this study, the initial technology allocation ratio w is set at 0.6. To systematically investigate its impact on game outcomes, we employ the control variable method to create five comparison groups with w values of 0.1, 0.2, 0.7, 0.8, and 0.9 for simulation analysis. Figure 5 illustrates how variations in parameter w affect the system’s evolutionary trajectory: when w = 0.1, indicated by the red plus sign curve, low-altitude economic enterprises converge towards cooperation slowly. As w increases to 0.7 (green circle dotted line), cooperative evolution accelerates significantly but begins to slow down thereafter. Importantly, once w exceeds the critical value of 0.7, convergence speed exhibits diminishing returns due to structural contradictions in technological factor allocation—this imbalance arises between research institutions and low-altitude economic enterprises. To foster a policy-guided tripartite collaborative innovation system, it is crucial to identify the dynamic balance point for technological factor allocation and establish a fair benefit distribution mechanism that ensures long-term stability in the innovation ecosystem.

Figure 5

Figure 5. Evolutionary trajectories under different technology upgrade benefit distribution ratios.

4.2 Policy costs C1

In this study, the benchmark policy cost C1 is set at 200 (in ten thousand yuan), with five gradient parameters of 50, 100, 400, 800, and 1,000 (in ten thousand yuan) for sensitivity analysis. As shown in Figure 6, while keeping other parameters constant, the simulation reveals how changes in policy costs affect the collaborative innovation strategies of the government, Low-Altitude Economy enterprises, and technology research institutions. The results indicate that when C1 = 50 (in ten thousand yuan), the government’s choice of independent R&D strategy remains relatively stable. As C1 increases to between 400 and 800 (in ten thousand yuan), a significant critical threshold effect occurs near C1 = 600 (in ten thousand yuan). When C1 exceeds 800 (in ten thousand yuan), the evolution rates of strategies for both government and enterprises align closely. Importantly, once policy costs surpass regional fiscal capacity, governmental willingness to collaborate on innovation reaches saturation. This highlights the need for decision-makers to implement a dynamic compensation mechanism that activates promptly when costs exceed critical thresholds to optimize fiscal resource allocation and sustain collaborative innovation among all parties involved.

Figure 6

Figure 6. Evolutionary trajectories under different policy costs.

4.3 The additional economic value of technology commercialization H1

In this study, the benchmark value of the additional economic value H1 for technology commercialization is set at 50 million yuan. To explore its influence on game equilibrium, five additional levels of H1—1,000, 2,000, 6,000, 8,000, and 10,000 million yuan—are analyzed. Figure 7 illustrates the evolution characteristics of collaborative innovation strategies among government entities, Low-Altitude Economy enterprises, and technology R&D institutions as H1 varies while other parameters remain fixed. Simulation data indicate that when H1 = 1,000 million yuan (red plus sign curve), Low-Altitude Economy enterprises show the fastest strategy evolution rate and lead in their adjustment process compared to government and R&D institutions. As H1 increases to 6,000 million yuan (green circle dotted line), all subjects’ evolution trajectories flatten; however, the overall system converges to collaborative innovation equilibrium more rapidly. Notably, within the range of 2,000 ≤ H1 ≤ 6,000 million yuan, Low-Altitude Economy enterprises exhibit a positive correlation between strategy response sensitivity and H1 value. However, after exceeding this threshold, the acceleration of evolution shows a slight diminishing trend. This may be due to the marginal benefit inflection point in technology collaborative innovation—when the commercial value of technology surpasses the elastic boundary for enterprise strategy adjustments, the saturation effect of economic incentives weakens the impetus for change. The research confirms that moderate regulation of technology commercialization’s value space can effectively stimulate collaboration among the three innovation subjects, allowing system evolution to achieve dynamic balance within maximum benefit intervals. Therefore, to enhance collaborative innovation speed among all parties, it is beneficial to jointly increase technology commercialization value while staying within a reasonable range.

Figure 7

Figure 7. Evolutionary trajectories under different additional economic values of technology commercialization.

4.4 R&D costs C4

In this study, the benchmark for independent R&D cost C4 of enterprises is set at 4 million yuan. To examine its impact on game equilibrium, five control parameters are introduced: C4 = 1 million, 2 million, 6 million, 8 million, and 10 million yuan. Figure 8 illustrates how variations in C4 affect the collaborative innovation strategies of government, Low-Altitude Economy enterprises, and technology R&D institutions while keeping other parameters constant. The analysis shows that when C4 = 1 million yuan (indicated by the red dotted line), Low-Altitude Economy enterprises exhibit a slight upward trend in collaboration; as C4 increases to between 2 and 10 million yuan, the rate of collaborative evolution accelerates significantly. This phenomenon indicates that a marginal increase in independent R&D costs reduces enterprises’ willingness for independent innovation while enhancing their preference for collaborative innovation. Notably, within the C4 range of 2–6 million yuan, the trajectory initially shows an opposite trend—high sunk costs encourage increased investment in independent R&D, as the scale effect of collaborative R&D is not yet fully realized. Once C4 surpasses a critical threshold, technology spillover effects and cost-sharing mechanisms drive enterprises to rethink their R&D decision-making framework. The research suggests establishing a dynamic cost adjustment mechanism to build a tripartite collaborative innovation ecosystem: optimizing the structure of independent R&D costs, guiding innovation entities to form resource-dependent networks, and managing risks associated with sunk costs to achieve optimal resource allocation and technological diffusion.

Figure 8

Figure 8. Evolutionary trajectories under different enterprises’ independent R&D costs.

4.5 The basic income of collaborative R&D R2

In this study, the initial basic benefit R2 for enterprise collaborative R&D is set at 8 million yuan. To assess the sensitivity of this parameter to system evolution, a comparative experimental group was created with five gradient values: R2 = 100, 200, 1,000, 2000, and 3,000 (in million yuan). Figure 9 illustrates the dynamic impact of R2 on the collaborative innovation strategies of government, Low-Altitude Economy enterprises, and technology R&D institutions. Simulation data indicate that when R2 = 100 (million yuan) (marked by the red plus sign and dotted line), Low-Altitude Economy enterprises exhibit significant lag in strategy convergence. As R2 increases exponentially, the convergence rate of the evolution trajectory accelerates markedly. Notably, in the medium benefit range of R2 = 200–1,000 (in million yuan), a phase reversal phenomenon occurs early in evolution. While increased basic benefits enhance collaborative motivation among enterprises, insufficient resource matching efficiency leads to significant nonlinear inhibitory effects, resulting in short-term oscillations within the system. When R2 exceeds 2,000 (in million yuan), the scale benefits of collaborative innovation greatly surpass coordination costs, leading to stable convergence in strategy choices among the three parties. Empirical results suggest that enhancing expected benefits of collaborative R&D through policy tools like fiscal incentives and intellectual property protection can effectively overcome bottlenecks in collaborative innovation and foster a virtuous cycle within the innovation ecosystem.

Figure 9

Figure 9. Evolutionary trajectories under different basic benefits of collaborative R&D.

4.6 Different allocation ratios

In this study, the initial benchmark for the revenue distribution ratio of technology commercialization was set as (a,b,c) = (0.3, 0.5, 0.2). To explore how different ratios affect game equilibrium, four additional parameter configurations were introduced: array 1(0.8, 0.1, 0.1), array 2(0.1, 0.8, 0.1), array 3(0.1, 0.1, 0.8), and array 4(0.33, 0.34, 0.33). Figure 10 illustrates the impact of dynamically adjusting the revenue distribution ratios a, b, and c on the collaborative innovation strategies of government entities, Low-Altitude Economy enterprises, and technology R&D institutions while keeping other parameters constant. The simulation results indicate that with configuration from array 1 adopted, the government’s participation strategy converges most rapidly; in Array 2’s context, Low-Altitude Economy enterprises show a steep upward trend in their collaborative strategy evolution; Array 3 significantly speeds up strategic evolution for technology R&D institutions; under array 4 conditions all three main bodies exhibit asymptotic convergence characteristics in their evolution trajectories. Based on the analysis, it is evident that a scientific allocation of revenue distribution coefficients a, b, and c for technology commercialization can significantly enhance collaborative innovation efficiency among the government, Low-Altitude Economy enterprises, and technology R&D institutions. When the distribution ratio falls within a reasonable range, all three parties exhibit an exponential growth trend in their probabilities of choosing collaborative strategies. This confirms a strong positive correlation between revenue distribution fairness and the dynamic stability of the collaborative innovation system. Therefore, during policy-making, the government should carefully consider the interest distribution among these parties to ensure sustainable development of the collaborative innovation ecosystem.

Figure 10

Figure 10. Evolutionary trajectories under different revenue distribution ratio combinations.

4.7 Different initial strategy evolution paths

Figure 11 shows how the stochastic beginning approach influences evolutionary stability in the E8(1,1,1) state. The point (1,1,1) represents the ultimate evolutionary outcome, irrespective of the players’ initial goals. At this stage, low-altitude economic enterprises yield higher net income than independent R&D. Technology R&D institutions adopt a synergy strategy based on technology commercialization allocation and fines, while the government maximizes net income through policy incentives. Consequently, the optimal strategies for the government, low-altitude economic enterprises, and technology R&D institutions are “participation,” “collaboration,” and “collaboration,” respectively.

Figure 11

Figure 11. Evolutionary paths from different initial strategies.

5 Conclusion

5.1 Policy implementation framework

Based on the key parameter thresholds and system dynamics from the simulation results, this study has developed a policy implementation framework to convert theoretical findings into practical governance tools. The framework focuses on four core mechanisms:

First, a contract-based benefit-sharing mechanism is proposed. Simulation results show optimal system stability when the distribution ratio of technological upgrade benefits is between 0.65 and 0.70. Therefore, it is recommended to implement a standardized “Collaborative Innovation Benefit Sharing Agreement” for government-funded R&D projects. Contractual terms should allocate 65 to 70% of technological value-added benefits to enterprises that bear commercial risks, with the remainder going to R&D institutions. Additionally, government subsidies can be linked to this agreement, and ownership rules for background and prospective intellectual property rights should be clearly defined to ensure long-term cooperation expectations from a systemic perspective.

Second, a dynamic fiscal incentive mechanism is proposed. Given the critical threshold of approximately 6 million yuan for policy costs, a stepped subsidy plan should be designed. This plan would provide higher start-up funds during the early stages of industrialization and implement a clear subsidy reduction mechanism based on technology maturity and market income. Additionally, a ‘co-innovation risk pool’ can be established to compensate when government coordination costs for individual projects approach this critical value, ensuring continued public sector participation.

Third, a precise triggering mechanism based on commercial value. The simulation revealed that the marginal effect is highest when the technology’s commercial value ranges from 20 to 60 million yuan. Therefore, a “Major Technology Commercialization Award for Low-Altitude Economy” can be established. Projects that successfully achieve commercialization and generate additional economic value within this range could receive a one-time bonus or tax credit. Measures like “Innovation Vouchers” and government priority procurement can directly enhance initial expectations for technology commercialization, creating effective market demand pull.

Fourth, an adaptive resource allocation mechanism is essential. As rising independent R&D costs drive enterprises towards collaboration, varying allocation ratios affect the evolution speed of each entity; thus, policies should be dynamically adjusted. During the industry incubation phase, resources should favor R&D institutions to promote core technology supply; in the scale-up phase, incentives should shift to application-end enterprises to boost market demand. Government investment in shared infrastructure like public test fields and universal certification platforms can systematically lower R&D and collaboration costs for all participants.

5.2 Research conclusions

This study reveals the dynamic mechanisms and stable conditions of low-altitude economic collaborative innovation through a tripartite evolutionary game model involving government, enterprises, and research institutions. The main conclusions are as follows: (1) Fair distribution of technological upgrade benefits is crucial for collaboration stability. When enterprises receive 65–70% of value-added benefits, the system is most likely to achieve and maintain an ideal equilibrium. (2) The additional economic value from technology commercialization serves as the core engine driving convergence towards an ideal state. Specifically, when this value ranges between 20 and 60 million yuan, it accelerates the convergence process. (3) There exists a clear efficiency boundary for government policy costs. A critical threshold around 60 million yuan marks a limit for sustainable public financial input; exceeding it may deter government participation and lead to collaborative failure. (4) Basic benefits from collaborative R&D and independent R&D costs are key factors influencing evolutionary paths and speeds. Increasing basic collaboration benefits or raising independent R&D challenges can effectively encourage enterprises to adopt cooperative strategies.

5.3 Model limitations and future research directions

The tripartite evolutionary game model developed in this study offers valuable insights into policy coordination and strategic interaction within the low-altitude economic innovation ecosystem, leading to several relevant policy conclusions. However, this study has inherent limitations that also suggest promising directions for future research.

First, this model analyzes strategic evolution in a stable technological-institutional environment but does not fully account for disruptive technological breakthroughs or sudden external shocks. Events such as increases in battery energy density, changes in airspace management regulations, or major safety incidents can quickly alter the game’s payoff structure and disrupt the current evolutionarily stable equilibrium. Future research could address this gap by introducing a stochastic game framework that models technological breakthroughs as random events with specific probability distributions to assess system robustness under uncertainty and inform adaptive policy design.

Second, the model lacks clarity in representing the temporal dynamics and evolution stages of the industry. While it describes the relative change rate of strategic proportions through replicator dynamics, it fails to align this with an actual time scale or distinguish between development stages such as “pilot demonstration,” “scale commercialization,” and “mature operation.” This ambiguity complicates accurate assessments of policy intervention urgency and timeliness. A promising direction for improvement is to develop a multi-stage evolutionary game model that divides the time axis into distinct periods corresponding to industry development stages, applying different parameter sets at each stage to better simulate the co-evolution path of policies and strategies throughout the industrial life cycle.

Thirdly, the research treats most key parameters as fixed and exogenous throughout the evolutionary process. However, in reality, factors like policy costs, technology commercialization value, and R&D costs are often endogenous and dynamic. For example, R&D costs decrease along the technology learning curve, while commercialization value fluctuates with market penetration rates. Future models could endogenize these parameters by linking commercialization value to market share or making R&D costs dependent on cumulative investment. This approach would better illustrate the endogenous feedback mechanisms among technology, market, and policy within the innovation ecosystem.

Fourth, this study focuses on the core analysis of three main actors in the Triple Helix model: government, enterprises, and research institutions. However, a complete innovation ecosystem also includes other key stakeholders like end-users and financial institutions. The attitudes and behaviors of these absent stakeholders can significantly impact the decisions of other actors. Therefore, future research could explore a multi-party game model that incorporates these roles to provide a more comprehensive system perspective.

Fifth, the model simplifies the treatment of environmental pressure. While this study focuses on sustainable development goals, it mainly views environmental benefits as part of technology’s commercial value or implicit government gains. It does not explicitly address how environmental regulation pressure or green market preferences influence strategic choices. To better align with the academic discourse on “environmental pressure driving innovation,” future research should incorporate an explicit environmental module—such as a carbon emission cap or linking enterprise profits to their environmental performance—to more accurately simulate the dynamics of green technological innovation and diffusion.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

QX: Writing – review & editing. QC: Writing – original draft. CH: Writing – review & editing. XX: Writing – review & editing.

Funding

The author(s) declare that no financial support was received for the research and/or publication of this article.

Acknowledgments

The author would like to express sincere gratitude to Professors Xu Qun, He Chan and Xu Xu. From the establishment of the research topic, the design of the research plan, to the revision of the simulation analysis, the three professors have always provided the author with meticulous guidance with their profound knowledge and rigorous academic attitude.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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