Stochastic evolutionary game and green transformation strategy selection for multi-agent systems under the digital economy

Abstract

In the process of green transformation under the context of the digital economy, stakeholders such as the government, producers, and consumers find it difficult to reach an effective consensus in the game process due to differences in their interest appeals. This study constructs a tripartite stochastic evolutionary game model among the government, producers, and consumers, and analyzes the evolutionary stable strategies and evolutionary processes of the three parties. The research finds that: The existence of stochastic disturbance factors will affect the evolutionary path and slow down the speed at which the three parties evolve to stable strategies. During the strategy evolution process, supply–demand imbalance phenomena will exist. Market product circulation volume has a significant impact on strategy selection, and differences in product circulation volume under different strategy combinations will directly affect the strategy selection process. In terms of digital green transformation input costs, producers are mainly affected by their own transformation costs, while consumers are affected not only by their own transformation costs but also by producers’ transformation investments. The development of the digital economy can reduce the difficulty for producers and consumers to evolve toward digital green strategies, and sensitivity to emission reduction effects is also an important factor influencing strategy selection. Therefore, the government should formulate reasonable subsidy–penalty policy combinations, dynamically regulate market product circulation volume, implement differentiated incentives for different industries and groups, and promote supply–demand synergy toward the evolution of digital green strategies.

Keywords

Digital economy producer consumer stochastic evolutionary game

1. Introduction

Against the background of increasingly severe global climate change and resource–environment constraints, traditional high-energy-consuming industries face multiple challenges. According to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, global greenhouse gas emissions continue to grow, and climate change has significant impacts on ecosystems and human society.¹ Taking China as an example, as the world’s largest manufacturing country, driven by the “dual carbon” goals, traditional high-energy-consuming enterprises face severe pressure for green transformation. In practice, governments need to balance fiscal pressure with transformation incentive effects in policy design.^2,3 In particular, when the participation rate of producers and consumers in digital construction is low, governments face higher regulatory costs and need to invest more resources in environmental monitoring, market supervision, violation penalties, etc., making policies difficult to implement effectively in practice. Producers face the dilemma of weighing high costs against various uncertainty factors in their decisions on whether and how to adopt digital technologies and green technologies.^4,5 As the main implementers of digital green transformation, producers need to bear the upfront input costs and transformation risks of technological upgrading, equipment renovation, and personnel training. Consumers’ willingness to pay for green products is influenced by green awareness and market recognition,⁶ but with the empowerment of digital technology, consumers can more conveniently obtain green product information and enjoy higher-quality green products and services, thereby becoming the main beneficiaries of digital green transformation. This asymmetric structure of “producers bearing more responsibilities while consumers gaining more benefits” makes how to design policy instruments to incentivize producers to transform proactively while mobilizing consumers’ enthusiasm for green consumption a core challenge in achieving strategic coordination among the three parties. More complicatedly, this multi-agent game process is continuously influenced by various uncertainty factors such as market fluctuations, policy changes, and technological iteration speed.^7,8 How to achieve strategic coordination among government, producers, and consumers in such a dynamic and complex environment becomes a key issue for accumulating theoretical support for green transformation practice.

The rapid development of the digital economy provides new possibilities for resolving the above challenges. According to statistics, China’s digital economy reached 50.2 trillion yuan in 2022, accounting for 41.5% of gross domestic product (GDP), and in 2023, China’s digital economy reached 53.9 trillion yuan, accounting for 42.8% of GDP.⁹ Digital green construction is a complex behavioral process involving technology adoption and capability enhancement, and its direct quantification is highly challenging. However, this behavior can be characterized through a series of observable manifestations: for producers, digital green construction behavior is manifested in that digital technologies such as the Internet of Things, big data, and artificial intelligence can help enterprises optimize resource allocation, reduce emission reduction costs, and improve employee work efficiency. For example, in manufacturing, through finite element simulation technology combined with intelligent algorithms such as genetic algorithms and particle swarm optimization, multi-objective optimization of product design can be achieved, improving product performance while reducing material consumption and energy costs,^10,11 and increasing the market penetration rate of green products through precision marketing.^12,13 For consumers, digital green construction behavior is manifested in the empowerment of digital technology on consumption decisions, reshaping the interaction between producers and consumers, promoting the popularization of green consumption concepts, and further facilitating producers’ green innovation practices.¹⁴ For regulators, digital green construction behavior is manifested in the enhancement of governance capabilities by digital technology, which has improved the government’s overall environmental governance capacity,¹⁵ such as real-time monitoring of carbon emissions through big data platforms. These empowerment effects have shown initial results in practice: high-energy-consuming industries improve total factor productivity through digital transformation and achieve emission reduction goals by enhancing innovation investment and improving management efficiency;¹⁶ cross-border e-commerce platforms utilize blockchain technology to construct product traceability frameworks, achieving full-chain traceability of products from production to consumption.¹⁷ However, how to achieve strategic coordination among the three parties in a complex environment where multiple factors such as digitalization level, policy instrument combinations, and market uncertainty are intertwined still lacks systematic theoretical guidance.

Academic research on green transformation issues has been relatively abundant. In terms of policy instruments, policy tools such as subsidies and carbon taxes effectively guide producers toward green transformation.¹⁸ In high-carbon emission industries, the coordination of economic benefits and carbon emission reduction goals can be achieved through optimizing decision indicators, but the optimal implementation timing is constrained by multiple factors,¹⁹ and policy implementation effects are often influenced by multi-party interactions.²⁰ The role of social cost internalization on production efficiency and social welfare has also received widespread attention.²¹ The constraints of multiple factors such as property rights protection, financing constraints, and market perception capabilities have also been systematically analyzed.²² The application of digital technology is considered capable of reducing carbon emission costs, decreasing carbon emissions, and enhancing environmental governance capacity,²³ and consumer behavior also plays an important role in environmental governance.^24–26

At the methodological level, evolutionary game theory has been widely applied to environmental governance and emission reduction strategy optimization analysis due to its ability to characterize the dynamic strategy adjustment process of boundedly rational agents.^27–29 In terms of game players, existing research covers various frameworks such as government–enterprise two-party games³⁰ and government–enterprise–public three-party games.³¹ In recent years, stochastic evolutionary game models have begun to receive attention, with researchers introducing stochastic disturbance terms to characterize the impact of internal and external factors such as market fluctuations, natural disasters, and unexpected events on strategy evolution.^7,32 These studies provide important references for the methodological choices of this paper.

Although existing research has laid the foundation for understanding the multi-agent interaction mechanisms of green transformation, there are still many deficiencies: the embedding of digital economy elements in green transformation game models is not yet systematic, and existing research mostly treats the digital economy as an exogenous variable, focusing on its unidirectional impact on production costs or consumption behavior. Table 1 summarizes the research content and methods of previous related studies, emphasizes the existing research progress in this field, and aims to fill the relevant research gaps.

Table 1.

Summary of representative literature.

Author	Research content	Method
Sun et al.³⁰	Evolutionary game analysis of resource integration of coal enterprises under government regulation; exploring how governments guide resource integration behavior of dominant-weak enterprises through tax incentives and pollution penalties; analyzing the impact of government subsidies and pollution penalties on enterprise green innovation.	Empirical analysis + two-party evolutionary game model
Chen et al.³¹	Evolutionary game relationship among government-enterprise–public tripartite in environmental governance; analyzing how government regulation and public participation influence enterprise pollution control behavior; exploring the strategic interaction mechanisms and evolutionary stable strategies of the three parties.	Three-party evolutionary game model + empirical research
Li et al.⁷	Government–enterprise stochastic evolutionary game research on renewable energy investment under dynamic carbon pricing; analyzing the impact of carbon price fluctuations, government subsidies, and market environment uncertainty on enterprise renewable energy investment decisions; exploring the effectiveness of government incentive policies.	Two-party stochastic evolutionary game model
Chen and Zou³²	Stochastic evolutionary game research on tripartite collaborative innovation among local government–manufacturing enterprises–research institutions under green innovation ecosystem; analyzing the impact of multi-source uncertainty environment (technological risks, market fluctuations, policy changes, etc.) on tripartite collaborative innovation strategies; exploring the effects of government reward and punishment mechanisms, cost–benefit sharing ratios, and collaboration coefficients on system evolution.	Three-party stochastic evolutionary game model

As shown in Table 1, previous contributions have provided an important foundation for understanding the roles of government, producers, and consumers in green transformation in multi-agent systems. However, no research has quantified the level of digital economy development into the model, and there is a lack of systematic analysis on policy instrument design for digital transformation. The impact mechanism of uncertainty factors on digital green transformation strategy evolution in the context of the digital economy remains unclear, with a lack of theoretical analysis and empirical or simulation evidence. Few studies provide a framework to systematically characterize the complexity of multi-agent interaction mechanisms in uncertain environments under the digital economy context. This study contributes to bridging this gap.

Based on the above research gaps, the research motivation and objectives of this paper are as follows:

Research motivation: In existing research, deterministic three-party games ignore uncertainty, stochastic two-party games cannot characterize supply–demand linkage, and traditional environmental regulation has not incorporated digital economy elements, making it difficult to systematically answer the practical question of “how governments coordinate tripartite green transformation in the face of uncertainty under the digital economy context.”

Research objectives: To reveal the strategy evolution mechanisms of government, producers, and consumers in uncertain environments and identify the interaction effects between digital economy development level and uncertainty factors. Specifically, three key questions are addressed: (1) How does the digital economy development level affect the equilibrium path of tripartite strategy selection? (2) At a certain digitalization stage, how do factors such as market commodity circulation volume and transformation input costs affect strategy evolution? (3) How do uncertainty factors affect the stability of strategy evolution? How should policymakers adjust strategy combinations?

The remainder of this paper is organized as follows: Section 2 constructs the three-party evolutionary game model; Section 3 conducts evolutionarily stable strategy (ESS) analysis; Section 4 constructs and analyzes the three-party stochastic evolutionary game model; Section 5 performs simulation analysis of the model; and finally, Section 6 concludes the paper.

2. Model description

In the context of the digital economy, how to reduce carbon emissions and achieve sustainable development has become a topic of great concern for all countries around the world. Reducing carbon emissions involves coordinated interactions among multiple parties, including the government, producers, and consumers. Traditional studies mostly focus on the bilateral interaction between the government and producers, or consider consumers as passive recipients, ignoring the fact that consumers are also an indispensable part in energy conservation and emission reduction efforts, and also neglecting the role of digital factors. Therefore, this paper constructs a three-party evolutionary game model including the government, producers, and consumers (as shown in Figure 1) and introduces the Digital Economic Development Index, which represents the current level of digital development, into the model. This is significantly different from previous research perspectives.

Figure 1.

Game relationship and organizational framework.

The Digital Economic Development Index of digital development level is derived from the initial letters of the four dimensions: Technology (T), Industry (I), Market (M), and Governance (G) (TIMG Index). Any year can be set as the base year to evaluate the digital progress after that year, thereby enabling cross-period comparative analysis.³³

2.1. Model parameter description and model assumptions

The tripartite game model is a complex dynamic system. The model constructed in this paper belongs to a three-party two-strategy evolutionary game, in which the government, producers and consumers all face a pair of mutually exclusive strategic choices.

(1) Government: When choosing between “supporting digital green transformation” and “not supporting digital green transformation,” the probabilities of strategy selection are $x$ and $1 - x$ , respectively, where $x \in [0, 1]$ .

(2) Producer: When choosing between “digital green production” and “traditional production,” the probabilities of choosing each strategy are $y$ and $1 - y$ , respectively, where $y \in [0, 1]$ .

(3) Consumer: When choosing between “digital green consumption” and “traditional consumption,” the probabilities of making the strategic choice are $z$ and $1 - z$ , respectively, where $z \in [0, 1]$ .

To reasonably apply the evolutionary game theory method to analyze the optimal choice strategies among the government, producers and consumers, the following assumptions are proposed:

Assumption 1. The digital emission reduction utility function is modeled as a variant of the exponential decay function: $θ (D) = α (1 - e^{- (β D / 100)})$ , where $α$ represents the maximum emission reduction potential, reflecting the maximum emission reduction effect that can be achieved after applying digital technologies; $β$ represents the sensitivity of digitalization to the emission reduction effect; $D / 100$ indicates the standardization of the digitalization level to the range [0, 1]. This function satisfies the mathematical properties of $θ^{'} (D) > 0$ and $θ^{″} (D) < 0$ , indicating that the digitalization level can continuously promote emission reduction but with diminishing marginal effects, and the function value is limited to the interval $[0, α]$ , reflecting the constraints on the emission reduction effect; the shape of the function reflects the regional heterogeneity of digital emission reduction, that is, the digital emission reduction effects in different regions with different levels of economic development vary.³⁴

Assumption 2. Assume that the government, producers, and consumers are all rational agents with limited capabilities. During the evolutionary process of the game, each of these entities continuously adjusts their strategies based on historical experience and current gains to maximize their own utility.

Assumption 3. When the government opts for “supporting the digital green transformation,” it has the necessary expenditures $C_{g 1}$ for the basic operation; investment $C_{g 2}$ in digital construction and promotion; subsidies $G_{s 1}$ and $G_{s 2}$ given to producers and consumers who are willing to undertake digital green construction; imposing punitive fines on producers and consumers who refuse to carry out digital green construction, $L_{2}$ and $L_{3}$ . Due to the environmental improvement and social recognition brought about by digital green construction, the government will receive utility income, and social recognition brought about by digital green construction. At the same time, the carbon tax revenue will change accordingly based on the strategy choices of producers and consumers. On the contrary, if the government chooses not to support the digital green transformation, although it does not need to bear the investment and subsidy costs for digital construction, due to external negative impacts such as environmental pollution, it will cause a loss of government credibility $L_{1}$ . Taking into account the government’s supervision over non-participating entities, additional regulatory costs $m_{p}$ and $m_{c}$ for producers and consumers are introduced. That is, assuming that the proportion of entities not participating in digital green production and sales increases, the government will need to invest more resources in supervision and law enforcement.

Assumption 4. When producers choose to adopt digital green production, in addition to the necessary expenditure $C_{p 1}$ for their own operation, they will also choose to provide additional expenditure $C_{p 2}$ for the digital green construction. If the government chooses to support the digital green transformation, producers can receive additional subsidies $G_{s 1}$ from the government. Due to the adoption of the digital green strategy, producers’ carbon emissions decrease, so they can pay less carbon tax based on the original carbon tax. Assuming there is a mismatch between supply and demand between producers and consumers,³⁵ if consumers also choose to purchase green products at this time, the producers will produce and sell more goods than they would if consumers chose to purchase traditional products but were forced to choose green products. Similarly, if the producers choose traditional production, they will have lower unit production costs because they are producing more familiar goods, and they do not need to spend funds on digital green construction. The government will not provide subsidies to the producers either, and the carbon emission level will remain unchanged. If consumers also choose to purchase traditional products at this time, the producers will produce and sell more goods than they would if consumers chose to purchase green products but had to purchase traditional products.

Assumption 5. When consumers opt for digital-driven green consumption, they will incur certain learning costs $G_{c}$ . As long as the government chooses to support the digital green transformation, consumers can receive additional subsidies $G_{s 2}$ from the government to reduce the burden of the transformation. Moreover, adopting digital-driven green consumption can lower carbon emission levels and reduce carbon tax expenditures. Similarly, if consumers choose non-digital traditional consumption, although they do not need to bear additional transformation costs and the unit expenditure when purchasing goods is relatively lower, they will face higher carbon emission levels and corresponding carbon tax expenditures, and will not be able to receive government subsidies.

Assumption 6. If producers and consumers adopt the same strategy, the unit utility gain from purchasing goods is greater than the unit expenditure for purchasing green products, the unit expenditure for purchasing green products is greater than the selling price of green products by producers, and the selling price of green products by producers is greater than the unit production cost of green products; when producers and consumers choose different strategies, the unit utility gain from purchasing goods is greater than the unit expenditure for purchasing traditional products, the unit expenditure for purchasing traditional products is greater than the selling price of traditional products by producers, and the selling price of traditional products by producers is greater than the unit production cost of traditional products, that is $i_{1} > p_{c 1} > s_{1} > p_{p 1}$ , $i_{2} > p_{c 2} > s_{2} > p_{p 2}$ .

Assumption 7. Assume $F + L_{1} > G_{s 1} + G_{s 2} + C_{g 2}$ , that is, the sum of the benefits from the government’s active support for digitalization and the loss of government credibility when the government is passive in supporting digitalization, is greater than the subsidies provided by the government to producers and consumers who choose the digitalization green strategy, as well as the construction costs invested by the government in digitalization. That is, the direct benefits of the government’s support for digital transformation and the avoided losses are greater than the maximum cost of the government’s support for digital transformation. This assumption is a sufficient condition for the government to actively promote the digitalization low-carbon transformation. Only when this assumption is met can the government take active actions to promote the development and application of digitalization low-carbon transformation.

To sum up, the model parameters and their corresponding meanings are shown in Table 2.

Table 2.

Parameter introduction.

Parameters	Explanation
$D$	Digital Economy Development Index—TIMG Index
$C_{g 1}$	Government operating costs
$C_{g 2}$	The government’s investment in digitalization construction expenses
$m_{p} / m_{c}$	Government’s additional investment for producers/consumers who do not participate in digitalized green production/consumption
$G_{s 1} / G_{s 2}$	Subsidies provided by the government to producers/consumers who adopt digital green strategies
$T_{p} / T_{c}$	Producer/consumer carbon tax rate
$E_{p} / E_{c}$	Producer traditional production/consumer traditional consumption carbon emissions
$θ (D)$	Digital emission reduction effect function
$F$	The government actively supports the income from digitalization
$L_{1}$	The government’s passive support for digitalization leads to a loss of credibility
$L_{2}$	The government imposes fines on producers who do not adopt digital green transformation
$L_{3}$	The government imposes fines on consumers who do not adopt digital green transformation
$C_{p 1}$	Daily production-related necessary expenditures of producers
$C_{p 2}$	The cost of digital green construction investment by producers
$C_{c}$	The cost of consumers’ digital green transformation
$Q_{1}$	When producers and consumers adopt the same strategy, the volume of market commodity circulation
$k$	The influence coefficient on the commodity circulation volume when the strategy does not match
$Q_{2} = {kQ}_{1}$	The volume of market commodity circulation when producers and consumers adopt different strategies
$p_{p 1} / p_{p 2}$	Producer’s green production/traditional production product unit cost
$s_{1} / s_{2}$	The selling prices of green products and traditional products by producers
$p_{c 1} / p_{c 2}$	The unit price of green products/traditional products purchased by consumers
$i_{1} / i_{2}$	When producers and consumers adopt the same strategy, the utility income per unit of goods for consumers
	When producers and consumers adopt different strategies, the utility income per unit of goods for consumers

2.2. Profit characterization and replication dynamic equation

Based on the above parameter descriptions and the depiction of benefits, the payoff matrix of this three-party evolutionary game model can be obtained, as shown in Table 3.

Table 3.

Revenue matrix.

	Consumer $z$		Consumer $1 - z$
	Producer $y$	Producer $1 - y$	Producer $y$	Producer $1 - y$
Government $x$	$G_{1}, P_{1}, C_{1}$	$G_{2}, P_{2}, C_{2}$	$G_{3}, P_{3}, C_{3}$	$G_{4}, P_{4}, C_{4}$
Government $1 - x$	$G_{5}, P_{5}, C_{5}$	$G_{6}, P_{6}, C_{6}$	$G_{7}, P_{7}, C_{7}$	$G_{8}, P_{8}, C_{8}$

Among them:

\begin{matrix} G_{1} = (T_{p} E_{p} + T_{c} E_{c}) (1 - θ (D)) + F - (C_{g 1} + C_{g 2}) \\ - (G_{s 1} + G_{s 2}) \end{matrix}

\begin{matrix} P_{1} = s_{1} Q_{1} + G_{s 1} - (C_{p 1} + C_{p 2} + p_{p 1} Q_{1}) - T_{p} E_{p} (1 - θ (D)) \\ C_{1} = i_{1} Q_{1} + G_{s 2} - (C_{c} + p_{c 1} Q_{1}) - T_{c} E_{c} (1 - θ (D)) \\ G_{2} = [T_{p} E_{p} + T_{c} E_{c} (1 - θ (D))] + (F + L_{2}) - (C_{g 1} + C_{g 2}) \\ - G_{s 2} - m_{p} \end{matrix}

\begin{matrix} P_{2} = s_{2} Q_{2} - (C_{p 1} + p_{p 2} Q_{2}) - T_{p} E_{p} - L_{2} \\ C_{2} = i_{2} Q_{2} + G_{s 2} - [C_{c} + p_{c 2} Q_{2}] - T_{c} E_{c} (1 - θ (D)) \\ G_{3} = [T_{p} E_{p} (1 - θ (D)) + T_{c} E_{c}] + (F + L_{3}) \\ - (C_{g 1} + C_{g 2}) - G_{s 1} - m_{c} \\ P_{3} = s_{1} Q_{2} + G_{s 1} - (C_{p 1} + C_{p 2} + p_{p 1} Q_{2}) - T_{p} E_{p} (1 - θ (D)) \\ C_{3} = i_{2} Q_{2} - p_{c 1} Q_{2} - T_{c} E_{c} - L_{3} \\ G_{4} = (T_{p} E_{p} + T_{c} E_{c}) + (F + 2 L_{2}) - (C_{g 1} + C_{g 2}) - m_{p} - m_{c} \end{matrix}

\begin{matrix} P_{4} = s_{2} Q_{1} - (C_{p 1} + p_{p 2} Q_{1}) - T_{p} E_{p} - L_{2} \\ C_{4} = i_{1} Q_{1} - p_{c 2} Q_{1} - T_{c} E_{c} - L_{3} \\ G_{5} = (T_{p} E_{p} + T_{c} E_{c}) (1 - θ (D)) - C_{g 1} - L_{1} \\ P_{5} = s_{1} Q_{1} - (C_{p 1} + C_{p 2} + p_{p 1} Q_{1}) - T_{p} E_{p} (1 - θ (D)) \\ C_{5} = i_{1} Q_{1} - (C_{c} + p_{c 1} Q_{1}) - T_{c} E_{c} (1 - θ (D)) \\ G_{6} = [T_{p} E_{p} + T_{c} E_{c} (1 - θ (D))] - C_{g 1} - L_{1} \\ P_{6} = s_{2} Q_{2} - (C_{p 1} + p_{p 2} Q_{2}) - T_{p} E_{p} \\ C_{6} = i_{2} Q_{2} - (C_{c} + p_{c 2} Q_{2}) - T_{c} E_{c} (1 - θ (D)) \\ G_{7} = [T_{p} E_{p} (1 - θ (D)) + T_{c} E_{c}] - C_{g 1} - L_{1} \\ P_{7} = s_{1} Q_{2} - (C_{p 1} + C_{p 2} + p_{p 1} Q_{2}) - T_{p} E_{p} (1 - θ (D)) \\ C_{7} = i_{2} Q_{2} - p_{c 1} Q_{2} - T_{c} E_{c} \\ G_{8} = (T_{p} E_{p} + T_{c} E_{c}) - C_{g 1} - L_{1} \\ P_{8} = s_{2} Q_{1} - (C_{p 1} + p_{p 2} Q_{1}) - T_{p} E_{p} \\ C_{8} = i_{1} Q_{1} - p_{c 2} Q_{1} - T_{c} E_{c} \end{matrix}

When the government decides to support digital transformation, the expected benefit is $E_{x}$ ; when it decides not to support digital transformation, the expected benefit is $E_{1 - x}$ ;. The average benefit is $\bar{E_{1}}$ .

{\begin{matrix} E_{x} = {yzG}_{1} + (1 - y) {zG}_{2} + y (1 - z) G_{3} + (1 - y) (1 - z) G_{4} \\ = - z [T_{c} E_{c} θ (D) + L_{3} + G_{s 2}] \\ - y [T_{p} E_{p} θ (D) + L_{2} + G_{s 1}] \\ + [T (E_{p} + E_{c}) + (F + L_{2} + L_{3}) - (C_{g 1} + C_{g 2})] \\ - m_{p} (1 - y) - m_{c} (1 - z) \\ E_{1 - x} = {yzG}_{5} + (1 - y) {zG}_{6} + y (1 - z) G_{7} + (1 - y) (1 - z) G_{8} \\ = - z [T_{c} E_{c} θ (D)] - y [T_{p} E_{p} θ (D)] \\ + [(T_{p} E_{p} + T_{c} E_{c}) - C_{g 1} - L_{1})] \\ \bar{E_{1}} = {xE}_{x} + (1 - x) E_{1 - x} \end{matrix}

(1)

When producers choose to support digital green production, the expected return is $E_{y}$ ; when they choose traditional production, the expected return is $E_{1 - y}$ . The average return is $\bar{E_{2}}$ .

{\begin{matrix} E_{y} = {xzP}_{1} + x (1 - z) P_{3} + (1 - x) {zP}_{5} + (1 - x) (1 - z) P_{7} \\ = {xG}_{s 1} + z [(s_{1} - p_{p 1}) (Q_{1} - Q_{2})] \\ + [s_{1} Q_{2} - (C_{p 1} + C_{p 2} + p_{p 1} Q_{2}) - T_{p} E_{p} (1 - θ (D))] \\ E_{1 - y} = {xzP}_{2} + x (1 - z) P_{4} + (1 - x) {zP}_{6} + (1 - x) (1 - z) P_{8} \\ = x (- L_{2}) + z [(s_{2} - p_{p 2}) (Q_{2} - Q_{1})] \\ + [s_{2} Q_{1} - (C_{p 1} + p_{p 2} Q_{1}) - T_{p} E_{p}] \\ \bar{E_{2}} = {yE}_{y} + (1 - y) E_{1 - y} \end{matrix}

(2)

When consumers choose digital green consumption, the expected benefit is $E_{z}$ ; when they choose traditional consumption, the expected benefit is also $E_{1 - z}$ ; the average benefit is $\bar{E_{3}}$ .

{\begin{matrix} E_{z} = {xyC}_{1} + x (1 - y) C_{2} + (1 - x) {yC}_{5} + (1 - x) (1 - y) C_{6} \\ = {xG}_{s 2} + y [(i_{1} - p_{c 1}) Q_{1} - (i_{2} - p_{c 2}) Q_{2}] \\ + [i_{2} Q_{2} - (C_{c} + p_{c 2} Q_{2}) - T_{c} E_{c} (1 - θ (D))] \\ E_{1 - z} = {xyC}_{3} + x (1 - y) C_{4} + (1 - x) {yC}_{7} + (1 - x) (1 - y) C_{8} \\ = x (- L_{3}) + y [(i_{2} - p_{c 1}) Q_{2} - (i_{1} - p_{c 2}) Q_{1}] \\ + (i_{1} Q_{1} - p_{c 2} Q_{1} - T_{c} E_{c}) \\ \bar{E_{3}} = {zE}_{z} + (1 - z) E_{1 - z} \end{matrix}

(3)

According to the replication dynamic equation theory proposed by Taylor and Jonker,³⁶ this study constructed a strategy evolution equation system for the three main entities—the government, producers, and consumers. When the return of a certain strategy is higher than the average return of the group, the proportion of the group choosing that strategy will gradually increase; conversely, when the return of the strategy is lower than the average return of the group, the proportion of the group choosing that strategy will decrease. Based on the framework of the replication dynamic equation theory, the replication dynamic equations for the three types of entities—the government, producers, and consumers—are as follows:

{\begin{matrix} F (x) = \frac{dx}{dt} = x (E_{x} - \bar{E_{1}}) \\ = x (1 - x) [yz (G_{1} - G_{5}) + (1 - y) z (G_{2} - G_{6}) \\ + y (1 - z) (G_{3} - G_{7}) + (1 - y) (1 - z) (G_{4} - G_{8})] \\ = x (1 - x) [- z (L_{3} + G_{s 2} - m_{c}) - y (L_{2} + G_{s 1} - m_{p}) \\ + (F + L_{2} + L_{3} - C_{g 2} + L_{1} - m_{p} - m_{c})] \\ F (y) = \frac{dy}{dt} = y (E_{y} - \bar{E_{2}}) \\ = y (1 - y) [xz (P_{1} - P_{2}) + (1 - x) z (P_{5} - P_{6}) \\ + x (1 - z) (P_{3} - P_{4}) + (1 - x) (1 - z) (P_{7} - P_{8})] \\ = y (1 - y) {x (L_{2} + G_{s 1}) + z \\ [(s_{1} + s_{2} - p_{p 1} - p_{p 2}) (Q_{1} - Q_{2})] \\ + [(s_{1} - p_{p 1}) Q_{2} - (s_{2} - p_{p 2}) Q_{1} - C_{p 2} + T_{p} E_{p} θ (D)]} \\ F (z) = \frac{dz}{dt} = z (E_{z} - \bar{E_{3}}) \\ = z (1 - z) [yz (C_{1} - C_{3}) + (1 - x) y (C_{5} - C_{7}) \\ + x (1 - y) (C_{2} - C_{4}) + (1 - x) (1 - y) (C_{6} - C_{8})] \\ = z (1 - z) {x (G_{s 2} + L_{3}) \\ + y [2 (i_{1} Q_{1} - i_{2} Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2})] \\ + [(i_{2} Q_{2} - i_{1} Q_{1}) - C_{c} - p_{c 2} (Q_{2} - Q_{1}) + T_{c} E_{c} θ (D)]} \end{matrix}

(4)

3. Analysis of evolutionary strategy stability

To analyze the equilibrium points of the model and their stability, let $F (x) = F (y) = F (z) = 0$ be set. Then, it can be obtained that this model has 15 equilibrium points.

According to Lyapunov’s first theorem, the stability of the system can be determined by analyzing the real part signs of the eigenvalues of the Jacobian matrix. Based on the replicated dynamic equation system constructed by Equation (4), the Jacobian matrix J of the system can be obtained. This matrix can reflect the local stability characteristics of the system near the equilibrium point. Its specific form is as follows:

J = (\begin{matrix} \frac{\partial F (x)}{\partial x} & \frac{\partial F (x)}{\partial y} & \frac{\partial F (x)}{\partial z} \\ \frac{\partial F (y)}{\partial x} & \frac{\partial F (y)}{\partial y} & \frac{\partial F (y)}{\partial z} \\ \frac{\partial F (z)}{\partial x} & \frac{\partial F (z)}{\partial y} & \frac{\partial F (z)}{\partial z} \end{matrix})

(5)

Among them:

{\begin{matrix} \frac{\partial F (x)}{\partial x} = (1 - 2 x) [- z (L_{3} + G_{s 2} - m_{c}) - y (L_{2} + G_{s 1} - m_{p}) \\ + (F + L_{2} + L_{3} - C_{g 2} + L_{1}) - m_{p} - m_{c}] \\ \frac{\partial F (y)}{\partial y} = (1 - 2 y) {x (L_{2} + G_{s 1}) \\ + z [(s_{1} + s_{2} - p_{p 1} - p_{p 2}) (Q_{1} - Q_{2})] \\ + [(s_{1} - p_{p 1}) Q_{2} - (s_{2} - p_{p 2}) Q_{1} - C_{p 2} \\ + T_{p} E_{p} θ (D)]} \\ \frac{\partial F (z)}{\partial z} = (1 - 2 z) {x (G_{s 2} + L_{3}) \\ + y [2 (i_{1} Q_{1} - i_{2} \cdot Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2})] \\ + [(i_{2} Q_{2} - i_{1} Q_{1}) - C_{c} - p_{c 2} (Q_{2} - Q_{1}) + T_{c} E_{c} θ (D)]} \end{matrix}

(6)

By making model assumptions and solving the characteristic equation $| λ I - J | = 0$ of the Jacobian matrix $J$ , the real part signs of the eigenvalues of the system at each equilibrium point can be obtained. Based on the above analysis and model assumptions, the real part signs of the equilibrium point eigenvalues are shown in Table 4.

Table 4.

Real part symbol.

Equilibrium point	Real part symbol
$E_{1} (1, 1, 1)$	(−,×,×)
$E_{2} (1, 1, 0)$	(−,×,×)
$E_{3} (1, 0, 1)$	(−,×,×)
$E_{4} (1, 0, 0)$	(−,×,×)
$E_{5} (0, 1, 1)$	(+,×,×)
$E_{6} (0, 1, 0)$	(+,×,×)
$E_{7} (0, 0, 1)$	(+,×,×)
$E_{8} (0, 0, 0)$	(+,×,×)
$E_{9} ~ E_{15}$	(×,×,×)

As shown in Table 3, the system may have stable points $E_{1} (1, 1, 1)$ , $E_{2} (1, 1, 0)$ , $E_{3} (1, 0, 1)$ , $E_{4} (1, 0, 0)$ , and $E_{9} - E_{15}$ .

4. Three-party stochastic evolutionary game model

4.1. Stochastic disturbance factors

Traditional evolutionary game theory simulates strategy evolution through deterministic replicator dynamics equations: $\frac{dx}{dt} = x (E_{x} - \bar{E_{ave}})$ , which assumes that environmental parameters are constant and evolutionary trajectories are predictable. However, in actual real-world decision-making processes, game players cannot make completely rational choices based on the information obtained, but are often influenced by uncertainty factors such as environmental uncertainty, information asymmetry, cognitive biases, and external disturbances. Conventional evolutionary game models have difficulty reflecting the impact of uncertainty factors on strategy selection. To describe the impact of uncertainty factors on the strategy selection of game players, stochastic evolutionary games address this limitation by introducing Gaussian white noise into the replicator dynamics equations, transforming them into stochastic differential equation (SDE):

dx = x (E_{x} - \bar{E_{ave}}) dt + σ xdW (t)

(7)

where $σ$ represents the uncertainty intensity, and $dW (t)$ is the Wiener process (Brownian motion).

This paper adopts Gaussian white noise to characterize parameter stochastic disturbances, with the following practical basis: uncertainty in digitalized green transformation stems from the combined effects of multiple heterogeneous factors. According to the central limit theorem, when a large number of independent or weakly correlated disturbances accumulate, their sum approaches a normal distribution. While real-world technological uncertainty, market uncertainty, policy uncertainty, and external shocks are difficult to model individually, their combined effects can be measured by noise intensity. Lower noise intensity indicates relative environmental stability, while higher noise intensity indicates multiple environmental uncertainties.

Following the research,³⁷ stochastic disturbance terms are introduced into the evolutionary game model, and the modified replicator dynamics equations are as follows:

\begin{matrix} dx (t) = {x (1 - x) [- z (L_{3} + G_{s 2} - m_{c}) - y (L_{2} + G_{s 1} - m_{p}) \\ + (F + L_{2} + L_{3} - C_{g 2} + L_{1} - m_{p} - m_{c})]} dt \\ + σ x (t) d ω (t) \end{matrix}

(8)

\begin{matrix} dy (t) = {y (1 - y) [x (L_{2} + G_{s 1}) \\ + z [(s_{1} + s_{2} - p_{p 1} - p_{p 2}) (Q_{1} - Q_{2})] \\ + [(s_{1} - p_{p 1}) Q_{2} - (s_{2} - p_{p 2}) Q_{1} \\ - C_{p 2} + T_{p} E_{p} θ (D)]} dt + σ y (t) d ω (t) \end{matrix}

(9)

\begin{matrix} dz (t) = {z (1 - z) [x (G_{s 2} + L_{3}) \\ + y [2 (i_{1} Q_{1} - i_{2} Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2})] \\ + [(i_{2} Q_{2} - i_{1} Q_{1}) - C_{c} - p_{c 2} (Q_{2} - Q_{1}) \\ + T_{c} E_{c} θ (D)]} dt + σ z (t) d ω (t) \end{matrix}

(10)

Here, $ω (t)$ represents one-dimensional standard Brownian motion, which is a kind of irregular random fluctuation phenomenon; $d ω (t)$ represents Gaussian white noise. When $t > 0$ and time step $h > 0$ are given, its increment $△ ω (t) = ω (t + h) - ω (t)$ follows a normal distribution $N (0, \sqrt{h})$ , and $σ > 0$ represents the random interference intensity.

4.2. Analysis of the existence and stability of equilibrium solutions

From Equations (8)–(10), it can be seen that when $(t) {|_{t = 0} = w^{'} (t) dt |}_{t = 0} = 0$ , the equation will have a zero solution, which represents the stable state of the system without the interference of Gaussian white noise. At this time, the evolution of the system will be completely dominated by the deterministic part, and the zero solution is the equilibrium solution of the equation. To better match more complex real situations, considering the random interference factors outside the system, based on the Lyapunov stability theory of SDEs,³⁸ the stability of Equations (8)–(10) is determined as follows:³⁹

Lemma 1. For an Itô stochastic differential equation:

dx (t) = f [t, x (t)] dt + g [t, x (t)] d ω (t), x (t_{0}) = x_{0}

(11)

Suppose there exists a continuously differentiable function $v (t, x)$ and a constant $c_{1}, c_{2}$ such that $c_{1} {| x |}^{p} \leq V (t, x) \leq c_{2} {| x |}^{p}, t \geq 0$ .:

(1) If there exists a positive constant $γ$ such that $LV (t, x) \leq - γ V (t, x), t \geq 0$ , then the zero solution of equation (11) is p^th moment exponentially stable, and condition $E {| x (t, x_{0}) |}^{p} < (\frac{c_{2}}{c_{1}}) {| x_{0} |}^{p} e^{- γ t}, t \geq 0$ holds.

(2) If there exists a positive constant $γ$ such that $LV (t, x) \geq γ V (t, x), t \geq 0$ , then the zero solution of Equation (11) has an unstable p^th moment exponent, and condition $E {| x (t, x_{0}) |}^{p} \geq (\frac{c_{2}}{c_{1}}) {| x_{0} |}^{p} e^{- γ t}, t \geq 0$ holds.

Among them: $LV (t . x) = V_{t} (t, x) + V_{x} (t, x) f (t, x) + \frac{1}{2} g^{2} (t, x) V_{xx} (t, x)$ .

According to Lemma 1, the stability criteria for Equations (8)–(10) can be obtained.

Corollary 1. (1) When $L_{2} + G_{s 1} - m_{p} > 0$ , $y \geq \frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} + 1}{2 (L_{2} + G_{s 1} - m_{p})}$ and $2 G_{s 1} + 2 z (L_{2} + G_{s 2}) - 2 (F - C_{g 2} + L_{1} + L_{3} - m_{c}) - σ^{2} \geq 1$ ; or when $L_{2} + G_{s 1} - m_{p} < 0$ , $y \leq \frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} + 1}{2 (L_{2} + G_{s 1} - m_{p})}$ and $2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} \leq - 1$ , Equation (8) is exponentially stable in the zero-th moment;

(2) When $L_{2} + G_{s 1} - m_{p} > 0$ , $y \leq \frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} - 1}{2 (L_{2} + G_{s 1} - m_{p})}$ and $2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} \geq 1$ ; or when $L_{2} + G_{s 1} - m_{p} < 0$ , $y \geq \frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} - 1}{2 (L_{2} + G_{s 1} - m_{p})}$ and $2 (F - C_{g 2} + L_{1} + L_{3} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} - 2 G_{s 1} \geq 1$ , Equation (8) is exponentially unstable in the zero-th moment.

Proof. (1) Let $V (t, x) = x^{2}$ , $c_{1} = c_{2} = 1$ , $p = 1$ and $γ = 1$ . Choose $V (t, x) = 2 x^{2} (1 - x) [- z (L_{3} + G_{s 2} - m_{c}) - y$ $(L_{2} + G_{s 1} - m_{p}) + (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c})] + σ^{2} x^{2}$ . Then when $2 x^{2} (1 - x) [- z (L_{3} + G_{s 2} - m_{c}) - y (L_{2} + G_{s 1} - m_{p}) + (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c})] + σ^{2} x^{2} \leq - x^{2}$ , it is exponentially stable in the zero-th moment.

Since the sign of $L_{2} + G_{s 1} - m_{p}$ is unknown, it is necessary to discuss by cases.

When $L_{2} + G_{s 1} - m_{p} > 0$ , then $y \geq \frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} + 1}{2 (L_{2} + G_{s 1} - m_{p})}$ . Also because $y \in [0, 1]$ , thus $\frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} + 1}{2 (L_{2} + G_{s 1} - m_{p})} \leq 1$ , solving yields $2 G_{s 1} + 2 z (L_{2} + G_{s 2}) - 2 (F - C_{g 2} + L_{1} + L_{3} - m_{c}) - σ^{2} \geq 1$ .

When $L_{2} + G_{s 1} - m_{p} < 0$ , then $y \leq \frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} + 1}{2 (L_{2} + G_{s 1} - m_{p})}$ . Also because $y \in [0, 1]$ , thus $\frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} + 1}{2 (L_{2} + G_{s 1} - m_{p})} \geq 0$ , solving yields $2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} \leq - 1$ .

(2) Let $V (t, x) = x^{2}$ , $c_{1} = c_{2} = 1$ , $p = 1$ and $γ = 1$ . Choose $V (t, x) = 2 x^{2} (1 - x) [- z (L_{3} + G_{s 2} - m_{c})$ $- y (L_{2} + G_{s 1} - m_{p}) + (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c})] + σ^{2} x^{2}$ . Then when $2 x^{2} (1 - x) [- z (L_{3} + G_{s 2} - m_{c})$ $- y (L_{2} + G_{s 1} - m_{p}) + (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c})] + σ^{2} x^{2} \geq x^{2}$ , it is exponentially unstable in the zero-th moment.

Since the sign of $L_{2} + G_{s 1} - m_{p}$ is unknown, it is necessary to discuss by cases.

When $L_{2} + G_{s 1} - m_{p} > 0$ , then $y \leq \frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} - 1}{2 (L_{2} + G_{s 1} - m_{p})}$ . Also because $y \in [0, 1]$ , thus $\frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} - 1}{2 (L_{2} + G_{s 1} - m_{p})} \geq 0$ , solving yields $2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} \geq 1$ .

When $L_{2} + G_{s 1} - m_{p} < 0$ , then $y \geq \frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} - 1}{2 (L_{2} + G_{s 1} - m_{p})}$ . Also because $y \in [0, 1]$ , thus $\frac{2 (F - C_{g 2} + L_{1} + L_{2} + L_{3} - m_{p} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} - 1}{2 (L_{2} + G_{s 1} - m_{p})} \leq 1$ , solving yields $2 (F - C_{g 2} + L_{1} + L_{3} - m_{c}) - 2 z (L_{2} + G_{s 2}) + σ^{2} - 2$ $2 G_{s 1} \geq 1$ .

Corollary 2. (1) Since $(s_{1} + s_{2} - p_{p 1} - p_{p 2}) (Q_{1} - Q_{2}) > 0$ , when $z \leq \frac{2 x (L_{2} + G_{s 1}) + 2 [(s_{1} - p_{p 1}) \cdot Q_{2} - (s_{2} - p_{p 2}) \cdot Q_{1} - C_{p 2} + T_{p} \cdot E_{p} \cdot θ (D)] - σ^{2} - 1}{2 (s_{1} + s_{2} - p_{p 1} - p_{p 2}) (Q_{1} - Q_{2})}$ and $2 x (L_{2} + G_{s 1}) + 2 [(s_{1} - p_{p 1}) \cdot Q_{2} - (s_{2} - p_{p 2}) \cdot Q_{1} - C_{p 2} + T_{p} \cdot E_{p} \cdot θ (D)] - σ^{2} \geq 1$ , Equation (9) is exponentially stable in the zero-th moment;

(2) Since $(s_{1} + s_{2} - p_{p 1} - p_{p 2}) (Q_{1} - Q_{2}) > 0$ , when $z \geq \frac{2 x (L_{2} + G_{s 1}) + 2 [(s_{1} - p_{p 1}) \cdot Q_{2} - (s_{2} - p_{p 2}) \cdot Q_{1} - C_{p 2} + T_{p} \cdot E_{p} \cdot θ (D)] - σ^{2} + 1}{2 (s_{1} + s_{2} - p_{p 1} - p_{p 2}) (Q_{1} - Q_{2})}$ and $- 2 x (L_{2} + G_{s 1}) - 4 (s_{2} - p_{p 2}) Q_{1} - 2 C_{p 2} + 2 T_{p} E_{p} θ (D) - 2 (s_{1} - p_{p 1}) Q_{1} + 2 (s_{2} - p_{p 2}) Q_{2} - σ^{2} \leq 1$ , Equation (9) is exponentially unstable in the zero-th moment.

Proof. Let $V (t, y) = y^{2}$ , $y \in [0, 1]$ , $c_{1} = c_{2} = 1$ , $p = 1$ and $γ = 1$ . The proof is the same as that of Corollary 1.

Corollary 3. (1) When $2 (i_{1} \cdot Q_{1} - i_{2} \cdot Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2}) \geq 0$ , $y \leq \frac{- 2 x (G_{s 2} + L_{3}) - 2 [(i_{2} \cdot Q_{2} - i_{1} \cdot Q_{1}) - C_{c} - p_{c 2} \cdot (Q_{2} - Q_{1}) + T_{c} \cdot E_{c} \cdot θ (D)] - σ^{2} - 1}{2 [2 (i_{1} \cdot Q_{1} - i_{2} \cdot Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2})]}$ and $- 2 x (G_{s 2} + L_{2}) - 2 [(i_{2} \cdot Q_{2} - i_{1} \cdot Q_{1}) - C_{c} - p_{c 2} \cdot (Q_{2} - Q_{1}) + T_{c} \cdot E_{c} \cdot θ (D)] - σ^{2} \geq 1$ ; or when $2 (i_{1} \cdot Q_{1} - i_{2} \cdot Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2}) \leq 0$ , $y \geq \frac{- 2 x (G_{s 2} + L_{3}) - 2 [(i_{2} \cdot Q_{2} - i_{1} \cdot Q_{1}) - C_{c} - p_{c 2} \cdot (Q_{2} - Q_{1}) + T_{c} \cdot E_{c} \cdot θ (D)] - σ^{2} - 1}{2 [2 (i_{1} \cdot Q_{1} - i_{2} \cdot Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2})]}$ and $- 2 x (G_{s 2} + L_{2}) + 2 i_{2} \cdot Q_{2} - 2 i_{1} \cdot Q_{1} + 2 C_{c} + 2 p_{c 1} (Q_{1} - Q_{2}) + 2 T_{c} \cdot E_{c} \cdot θ (D) - σ^{2} \geq 1$ , Equation (10) is exponentially stable in the zero-th moment;

(2) When $2 (i_{1} \cdot Q_{1} - i_{2} \cdot Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2}) \geq 0$ , $y \geq \frac{- 2 x (G_{s 2} + L_{3}) - 2 [(i_{2} \cdot Q_{2} - i_{1} \cdot Q_{1}) - C_{c} - p_{c 2} \cdot (Q_{2} - Q_{1}) + T_{c} \cdot E_{c} \cdot θ (D)] - σ^{2} + 1}{2 [2 (i_{1} \cdot Q_{1} - i_{2} \cdot Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2})]}$ and $- 2 x (G_{s 2} + L_{2}) - 2 [- C_{c} + T_{c} \cdot E_{c} \cdot θ (D)] - σ^{2} + 2 (p_{c 1} (Q_{1} - Q_{2}) \leq - 1$ ; or when $2 (i_{1} \cdot Q_{1} - i_{2} \cdot Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2}) \leq 0$ , $y \leq \frac{- 2 x (G_{s 2} + L_{3}) - 2 [(i_{2} \cdot Q_{2} - i_{1} \cdot Q_{1}) - C_{c} - p_{c 2} \cdot (Q_{2} - Q_{1}) + T_{c} \cdot E_{c} \cdot θ (D)] - σ^{2} + 1}{2 [2 (i_{1} \cdot Q_{1} - i_{2} \cdot Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2})]}$ and $- 2 x (G_{s 2} + L_{2}) - 2 [(i_{2} \cdot Q_{2} - i_{1} \cdot Q_{1}) - C_{c} - p_{c 2} \cdot (Q_{2} - Q_{1}) + T_{c} \cdot E_{c} \cdot θ (D)] - σ^{2} \leq - 1$ , Equation (10) is exponentially unstable in the zero-th moment.

Proof. Let $V (t, z) = z^{2}$ , $z \in [0, 1]$ , $c_{1} = c_{2} = 1$ , $p = 1$ and $γ = 1$ . The proof is the same as that of Corollary 1.

Corollaries1–3 indicate that, when condition (1) is satisfied, Equations (8)–(10) exhibit zero-order moment exponential stability, a unique evolutionarily stable strategy exists within the system, and the system tends to favor the non-green strategy. Conversely, when condition (2) is satisfied, Equations (8)–(10) are not exponentially stable in the zero-order moment sense, and the system tends to select the green strategy. Moreover, strategy selection is influenced not only by intrinsic factors, but also by uncertainties; both types of factors affect the strategic choices of the players in the game.

4.3. Numerical solution of the stochastic evolutionary game model

Since Equations (8)–(10) are nonlinear SDEs and their analytical solutions cannot be obtained directly, numerical methods are employed in this paper to obtain approximate solutions, specifically using the stochastic Taylor expansion method.⁴⁰ For the Itô stochastic differential Equation (11), the higher-order Milstein method is used to obtain:

\begin{matrix} x (t_{n + 1}) = x (t_{n}) + f [x (t_{n})] h + g [x (t_{n})] △ ω_{n} \\ + \frac{1}{2} g^{'} [x (t_{n})] g [x (t_{n})] [{(△ ω_{n})}^{2} - h] \end{matrix}

(12)

where $h$ denotes the simulation step size. Using this method, Equations (8)–(10) are numerically solved in this paper, yielding:

{\begin{matrix} x (t_{n + 1}) = x (t_{n}) + {x (1 - x) [- z (L_{3} + G_{s 2} - m_{c}) \\ - y (L_{2} + G_{s 1} - m_{p}) \\ + (F + L_{2} + L_{3} - C_{g 2} + L_{1} - m_{p} - m_{c})]} h \\ + σ x (t) △ ω (t) + 0.5 μ^{2} x (t) [{(△ ω_{n})}^{2} - h] \\ y (t_{n + 1}) = y (t_{n}) + {y (1 - y) [x (L_{2} + G_{s 1}) \\ + z [(s_{1} + s_{2} - p_{p 1} - p_{p 2}) (Q_{1} - Q_{2})] \\ + [(s_{1} - p_{p 1}) Q_{2} - (s_{2} - p_{p 2}) Q_{1} - C_{p 2} \\ + T_{p} E_{p} θ (D)]} h \\ + σ y (t) △ ω (t) + 0.5 μ^{2} y (t) [{(△ ω_{n})}^{2} - h] \\ z (t_{n + 1}) = z (t_{n}) + {z (1 - z) [x (G_{s 2} + L_{3}) \\ + y [2 (i_{1} Q_{1} - i_{2} Q_{2}) - (p_{c 1} + p_{c 2}) (Q_{1} - Q_{2})] \\ + [(i_{2} Q_{2} - i_{1} Q_{1}) - C_{c} - p_{c 2} (Q_{2} - Q_{1}) \\ + T_{c} E_{c} θ (D)]} h \\ + σ z (t) △ ω (t) + 0.5 μ^{2} z (t) [{(△ ω_{n})}^{2} - h] \end{matrix}

(13)

5. Numerical simulations

To verify the previous theoretical analysis, further analyze the evolutionary trend of the system, intuitively demonstrate the stable results of system evolution, and validate the dynamic evolutionary behavior, stability characteristics, and parameter sensitivity of the system under given parameter conditions, this study employs numerical simulation methods for analysis. Python simulation is used to simulate the dynamic evolutionary process of strategy selection by three participants under different initial states.

Sterman⁴¹ pointed out that model simulation is not about complete realization of reality, but about the degree to which it reveals system behavior and patterns of change. Regarding the setting of model parameters, accurate data are often lacking in many cases, and evolutionary game models primarily focus on overall system behavioral trends and the impact of intermediate parameter changes on the system.⁴² To further analyze the strategy selection of the government, producers, and consumers in the game, this paper refers to the assignment methods in references^43,44 to assign values to the variables involved in the model.

5.1. Simulation analysis of evolutionary game strategy stability

(1) The following parameters are selected for simulation: $C_{g 1} = 5$ , $C_{g 2} = 3$ , $G_{s 1} = 2$ , $G_{s 2} = 1$ , $T_{p} = 6$ , $T_{c} = 3$ , $E_{p} = 2$ , $E_{c} = 1$ , $F = 5$ , $L_{1} = 2$ , $L_{2} = 2$ , $L_{3} = 0.2$ , $m_{p} = 0.6$ , $m_{c} = 0.4$ , $C_{p 1} = 5$ , $C_{p 2} = 3$ , $C_{c} = 2$ , $Q_{1} = 10$ , $k = 0.8$ , $p_{p 1} = 3$ , $p_{p 2} = 1$ , $s_{1} = 4$ , $s_{2} = 2$ , $p_{c 1} = 5$ , $p_{c 2} = 4$ , $i_{1} = 6$ , $i_{2} = 5$ , $D = 80$ , $α = 0.6$ , $β = 1.0$ , Under these conditions, the system eventually converges to the stable point $E_{1} (1, 1, 1)$ , where the government chooses to support digital green transformation, producers adopt digital green production, and consumers engage in digital green consumption. The evolutionary process is shown in Figure 2.

(2) The following parameters are selected for simulation: $C_{g 1} = 5$ , $C_{g 2} = 3$ , $G_{s 1} = 2$ , $G_{s 2} = 1$ , $T_{p} = 6$ , $T_{c} = 3$ , $E_{p} = 2$ , $E_{c} = 1$ , $F = 5$ , $L_{1} = 2$ , $L_{2} = 2$ , $L_{3} = 0.2$ , $m_{p} = 0.6$ , $m_{c} = 0.4$ , $C_{p 1} = 5$ , $C_{p 2} = 3$ , $C_{c} = 20$ , $Q_{1} = 10$ , $k = 0.8$ , $p_{p 1} = 3$ , $p_{p 2} = 1$ , $s_{1} = 4$ , $s_{2} = 2$ , $p_{c 1} = 5$ , $p_{c 2} = 4$ , $i_{1} = 6$ , $i_{2} = 5$ , $D = 80$ , $α = 0.6$ , $β = 1.0$ , Under these conditions, the system eventually converges to the ESS, namely the equilibrium point $E_{2} (1, 1, 0)$ , where the government chooses to support digital low-carbon transformation, producers opt for green production, and consumers choose traditional consumption. In this case, consumers may be forced to choose non-ideal products to meet their daily needs, or may be willing to choose imported goods or alternative goods, resulting in a situation where products produced by domestic producers rely on export or are forced into stagnation. The evolutionary path is shown in Figure 3.

(3) The following parameters are selected for simulation: $C_{g 1} = 5$ , $C_{g 2} = 3$ , $G_{s 1} = 2$ , $G_{s 2} = 1$ , $T_{p} = 6$ , $T_{c} = 3$ , $E_{p} = 2$ , $E_{c} = 1$ , $F = 5$ , $L_{1} = 2$ , $L_{2} = 2$ , $L_{3} = 0.2$ , $m_{p} = 0.6$ , $m_{c} = 0.4$ , $C_{p 1} = 5$ , $C_{p 2} = 30$ , $C_{c} = 2$ , $Q_{1} = 1$ , $k = 0.8$ , $p_{p 1} = 3$ , $p_{p 2} = 1$ , $s_{1} = 4$ , $s_{2} = 2$ , $p_{c 1} = 5$ , $p_{c 2} = 4$ , $i_{1} = 6$ , $i_{2} = 5$ , $D = 80$ , $α = 0.6$ , $β = 1.0$ , Under these conditions, the system eventually converges to the ESS, namely the equilibrium point $E_{3} (1, 0, 1)$ , where the government chooses to support digital low-carbon transformation, producers opt for traditional production, and consumers choose green consumption. In this scenario, consumers may be forced to select non-ideal products to meet their daily needs, or may become more willing to choose imported goods, leading to a situation in which products produced by domestic producers rely on exports or are forced into stagnation. The evolutionary path is shown in Figure 4.

(4) The following parameters are selected for simulation: $C_{g 1} = 5$ , $C_{g 2} = 3$ , $G_{s 1} = 2$ , $G_{s 2} = 1$ , $T_{p} = 6$ , $T_{c} = 3$ , $E_{p} = 2$ , $E_{c} = 1$ , $F = 5$ , $L_{1} = 2$ , $L_{2} = 2$ , $L_{3} = 0.2$ , $m_{p} = 0.6$ , $m_{c} = 0.4$ , $C_{p 1} = 5$ , $C_{p 2} = 30$ , $C_{c} = 2$ , $Q_{1} = 10$ , $k = 0.8$ , $p_{p 1} = 3$ , $p_{p 2} = 1$ , $s_{1} = 4$ , $s_{2} = 2$ , $p_{c 1} = 5$ , $p_{c 2} = 4$ , $i_{1} = 6$ , $i_{2} = 5$ , $D = 80$ , $α = 0.6$ , $β = 1.0$ , Under these conditions, the system eventually converges to the ESS, namely the equilibrium point $E_{4} (1, 0, 0)$ , where the government chooses to support digital low-carbon transformation, but producers opt for traditional production and consumers choose traditional consumption. In this scenario, although the government is willing to support digital green strategies and has introduced corresponding incentive policies, these policies fail to attract producers and consumers. The evolutionary path is shown in Figure 5.

Figure 2.

Evolutionary results of equilibrium point $E_{1} (1, 1, 1)$ under disturbance-free and disturbed conditions.

Figure 3.

Evolutionary results of equilibrium point $E_{2} (1, 1, 0)$ under disturbance-free and disturbed conditions.

Figure 4.

Evolutionary results of equilibrium point $E_{3} (1, 0, 1)$ under disturbance-free and disturbed conditions.

Figure 5.

Evolutionary results of equilibrium point $E_{4} (1, 0, 0)$ under disturbance-free and disturbed conditions.

The simulation results indicate that, influenced by the stochastic disturbance intensity, the behavioral strategies of the government, producers, and consumers exhibit obvious fluctuation characteristics during the evolutionary process. The stability of this system’s evolutionary process shows a significant negative correlation with the stochastic disturbance intensity, that is, the greater the degree of stochastic disturbance, the more unstable the evolutionary process of each agent in the system. Furthermore, the increase in stochastic disturbance intensity significantly prolongs the time required for the system to reach convergence. Under high-intensity noise, the system may even have difficulty stably converging to the desired equilibrium, indicating that when external uncertainty exceeds a certain threshold, relying solely on spontaneous market forces may be insufficient to achieve stable transformation. It is noteworthy that when the strategy proportion of producers or consumers approaches zero, the fluctuation amplitude of the curve also decreases substantially. This stems from the noise form adopted in the model being $g (x) = σ (x) x$ . When participants adopt passive behavior, the impact of external uncertainty on them becomes limited.

From a practical perspective, the government’s choice of different planning strategies, the varying difficulty of implementing digital low-carbon transformation in different types of enterprises, differences in environmental awareness among different consumer groups due to variations in age, income, and education level, and differences in individuals’ acceptance of digital green consumption all generate stochastic disturbances in the strategy selection process. Any change in decision-making can easily disrupt the overall situation. To prevent strategy selection from failing to flow toward the ideal state, all agents should actively take measures to reduce stochastic disturbance intensity. If producers and consumers tend to adopt passive strategies, then no matter how intense the policy regulation or market stimulation, the actual effectiveness will be significantly weakened. Therefore, policymakers and industry managers should establish medium- to long-term strategic roadmaps, implement differentiated strategies for different agents, attach importance to the proactivity and stability of industry decisions, and avoid taking intervention measures only when the market tends toward passivity.

5.2. Impact of parameters on the evolutionary path of strategies

5.2.1. Producer–consumer equilibrium strategy and the impact of commodity circulation volume under different strategies on strategy selection

To further explore the impact of market commodity circulation volume on strategy selection, heatmaps of $k$ and $Q_{1}$ under different initial states were generated, as shown in Figure 6. The parameter values of $C_{g 1} = 5$ , $C_{g 2} = 3$ , $G_{s 1} = 2$ , $G_{s 2} = 1$ , $T_{p} = 6$ , $T_{c} = 3$ , $E_{p} = 2$ , $E_{c} = 1$ , $F = 5$ , $L_{1} = 2$ , $L_{2} = 2$ , $L_{3} = 0.2$ , $m_{p} = 0.6$ , $m_{c} = 0.4$ , $C_{p 1} = 5$ , $C_{p 2} = 3$ , $C_{c} = 2$ , $p_{p 1} = 3$ , $p_{p 2} = 1$ , $s_{1} = 4$ , $s_{2} = 2$ , $p_{c 1} = 5$ , $p_{c 2} = 4$ , $i_{1} = 6$ , $i_{2} = 5$ , $D = 80$ , $α = 0.6$ , $β = 1.0$ , $σ_{x} = σ_{y} = σ_{z} = 0$ were selected, and three sets of initial system states were chosen: [0.2, 0.2, 0.2], [0.5, 0.5, 0.5], and [0.8, 0.8, 0.8].

Figure 6.

The influence of commodity circulation volume on strategy selection.

From the simulation results of the model, it can be observed that the coefficient $k$ , which represents the impact on the commodity circulation volume, is a key factor influencing the strategies chosen by producers and consumers. When the initial participation degree in the digital green transformation is low, the system is highly sensitive to the value of $k$ . A smaller value of $k$ will lead producers and consumers to be more inclined to choose traditional strategies, while a larger value of $k$ can significantly promote the transformation of producers and consumers toward digital green strategies. In the low $k$ value region, the strategy transformation will only promote the transition to digital green strategies when the commodity circulation volume $Q_{1}$ is relatively small. However, when $Q_{1}$ is larger, both parties tend to choose traditional strategies. In the high $k$ value region, that is, when the impact of strategy transformation on the market commodity circulation volume is relatively small, both producers and consumers tend to choose digital green strategies, which is conducive to the transformation of producers and consumers. When the initial participation degree in the digital green transformation is high, producers and consumers tend to choose digital green strategies, and at this time, the commodity circulation volume has no impact on the final choice of strategies. Thus, it can be seen that the initial distribution of producers’ and consumers’ strategies has a significant impact on the evolution path of the system. Especially in the case of a low initial participation degree, maintaining the balance of market supply and demand under different strategy choices is of great significance for promoting the stable transformation of the system.

5.2.2. The impact of digital low-carbon transformation investment costs on the strategic choices of producers and consumers

To further explore the impact of low-carbon transformation input costs on strategy selection, heatmaps are plotted regarding the producers’ input cost for digital green transformation and the consumers’ input cost for digital transformation under initial states [0.2, 0.2, 0.2], [0.5, 0.5, 0.5], and [0.8, 0.8, 0.8], showing the strategies of producers and consumers at different transformation input costs. Among them, the area close to 1 represents the digital green strategy, while the area close to 0 represents the non-digital traditional strategy, and the evolutionary results are shown in Figure 7. Let $C_{g 1} = 5$ , $C_{g 2} = 3$ , $G_{s 1} = 2$ , $G_{s 2} = 1$ , $T_{p} = 6$ , $T_{c} = 3$ , $E_{p} = 2$ , $E_{c} = 1$ , $F = 5$ , $L_{1} = 2$ , $L_{2} = 2$ , $L_{3} = 0.2$ , $m_{p} = 0.6$ , $m_{c} = 0.4$ , $C_{p 1} = 5$ , $Q_{1} = 10$ , $k = 0.8$ , $p_{p 1} = 3$ , $p_{p 2} = 1$ , $s_{1} = 4$ , $s_{2} = 2$ , $p_{c 1} = 5$ , $p_{c 2} = 4$ , $i_{1} = 6$ , $i_{2} = 5$ , $D = 80$ , $α = 0.6$ , $β = 1.0$ , $σ_{x} = σ_{y} = σ_{z} = 0$ .

Figure 7.

Cost input and final strategy choice for green transformation.

From the model simulation results, the following observations can be made: First, the producers’ strategy is highly sensitive to their own transformation cost input $C_{p 2}$ , exhibiting abrupt strategy transitions within specific cost intervals, but is insensitive to consumers’ transformation cost input $C_{c}$ , with the boundary being nearly vertical; The consumers’ strategy demonstrates significant sensitivity to both cost parameters, with the boundary of the area close to 1 presenting a curve characteristic that protrudes toward the upper right, indicating that both $C_{p 2}$ and $C_{c}$ have substantial impacts on their strategy selection, and both parameters must be at relatively low levels simultaneously to maintain digital green consumption. Second, the sensitivity analysis of initial states reveals a significant path dependence effect. As the initial digital green propensity increases from 0.2 to 0.8, the area close to 1 gradually expands from the narrow left-side area in Figure 7(a1)–(a2) to the larger region in Figure 7(c1)–(c2), with the boundary clearly shifting to the right, and identical cost combinations can produce completely opposite evolutionary results under different initial conditions. Third, there exists obvious asymmetry in the digital green strategies between the supply and demand sides. Under the same initial state, the area close to 1 for producers is consistently larger than that for consumers, and this gap widens as the initial state increases.

5.2.3. The relationship between the sensitivity of emission reduction effect and the level of digitalization

To investigate the relationship between emission reduction sensitivity coefficient $β$ and digitalization level $D$ , the system’s final evolutionary outcomes under different values of these parameters were plotted, as shown in Figures 8 –10. The following parameters were used for the simulations: $C_{g 1} = 5$ , $C_{g 2} = 3$ , $G_{s 1} = 2$ , $G_{s 2} = 1$ , $T_{p} = 6$ , $T_{c} = 3$ , $E_{p} = 2$ , $E_{c} = 1$ , $F = 5$ , $L_{1} = 2$ , $L_{2} = 2$ , $L_{3} = 0.2$ , $m_{p} = 0.6$ , $m_{c} = 0.4$ , $C_{p 1} = 5$ , $Q_{1} = 10$ , $k = 0.8$ , $p_{p 1} = 3$ , $p_{p 2} = 1$ , $s_{1} = 4$ , $s_{2} = 2$ , $p_{c 1} = 5$ , $p_{c 2} = 4$ , $i_{1} = 6$ , $i_{2} = 5$ , $α = 0.6$ , $σ_{x} = σ_{y} = σ_{z} = 0$ . Among them, in Figure 8, $C_{p 2} = 3, C_{c} = 3$ ; in Figure 9, $C_{p 2} = 7, C_{c} = 3$ ; in Figure 10, $C_{p 2} = 3, C_{c} = 7$ .

Figure 8.

The influence diagram of $β$ and $D$ on strategy selection ( $C_{p 2} = 3, C_{c} = 3$ ).

Figure 9.

The influence diagram of $β$ and $D$ on strategy selection ( $C_{p 2} = 7, C_{c} = 3$ ).

Figure 10.

The influence diagram of $β$ and $D$ on strategy selection ( $C_{p 2} = 3, C_{c} = 7$ ).

From the simulation results of the model, it can be observed that the Digital Economic Development Index $D$ and the sensitivity $β$ of emission reduction effect are the key factors influencing the strategic choices of producers and consumers. When the parameter selections are all within a reasonable range and external uncertainties are not considered, at a higher level of digital economic development, even under a lower sensitivity of emission reduction effect, producers and consumers can reach the convergence state of digital green strategies, while at a lower level of digital economic development, a higher sensitivity of emission reduction effect is required to achieve the selection of digital green strategies. It is noteworthy that, as shown in Figure 7, producers and consumers will only evolve toward digital green strategies when appropriate transformation costs are invested by both parties. As can be seen from Figure 8 –10, the system exhibits clear phase transition characteristics under different parameter conditions. When the consumers’ cost input $C_{c}$ is low, an increase in the producers’ cost input $C_{p 2}$ will increase the difficulty of transformation, meaning that producers must have a higher sensitivity to emission reduction effects to promote transformation, and the evolutionary results of strategy selection exhibit obvious first-order phase transition characteristics. Once a certain critical value is breached, the strategy selection will rapidly shift from traditional strategies to digital green strategies. However, when producers invest lower transformation costs and consumers invest higher transformation costs, the system exhibits second-order phase transition behavior, with the strategy transition process being continuous.

6. Conclusion and prospect

6.1. Conclusion

To deeply analyze the behavioral strategic relationships among multiple agents under the broad context of the digital economy, this paper establishes a tripartite evolutionary game model among the government, producers, and consumers. By introducing the digital economy development index, which represents the level of digital economy development, into the model, comprehensively considering the product circulation volume when the strategy choices of producers and consumers are the same or different, and combining multiple factors such as carbon tax, carbon emissions, sensitivity to emission reduction effects, and stochastic disturbances, this paper deeply analyzes the evolutionary trajectories and stable states of strategy selection by game participants. Through data simulation, the following conclusions are obtained:

(1) Stochastic disturbance intensity has a significant impact on system stability:

Stochastic disturbance factors slow down the speed at which strategic agents evolve to stable strategies. The greater the stochastic disturbance intensity, the slower the speed at which game players evolve to stable strategies.

(2) Product circulation volume plays a critical role in final strategy selection:

The impact coefficient k on product circulation volume is a key factor determining the strategies of both parties;

The low initial participation stage is highly sensitive to the k value, while the influence of the k value weakens in the high initial participation stage.

(3) The constraining effect of cost input on strategy selection:

Producers’ cost input is highly sensitive to the strategy selection of both producers and consumers;

Consumers’ cost input has limited impact on producers’ strategy selection.

(4) The synergistic mechanism between digital development and sensitivity to emission reduction effects:

The digital economy development index $D$ and the sensitivity to emission reduction effects $β$ have a substitute-complementary relationship;

A high level of digitalization can reduce the demand for sensitivity to emission reduction effects, and vice versa;

Under different cost structures, the system exhibits different phase transition characteristics.

6.2. Contribution and application

In terms of methodological generalization, this paper employs SDEs and the Milstein high-order numerical method. By adding noise to the three parties, it simultaneously examines the impact of changes in internal factors and external factor influences faced by the government, producers, and consumers on the system’s evolutionary path. The model framework proposed in this paper has strong potential for cross-disciplinary application. For instance, in energy transition, it can be applied to the renewable energy adoption game among regulatory authorities–power generation enterprises–electricity users; in supply chain sustainability, it can be applied to the green supply chain game among certification agencies–suppliers–brand owners. The core logic of this method has universal applicability; in the environmental domain, it can be used to study the behavioral evolution of government–emitting enterprises–green consumers under carbon reduction policies; in the economic field, it can be applied to multi-agent collaborative problems in industrial digital upgrading. The strategy evolution conditions and policy tool combinations identified by the model can provide theoretical basis for formulating transformation policies in these fields.

6.3. Recommendations

Based on the above research conclusions and actual conditions, this paper proposes the following recommendations: (1) When formulating policies, the government should comprehensively consider the synergistic effect of explicit policy instruments (such as subsidies and corresponding penalties) and implicit influencing factors (such as reputation gains and credibility losses), and through scientifically designed policy combinations and optimized reward and punishment mechanisms, ensure the comprehensiveness and sustainability of policy effects; (2) improve the market monitoring system, and through regular surveys and data analysis, systematically assess costs and prices, product circulation and supply–demand matching, to better respond to the impacts brought by external uncertainties; (3) establish cost-sharing mechanisms. For producers, implement differentiated fiscal subsidies based on enterprise scale and industry characteristics; for consumers, reduce user costs through price subsidies and consumption vouchers; (4) strengthen the construction of digital infrastructure and related technical training, enhance market entities’ cognition and application capabilities of digital technologies, and promote the deep integration of digitalization with relevant industries.

6.4. Limitations and future work

The limitations of this study lie in the fact that, as a new form of economic development, the digital economy not only permeates multiple economic activity links such as production, circulation, and consumption but is also closely related to various aspects such as technological innovation, industrial upgrading, and efficiency improvement. However, this paper primarily focuses on the impact of digital economy development on carbon emissions and has not yet fully considered its role in enhancing production efficiency, strengthening consumption capacity, and other aspects. Future research can further consider constructing a more complete theoretical framework to deeply explore the impacts of digital economy development in more aspects including carbon emissions. The empirical validation of model assumptions needs to be strengthened. Limited by the data availability of digital green transformation, future research may consider collecting and organizing relevant data to empirically test the impacts of different strategies on market product circulation volume and subsidy structures, so as to enhance the model’s predictive capability and policy applicability. Meanwhile, consideration should be given to the impacts of different types of noise, in addition to Gaussian noise, on the evolutionary outcomes of game players in the system.

Footnotes

ORCID iD

Dun Han

Author contributions

Dun Han: Conceptualization; Supervision; Funding acquisition; Writing—review and editing.

Yanyi Lu: Methodology; Formal analysis; Software; Writing—original draft.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: They appreciate partial funding from the National Natural Science Foundation of China (Grant Nos. 72403103 and 72243005).

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Author biographies

Dun Han is an associate professor at the School of Mathematical Sciences, Jiangsu University, China.

Yanyi Lu is a graduate student at the School of Mathematical Sciences, Jiangsu University, China.

References

Lee

Calvin

Dasgupta

, et al. Climate change 2023 synthesis report summary for policymakers, 2024, https://www.preventionweb.net/publication/climate-change-2023-synthesis-report-summary-policymakers

Owen

Brennan

Lyon

. Enabling investment for the transition to a low carbon economy: government policy to finance early stage green innovation. Curr Opin Environ Sustain 2018; 31: 137–145.

Deng

Feng

. Incentive effect of government preferential policies on enterprises’ green technology innovation decision. Chin J Manag Sci 2025: 1–28. https://www.zgglkx.com/CN/abstract/article_19350.shtml

Zhou

. Environmental policy uncertainty and green transformation dilemma of Chinese enterprises. J Environ Manag 2024; 370: 122891.

. Digital economy and Chinese modernization: contemporary significance, opportunities and challenges, and exploration of paths. Economic Review 2023; 240: 3–14.

Gao Lh Cai

Zhao

, et al. A tripartite evolutionary game for marine economy green development with consumer participation. Environ Develop Sustain 2024; 26: 4197–4228.

Yang

Ren

, et al. Stochastic evolutionary game between governments and enterprises in renewable energy investment in view of dynamic carbonprice perspective. China Environ Sci 2024; 44: 567–580.

Cobb

Ragade

. Applications of catastrophe theory in the behavioral and life sciences. Behav Sci 1978; 23: 291–419.

Jiang

Feng

. Spatio-temporal characteristics of China’s digital economy’s ability to absorb and drive employment. Econ Geograph 2024; 44: 117–125.

10.

Belhocine

Bouchetara

. Simulation of fully coupled thermomechanical analysis of automotive brake discs. Simulation 2012; 88: 921–935.

11.

Belhocine

Shinde

Patil

. Thermo-mechanical coupled analysis based design of ventilated brake disc using genetic algorithm and particle swarm optimization. JMST Adv 2021; 3: 41–54.

12.

Tang

Liu

, et al. Effect of digital transformation on enterprises’ green innovation: empirical evidence from listed companies in China. Energy Econ 2023; 128: 107135.

13.

. Integrated analysis of employee cooperation and conflict behaviors in the context of digital technology. Simulation 2023; 99: 969–985.

14.

Xie

Xiao

. Digital transformation of production and adaptive innovation: the innovation logic of digital economy(5). J Beijing Jiaotong Univ 2021; 20: 1–10.

15.

Jiang

Zhu

. The effect of digital technology development on the improvement of environmental governance capacity: a case study of China. Ecolog Indicat 2024; 165: 112162.

16.

You

Chen

Lee

. Driving total factor productivity through digitalization? Evidence from high energy consuming enterprises in China. J Asian Econ 2025: 102022.

17.

Liu

. A blockchain-based framework of cross-border e-commerce supply chain. Int J Inform Manag 2020; 52: 102059.

18.

Kotchen

Maggi

. Carbon taxes and green subsidies in a world economy. Technical Report, National Bureau of Economic Research, Cambridge, MA, 2025.

19.

Zhang

, et al. Improved co-optimal and evaluable index of carbon sequestration and enhanced oil recovery. Simulation 2021; 97: 145–154.

20.

Han

Mao

, et al. Government environmental protection subsidies and corporate green innovation: evidence from Chinese microenterprises. Jour Innovat Knowl 2024; 9: 100458.

21.

Mardones

. Internalization of the social cost of carbon in each of the countries of the world—an economic assessment of its impacts. Energy Efficien 2024; 17: 79.

22.

Fang

Zhang

. The impact of digital development on corporate green innovation. Chin J Manag Sci 2023; 31: 350–360.

23.

Liu

Chen

, et al. The impact of digital technology development on carbon emissions: a spatial effect analysis for China. Resourc Conserv Recycl 2022; 185: 106445.

24.

Liu

. Behavior-based pricing of green product supply chain in different channels considering consumer preference differences. Chin J Manag Sci 2024; 32: 279–286.

25.

Yang

. Impacts of carbon emission regulation and consumer environmental consciousness on green innovation. Syst Eng Theory Pract 2021; 41: 702–712.

26.

D’Souza

Taghian

Khosla

. Examination of environmental beliefs and its impact on the influence of price, quality and demographic characteristics with respect to green purchase intention. J Target Meas Anal Market 2007; 15: 69–78.

27.

Huang

Bao

Chen

, et al. The evolution game analysis of enterprise’s strategic environmental governance with the effect of different combinations of reward policies and punishment policies. China Environ Sci 2023; 43: 3808–3820.

28.

Sun

Gao

. Evolutionary game analysis of enterprise carbon emission regulation based on prospect theory. Soft Comput 2022; 26: 13357–13368.

29.

Zhang

. Evolutionary game analysis of air pollution co-investment in emission reductions by steel enterprises under carbon quota trading mechanism. J Environ Manag 2022; 317: 115376.

30.

Sun

Wang

Zhu

, et al. Evolutionary game analysis of coal enterprise resource integration under government regulation. Environ Sci Pollut Res 2022; 29: 7127–7152.

31.

Chen

Zhang

Tadikamalla

, et al. The relationship among government, enterprise, and public in environmental governance from the perspective of multi-player evolutionary game. Int J Environ Res Public Health 2019; 16: 3351.

32.

Chen

Zou

. Stochastic evolutionary game of government industry-academia-research collaborative innovation in green innovation ecosystems. J Green Sci Tech 2025; 27: 272–280.

33.

Wang

Jiang

Zhang

. Seeking new location advantages: analysis of emerging digital cross-border M&As—based on TIMG index. Int Bus Rev 2024; 33: 102243.

34.

Wang

Dong

, et al. Assessing the digital economy and its carbon-mitigation effects: the case of China. Energ Econ 2022; 113: 106198.

35.

Wikström

Jönsson

Decosta

. A clash of modernities: developing a new value-based framework to understand the mismatch between production and consumption. J Consumer Cult 2016; 16: 824–851.

36.

Taylor

Jonker

. Evolutionary stable strategies and game dynamics. Mathem Biosci 1978; 40: 145–156.

37.

Jia

Zhou

. Stochastic evolutionary decision analysis of collaborative prevention and control strategies for public health emergencies. Chin J Manag Sci 2024; 32: 237–247.

38.

Baker

Buckwar

. Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations. J Comput Appl Math 2005; 184: 404–427.

39.

Kloeden

Platen

Kloeden

, et al. Stochastic differential equations. New York: Springer, 1992.

40.

Tian

Burrage

. Implicit Taylor methods for stiff stochastic differential equations. Appl Numer Math 2001; 38: 167–185.

41.

Sterman

Zhu

Zhong

. Business dynamics: systems thinking and modeling for a complex world. Beijing, China: Tsinghua University Press, 2008.

42.

Shen

Wang

. Collaboration of public service contracting: an evolutionary game analysis. Manag Rev 2017; 29: 219–230.

43.

Liu

Zou

Qin

, et al. An evolutionary game analysis of digital transformation of multiagents in digital innovation ecosystems. PLoS ONE 2023; 18: e0289011.

44.

Sun

. Evolutionary game analysis of environmental NGOs’ participation in corporate carbon emission reduction under low carbon economy. Oper Res Manag 2016; 25: 113–119.