Study on the Evolutionary Mechanism of Double-Round Monopoly of Super Platforms in China—Based on Four-Party Evolutionary Game

: In recent years, the regulation of double-round monopoly for super platforms has rapidly become a key instrument of the anti-monopoly practice for Internet platforms in China. This paper aims to explore the evolutionary mechanism of double-round monopoly by using evolutionary game theory and constructing a four-party evolutionary game model that includes a super platform, an incumbent platform, a startup platform, and a government regulator, based on considering the micro-dynamic interactions between subjects and the main inﬂuencing factors of the evolutionary mechanism using numerical simulation. This study illustrates that the decisions made by each subject and its probability changes affect the evolution rate of double-round monopoly. Depending on the probability of double-round monopoly in the market, government regulators need to be ﬂexible in switching between strict and less-strict regulation strategies to evolve the system to a more desirable, stable state. As well, we conclude that the regulatory strength of the government regulator, the cross-border magnitude of the super platform, the innovation incentive subsidy of the startup platform, and the synergistic risk cost of the incumbent platform have more signiﬁcant effects on the evolutionary process of double-round monopoly. Therefore, it is suggested that government regulators should focus on the cross-border magnitude of super platforms to safeguard the rights and interests of incumbent platforms and startup platforms in order to allow for fair competition. At the same time, they should also adjust the regulation intensity by the evolution of double-round monopoly in the market, in order to realize real-time dynamic regulation with the mutual coordination of multiple entities.


Introduction
The platform economy has rapidly developed in recent years and has now become a new force behind China's economic expansion [1].The growth of platform enterprises depends on support by capital and traffic, and realizes the two-way interaction cycle of "traffic attracts capital" and "capital maintains traffic" [2].At the beginning stage, because of particular characteristics like network impact, scale effect, and user lock-in effect, the platform economy market is more likely to result in a "winner-takes-all" competitive pattern, which will eventually lead to the initial monopoly of platform firms [3].Platform firms continue to grow their industry chains horizontally and vertically based on their initial monopoly.Since they have amassed a number of users and data information through their Systems 2023, 11, 492 2 of 28 original domains, the sustained maintenance of their advantageous positions in new sectors or industries is facilitated, leading to the emergence of double-round monopoly [4].Super platforms are more prone to leading to the emergence of double-round monopoly.However, various problems caused by double-round monopoly have still not been effectively regulated, such as inhibiting mergers and acquisitions, restricted transactions, algorithmic collusion, etc., which will provide the market with significant potential risks.Therefore, we need to understand why double-round monopoly occurs.
The emergence of double-round monopoly is due to the cross-border expansion of super platforms.What steps have been taken in China to address this problem, and what are the practical shortcomings?In addition, what scientific advances have been made?From a legislative perspective, China's first revised Anti-Monopoly Law, which came into effect on 1 August 2022, offers valuable guidance for China to continue promoting highquality economic development and fair competition in the platform market [5].However, the market environment, subject behavior, and development model of the Internet platform economy are all changing quickly, which leads to it outpacing the formulation of legal policies [6].In particular, the rules and regulations have not included specific measures for double-round monopoly.From a practice perspective, although the Chinese government has penalized and regulated the monopolistic behavior of Internet "giants" like Alibaba and Meituan for their monopolistic conduct, the monopolistic status of these enterprises in cross-border markets has not been formally recognized, despite the consideration of their competitive advantages in related markets [7].From a theoretical research perspective, current studies on double-round monopoly include monopoly behaviors [8,9], monopoly causes [10,11], potential hazards [12,13], and regulatory pathways [14,15].However, previous studies have not dealt with the evolutionary mechanism of double-round monopoly, especially in relation to the behavioral dynamics of various stakeholders and the conflicts and coordination processes among multiple entities.For example, how do the behaviors of the incumbent platform, the startup platform, and the government regulator affect the cross-border expansion of the super platform?How do their behaviors affect the evolution of double-round monopoly?Furthermore, the research methods in this field are predominantly theoretical and analytical.
Therefore, there is an urgent need for research into the double-round monopoly of super platforms to guide anti-monopoly practices in China's Internet sector.As a result, this research addresses both theoretical and practical gaps.It fills a theoretical research gap on the evolutionary mechanism of double-round monopoly and enriches the theory of platform monopoly in the digital economy environment, providing scholars with valuable references for future research on the monopolistic practice of super platforms.Additionally, we have developed a four-party evolutionary game model of the super platform, the incumbent platform, the startup platform, and the government regulator that can be used to analyze double-round monopoly.The model can also be used to depict double-round monopoly's evolution in other countries by modifying its assumptions and parameters, even though our primary focus is on the Chinese environment.In terms of practical application, we employ model simulation to identify the key factors influencing doubleround monopoly and present actionable policy proposals that might assist government regulators in improving legislation and enforcement.
Based on the above analysis, the rest of the paper is arranged as follows.In Section 2, we sum up the concept, the evolutionary motivation, and the application of the evolutionary game approach to double-round monopoly.In Section 3, we analyze the strategy stability of the super platform, the incumbent platform, the startup platform, and the government regulator, respectively, and we discuss various scenarios of quadripartite strategy stability.In Section 4, we explore the impact of the regulatory strength of government regulators, the cross-border magnitude of super platforms, the innovation incentive subsidy of startup platforms, and the cost of the synergistic risk of incumbent platforms on the evolutionary process of double-round monopoly, using numerical simulation methodology.In Section 5, we summarize the main conclusions, including regulatory recommendations to the government, as well as the shortcomings of this study and future research directions.

Research on the Concept and Evolutionary Motivation of Double-Round Monopoly
Scholars have successively proposed concepts related to double-round monopoly over the past years.While the emphases of these concepts may differ to some extent, such as the "data + algorithm" dual-drive [4], ecological monopolies [7], and integrated monopolies [16], they all delineate the phenomenon of the cross-market transmission of monopolistic power by super platforms.Some scholars [17] have suggested that the essence of double-round monopoly is the leverage effect from the standpoint of regulation, which is a typical expression of leverage theory in the platform economy.
To investigate the underlying drivers of double-round monopoly evolution, scholars have primarily researched different aspects.From the perspective of ecosystem construction, the process of horizontal and vertical cross-border expansion of super platforms in the digital economy has accelerated significantly [18], gradually building a diversified platform ecosystem by extending upstream and downstream industrial chains, etc., and further amplifying the network effect by utilizing mature markets [19], which will continuously reduce the innovation willingness and competitive ability of other enterprises [20].From the perspective of data algorithm drive, the super platform enhances the "leverage function" through an interactive feedback mechanism between data, algorithms, and traffic, which makes it simpler for the super platforms to create their own "closed loop" of interests.The "traffic pool" created by traffic conduction will also increase the hurdles to entrance for potential competitors [21].From the perspective of user lock-in, the double accumulation of users and attention paid to the super platforms has been completed, prompting the platforms to gradually change from open to closed.The coupling of closed ecology and "double-round monopoly" will intensify the locking and clustering effect and gradually release the value of users in the new market by establishing a relationship between the users and the platforms [22].From the perspective of the estrangement of market power, the estrangement of market power will hasten the transmission of monopoly power and the formation of double-round monopoly [6].The super platforms adopt a number of competitive measures such as refusal to deal, self-preference, competition restriction, and stifling mergers and acquisitions in order to consolidate their competitive advantages and raise market entry barriers [23], which will harm competition even if they do not constitute monopolies [24].According to Coveri et al. [25], these forms of market power can also be categorized as market expansion and diversification, data and technology exploitation, labor specialization, and strong negotiating power over the government.
Current studies place a greater emphasis on market behavior and the dominance effect of the super platforms; however, the interaction relationship among subjects in the economic market has been ignored.Additionally, while previous studies highlight the risks of double-round monopoly and suggest governance strategies, they do not pay attention to the role of government regulators in coordinating different subjects in the overall market environment.

Research on the Application of Evolutionary Game Approach in Double-Round Monopoly
Platforms are essentially complex ecosystems in which multiple subjects interact with each other.At a microscopic level, the dynamic evolution between multiple subjects and the environment leads to changes in the competitive structure of the market, such as the formation process of platform monopolies [26].The evolutionary game approach combines game theory with dynamic evolutionary analysis, which is suitable for analyzing such multi-subject dynamic interaction mechanisms [27,28].Tirole Jean and other scholars have applied game theory to the field of Internet monopoly, including platform competition and cooperation, market power regulation, and monopolistic competition policy [29,30].Yang and other scholars have proposed that game theory is an important fundamental theory in the field of digital economy [31], while other scholars have also explored the microscopic issues of platform economic monopoly from the perspective of game theory, such as the trend of monopoly formation on digital platforms [32], the competitive effect of exclusive transactions and monopoly regulation [33], the market structure formation mechanism of monopolies on Internet platforms, and the effectiveness of platform regulation [34].Therefore, by studying the game relationship between each platform subject and government regulators in the Internet market, the micro-logic of the double-round monopoly of super platforms is easier to sort out.
However, the aforementioned studies usually involve only two or three subjects among super platforms, startup platforms, merchants, consumers, and government departments.The formation of monopolies in the platform economy is actually a complex process in which multiple subjects act mutually.Moreover, when researchers use the evolutionary game approach to explore the monopoly mechanism, they mainly focus on the competition in the initial monopoly market, and pay less attention to the cross-border monopoly problem of super platforms.Therefore, the evolutionary trends and micro-mechanisms of double-round monopoly for super platforms are worth exploring in depth.Based on the above analysis, this paper uses the theory and method of the evolutionary game approach to explore the evolutionary mechanism of double-round monopoly for the super platforms, investigating the micro-dynamic interactions among the super platform, the incumbent platform, the startup platform, and the government regulator.

Evolutionary Game Model
Before constructing the model, this study addresses the model's scope of applicability with the following brief explanation.Traditional games are primarily categorized as either cooperative or non-cooperative games [35].Evolutionary games, unlike traditional games, gradually evolve from non-cooperative games by relaxing the assumption of rationality [35].In non-cooperative games, competition and collaboration co-exist, and the formulation of winning strategies may or may not involve mutual competition [36,37].Furthermore, there is a degree of strategic uncertainty in how certain games progress [38].In the real world, platform businesses must consider government policies, dynamic changes in the market environment, and other factors in addition to the strategic decisions made by competitors [39][40][41][42].In the evolutionary game model used in this study, game parties are considered to be finite rational actors who adjust their strategic choices in response to the choices of other game parties.In practice, they may not always make the optimal choice [43,44].Therefore, strategic uncertainty is inherent in the evolutionary game model employed in this study, but it aligns with real-world scenarios.Consequently, we assume the evolutionary scenario of the platform economy market in China.
After proposing this assumption, it is essential to determine whether the model can be applied to more general situations, including other countries.In reality, double-round monopolies have occurred worldwide, prompting governments to implement relevant regulations [45].These regulations can significantly influence the design of the model, as market and governmental environments vary globally.Firstly, different countries exhibit varying numbers and sizes of super platforms based on their unique market conditions.For example, the European Union has fewer super Internet platforms, mainly because super platform firms of the United States exert a more substantial influence on the local small and medium-sized platform firms of the European Union [46].In contrast, the United States has long held a dominant position in the Internet industry [47], led by companies like Google, Amazon, Facebook, and Apple.China, on the other hand, boasts a sizable and thriving platform economy [48].These differing market environments necessitate adjustments in how relevant parameters for super platforms are set.Secondly, each nation maintains a distinct policy environment shaped by its economic structure, level of incentives, and political system.In terms of regulatory frameworks, the United States has transitioned from lenient to stringent regulation, while the European Union has consistently enforced strict anti-trust rules [49].China, meanwhile, has increased its enforcement of platform monopoly regulations since 2020 [1] and exercises greater macro-economic control.It is important to note that China lags behind Western countries in terms of incentives [6].These policy environments have a profound impact on parameter settings for other platforms and government regulators [24].In summary, we have limited the model's applicability to China because the model's configuration largely aligns with the actual conditions in China.However, by adjusting the assumptions and parameters to reflect the unique circumstances of other countries, the model can be adapted to investigate double-round monopolies in those regions.

Problem Description
This study focuses on the market environment of the platform economy, where super platforms are equipped with dominant market power and significant influence, and are also typically the leading businesses in their respective sectors [3].The incumbent platform refers to the platform that has reached maturity in the market that the super platform has penetrated [50].However, the incumbent platform and the super platform still have some scale differences; thus, the concept of duopoly is not taken into consideration in this research [51].The startup platform is one that has not yet entered the market or matured but exhibited traits like a great capacity for innovation and a high degree of development potential [52].The government regulator includes all levels of government entities engaged in anti-trust enforcement, including the State Antimonopoly Bureau, the State Administration of Market Supervision, and related entities [41].Figure 1 illustrates the tactical decisions and related interactions of several subjects during the evolutionary process of double-round monopoly.
Systems 2023, 10, x FOR PEER REVIEW 5 of 31 and political system.In terms of regulatory frameworks, the United States has transitioned from lenient to stringent regulation, while the European Union has consistently enforced strict anti-trust rules [49].China, meanwhile, has increased its enforcement of platform monopoly regulations since 2020 [1] and exercises greater macro-economic control.It is important to note that China lags behind Western countries in terms of incentives [6].These policy environments have a profound impact on parameter settings for other platforms and government regulators [24].In summary, we have limited the model's applicability to China because the model's configuration largely aligns with the actual conditions in China.However, by adjusting the assumptions and parameters to reflect the unique circumstances of other countries, the model can be adapted to investigate doubleround monopolies in those regions.

Problem Description
This study focuses on the market environment of the platform economy, where super platforms are equipped with dominant market power and significant influence, and are also typically the leading businesses in their respective sectors [3].The incumbent platform refers to the platform that has reached maturity in the market that the super platform has penetrated [50].However, the incumbent platform and the super platform still have some scale differences; thus, the concept of duopoly is not taken into consideration in this research [51].The startup platform is one that has not yet entered the market or matured but exhibited traits like a great capacity for innovation and a high degree of development potential [52].The government regulator includes all levels of government entities engaged in anti-trust enforcement, including the State Antimonopoly Bureau, the State Administration of Market Supervision, and related entities [41].Figure 1 illustrates the tactical decisions and related interactions of several subjects during the evolutionary process of double-round monopoly.Cross-border expansion drives the emergence of a double-round monopoly.In this study, the term "cross-border" is defined as the expansion of platform enterprises from their main business area to an associated market or a non-associated market, including horizontal cross-border, vertical cross-border, and diversified cross-border expansion [3,4,53].Horizontal cross-border expansion refers to the expansion of platform enterprises Cross-border expansion drives the emergence of a double-round monopoly.In this study, the term "cross-border" is defined as the expansion of platform enterprises from their main business area to an associated market or a non-associated market, including horizontal cross-border, vertical cross-border, and diversified cross-border expansion [3,4,53].Horizontal cross-border expansion refers to the expansion of platform enterprises to the associated markets where the business type in the market is the same as the original basic business [54,55].For example, Facebook has outlined social media services including Facebook, Messenger, WhatsApp, Instagram, etc.Although these services belong to the same category, these social services can meet various user needs, providing Facebook with a greater competitive advantage in the social service industry.Vertical cross-border expansion refers to the expansion of platform enterprises Systems 2023, 11, 492 6 of 28 to the associated market, where the business in this market is complementary to the original basic business [56].For example, Jingdong's main business is e-commerce, in addition, it also has Jingdong Logistics, Jingdong Finance, Jingdong Intelligence, and other businesses, which, together, constitute a complete industry chain.Diversified cross-border expansion refers to the expansion of platform enterprises into associated or non-associated markets.In some cases, it is a combination of horizontal and vertical cross-border expansion; in other cases, it is the expansion to new industries and of other businesses in new domains [57,58].For example, Tencent initially started with a social platform service as its basic business, and then gradually expanded into digital content, financial technology, online advertising, and cloud computing.It has now developed a dominant position within multiple markets.
Super platforms have already held a dominant market position in the initial business market, and they are further consolidating their position with the use of data and algorithms [59].They are inclined to extend their advantages from the original market to new ones, by horizontal cross-border, vertical cross-border, and diversified cross-border strategies [4,60].The platform ecosystem will then be strengthened.Therefore, cross-border expansion is a crucial requirement before double-round monopoly formation.The super platform makes differentiated investments in various target markets by thoroughly analyzing its resource advantages, cross-border challenges, and potential returns [61].When a super platform enters a new market with its huge advantages in data, algorithms, traffic, and user scale, the incumbent platform in that market will face huge competitive pressure and urgently needs to adjust its strategy to cope with the super platform's expansion trend.In terms of business sectors, the super platform and the incumbent platform might be rivals or allies [42], and their synergistic cooperation would indicate that benefits and hazards would co-exist [62].The super platform and the incumbent platform will both experience larger innovation impacts and competitive pressure when the startup platform is able to amass users quickly and has the ability to infiltrate the super platform's primary business sector [63].Therefore, the super platform may engage in unfair competitive practices to block the entry of the startup platform [64].The Chinese government promotes the inventive growth of startup platforms through innovation subsidies and other means, and uses them as a powerful tool to counterbalance super platforms and preserve a healthy level of competition in the platform economy market [65].In order to prevent other market participants from losing their ability to compete effectively, government regulators must also regulate the super platforms promptly and effectively, but in practice, negative regulation and regulatory failure may still occur.

Basic Assumptions
The super platform, the incumbent platform, the start-up platform, and the government regulator interact with each other and are also the main game subject in the evolution of the double round of monopoly.Based on the problem description, the following fundamental presumptions of the four-party evolutionary game model are provided as follows.
Assumption 1.In a complicated platform economics market environment, all parties in the game are provided with incomplete information and are finitely rational in their decisionmaking.The game parties continuously learn and develop throughout the evolutionary process, and they eventually reach the ideal strategy by dynamically altering their approach, which will also generate multiple evolutionary equilibria.Assumption 2. The strategy selection of the super platform involves a "high cross-border magnitude expansion strategy" and a "low cross-border magnitude expansion strategy."Financial investment is necessary, as well as other factors like resources, abilities, and facilities, for cross-border expansion.When the super platform grows, it must invest in data resources, marketing funds, user migration costs, and other cross-border costs, which are set as C 1 .For instance, Alibaba is one of China's super platform companies.Alibaba has made significant financial investments to pursue cross-border acquisition activities across a range of industries, including e-commerce, logistics, cloud computing, finance and insurance, artificial intelligence, film and television, and more.The cross-border range is r 1 (0 < r 1 < 1).The cross-border cost of choosing low cross-border margin expansion for a super platform is r 1 * C 1 .It is clear that platform companies have gained more crossborder benefits as a result of cross-border expansion.Due to Alibaba's development into other businesses, it has achieved great financial success, with turnover rising sharply on an annual basis.The cross-border benefits of selecting high cross-border margin expansion and low cross-border margin expansion are R 1 and R 2 , respectively.However, platform enterprises may be penalized as a result of the harm done to ethical business practices by their cross-border expansion strategy.For instance, Alibaba must pay a fine of 18.228 trillion yuan, according to a decision made by Market Supervision and Administration on 24 December 2020.These are the penalty costs.When the government regulator puts stringent controls on the super platform, the penalty costs corresponding to high and low cross-border magnitude increases are P 1 and P 2 , respectively.When the startup platform enters the market, the super platform with high and low cross-border magnitude expansion will suffer innovation impact effects N 1 and N 2 , respectively.The technological shock effect refers to the chance that super platforms may lose some of their market share.Assumption 3. The strategy selection of the incumbent platform involves a "synergistic collaboration strategy" and a "non-synergistic collaboration strategy."The incumbent platform must pay collaborative costs C 2 in terms of data sharing, business connectivity, and resource investment when it chooses a collaborative cooperation strategy.Collaborative cooperation may also be fraught with hazards including contract breach, conflict of interest, and failure to cooperate, so the associated risk cost is C 3 .The benefit of cooperative collaboration is R 3 .If the startup platform chooses the market entry strategy, the incumbent platform with synergistic and non-synergistic collaboration will suffer innovation impact effects N 3 and N 4 , respectively.Assumption 4. The strategy selection of the startup platform involves a "market entry strategy" and a "market exclusion strategy."Prior to entering the market, the startup platform typically has to boost R&D costs C 4 to accomplish technological innovation and overcome the market's entry barrier.The startup platform must also consider the risk of prospective market rivalry, and this risk cost is C 5 .Additionally, the startup platform will face a higher entry barrier when the super platform collaborates with the incumbent platform, which will result in additional entry costs C 6 .The benefit of entering the market is R 4 .
Assumption 5.The strategy selection of the government regulator involves a "strict regulation strategy" and a "loose regulation strategy."The government regulator's regulatory strength against the super platform is r 2 and its regulatory cost is C 7 .The startup platform's innovation incentive subsidy given by the government regulator is S 1 , while the innovation revenue generated by the startup platform for the government regulator is R 5 .Assumption 6.If the super platform decides to expand at a high cross-border margin, a double-round monopoly may occur, which has a probability of r 3 .If the government regulator does not strictly regulate the double-round monopoly, the incumbent platform, the startup platform, and the government regulator will all experience losses in economic efficiency as a result; these losses are L 1 , L 2 , and L 3 , respectively.In contrast, if the government regulator strictly regulates the double-round monopoly, the incumbent platform, the startup platform, and the government regulator will all receive the benefits of economic efficiency; these benefits are R 6 , R 7 , R 8 , respectively.However, if the government regulator tightly controls the super platform when it expands with a low cross-border margin, "overregulation" may happen, with a likelihood that the super platform, the incumbent platform, the startup platform, and the government regulator will have to pay the innovation losses.Therefore, the probability of over-regulation is r 4 , and the innovation losses are F 1 , F 3 , and F 4 , respectively.Assumption 7. We assume that the probability of the super platform choosing the strategy of "high cross-border magnitude expansion" is p (0 < p < 1), and the probability of choosing the strategy of "low cross-border magnitude expansion" is 1 − p.The probability of the incumbent platform choosing the strategy of "synergistic collaboration" is x (0 < x < 1), and the probability of the incumbent platform choosing the strategy of "non- synergistic collaboration" is 1 − x.The probability of the startup platform choosing the strategy of "market entry" is y (0 < y < 1), and the probability of the startup platform choosing the strategy of "market exclusion" is 1 − y.The probability of the government regulator choosing the strategy of "strict regulation" is z (0 < z < 1), and the probability of the government regulator choosing the strategy of "loose regulation" is 1 − z.

Payoff Matrix
Based on the above assumptions and parameter settings, the payoff matrix of the super platform, the incumbent platform, the startup platform, and the government regulator is shown in Table 1.Market entry (y) Market entry (y) Market entry (y) Market entry (y)

Analysis of Evolutionary Stability Strategy
Each subject continuously alters its strategy choice in reaction to the activities of the subjects around it as the double-round monopoly develops because of the subjects' weak reasoning and lack of complete information [66].Therefore, by calculating the expected Systems 2023, 11, 492 9 of 28 return function of each subject, the replication dynamic equation can be obtained [67][68][69], so as to analyze the stability of the strategy of each subject [70].

Evolutionary Stability Strategy Analysis of the Super Platform
When the super platform chooses the strategy of high cross-border margin expansion, its expected benefit is: When the super platform chooses the strategy of low cross-border margin expansion, its expected benefit is: The average expected benefit when the super platform chooses different strategies is: Therefore, the replication dynamic equation of the super platform is: Thus, the super platform's decision depends not only on the costs and benefits of selecting various strategies, but also on the decision probabilities of the startup platform and the government regulator.According to the stability theorem of the differential equation [71][72][73][74], if the super platform reaches a stable state, then: F(p) = 0, F (p) < 0. Proposition 1.If y < y 1 , then the super platform chooses the strategy of low cross-border margin expansion; if y > y 1 , then the super platform chooses the strategy of high crossborder margin expansion; if y = y 1 , then the stabilization strategy of the super platform cannot be determined.The threshold value is y 1 , y Proof.If ∂A(y, z)/∂y > 0, then A(y, z) is the increasing function of y.If y < y 1 , then A(y, z) < 0, F(p)| p =0 = 0, F (p) p =0 < 0; thus, the point of p = 0 is locally stable; if y > y 1 , then A(y, z) > 0, F(p)| p =1 = 0, F (p) p =1 < 0; thus, the point of p = 1 is locally stable; if y = y 1 , then A(y, z) = 0, F(p) = 0, F (p) = 0; thus, the stabilization strategy cannot be determined at this point.
According to Proposition 1, when the probability of the startup platform entering the market rises, the super platform's strategy will change from low to high cross-border margin expansion.The super platform can only adopt a more significant expansion strategy to squeeze the startup platform's survival space in order to lessen the startup platform's impact on it in terms of innovation.In contrast, the super platform experiences less competitive pressure as the probability of the startup platform entering the market decreases, hence it tends to reduce the cross-border magnitude suitably.
Then, we suppose that the probability of the super platform choosing the low crossborder magnitude expansion strategy is V p0 , and the probability of the super platform choosing high cross-border magnitude expansion strategy is V p1 : Corollary 1.The super platform is more likely to choose high cross-border expansion when the cost is lower and the revenue is higher, or when the revenue of low cross-border expansion gradually declines.As a result, the super platform is willing to expand crossborder to emerging markets with promising development prospects.The super platform will lower its cross-border expansion margin as regulatory efforts and corresponding penalties increase, demonstrating how the penalty mechanism can significantly limit the super platform's desire to expand.However, if the government regulator continues to enact severe regulatory measures, the probability of over-regulation will increase, so that the super platform may experience more losses from an innovation environment.
Proof.If we calculate the partial derivative V p1 with respect to C 1 , R 1 , R 2 , P 1 , r 2 , r 4 , and F 1 , then we obtain

Evolutionary Stability Strategy Analysis of the Incumbent Platform
When the incumbent platform chooses the strategy of synergistic collaboration, its expected benefit is: When the incumbent platform chooses the strategy of non-synergistic collaboration, its expected benefit is: The average expected benefit when the incumbent platform chooses different strategies is: Therefore, the replication dynamic equation of the incumbent platform is: Therefore, the incumbent platform's decision depends not only on the costs and benefits of selecting various strategies, but also on the decision probabilities of the startup platform.According to the stability theorem of the differential equation, if the incumbent platform reaches a stable state, then: F(x) = 0, F (x) < 0. Proposition 2. If y < y 2 , then the incumbent platform chooses the strategy of nonsynergistic collaboration; if y > y 2 , then the incumbent platform chooses the strategy of synergistic collaboration; if y = y 2 , then the stabilization strategy of the incumbent platform cannot be determined.The threshold value is y 2 , y 2 = (C 2 + C 3 − R 3 )/(N 4 − N 3 ).
According to Proposition 2, as the probability of the startup platform entering the market rises, the incumbent platform will switch to the strategy of synergistic cooperation.The incumbent platform will continuously optimize its own resource allocation by working with the super platform in order to counteract the innovation impact produced by the startup platform.On the other hand, when the probability of the startup platform entering the market decreases, the incumbent platform suffers a smaller competitive impact and has less willingness to choose the strategy of synergistic cooperation.
Then, we suppose that the probability of the incumbent platform choosing the nonsynergistic collaboration strategy is V x0 , and the probability of the incumbent platform choosing the synergistic collaboration strategy is V x1 : Corollary 2. The incumbent platform's decision is mostly influenced by its cost and benefit.When the cost and risk cost of collaboration is lower, or the benefit is higher, the incumbent platform will tend to choose the synergistic collaboration strategy.Other factors such as the innovation impact effect also influence its strategy decision to some extent.More importantly, the incumbent platform should use the synergistic cooperation strategy to promote its own innovative development.

Evolutionary Stability Strategy Analysis of the Startup Platform
When the startup platform chooses the strategy of market entry, its expected benefit is: When the startup platform chooses the strategy of market exclusion, its expected benefit is: The average expected benefit when the startup platform chooses different strategies is: Therefore, the replication dynamic equation of the startup platform is: Thus, the startup platform's decision depends not only on the costs and benefits of selecting various strategies, but also on the decision probability of the super platform, the incumbent platform, and the government regulator.According to the stability theorem of the differential equation, if the startup platform reaches a stable state, then: F(y) = 0, F (y) < 0. Proposition 3. If x > x 1 , then the startup platform chooses the strategy of market exclusion; if x < x 1 , then the startup platform chooses the strategy of market entry; if x = x 1 , then the stabilization strategy of the startup platform cannot be determined.The threshold value is Proof.∂C(p, x, z)/∂x < 0, then C(p, x, z) is the decreasing function of x.If x < x 1 , then C(p, x, z) > 0, F(y)| y =1 = 0, F (y) y =1 < 0; thus, the point of y = 1 is locally stable; if x > x 1 , then C(p, x, z) < 0, F(y)| y =0 = 0, F (y) y =0 < 0; thus, the point of y = 0 is locally stable; if x = x 1 , then C(p, x, z) = 0, F(y) = 0, F (y) = 0; thus, the stabilization strategy cannot be determined at this point.
According to Proposition 3, as the probability of the incumbent platform choosing the synergistic cooperation strategy rises, the startup platform will switch to the strategy of market exclusion.Strong cooperation between the super platform and the incumbent platform will greatly raise the entry barrier for the startup platform and decrease its willingness to participate in competition.In contrast, the lower the probability of synergistic collaboration between the incumbent platform and the super platform, the greater the willingness of the startup platform to enter the market.
Then, we suppose that the probability of the startup platform choosing the market exclusion strategy is V y0 , and the probability of the startup platform choosing the market entry strategy is V y1 : Corollary 3. The startup platform is more likely to adopt the market entry strategy when the cost of entry is lower, revenue is higher, and the innovation incentive subsidy is larger.If the chance of double-round monopoly in the market continuously rises and the monopolistic power is not appropriately regulated, the probability of market entry for the startup platform will be drastically decreased.It is clear that a trend of the super platform expanding would increase entry barriers and the level of market rivalry, obstructing the creative growth of the startup platform.In addition, the startup platform will be less eager to enter the market if excessive government regulation results in a generally low efficiency of market innovation.

Evolutionary Stability Strategy Analysis of the Government Regulator
When the government regulator chooses the strategy of strict regulation, its expected benefit is: When the government regulator chooses the strategy of loose regulation, its expected benefit is: The average expected benefit when the government regulator chooses different strategies is: Therefore, the replication dynamic equation of the government regulator is: Thus, the government regulator's decision depends not only on its own costs and benefits from selecting various strategies, but also on the decision probability of the super platform, the incumbent platform, and the startup platform.According to the stability theorem of the differential equation, if the government regulator reaches a stable state, then: F(z) = 0, F (z) < 0. Proposition 4. If x > x 2 , y < y 3 , then the government regulator chooses the strategy of loose regulation; if x < x 2 , y > y 3 , then the government regulator chooses the strategy of strict regulation; if x = x 2 , y = y 3 , then the stabilization strategy of the government regulator cannot be determined.The threshold value is x 2 , y 3 , Proof.∂ D(p, x, y)/∂x < 0, then D(p, x, y) is the decreasing function of x.If x < x 2 , then D(p, x, y) > 0, F(z)| z =1 = 0 , F (z) z =1 < 0; thus, the point of z = 1 is locally stable; if x > x 2 , then D(p, x, y) < 0, F(z)| z =0 = 0, F (z) z =0 < 0; thus, the point of z = 0 is locally stable; if x = x 2 , then D(p, x, y) = 0, F(z) = 0, F (z) = 0; thus, the stabilization strategy cannot be determined at this point.
According to Proposition 4, the government regulator will likely choose the strict regulation strategy if the probability of the incumbent platform cooperating reduces or the probability of the startup platform entering the market rises.To safeguard the market's smooth functioning and address competition issues brought on by the super platform's excessive growth, the government regulator should impose strong regulatory measures for the super platform.
Then, we suppose that the probability of the government regulator choosing the loose regulation strategy is V z0 , and the probability of the government regulator choosing the strict regulation strategy is V z1 : Corollary 4. The government regulator's willingness to regulate strictly is stronger when the cost of strict regulation is lower, or the penalty revenue and other indirect benefits are higher.Meanwhile, the extent to which the market's competitive environment and regulation strength are aligned affects the regulatory losses.When the potential overregulation loss is low, the government regulator tends to choose the strict regulation strategy, which is the case with positive regulation.However, as the probability of doubleround monopoly gradually increases in the market environment, the government regulator also tends to choose the strict regulation strategy, which is the case with passive regulation.

Quadripartite Evolutionary Stability Strategy Analysis
In order to analyze the evolutionary stability strategy of the four parties, a system of equations is constructed and solved, then multiple groups of stable solutions can be obtained.The equations are: However, rigorous Nash equilibria, which must be pure strategy equilibrium points, are required for stable solutions in multiple group evolutionary games [67,74], and mixed strategy points cannot reach evolutionarily stable equilibria in this model.As a result, only 16 pure strategy equilibrium points are examined in this study's evolutionary stability strategy analysis.According to Lyapunov's first law [70], a Jacobian matrix based on the repeated dynamic equations is, first, constructed, and then the characteristic roots of the Jacobian matrix are solved to establish the asymptotic stability of the equilibrium points.The Jacobian matrix of the system is as follows and quadripartite evolutionary stability strategy analysis is listed in Table 2:

Equilibrium Point Equilibrium Condition Evolutionary Stability Strategy Analysis Situation
Only the super platform and the incumbent platform compete in the market, and the startup platform is unable to penetrate.The overall innovation level is low.
Undesirable situation E 2 (0, 0, 0, 1) While the government regulator strictly controls the cross-border expansion of the super platform, protecting the rights of the incumbent platform to compete, this also results in a persistent decline in the degree of innovation as a whole.
Undesirable situation When the super platform, the incumbent platform, and the startup platform can fully compete with each other, the probability of double-round monopoly is extremely low.The government regulator can achieve a long-term equilibrium without stringent regulation.
Ideal situation E 4 (0, 0, 1, 1) Although the super platform, the incumbent platform, and the startup platform can fully compete with each other, the government regulator still decrease the probability of double-round monopoly by strict regulation.
Relatively ideal situation The super platform and the incumbent platform collaborate, preventing the startup platform from entering the market.The super platform and the incumbent platform may jointly monopolize the market once the government regulator fails to strictly regulate them.
Undesirable situation Although the government regulator strictly regulates the cross-border expansion of the super platform, reducing the probability of double-round monopoly in the market, this is insufficient to spur the entry of the startup platform, and the level of innovative activity within each platform subject is still low.
Relatively undesirable situation E 7 (0, 1, 1, 0) The startup platform can still overcome the barrier to entry into the market to compete even when the super platform collaborates with the incumbent platform.At this time, there is effective market competition, and double-round monopoly is not likely to happen.The government regulator can keep the balance without tightly regulating the super platform.
Relatively ideal situation E 8 (0, 1, 1, 1) When there is effective market competition, the government regulator's strict regulatory strategy will have an impact on the market's overall degree of innovation.
Relatively undesirable situation Only the super platform and the incumbent platform compete in the market, but the super platform's competitive strength much outweighs that of the incumbent platform, increasing the probability of double-round monopoly and accelerating its tendency by the government regulator's loose regulation.
Undesirable situation The government regulatory actions can successfully limit the probability of double-round monopoly when the super platform expands with huge cross-border margins, but other market forces are also required to keep the super platform in check.

Equilibrium Point Equilibrium Condition Evolutionary Stability Strategy Analysis Situation
The super platform, the incumbent platform, and the startup platform can fully compete with each other, but at this moment the super platform has a larger competitive edge.As a result, there is still a chance that the double-round monopoly may arise, necessitating prompt regulatory action from the government regulator.
Relatively undesirable situation At this point, both effective competition patterns can be formed in the market and the risk of double-round monopoly can be avoided to a certain extent.
Relatively ideal situation When the super platform expands with a high cross-border magnitude and synergizes with the incumbent platform, the startup platform cannot enter the market to balance the competitive landscape.And when the government regulator chooses the loose regulation strategy, the probability of double-round monopoly is extremely high, which may have a significant impact on the stability of the market environment later.
Undesirable situation Compared to the situation of E 13 , the strict regulatory strategy by the government regulator can appropriately reduce the probability of double-round monopoly in the market.
Relatively undesirable situation Although the entry of the startup platform into the market may have some effect on the current competitive environment, the government regulator must play a role in it to effectively decrease the probability of double-round monopoly.
Relatively undesirable situation The probability of double-round monopoly in the market is strong due to the super platform's high levels of cross-border expansion and its synergistic collaboration with the incumbent platform.At this point, the strict regulatory strategy by the government regulator and the entry of the startup platform into the market are both required in order to limit the risk of double-round monopoly.

Relatively ideal situation
It can be seen from Table 1 that, in the evolution of the double-round monopoly of the super platform, the ideal stability situation needs to meet certain conditions.That is, the market can form a dynamic competition pattern and effectively avoid the double-round monopoly risk.Therefore, the government regulator should adopt a regulatory philosophy that both promotes development and monopolistic regulation.Five common scenarios are chosen for discussion, as the model's parameters are numerous and complex, and equilibrium points can achieve the stable state when they match specific conditions.
At this point, the benefits and innovation incentive subsidies of the startup platform entering the market are higher than the costs; therefore, the startup platform adopts the strategy of entering the market.The net benefits of the incumbent platform choosing to collaborate are lower than the innovation impact of the startup platform on it, so the incumbent platform chooses the non-collaboration strategy.The gains obtained by the super platform choosing high cross-border-magnitude expansion are lower and do not exceed the sum of the cross-border cost, penalty cost, and innovation shock effect, which may be due to the higher innovation shock generated by the startup platform to the super platform or the higher competitive ability of the incumbent platform in that market.In this case, the super platform, the incumbent platform, and the startup platform can fully compete with each other, and the super platform cannot easily form double-round monopoly, so the government regulator can reach a long-term equilibrium without strict regulation.In the actual environment, this strategy combination is the ideal strategy combination.
Case 2: ) is asymptotically stable, and the corresponding evolutionary stability strategy combination is {low cross-border margin, synergistic collaboration, market entry, strict regulation}.
Compared with case 1, the incumbent platform actively pursues the synergistic strategy since it has a bigger net advantage in choosing to synergize.Although the incumbent platform's synergistic relationship with the super platform will unavoidably make it more challenging for the startup platform to enter the market, the government regulator's strict regulation strategy will have an impact on the market's competitive structure, making it possible for the startup platform to enter the market smoothly.The government regulator typically continues to strictly regulate the super platform since the penalty income is larger than its regulation cost and loss.At this point, the super platform can only maintain its trend of low cross-border expansion.In this case, double-round monopoly is less likely to arise, notwithstanding the general competitive dynamics in the market environment.However, if the government regulator over-regulates the market, all platforms will experience losses in innovation efficiency, with the super platform suffering the most immediately and the incumbent and the startup platform suffering one after the other in the long run.On the whole, the market's innovative activity will unavoidably decrease as a result of excessive regulation.This strategy combination is not ideal in a real-world setting.
The super platform gradually grows its cross-border magnitude as long as the benefits of its high cross-border-magnitude expansion outweigh the total of its expansion costs and penalty costs.The living spaces of both the incumbent platform and the startup platform are gradually reduced in the market as the super platform expands at a high cross-border rate.Even if the incumbent platform chooses to cooperate with the super platform, the benefit of cooperation is lower than the cost, so the incumbent platform can only passively choose the non-cooperative cooperation strategy.Similarly, even if the startup platform has certain innovative advantages, it tends to forego entering the market or remain on the sidelines because doing so carries a high risk.The situation of negative regulation results when the benefits of stringent government regulation outweigh its costs.In this case, due to the high likelihood of double-round monopoly developing, each platform's decision-making will exacerbate the monopolistic trend.At this point, the double-round monopoly will have a significant impact on the efficient operation of the market and society if the government regulator does not adequately and promptly limit the expansion of the super platform.This strategy combination is undesirable in a real-world setting.
Compared with case 3, the incumbent platform actively collaborates with the super platform when the net benefit of choosing to collaborate is greater than zero.The synergy between the super platform and the incumbent platform raises the entry risk for the startup platform, and the government regulator's inaction leaves the startup platform's competitive rights unprotected.Since only the super platform and the incumbent platform exist in the market, the startup platform decides not to enter it.Although the synergistic strategy will make the incumbent platform gain more market share, as the competitive power of the super platform gradually expands, the incumbent platform's interest space is likely to be compressed, and it can even face the risk of the platform being acquired or merged.In case 3 and case 4, a double-round monopoly is highly likely to occur, and the government regulator's loose regulation would only accelerate the double-round monopoly's evolution.This strategy combination is undesirable in a real-world setting, so the government regulator should monitor the competition dynamics in order to prevent these cases from reaching stability.
Case 5: ) is asymptotically stable, and the corresponding evolutionary stability strategy combination is {high cross-border margin, synergistic collaboration, market entry, strict regulation}.
Compared with case 4, the startup platform chooses the market entry strategy because the net benefit is larger than zero and it can obtain extra market efficiency gains from a strict regulation strategy of the government regulator.The government regulator starts to strictly regulate the super platform that expands at high rates and gradually increase the amount of penalty.Even though the amount of penalty in a realistic scenario is typically much lower than the super platform's cross-border gain, it can still have an alarming effect on the super platform.As the super platform develops and becomes governed by the government, it gradually begins to focus on both reputation and credibility management.In this case, even though the super platform, the incumbent platform, and the startup platform can compete with each other, the super platform's dominance in the initial market will continue to affect the competitive environment, and there is still a possibility of double-round monopoly developing in the existing market.Therefore, the government regulator must continue to macro-regulate the market, and the level of that regulation must be continually changed in light of actual platform competition.

The Numerical Simulation
In order to verify the validity of the above analysis results and to more intuitively demonstrate the main parameters' influence on the evolutionary process, this study uses Matlab R2022a software to conduct the numerical simulation.According to the model parameter settings, the initial values of each parameter are assigned as follows: Based on the initial values, four arrays are selected to analyze the influence of the main parameters on the evolutionary process according to the stability point constraints.

The Impact of the Regulatory Strength of the Government Regulator
In order to explore the impact of the government regulator's regulation strength on the strategy evolution of all parties, array I is selected, setting r 2 = 0.3 and r 2 = 0.7 for simulation, respectively.The results are shown in Figure 2. Figure 2 illustrates that the regulatory strength has a significant impact on how the super platform's decision develops.When r = 0.3, both the super platform and the government regulator's strategic choices show oscillations.The super platform continuously adjusts its willingness to expand throughout this process in accordance with the govern- Figure 2 illustrates that the regulatory strength has a significant impact on how the super platform's decision develops.When r 2 = 0.3, both the super platform and the government regulator's strategic choices show oscillations.The super platform continuously adjusts its willingness to expand throughout this process in accordance with the government regulator's regulatory strength.When the likelihood of strict regulation by the government regulator is higher, the super platform also reduces the possibility of high cross-border expansion.As the government regulator gradually slackens off on regulation, the super platform quickly seizes the regulation slack period in order to expand, so double-round monopoly in the market is highly uncertain.When r 2 = 0.7, the government regulator's regulation is stronger and remains strict, so the super platform could only lower the likelihood of cross-border expansion to a minimal level and make it stable at that point, so the double-round monopoly is under control.
To discuss the regulation strength more sensibly, we continue to investigate the impact of the likelihood of the government regulator over-regulating.We set up r 4 = 0.1 and r 4 = 0.5 for simulation, respectively.The results are shown in Figure 3. ment regulator's regulatory strength.When the likelihood of strict regulation by the government regulator is higher, the super platform also reduces the possibility of high crossborder expansion.As the government regulator gradually slackens off on regulation, the super platform quickly seizes the regulation slack period in order to expand, so doubleround monopoly in the market is highly uncertain.When r = 0.7, the government regulator's regulation is stronger and remains strict, so the super platform could only lower the likelihood of cross-border expansion to a minimal level and make it stable at that point, so the double-round monopoly is under control.
To discuss the regulation strength more sensibly, we continue to investigate the impact of the likelihood of the government regulator over-regulating.We set up r = 0.1 and r = 0.5 for simulation, respectively.The results are shown in Figure 3. Figure 3 demonstrates that the higher the probability of the government regulator over-regulating, the more stable the probability of its decision to choose the strict regulation strategy is, at 1.The likelihood that the super platform will decide to choose a high level of cross-border expansion does not decline as the likelihood of over-regulation rises, but instead progressively climbs to a stable state.However, the decision probability is always less than 1.A possible reason for this is that the super platform can only increase the cross-border revenue by increasing the cross-border magnitude, which will compensate for the loss of additional costs brought on by over-regulation.For instance, despite the global anti-trust boom, the expansion trend of a number of American technology giants has not slowed down; rather, their market capitalization has increased more quickly and broken records.This shows that there is still a possibility of double-round monopoly formation in the market when the probability of over-regulation is high.The aforementioned findings demonstrate that the government regulator's strict regulatory strategy has a beneficial impact on double-round monopoly prevention.However, this regulatory strategy can only be effective to a fair extent of regulation, reducing the likelihood of double-round monopoly while controlling the risk.

The Impact of Cross-Border Magnitude of the Super Platform
In order to explore the impact of the super platform's cross-border magnitude on the strategy evolution of all parties, array II is selected, setting r 1 = 0.2 and r 1 = 0.7 for simulation, respectively.The results are shown in Figure 4.
regulator.(b) High likelihood of over-regulation by government regulator.Figure 3 demonstrates that the higher the probability of the government regulator over-regulating, the more stable the probability of its decision to choose the strict regulation strategy is, at 1.The likelihood that the super platform will decide to choose a high level of cross-border expansion does not decline as the likelihood of over-regulation rises, but instead progressively climbs to a stable state.However, the decision probability is always less than 1.A possible reason for this is that the super platform can only increase the cross-border revenue by increasing the cross-border magnitude, which will compensate for the loss of additional costs brought on by over-regulation.For instance, despite the global anti-trust boom, the expansion trend of a number of American technology giants has not slowed down; rather, their market capitalization has increased more quickly and broken records.This shows that there is still a possibility of double-round monopoly formation in the market when the probability of over-regulation is high.The aforementioned findings demonstrate that the government regulator's strict regulatory strategy has a beneficial impact on double-round monopoly prevention.However, this regulatory strategy can only be effective to a fair extent of regulation, reducing the likelihood of double-round monopoly while controlling the risk.

The Impact of Cross-Border Magnitude of the Super Platform
In order to explore the impact of the super platform's cross-border magnitude on the strategy evolution of all parties, array II is selected, setting r 1 = 0.2 and r 1 = 0.7 for simulation, respectively.The results are shown in Figure 4. Figure 4 shows that, when r 1 = 0.2, the probability of double-round monopoly is low, and the government regulator tends not to strictly regulate in order to ensure a reasonable allocation of governance resources at the time.When r 1 = 0.7, as the super platform's cross-border magnitude increases, the government regulator gradually reduces the transition rate of choosing a loose regulatory strategy.The market's competitive environment has undergone major change at this time.The super platform and the incumbent platform share the market share jointly after the startup platform leaves the market, but the super platform holds a more advantageous competitive position.Therefore, the likelihood of a double-round monopoly in the market is high at this time, which is consistent with the analysis in case 4. It is clear that the evolution of double-round monopoly is directly catalyzed by the growth in the super platform's cross-border magnitude.Figure 4 shows that, when r 1 = 0.2, the probability of double-round monopoly is low, and the government regulator tends not to strictly regulate in order to ensure a reasonable allocation of governance resources at the time.When r 1 = 0.7, as the super platform's crossborder magnitude increases, the government regulator gradually reduces the transition rate of choosing a loose regulatory strategy.The market's competitive environment has undergone major change at this time.The super platform and the incumbent platform share the market share jointly after the startup platform leaves the market, but the super platform holds a more advantageous competitive position.Therefore, the likelihood of a double-round monopoly in the market is high at this time, which is consistent with the analysis in case 4. It is clear that the evolution of double-round monopoly is directly catalyzed by the growth in the super platform's cross-border magnitude.

The Impact of Innovation Incentive Subsidy for the Startup Platform
In order to explore the impact of innovation incentive subsidies for the startup platform on the strategy evolution of all parties, array III is selected, setting S 1 = 10 and S 1 = 20 for simulation, respectively The results are shown in Figure 5. Figure 5 shows that an innovation incentive subsidy can not only balance the gap between the startup platform's innovation input and revenue, increasing the startup platform's willingness to enter the market, but can also change the super platform's and the government regulator's strategy stability.When s 1 = 10, the super platform is steady with low cross-border-magnitude expansion, and the government regulator is stable with loose regulation.When s 1 = 20, there is an oscillation in the government regulator's and Figure 5 shows that an innovation incentive subsidy can not only balance the gap between the startup platform's innovation input and revenue, increasing the startup platform's willingness to enter the market, but can also change the super platform's and the government regulator's strategy stability.When s 1 = 10, the super platform is steady with low cross-border-magnitude expansion, and the government regulator is stable with loose regulation.When s 1 = 20, there is an oscillation in the government regulator's and the super platform's strategic decisions.At this point, the startup platform's competitive potential increases with the amount of the innovation incentive subsidy it receives.Therefore, the super platform could only increase its cross-border magnitude in order to maintain its dominant position.However, as the probability of the government regulator choosing strict regulation has gradually increased, the competitive ecology of the market has been altered, which forces the super platform to decrease its cross-border magnitude.The development rate of double-round monopoly in the market is effectively controlled by the government regulator, which provides innovation incentives for the startup platform to enter the market.

The Impact of Synergy Risk Cost of the Incumbent Platform
In order to explore the impact of the synergy risk cost of the incumbent platform on the strategy evolution of all parties, array IV is selected, setting C 3 = 5 and C 3 = 15 for simulation, respectively.The results are shown in Figure 6. the super platform's strategic decisions.At this point, the startup platform's competitive potential increases with the amount of the innovation incentive subsidy it receives.Therefore, the super platform could only increase its cross-border magnitude in order to maintain its dominant position.However, as the probability of the government regulator choosing strict regulation has gradually increased, the competitive ecology of the market has been altered, which forces the super platform to decrease its cross-border magnitude.
The development rate of double-round monopoly in the market is effectively controlled by the government regulator, which provides innovation incentives for the startup platform to enter the market.

The Impact of Synergy Risk Cost of the Incumbent Platform
In order to explore the impact of the synergy risk cost of the incumbent platform on the strategy evolution of all parties, array Ⅳ is selected, setting C 3 = 5 and C 3 = 15 for simulation, respectively.The results are shown in Figure 6.As demonstrated in Figure 6, as the synergy risk cost of the incumbent platform rises from C 3 = 5 to C 3 = 15, the incumbent platform's readiness to cooperate gradually decreases until it stabilizes in a non-cooperative condition.In actuality, the super platform's high magnitude expansion is the main source of the incumbent platform's risk cost.The stronger the super platform's potential monopoly power, the greater the risk of the incumbent platform in collaborating.The evolution rate of the other subjects' strategies is As demonstrated in Figure 6, as the synergy risk cost of the incumbent platform rises from C 3 = 5 to C 3 = 15, the incumbent platform's readiness to cooperate gradually decreases until it stabilizes in a non-cooperative condition.In actuality, the super platform's high magnitude expansion is the main source of the incumbent platform's risk cost.The stronger the super platform's potential monopoly power, the greater the risk of the incumbent platform in collaborating.The evolution rate of the other subjects' strategies is also impacted, with the startup platform's evolution rate increasing as it implements the market entry strategy, the super platform's evolution rate decreasing as it implements the high cross-border-magnitude expansion strategy, and the government regulator's evolution rate gradually increasing as it implements the strict regulation strategy.It is evident that the double-round monopoly's evolution rate is also affected by the decision parameters of the incumbent platform.As a result, the government regulator must safeguard the synergistic rights and interests of the incumbent platform, prevent speculation and unfair competition between the two parties, and limit the super platform's desire to grow by collaboration.

Discussion and Conclusions
This study focuses on the double-round monopoly of a super platform.In order to analyze double-round monopoly's evolutionary mechanism, we first build a four-party evolutionary game model of a super platform, an incumbent platform, a startup platform, and a government regulator, considering the micro-dynamic interactions among the four parties.Then, we discuss the factors influencing the evolutionary stability strategy of each party and the conditions for multiple parties to achieve stability of their strategy combination.Finally, the key parameters' impact on the double-round monopoly's evolution mechanism is examined by varying the values of the parameters, including the government regulator's regulatory strength, the super platform's cross-border magnitude, an innovation incentive subsidy for the startup platform, and the incumbent platform's synergy risk cost.As a result, the following conclusions are drawn in this study, along with some pertinent recommendations for government regulators about the regulation of double-round monopoly.

Main Conclusions
This study clarifies the influencing variables and the mechanism of the four parties' decision-making during double-round monopoly's evolution.The behavioral decision and strategy choice probabilities of each party are closely related to the evolutionary process of double-round monopoly.In addition, changes in one party's decision probability also affect those of the other parties, which can either promote or inhibit the evolution of double-round monopoly.The super platform's strategy to expand cross-border is mostly influenced by its net revenue as well as the startup platform's innovation impact and the government regulator's regulatory strength on it.The incumbent platform's strategy to synergistically cooperate is mostly influenced by the synergy cost, risk cost, synergy benefit, and innovation impact.The startup platform's strategy to enter the market is primarily influenced by its net revenue, the innovation incentive subsidy, the innovative environment loss, and the entry cost resulting from synergies between the incumbent platform and the super platform.The government regulator's strategy to strictly regulate is primarily influenced by the regulation cost, penalty revenue, efficiency loss, and regulation loss.In the evolution of the double-round monopoly, the cross-border magnitude of the super platform plays a direct catalytic role, the decision-making choices of the startup platform and the incumbent platform play a role in regulating the evolution rate, and the government regulatory department plays an important regulatory role.
The evolutionary stability of strategy combinations and their conditions are investigated in the evolution of double-round monopoly.Due to the complicated model and a wide range of factors, the evolutionary stability of strategy combinations can reach a stable state when specific pre-conditions are achieved.However, these strategy combinations might not be ideal in the actual environment.When the probability of double-round monopoly in the market environment is low, the government regulator should appropriately reduce the regulatory strength.For example, when the evolutionary stability of the strategy combination is {low cross-border margin, non-synergistic collaboration, market entry, loose regulation}, the government regulator's persistent efforts to tighten regulation will only result in innovation efficiency losses for all parties, impeding the healthy growth of the Internet platform economy market.When the probability of double-round monopoly in the market environment is high, the government regulator should promptly increase the regulatory strength.For example, when the evolutionary stability strategy combination is {high cross-border margin, non-synergistic collaboration, market exclusion, loose regula-tion}, the government regulator should increase the regulation strength and set differential penalties, then it will achieve the evolution of strict regulation in decision-making.It is obvious that the government regulator must pay attention to the adaptive transformation of regulatory methods in order to achieve the mutual coordination of promoting development and monopolistic regulation.

Recommendations
Some feasible suggestions for the government regulator to regulate double-round monopoly are provided as follows.
Firstly, the super platform is the core subject of double-round monopoly regulation.The government regulator should concentrate on the super platform's market behavior, particularly on its cross-border margin.Besides, the government regulator should carry out whole-process supervision and strengthen punishment to prevent the super platform from using the resource advantages of the original market to carry out a series of acts of unfair competition against other platform subjects, and prevent them from taking advantage of the slack regulation period of government regulatory departments to increase their cross-border range.
Secondly, the incumbent platform should act as a check and balance on the super platform.Therefore, the government regulator should concentrate on defending the competitive rights and interests of the incumbent platform, lowering the risk cost of collaborative cooperation for the incumbent platform, and preventing the super platform from restricting the effective competition of the incumbent platform through collaborative cooperation.
Thirdly, the startup platform is a crucial component of the market innovation environment.The government regulator should not only provide the startup platform with the appropriate policy inclination and innovation subsidy, but they should also protect the startup platform's competition rights to prevent the super platform from "strangling mergers and acquisitions".At the same time, in order to prevent excessive regulation from negatively affecting the innovation environment, the government regulator must strike a balance between regulation and innovation incentives.
Fourthly, in order to reduce the risk and harm associated with the double-round monopoly of the super platform, the government regulator should also well-perform dynamic monitoring of the market environment, scientifically and effectively monitor the competition dynamics of each subject, and promptly adjust the regulation strategy to realize the entire life cycle of the regulation process.

Limitations and Future Research
This study focuses on examining the theoretical underpinnings of the double-round monopoly evolutionary mechanism and building an evolutionary game model of the four parties, which examines the corresponding regulatory mechanism and analyzes the primary theoretical influences on double-round monopoly evolution.However, this study still has several limitations.
A limitation of this study is that the model construction primarily takes into account each subject's individual market behavior and designs the model parameters based on their interactions, but it neglects to consider the complex market behavior of these subjects in their actual environments (such as mergers and acquisitions, competitive collusion, etc.), as well as the influence of changes in the external environment on each subject's decisionmaking (such as the financial environment, the policy environment, etc.).In addition, the model in this study is primarily relevant to the double-round monopoly issue in China, but it can also serve as a basis for future comparisons with similar issues in other countries, enhancing the model's generalizability.
In addition, some realistic samples are not chosen for support, and there is a lack of an empirical validity test, even though this study uses the numerical simulation method to imitate the real-world issue.In order to further enhance the model settings and optimize the simulation process, future research can choose heterogeneous company data and government data from other sources.This will allow the study findings to better inform policy practice.Future research can also increase the subject's dynamics and take the subject's decision-making bias in the real world into account, improving the model's ability to explain reality.
Finally, a limitation of the evolutionary game approach is that it can solely depict the dynamics of simplified interactions portrayed by model players and it can only achieve the formulation of co-evolutionary rules.To better represent the interaction process, we plan to enhance the model's complexity in the future.By introducing different scenarios within the model, we aim to analyze a wider range of evolutionary situations.In the future, by identifying the characteristics of each party and establishing their rules of action, we intend to construct a simulation system based on complex system theory and ABM (Agent-Based Model) methodology.This will provide a more comprehensive perspective on the evolution of double-round monopoly.Additionally, the issue of strategic uncertainty has not yet been sufficiently addressed, despite the fact that the four-party evolutionary game model can partially depict realistic scenarios of double-round monopoly.We will attempt to incorporate the Markov transfer probability matrix into the construction of the evolutionary game model based on the stochastic characteristics of the evolutionary system in future research, in order to better depict the dynamic switching process of the evolutionary system under the influence of strategy uncertainty.

Figure 1 .
Figure 1.Interactions and strategy selections of multiple subjects during double-round monopoly.

Figure 1 .
Figure 1.Interactions and strategy selections of multiple subjects during double-round monopoly.
Array I: array of initial values.Array II: C 1 = 20, C 7 = 20, R 4 = 10, P 1 = 12, S 1 = 5, the other parameters are the same as array I. Array III: C 4 = 15, C 5 = 10, C 6 = 10, the other parameters are the same as array I. Array IV: C 1 = 6, R 3 = 5, the other parameters are the same as array I.

Figure 2 .
Figure 2. The impact of the government regulator's regulatory strength on the evolution of each party's strategy.(a) Low regulatory strength.(b) High regulatory strength.

Figure 2 .
Figure 2. The impact of the government regulator's regulatory strength on the evolution of each party's strategy.(a) Low regulatory strength.(b) High regulatory strength.

Figure 3 .
Figure 3.The impact of the likelihood of the government regulator over-regulating on the evolution of each party's strategy.(a) Low likelihood of over-regulation by government regulator.(b) High likelihood of over-regulation by government regulator.

Figure 4 .
Figure 4.The impact of the super platform's cross-border magnitude on the evolution of each party's strategy.(a) Low cross-border magnitude.(b) High cross-border magnitude.

Figure 4 .
Figure 4.The impact of the super platform's cross-border magnitude on the evolution of each party's strategy.(a) Low cross-border magnitude.(b) High cross-border magnitude.

Figure 5 .
Figure 5.The impact of an innovation incentive subsidy for the startup platform on the evolution of each party's strategy.(a) Low innovation incentive subsidy.(b) High innovation incentive subsidy.

Figure 5 .
Figure 5.The impact of an innovation incentive subsidy for the startup platform on the evolution of each party's strategy.(a) Low innovation incentive subsidy.(b) High innovation incentive subsidy.

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Figure 6 .
Figure 6.The impact of the incumbent platform's synergy risk cost on the evolution of each party's strategy.(a) Low synergy risk cost.(b) High synergy risk cost.

Figure 6 .
Figure 6.The impact of the incumbent platform's synergy risk cost on the evolution of each party's strategy.(a) Low synergy risk cost.(b) High synergy risk cost.

Table 1 .
The payoff matrix of the four parties.