The rapid development of the sharing economy not only has a huge impact on the economy, environment, society, government and other aspects but also further reshapes people's daily life, subverts the traditional business model and affects the existing system and mechanism by ‘destroying regulation.’ Therefore, the aspects of ‘whether’ and ‘how’ to effectively supervise and govern the sharing economy have become a global governance problem. In response to this problem, scholars from all over the world have put forward many regulatory ideas, and different regulatory models have been adopted in practice. For the regulation and governance of the sharing economy, many experts and practitioners around the world have proposed three main strategies [1,2]. First, stop and delay, i.e. to prevent and delay the development of sharing economy regulation through mandatory administrative measures to avoid possible negative problems. Policy regulations that adopt the strategy of blocking and delaying often use traditional policy norms to measure regulatory practices, which will inevitably hinder the healthy development of the new sharing economy. Therefore, many public policy scholars have criticised the government as ‘lazy government [3]. Second, modify or establish a new regulatory framework to accommodate regulation. Scholars such as Koopman believe that the public policies formulated by the government should be continuously improved so as to better coordinate with the development of the sharing economy market [4]. Peng Yue advocates reducing administrative intervention, adopting responsive regulation strategies, gradually solving various conflicts and problems caused by the sharing economy, as well as seeking a dynamic balance between supervision and regulation [5]. This strategy has been accepted by many experts, scholars and local governments. Third, to apply government threats: that is to say, the government does not adopt a clear attitude or measures but conducts supervision through relatively broad and flexible means such as acquiescence or intimidation, which is a vague regulatory strategy [6]. This strategy is often applied to the initial stage of the sharing economy, and the government adopts a waitandsee attitude, neither prohibiting new forms of sharing economy nor allowing them to be legalised. Some scholars believe that although the threat strategy is suspected of inaction, it also has certain advantages. It can not only provide regulatory space for the development of new formats but also restrain them and prevent their barbaric development [7].
Research on industrial collaborative innovation mainly focusses on hightech industries and resourcebased industries [8]. Osborne
On the basis of the theory of collaborative governance, the theory of responsive regulation and the theory of coproduction, this study draws on the existing research results to analyse how to encourage the public to participate in the collaborative regulatory governance of the sharing economy. The research mainly adopts a combination of field research and literature research, qualitative analysis and quantitative research, normative research and empirical research. Second, we use inductive and deductive methods, and in the third step, we apply game analysis. The supervision of new forms of sharing economy requires the cooperation of enterprises, government departments and the public.
First, multiple coordination and cooperative supervision and governance are the inevitable trends in the regulation of the sharing economy in the future. The sharing economy realises the separation of ownership and use rights and also makes the traditional government supervision model completely invalid. In the supervision of the sharing economy, the government's regulatory role must rely on the enterprise platform to regulate and guide users, which requires the enterprise to assume the responsibility for operation management services; users or the public can provide feedback information to the enterprise to cooperate with the management of the enterprise or to the government. Complaints and protests are used to supervise corporate violations. Therefore, the government, companies and the public have become the main stakeholders involved in sharing economic activities. The three interact, restrict each other and are indispensable to each other; they then build a multidimensional synergy with the media and social organisations, yielding the modern regulatory governance system.
Second, the current practice of collaborative supervision and governance in my country is still in its infancy. The supervision of the sharing economy involves multiple subjects, such as the government, enterprises and society. It has become the theoretical consensus of the whole society to build a multiple cooperative supervision and governance system. However, judging from the past practice of collaborative supervision and governance of the sharing economy in my country, due to the large influx of venture capital, the longterm goal of ‘making the city better’ of bicycle companies has succumbed to the pressure of capital, resulting in a pattern of collaborative supervision and governance exploration, thus resulting in a completely broken system. After the end of the ‘bike melee’, although the government and enterprises have started new explorations of cooperative supervision, they have not yet established a standardised and effective communication system and mechanism [16]. At the same time, it is always difficult for bicycle companies to find an effective profit model, and the value cocreation role of the public has always been seriously ignored, which has become an important obstacle to the construction of a collaborative supervision and governance system.
Third, the role of the public has not been valued and brought into play. In the process of supervision and governance of shared bicycles in my country, enterprises and the government have successively played the leading roles of collaborative governance at different stages, but the role of the public in collaborative governance has always been ignored; multiple collaborative cooperation is often simplified in practice as a government–enterprise dual regulatory model. In the interviews and questionnaires, we also found that due to the lack of guidance and assistance from the government or enterprises, most of the public can – at the most – consciously follow the shared bicycle code of conduct, and it is difficult to truly cooperate with enterprises or the government and participate in the supervision and maintenance of shared bicycles in action. Among them, 36.4% of the interviewees said that they basically did not participate in any supervision and maintenance activities of shared bicycles, 39.6% of the interviewees had actively participated in the supervision and maintenance of shared bicycles but did not enjoy the material rewards given by the company and 51.8% of the interviewees have actively participated in the supervision and maintenance of shared bicycles but have not been commended by the government or enterprises [17, 18]. This shows that neither the enterprise nor the government pays attention to the public's coproduction role in the supervision of shared bicycles. On the one hand, bicycle companies regard the public as a pure consumer, blindly hoping to regulate user behaviour through new technologies and systems, ignoring the role of the public as a value cocreator or coproducer of the sharing economy. On the other hand, the government is accustomed to using traditional regulatory thinking and models to supervise bicycle companies on behalf of users and the public, which tends to ignore the real needs of the public. In the supervision and governance of the sharing economy, the government, enterprises and society are all stakeholders, which cannot be ignored. Losing the participation of the public will inevitably lead to a sharp increase in the maintenance and scheduling costs of enterprises, and reduce the government's supervision performance of enterprises. It is difficult to establish a collaborative supervision and governance system. Therefore, paying attention to the role of the public as coproducers in the sharing economy, creating corresponding platforms and channels, as well as guiding the public to participate in the coconstruction, cogovernance and sharing of the sharing economy have become inevitable requirements for the realisation of collaborative supervision and governance in the future.
The main difference between the gas generator set and the cogeneration unit is that it will use the chemical substances after burning natural gas to generate electricity, improving the efficiency of the starting point [19]. There is a certain relationship between the natural gas flow used by the unit and the generated electric power during operation, which can be expressed as follows.
The benefit matrix of sharing economyrelated enterprises, local governments and the public under eight strategies
A  Business benefit 
F  Corporate punishment 
P  Probability of violation 
U  Government budget 
H  Government cost 
B  Cost of government remediation 
Q  Government oversight probability 
T  Business loss 
D  Government losses 
I  Social regulation probability 
Definition of model parameters and their meanings
Active participation, supervision, active participation  − 
− 

Active participation, supervision, passive participation  − 

Active participation, no regulation, active participation  0  − 

Active participation, no supervision, passive participation  − 
0  
Passive participation, supervision, active participation  − 
− 

Passive participation, regulation, passive participation  − 

Passive participation, no supervision, active participation  0  − 

Passive participation, no regulation, passive participation  0  0 
In order to explore the important role of public participation in cooperative supervision and to better build a multidisciplinary cooperative supervision governance system, this study continues to add the public as a subject to the game model [20]. In this case, the extended form of the tripartite game between government departments, enterprises and the public is shown in Figure 1:
Assuming that the expected revenue of sharing economyrelated companies choosing the ‘active participation’ strategy is
Assuming that the expected return of the local government choosing the ‘regulatory’ strategy is
Assuming that the expected benefit of the public choosing the ‘active participation’ strategy is
Based on the above assumptions and calculations of the model, a replication dynamic analysis is carried out on the proportion of the government adopting the ‘regulatory’ strategy and the proportion of the sharing economyrelated enterprises and the public adopting the ‘active participation’ strategy.
The replication dynamic equation of sharing economyrelated enterprises is as follows:
The replication dynamic equation of the local government is as follows:
The replication dynamic equation of the public is as follows:
In Eq. (6),
Gas boilers are different from electric boilers. Electric boilers use electricity as the main energy source, while gas boilers use natural gas as the main energy source. The operating principle is as follows: after natural gas is input to the gas boiler, the boiler will burn the natural gas. When the water temperature reaches the standard value, it will form hot water or steam, and the gas boiler will send this hot water and steam to the outside. Gas boilers not only have higher heating efficiency than electric boilers (the conversion rate even reaches 99% [21]) but also can reduce pollutant emissions, so they are more suitable than electric boilers. The relationship between the amount of natural gas consumed by the gas boiler per unit time and the output thermal power is as follows:
There are two kinds of coordinates in the nonlinear power flow equation of the power system (the rectangular coordinates and the polar coordinates), but the active power
In Eq. (14),
Gas pipelines, compressors, gas sources and gas loads together form a natural gas system [22]. Gas sources are classified into gas fields and shale gas. Gas pipelines are responsible for natural gas transportation. Longdistance gas pipelines are usually buried underground. The compressor exists to make up for the pressure loss of natural gas during the transmission process, and it plays the role of boosting or reducing pressure at different positions. The network model of natural gas mainly considers the pipeline flow model and the compressor model, and the pipeline flow
In the formula,
In the expressions,
The calculation formula of the flow of the injected gas
In the formula,
At time
At time
According to the public replication dynamic expression in Eq. (17) constructed above, the game strategies adopted by the public are as follows:
At time
At time
At time
At time
From the above analysis, it can be seen that adjusting the relevant parameters makes the tripartite game between enterprises, local governments and the public in the sharing economy evolve to a relatively ideal stable state. Take (active participation, supervision, active participation) as an example: when the value of
At time
At time
Combining the above formulas, the replication dynamic system of sharing economyrelated enterprises, local governments and the public is obtained as follows:
The Jacobian matrix of the system can be obtained as follows:
According to the system solution, the local equilibrium points are (0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 0, 1), (1), 1, 0), (0, 1, 1) and (1, 1, 1). In evolutionary game theory, the equilibrium point where all the eigenvalues of the Jacobian matrix are nonperiodic is the evolutionary stable point of the system.
The Jacobian matrix of the system at the equilibrium point (1, 1, 1) is as follows:
From the above formula, the eigenvalues of the Jacobian matrix are the following:
When the local government chooses the supervision strategy, and the sharing economyrelated enterprises and the public choose the active participation strategy, the equilibrium point (1, 1, 1) of the system is the equilibrium point of evolution and stability, and
In the development process of an evolutionary game event, many influencing factors affect the strategic choices of enterprises, local governments and the public in the sharing economy. The change of a certain influencing factor will lead to changes in the strategic choice of at least one participant and the enterprise. The behaviour strategies of the government and the public are related, which will lead to continuous adjustment of the strategic choices of the corporate entity, government and the public. From the direction of maximising the interests of all game subjects, we can analyse the strategy (1, 1, 1). It is a situation that is beneficial to all three types of participants. Due to the needs of the development of circular economy, my country has strictly restricted the import of solid waste, and industries related to the sharing economy want to make up for the gap in supply and stabilise the price of upstream raw materials, and they urgently need to actively conduct technical supervision. The supervision of industrial operation management with the help of a new generation of Internet technology is one of the development trends. At the same time, in order to provide public recognition, local governments will actively participate in collaborative supervision by increasing financial subsidies to enterprises related to the sharing economy. Many cities across the country have begun to establish a sharing economic system. Therefore, the equilibrium point (1, 1, 1) is the most ideal development state, and its corresponding evolutionary game strategy is (active participation, supervision, active participation).
Through the above analysis, we can see that under the condition of bounded rationality, the most suitable strategy choice for the evolutionary game enterprise, government and the public is (active participation, supervision, active participation).
In general, the government's subsidy policy has a significant impact on the evolution of the public's strategy, and the local government's subsidy intensity is proportional to the speed at which sharing economyrelated enterprises and the public converge to actively participate in the collaborative supervision strategy. Different initial probabilities also have a clear impact on the convergence speed of government, enterprises and the public. When the local government subsidy is large, although the initial probability of public participation in collaborative management is low, it will eventually tend to actively participate in collaborative management.
It can be seen from Figure 3 that changes in the social benefits of local governments have a significant impact on the strategic evolution of the corporate, government and the public, and they are positively correlated.
From the overall analysis of the data, it can be found that the choice of strategies between government, enterprises and the public is interactive and has a certain correlation mechanism.
However, because the actual economic strength and membership size of each participant are different, when a parameter change causes a change in the strategy choice of one of the participants, the evolutionary stable strategy will only be closer to (1, 1, 1), and the public will not respond to the parameter change. The sensitivity is higher, and different initial probabilities will affect the convergence speed of the participants and are positively correlated.
First of all, the strategic choices between the game subjects in the game model will affect each other, and the threeparty relationship is developed in coordination, i.e. the evolutionary stable strategy is (active participation, supervision, active participation).
To encourage and attract the public to actively participate in collaborative supervision, the government can spread the relevant policies of the sharing economy through multiple channels, so that the public can better participate in the sharing economy system. Companies should also vigorously promote their own sharing economy platforms to encourage public participation. Therefore, local governments should take the lead, be led by enterprises, and the public should cooperate and assist local governments and enterprises to participate in the coordinated supervision of the sharing economy.
The benefit matrix of sharing economyrelated enterprises, local governments and the public under eight strategies
A  Business benefit 
F  Corporate punishment 
P  Probability of violation 
U  Government budget 
H  Government cost 
B  Cost of government remediation 
Q  Government oversight probability 
T  Business loss 
D  Government losses 
I  Social regulation probability 
Definition of model parameters and their meanings
Active participation, supervision, active participation  − 
− 

Active participation, supervision, passive participation  − 

Active participation, no regulation, active participation  0  − 

Active participation, no supervision, passive participation  − 
0  
Passive participation, supervision, active participation  − 
− 

Passive participation, regulation, passive participation  − 

Passive participation, no supervision, active participation  0  − 

Passive participation, no regulation, passive participation  0  0 
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digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edgebased finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of ecommerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on ecommerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long shortterm memory networks A study of local smoothnessinformed convolutional neural network models for image inpainting Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Crossborder Ecommerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRABiLSTMCRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model ChineseEnglish Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithmbased congestion control optimisation for mobile data network