Evolutionary Game Analysis of Participations in Museum Public–Private Partnership Projects Based on System Dynamics

Abstract

In China, the number of public–private partnership (PPP) projects in the museum field is relatively small, and the success rate is quite low, which highlights the drawback of an insufficient depth of overall promotion. To break through this dilemma, this paper objectively analyzes the interest demands and behavior-oriented conflicts between two stakeholders and constructs a game model based on limited rationality for the evolution of participation behaviors in museum PPP projects. In addition, different levels of government incentives are introduced along with the modeling and simulation of system dynamics (SD) to further clarify the dynamic evolution process of participants’ behavioral strategies. There are multiple complex scenarios associated with the system evolution according to the study. In a museum PPP project, the strategic choices of social forces and museums may have a mutual effect under different scenarios. Under the premise of guaranteeing the benefits of both sides of the game, there are two evolutionary stability strategies. Incentives from the government have a significant impact on the evolution of the game system at different stages of implementation. This paper hopes to provide reference and guidance to facilitate the formation of an appropriate action mechanism for participants and then ensure the sustainable promotion and development of the projects.

1. Introduction

Public–private partnership (PPP) projects have increasingly become a practical and important method for developing cultural spaces [1,2]. Essentially, it is an inter-organizational cooperation mechanism through which the public sector and one or more social forces (especially enterprises) jointly commit themselves to further align resources to provide public services [3,4]. One of these is the promotion and application of PPP project models in the field of public infrastructure, which plays a crucial role in promoting museum project construction and improving museum governance and capabilities [5]. Since 2005, the State Council has issued relevant regulations on cultural heritage and museums. As stated in Articles 4 and 5 of the Regulation on Museums, “The state encourages the formation of museums by enterprises, public institutions, social organizations, citizens, and other social forces in accordance with the law”. “The state encourages the establishment of public welfare funds to provide funds to museums and encourages museums to raise funds by multiple means to promote their own development”. Article 6 of the “Regulation” once again emphasizes, “Museums shall enjoy tax preferences in accordance with the law” [6]. It is crucial to encourage all social forces to invest in museum practices through cooperation, joint ventures, donations, and bidding. Companies, museum associations, and other organizations have gradually become involved in museum construction, exhibitions, cultural products, and education under the guidance of national policies in recent years. Using the PPP model, social forces and government departments can collaborate on the financing, construction, and operation of projects, which not only optimizes the project’s financing structure but also relieves the government’s financial burden after the pandemic. Additionally, the unique management system for social forces and the use of sophisticated technology can improve the project’s performance [7,8,9]. There are, however, many challenges that PPP projects often face, such as conflicting relationships between collaborators and project users [10], the noncommitment of some collaborators [11,12], and opposition from stakeholders [13,14]. All of the above factors can lead to the failure of PPP projects [15,16,17]. Developing an understanding of the motives and reasons for the behavior of the subjects involved in a PPP project is one of the keys to its success.

The cultural sector is currently less involved in public welfare undertakings than the fields of education, health, and environmental protection, and a multi-participation mechanism of social forces has not yet been established. There are numerous reasons for this, including a lack of alignment between the relevant laws, regulations, and policies, which results in no apparent incentive effect [18,19]. The participation of social forces in PPP projects within the museum sector is also a manifestation of structural problems, such as a lack of self-awareness and autonomy, as well as an absence of mature institutional guarantees. Compensation is one of the most compelling reasons for the limited participation of social forces, whose implementation and refinement pose significant challenges [20]. The use of practical cultural relic compensation mechanisms can provide specific compensation and rewards for individuals and organizations that actively support and participate in museum practices. To encourage social forces into the museum field, specific regulations regarding the fiscal and taxation system should be enacted, and additional tax reductions and exemptions should be made available. In particular, there are some representative fields where it would be more beneficial to develop and promote some operational tax relief measures rather than simple principles. In the case of museum collection donations involving the rewards of corporate donations, in addition to the prestigious benefits, there should be a straightforward tax deduction so that the enterprise is able to benefit substantively. Moreover, there are concerns relating to intellectual property rights and profit distribution in the cultural and creative industries which need to be addressed, and a transparent reward scheme may be established [21]. It is true that PPP projects in the museum sector have received some attention. While their application is limited by the concept, they are still in the early stages of implementation. In this regard, it is essential to have an in-depth discussion of the promotion strategies of each participant in the museum PPP project, as well as their relationship game.

An incentive mechanism for public participation in the museum PPP project is a multifaceted process that includes process control, attention to results, and planning for the future. There are few studies examining social forces’ participation in museums. Liu discussed the feasibility of social forces participating in the operation of museums and obtained countermeasures for reforming the mode of the operation of museums [22]. With the use of private museums in Zhejiang Province as a case study and drawing on the valuable experience of managing foreign museums, Wu investigated the necessity and feasibility of absorbing social forces and developing cultural private non-enterprises within China’s public welfare cultural undertakings as well as the policy environment and development status. This led to a framework for absorbing social forces and developing non-profit cultural initiatives [23]. Zheng used the development and construction of the “Famous Hao River” project of the Hao River circle museum group led by the Nantong Museum as an example of how private capital can participate in public cultural activities. In addition, the Ningbo Shili Hongzhuang Museum is used as another example to illustrate the cooperative model of “national assistance and private management” [24]. Based on the extensive practice of social forces’ participation in public cultural services such as museums, Ma summarized the participating subjects, participation methods, and contents of social forces in public cultural services and provided the theoretical basis and reference for subsequent research on promoting and mobilizing social forces’ continuous participation in public cultural services [25]. Li and Gu pointed out that the vigorous development of museums not only requires the active participation of social forces, but also requires the introduction of advanced and mature management concepts. With the help of professional management thinking and the means to facilitate cooperation between government and enterprises, the interests of both sides can be achieved [26].

A significant change has occurred in the mode of providing public cultural services as the role of social forces has changed to that of “collaborators”. According to Wu, social forces can be divided into three categories in supply practice: producers of public cultural services, fund providers, and supervisors [27]. In particular, enterprises hold a significant amount of power. It has been pointed out by many scholars that the PPP cooperation model will only be successful if all the participants are committed and loyal to each other [14]. In any other case, such a partnership would be extremely costly [11]. Additionally, the project’s performance mechanism will also be ineffective [28]. There will be a lack of interest on the part of stakeholders in cooperating with the PPP project or actively maintaining their own relationships with it [29]. Given the high failure rate of PPP projects in the public cultural sector worldwide [30,31,32,33], understanding the drivers of stakeholder commitment and the forms of incentive compensation is a fundamental foundation, such as considering government compensation as a real option [34,35], constructing dynamic compensation models [36], improving social force benefits [37], government guarantees, revenue guarantees, and government risk compensation [38]. Based on this, analyzing the mode of participation of social forces in the operation of PPP projects has also become an important issue [39]. Since the benefits and risks of different PPP projects differ, the same incentive compensation mechanism and operation mode may not apply to all projects, so the public sector may be required to design different mechanisms. Different incentive compensation mechanisms and operating models determine not only their effects on social forces but also the size of the public sector’s cooperation costs. Game theory has been used by scholars to study the relevant incentive mechanism [40,41], compensation mechanisms [36], reward models [42,43], income distribution [44], etc., between the public and private sectors in PPP projects. In light of this, this paper examines the current situation of domestic and international PPP projects in the museum field by focusing on museum service practices. Additionally, the evolutionary game model combined with the system dynamics modeling simulation is used to reveal the behavioral strategy evolution process of museums and social forces under different levels of government incentives so as to provide a scientific incentive mechanism for museum PPP projects and to provide a theoretical basis and practical guidance for further achieving the efficient and sustainable participation of social forces.

2. Conceptual Framework

2.1. Operational Modes for Museum PPP Projects

Museum PPP projects in China are still in their infancy. According to the Statistical Yearbook of Chinese Culture, Cultural Relics, and Tourism 2021, public museums nationwide generated a total revenue of CNY 26.786 billion in 2020. It is estimated that 89.8% of the total revenue is derived from financial appropriation, whereas only 4.2% is derived from social revenue (operating income, income from affiliated units, and other income) [45]. Taking a closer look at the financial allocation and the proportion of social revenue in the total revenue may provide insight into the context of social forces, especially the lack of enthusiasm of businesses to participate in the provision of museum services. In China, the involvement of social forces in the museum field is primarily concentrated on the construction of large museums, which are long-term projects with high risks; for instance, the Sarawusu National Archaeological Park Site Museum, Ordos, Inner Mongolia Autonomous Region, Baotou New Metropolitan Area Urban Exhibition and Museum, Cultural and Art Center PPP Project, Shenzhen Jewelry Museum of Guangdong Province, the PPP project of cultural and sports venues in Tongchuan, Shaanxi Province, and the PPP project of “GLAM” in Hezhou, Guangxi Province, etc. [46]. Generally, PPP projects are of a large scale and are associated with multiple cultural institutions (such as libraries, archives, cultural centers, etc.). However, there are relatively few PPP projects, the success rate is low, or they remain stagnant. On the other hand, internationally, PPP projects in the museum field have been underway since relatively early times, and a number of paths have been developed that can serve as references. The following categories summarize the forms of public participation in international museum communities and the models of public–private partnerships.

2.1.1. The Museum’s “Outsourcing” Model

A number of modes are currently in use, including Operate–Transfer (OT), Build–Operate–Transfer (BOT), Build–Transfer–Operate (BTO), Rehabilitate–Operate–Transfer (ROT), and Build–Own–Operate (BOO). For example, the Taiwan Museum of Marine Biology and Aquarium, which opened in February 2000, played a leading role in promoting civic engagement in public construction. By entrusting the aquarium department of exhibitions and public space to private companies, the government has increased its financial resources. This is the first case in Taiwan in which public museums are operated by private companies and is considered a benchmark for this form of operation. In terms of finance, apart from receiving a fixed amount of “royalty”, the public sector will also share operating profits with the private sector in charge of operations.

2.1.2. Museum IP Authorization for Cultural and Creative Industries

Specifically, authorized brokers are responsible for managing all aspects of cultural authorization on behalf of museums, including communication, negotiation, signing, supervision, and collection of royalties. In the case of the Guggenheim Museum, for example, 360 ep is in charge of the museum’s authorization. The authorization process can also be performed using a centralized authorization platform. In most cases, nonprofit organizations represent museums that share membership to exercise the authorization function and manage the authorization process effectively. For example, the Réunion des Musées Nationaux received authorization for 32 state-owned museums in France and was responsible for their external authorization and management.

2.1.3. Tax System Reform

France, for example, offers tax incentives for museum patrons while simultaneously promoting a unique mechanism to replace taxes with in-kind payments. It is possible, for instance, to substitute art donations for cash payments in inheritance taxes. The French government has established an inspection and approval committee in conjunction with this mechanism. This is one way to enrich the national public collections, and it was precisely due to this policy that the Picasso Museum was established in Paris. Taiwan also offers tax concessions, including business taxes, entertainment taxes (Culture and the Arts Reward and Promotion Act and its sub-laws Measures for Reduction and Exemption of Business Tax and Entertainment Tax for Cultural and Arts Enterprises), land and housing tax (Culture and the Arts Reward and Promotion Act), and income tax (Applicable Standards for Income Tax Exemption for Education, Culture, Public Welfare and Charity Organizations). An individual donation is credited with 20%, whereas a corporate donation is credited with 10%. Exhibitors are exempt from commodity tax (Commodity Tax Act), and donation receipts are exempt from stamp duty (Stamp Act), estate duty, and gift tax.

2.1.4. Build Foundations and Establish Think Tanks

Support the establishment of public welfare funds for museums, as in South Korea. As part of this effort, the fund will offer customized savings products based on a portion of its income, as well as establish preferential policies for predeposits. Meanwhile, through the “Arts Grants” and “Art Aids Matching Funds” offered by the Arts Council Korea–Korea Mecenat Association, backbone enterprises have become more frequent participants in museums and art galleries [47].

2.1.5. Develop an Efficient Donation Funding System That Gives Priority to Social Donors

The United Kingdom’s Gift Aid mechanism provides that museums under the jurisdiction of the Department for Digital, Culture, Media, and Sport (hereafter referred to as DCMS) are all classified as charitable organizations and may be exempt from taxation. This is a highly innovative mechanism by which the British government promotes charitable donations. Gift Aid was activated once the UK taxpayer had applied and signed the application form. Whenever a taxpayer donates GBP 100, the UK government increases the total amount donated by 25%, that is, the value of charitable donations is increased by 25% without the taxpayer incurring any additional fees. Accordingly, if a taxpayer donates GBP 100 to a museum, the museum will actually receive GBP 125 from the government, of which GBP 25 is obtained through Gift Aid. Through Gift Aid, charitable organizations received a total of GBP 1.4 billion in donations from 2020 to 2021. If museums and their affiliated companies are registered as charitable organizations, admission fees, membership fees, and other donations paid by visitors can be benefited through the Gift Aid mechanism, which can increase the museum’s revenue as well as increase the value of the donation of taxpayers, whose rates start at 20% and go up to 40% and 45%. In addition to increasing museum funds, this tax mechanism can also encourage companies and individuals to make contributions. Applicants for the Gift Aid mechanism must be British taxpayers and must apply to join, as well as sign the appropriate Gift Aid form provided by the donor.

2.2. Participants in Museum PPP Projects

The government, museums, and social forces constitute the main participants in museum PPP projects. A number of policy guarantees are provided by the government to museums and social forces. By offering projects and activities, the museums seek to engage various social forces in their realization and efficient operation. As is the case with PPP projects, museums conduct the performance evaluation and provide their resources, such as collections and research, mainly covering various levels of museums. In 2021, there were 6183 Chinese museums with 90% offering free admission. Among them, a number of small and medium-sized museums have embarked on PPP projects in order to develop their own museum practices. Small and medium-sized museums lack financial support and technological capabilities; therefore, they are more likely to attract social forces to assist with their healthy development.

Social forces attempt to act as a “co-operator” on an all-round basis. There is no dependency relationship with the government, and there is also a certain degree of autonomy in production, which mainly consists of the following:

One is the producer and executor of museum practices. Social forces, particularly enterprises, can leverage their own professional advantages for the production of museum services.

The second is the source of funding. Through self-funding, social forces can meet the funding needs of museums. In particular, enterprises with a strong fundraising capability may be able to obtain certain income-sharing rights through capital investments. A market-oriented operation can not only increase economic benefits but also relieve governmental and museum financial pressures.

Third, the supervisor of the museum practices mainly refers to the public. The public awareness and appeal feedback mechanisms have been continuously improved, enabling the public to better supervise museum services and ultimately enhance their effectiveness.

As a public service product of museums, museum PPP projects are initiated and promoted by museums in partnership with social forces (primarily enterprise capital) under the guidance, authorization, and incentives of the government. As is the case with PPP projects, museums conduct a performance evaluation and pay the corresponding service fee, while the social forces that won the bid invest in the project by providing technology or funds. Ultimately, the PPP project is intended to result in a win–win situation for both parties. Meanwhile, financial institutions and project consumers (individuals or institutions) provide financial support; insurance companies assume risks; museum departments or research institutions provide professional services, operations, and evaluation; while the public, as well as professional agencies (such as museum associations), perform supervision. In all respects, they “performed their duties” and contributed to the smooth progress of the museum’s PPP project. Therefore, effective mechanisms for risk sharing, benefit sharing, supervision, and performance evaluation are crucial to the success of museum PPP projects (Figure 1).

3. Research Methodology

Museum PPP projects are particularly vulnerable to investment uncertainty, construction difficulties, high communication and cooperation costs, as well as long revenue cycles. The museums and social forces are key participants in museum PPP projects. In this study, collaboration between museums and social forces is conceptualized as a dynamic game process. Thus, this paper develops an evolutionary game model to further explore the strategic decisions and adjustments made by social forces and museums.

3.1. Model Assumptions and Descriptions

Assumption 1. 

A complete system exists between social forces and museums. Both parties are assumed to be boundedly rational, and their information is incompletely symmetric. It is challenging to develop an optimal strategy by making only one decision. In the course of the long franchise period, these participants continue to adapt and improve their strategies until they reach a state of evolutionary stability. Other possible effects on the game system in the PPP project are not considered during the process. Museums hope to promote cooperation with social forces. Museums have the option to choose either active or passive behaviors to encourage social forces to improve the project’s performance. Meanwhile, social forces are engaged in (effort, speculation) behaviors in museum PPP projects.

Assumption 2. 

Since both types of groups remain relatively stable, a normalized group size of 1 can be used for both types. The proportion of social forces choosing to make an effort is x, and the proportion of museums that choose active promotion is y, where x, y is both related to time t and $0\le x\le 1$, $0\le y\le 1$.

Assumption 3. 

Social forces who speculate to participate in a PPP project do not incur any costs, and the basic benefits they receive at this point are$R_s$. An additional revenue streamed from a “free rider” is$\mathsf{\Delta }{R}_s$. By passively promoting the PPP project, museums do not incur the expense of participating, and the basic benefit received is$R_m$. The additional revenue generated by the “free rider” is$\mathsf{\Delta }{R}_m$.

Assumption 4. 

A PPP project will be promoted to a certain extent if only social forces make an effort, and the benefits of these forces are:$(1+{\beta }_0)R_s-C_s$, where${\beta }_0$is the ratio of increased investment income of social forces and$C_s$is the operating cost of social forces. Only if museums actively promote the project will museums’ revenue be$(1+{\gamma }_0)R_m-C_m$, where${\gamma }_0$is the ratio of increased investment income of museums and$C_m$is the operating cost of museums.

Assumption 5. 

If social forces and museums choose positive behaviors at the same time, the PPP project is effectively promoted, and the benefits of social forces are$(1+{\beta }_1)R_s-C_s$, where${\beta }_1$is the ratio of increased investment income of social forces${\beta }_1>{\beta }_0$; the revenue of museums is$(1+{\gamma }_1)R_m-C_m$, where${\gamma }_1$is museums’ investment income increase rate${\gamma }_1>{\gamma }_0$.

Assumption 6. 

To encourage social forces to establish museums, guide public resources toward the development of museums, enrich museum categories, and optimize the system layout, the local government has created a social forces–museums support fund, which contributes to reducing the operating costs associated with museum PPP projects. Assume that the reduction ratio is$\alpha$. Additionally, if the social forces make efforts during the project, they will receive government incentives (such as tax deductions, rewards, etc.), that is,$P$.

Assumption 7. 

When only the social forces choose to make efforts, the increase in the input income obtained by the social forces is less than the operating cost of the project, that is,${\beta }_0{R}_s<C_s$. When only the museums actively promote, the museums’ investment income is less than the project’s operating costs, that is,${\gamma }_0{R}_m<C_m$.

According to the above assumptions, both parties in the game evolve through different strategic combinations to achieve the optimal outcome.

Table 1. Descriptions of each parameter.

Parameters	Descriptions	Notes
$x$	The proportion of social force groups that make an effort	$0\le x\le 1$
$y$	The proportion of museum groups that choose an active promotion	$0\le y\le 1$
$R_s$	The basic benefits received by social forces that participate in a museum PPP project	$R_s\ge 0$
$\mathsf{\Delta }{R}_s$	The additional revenue streamed from a “free rider” for social forces	$\mathsf{\Delta }{R}_s\ge 0$
$R_m$	The basic benefits received by museums that promote the museum PPP project	$R_m\ge 0$
$\mathsf{\Delta }{R}_m$	The additional revenue generated by the “free rider” for museums	$\mathsf{\Delta }{R}_m\ge 0$
${\beta }_0$	The ratio of increased investment income of social forces when only social forces make an effort	${\beta }_0\ge 0$
${\gamma }_0$	The ratio of increased investment income of museums when only museums actively promote the project	${\gamma }_0\ge 0$
${\beta }_1$	The ratio of increased investment income of social forces when social forces and museums choose positive behaviors at the same time	${\beta }_1>{\beta }_0$
${\gamma }_1$	The ratio of increased investment income of museums when social forces and museums choose positive behaviors at the same time	${\gamma }_1>{\gamma }_0$
$C_s$	The operating costs incurred by social forces when they make an effort for the project	$C_s\ge 0$
$C_m$	The operating costs incurred by museums when they actively promote the project	$C_m\ge 0$
$\alpha$	The reduction ratio of operating costs of the project when the local government offers a social forces–museums support fund	$0\le \alpha \le 1$
$P$	The government incentives (such as tax deductions, rewards, etc.) when the social forces make an effort	$P\ge 0$

shows the descriptions of each parameter used in our game.

3.2. Replication Dynamic Equations

There are four strategy combinations available to social forces and museums: {effort, active promotion}, {effort, passive promotion}, {speculation, active promotion}, and {speculation, passive promotion}. The two parties play the game under the condition of bounded rationality, and

Table 2. Payoff matrix of the museums and social forces.

Social Forces (S)	Museums (M)
	Active Promotion y	Passive Promotion $1-\mathit{y}$
Effort x	$(1+{\beta }_1)R_s-(1-\alpha )C_s+P,(1+{\gamma }_1)R_m-(1-\alpha )C_m$	$(1+{\beta }_0)R_s-C_s+P,R_m+\mathsf{\Delta }{R}_m$
Speculation $\mathbf{1}\mathbf{-}\mathit{x}$	$R_s+\mathsf{\Delta }{R}_s,(1+{\gamma }_0)R_m-C_m$	$R_s,R_m$

shows the payoff matrix for various strategy combinations.

$U_s^1$ and $U_s^2$ represent the benefits of social force groups adopting different strategies (effort, speculation), which can be expressed as follows:

$$U_s^1=y[(1+{\beta }_1)R_s-(1-\alpha )C_s+P]+(1-y)[(1+{\beta }_0)R_s-C_s+P]$$

(1)

$$U_s^2=y(R_s+\mathsf{\Delta }{R}_s)+(1-y)R_s$$

(2)

Thus, based on Equations (1) and (2), the average benefits of social force groups can be expressed as:

$${\overline{U}}_s=x{U}_s^1+(1-x)U_s^2$$

(3)

Similarly, $U_m^1$ and $U_m^2$ represent the benefits of museum groups using different strategies (active promotion, passive promotion), which can be expressed as follows:

$$U_m^1=x[(1+{\gamma }_1)R_m-(1-\alpha )C_m]+(1-x)[(1+{\gamma }_0)R_m-C_m]$$

(4)

$$U_m^2=x(R_m+\mathsf{\Delta }{R}_m)+(1-x)R_m$$

(5)

Therefore, based on Equations (4) and (5), the average benefits of the museum groups can be expressed as:

$${\overline{U}}_m=y{U}_m^1+(1-y)U_m^2$$

(6)

Due to the asymmetric nature of knowledge acquisition between museums and social forces, both parties in the game will continue to learn and gain historical experience. This will enable them to formulate their strategies. When the two parties of the game dynamically adjust their strategies, x and y show the dynamic replication process described by evolutionary game theory. According to the Malthusian equation [48], the growth rate of the proportion of social forces choosing effort is proportional to the difference between the gain obtained by choosing this strategy and the group average gain; the growth rate of the proportion of museums choosing active promotion is proportional to the difference between the gain obtained by choosing this strategy and the group average gain. The two-dimensional dynamic system (I) is obtained from the replication dynamic equations for the two groups.

$$\left\{\begin{array}{ll}S(x,y)\hfill & =\frac{dx}{dt}=x(U_s^1-{\overline{U}}_s)=x\left(1-x\right)(U_s^1-U_s^2)\\ & =x\left(1-x\right)\{({\beta }_0{R}_s-C_s+P)+y[({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s]\}\hfill \\ M(x,y)\hfill & =\frac{dy}{dt}=y(U_m^1-{\overline{U}}_m)=y\left(1-y\right)(U_m^1-U_m^2)\\ & =y\left(1-y\right)\{({\gamma }_0{R}_m-C_m)+x[({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m]\}\hfill \end{array}\right.$$

(7)

3.3. Evolutionary Stable Strategy Aanalysis

According to the equilibrium theory, the necessary conditions for a game system to have an evolutionary stability strategy are $S(x,y)=0,M(x,y)=0$. The solution is as follows:

$$x_1=0,x_2=1,x^{*}=\frac{C_m-{\gamma }_0{R}_m}{({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m};$$

$$y_1=0,y_2=1,y^{*}=\frac{C_s-{\beta }_0{R}_s-P}{({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s}.$$

System (I) necessarily has four equilibrium points containing two groups adopting pure strategies:$(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$. When $0<\frac{C_m-{\gamma }_0{R}_m}{({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m}<1$ and $0<\frac{C_s-{\beta }_0{R}_s-P}{({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s}<1$, that is, $0<x^{*},y^{*}<1$, $(x^{*},y^{*})$ is considered to be the mixed-strategy equilibrium point.

According to the evolutionary stability strategy (ESS), the analytical ideas are as follows: First, Jacobian matrix J is established. Then, for each equilibrium point, determine whether conditions $\det J>0$ and $trJ<0$ are satisfied. If yes, the point is an ESS [49]. The ESS of a multi-group evolutionary game must be a pure-strategy Nash equilibrium, i.e., a mixed-strategy equilibrium in an asymmetric game must not be an evolutionary stable strategy [50,51]. Therefore, the subsequent study analyzes only four pure strategy equilibrium points.

According to Equation (7), the Jacobian matrix is expressed as follows:

$$\begin{array}{ll}J& =\left(\begin{array}{cc}\frac{\partial S(x,y)}{\partial x}& \frac{\partial S(x,y)}{\partial y}\\ \frac{\partial M(x,y)}{\partial x}& \frac{\partial M(x,y)}{\partial y}\end{array}\right)\\ & =\left(\begin{array}{cc}(1-2x)\{({\beta }_0{R}_s-C_s+P)+y[({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s]\}& x(1-x)[({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s]\\ y(1-y)[({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m]& (1-2y)\{({\gamma }_0{R}_m-C_m)+x[({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m]\}\end{array}\right).\end{array}$$

(8)

Four pure-strategy equilibrium points are distinguished as follows:

(1) For $(0,0)$, $J=\left(\begin{array}{cc}{\beta }_0{R}_s-C_s+P& 0\\ 0& {\gamma }_0{R}_m-C_m\end{array}\right)$;

(2) For $(0,1)$, $J=\left(\begin{array}{cc}-(1-\alpha )C_s+P+{\beta }_1{R}_s-\mathsf{\Delta }{R}_s& 0\\ 0& -{\gamma }_0{R}_m+C_m\end{array}\right)$;

(3) For $(1,0)$, $J=\left(\begin{array}{cc}-{\beta }_0{R}_s+C_s-P\hfill & 0\\ 0& -(1-\alpha )C_m+{\gamma }_1{R}_m-\mathsf{\Delta }{R}_m\end{array}\right)$;

(4) For $(1,1)$, $J=\left(\begin{array}{cc}(1-\alpha )C_s-P-{\beta }_1{R}_s+\mathsf{\Delta }{R}_s& 0\\ 0& (1-\alpha )C_m-{\gamma }_1{R}_m+\mathsf{\Delta }{R}_m\end{array}\right)$.

The obtained $\det J$ and $trJ$ values at the four equilibrium points are shown in

Table 3. Stability analysis of equilibrium point.

Equilibrium Point	$\mathit{D}\mathit{e}\mathit{t}\mathit{J}$	$\mathit{T}\mathit{r}\mathit{J}$
$(0,0)$	$({\beta }_0{R}_s-C_s+P)({\gamma }_0{R}_m-C_m)$	${\beta }_0{R}_s-C_s+P+{\gamma }_0{R}_m-C_m$
$(0,1)$	$\left[-(1-\alpha )C_s+P+{\beta }_1{R}_s-\mathsf{\Delta }{R}_s\right](-{\gamma }_0{R}_m+C_m)$	$-(1-\alpha )C_s+P+{\beta }_1{R}_s-\mathsf{\Delta }{R}_s-{\gamma }_0{R}_m+C_m$
$(1,0)$	$(-{\beta }_0{R}_s+C_s-P)\left[-(1-\alpha )C_m+{\gamma }_1{R}_m-\mathsf{\Delta }{R}_m\right]$	$-{\beta }_0{R}_s+C_s-P-(1-\alpha )C_m+{\gamma }_1{R}_m-\mathsf{\Delta }{R}_m$
$(1,1)$	$\begin{array}{l}\left[(1-\alpha )C_s-P-{\beta }_1{R}_s+\mathsf{\Delta }{R}_s\right]\cdot \\ \left[(1-\alpha )C_m-{\gamma }_1{R}_m+\mathsf{\Delta }{R}_m\right]\end{array}$	$\begin{array}{l}(1-\alpha )C_s-P-{\beta }_1{R}_s+\mathsf{\Delta }{R}_s\\ +(1-\alpha )C_m-{\gamma }_1{R}_m+\mathsf{\Delta }{R}_m\end{array}$

.

Scenario 1: In a museum PPP project, the government incentive P obtained by social forces’ efforts reaches a certain level, and its value range is $P>C_s-{\beta }_0{R}_s$.

In this scenario, there are ${\beta }_0{R}_s-C_s+P>0$ and ${\gamma }_0{R}_m-C_m<0$ (Assumption 7). Based on the comparison of the additional benefits that the participants can obtain by choosing the positive behavior or the “free-riding” behavior, the discussion is divided into four sub-scenarios.

① If social forces satisfy ${\beta }_1{R}_s-(1-\alpha )C_s+P>\mathsf{\Delta }{R}_s$ and museums satisfy ${\gamma }_1{R}_m-(1-\alpha )C_m>\mathsf{\Delta }{R}_m$, the stability results of four equilibrium points can be obtained, as shown in

Table 4. The states of the four equilibrium points in sub-scenario ①.

Equilibrium Point	$\mathit{D}\mathit{e}\mathit{t}\mathit{J}$	$\mathit{T}\mathit{r}\mathit{J}$	Stability
$(0,0)$	$-$	Uncertain	Saddle point
$(0,1)$	$+$	$+$	Unstable point
$(1,0)$	$-$	Uncertain	Saddle point
$(1,1)$	$+$	$-$	ESS

.

② If social forces satisfy ${\beta }_1{R}_s-(1-\alpha )C_s+P>\mathsf{\Delta }{R}_s$ and museums satisfy ${\gamma }_1{R}_m-(1-\alpha )C_m<\mathsf{\Delta }{R}_m$, the stability results of four equilibrium points can be obtained, as shown in

Table 5. The states of the four equilibrium points in sub-scenario ②.

Equilibrium Point	$\mathit{D}\mathit{e}\mathit{t}\mathit{J}$	$\mathit{T}\mathit{r}\mathit{J}$	Stability
$(0,0)$	$-$	Uncertain	Saddle point
$(0,1)$	$+$	$+$	Unstable point
$(1,0)$	$+$	$-$	ESS
$(1,1)$	$-$	Uncertain	Saddle point

.

③ If social forces satisfy ${\beta }_1{R}_s-(1-\alpha )C_s+P<\mathsf{\Delta }{R}_s$ and museums satisfy ${\gamma }_1{R}_m-(1-\alpha )C_m>\mathsf{\Delta }{R}_m$, the stability results of four equilibrium points can be obtained, as shown in

Table 6. The states of the four equilibrium points in sub-scenario ③.

Equilibrium Point	$\mathit{D}\mathit{e}\mathit{t}\mathit{J}$	$\mathit{T}\mathit{r}\mathit{J}$	Stability
$(0,0)$	$-$	Uncertain	Saddle point
$(0,1)$	$-$	Uncertain	Saddle point
$(1,0)$	$-$	Uncertain	Saddle point
$(1,1)$	$-$	Uncertain	Saddle point

.

④ If social forces satisfy ${\beta }_1{R}_s-(1-\alpha )C_s+P<\mathsf{\Delta }{R}_s$ and museums satisfy ${\gamma }_1{R}_m-(1-\alpha )C_m<\mathsf{\Delta }{R}_m$, the stability results of four equilibrium points can be obtained, as shown in

Table 7. The states of the four equilibrium points in sub-scenario ④.

Equilibrium Point	$\mathit{D}\mathit{e}\mathit{t}\mathit{J}$	$\mathit{T}\mathit{r}\mathit{J}$	Stability
$(0,0)$	$-$	Uncertain	Saddle point
$(0,1)$	$-$	Uncertain	Saddle point
$(1,0)$	$+$	$-$	ESS
$(1,1)$	$+$	$+$	Unstable point

.

Following the above analysis, it is concluded that, when $P>C_s-{\beta }_0{R}_s$, if the sum of the additional benefits that social forces can gain by making an effort (${\beta }_1{R}_s-(1-\alpha )C_s$) and the government’s incentives ($P$) is higher than the additional benefits brought by the “free-riding” behavior ($\mathsf{\Delta }{R}_s$), and museums are able to obtain more benefits by actively promoting compared to the additional benefits they receive from participation in “free rider” activities (${\gamma }_1{R}_m-(1-\alpha )C_m>\mathsf{\Delta }{R}_m$), (1, 1) represents an evolutionary stable strategy (ESS) at this moment, indicating that the most desirable state (effort, active promotion) will occur. The phase diagram of dynamic evolution is shown in Figure 2a. Once museums choose to promote actively and obtain less income than what would have been generated by “free rider” behavior, that is, ${\gamma }_1{R}_m-(1-\alpha )C_m<\mathsf{\Delta }{R}_m$, museums will choose passive behavior, while the social forces will continue to make efforts, thus forming an evolutionary stable strategy (1, 0). This shows that government incentives play a key role in this process and can consistently and effectively guide social forces in their efforts to participate in the construction of museum PPP projects.

Scenario 2: In a museum PPP project, there are few incentives provided by the government for social forces that make an effort, and the value range is $0<P<C_s-{\beta }_0{R}_s$.

In this scenario, there are ${\beta }_0{R}_s-C_s+P<0$ and ${\gamma }_0{R}_m-C_m<0$ (Assumption 7). Considering the comparison between the additional benefits that participants can acquire by choosing the positive behavior or the “free rider” behavior, four sub-scenarios are proposed.

① If social forces satisfy ${\beta }_1{R}_s-(1-\alpha )C_s+P>\mathsf{\Delta }{R}_s$ and museums satisfy ${\gamma }_1{R}_m-(1-\alpha )C_m>\mathsf{\Delta }{R}_m$, the stability results of four equilibrium points can be obtained, as shown in

Table 8. The states of the four equilibrium points in sub-scenario ①.

Equilibrium Point	$\mathit{D}\mathit{e}\mathit{t}\mathit{J}$	$\mathit{T}\mathit{r}\mathit{J}$	Stability
$(0,0)$	$+$	$-$	ESS
$(0,1)$	$+$	$+$	Unstable point
$(1,0)$	$+$	$+$	Unstable point
$(1,1)$	$+$	$-$	ESS

.

② If social forces satisfy ${\beta }_1{R}_s-(1-\alpha )C_s+P>\mathsf{\Delta }{R}_s$ and museums satisfy ${\gamma }_1{R}_m-(1-\alpha )C_m<\mathsf{\Delta }{R}_m$, the stability results of four equilibrium points can be obtained, as shown in

Table 9. The states of the four equilibrium points in sub-scenario ②.

Equilibrium Point	$\mathit{D}\mathit{e}\mathit{t}\mathit{J}$	$\mathit{T}\mathit{r}\mathit{J}$	Stability
$(0,0)$	$+$	$-$	ESS
$(0,1)$	$+$	$+$	Unstable point
$(1,0)$	$-$	Uncertain	Saddle point
$(1,1)$	$-$	Uncertain	Saddle point

.

③ If social forces satisfy ${\beta }_1{R}_s-(1-\alpha )C_s+P<\mathsf{\Delta }{R}_s$ and museums satisfy ${\gamma }_1{R}_m-(1-\alpha )C_m>\mathsf{\Delta }{R}_m$, the stability results of four equilibrium points can be obtained, as shown in

Table 10. The states of the four equilibrium points in sub-scenario ③.

Equilibrium Point	$\mathit{D}\mathit{e}\mathit{t}\mathit{J}$	$\mathit{T}\mathit{r}\mathit{J}$	Stability
$(0,0)$	$+$	$-$	ESS
$(0,1)$	$-$	Uncertain	Saddle point
$(1,0)$	$+$	$+$	Unstable point
$(1,1)$	$-$	Uncertain	Saddle point

.

④ If social forces satisfy ${\beta }_1{R}_s-(1-\alpha )C_s+P<\mathsf{\Delta }{R}_s$ and museums satisfy ${\gamma }_1{R}_m-(1-\alpha )C_m<\mathsf{\Delta }{R}_m$, the stability results of four equilibrium points can be obtained, as shown in

Table 11. The states of the four equilibrium points in sub-scenario ④.

Equilibrium Point	$\mathit{D}\mathit{e}\mathit{t}\mathit{J}$	$\mathit{T}\mathit{r}\mathit{J}$	Stability
$(0,0)$	$+$	$-$	ESS
$(0,1)$	$-$	Uncertain	Saddle point
$(1,0)$	$-$	Uncertain	Saddle point
$(1,1)$	$+$	$+$	Unstable point

.

In the sub-scenario ① of Scenario 2, that is, when $\mathsf{\Delta }{R}_s-\left[{\beta }_1{R}_s-(1-\alpha )C_s\right]<$$P<C_s-{\beta }_0{R}_s$ and, for museums, choosing to actively promote is more beneficial than engaging in “free-riding” behavior (${\gamma }_1{R}_m-(1-\alpha )C_m>\mathsf{\Delta }{R}_m$), the equilibrium state emerges at this time, which is related to the initial state of two parties’ decision-making in the game. When the initial state (x, y) of the two parties in the game is located in the OADC region, the game result of the system will gradually converge to the equilibrium point (0, 0). When the initial state (x, y) of the two parties in the game is located in the DABC region, the game result of the system will progressively converge to the equilibrium point (1, 1). The phase diagram of dynamic evolution is shown in Figure 2b. Therefore, both (effort, active promotion) and (speculation, passive promotion) may occur in the long-term evolutionary process. In reality, if social forces and museums are simultaneously positive in the construction of the museum PPP project, both parties will benefit most, and (effort, active) is the best state to be in at this time.

Saddle point D $(x^{*},y^{*})$ = ($\frac{C_m-{\gamma }_0{R}_m}{({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m}$,$\frac{C_s-{\beta }_0{R}_s-P}{({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s}$) is the equilibrium solution of the mixed strategy in the game. By changing the position of this point, the relative area size of the OADC and DABC regions will change, which in turn leads to a shift in system evolution toward a different equilibrium strategy. Accordingly, for the game outcome to converge to the ideal state (1, 1), it is necessary to enlarge the area of region DABC to increase the probability that the initial states of the two parties in the game fall into it.

The areas of OABC, OADC, DABC are denoted by S, S₁, S₂, respectively; then,

$$\begin{array}{cc}{S}_2& =S-S_1\hfill \\ & =1-\frac{x^{*}+y^{*}}{2}\hfill \\ & =1-\frac{1}{2}\left[\frac{C_m-{\gamma }_0{R}_m}{({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m}+\frac{C_s-{\beta }_0{R}_s-P}{({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s}\right]\hfill \end{array}$$

(9)

When other parameters remain unchanged, it is evident that $\frac{\partial S_2}{\partial P}>0$. This shows that the government incentive P is positively correlated with the area (S₂) of region DABC. If the government incentive P is increased, the area of region DABC will expand, the probability of the initial state of the two parties in the game falling into it will lift, and then the likelihood of the game’s evolution converging to (1, 1) will increase accordingly. In other words, increasing government incentives lifts the likelihood that (effort, active promotion) will occur. Alternatively, when government incentive is reduced, it is more likely that the game system will approach the equilibrium point (0, 0) and then converge to the strategy (speculation, passive promotion).

From another perspective, if the initial state of the two parties is close to the equilibrium point (1, 1), the government can reduce the level of incentives accordingly. While easing the financial burden, the results can still be optimized to achieve a win–win outcome. Reflected in reality, this means a smooth transition to a mature stage of museum PPP projects, when the proportion of museums and social forces choosing to behave positively has accordingly reached a certain level.

In summary, in the first sub-scenario of Scenario 1 and Scenario 2, an effective result will be achieved, primarily due to government incentives (P), which play a vital part in the evolution of the system. As the actual museum PPP project promotion cycle is long, the government needs to adjust the incentive level in conjunction with the project implementation progress, so as to promote the operation and development of the project with maximum efficiency and guarantee high-quality achievements.

4. Simulation Analysis

4.1. System Dynamics (SD) Model Establishment

On the basis of the above theoretical analysis, a system dynamics model of the game behavior of PPP projects in the museum field is further constructed and simulated, so as to visually portray the evolutionary trends of the decision-making behavior of both game parties.

The game system of museum PPP projects contains two types of participants: social forces and museums. Therefore, the system dynamics model of the evolutionary game of museum PPP projects constructed by Vensim PLE contains two sub-models accordingly: the social forces SD sub-model and the museums SD sub-model, as shown in Figure 3.

The model contains two level variables, two rate variables, four auxiliary variables, and twelve constants, as shown in

Table 12. The variables of the system.

Variable Type	Specific Variable
Level variable	The proportion of the efforts of social forces (x); The proportion of active promotion of museums (y)
Rate variable	The changing rate of the efforts of social forces ($\frac{dx}{dt}$); The changing rate of active promotion of museums ($\frac{dy}{dt}$)
Auxiliary variable	Benefits of social forces under different behavior strategies ($U_s^1$$,U_s^2$); Benefits of museums under different behavior strategies ($U_m^1$$,U_m^2$)
Constant	$R_s$$,R_m$$,\mathsf{\Delta }{R}_s$$,\mathsf{\Delta }{R}_m$$,{\beta }_0$$,{\beta }_1$$,{\gamma }_0$$,{\gamma }_1$$,C_s$$,C_m$$,\alpha$, P

.

4.2. Parameter Setting

The original parameters of the model are set as follows: INITIAL TIME = 0, FINAL TIME = 30, TIME STEP = 0.125, and Units for Time = Month. The level and rate variables as well as the auxiliary variables can be determined by Equations (1)–(7). To ensure the accuracy and reliability of the study results, the constant parameters are set by expert consultation and press reports [52], so that $R_s$ = 16, $R_m$ = 12, $\mathsf{\Delta }{R}_s$ = 4, $\mathsf{\Delta }{R}_m$ = 1, ${\beta }_0$ = 0.08, ${\beta }_1$ = 0.35, ${\gamma }_0$ = 0.05, ${\gamma }_1$ = 0.25, $C_s$ = 3, $C_m$ = 1.5, and $\alpha$ = 0.4. In addition, drawing on the setting of the initial value of the system in an existing study [53], the initial states of social forces’ efforts and the active promotion of museums are set as low, medium, and high levels, i.e., $x_0,y_0\in \mathsf{\Omega }(0.2,0.5,0.8)$.

4.3. Results and Discussion

(1) Sub-scenario ① in Scenario 1, whose conditions are as follows:

$$\left\{\begin{array}{c}P>C_s-{\beta }_0{R}_s\\ {\beta }_1{R}_s-(1-\alpha )C_s+P>\mathsf{\Delta }{R}_s\\ {\gamma }_1{R}_m-(1-\alpha )C_m>\mathsf{\Delta }{R}_m\end{array}\right.$$

According to the need for scenario simulation analysis, and combined with the parameter values set above, P = 2 is formulated here. In addition, $x_0,y_0\in \mathsf{\Omega }(0.2,0.5,0.8)$ is taken as the low, medium, and high levels of the initial state of the game parties, respectively, and substituted into the SD model for simulation analysis. The results are shown in Figure 4.

The simulation results show that under different initial states, the social forces eventually tend to make efforts, and the museums tend to promote actively, i.e., the system ultimately converges to the equilibrium point (1, 1). The conclusion drawn from sub-scenario ① in scenario 1 is confirmed. The speed of curve convergence is highly correlated with the initial state of the social forces and museums, namely, the closer the initial state is to equilibrium point (1, 1), the faster the rate of convergence.

(2) Sub-scenario ② in Scenario 1, whose conditions are as follows:

$$\left\{\begin{array}{c}P>C_s-{\beta }_0{R}_s\\ {\beta }_1{R}_s-(1-\alpha )C_s+P>\mathsf{\Delta }{R}_s\\ {\gamma }_1{R}_m-(1-\alpha )C_m<\mathsf{\Delta }{R}_m\end{array}\right.$$

According to the need for scenario simulation analysis, the other parameters are kept unchanged, and only $\mathsf{\Delta }{R}_m$ is adjusted to 2.5. Furthermore, $x_0,y_0\in \mathsf{\Omega }(0.2,0.5,0.8)$ is also taken as the low, medium, and high levels of the initial state of the game parties, respectively, and substituted into the SD model for simulation analysis. The results are indicated in Figure 5.

The simulation results show that, under different initial states, social forces eventually still tend to make efforts. However, driven by interests, the behavioral paths of the museums shift and gradually tend to be promoted passively. Consequently, the system converges to the equilibrium point (1, 0), and the conclusion drawn from sub-scenario ② in scenario 1 is confirmed.

Comparing Figure 4 and Figure 5, it can be found that the results of the behavioral strategy evolution of social forces in the two sub-scenarios, although converging, differ somewhat in the speed of convergence. Under sub-scenario ②, the speed of social forces converging to effortful behavior shows a significant weakening trend, and the lower the initial state of social forces’ efforts, the slower the convergence speed is, which indicates that the passive promotion behavior of museums will have a considerable impact on the evolution of social forces’ behavioral strategies. Therefore, in the early stage of the construction of the museum PPP project, the government should especially increase the incentives for social forces, stimulate their determination to participate in the project, and strive to preserve the healthy and continuous operation of the project.

(3) Sub-scenario ① in Scenario 2, whose conditions are as follows:

$$\left\{\begin{array}{c}\mathsf{\Delta }{R}_s-\left[{\beta }_1{R}_s-(1-\alpha )C_s\right]<P<C_s-{\beta }_0{R}_s\\ {\gamma }_1{R}_m-(1-\alpha )C_m>\mathsf{\Delta }{R}_m\end{array}\right.$$

According to the need for scenario simulation analysis, other parameters are kept unchanged, and only the government incentive P is reduced to 0.5, at which point D $(x^{*},y^{*})$ = ($\frac{C_m-{\gamma }_0{R}_m}{({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m}$,$\frac{C_s-{\beta }_0{R}_s-P}{({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s}$) = (0.41,0.62).

(1) When $\frac{C_m-{\gamma }_0{R}_m}{({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m}<x\le 1$, namely, $0.41<x\le 1$, y = 1 is an evolutionary stable strategy (ESS). When $0\le x<\frac{C_m-{\gamma }_0{R}_m}{({\gamma }_1-{\gamma }_0)R_m-\mathsf{\Delta }{R}_m+\alpha C_m}$, that is, $0\le x<0.41$, y = 0 is an evolutionary stable strategy (ESS).

First, the initial state x of the social forces is fixed and assumed to be 0.6. Second, the initial value of y is assumed to be 0.1, and y changes every 0.15 interval until y = 1. The model is simulated seven times to observe the evolutionary trend of the behavioral strategy of museums, and the results are shown in Figure 6a. Keeping the other settings unchanged, only the fixed value of x is changed and assumed to be 0.4. The simulation is continued by substituting into the model, and the results are depicted in Figure 6b.

According to the simulation results, when x is 0.6 and the initial value of y is within [0.1, 0.4], y converges in the direction of active promotion at the beginning. However, after a certain time step, its behavioral path starts to change and eventually converges to a stable state of passive promotion. When x is 0.6 and the initial value of y is within [0.55, 1], y gradually converges in the direction of active promotion. When x is 0.4 and the initial value of y is within [0.1, 0.7], y converges rapidly in the direction of passive promotion. When x is 0.4 and the initial value of y is within [0.85, 1], y ultimately reaches a steady state at the active promotion.

(2) When $\frac{C_s-{\beta }_0{R}_s-P}{({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s}<y\le 1$, namely, $0.62<y\le 1$, x = 1 is an evolutionary stable strategy (ESS). When $0\le y<\frac{C_s-{\beta }_0{R}_s-P}{({\beta }_1-{\beta }_0)R_s-\mathsf{\Delta }{R}_s+\alpha C_s}$, that is, $0\le y<0.621$, x = 0 is an evolutionary stable strategy (ESS).

First, the initial state y of museums is fixed and assumed to be 0.7. Second, the initial value of x is assumed to be 0.1, and x changes every 0.15 interval until x = 1. The model is simulated seven times to observe the evolutionary trend of the behavioral strategies of social forces, and the results are shown in Figure 7a. Keeping the other settings unchanged, only the fixed value of y is changed and assumed to be 0.5. The simulation is continued by substituting into the model, and the results are displayed in Figure 7b.

According to the simulation results, when y is 0.7 and the initial value of x is within [0.1, 0.4], then x finally converges to the steady state of speculation. When y is 0.7 and the initial value of x is within [0.55, 1], x eventually converges to the steady state of effort. When y is 0.5 and the initial value of x is within [0.1, 0.55], x converges rapidly in the direction of speculation. When y is 0.5 and the initial value of x is within [0.7, 1], x eventually reaches a steady state at the behavior strategy of effort.

In summary, the convergence path of the game model in different initial states is correspondingly different, but ultimately, there are only two equilibria, namely, (speculation, passive promotion) and (effort, active promotion). The higher the initial probability that both parties of the game choose positive behavior, the greater the system tends to converge to the strategy (effort, active promotion).

In sub-scenario ① of scenario 2, the other parameters should be kept unchanged, and the initial states of social forces and the museums should be fixed at 0.6 and 0.7, respectively. Next, the government incentive P is set as P₁ = 0.2, P₂ = 0.5, and P₃ = 1.5 and brought into the model for simulation to observe how the government incentive (P) influences the evolution results of the system. The simulation results are presented in Figure 8.

The simulation results show that the higher the level of government incentives, the more social forces and museums tend to adopt positive behaviors, and therefore, the more inclined the game evolution result of the system is to the equilibrium strategy (effort, active promotion). Conversely, the lower the government incentive level is, the greater the system evolution tends to converge to the equilibrium strategy (speculation, passive promotion). Combining Figure 8a,d, it is found that as long as the initial probability of both parties of the game adopting positive behaviors reaches a certain level, i.e., a smooth transition to maturity for a museum PPP project, government incentives, even at a low level, can still make the system’s game evolutionary outcome converge to the ideal state of (effort, active promotion).

5. Conclusions and Management Insights

In order to break through the traditional game theory, a game model for the evolution of behavioral strategies between social forces and museums in museum PPP projects under the assumption that all participating subjects are finitely rational is constructed, so as to clarify the interactions between subjects’ participation behaviors. Furthermore, different levels of government incentives are combined with the SD model of the evolutionary game, and through the simulation of Vensim PLE software, the dynamic evolution process of behavioral strategies is explored in depth. Based on theoretical and simulation analysis, the following conclusions are drawn:

(1)

In comparison with the current status of museum PPP projects around the world, China is still at an early stage, making it imperative to establish a multi-participant mechanism that ensures high-quality promotion and sustainable development;

(2)

In the museum PPP project, the strategic decisions made by the museums and social forces will interact. As long as both parties benefit from the game, that is, if the additional benefits of positive behavior exceed those of “free-riding” behavior, there are two stable strategies—(effort, active promotion) and (speculation, passive promotion);

(3)

Government incentives can have a critical impact on the evolution of museum PPP projects at varying phases of their implementation. In the early stage of project implementation, government incentives are required to reach a certain level to achieve the ideal state of evolution (effort, active promotion). As project implementation transitions to maturity, the level of government incentives can be appropriately reduced to alleviate financial pressures while still ensuring the desired state of system evolutionary convergence (effort, active promotion).

The theoretical contributions of the findings of this paper are as follows:

(1)

This paper clarifies the interest demands and behavior-oriented conflicts among participants in museum PPP projects, and then reveals the dynamic evolutionary characteristics of their behavioral strategies;

(2)

The influence of government incentives on the participation behavior of museum PPP projects is analyzed, and it is found that government incentives play an important regulatory role;

(3)

The combination of evolutionary game and system dynamics provides more quantitative evidence for the research and promotion of museum PPP projects based on numerous existing theoretical studies.

To improve the construction of museum PPP projects, the following countermeasures are recommended:

(1)

The formulation of scientific long-term incentive policies is essential for optimizing the role of incentives. The government should take reasonable measures to control the level of incentives, optimize the structure of incentive policies, enrich the approaches to incentives, and improve the flexibility of incentives in involving social forces through financing, funds, donations, and technological inputs. Given that the current PPP projects in the domestic museum sector are still in their infancy, the government should increase the level of incentives to further accelerate the transition to the mature stage and promote the construction of the projects with maximum efficiency;

(2)

With a focus on improving the quality and effectiveness of museum PPP project operations, a reasonable operating model must be designed to create favorable conditions for the smooth operation of the project. Additionally, performance evaluation and benefits distribution systems must be established and optimized to ensure high-quality project accomplishments while better safeguarding and enhancing the overall benefits of social forces and museums;

(3)

In order to reduce project risks and ease operational pressures, it is necessary to effectively leverage and coordinate professional support for museum PPP projects among the public or intermediaries, professional agencies, museum departments, research institutions, insurance companies, consumers (individuals or institutions), and financial institutions. In addition, it is also necessary to strengthen the supervision of the museums and social forces involved in the project to minimize the “free-riding” behavior.

(4)

However, it should be noted that the presented research is by no means free of limitations. In this paper, only two subjects in museum PPP projects are considered, that is, museums and social forces. There are, however, more subjects interacting in reality, such as the general public. The public, the real beneficiaries of public cultural services, can monitor the development and implementation of projects and make demands, thus ensuring the quality of the projects. In addition, the participation behavior of subjects will not only be influenced by government incentives, but also be profoundly affected by other factors, such as various compensation mechanisms, benefit distribution, and risk sharing issues, so the boundaries of the system will be expanded and the interaction relationships within the system will be more complex. Hopefully, future research will expand the types of subjects and factors affecting museum projects, expand the assumptions, and increase the relevance of the research, thereby improving the stability and depth of museum PPP projects.