Sustainable Development of State-Owned Enterprises: Research on the Management Transformation Path of Mixed-Ownership Companies from the Perspective of Shareholders Relationship

Abstract

In recent years, the development of China’s state-owned companies (SOEs) has slowed dramatically due to the improvement of the government–enterprise relationship and maturity in the market system. To accomplish the market-oriented transformation of the management model and promote sustainable development, some SOEs have incorporated private capital to conduct mixed-ownership reforms. Nonetheless, the emergence of heterogeneous shareholder conflicts seriously hampers the transformation of businesses. This paper proposed a two-party evolutionary game model between state-owned and private shareholders in the management transformation of mixed-ownership companies. Based on the proposed model, the evolutionary stability of heterogeneous shareholders’ action strategies was analyzed to obtain the evolutionary stability strategies for the system. The crucial factors of the ideal equilibrium strategies are studied at the same time. The analysis results show that the probability of “change” in private shareholders is positively proportional to factors such as the success rate of change, change dividend, control gain, and policy burden, and it is inversely proportional to the factors including the cost of change, contractual cost of private shareholders, the additional cost of change, hidden income, state-owned shareholders’ shareholding ratio, and loss of change failure. Finally, the findings of this study provide a theoretical foundation for the transition of mixed-ownership enterprises’ management systems.

1. Introduction

State-owned enterprises (SOEs) are the backbone of China’s national economy, dominating key areas and important sectors and playing a significant role in ensuring sustained rapid and healthy development. At the same time, they influence the development of the world economy through their foreign investments. Nasrollahi et al. found that weak and strong sustainability is affected by population, industrialization, economy, and national agreements in certain regions [1]. In addition, the sustainability in each region showed a positive spillover effect on sustainability in other regions [2]. SOEs can transform their management systems and promote the sustainable development of China’s national economy through mixed-ownership reforms. In addition, it can exert positive spillover effects to encourage global flow-based governance and improves sustainability in a peaceful and integrated environment through international agreements and partnerships [2]. Therefore, it is crucial to explore the transformation path of the management system of SOEs for the sustainable development of both China and the world economy.

The company management models can be divided into the administrative-type model and the market-based model. Generally, state-owned enterprises apply the administrative-type model [3], whereas private companies utilize the market-type model. The academic community has conducted extensive research on the properties of the two management models and the relationship between management models and company performance, yielding a wealth of research results. For example, Li Weian argued that the main difference between the two models lies in corporate objectives, resource allocation methods, and executive appointments and dismissals [4]. Liu Xiaoxuan and Wu Yanbin thought that the inefficiency of the administrative-type management model of SOEs firms is the main reason why SOEs’ performance is inferior to that of private firms [5,6]. However, few research results have been conducted on the management mode transition of SOEs from an administrative to a market-based model. Only Li Weian et al. conducted a preliminary investigation on corporate governance transformation and constructed a mathematical model to study the dynamic game process of corporate governance transformation and its equilibrium state [7,8,9].

China has the most significant number, the most comprehensive types, the largest scale, and the highest proportion of SOEs worldwide. The practice has demonstrated that the transformation of the governance system of SOEs with the market approach as the primary means is in the right direction [10]. According to Wang Chen, although resource-based theory emphasized the impact of resource advantages on firm performance and sustainability, resources were only the starting point, not the endpoint, and a transformation process was needed from the starting point to the endpoint, which is the management change and management innovation of enterprises [11]. Zhao Binbin et al. argued that management change could significantly enhance the developmental effects of the redistribution of rights in the subject structure and the shaping of the subject’s governance capacity, thus improving the company’s sustainability [12]. For China, the transition economy with increasing marketization, SOEs must undergo a market-oriented management transformation to improve corporate performance and sustainability. Moreover, Li Zhanxiang and Liu Ligang also regarded enterprises’ transformation and innovation capabilities as sustainable development capabilities [13,14]. Thus, it is essential to explore the effective management transformation of SOEs.

Since the Third Plenary Session of the 11th CPC Central Committee in 1978, China’s SOEs have been carrying out management mechanism reform. After several self-transformations of the management mechanisms of SOEs, the state started to implement large-scale mixed-ownership reforms of SOEs in 2013, attempting to thoroughly transform the management model of state-owned companies into a market-oriented one by introducing private capital. Li Jianbiao et al. pointed out that mixed-ownership reform is not an isolated event but is closely related to the transformation of SOEs from administrative to market-based governance [15]. However, the complicated co-opetition relationship between heterogeneous shareholders, state-owned shareholders (SS), and private shareholders (PS) substantially impacts the management transformation of mixed-ownership companies [16]. Thus, it is reasonable and necessary to study the management reform path of mixed-ownership enterprises from the perspective of heterogeneous shareholder relationships.

The proxy theory is the dominant theory in the study of shareholder relations, which holds that major shareholders and minority shareholders are in a state of antagonism due to agency conflicts [17,18]. However, this theory cannot explain the synergistic relationship between heterogeneous shareholders. Yang Songling et al. applied symbiosis theory to study the relationship between major shareholders and minority shareholders and claimed that the relationship between major shareholders and minority shareholders is not completely opposite but coordinated symbiosis [19]. Compared with agency theory, which concentrates on the principal-agent problem between major and minority shareholders, symbiosis theory focuses on the opposition and unity of interests between major shareholders and minority shareholders. It provides a more comprehensive study of shareholder relationships. However, the above theories only explore the relationship between shareholders with various ownership percentages without considering various shareholders’ resources difference in other areas. After analyzing the resource differentiation between SS and PS in mixed-ownership companies, Li Mingmin et al. argued that SS and PS should establish a symbiotic relationship based on the relevance and complementarity of their resources [20]. Nevertheless, how can heterogeneous shareholders of mixed-ownership enterprises build symbiotic relationships? It is not covered in the study of Ming-Min Li et al., and we have yet to find relevant research results. The symbiotic relationship between resource heterogeneous shareholders is the most critical feature of shareholder relationships in mixed-ownership companies, which significantly impacts the company’s management transformation.

Therefore, aiming at the practical background of the management transformation of mixed-ownership companies, this paper explores how mixed-ownership companies can achieve the synergism of heterogeneous shareholders to promote the corporate management transformation based on the institutional advantages of PS [21] and the disadvantages of government intervention by SS [22]. The research findings deepen the study of heterogeneous shareholder relationships and provide a theoretical foundation for implementing mixed-ownership reforms in SOEs.

Evolutionary game theory is a crucial method for studying the evolution of action strategies of multiple subjects. Consequently, this paper constructs an evolutionary game model to analyze the stability of heterogeneous shareholders’ equilibrium strategies in the management transformation of mixed-ownership companies, investigate the path of management transformation, and promote the successful transformation of the management model of mixed-ownership companies.

The innovation and contribution of this article lie in the following:

(1)

Capturing the fundamental contradiction between SS and PS, two important interest subjects in the management transformation process of mixed-ownership companies, the evolutionary stability strategies and factors affecting their action strategies are analyzed to provide a reliable basis for choosing the transformation path.

(2)

This paper explores the optimal shareholding ratio of SS by establishing the relationship between the shareholding ratio and the income and expenditure of the two types of shareholders, and it analyzes the impact of the shareholding ratio on the action strategies of both parties, which provides a new idea and method for determining the optimal shareholding ratio of SS in mixed-ownership companies.

(3)

This paper investigates the impact of heterogeneous shareholder synergy on the management transformation of mixed-ownership companies from a symbiotic relationship between the two types of shareholders. It offers a new perspective for determining the status and role of PS and SS in mixed-ownership companies.

The rest of the study unfolds as follows: Section 2 reviews the literature on reforming the mixed-ownership system in SOEs and the characteristics, strategies, and relationships of heterogeneous shareholders. Section 3 builds the two-party evolutionary game model between SS and PS. Section 4 analyzes the evolutionary stability strategies of the individual game players and the game system. Section 5 carried out numerical simulations of the main parameters. Finally, Section 6 summarizes our findings and proposes policy recommendations.

2. Literature Review

2.1. Research on Mixed-Ownership Reform of SOEs

Mixed-ownership means that SS and PS jointly own an enterprise [23]. It is a crucial method for combining public ownership with the market economy in the current environment, and it is also an objective requirement for further advancing the reform of SOEs and accomplishing economic transformation [24]. SOEs implement mixed-ownership reform not simply by mixing equity but by introducing private capital to cross-hold equity, removing institutional obstacles between SOEs and private enterprises, and promoting SOEs to transform their business management mechanism [25]. Company management mechanisms are optimized in various aspects to improve firm performance, such as corporate governance [26], management incentives [27], technology innovation [28], investment efficiency [29], risk-taking [30], redundancy, and employee efficiency [31].

The key to improving company performance through mixed-ownership reform is to realize the complementary resource advantages of two types of heterogeneous property rights subjects, SS and PS. According to Wernerfelt’s resource-based theory, the source of a firm’s competitive advantage was heterogeneous resources, and the inimitable nature of heterogeneous resources determined the sustainability of competitive advantage [32]. Crook contended that different resource elements are integrated into a firm to provide a competitive advantage and thus create firm value [33]. The resource heterogeneity between PS and SS is distinct, and heterogeneous shareholder relationships are the foundation of the corporate governance structure of mixed-ownership reform and also the critical factor in determining the effectiveness of mixed-ownership reform in SOEs [20]. Hao Yang et al. claimed that “mixed-ownership” helps to compensate for the inadequacies of marketization, and the “complementarity” of heterogeneous shareholders enables businesses to capitalize on the strengths of various ownership capital [34].

The conflict between PS and SS is primarily over the company’s control based on the shareholding ratio. Hence, the reasonable determination of the shareholding ratio of SS is the key to the success of mixed-ownership reform. Matsumura previously discussed the optimal shareholding ratio of SS after the privatization of SOEs [35]. Sun Qunyan et al. studied the impact of changes in the shareholding ratio of state-owned shares on social welfare by constructing a game model of mixed oligopolistic competition [36]. Yin Jun et al. explored the optimal shareholding ratio of SOEs in mixed-ownership reform by constructing a mixed oligopoly model [37]. Ma Lianfu et al. found that the best business performance occurred when the proportion of non-state shares was between 30% and 40% [24]. Shen Hao et al. argued that the impact of state-owned shares on business performance is not absolute, and the key is to allow the market to play a decisive role in resource allocation [38]. Liu Xiaoxuan et al. discovered that corporate efficiency is optimum when the proportion of state-owned shares is between 10% and 50%, notably around 30% [39]. Zhang Jide et al. emphasized that the control of state-owned capital over the firm must be maintained when introducing strategic investors such as private enterprises [40]. According to Chen Qian et al., the ownership structure of a corporation with mixed-ownership must be chosen based on the enterprise’s actual condition, and there is no absolute norm [41].

2.2. Research of the Characteristics, Strategies, and Relationships of Heterogeneous Shareholders

The advantages of SS are high-quality human resources, solid financial resources, a positive social reputation, and government resources [42]. At the same time, the disadvantage is the lack of corporate independence due to the specific government–enterprise connection [22]. The superiority of PS is the flexible market mechanism [21], whereas the drawback of private property rights owners is the tunneling effect in the firm’s management [43].

PS and SS have different objective functions, behavior patterns and interest demands, and the crucial component to the success of the mixed-ownership reform of SOEs is to address the link between the interests of the various ownership subjects [44]. Resource acquisition and property rights protection motivate private enterprises to participate in mixed-ownership reforms [22,45]. Wu Wenfeng et al. contended that private firms are in a discriminated position under the ownership structure in which the state-owned economy predominates [22]. Liu Hanmin et al. elucidated [16] the ambivalent mentality of two types of capital. From the view of state-owned capital, giving up control would result in the loss of state-owned assets, and it is challenging to introduce private capital without abandoning control. From the perspective of private capital, it is difficult to obtain control by participating in mixed-ownership reform, and surrendering control cannot guarantee that legitimate rights and interests are not violated. Li Jianbiao et al. stated that the super-shareholder status of state-owned capital would increase its bargaining strength [15]. Huang et al. argued that government should adequately protect the property rights of private capital and the interests of PS [46]. Cai Guilong et al. contended that if private capital is not given a special right, it will damage their willingness to participate in mixed-ownership reform and cause them to lose the essential significance of mixed-ownership reform [47].

The academic community has initially explored heterogeneous shareholder relationships. Qi Ping et al. argued that PS and SS, as heterogeneous economic agents, are bound to engage in dynamic economic behavior games during mixed-ownership reform [48]. Li et al. investigated the behavioral game process between PS and SS in reforming mixed-ownership in monopolistic industries [15]. Sun Kunpeng et al. constructed a full-information dynamic game model with the participation of four parties, including SS, PS, foreign shareholders, and managers, and deduced the relationship between the equity portfolio model and the corporate performance of mixed-ownership companies [44]. Li Mingmin et al. applied subjective game theory to study the behavioral logic of the strategy selection of two game players, state and non-state capital, in the context of the mixed-ownership reform [20]. Ren Guangqian et al. constructed the evolutionary game model to explore the relationship between government incentive behavior and the strategic choice of non-state capital in mixed-ownership transition [49].

3. Model Background and Building

3.1. Model Background

China’s economic system belongs to the transition economy from the planned economy to the market economy. The market-oriented transformation of the management mechanism of SOEs is the most difficult of economic system transformation. Since the third session of the 11th Central Committee (1978), SOEs have undergone four large-scale changes [50]: decentralization, contract management, shareholding reform, and mixed-ownership reform. The first three reforms were self-imposed improvements within the SOE structure that took 35 years to implement and were unsuccessful. The 18th CPC Central Committee’s Third Plenary Session in November 2013 approved the Decision on Several Major Issues of Comprehensively Deepening Reform. The Decision took mixed-ownership reform with integrating state-owned and private capital as an essential direction of economic system reform. It was the first attempt to involve private enterprises in reforming state-owned companies. By the end of 2020, the number of centrally mixed-ownership companies accounted for more than two-thirds, and the number of locally mixed-ownership companies made up more than fifty percent. The reform involves significant industries and critical sectors, such as electric power, petroleum, petrochemicals, telecommunications, aviation, the military, etc.

The primary goal of implementing the mixed-ownership reforms in SOEs is to drive the management mechanism of SOEs from the administrative-type model to the market-based model by introducing private capital to improve the operational efficiency and overall performance of the company [43,51]. PS and SS have substantial conflicts of interest and divergent viewpoints on firm management processes due to disparities in their histories, resources, and ambitions, making it difficult to make synergistic reforms [52]. Therefore, giving full play to the advantages of heterogeneous shareholders and guiding them to promote the market-oriented transformation of corporate management mode is the key to the success of mixed ownership reform.

3.2. Model Building

Evolutionary game theory, which originated from biological evolution [53], abandoned the hypothesis of sufficient information and complete rationality of participants in the classical game theory and held that the continuous evolution of individuals forms the stability of group strategies through learning and imitation. During the evolutionary process, poorly adapted strategies are removed, whereas highly adaptive tactics are more likely to be mimicked and accepted by other participants, leading to eventual stability [54]. The assumptions of limited rationality and information of participants are more consistent with economic and social reality. Consequently, evolutionary game theory is widely used in economic management research and has become the principal research method for solving multi-party cooperation problems [55]. It is impossible to achieve total rationality in the strategic decisions of SS and PS due to the information asymmetry between the parties involved. The decision of either party is based on the analysis of the other party’s strategies and will be adjusted as the other party’s strategies change. Therefore, this paper applies the evolutionary game methods to analyze the strategic choices of SS and PS in the corporate management transformation to explore the path of mixed-ownership corporate management transformation.

By introducing private capital, the primary aim of the mixed-ownership reform of SOEs is to enable SOEs to become independent market actors and to play a decisive role in the market in allocating resources [38]. According to Lu Feng (2001), under the condition that market forces were increasingly determining the fate of enterprises, the legitimacy basis of management authority was progressively shifting from state authorization to special management competence, which was reflected by management competence that determines enterprise performance, and management competence can only gradually develop and grow in the process of management practice and in the adjustment of organizational structure [56]. The flexible market mechanisms are the most significant advantage of PS in terms of the characteristics of diverse owners [21]. The purpose of introducing private capital into SOEs is also to leverage the advantages of private shareholders in the market mechanism and promote the transformation of the enterprises management mechanism. To sum up, PS more appropriately lead the management transition of mixed-ownership companies. However, PS may also decide to retain the administrative-type model due to the factors such as enormous hidden income and the dangers associated with the transformation under the administrative-type model [15]. Therefore, there are two possibilities for “change” and “no change” in the decision-making of PS. In SOEs, SS have some special rights beyond the rights of shareholders, and such special rights will become less and less in mixed-owned companies [10]. Thus, as a response to the loss of such rights, SS may coexist with PS in harmony or confrontation. Figure 1 illustrates heterogeneous shareholders’ characteristics, relationships, and action strategies in mixed-ownership enterprises.

3.2.1. Parameters Setting

The parameters settings involved in this evolutionary game model are shown in

Table 1. Definition of model parameters.

Symbol	Meaning
p	Probability of change for PS,
q	Probability of synergy for SS,
b	Proportion of shares held by SS,
α	Success rate of company reform when SS is synergistic, $0\le \alpha \le 1$
β	Success rate of company reform when SS is not synergistic, $0\le \beta \le 1$
R1	Company dividends from the successful transformation
C1	Cost of the company change
C2	Corporate losses from failed transformation
C3	Policy burden of the company arising from the “no change” of PS
C4	Loss of intra-company fighting due to “no synergy” of SS
R2	Hidden income of the company due to “no change” of PS
$C_1^G$	Contractual costs for SS due to corporate reform
$C_2^G$	Sunk costs and loss of control incurred by SS as a result of corporate transformation
$C_3^G$	Loss of SS fighting due to “no synergy” of SS
$R_1^G$	Control gains for SS under the administrative-type model
$C_1^M$	Contractual costs incurred by PS as a result of corporate change
$R_1^M$	Enhanced control gains for PS resulting from enterprise transform
$C_2^M$	Additional cost of reform for PS
$C_3^M$	Loss of PS fighting due to “no synergy” of SS

.

3.2.2. Assumptions

Li Weian [57] pointed out that external subjects determine the internal decision-making of SOEs, and SOEs do not have substantive decision-making power, which hinders the reform process of SOEs. He believed that through the mixed-ownership reform, state-owned capital and private capital state mutual integration could reverse the dilemma of the lack of separation between government and enterprises, thus achieved by the company insiders making their own decisions. Yu Jing argued [58] that developing the mixed-ownership economy requires establishing the synergy mechanism in which the advantages of the property rights subjects complement each other and takes advantage of each other’s strengths. Zhang Jide et al. considered [40] that if a mixed-ownership company gives SS the advantage in equity structure and endows PS the superiority in control, it can ensure both the earnings of SS and the control of private capital to achieve governance synergy. Accordingly, the following game players and their action strategies are hypothesized:

Hypothesis 1:

The game players and mechanism change basic assumptions. SS and PS are two-game players involved in the management transformation of mixed-ownership companies. The strategic space of PS is {change, no change}, and the strategic space of SS is {synergy, no synergy}. The probability of PS choosing “change” is $p$ and “no change” is $1-p$. The probability of SS choosing “synergy” is $q$ and “non-synergy” is  $1-q$ . The success rate of corporate change is $\alpha$ when SS is synergistic, and when SS is not synergistic, the success rate of corporate change is $\beta$ . The percentage of shares held by SS is $b$ , whereas the percentage held by PS is $1-b$ .

Ping Xinqiao et al. calculated that the agency cost of Chinese SOEs is equivalent to 60%~70% of the profit potential [59]. The studies of Hu Yifan et al. [60], Bai Chongen et al. [61], Guo et al. [62], and Yang Jijun et al. [63] confirmed the remarkable performance of SOEs in economic benefits after the privatization reform. Therefore, the management transformation of the mixed-ownership companies will create the reform dividend. Liu Hang et al. [64] argued that SOEs bear both policy burdens and enjoy hidden income given by the government. Qi Hao Dong et al. [52] considered that non-state capital entering monopoly areas that SOEs formerly owned would share a portion of monopoly rents that should have belonged to SS. According to Li Jianbiao et al. [15], state-owned and non-state-owned capital inevitably appeared as contradictions in the integration process due to different objective functions and ideological conflicts, resulting in internal fighting loss. Therefore, this paper presents the benefit and cost assumptions for managing change in mixed-ownership companies.

Hypothesis 2:

Company revenue and cost assumptions. The company’s income includes the following: (1) Corporate dividends from the successful transformation is $R_1$. (2) Hidden income due to “no change” of PS is $R_2$ [65]. SS exclusively shared $R_2$ before the mixed-ownership reform and in proportion to their shareholdings after the mixed-ownership reform. Therefore, the revenue shared by PS equals the implicit revenue reduced by SS. Company costs and losses include the following: (1) Cost of the company change is $C_1$ . (2) Corporate loss from failed transformation is $C_2$ . (3) Cost of the policy burden arising from the “no change” of PS is $C_3$ [66]. Similarly to hidden income, the policy burden shared by PS is equal to the reduced policy burden of SS. (4) Loss of intra-company fighting due to “no synergy” of SS is $C_4$. Company revenues and costs are distributed between SS and PS according to the shareholding ratio because the hidden income and policy burden existed before the mixed-ownership reform and were exclusively shared by SS and need to be distributed after the mixed-ownership reform. Therefore, the hidden income and policy burden shared by PS after the mixed-ownership reform is equal to the hidden income and policy burden reduced by the SS.

Bai Chongen et al. [66], Liao Li et al. [67], and Yanmada et al. [68] pointed out that transactions between PS and SS in mixed-ownership reforms inevitably incur transaction costs and default risks. Therefore, this paper proposes the benefit and cost assumptions for each party of the game, including Hypothesis 3 and Hypothesis 4.

Hypothesis 3:

Assumptions about costs and benefits for players during the change. “Synergy” and “no synergy” are two possible strategic choices for SS. In the case of SS “synergy,” both parties negotiate and agree on the reform plan, and both parties incur contractual costs. The contractual cost for SS is $C_1^G$and for PS is $C_1^M$. SS not only need to make significant adjustments to their original management system, occurring sunk costs, but also need to transfer the management control to PS, resulting in a loss of control, that the total is $C_2^G$. As the dominant party of change, PS unilaterally bear the additional change cost $C_2^M$ while gaining control earnings $R_1^M$ . In the case of SS “no synergy,” both parties’ contractual costs and PS’ additional change costs are the same as in the case of “synergy.” PS incur the same contract costs and change expenses as SS.

Hypothesis 4:

The assumptions of the benefits and costs of players without reform. “Synergy” and “no synergy” are two possible strategic choices for SS. In the scenario of SS “synergy,” the company is controlled by SS, keeping the original administrative model unchanged, and SS obtains the gaining control, which is $R_1^G$. In the case of SS “no synergy,” both sides of the game produce internal fighting losses due to mutual conflicts. Based on the fighting among heterogeneous shareholders, the loss of PS is $C_3^M$, and the loss of SS is $C_3^G$.

Based on the above assumptions, the payoff matrix of both parties in the game is constructed, as shown in

Table 2. The payoff matrix for the SS and PS.

		PS
		Change (p)	No Change (1 − p)
SS	Synergy $\left(q\right)$	$b\alpha R_1-b{C}_1-C_1^G-\left(1-\alpha \right)b{C}_2-C_2^G$	$R_1^G+(1-b)C_3-\left(1-b\right)R_2$
		$$\begin{gathered} \alpha \left(1-b\right)R_1+R_1^M-\left(1-b\right)C_1-C_1^M- \\ \hspace{1em}\left(1-\alpha \right)1-b{C}_2-C_2^M \end{gathered}$$	$(1-b)R_2-(1-b)C_3$
	No Synergy $\left(1-q\right)$	$$\begin{gathered} b\beta R_1-C_1^G-b{C}_1-\left(1-\beta \right)b{C}_2-b{C}_4- \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{C}_2^G-C_3^G \end{gathered}$$	$(1-b)C_3+R_1^G-(1-b)R_2-b{C}_4-C_3^G$
		$$\begin{gathered} \beta \left(1-b\right)R_1+R_1^M-\left(1-\beta \right)\left(1-b\right)C_2- \\ \left(1-b\right)C_4-\left(1-b\right)C_1-C_1^M-C_2^M-C_3^M \end{gathered}$$	$(1-b)R_2-(1-b)C_3-(1-b)C_4-C_3^M$

.

4. Stability Analysis of Game Model

4.1. Analysis on the Evolution of PS Strategies

According to the payoff matrix, Equations (1)–(3) display the expected return $U_{e1}$ for PS choosing the “change” strategy, the expected return $U_{e2}$ for the “no change” strategy, and the average expected return $U_e$ .

$$\begin{gathered} U_{e1}=q\left[\alpha \left(1-b\right)R_1+R_1^M-\left(1-b\right)C_1-C_1^M-\left(1-\alpha \right)(1-b)C_2-C_2^M\right] \\ +\left(1-q\right)\left[\beta \left(1-b\right)R_1+R_1^M-\left(1-\beta \right)\left(1-b\right)C_2-\left(1-b\right)C_4-\left(1-b\right)C_1-C_1^M-C_2^M-C_3^M\right] \end{gathered}$$

(1)

$$U_{e2}=q\left[(1-b)R_2-(1-b)C_3\right]+\left(1-q\right)\left[(1-b)R_2-(1-b)C_3-(1-b)C_4-C_3^M\right]$$

(2)

$$U_e=p{U}_{e1}+\left(1-p\right)U_{e2}$$

(3)

The dynamic replication equation of PS is as shown in Equation (4).

$$\begin{gathered} F\left(p\right)=\frac{dp}{dt}=p\left(U_{e1}-U_e\right)=p\left(1-p\right)(U_{e1}-U_{e2}) \\ =p\left(1-p\right)\left\{q\left[\left(\alpha -\beta \right)\left(1-b\right)R_1+\left(\alpha -\beta \right)\left(1-b\right)C_2\right] \\ +\left[\beta \left(1-b\right)R_1+R_1^M-\left(1-\beta \right)\left(1-b\right)C_2-\left(1-b\right)C_1-C_1^M-C_2^M-(1-b)R_2+(1-b)C_3\right]\right\} \end{gathered}$$

(4)

The first order partial derivative of $F\left(p\right)$ concerning $p$ is expressed as Equation (5).

$$\begin{gathered} \frac{dF\left(p\right)}{dp}=\left(1-2p\right)\left\{q\left[\left(\alpha -\beta \right)\left(1-b\right)R_1+\left(\alpha -\beta \right)\left(1-b\right)C_2\right] \\ +\left[\beta \left(1-b\right)R_1+R_1^M-\left(1-\beta \right)\left(1-b\right)C_2-\left(1-b\right)C_1-C_1^M-C_2^M-(1-b)R_2+(1-b)C_3\right]\right\} \end{gathered}$$

(5)

Solving the equation $F\left(p\right)=0,$ we obtain $q_0=\frac{-\beta \left(1-b\right)R_1-R_1^M+\left(1-\beta \right)\left(1-b\right)C_2+\left(1-b\right)C_1+C_1^M+C_2^M+(1-b)R_2-(1-b)C_3}{\left(\alpha -\beta \right)\left(1-b\right)R_1+\left(\alpha -\beta \right)\left(1-b\right)C_2}$.

According to the stability principle of replicated dynamic differential equations and the nature of the evolutionary stabilization strategy, it is known that $p_i$ is an evolutionary stability strategy when $Fp=\frac{dp}{dt}=0$ and $F^{\text{'}}\left(p_i\right)<0$.

The details can be discussed in the following scenarios:

Scenario 1: If $q=q_0$, then $F\left(p\right)=0$ and $\frac{dF\left(p\right)}{dp}{|}_{p=1}=0$. At this point, the evolutionary stability conditions are unsatisfied, and PS have no evolutionary stability state.

Scenario 2: If $0<q_0<q<1$, then $F\left(p=1\right)=0$, and $\frac{dF\left(p\right)}{dp}{|}_{p=1}<0$, which satisfies the evolutionary stability conditions, and therefore, $p^{*}=1$ is the evolutionary stability state of PS.

Scenario 3: If $0<q<q_0<1$,then $F\left(p=0\right)=0$, and $\frac{dF\left(p\right)}{dp}{|}_{p=0}<0$, which satisfies the evolutionary stability conditions, and therefore, $p^{*}=0$ is the evolutionary stability state of PS.

Figure 2 depicts the relationship between the evolutionary trend of PS’ strategies and the threshold value of the probability of “synergy.”.

The area $S_1$ in Figure 2 indicates the probability of PS “change,” whose size depends on the threshold value $q_0$. The lower the crucial value $q_0$, the bigger the area $S_1$, and the greater the likelihood of “change” for PS. Analysis of $q_0=\frac{-\beta \left(1-b\right)R_1-R_1^M+\left(1-\beta \right)\left(1-b\right)C_2+\left(1-b\right)C_1+C_1^M+C_2^M+(1-b)R_2-(1-b)C_3}{\left(\alpha -\beta \right)\left(1-b\right)R_1+\left(\alpha -\beta \right)\left(1-b\right)C_2}$ reveals that holding all other parameters constant, $q_0$ becomes smaller, as $\alpha ,R_1,R_1^M,$ and $C_3$ increase, as well as $C_1,C_1^M,C_2^M,R_2$ and $b$ decrease.

4.2. Analysis on the Evolution of SS Strategies

According to the payoff matrix, Equations (6)–(8), respectively, represent the expected return $U_{g1}$ for the “synergy” strategy, the expected return $U_{g2}$ for the “no change” strategy, and the average expected return $U_g$ for SS.

$$U_{g1}=p\left[b\alpha R_1-b{C}_1-C_1^G-\left(1-\alpha \right)b{C}_2-C_2^G\right]+\left(1-p\right)\left[R_1^G+(1-b)C_3-\left(1-b\right)R_2\right]$$

(6)

$$\begin{gathered} U_{g2}=p\left[b\beta R_1-C_1^G-b{C}_1-\left(1-\beta \right)b{C}_2-b{C}_4-C_2^G-C_3^G\right] \\ +\left(1-p\right)\left[(1-b)C_3+R_1^G-(1-b)R_2-b{C}_4-C_3^G\right] \end{gathered}$$

(7)

$$U_g=q{U}_{g1}+\left(1-q\right)U_{g2}$$

(8)

The dynamic replication equation of SS can be obtained as Equation (9).

$$F\left(q\right)=\frac{dq}{dt}=q\left(U_{g1}-U_g\right)=q\left(1-q\right)U_{g1}-U_{g2}=q\left(1-q\right)\left\{p\left[\left(\alpha -\beta \right)b{R}_1+\left(\alpha -\beta \right)b{C}_2\right]+b{C}_4+C_3^G\right\}$$

(9)

The first order partial derivative of $F\left(q\right)$ with respect to $q$ is shown in Equation (10).

$$\frac{dF\left(q\right)}{dq}=\left(1-2q\right)\left\{p\left[\left(\alpha -\beta \right)b{R}_1+\left(\alpha -\beta \right)b{C}_2\right]+b{C}_4+C_3^G\right\}$$

(10)

Solving the equation $F\left(q\right)=0,$ we obtain $p_0=\frac{-b{C}_4-C_3^G}{\left(\alpha -\beta \right)b{R}_1+\left(\alpha -\beta \right)b{C}_2}<0$.

According to the stability principle of replicated dynamic differential equations and the nature of the evolutionary stabilization strategy, it is known that $q_i$ is an evolutionary stabilization strategy when $Fq=\frac{dq}{dt}=0$ and $F^{\text{'}}\left(q_i\right)<0$.

When $q=q_0$, then $F\left(p\right)=0$ and $\frac{dF\left(p\right)}{dp}=0;$ when $q=0,$ then $F\left(q=0\right)=0$ and $\frac{dF\left(q\right)}{dq}{|}_{q=0}>0$; when $q=1,$ then $F\left(q=1\right)=0$ and $\frac{dF\left(q\right)}{dq}{|}_{q=1}<0$. Therefore, according to the judgment condition of the evolutionary stability strategy, $q^{*}=1$ is the only evolutionary stable state for the SS.

4.3. Stability Analysis of Evolutionarily Equilibrium Points

According to the method proposed by Friedman [69], the Jacobian matrix can be applied to verify the local stable points of the system and obtain the evolutionary stability strategy (ESS) of the system. The expression of the Jacobian matrix J can be obtained as follows:

$$J=\left[\begin{array}{cc}\frac{dp/dt}{dp}& \frac{dp/dt}{dq}\\ \frac{dq/dt}{dp}& \frac{dq/dt}{dq}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$$

The constituent elements of the Jacobian matrix are as follows:

$$\begin{gathered} a_{11}=\left(1-2p\right)\left\{q\left[\left(\alpha -\beta \right)\left(1-b\right)R_1+\left(\alpha -\beta \right)\left(1-b\right)C_2\right] \\ +\left[\beta \left(1-b\right)R_1+R_1^M-\left(1-\beta \right)\left(1-b\right)C_2-\left(1-b\right)C_1-C_1^M \\ -C_2^M-(1-b)R_2+(1-b)C_3\right]\right\} \end{gathered}$$

$$a_{12}=p\left(1-p\right)\left[\left(\alpha -\beta \right)\left(1-b\right)R_1+\left(\alpha -\beta \right)\left(1-b\right)C_2\right]$$

$$a_{21}=q\left(1-q\right)\left[\left(\alpha -\beta \right)b{R}_1+\left(\alpha -\beta \right)b{C}_2\right]$$

$$a_{22}=\left(1-2q\right)\left\{p\left[\left(\alpha -\beta \right)b{R}_1+\left(\alpha -\beta \right)b{C}_2\right]+b{C}_4+C_3^G\right\}$$

According to the evolutionary equilibrium theory, the stability of the equilibrium points can be judged by whether the following two conditions are satisfied.

Condition 1: $detJ=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|>0$

Condition 2: $trJ=a_{11}+a_{22}<0$

The respective evolutionary findings of the game players indicate that the system has two local equilibrium points, $E_1\left(0,1\right)$ and $E_2\left(1,1\right)$, with the associated Jacobi matrix determinant and trace represented in

Table 3. Values within the Jacobi matrix at the equilibrium points.

Equilibrium Points	$\mathit{det}(J)$	$tr\left(J\right)$
$E_1\left(0,1\right)$	$$\begin{gathered} -\left[\alpha \left(1-b\right)R_1+\left(\alpha -1\right)\left(1-b\right)C_2+R_1^M- \\ \left(1-b\right)C_1-C_1^M-C_2^M-(1-b)R_2+(1- \\ b)C_3\right]\left[b{C}_4+C_3^G\right] \end{gathered}$$	$$\begin{gathered} \left[\alpha \left(1-b\right)R_1+\left(\alpha -1\right)\left(1-b\right)C_2+R_1^M-\left(1-b\right)C_1- \\ {C}_1^M-C_2^M-(1-b)R_2+(1-b)C_3\right]-\left[b{C}_4+C_3^G\right] \end{gathered}$$
$E_2\left(1,1\right)$	$$\begin{gathered} -\left[-\alpha \left(1-b\right)R_1-\left(\alpha -1\right)\left(1-b\right)C_2-R_1^M+ \\ \left(1-b\right)C_1+C_1^M+C_2^M+(1-b)R_2-(1- \\ b)C_3\right]\left[\left(\alpha -\beta \right)b{R}_1+\left(\alpha -\beta \right)b{C}_2+b{C}_4+C_3^G\right] \end{gathered}$$	$$\begin{gathered} -\alpha \left(1-b\right)R_1-\left(\alpha -1\right)\left(1-b\right)C_2-R_1^M+\left(1-b\right)C_1+ \\ {C}_1^M+C_2^M+(1-b)R_2-(1-b)C_3-\left(\alpha -\beta \right)b{R}_1- \\ \left(\alpha -\beta \right)b{C}_2-\left(b{C}_4+C_3^G\right) \end{gathered}$$

.

The analysis of $E_1\left(0,1\right)$ shows that when $\alpha \left(1-b\right)R_1+\left(\alpha -1\right)\left(1-b\right)C_2+R_1^M+(1-b)C_3<\left(1-b\right)C_1+C_1^M+C_2^M+(1-b)R_2$, both conditions 1 and 2 are satisfied, indicating that $\left(0,1\right)$ is an ESS of the system and the corresponding strategy combination is (no change, synergy).

When $\alpha \left(1-b\right)R_1+\left(\alpha -1\right)\left(1-b\right)C_2+R_1^M+(1-b)C_3>\left(1-b\right)C_1+C_1^M+C_2^M+(1-b)R_2$, both condition 1 and condition 2 are satisfied, demonstrating that $\left(1,1\right)$ is an ESS of the system, and the corresponding strategy combination is (change, synergy).

4.4. Analysis of Influencing Factors of the Ideal ESS

The strategy (1,1) accomplishes the synergistic reform in the management mechanism of mixed-ownership reform enterprises, which is undoubtedly the optimal ESS. The following are the analysis of the ideal ESS influencing mechanisms and factors.

Let $$\begin{gathered} H=\alpha \left(1-b\right)R_1-\left(1-\alpha \right)\left(1-b\right)C_2+R_1^M+(1-b)C_3-\left(1-b\right)C_1-C_1^M-C_2^M- \\ (1-b)R_2=[(R_2-C_3)-\left[\alpha R_1-C_1-\left(1-\alpha \right)C_2\right]b-\left[\left(1-\alpha \right)C_2+C_1+C_1^M+C_2^M+R_2 \\ -\alpha R_1-R_1^M-C_3\right]=\mu b-\pi \end{gathered}$$.

Among them, $\mu =[(R_2-C_3)-\left[\alpha R_1-C_1-\left(1-\alpha \right)C_2\right]$, $\pi =\left(1-\alpha \right)C_2+C_1+C_1^M+C_2^M+R_2-\alpha R_1-R_1^M-C_3$.

When $H$ > 0, the evolutionary stability strategy is (1,1). From this inequality, the factors that affect the evolution of (1,1) into ESS include $R_1$, $R_1^M$, $C_3$, $\alpha$, $C_1$, $C_1^M$, $C_2^M$, $R_2$, $C_2$ and b.

Find the partial derivatives of $H$ concerning each parameter, respectively, where the parameters with partial derivatives greater than zero are $R_1$, $R_1^M$, $C_3$ and $\alpha$, and those with partial derivatives less than zero are $C_1$, $C_1^M$, $C_2^M$, $R_2$ and $C_2$.

$$\frac{\partial H}{\partial b}=R_2-C_3-\left[(\alpha R_1-C_1-\left(1-\alpha \right)C_2\right]$$

Since the sign of the partial derivative of $H$ concerning the parameter b is uncertain, the following two scenarios are analyzed in detail.

Scenario 1: When $dH/db=\mu <0$, if $0<b<\frac{\pi }{\mu }$, then the value of the function $H$ is greater than zero, and ESS is (1,1).

Scenario 2: When $dH/db=\mu >0,$ if $\frac{\pi }{\mu }<b<1$, then the value of the function $H$ is more than zero, and ESS is (1,1).

5. Numerical Stimulation Analysis

MATLAB is used to simulate the dynamic evolutionary process by assigning a value to each parameter to visually verify the evolutionary stability of this system and the influence of the shareholding ratio for ESS.

5.1. The Test of Evolutionary Stability of the System

Assume that Company A is a local state-owned mixed-ownership company with more significant social functions. Before the mixed-ownership reform, Company A had a heavy policy burden. Relevant data of Company A, such as hidden income, policy burden, and mechanism reform, are shown in Array 1. Suppose that situations in Company A are similar to those in Company B, except that Company B has a more considerable expectation for the cost of mechanism reform and change control gains, as shown in array 2. Simulations are performed for the values related to Company A and Company B. The results are shown in Figure 3 and Figure 4.

Array 1: $$\begin{gathered} p=0.5,q=0.5,\alpha =0.8,b=0.4,\beta =0.1,R_1=200,R_2=150,C_1= \\ 80,C_2=100,C_3=150,C_4=60,R_1^G=100,C_1^G=70,C_2^G=80,C_3^G=120,R_1^M= \\ 150,C_1^M=75,C_2^M=100,C_3^M=80 \end{gathered}$$.

Array 2: $C_1=200,R_1^M=200$, and other parameters are the same as in array 1.

Based on Figure 3 and Figure 4, the evolutionary stability strategy of Company A is $\left(1,1\right)$, indicating that SS and PS reach a stable strategy combination of synergistic reform. Whereas the evolutionary stability strategy of Company B is $\left(0,1\right)$, implying that PS and SS work together to maintain the original administrative model. The simulation findings are compatible with the conclusions of the analysis.

5.2. The Influence of SS’ Shareholding on Evolutionary Equilibrium

The ownership ratio is the basis for allocating the company’s gains, costs, and losses between the two game players. It influences heterogeneous shareholder action strategies and the rates and consequences of evolutionary equilibrium.

To verify the influence of parameter $b$ on ESS under Scenario1 of Section 4.4, based on array 1, the simulation is performed from both positive and negative directions by taking different values of b. Under array 1, when $\mu <0$, $\frac{\pi }{\mu }=0.58$, taking $b=0.2, 0.3, 0.4 (b<\frac{\mathsf{\pi }}{\mathsf{\mu }})$ respectively, the operation results are depicted in Figure 5. It indicates that the shareholding percentage of SS is lower than the threshold value when the net return of the administrative model is below the net return of the economic model. At this time, the ESS of the system is $\left(1,1\right)$. In addition, the conclusion of the research under Scenario 1 of Section 4.4 is confirmed by the fact that the faster ESS is reached as the shareholding ratio of SS decreases. By contrary, taking $b=0.7, 0.8, 0.9\left(b>\frac{\pi }{\mu }\right)$ separately, the operation results are shown in Figure 6. It demonstrates that the system does not converge to the ideal ESS $\left(1,1\right)$ when the net return of the administrative model is less than the net return of the economic model. Simultaneously, the shareholding percentage of SS exceeds the critical value. Therefore, the analysis conclusion of Scenario 1 is verified from the opposite side. The simulation results are shown in Figure 5 and Figure 6.

To verify the influence of parameter $b$ on ESS under Scenario 2 of Section 4.4, based on array 2 of Company B, the simulation is performed from both positive and negative directions by taking different values of b. Under array 2, when $\mu >0$, $\frac{\pi }{\mu }=0.58$, taking $b=0.7, 0.8, 0.9 (b>\frac{\mathsf{\pi }}{\mathsf{\mu }})$, respectively, the operation results are depicted in Figure 7. It attests that the shareholding percentage of SS is higher than the threshold value when the net return of the administrative model exceeds the net return of the economic model. At this time, the ESS of the system is $\left(1,1\right)$. In addition, the conclusion of the research in Scenario 2 is confirmed by the fact that the faster ESS is reached as the shareholding ratio of SS increases. On the contrary, taking $b=0.2,0.3,0.4\left(b<\frac{\pi }{\mu }\right)$, respectively, the operation results are shown in Figure 8. It demonstrates that the system does not converge to the ideal equilibrium stability strategy $\left(1,1\right)$ when the net return of the administrative model is greater than the net return of the economic model. Simultaneously, the shareholding percentage of SS is below the critical value. Therefore, the analysis conclusion of Scenario 2 under Section 4.4 is verified from the opposite side. The simulation results are shown in Figure 7 and Figure 8.

6. Discussion and Suggestions

6.1. Discussion

This paper constructs the evolutionary game model for the action strategies of heterogeneous shareholders in SOEs in management mode transformation after the introduction of private capital, analyzes their equilibrium strategies, the stability of evolutionary strategies, and their influencing factors, and obtains the following three research results.

(1)

Discussion on the results of the evolution of the strategies of PS and SS

The evolution of PS depends on the “synergy” probability of SS. There is a threshold value $q_0$ for the probability of synergy of SS. When the probability of “synergy” of SS is greater than, equal to, and less than $q_0$, the strategy evolution of PS is “change,” “no change,” and instability, respectively.

From the analysis of the factors influencing of $q_0$, it demonstrates that as long as either of the following two conditions is satisfied, then $q_0$ becomes smaller, and the possibility of PS reform becomes greater. The specific conditions are shown as follows: (1) change dividend, reform success rate, control gains, and policy burden, and any one of the above factors increase; (2) cost of change, contractual cost of PS, and additional cost of reform, and any one of the above factors decrease. In effect, these factors increase the net reform benefits by improving the expected revenue and reducing the reform costs, positively incentivizing PS to “change” or reducing policy revenue and increasing policy burden to force PS to “change.” This is consistent with the viewpoint of protecting the property rights of PS and giving them a voice, as Cai Guilong et al. proposed [47].

“Synergy” is the only stabilization strategy for SS. This finding contradicts Li Jianbiao et al., who argued that the super-shareholder status of state-owned capital would give it a higher bargaining power [15]. The main reason for this opposed conclusion is that the study in this paper is based on the hypothesis that SS and PS pursue profit maximization as economic agents, while Li et al. based their results on an analysis of the particular political and economic environment in which Chinese SOEs are located.

The results of this study provide the theoretical basis for the policy orientation of change in the management system of mixed-ownership companies. It proves that although the special status of SS gives them an advantage in the game of change, in terms of the outcome, the interests of PS should be highly valued, and PS should be guided to choose the strategy of “change” because there is only one stable strategy of “synergy” for SS.

(2)

Discussion of the evolutionary results of the game system

Strategies (0,1) and (1,1) can be ESS for this system. Strategy (0,1) tends to occur in mixed-ownership enterprises in public services or monopolistic industries, as these companies can jointly share bigger implicit government advantages than the change dividend if they do not change. Strategy (0,1) generally comes from competitive mixed-ownership firms because they cannot meet their investment objectives through implicit government benefits and must synergistically change to capture the reform dividend. Although (0,1) results from an evolutionary equilibrium between PS and SS, it is not the goal the state hopes to achieve through the mixed-ownership reform. In contrast, (1,1) satisfies the common reform objectives of the PS, SS, and the state simultaneously, which is the ideal result of the system evolutionary equilibrium.

The results of this study provide a theoretical explanation for the phenomenon of “the mixed-ownership reform without change,” namely, “the passive mixed-ownership reform,” which is common in the practice of mixed ownership reform. Therefore, mixed-ownership reform is only suitable for those in competitive fields, while public services and monopolistic industries are not suitable for now.

(3)

Analysis of factors influencing the ideal equilibrium strategy

The system is conducive to evolving to the ideal evolution equilibrium strategy by satisfying any of the following requirements: (1) change dividends, PS’ control gains, policy burdens, and the success rate of change under synergy, and any one of the above factors increase; (2) cost of corporate reform, loss from the failure of change, cost of private shareholder contracts, additional cost of change for PS, and the implicit revenue under the administrative model, and any one of the above factors decrease. Among them, the effect of the percentage of SS’ ownership on the evolution of the equilibrium strategy is more complex. It depends on the specifics of the net benefit of “change” and the net benefit of “no change.” If the net revenue from “change” exceeds the net income from “no change,” SS’ shareholding should be below a specific threshold value; otherwise, it should be above the threshold value. It is evident that the synergistic transformation of mixed-ownership reform enterprises is the consequence of the combined influence of the equity structure and the difference in net income of companies operating under various types of mechanisms. Suppose the net returns under the market-based model are greater than those under the administrative model. In that case, PS must be given more control to motivate them to lead the mechanism change and achieve a win–win situation for heterogeneous property rights subjects by sharing net returns from the change. Suppose the net gains under the market-based model are less than that under the administrative model. In that case, SS will need a higher shareholding to obtain more dividends from the change to offset the loss from the mechanism shift and accomplish the synergistic change.

The findings of this paper also support the following scholars’ views. The study by Chen Qian et al. [41] showed that the ownership structure of mixed-ownership companies has to be determined according to the company’s actual situation, and there is no absolute standard. Ren Guangqian et al. [49] found that when the expected gains from PS participation in the reform are lower than the losses from the reform, it will discourage their participation and lead to their withdrawal from the mixed-ownership reform. This also explains, to some extent, the phenomenon of different research findings on the optimal shareholding ratio of state-owned shareholders [24,39]. This paper argues that the main reason is that these studies are based on the results obtained under different cost-benefit scenarios.

6.2. Suggestions

Based on the analysis of the game equilibrium and its influencing factors, we propose the following management transformation path for mixed-ownership companies.

(1)

Mixed-ownership should be implemented as the classified reform

Competitive SOEs have no hidden revenue and no policy burden, which are favorable to forming the ideal ESS, (1,1). Therefore, priority should be given to implementing the mixed-ownership reforms in competitive SOEs. For public welfare or monopolistic SOEs, the policy burden or hidden income may cause (0,1) to become an ESS, and the mixed-ownership reform loses its significance. Therefore, policy modifications should be implemented for such SOEs to minimize policy burdens or establish settlement norms.

(2)

Adhere to the principle of priority of PS

“Synergy” is the only evolution stability state for PS, and the ESS of the system depends on PS. Therefore, in heterogeneous shareholder conflicts, we must prioritize meeting PS’ requirements, maintaining PS’ interests, and reducing SS’ reform resistance.

(3)

Separation of control and ownership, giving private shareholders greater control

The ownership of shareholders is mainly related to the distribution of profits, but the control right is primarily articulated as the decision-making powers, which can be completely separated. “Change” probability for PS can be improved by increasing the success rate of reform, improving the gain of control, reducing the loss of internal conflict, and other factors, thus giving PS the control power.

(4)

Providing policy support for change

Government policy support can not only increase the benefits of the success rate of reform, but also reduce the cost of transformation. For example, the firm provides resettlement and compensation for the surplus staff and facilitates financing related to the change.

(5)

Reasonable determination of the shareholding ratio of SS

The SS’ shareholding ratio determines the sharing of benefits, costs, and losses of change between PS and SS. It has a significant impact on the evolutionary stability strategy of the system. Therefore, the shareholding ratio of SS should not only be determined based on the shareholders’ financial strength or negotiating power, but also oriented by the systematic and stable equilibrium strategy and based on the actual situation of the mixed-ownership company.

The study is not free from limitations. Except for PS and SS, this paper does not consider the action strategies of core employees of SOEs, especially managers, in the reform process. The action strategies of PS and SS mainly affect the decision-making of management reform, while the managers are the executors of management reform and have a substantial impact on the progress and result of reform implementation. It is suggested that in future research, tripartite evolution game model among PS, SS, and managers can be developed to analyze their evolutionary stability strategies and influencing factors, etc. Such a research design is more consistent with the actual situation in mixed-ownership companies, and the research results are more reliable. It can better guide the practice of management system change in mixed-ownership companies.