2.1. VC

Research on the VC influence on VC-backed firms generally focuses on three fields: (1) the distribution of management and control rights between VC entities and VC-VC-backed firms. The previous evidence has shown that VC support financially supports start-ups and partially controls the invested companies [23–26]. (2) The second field examines the impact of VC investment on a start-up’s initial public offering (IPO). In this case, some scholars have found that a lower market price in the initial offering may be conducive to shortening a supported firm’s time to go public, improving the reputation of the VC entity. In consequence, on facilitating further access to funds [27, 28], additional evidence also shows that VC-backed start-ups can present better IPO performance [29, 30]. (3) The third field of research explored the impact of VC on start-up performance, where previous studies found that VC support implies a certain degree of control of the start-up decision-making structure through professional training and talent recruiting, among other services while seeking performance improvements [25, 31–33].

The relationship between VC and corporate green innovation has also garnered significant academic attention. Mrkajic et al. [34] examined whether being born-to-be-green sends a credible signal to potential VC investors. Their findings suggest that this signal is only reliable when entrepreneurs engage in activities based on green technologies or products while simultaneously positioning their companies within green industries. Bellucci et al. [35] explored the role of green innovation in attracting VC financing, finding that, compared to other equity financing methods, firms engaging in green innovation are more likely to secure VC funding. Furthermore, the larger the share of green patents in a company’s patent portfolio relative to nongreen patents, the higher the likelihood of obtaining VC investment. Maiti [36], through empirical research, demonstrated that the growth of VC investment positively impacts environment-related technological innovations and the share of renewable energy supply by reducing debt-related financial burdens and mitigating firms’ tendencies to take excessive risks. Similarly, Lin and Xie [37] indicated that VC significantly enhances the green technological innovation performance of Chinese renewable energy firms by increasing monetary capital, alleviating financing constraints, and strengthening corporate R&D investments; VC indirectly promotes green technological innovation.

Likewise, the concept of green venture capital (GVC) has gained widespread attention. GVC refers to VC investments that focus on environmental protection, particularly in sectors such as clean technology, sustainable entrepreneurship, technological innovation, green innovation, and sustainability [38]. Wei et al. [39] clarified the role of GVC by providing empirical evidence of its impact on the adoption of renewable energy in the clean technology sector, proposing that GVC can effectively drive green innovation. Dong et al. [40] examined the relationship between the green innovation parameters of environmentally friendly enterprises and the practical efficiency of GVC. Dhayal et al. [41] further explored why GVC can serve as a catalyst for achieving the United Nations’ 2030 Sustainable Development Goals, highlighting the transition of VC research toward GVC, as well as its connections with clean technology, sustainability, and policy-driven green finance initiatives. Bocken [42] provided an in-depth analysis of how venture capitalists contribute to the success of sustainable enterprises by examining their roles, motivations, investment rationales, and the barriers and enablers for achieving sustainability.

The relationship between VC and CSR behavior has been widely discussed in recent studies [9, 11, 12, 20, 21]. However, there remains no clear consensus on whether VC investment promotes or inhibits CSR behavior in VC-backed firms. One strand of research suggests that VC involvement negatively impacts CSR performance, with studies indicating that start-ups backed by VC investors tend to have lower social responsibility scores. For instance, the authors in [22] found that VC participation may reduce CSR performance in portfolio companies, particularly small- and medium-sized enterprises (SMEs). Their study also identified internal control quality (ICQ) as a key mechanism through which VC influences CSR outcomes.

Conversely, another line of research argues that VC investment can positively influence CSR behavior in supported firms [12]. For example, Li et al. [12] and Cheng et al. [10] suggest that VCs provide a resource endowment, equipping portfolio firms with financial and strategic support that enhances their CSR engagement. Their findings indicate that VC-backed start-ups may experience sustained CSR performance, particularly when investors emphasize long-term value creation, reputation-building, and sustainable business practices.

2.2. CSR

CSR extends the field of decision responsibility beyond the shareholder interest by also considering society, the environment, employees, suppliers, and customers while generating a value [43]. The research on CSR focuses on the enterprise’s motivation to undertake SRP and the factors affecting its practice.

Existing studies have shown that active involvement in SRP may be conducive to enterprise rent-seeking [44], legitimacy gain [45], competitive advantage [46–48], consumer buying willingness [49], and may promote growth and innovation [50]; among other benefits stimulated by government regulations, firms may undertake SRP also to access government subsidies [51] and tax relief measures [52], among other preferential policies.

On the impact of the external environment on CSR, some scholars found that market competition level, national development, and economic openness may impact CSR behavior [53]. Additional evidence also found that other social-related factors may influence CSR attitudes and behavior, such as family upbringing [54] and religious beliefs [55].

Empirical studies have studied the impact of internal governance on CSR behavior, finding that factors such as corporate equity structure [56, 57], management compensation structure, institutional investor shareholding [58], and board composition [59] may constitute influencing factors. Some researchers have indicated that institutional investors can promote CSR [60] and inhibit excessive investment in CSR and self-interested management behavior [61].

2.3. EGT

Originally used by genetic ecologists to analyze the game between cooperative and conflicting behavior among animals [62], EGT’s field of action also extended to economics, sociology, and management, among other fields. Scholars also applied it to study the decision-making process in VC-related areas; for instance, Zheng et al. [63] used EGT to analyze the evolutionary process of VC strategy selection, assuming a framework of limited rationality and exploring the applications of the EGT to the problem of adverse selection. In addition, Chen and Bian [64] used a three-party evolutionary game method to develop a noncooperative evolutionary game model, including VC entities, start-ups, and the government, to study the effect of the government’s direct subsidy policy on VC entities.

Other scholars applied EGT while studying CSR strategy selection [65–67]; for instance, Li and Lai [68] constructed an EGT-based theoretical model for CSR realization in the energy industry, finding that cooperation constitutes a key premise to achieve CRS sustainability and stability. In another study, Nie et al. [69] combined an EGT-based model with system dynamics to perform a simulation analysis to study the dynamic processes between local government, mining enterprises, and the local community.

2.4. Research Gap and Contributions of This Study

3.1. Basic Assumptions

More specifically, each player has two respective choices: A VC entity supports SRP when it utilizes its financial resources, expertise, networks, or other assets to encourage the VC-backed firm’s engagement in CSR activities. Alternatively, a VC entity withholds support by choosing not to provide resources or actively discouraging SRP implementation. From the VC-backed firm’s perspective, performing SRP involves engaging in activities that benefit employees, shareholders, consumers, suppliers, and society at large, while withholding participation in SRP includes actions such as neglecting stakeholder interests, contributing to environmental degradation, or remaining passive on CSR issues.

The cost of implementing SRP is represented by C, which is borne by both the VC entity and the VC-backed firm. Assuming a cost-sharing coefficient β, the cost assumed by the VC entity is βC, while the cost borne by the VC- backed firm is (1 − β)C.

Conversely, if the VC entity withholds support, but the VC-backed firm performs SRP, a reverse free-rider problem arises, where CSR benefits accrue disproportionately to the VC entity while reducing the VC-backed firm’s immediate profits. If a free-rider scenario emerges, the variable L represents the magnitude of CSR-related gains or losses for both players.

The additional benefits provided by the government to both players are denoted by H, with the VC entity receiving αH and the VC-backed firm receiving (1 − α)H, where α represents the VC entity’s shareholding ratio, determining its proportion of government incentives relative to its ownership and influence over the VC-backed firm.

3.2. Model Construction

3.2.1. Evolutionary Game Model Without Government Intervention

Based on the assumptions described in the previous section, the evolutionary game return matrix of the VC entity and the VC-backed firm can be shown in Table 2 as follows.

Table 2. Payment matrix without government supervision.

		VC-backed firm (Player B)
		Perform SRP (y)	Withhold Participation (1 − y)
VC entity (Player A)	Support SRP (x)	R1 + α∆R − βC R2 + (1 − α)∆R − (1 − β)C	R1 − βC − L R2 + L
	Withhold Support (1 − x)	R1 + L R2 − (1 − β)C − L	R1 R2

Drawing from the principles of the evolutionary game [62], when a determined VC entity (Player A) supports the SRP strategy on a determined VC-backed firm (Player B), the expected return E_v1 can be formulated as follows:

$$ ()

Alternatively, when VC does not support the VC-backed firm SRP strategy, the formulation of the expected return E_v2 is shown in the following equation:

$$ ()

The average expected return of the VC mixed strategy E_v is defined as follows:

$$ ()

Therefore, the dynamic replication equation for the VC strategy is expressed as follows:

$$ ()

Comparably, the expected benefit of the VC-backed firm when performing SRP can be formulated as follows:

$$ ()

The expected benefit obtained by the VC-backed firm when it does not perform SRP is expressed as follows:

$$ ()

By combining the two factors mentioned above, the average expected return of the VC-b acked firm mixed strategy E_f can be formulated asfollows:

$$ ()

Thus, the dynamic replication equation of the VC-backed firm strategy can be formulated as follows:

$$ ()

Having defined the dynamic replication equations for VC (4) and the VC-backed firm, equation (8) has the possibility to construct the two-dimensional dynamic system as follows:

$$ ()

3.2.1.1. Evolutionary Stability Analysis: VC Entity Strategy

Considering the evolutionary stability of the player, VC entity strategy, its function is validated for F(x) = dx/dt and is expressed as follows:

$$ ()

Then, y = (βC + L)/α∆R and F(x) = 0 can be calculated, and the system will be in a stable state for any value of x; otherwise, when y ≠ (βC + L)/α∆R, then for F(x) = 0, x = 0 and x = 1 can be the steady state of x. According to the stability theorem of differential equations, when $$, then x^∗ is the evolutionary stable strategy; in combination with equation (10) being possible to affirm that since α∆R > 0, the stable strategy of VC depends on y when y < (βC + L)/α∆R due to dF(x)/dx|_x=0 < 0, dF(x)/dx|_x=1 > 0; therefore, x = 0 can be defined as an evolutionary stable strategy.

When y > (βC + L)/α∆R due to dF(x)/dx|_x=0 > 0, then dF(x)/dx|_x=1 < 0; therefore, x = 1 can also be defined an evolutionary stable strategy.

3.2.1.2. Evolutionary Stability Analysis: VC-Backed Firm Strategy

The stability of the VC-backed firm strategy is validated through F(y) = dy/dt and by expressing its function as follows:

$$ ()

When x = [(1 − β)C + L]/[(1 − α)∆R], then F(y) = 0 can be calculated, and the system will be in a stable state for any value of y; on the other hand, when x ≠ [(1 − β)C + L]/[(1 − α)∆R], for F(y) = 0, then y = 0 and y = 1 can be the steady state of y, and according to the stability theorem of differential equations, when $$, then y^∗ is the evolutionary stable strategy that in combination with equation (11) being possible to define the stable strategy as follows:

When [(1 − β)C + L]/[(1 − α)∆R] > 0, the stable strategy of VC (VC) depends on x; when x < [(1 − β)C + L]/[(1 − α)∆R], due to dF(y)/dy|_y=1 > 0, then dF(y)/dy|_y=0 < 0; therefore, y = 0 can be defined as an evolutionary stable strategy.

When x > [(1 − β)C + L]/[(1 − α)∆R], due to dF(y)/dy|_y=1 > 0, then dF(y)/dy|_y=0 > 0; therefore, y = 1 is also an evolutionary stable strategy.

3.2.1.3. Evolutionary Stability Analysis for the System S₁

This analysis will consider two scenarios, the first when T > t and the second when T < t.

3.2.1.3.1. When T > t

As seen in the analysis in the previous section, depending on the initial conditions, the system will display distinct evolutionary strategies with five equilibrium points:

$$ ()

Since the local equilibrium point is solved through the dynamic equation, it does not always determine the system’s evolutionary stability strategy (ESS); thus, as proposed by the authors in [68, 70], the stable equilibrium point of the system is obtained by analyzing the local stability of the system by the Jacobian matrix that can be formulated as follows:

$$ ()

Therefore, the equilibrium point of the replicated dynamic equation will determine the system’s ESS if the following two conditions are satisfied:

• i.

  $$

• ii.

  tr(j) = C₁₁ + C₂₂ < 0

Based on the Jacobian matrix local stability analysis, the values (negative or positive) of det(j) and tr(j) allow the stability identification for each of the five equilibrium points, as shown in Table 3.

Table 3. Partial stability of the CSR strategy evolution system when T > t without supervision.

Equilibrium point	det(j)	tr(j)	Stability
A(0, 0)	> 0 (+)	< 0 (−)	ESS
B(1, 0)	> 0 (+)	> 0 (+)	Unstable
C(1, 1)	> 0 (+)	< 0 (−)	ESS
D(0, 1)	> 0 (+)	> 0 (+)	Unstable
O(a, b)	< 0 (−)	= 0 (0)	Saddle point

As shown in Table 3, the equilibrium points, where det(j) > 0 and tr(j) < 0, satisfy the conditions for the system to reach a stable equilibrium. These correspond to two Pareto-optimal strategies: (Withhold Support, Withhold Participation) and (Support SRP, Perform SRP). The remaining points represent unstable states, indicating that the strategies (Withhold Support, Perform SRP) and (Support SRP, Withhold Participation) cannot lead to a stable system equilibrium. As a result, the system undergoes dynamic adjustments until it converges toward one of the two stable equilibrium points (A or B). Additionally, O points serve as saddle points, representing critical states in EGT that determine the system’s trajectory.

Figure 1 presents the phase diagram of the system’s evolution in the absence of government regulation, illustrating how the system evolves when VC entities favor long-term or responsible investment strategies.

As shown in Figure 1, the segments connecting points B, O, and D form the critical trajectory that determines the system’s convergence toward one of two stable equilibrium states. The prevalence of the lower-left region (S_ABOD) may drive the system toward the stable equilibrium A(0, 0), where both players ultimately adopt the strategy (Withhold Support, Withhold Participation). Conversely, the dominance of the upper-right region (S_BCDO) may lead the system to converge toward the stable equilibrium C(1, 1), where both players adopt the strategy (Support SRP, Perform SRP).

Thus, the final state of the evolutionary game between the VC entity and the VC-backed firm depends on the relative sizes of S_ABOD and S_BCDO. A larger S_ABOD area increases the probability of both players converging toward (Withhold Support, Withhold Participation), while a larger S_BCDO area corresponds to a higher likelihood of (Support SRP, Perform SRP) convergence.

The analysis of the changes of parameters in the payment matrix of the game model allows us to define its influence on the evolutionary strategy as follows:

$$ ()

Four main factors influence the changes in the S_ABOD area: (1) excess return ∆R, (2) SRP cost C, (3) VC shareholding ratio α, and (4) variation of SRP benefits when there is “free rider” behavior L. Discussion on each one is detailed below:

• 1.

  Excess return ∆R

•  

  The derivative of ∆R through the formula of the area for S_ABOD results in a monotonic subtractive function of d∆R:

  $$ ()

•  

  Any increase in ∆R implies a decrease in S_ABOD, the respective increase in S_BCDO and the probability that both players evolve into (Support SRP, Perform SRP). In contrast, any decrease in ∆R will be related to an increase in S_ABOD, a decrease in S_BCDO, and the increase in the probability that the game scenario evolves to the SRP negative setting (Withhold Support, Withhold Participation). Then, the increase in excess returns can effectively promote VC to actively support the VC-backed firm in their CSR activities boosting any related initiative that may emerge within the start-up.

• 2.

  SRP cost C

•  

  The derivative of C through the formula of the area for S_ABOD results in a monotonically increasing function for DC:

  $$ ()

•  

  In other words, an increase in C implies an expansion of the area S_ABOD; thus, the corresponding reduction of S_BCDO and the probability that the game will evolve into the (Withhold Support, Withhold Participation) setting. On the other hand, in the opposite scenario, a decrease in C will reduce the area S_ABOD, increasing the corresponding area for S_BCDO, increasing the probability for the game players to evolve into the (Support, Perform SRP) setting; thus, to seek this scenario, the VC-backed firm should effectively control the cost of SRP.

• 3.

  VC shareholding ratio α

•  

  The derivative of α through the formula of the area for S_ABOD results in a nonmonotonic function for dα:

  $$ ()

•  

  Nevertheless, the quadratic derivative for α does result in a monotonically increasing function:

  $$ ()

•  

  Therefore, S_ABOD is a concave function of α with a specific point where S_ABOD can be minimized, maximizing the probability of the system evolving into (Support, Perform SRP) setting.

•  

  Furthermore, when

  $$ ()

•  

  then,

  $$ ()

•  

  In this case, the probability of the players reaching the ideal evolution state is the largest. Therefore, there is an optimal VC shareholding ratio able to maximize the possibility of the game achieving the (Support, Perform SRP) strategy.

• (4)

  Variation of SRP benefits when there is “free rider” behavior L

•  

  The derivative of L through the formula of the area for S_ABOD results in a monotonically increasing function for dL:

  $$ ()

It means that an increase for L will be subsequently linked to an increase for S_ABOD; thus, to decrease the area for S_BCDO as well as the probability of both players evolving to (Withhold Support, Withhold Participation). In contrast, a decrease for L will imply a reduction in S_ABOD, an increase in S_BCDO, and the subsequent probability for the game to evolve into (Support, Perform SRP) strategy.

As seen in the function, the emergency of a free-riding behavior is not conducive for the system to evolve into the ideal stable state, meaning that to make the system converge into the (Support, Perform SRP) stable state, it is necessary to formulate scientific and reasonable measures that may reduce the occurrence of “free riding” behavior.

3.2.1.3.2. When T < t

In this situation, VC cannot perceive the returns of CSR activities in its investment cycle. Without the effect of government supervision, the system stability solutions with their corresponding det(j) and tr(j) values (negative or positive) for each one of the four equilibrium points are shown in Table 4 as follows:

Table 4. Partial stability of CSR strategy evolution system when T < t without supervision.

Equilibrium point	det(j)	tr(j)	Stability
A(0, 0)	> 0 (+)	< 0 (−)	ESS
B(1, 0)	< 0 (−)		Saddle point
C(1, 1)	> 0 (+)	> 0 (+)	Unstable
D(0, 1)	< 0 (−)		Saddle point

As shown in Table 4, the equilibrium point A(0, 0) satisfies the conditions for a stable equilibrium, as its det(j) is greater than 0 and tr(j) is less than 0. This means that A(0, 0) represents a stable equilibrium strategy for the system, corresponding to the (Withhold Support, Withhold Participation) strategy adopted by both players.

On the other hand, C(1, 1) is unstable, meaning that the strategies (Withhold Support, Perform SRP) and (Support SRP, Withhold Participation) do not lead the system to stability. This indicates that when the VC entity operates under a short investment cycle, it becomes unable to perceive the long-term returns of CSR engagement by the VC-backed firm. As a result, both players are more likely to evolve toward the (Withhold Support, Withhold Participation) equilibrium state.

3.2.2. CSR Evolutionary Game Model Under Government Supervision

Based on the assumptions in the previous section, the evolutionary game return matrix for each player in the scenario with government supervision is shown in Table 5 as follows:

Table 5. Payment matrix under government supervision.

		VC-backed firm
		Perform SRP (y)	Withhold Participation (1 − y)
VC entity	Support SRP (x)	R1 + α∆R − βC + αH R2 + (1 − α)∆R − (1 − β)C + (1 − α)H	R1 − βC − L R2 + L
	Withhold Support (1 − x)	R1 + L R2 − (1 − β)C − L	R1 R2

By following a similar procedure as the one described in Section 3.2.1, the two-dimensional dynamic system S₂ for the game under government supervision can be developed as follows:

$$ ()

From this step, it is possible to determine the five equilibrium points corresponding to the system S₂:

E(0, 0); F(1, 0);  G(0, 1); H(1, 1); and P(a₁, b₁) where a₁ = [(1 − β)C + L]/[(1 − α)(∆R + H)] and b₁ = [βC + L]/[α(∆R + H)]; then, the Jacobian matrix J₂ for the system S₂ is calculated as follows:

$$ ()

Moreover, the corresponding values of det(J₂) and tr(J₂) are therefore calculated as follows:

$$ ()

With these calculations, the local stability of the investment on CSR strategy evolution in a system under government supervision S₂ is discussed for two scenarios: when T > t and when T < t.

3.2.2.1. When T > t

The local stability for this specific case is shown in Table 6, displaying two stable points for the system S₂ corresponding to the Pareto optimal solutions (Withhold Support, Withhold Participation) and (Support SRP, Perform SRP). F(1, 0) and H(0, 1) are unstable system points, and P(a₁, b₁) represent the corresponding saddle point.

Table 6. Partial stability of the CSR strategy evolution system when T > t under government supervision.

Equilibrium point	det(j)	tr(j)	Stability
E(0, 0)	> 0 (+)	< 0 (−)	ESS
F(1, 0)	> 0 (+)	> 0 (+)	Unstable
G(1, 1)	> 0 (+)	< 0 (−)	ESS
H(0, 1)	> 0 (+)	> 0 (+)	Unstable
P(a1, b1)	< 0 (−)	= 0 (0)	Saddle point

As shown in Table 6, the equilibrium points E(0, 0) and G(1, 1) satisfy the conditions for a stable equilibrium, as their det(j) values are greater than 0 and tr(j) values are less than 0. This means that these two points serve as the system’s stable equilibrium states, corresponding to two Pareto-optimal strategies: (Withhold Support, Withhold Participation) and (Support SRP, Perform SRP).

Conversely, F(1, 0) and H(0, 1) are unstable points, indicating that the strategies (Withhold Support, Perform SRP) and (Support SRP, Withhold Participation) do not lead the system to stability. In such cases, the system will continue dynamic adjustments until it eventually converges to the stable equilibrium points E or G. Additionally, P(a₁, b₁) functions as a saddle point, representing a critical threshold in the evolutionary game where the system’s trajectory is highly sensitive to initial conditions.

Similarly, based on these findings, the phase diagram of the evolutionary game between VC entities and VC-backed firms under government supervision can be derived, as illustrated in Figure 2.

As shown in Figure 2, the line connecting points H, P, and F forms the critical boundary that determines the system’s convergence toward one of the two stable equilibrium states. In the lower-left region (S_EFPH), the system converges toward equilibrium point E(0, 0), where both players ultimately adopt the strategy (Withhold Support, Withhold Participation). Conversely, in the upper-right region (S_HPFG), the system converges toward equilibrium point G(1, 1), where both players eventually adopt the strategy (Support SRP, Perform SRP).

By comparing Figure 2 with Figure 1, it is evident that S_BCDO < S_HPFG. The primary reason for this difference lies in the role of government supervision. Compared to the system without government supervision (S₁) and its saddle point O(a, b), the system under government supervision (S₂) introduces the incentive factor H, which determines the location of the saddle point P(a₁, b₁). As a result, the presence of government incentives alters the evolutionary dynamics, increasing the probability that both players will ultimately adopt (Support SRP, Perform SRP).

Therefore, the impact of government supervision on the evolutionary game strategy can be analyzed by examining the effect of H on the system’s equilibrium states. This highlights the regulatory role of government intervention in influencing CSR engagement within VC-backed firms and shaping long-term investment strategies for VC entities.

Since the government rewards firms involved in CSR, it is reasonable to assume that H > 0 and its consequent effect while calculating a₁:

$$ ()

The introduction of government supervision leads to S_ABOD > S_EFPH which reduces the likelihood of the system evolving toward the (Withhold Support, Withhold Participation) equilibrium. Instead, the system becomes more likely to converge toward the alternative stable solution (Support SRP, Perform SRP). This indicates that as the government’s incentive (H) for firms actively engaging in CSR increases, the probability of both players adopting (Support SRP, Perform SRP) also rises.

3.2.2.2. When T < t

It is assumed that the government rewards firms actively involved in CSR activities. However, their role is usually limited to a guiding position, and the amount of this reward is often smaller than the cost assumed by VC to stimulate the VC-backed firm to perform SRP, implying that αH < βC; (1 − α)H < (1 − β)H.

Having included the government supervision effect in the game system S₂, the system stability solutions from the corresponding det(j) and tr(j) values are shown in Table 7 as follows:

Table 7. Partial stability of the CSR strategy evolution system when T < t under government supervision.

Equilibrium point	det(j)	tr(j)	Stability
E(0, 0)	> 0 (+)	< 0 (−)	ESS
F(1, 0)	< 0 (−)		Saddle point
G(1, 1)	> 0 (+)	> 0 (+)	Unstable
H(0, 1)	< 0 (−)		Saddle point

As shown in Table 7, the equilibrium point E(0, 0) satisfies the conditions for a stable equilibrium, as its det(j) is greater than 0 and tr(j) is less than 0. This means that E(0, 0) serves as the system’s stable equilibrium state, corresponding to the strategy (Withhold Support, Withhold Participation) adopted by both players.

Conversely, F(1, 0) and H(0, 1) are saddle points, indicating that the strategies (Withhold Support, Perform SRP) and (Support SRP, Withhold Participation) do not lead the system to stability. This suggests that, regardless of government support, the evolutionary trajectory remains unchanged compared to the system without government supervision (S₁). In other words, even if the government provides incentives for CSR engagement and if the VC entity operates under a short investment cycle that prevents it from realizing the benefits of supporting SRP, the system will still evolve toward (Withhold Support, Withhold Participation).

4.1. Effect of Players’ Initial CSR Preference on the System Evolution

The influence of the players’ initial willingness to undertake SRP on the evolution of a system without government supervision (H = 0) is illustrated in Figure 3 below. In the figure, VC entities are represented by red lines, while VC-backed firms are represented by green lines.

Keeping the above parameters unchanged, we assume that the initial willingness of both the VC entity and the VC-backed firm to undertake CSR is the same.

The results displayed in Figure 3 indicate that when both players have a low initial preference for SRP (below 0.3 in this case), the system tends to evolve toward (0, 0), meaning (Withhold Support, Withhold Participation). However, as the initial preference increases (up to 0.4 in this case), the system’s evolutionary trend shifts. While the VC-backed firm’s willingness to engage in SRP steadily increases, the VC entity’s willingness to support SRP initially declines before stabilizing, ultimately leading the system to converge toward (1, 1) or (Support SRP, Perform SRP). Furthermore, as initial preferences increase further, the system converges toward the (1, 1) stable state at a faster rate.

The primary reason for this behavior is that in the system, the VC entity primarily provides capital, management, and resource support, while the VC-backed firm is responsible for implementing SRP. Since CSR investment incurs immediate costs for the VC-backed firm while offering delayed returns, its willingness to engage in CSR is relatively low in the early stages, especially when it faces financial constraints and long CSR investment cycles. However, as the VC entity becomes involved and influences the VC-backed firm through its investment philosophy, the firm’s CSR willingness is affected accordingly. If the VC entity does not support SRP, the VC-backed firm’s motivation for CSR diminishes, leading the system to evolve toward (Withhold Support, Withhold Participation). Conversely, if the VC entity actively provides financial and resource support, the VC-backed firm’s willingness to undertake CSR gradually strengthens, driving the system toward (Support SRP, Perform SRP).

These findings demonstrate that VC support plays a crucial role in influencing the willingness of VC-backed firms to engage in CSR. Financial and resource assistance enhances a firm’s motivation and capacity, encouraging it to adopt more proactive CSR practices.

Figure 4 shows the simulation results by including government supervision (H > 0) in the evolutionary system.

As shown in Figure 4, in comparison with Figure 3, it becomes evident that when both players have a low initial preference for CSR (0.3, 0.3), government supervision effectively stimulates the system, increasing the players’ willingness to engage in SRP to a level that drives the system toward the stable state (Support SRP, Perform SRP). On the other hand, when the initial CSR preference is relatively high (above 0.4), the presence of government supervision accelerates the system’s convergence toward the optimal equilibrium (1, 1).

When both the VC entity and the VC-backed firm initially have a low willingness to undertake CSR, such as at x = 0.3 and y = 0.3, the introduction of government supervision gradually strengthens their motivation to engage in CSR practices. Consequently, the system ultimately evolves toward the ideal stable state (1, 1), corresponding to (Support SRP, Perform SRP). A comparative analysis of Figures 3 and 4 indicates that when both players have a higher initial CSR preference, government supervision accelerates the system’s evolution toward the optimal equilibrium (1, 1).

These findings demonstrate that government regulatory policies—such as reward and penalty mechanisms, financial subsidies, and other incentive structures—can significantly influence stakeholders’ strategic choices. By guiding investment behavior, such policies encourage both VC entities and firms to adopt strategies that support and implement CSR. This highlights the crucial role of institutional environments in shaping CSR engagement and suggests that well-designed regulatory incentives and constraints can effectively promote responsible behavior in both firms and capital markets.

4.2. The Influence of VC Preference Variations on the System Evolution

Keeping other parameters unchanged, Figures 5 and 6 illustrate the evolutionary path of the VC-backed firm’s CSR engagement under two scenarios: with and without government supervision, when the VC entity’s preference for CSR practices changes.

As shown in Figure 5, under the evolutionary game scenario without government supervision, the VC-backed firm’s initial willingness (y) to undertake SRP is set at a fixed value (y₀ = 0.4). To evaluate the influence of different levels of the VC entity’s initial preference for CSR on the system’s final convergence, the simulation considers three different values: x₀ = 0.1, x₀ = 0.3, and x₀ = 0.6.

When the VC entity’s preference for supporting SRP is extremely low (e.g., x = 0.1), the system evolves toward (0, 0), meaning (Withhold Support, Withhold Participation). However, as x increases, the VC-backed firm’s willingness to engage in SRP also strengthens. For instance, when the initial x-value is set to 0.3, the firm’s willingness (y) first increases, but as x gradually declines, y also decreases until reaching zero. This indicates that the VC entity, as a resource allocator, plays a crucial role in guiding CSR engagement. Without external support, firms may abandon CSR initiatives due to resource constraints or cost-benefit considerations.

Conversely, when the VC entity demonstrates a stronger preference for CSR (e.g., x = 0.6), the VC-backed firm’s CSR engagement (y) steadily increases and approaches 1. Meanwhile, x initially declines but later increases again, leading the system to ultimately evolve toward (1, 1), representing (Support SRP, Perform SRP).

These results demonstrate that the VC entity’s CSR preference significantly influences the CSR engagement of VC-backed firms. The relationship between VC support and corporate CSR behavior forms a positive feedback loop—proactive investment in CSR encourages firms to engage in socially responsible practices, and in turn, firms’ active CSR participation strengthens their collaboration with VC entities.

As shown in Figure 6, when government supervision is introduced, the system’s evolutionary speed significantly increases compared to the scenario without government regulation. Additionally, when the initial values of x and y remain the same, the probability of the system converging to (1, 1) is higher under government supervision.

For instance, when the VC entity’s willingness to support CSR is relatively low (x = 0.3), the system no longer evolves toward the (0, 0) stable state, as observed in the absence of government supervision. Instead, it gradually converges to (1, 1), representing (Support SRP, Perform SRP). This reinforces the idea that government supervision plays a guiding role in encouraging CSR engagement, promoting VC entities to actively support CSR implementation in VC-backed firms. Even if a VC entity initially exhibits a low willingness to support CSR, government incentives and regulatory guidance can gradually drive it toward a more proactive CSR approach. In response, the VC-backed firm’s willingness to engage in CSR also increases, ultimately leading the system to stabilize at (Support SRP, Perform SRP).

However, when the VC entity’s initial willingness to support CSR is extremely low (e.g., x = 0.1), the VC-backed firm eventually chooses not to engage in CSR (y = 0). This suggests that while government supervision can encourage CSR adoption among firms and investors, its influence has limitations. If a VC entity completely lacks the willingness to support CSR, government incentives and penalties alone may be insufficient to alter the system’s evolutionary trajectory. To prevent the system from falling into the undesirable (0, 0) equilibrium, policy design should focus on enhancing VC entities’ willingness to support CSR, ensuring a more robust and sustainable CSR ecosystem.

4.3. The Influence of Start-Up Preference Variations on the System Evolution

Figures 7 and 8 illustrate the evolutionary path of strategy selection for both players under two scenarios—with and without government supervision—when the initial willingness (y) of the VC-backed firm to undertake CSR varies, while keeping other parameters unchanged.

The simulation results in Figure 7 indicate that when the VC entity’s initial willingness to support CSR remains unchanged, the absence of government supervision leads to varied system evolution outcomes. Even when the VC entity has a moderate initial preference for supporting CSR (x = 0.4), if the VC-backed firm’s willingness to undertake CSR is low (e.g., y = 0.1 or y = 0.3), the system still evolves toward the (Withhold Support, Withhold Participation) stable state. However, when the VC-backed firm’s initial willingness to engage in CSR is relatively high (e.g., y = 0.6), the system converges toward the ideal equilibrium (Support SRP, Perform SRP).

These findings suggest that even if the VC entity’s initial willingness to support CSR is moderate, a strong intrinsic motivation from the VC-backed firm can drive the system toward the positive evolution. This implies that VC-backed firms must exhibit sufficient proactive engagement in CSR from the early stages to fully leverage the support provided by VC entities. When such synergistic interactions occur, they create a reinforcing feedback loop that drives the system toward the optimal stable state (Support SRP, Perform SRP).

The simulation results in Figure 8 indicate that under government supervision, the VC-backed firm’s willingness to engage in CSR gradually increases. Although in cases where the initial willingness is extremely low (e.g., y = 0.1), the system still ultimately evolves toward (0, 0), the probability of reaching the (1, 1) stable state increases compared to the scenario without regulation.

By comparing the results of Figures 7 and 8, it becomes evident that when the VC-backed firm’s initial willingness to undertake CSR is relatively low (e.g., y = 0.3), the system without supervision is more likely to fall into the undesirable equilibrium (0, 0). However, when government supervision is introduced, the system tends to evolve toward the optimal stable state (1, 1). This demonstrates that government incentive mechanisms, such as rewards, penalties, and subsidies, can help firms overcome the initial lack of motivation for CSR, thereby enhancing the overall stability of the system.

However, when the initial willingness is extremely low (e.g., y = 0.1), the system may still remain at (0, 0) despite government intervention. This reflects a threshold effect in government regulation—if the VC-backed firm’s initial willingness to engage in CSR falls below a critical level, regulatory measures may not be sufficient to completely alter the system’s evolutionary trajectory.

These simulation results highlight the long-term positive effects of government supervision in promoting CSR engagement across the system. By guiding VC-backed firms toward active CSR participation, government supervision not only improves short-term evolutionary trends but also establishes a foundation for long-term sustainable development.