Research on the Path of Policy Financing Guarantee to Promote SMEs’ Green Technology Innovation

Abstract

In the process of policy financing guaranteeing help to SMEs to make innovations in green technologies, multiple parties continue to play strategic games for their interests. Evolutionary game theory is a practical tool for analyzing multi-agent strategies, which can help us to explore how policy financing guarantees help to SMEs to achieve effective credit enhancement. This paper constructs a four-party evolutionary game model among SMEs, banks, guarantee agencies, and the government, and obtains four evolutionary stable strategies by analyzing various players’ replicator dynamics. In addition, we carry out numerical simulations on the key parameters affecting the stability of the game system. The findings suggest that keeping the fixed risk-ratio between guarantee agencies and banks constant reduces the government’s financial burden and strengthens the re-guarantee system’s construction at the initial stage of SME financing, which can indirectly increase the enthusiasm for cooperation between banks and guarantee agencies. The interest subsidy policy is more effective in promoting SMEs’ compliance and bank–guarantee cooperation in the short term. Meanwhile, the government should increase the supervision of defaulting SMEs and cooperate with financial institutions to improve the credit system for SMEs.

1. Introduction

Green technology is the general term for methods, processes, and products that can reduce environmental pollution, raw material use, and energy consumption [1]. Many research results show that green technology innovation (GTI) plays an important role in promoting carbon emission reduction [2], improving resource utilization [3], reducing production costs [4], and enhancing the financial performance [5] and reputation [6] of enterprises. GTI is a critical way to reconcile environmental protection and economic growth. However, higher innovation costs [7], “double externalities” [8], and higher financing costs [9] reduce enterprises’ willingness and motivation to engage in GTI. Meanwhile, SMEs are an important part of the world economy, but in most developing countries, they have a low technology level and serious energy consumption and pollution in the production process. SMEs are an essential part of the world economy, but in most developing countries, the low technology level in SMEs leads to serious energy consumption and pollution in the production process [10,11]. Therefore, the importance of GTI by SMEs to social development is self-evident; however, SMEs currently face serious financing difficulties in GTI. There is severe information asymmetry between banks and SMEs, because SMEs are asset-light, lack collateral and pledges, and their credit is difficult to assess. In addition, the R&D of green technology has high risks, so it is difficult for SMEs to obtain green finance [12]. Therefore, achieving effective credit enhancement for SMEs is the key to promoting green innovation in SMEs.

Given SMEs’ “quasi-public goods” nature, it is necessary to apply the “government and market sharing” principle to solve their financing problems. It is necessary for the government, which is responsible for public services, to provide moderate credit support [13,14]. The long-standing unbalanced risk–profit ratio between banks and guarantee agencies has seriously hindered the further development of guaranteeing businesses [15]. However, the policy-financing guarantee endorsed by government credit can further improve the credit rating of SMEs and reduce the probability of systemic risk for banking and guarantee agencies [16]. International experience proves that policy financing guarantee is an effective way to enhance credit [17]. Many countries, such as the United States of America, Japan, and Germany, have developed policy credit guarantees [18]. Since 2014, China has gradually established a policy financing guarantee system to reduce the cost and threshold of SME loans through tripartite cooperation between the government, banks, and guarantee agencies, which has significantly increased banks’ confidence in SME credit support and effectively solved the “financial exclusion” of SMEs by banks and guarantee agencies [19]. To sum up, the policy financing guarantee is a crucial way to address the financing problems of SMEs. At present, there is less research and practice on using policy financing guarantee mechanisms to support the GTI of SMEs. This paper studies the specific cooperation model of government–banks–guarantee agencies to help SMEs apply and innovate green technologies, which is essential to make up for the shortcomings of GTI in the market economy.

Evolutionary game theory originated from biological evolution theory [20], and the core concepts are the evolutionary stable strategy [21] and replicator dynamic [22]. The theory believes that a group’s decision-making can be achieved through dynamic behaviors such as imitation, communication, and learning among individuals, and strategies with high adaptability are more likely to be imitated by other participants; otherwise, the strategies will be eliminated [23]. Classical game theory assumes that the participants are completely rational and have complete information. In contrast, evolutionary game theory is based on bounded rationality and limited information, which is more in line with economic reality. Evolutionary game theory is widely used in the economic field and has become a critical mathematical tool for solving complex multi-agent game problems. There are many participants with limited rationality and limited information in policy financing guarantees helping SMEs’ GTI. Moreover, the subjects in the policy financing guarantee do not make decisions in isolation, but rather are in the same system of credit enhancement and risk sharing, working together to solve the financing problems of SMEs. Evolutionary game theory will help us to better analyze the key factors that affect financial support for SMEs’ innovation strategies and provide some inspiration for alleviating SMES’ financing difficulties.

According to the above analysis and China’s current policy financing guarantee model, we construct a four-party evolutionary game model among the government, banks, policy guarantee agencies, and SMEs. We aim to find the optimal risk-sharing ratio between the government, banks, guarantee agencies, and re-guarantee agencies and leverage the policy financing guarantee to help solve SMEs’ credit dilemma in GTI.

The main contributions of this paper are as follows:

(1) Current research focusing on measuring the risk-sharing ratios between banks and guarantee agencies [24,25], or between guarantee agencies and re-guarantee agencies [26], lacks an overall analysis of the behavior of the government, guarantee agencies, re-guarantee agencies, banks, and enterprises, and their mutual influence. However, we study the different strategic choices and possible interactions between the main stakeholders in the process of SMEs seeking financial guarantees for GTI, i.e., SMEs, banks, financial guarantee agencies, and the government, and analyze how each game player uses policy financing guarantees to promote SMEs’ green transformation.

(2) Most of the literature focuses on the impact of green finance and government subsidies on enterprises’ GTI, but we study specific guarantee risk-sharing approaches to alleviate the financing dilemma of enterprises’ green technology upgrading.

(3) Existing research points out that compared with interest subsidies, guarantee fee subsidies are not conducive to promoting green investment [27]. However, we find that the differences between interest subsidies and guarantee fee subsidies are slight in the long term. In the short term, interest subsidies are more effective in promoting compliance and cooperation between banks and guarantee agencies. Moreover, under the condition that the guarantee agency and the bank bear a fixed risk ratio, reducing the government’s risk-sharing ratio and strengthening the re-guarantee system’s construction can effectively promote the cooperation enthusiasm of banks and guarantee agencies.

The remainder of this paper is organized as follows: Section 2 reviews the literature related to financial support for SMEs’ GTI and the policy financing guarantee system development. Section 3 constructs a four-party evolutionary game model among SMEs, banks, financial guarantee agencies, and the government. Evolutionary stability analyses of the individual game players and the game system are provided in Section 4. In Section 5, we describe numerical simulations of the main parameters. Finally, in Section 6, we summarize our findings and policy recommendations.

2. Literature Review

This paper is related to two streams: the financing of GTI and policy credit guarantee systems. We review the related literature in the following.

2.1. The Current State of Financing for Green Technology Innovation

GTI cannot develop without financial support, and green innovation by companies is also conductive to obtaining loans [28]. Although listed companies can carry out green innovation through debt financing and equity financing [29], it is difficult for most SMEs to obtain traditional financial support due to the lack of guarantees and collateral and banks’ risk-aversion to green technologies and projects [30]. Hong et al. [31] analyzed the relevant data of 2825 listed companies in China and found that green credit can promote the GTI of enterprises. However, green credit guidelines do not significantly promote the GTI of SMEs and non-state-owned enterprises. Moreover, the current structure of China’s green financial market is relatively homogeneous [32] and lacks a corresponding green guarantee system, which cannot meet the needs of the GTI market. In addition, the existing literature has different views on the relationship between government subsidies and GTI. Liu et al. [33] found that government subsidies can alleviate enterprises’ financing difficulties and effectively promote green innovation. However, Howell et al. [34] pointed out that government subsidies reduce innovation inputs in low-tech industries. Bai et al. [35] found that the effect of government subsidies on firms’ green innovation is heterogeneous.

As the bridge between financing enterprises and banks, guarantee agencies provide credit guarantees for SMEs, which enables banks to transfer most of the credit risk [36] and increase the loan acquisition rate of SMEs. However, SMEs still face the following dilemmas when seeking green financing guarantees. First, the form of guarantee for SMEs is dominated by the parent companies providing guarantees for their subsidiaries, and enterprises with insufficient collateral and weak counter-guarantee capacity cannot obtain credit guarantee services from third-party guarantee agencies [30]. Second, the guarantee rates of commercial guarantee agencies and banks’ loan rates will all increase to control credit risks, which may aggravate SMEs’ debt burden. Third, cyclical industries such as equipment manufacturing are prone to concentrated defaults when the market boom declines [37]. Thus, guarantee agencies’ compensatory risk and banks’ bad debt rate will increase, which will reduce the enthusiasm of banks and guarantee agencies to serve SMEs in concert.

Green technology financing has the characteristics of large scale, long term, high cost, and high risk, which inhibits investors’ enthusiasm. Meanwhile, SMEs are the backbone of social development. In response to the above financing problems SMEs face, it is necessary to seek new credit enhancement and risk-sharing mechanisms to help the government and financial institutions better serve SMEs’ green transformation.

2.2. Policy Credit Guarantee Development Status

Due to the information asymmetry between banks and enterprises, there are adverse selections and moral hazards in the financing of SMEs [38], which leads to the problem of “financing [being] difficult and expensive” for SMEs in all countries. The key to improving the availability of loans is to improve SMEs’ credit levels. Since the 20th century, governments worldwide have established financing guarantee mechanisms for enhancing SMEs’ credit levels. The literature analyzed credit guarantee policies in countries and regions such as the United States of America [39], Japan [40], the Middle East and North Africa region [41], South Korea [42,43], and China [18]. Despite significant differences in loan pricing, guarantee ratios, risk-sharing ratios, and industry and regional restrictions [44], credit guarantee systems indeed share banks’ loan risk and reduce information asymmetry and SMEs’ collateral burden. Moreover, the credit guarantee system plays an essential role in helping SMEs to obtain loan supply [24], promoting technological innovation in SMEs [45] and improving the efficiency of green investment by banks and SMEs [46].

Credit guarantee policies have significant social benefits, but the persistent unbalanced risk–profit ratio between banks and guarantee institutions hinders the deep development of credit guarantee schemes [15]. Government involvement is necessary to guarantee credit guarantee schemes’ sustainable operation [47]. The financing guarantee systems of the United States of America, Japan, and Germany all reveal that government finance plays a vital role in risk compensation in the SME financing guarantee system. Additionally, China has established a financing guarantee system based on government-supported financing guarantee agencies and re-guarantee agencies, and piloted different models of risk-sharing mechanisms. The importance of government participation in credit risk sharing has been proven [18]. However, the existing research on the risk-sharing ratio of various guarantee entities focuses on financial institutions [26,48], and does not consider the impact of the government’s risk-sharing ratio on the guarantee system. This paper uses evolutionary game theory and sets the risk-sharing ratio as a variable to find the optimal risk-sharing ratio among the government, guarantee agencies, banks, and re-guarantee agencies.

In summary, it is feasible to use a policy financing guarantee to help SMEs to carry out GTI, but the financing guarantee system’s risk-sharing and benefit compensation mechanism need to be improved urgently. Coordinating the risks and benefits of each stakeholder of the policy credit guarantee system and realizing the long-term and stable cooperation between government, banks, and guarantee agencies are key to solving the financing problems of GTI of SMEs.

Table 1. Papers that are most related to our research on SMEs financing and guarantees.

Literature	Government	Bank	Guarantee Agency	Re-Guarantee Agency	SMEs
[24]		√	√
[25]		√	√
[26]	√	√	√
[48]		√	√	√
[49]		√			√
[50]	√	√	√		√
[51]	√	√	√		√
This paper	√	√	√	√	√

Source: Authors’ compilation.

shows the differences between this study and other studies.

3. System Background and Hypothesis Description

3.1. System Background

China leads the world in the number of SMEs and the growth rate of green patents. By the end of 2020, the number of SMEs in China has reached 42 million, but the number of SMEs with green patents does not exceed 3% of the total number of SMEs. In April 2019, China’s National Development and Reform Commission and Ministry of Science and Technology issued the “Guidance on Building a Market-Oriented Green Technology Innovation System”, proposing increasing financial support for GTI and encouraging local governments to support GTI through guarantee funds or entrusting professional guarantee companies, etc. (Chinese government website. Available online: http://www.gov.cn/zhengce/zhengceku/2019-09/29/content_5434807.htm (accessed on 26 December 2021)). In December 2021, the “14th Five-Year Plan for the Development of SMEs” again emphasized supporting SMEs to carry out GTI (Chinese government website. Available online: http://www.gov.cn/zhengce/2021-12/20/content_5662036.htm (accessed on 26 December 2021)). To help SMEs, several provinces and cities in China have gradually established unique policy financing guarantee systems and experimented with different risk-sharing mechanisms: Anhui Province proposed that the guarantee agency, provincial re-guarantee agency, bank, and local government bear the responsibility for repayment a the ratio of 4:3:2:1, respectively. Shanxi Province proposed that the bank, re-guarantee agency, financing guarantee agency, and national financing guarantee fund share the responsibility for repayment in the ratio of 2:2:4:2, respectively. Jiangmen City, Guangdong Province, implemented a model in which the risk guarantee fund, cooperative bank, and financial guarantee agency share the risk in the ratio of 2:1:7, respectively [52].

A synthesis of the operating model of the policy financing guarantee system in most Chinese provinces reveals that the policy financing guarantee system integrates all levels of government, banks, guarantee agencies, and re-guarantees into the same system. This approach can provide a layer-by-layer credit enhancement for SMEs, collaborative risk control and mutual benefit. The policy financing guarantee system can play an essential role in the GTI of SMEs. Relevant government departments can guide and supervise the guarantee process, enhancing the overall development level and risk identification ability of the policy financing guarantee system. Simultaneously, the government–bank–guarantee agency cooperation can reduce the leverage ratio of SMEs and promote the green development of SMEs. Based on the above analysis, we construct the logic relationship diagram in Figure 1 to show how policy financing guarantee helps credit enhancement.

Figure 1 shows that SMEs need substantial financial support for GTI, and those that cannot meet the lending criteria seek credit enhancement from the guarantee agency. The guarantee agency assesses the SME credit and green project risks and guarantees qualified similar companies in bulk. When the credit magnification of the guarantee agency is too high, the guarantee agency will apply for a fixed-ratio re-guarantee from the re-guarantee agency. The bank signs a guaranteed contract with the guarantee agency and approves the SME’s financing application. In recent years, many Chinese cities have implemented a “whitelist” policy for financing guarantees, giving particular support to enterprises on the list regarding the guarantee amount, guarantee fees, and approval process, which has achieved good social benefits. Based on this situation, we assume the companies that abide by the contract to be automatically included in the “whitelist” of financing guarantees, so that they can enjoy preferential loan interest rates and guarantee fees. If the SME defaults, the guarantee agency will compensate and recover the principal and interest of the loan following the agreed proportion. Meanwhile, the guarantee agency will apply to the re-guarantee agency for reimbursement and the government for specific risk compensation.

During the operation of this system, the government regulates, subsidizes, and stimulates the cooperation between the bank and guarantee agency. The bank gradually learns about the guarantee agency’s capabilities in the collaboration. The guarantee agency consolidates various SME needs and promotes bank products’ innovation. The re-guarantee agency provides a solid guarantee for the guarantee agency. The four parties work together to promote the sustainable development of SMEs.

3.2. Model Assumption

Assumption 1.

The policy financing guarantee system involves five bounded rational players, namely the government, SMEs, bank, policy guarantee agency, and re-guarantee agency. To simplify the calculation and highlight the focus of our research, we only analyze the first four evolutionary players and add the re-guarantee agency to the game model as a critical parameter.

Assumption 2.

To help SMEs to make innovations in green technologies, banks, guarantee institutions, and the government can choose cooperation or non-cooperation strategies. SMEs also decide whether to repay the loan according to their actual business conditions. Specifically, the SME’s strategies are compliance or default, the bank’s strategies are lending or reject lending, the guarantee agency’s strategies are guarantee or reject guarantee, and the government’s strategies are active participation or passive participation. Let $m,n,p,q$ represent the probability of compliance, lending, guarantee, and active participation, respectively. Additionally, let $1-m,1-n,1-p,1-q$ represent the probability of default, non-lending, non-guarantee, and passive participation, respectively, $\mathrm{m},\mathrm{n},\mathrm{p},\mathrm{q}\in [0,1]$.

Assumption 3.

Assume that to promote the construction of the policy financing guarantee system, the government has invested in establishing a financing guarantee fund. Based on the amount of guarantee, the government provides a certain percentage of the premium subsidies to the guarantee agency for each business. Under the risk compensation mechanism, guarantee agencies share risks with banks, governments, and re-guarantee agencies. However, guarantee agencies perform their compensatory responsibilities firstly when SMEs default. SMEs provide stable and realizable counter-collateral to obtain a short-term loan for 1 year, with the principal and interest repaid at maturity. Guarantee agencies provide a full guarantee of principal and interest to SMEs and charge a variable guaranteed rate according to the risk level of the green project. SMEs that perform on time automatically join the “whitelist” of policy financing guarantees shared by the government, banks, and guarantee agencies. SMEs that join the “whitelist” can receive certain interest rate concessions, guarantee rate concessions, and government interest rate subsidies.

3.3. Profit and Loss Variables for Each Player

(1) The SME obtains a loan amount $L$ with a maturity of 1 year at an annualized interest rate $r_1$. SMEs provide counter-collateral with the value of $V$ (the value after deducting recovery costs from the bank and guarantee agency), and $V<L$. SMEs’ innovative green technology will reduce the enterprises’ production costs and energy consumption, which will increase the revenue of $K$ for SMEs, where $K$ represents the capital utilization rate of the SME.

(2) The total cost of the bank’s credit assessment of the SME and the risk assessment of the green project is $C_{11}$, and the bank shares the compensation risk of ${\beta }_1$ proportion. If the SME repays the principal and interest on time, the bank will receive $r_{1}L$’s loan proceeds. If the SME defaults, the bank will ask the guarantee institution to compensate, and the bank will lose $(1+r_1){\beta }_{1}L$.

(3) The total cost of credit screening for SMEs and the value of counter-guaranteed items assessed by guarantee institution is $C_{21}$, and the guarantee agency charges a guaranteed rate of $r_2$ for SMEs. After sharing the risk with the bank, the guarantee agency’s risk-sharing ratio is ${\beta }_2$ (${\beta }_2=1-{\beta }_1$), the average supervision cost for SMEs is $C_{22}$, and the re-guarantee rate paid to the re-guarantee agency is $r_3$ (based on the guarantee premium). If the SME defaults, the guarantee agency’s insured balance is $(1+r_1)L$. The guarantee agency will pay $(1+r_1){\beta }_{2}L$ to the bank, recover the SME counter-guarantee items, and apply to the re-guarantee agency for compensation.

(4) The government subsidizes ${\alpha }_1$ percentage of the premium for the guarantee agency’s each business and shares ${\beta }_3$ proportion of the reimbursement amount for the guarantee agency (i.e., ${\beta }_3(1+r_1)L$). The government supervises SMEs’ financing and the green-technology upgrading process, and the total cost of supervision is $C_{31}$. The government subsidizes the interest rate of $w$ percentage for SMEs included in the “whitelist” and imposes the fine $f$ for defaulting enterprises. Improving the policy financing guarantee system and developing green technology will benefit the government by providing potential social and economic benefits, totaling $E$. If the government negatively participates in constructing the policy financing guarantee system, it will stop subsidizing banks and guarantee agencies, and relax the supervision of SMEs ($w=0,f=0$). The government’s slack participation in building the policy financing guarantee system will face performance loss, set as $\delta$. In addition, the re-guarantee agency shares the compensation amount of ${\beta }_4$ proportion for the guarantee agency (i.e., ${\beta }_4(1+r_1)L$). Thus, the actual amount of compensation borne by the guarantee agency is $({\beta }_2-{\beta }_3-{\beta }_4)(1+r_1)L$.

Based on the above assumptions, we derive the payoff matrix of the game model among SMEs, banks, policy financing guarantee agencies, and the government (see

Table 2. Payoff matrices for both players.

		Bank		Government
			Guarantee Agency	Active Participation q		Positive Participation 1 − q
				Guarantee p	Reject Guarantee 1 − p	Guarantee p	Reject Guarantee 1 − p
SMEs	Compliancem	Lending n	$K-(1+r_1+r_2-w)L$		$K-(1+r_1)L$	$K-(1+r_1+r_2)L$	$K-(1+r_1)L$
			$r_{1}L-C_{11}$		$r_{1}L-C_{11}$	$r_{1}L-C_{11}$	$r_{1}L-C_{11}$
			$(r_2-r_2{r}_3+{\alpha }_1)L-C_{21}-C_{22}$		$0$	$r_2(1-r_3)L-C_{21}-C_{22}$	$0$
			$E-{\alpha }_{1}L-C_{31}-wL$		$-C_{31}$	$E-\delta$	$-\delta$
		Reject lending 1 − n	$0$		$0$	$0$	$0$
			$-C_{11}$		$-C_{11}$	$-C_{11}$	$-C_{11}$
			$-C_{21}$		$-C_{21}$	$-C_{21}$	$-C_{21}$
			$0$		$0$	$0$	$0$
	Default 1 − m	Lending n	$(1-r_2)L-V-f$		$L-V-f$	$(1-r_2)L-V$	$L-V$
			$-(1+r_1){\beta }_{1}L-C_{11},$		$-(1+r_1)L+V-C_{11}$	$-(1+r_1){\beta }_{1}L-C_{11}$	$-(1+r_1)L+V-C_{11}$
			$\begin{array}{l}(r_2-r_2{r}_3+{\alpha }_1)L-(1+r_1)({\beta }_2\\ -{\beta }_3-{\beta }_4)L+V-C_{21}-C_{22}\end{array}$		$0$	$\begin{array}{l}{r}_2(1-r_3)L-(1+r_1)({\beta }_2\\ -{\beta }_4)L+V-C_{21}-C_{22}\end{array}$	$0$
			$E-({\alpha }_1+{\beta }_3)L-C_{31}+f$		$f-C_{31}$	$E-\delta$	$-\delta$
		Reject lending 1 − n	$0$		0	$0$	$0$
			$-C_{11}$		$-C_{11}$	$-C_{11}$	$-C_{11}$
			$-C_{21}$		$-C_{21}$	$-C_{21}$	$-C_{21}$
			$0$		$0$	$0$	$0$

).

4. Dynamic Evolutionary Path and Stability Analysis of the Game Model

Analyzing the evolution stability strategy of the policy financing guarantee game system is the key to helping us solve the financing problems of SMEs. According to the properties of the differential equation stability theorem and evolutionary stability strategy, for the differential equation $F(x(t))=x(t)$, if $x^{*}(t)$ is the solution of $F(x(t))=0$, and $F^{\prime }(x(t))|{}_{x(t)=x^{*}(t)}<0$, we can obtain $x^{*}(t)$, the local stability point of the single-group evolutionary game model. Considering the multi-group evolutionary game model, whether the strategy of the whole game system is stable can be judged according to Lyapunov’s first method. Assuming that $x=(x_1,x_2)$, and $x^{*}(t)$ is the general solution of the system $F(x(t))=x(t)$, the necessary and sufficient condition for the stability of a linear system is that the real parts of the roots of the characteristic equations of the system are all negative [53,54,55,56,57]. According to the above theorem, we analyze the stability of the game strategy of each player and the stability of the game system, respectively.

4.1. Stability Analysis of the SME

Assuming that the expected profit when the SME chooses to comply is $U_m$, the expected profit when the SME defaults is $U_{1-m}$, and the average expected profit is $\overline{U_m}$, we have:

$$\{\begin{array}{l}{U}_m=n[K-L(1+r_1+p{r}_2-pqw)]\\ U_{1-m}=n[L(1-p{r}_2)-V-fq]\\ \overline{U_m}=m{U}_m+(1-m)U_{1-m}\end{array}$$

(1)

Therefore, the replication dynamic equation of the SME and its derivative are:

$$\{\begin{array}{l}F(m)=\frac{dm}{dt}=m(U_m-\overline{U_m})=m(1-m)A(n,p,q)\\ A(n,p,q)=n[K+fq-L(2+r_1-pqw)+V]\end{array}$$

(2)

$$\frac{d(F(m))}{dm}=(1-2m)A(n,p,q).$$

(3)

According to the stability theorem of differential equations, if the probability of SME repayment on time is stable, we have $F(m)=0$ and $\frac{d(F(m))}{dm}<0$. When $n\ne 0$, we have $\partial A(n,p,q)/\partial q>0$, so $A(n,p,q)$ is an increasing function of $q$. Let $q=\frac{2L+L{r}_1-V-K}{f+pwL}=q^{*}$; if $0<q<q^{*}<1$, we have $A(n,p,q)<0$, $F(m)|{}_{m=0}=0$ and $F^{\prime }(m)|{}_{m=0}<0$, so $m=0$ is the SME’s evolutionary stable strategy. If $1>q>q^{*}>0$, we have $A(n,p,q)>0$, $F(m)|{}_{m=1}=0$ and $F^{\prime }(m)|{}_{m=1}<0$, so $m=1$ is the SME’s evolutionary stable strategy. If $q=q^{*}$ or $n=0$, we have $A(n,p,q)=0$ and $F(m)\equiv 0$, so both strategies of the SME are stable. We show the evolution process of the SME’s strategy in Figure 2.

In Figure 2, the volume of space $V_{m0}$ represents the probability of the SME repaying the debt on time, and the volume of space $V_{m1}$ represents the probability of the SME defaulting, and we can obtain that:

$$V_{m0}={\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\frac{L(r_1+2)-V-K}{f+pw}}}dmdp=\frac{\log (1+\frac{wL}{f})[L(r_1+2)-V-K]}{wL}$$

(4)

$$V_{m1}=1-V_{m0}$$

(5)

Based on the above analysis, we obtain Conclusion 1: an increase in the probability of active participation in system construction promotes on-time repayment by SMEs; conversely, the negativity of government participation in the system construction reduces the likelihood of SME compliance. Moreover, the probability of SME default is positively correlated with the loan amount and loan interest rate but negatively associated with the value of counter-guarantee items, the increase in revenue from green technology innovation, and the number of government incentives and penalties. The debt burden of SMEs increases when the amount of SMEs financing is too large, or the interest rate of the loans is too high, which increases the default rate of SMEs, if the value of the counter-collateral items provided by SMEs is high enough, that is, they have strong solvency. In that case, banks may moderately relax their lending restrictions on such SMEs. The government must strengthen supervision and financial support for SMEs, which will increase the probability of SMEs’ compliance.

4.2. Stability Analysis of Bank

Assuming that the expected profit when the bank chooses to lend to SMEs is $U_n$, the expected profit when the bank refuses to lend is $U_{1-n}$, and the average expected profit is $\overline{U_n}$, we have:

$$\{\begin{array}{l}{U}_n=-C_{11}-L(1+r_1)[1+p({\beta }_1-1)]+mL[1+2r_1+p({\beta }_1-1)(1+r_1)]\\ +(m-1)(p-1)V\\ U_{1-n}=-C_{11}\\ \overline{U_n}=n{U}_n+(1-n)U_{1-n}\end{array}$$

(6)

Therefore, the replication dynamic equation of bank and its derivative are:

$$\{\begin{array}{l}F(n)=\frac{dn}{dt}=n(U_n-\overline{U_n})=n(1-n)B(m,p,q)\\ B(m,p,q)=-L(1+r_1)[1+p({\beta }_1-1)]+mL[1+2r_1+p({\beta }_1-1)(1+r_1)]\\ +(m-1)(p-1)V\end{array}$$

(7)

$$\frac{d(F(n))}{dn}=(1-2n)B(m,p,q)$$

(8)

According to the stability theorem of differential equations, if the strategy of the bank choosing to lend is stable, we have: $F(n)=0$ and $\frac{d(F(n))}{dn}<0$. As $\partial B(m,p,q)/\partial p>0$, $B(m,p,q)$ is an increasing function of $p$. Let $p=\frac{[1+r_1-m(1+2r_1)]L+(m-1)V}{(m-1)[({\beta }_1-1)(1+r_1)L+V]}=p^{*}$; when $0<p<p^{*}<1$, we have $B(m,p,q)<0$, $F(n)|{}_{n=0}=0$ and $F^{\prime }(n)|{}_{n=0}<0$, so $n=0$ is the bank’s evolutionary stable strategy. If $1>p>p^{*}>0$, we have $B(m,p,q)>0$, $F(n)|{}_{n=1}=0$ and $F^{\prime }(n)|{}_{n=1}<0$, so $n=1$ is the bank’s evolutionary stable strategy. If $p=p^{*}$, we have $B(m,p,q)=0$ and $F(n)\equiv 0$, and so both strategies of the bank are stable. We show the evolution process of the bank’s strategy in Figure 3.

In Figure 3, the volume of space $V_{n0}$ represents the probability that the bank lends to SMEs, and the volume of space $V_{n1}$ represents the probability that the bank refuses to lend, and the calculation step is the same as Equations (4) and (5).

Based on the above analysis, we obtain Conclusion 2: the active participation of the guarantee agency will increase the probability of bank lending to SMEs. Meanwhile, the probability of banks rejecting loans is positively related to the loan amount and the proportion of risk-sharing (${\beta }_1$) and negatively associated with the loan interest rate and the value of counter-guaranteed items. When the SME’s financing amount is too large, and the value of the collateral is low, banks will be less likely to lend to the SME out of concern for the risk of default. In addition, the risk-sharing ratio between the government and banks is an essential factor influencing banks’ approval of SME loan applications, and banks will be more likely to reject applications when ${\beta }_1$ is too high.

4.3. Stability Analysis of Guarantee Agency

Assuming that the expected profit when the guarantee agency agrees to guarantee for SME is $U_p$, the expected profit when the guarantee agency refuses to guarantee is $U_{1-\mathrm{p}}$, and the average expected profit is $\overline{U_p}$, we have:

$$\{\begin{array}{l}{U}_p=-C_{21}+n\{-C_{22}+L[{\beta }_4+q({\alpha }_1+{\beta }_3)+{\beta }_2(m-1)(1+r_1)-({\beta }_4+q{\beta }_3)\\ (m+(m-1)r_1)+r_2-r_2{r}_3]+V(1-m)\}\\ U_{1-p}=C_{21}(n-1)\\ \overline{U_p}=p{U}_p+(1-p)U_{1-p}\end{array}$$

(9)

Therefore, the replication dynamic equation of the guarantee agency and its derivative are:

$$\{\begin{array}{l}F(p)=\frac{dp}{dt}=p(U_p-\overline{U_p})=p(1-p)C(m,n,q)\\ \mathrm{C}(m,n,q)=n\{-C_{21}-C_{22}+L[{\beta }_4+q({\alpha }_1+{\beta }_3)+{\beta }_2(m-1)(1+r_1)\\ -({\beta }_4+q{\beta }_3)(m+(m-1)r_1)+r_2-r_2{r}_3]+V(1-m)\}\end{array}$$

(10)

$$\frac{d(F(p))}{dp}=(1-2p)C(m,n,q)$$

(11)

According to the stability theorem of differential equations, if the strategy of the guarantee agency choosing to guarantee is stable, we have $F(p)=0$ and $\frac{d(F(p))}{dp}<0$. When $n\ne 0$, we have $\partial C(m,n,q)/\partial m>0$, so $C(m,n,q)$ is an increasing function of $m$. Let $m=1+\frac{C_{21}+C_{22}-L(q{\alpha }_1+r_2-r_2{r}_3)}{L({\beta }_2-{\beta }_{3}q-{\beta }_4)(1+r_1)-V}=m^{*}$; if $1>m>{\mathrm{m}}^{*}>0$, we have $C(m,n,q)>0$, $F(p)|{}_{p=1}=0$ and $F^{\prime }(p)|{}_{p=1}<0$, so $p=1$ is the guarantee agency’s evolutionary stable strategy. If $0<m<m^{*}<1$, we have $C(m,n,q)<0$, $F(p)|{}_{p=0}=0$ and $F^{\prime }(p)|{}_{p=0}<0$, so $p=0$ is the guarantee agency’s evolutionary stable strategy. When $m=m^{*}$ or $n=0$, we can obtain $C(m,n,q)=0$ and $F(p)\equiv 0$, so both strategies of the guarantee agency are stable. We show the evolution process of the guarantor’s strategy in Figure 4.

In Figure 4, the volume of space $V_{p0}$ represents the probability that the guarantee agency services SMEs, and the volume of space $V_{p1}$ represents the probability that the guarantee agency refuses to guarantee, and the calculation principle is the same as Equations (4) and (5).

Based on the above analysis, we obtain Conclusion 3: SMEs’ increased probability of compliance can shift the stability strategy of guarantee agencies from not guarantee to guarantee. In addition, the probability of the guarantee agency choosing to guarantee is positively correlated with the guarantee fee rate and the value of counter-collateral items and negatively correlated with the guarantee amount, risk-sharing ratio (${\beta }_2$), various costs, and the guarantee fee rate. The value of counter-collateral goods is an essential reference for guarantee agencies to verify the credit level of SMEs. However, most SMEs have fewer fixed assets, and the risk of the guarantee agency to guarantee them is higher. The risk losses of guarantee agencies can be reduced by reasonably adjusting the risk-sharing ratio. Moreover, the government should increase subsidies for guarantee businesses to increase the guarantee agencies’ incentive to service SMEs.

4.4. Stability Analysis of Government

Assuming that the expected profit when the government participates actively is $U_q$, the expected profit when the government participates negatively is $U_{1-\mathrm{q}}$, and the average expected profit is $\overline{U_q}$, we have:

$$\{\begin{array}{l}Uq=-n\{f(m-1)+pL[({\alpha }_1+{\beta }_3-m{\beta }_3+mw)-Ep]+C_{31}\}\\ U_{1-q}=n(pE-\delta )\\ \overline{U_q}=q{U}_q+(1-q)U_{1-q}\end{array}$$

(12)

Therefore, the replication dynamic equation of government and its derivative are:

$$\{\begin{array}{l}F(q)=\frac{dq}{dt}=q(U_q-\overline{U_q})=q(1-q)D(m,n,p)\\ D(m,n,p)=-n\{f(m-1)+pL[{\alpha }_1-{\beta }_3(m-1)+mw]+C_{31}-\delta \}\end{array}$$

(13)

$$\frac{d(F(q))}{dq}=(1-2q)D(m,n,p)$$

(14)

According to the stability theorem of differential equations, if the strategy of the government choosing to participate actively is stable, we have $F(q)=0$ and $\frac{d(F(q))}{dq}<0$. When $n\ne 0$, we have $\partial D(m,n,p)/\partial p<0$, so $D(m,n,p)$ is a decreasing function of $p$. Let $p=\frac{f-fm+\delta -C_{31}}{L[{\alpha }_1-{\beta }_3(m-1)+mw]}=p^{**}$; if $0<p<p^{**}<1$, we have $D(m,n,p)>0$, $F(q)|{}_{q=1}=0$ and $F^{\prime }(q)|{}_{q=1}<0$, so $q=1$ is the government’s evolutionary stable strategy. If $1>p>p^{*}{}^{*}>0$, we have $D(m,n,p)<0$, $F(q)|{}_{q=0}=0$ and $F^{\prime }(q)|{}_{q=0}<0$, so $q=0$ is government’s evolutionary stable strategy. If $p=p^{**}$ or $n=0$, we have $D(m,n,p)=0$ and $F(q)\equiv 0$, so both strategies of the government are stable. We show the evolution process of the government’s strategy in Figure 5.

In Figure 5, the volume of space $V_{q0}$ represents the probability that the government participates actively, the volume of space $V_{q1}$ represents the probability that the government participates negatively, and the calculation principle is the same as Equations (4) and (5).

Based on the above analysis, we obtain Conclusion 4: increasing the probability of the guarantee agency choosing the guarantee strategy will encourage the government to shift from passive participation to active participation. In addition, the probability of active government participation is positively correlated with fines for defaulting enterprises and loss of performance due to passive involvement, while it is negatively correlated with the amount of SME loans, the percentage of guarantee fee subsidies, the rate of interest subsidies for SMEs on the “whitelist”, and the percentage of risk-sharing (${\beta }_3$). The losses incurred by the government’s negative participation and the fines paid by defaulting enterprises are incentives for the government to improve the policy financing guarantee system. If the social benefits of active participation are higher than the number of government subsidies to guarantee agencies, banks, and SMEs, the probability that the government participates actively will increase.

4.5. Stability Analysis of Evolutionary Game Systems

SMEs, banks, policy finance guarantee agencies, and the government form a replicated dynamic system. According to the description in Section 4, we can judge the stability of the quadratic game system by Lyapunov’s first rule, which states that an equilibrium is stable when the eigenvalues of the equilibrium are all less than 0. According to Selten [58], the stable solution in a multi-group evolutionary game must be pure strategy equilibrium solutions. Therefore, we only analyze the pure strategy equilibrium points. Let $F(m)=0,F(n)=0,F(p)=0,F(q)=0$; we have $E_1-E_{16}$, which are 16 pure strategy local equilibrium points. From the replicated dynamic equations of each game subject, we can derive the Jacobian matrix of the game system:

$$J=\left[\begin{array}{cccc}\partial F(m)/\partial m& \partial F(m)/\partial n& \partial F(m)/\partial p& \partial F(m)/\partial q\\ \partial F(n)/\partial m& \partial F(n)/\partial n& \partial F(n)/\partial p& \partial F(n)/\partial q\\ \partial F(p)/\partial m& \partial F(p)/\partial n& \partial F(p)/\partial p& \partial F(p)/\partial q\\ \partial F(q)/\partial m& \partial F(q)/\partial n& \partial F(q)/\partial p& \partial F(q)/\partial q\end{array}\right]$$

(15)

The government’s enthusiasm to participate in SME financing guarantees is a crucial factor affecting the stability of the game system. The equilibrium points obtained are divided into two groups for discussion according to government participation.

4.5.1. Stability Analysis of Gaming Systems with Active Government Participation

When the government’s stabilization strategy is “active participation,” namely $f(m-1)+pL[{\alpha }_1-{\beta }_3(m-1)+mw]+C_{31}-\delta <0$ is satisfied, the stability analysis of the equilibrium point in the game system is shown in

Table 3. The pure strategy equilibrium points and eigenvalues under active government participation.

Equilibrium Point	Eigenvalue ${\mathit{\lambda }}_1,{\mathit{\lambda }}_2,{\mathit{\lambda }}_3,{\mathit{\lambda }}_4$	Asymptotically Stable
$E_1(0,0,0,1)$	$0,V-L(r_1+1),0,0$	Unstable
$E_2(0,1,0,1)$	$\begin{array}{l}f+K-L(r_1+2)+V,L(r_1+1)-V,L[{\alpha }_1-{\beta }_2(r_1+1)+(1+r_1)({\beta }_3+{\beta }_4)\\ -r_2{r}_3+r_2]-C_{21}-C_{22}+V,C_{31}-f-\delta \end{array}$	Unstable
$E_3(0,0,1,1)$	$0,0,0,0$	Unstable
$E_4(1,0,0,1)$	$0,L{r}_1,0,0$	Unstable
$E_5(1,0,1,1)$	$0,L{r}_1,0,0$	Unstable
$E_6(1,1,0,1)$	$L(r_1+2)-f-V-K,-L{r}_1,L({\alpha }_1-r_2{r}_3+r_2)-C_{21}-C_{22},C_{31}-\delta$	Condition 1
$E_7(0,1,1,1)$	$\begin{array}{l}f+K-L(r_1+2-w)+V,{\beta }_1(1+r_1)L,C_{21}+C_{22}-L[{\alpha }_1-{\beta }_2(r_1+1)\\ +(1+r_1)({\beta }_3+{\beta }_4)-r_2{r}_3+r_2]-V,({\alpha }_1+{\beta }_3)L+C_{31}-\delta -f\end{array}$	Unstable
$E_8(1,1,1,1)$	$\begin{array}{l}L(r_1+2-w)-f-K-V,-L{r}_1,-L({\alpha }_1-r_2{r}_3+r_2)+C_{21}+C_{22},\\ C_{31}+({\alpha }_1+w)L-\delta \end{array}$	Condition 2
$\mathrm{Condition} 1 :L(r_1+2)-f-V-K<0,L({\alpha }_1-r_2{r}_3+r_2)-C_{21}-C_{22}<0,C_{31}-\delta <0$ $\mathrm{Condition} 2: L(r_1+2-w)-f-K-V<0,-L({\alpha }_1-r_2{r}_3+r_2)+C_{21}+C_{22}<0,C_{31}+({\alpha }_1+w)L-\delta <0$

.

As shown in Table 3, there are two possible stability points in the gaming system with active government participation in constructing the policy financing guarantee system: $E_6(1,1,0,1)$ and $E_8(1,1,1,1)$. The following discussion is divided into scenarios.

If condition 1 is met, that is, the net loss of SME when default ($L-f-V$) is less than the net revenue brought by GTI ($K-L(r_1+1)$), the guarantee fee earned by the guarantee institution ($L({\alpha }_1+r_2)$) is less than the sum of the various costs ($L{r}_2{r}_3+C_{21}+C_{22}$), and the cost of government regulation ($C_{31}$) is lower than the loss of sloth administration ($\delta$), we can conclude that $E_6(1,1,0,1)$ is an evolutionary stable strategy (ESS). Under these conditions, the guarantee agency would not be able to make ends meet if they provided guarantees to SMEs, so the guarantee agency refuses to provide guarantee services. Banks and enterprises cooperate with the government’s active involvement, and SMEs will repay loans on time.

If condition 2 is met, that is, the net loss in the event of default of the SME that joins the “whitelist” of financial guarantees ($L-f-V$) is less than the net profit from GTI ($K-L(r_1+1-w)$), the cost paid by the guarantee agency ($C_{21}+C_{22}+L{r}_2{r}_3$) is less than the net profit ($L({\alpha }_1+r_2)$), and the total cost of the government’s active participation ($({\alpha }_1+w)L+C_{31}$) is less than the loss of sloth administration ($\delta$), we can gather that $E_8(1,1,1,1)$ is an ESS. Under these conditions, the government, the bank, and the guarantee agency cooperate to serve SMEs, and SMEs can achieve technological innovation and repay loans on time, which is also the expected goal of developing the policy financing guarantee system.

4.5.2. Stability Analysis of Gaming Systems with Negative Government Participation

When the government’s stabilization strategy is “negative participation,” namely, $f(m-1)+pL[{\alpha }_1-{\beta }_3(m-1)+mw]+C_{31}-\delta >0$ is satisfied, the stability analysis of the equilibrium point in the game system is shown in

Table 4. The pure strategy equilibrium points and eigenvalues under negative government participation.

Equilibrium Point	Eigenvalue ${\mathit{\lambda }}_1,{\mathit{\lambda }}_2,{\mathit{\lambda }}_3,{\mathit{\lambda }}_4$	Asymptotically Stable
$E_9(0,0,0,0)$	$0,V-L(r_1+1),0,0$	Unstable
$E_{10}(0,1,0,0)$	$\begin{array}{l}K-L(r_1+2)+V,L(r_1+1)-V,-L[{\alpha }_1-{\beta }_2(r_1+1)+r_1({\beta }_3+{\beta }_4)\\ +{\beta }_3+{\beta }_4-r_2{r}_3+r_2]+C_{21}+C_{22}-V,f+\delta -C_{31}\end{array}$	Unstable
$E_{11}(0,0,1,0)$	$0,-{\beta }_1(1+r_1)L,0,0$	Unstable
$E_{12}(1,0,0,0)$	$0,L{r}_1,0,0$	Unstable
$E_{13}(1,0,1,0)$	$0,L{r}_1,0,0$	Unstable
$E_{14}(1,1,0,0)$	$L(r_1+2)-V-K,-L{r}_1,L(-r_2{r}_3+r_2)-C_{21}-C_{22},\delta -C_{31}$	Condition 3
$E_{15}(0,1,1,0)$	$\begin{array}{l}K-L(r_1+2)+V,{\beta }_1(1+r_1)L,-L[-{\beta }_2(r_1+1)+r_1{\beta }_4+{\beta }_4-r_2{r}_3+r_2]\\ +C_{21}+C_{22}-V,f-({\alpha }_1+{\beta }_3)L-C_{31}+\delta \end{array}$	Unstable
$E_{16}(1,1,1,0)$	$L(r_1+2)-V-K,-L{r}_1,L(-r_2{r}_3+r_2)-C_{21}-C_{22},-C_{31}-L({\alpha }_1+w)+\delta$	Condition 4
$\mathrm{Condition} 3:L(r_1+2)-V-K<0,L(-r_2{r}_3+r_2)-C_{21}-C_{22}<0,\delta -C_{31}<0$ $\mathrm{Condition} 4: L(r_1+2)-V-K<0,L(-r_2{r}_3+r_2)-C_{21}-C_{22}<0,-C_{31}-L({\alpha }_1+w)+\delta <0$

.

As shown in Table 4, there are two possible stability points in the gaming system with negative government participation in constructing the policy financing guarantee system: $E_{14}(1,1,0,0)$, $E_{16}(1,1,1,0)$. The following discussion is divided into scenarios.

If condition 3 is met, that is, the net loss to the SME in the event of default ($L-V$) is less than the net profit from GTI ($K-L(r_1+1)$), the guarantee fee earned by the guarantee agency ($L{r}_2$) is less than the sum of the costs ($L{r}_2{r}_3+C_{21}+C_{22}$), and the cost of government regulation of the financing process ($C_{31}$) is greater than the loss of sloth administration ($\delta$), we can gather that $E_{14}(1,1,0,0)$ is an ESS. Under these conditions, neither the guarantee agency nor the government is involved in the SME financing process due to cost considerations. However, because of the higher innovation benefits of developing green technologies, the probability of SMEs voluntarily repaying their loans increases significantly, and the likelihood of banks refinancing SMEs increases. The probability of SMEs voluntarily repaying their loans increases significantly, and the likelihood of banks refinancing SMEs increases. This is only possible if SMEs have high credit ratings, which is inconsistent with the reality that most SMEs have low credit ratings and own low-value counter-collateral items.

If condition 4 is met, that is, the net loss to the SME in the event of default ($L-V$) is less than the net profit from GTI ($K-L(r_1+1)$), the guarantee fee earned by the guarantee agency ($L_{r2}$) is greater than the sum of the costs ($L{r}_2{r}_3+C_{21}+C_{22}$), and the total cost of the government’s active participation ($({\alpha }_1+w)L+C_{31}$) is greater than the loss of sloth administration ($\delta$), we can surmise that $E_{16}(1,1,1,0)$ is an ESS. Under these conditions, the benefits of GTI for the SME fully cover the default cost, and both the guarantee agency and the bank obtain considerable profits, which is in line with the characteristics of a relatively mature development of the policy financing guarantee system.

The above analysis of the game system’s stability shows that the government’s active participation is a necessary precondition for solving the problems of green innovation financing for SMEs. $E_8(1,1,1,1)$ is more in line with the current status quo of SMEs financing, and the following presents simulation analyses of the relevant parameters based on the stability conditions of $E_8(1,1,1,1)$.

5. Stimulation Analysis

In this paper, we set each player’s probabilities of the initial strategy selection as: $m=0.4,n=0.5,p=0.5,q=0.5$. The SME applies for guarantees from guarantee agency for GTI. The value of the counter-collateral items provided by the SME is $V=60$, and the increase in revenue that the GTI will generate for the SME is $K=150$. The guarantee agency’s guarantee rate is $r_2=2.5\%$, the cost of the audit is $C_{21}=0.5$, the cost of supervision and recovery is $C_{22}=0.5$, and the re-guarantee rate is $r_3=0.01$. The actual amount lent by the bank is $L=100$, and the annualized interest rate is $r_1=0.06$. The risk-sharing ratio agreed between the bank and the guarantee agency is ${\beta }_1=0.2, {\beta }_2=0.8$. The risk-sharing ratio shared by the re-guarantee agency is ${\beta }_4=0.2$. The cost of government regulation is $C_{31}=10$, the loss of passive participation is $\delta =30$, and the proportion of risk-sharing is ${\beta }_3=0.2$. Moreover, the amount of government fines for defaulting firms is $f=30$, the proportion of interest rate subsidy for SMEs is $w=0.01$, and the proportion of guaranteed fee subsidy is ${\alpha }_1=0.01$. The development of SMEs will inevitably require long-term support from banks and the government; applying for loans currently involves multiple attempts, although China’s policy financing guarantee system is in a state of gradual development and improvement. Therefore, we divide the evolution time $t$ into two stages—short-term ($t\in [0,5]$) and long-term ($t\in [0,300]$)—to observe the changing trend of the evolution path of the entire game system, and then help us to make more targeted recommendations.

Combined with the previous analysis, we used the MATLAB 2019 software to perform numerical simulations of key factors to find the specific effects of parameter changes on the game system:

• The ratio of risk-sharing among the government, bank, guarantee agency, and re-guarantee agency.
• The intensity of government penalties.
• Government interest subsidies and guarantee fee subsidies.
• The capital utilization rate of SMEs.
• Interest rates and guarantee fee rates.

5.1. The Impacts of Risk-Sharing Ratio on the Evolution

Combined with China’s existing risk-sharing model, we compared (1) the Shanxi Province risk-sharing model (bank, re-guarantee agency, financial guarantee agency and the State Financing Guarantee Fund share the amount of repayment in the ratio of 2:2:4:2), (2) the Anhui province risk-sharing model (the guarantee agency, re-guarantee agency, commercial bank and local government share the amount of reimbursement in the ratio of 4:3:2:1), (3) the Guangdong province risk-sharing model (risk guarantee fund, bank and financial guarantee institution share the amount of repayment in the ratio of 2:1:7), and (4) the traditional risk-sharing model (bank and guarantee institution share the amount of repayment in the ratio of 2:8).

In conjunction with the previous assumptions, we set four groups of parameters corresponding to each mode: ${\beta }_1=0.2,{\beta }_2=0.8,{\beta }_3=0.2,{\beta }_4=0.2$, ${\beta }_1=0.2,{\beta }_2=0.8,{\beta }_3=0.1,{\beta }_4=0.3$, ${\beta }_1=0.1,{\beta }_2=0.9,{\beta }_3=0.2,{\beta }_4=0$, ${\beta }_1=0.2,{\beta }_2=0.8,{\beta }_3={\beta }_4=0$. We show the simulation results in Figure 6.

According to the evolution of Figure 6, we can see that under the traditional risk-sharing model, despite the government’s high motivation to participate, the guarantee agency’s willingness to guarantee is extremely low due to the high risk-sharing ratio. The low guarantee willingness also accelerates the evolution rate of the bank to “reject lending,” which verifies conclusion 2. In the long term ($t\in [0,300]$), the 2:2:4:2 risk-sharing model evolves to be stable significantly faster than the other two models. In terms of short-term game results ($t\in [0,5]$), the bank and the government both bear the lowest risk ratios (i.e., 10% and 0%, respectively) under the 2:1:7 risk-sharing model compared to the other two models. Therefore, the bank evolves to the “lending” strategy and the government evolves to “active participation” at the fastest rate. However, in this model, the guarantee agency bears a high compensation risk, and the guarantee agency shows a trend to the “reject guarantee” strategy during the evolution time of $[0,1]$.

Under both the 2:2:4:2 risk-sharing model and the 4:3:2:1 risk-sharing model, the bank and the guarantee agency bear the same risk-sharing ratio (i.e., 20% and 40%, respectively), and both evolve at higher rates under the 4:3:2:1 risk-sharing model than under the 2:2:4:2 risk-sharing model. This suggests that reducing the risk-sharing ratio of government can effectively encourage the government to construct the guarantee system, and can further increase the willingness of the bank and guarantee agency to collaborate in serving the green development of SMEs, which verifies conclusions 2 and 3. Based on the above analysis, in the short term, the government’s financial pressure should reasonably reduce to motivate the government to participate in the guarantee system. In the long run, the government should accelerate the improvement of the re-guarantee system, debt–equity combination financing, and insurance mechanisms to stimulate market players to support SMEs’ GTI.

5.2. The Impacts of Government Punishment Intensity on the Evolution

To analyze the impacts of different government punishment intensities on the evolutionary path, we set four different groups of values from low to high: $f=0,20,30,40$. We show the simulation results in Figure 7.

From Figure 7, we can see that if the government does not penalize defaulting firms ($f=0$), the default cost of SMEs is relatively low. During the evolution time of [0,0.5], the SME’s compliance probability remains stable below 0.5 after increasing slightly initially, but the bank accelerates evolution to the “reject lending” strategy and eventually stabilizes at 0. When the government imposes less punishment on defaulting companies ($f=20$), although the probability of the government choosing the “active participation” strategy and the likelihood of SME compliance have increased significantly, the guarantee agency’s guarantee willingness does not change considerably, and the bank’s strategic choice is still stable at “reject lending.” Conclusion 2 is verified above. With the increase in government punishment ($f=30,40$), the strategy choice of the four players changed significantly and gradually stabilized at $E_8(1,1,1,1)$, and the higher the penalty, the faster the rate of evolution, which verifies conclusions 1 and 4. Regardless of the length of the evolution time, the government’s indifference to defaulting companies or insufficient punishment is not conducive to solving the financing problems of SMEs. Therefore, rationally increasing government punishment is an inevitable choice to maintain the stability of the game system.

5.3. The Impacts of Interest Subsidy and Guarantee Fee Subsidy on the Evolution

Government subsidies for loans can improve firms’ willingness to innovate with regard to technology [59], and grants for guarantee fees can mitigate the asymmetry between the risks and benefits of guarantee agencies [60]. To explore the impact of government interest subsidies and guarantee fee subsidies on the game system, we set four groups of different values: $w=0,{\alpha }_1=0;w=0.01,{\alpha }_1=0;w=0,{\alpha }_1=0.01;w=0.015,{\alpha }_1=0.015$. We show the simulation results in Figure 8.

As shown in Figure 8, the government subsidizes interest for SMEs only ($w=0.01,{\alpha }_1=0$) or subsidizes the premium for guarantee agency only ($w=0,{\alpha }_1=0.01$), which can increase the probability of bank choosing the “lending” strategy. During the evolution time [0,2], the change in the strategy of the guarantee agency after receiving the premium subsidy is not significant compared to the interest-only subsidy. However, during the evolution time [2,5], the interest-only subsidy increases the probability of the guarantee agency to choose the “guarantee” strategy. This is because the possibility of the SME choosing “compliance” increases after receiving the interest subsidy, and thus motivates the guarantor to guarantee, which confirms conclusion 3. For the whole game system, in the short run, the policy effects of interest subsidies are significantly better than those of premium subsidies. However, in the long run, the difference in policy effects between the two is more negligible, regardless of the length of evolution; the simultaneous introduction of interest and premium subsidies is the best option to facilitate the four players evolving towards $E_8(1,1,1,1)$.

5.4. The Impacts of SME Capital Utilization Rate on the Evolution

To analyze the impacts of different SME capital utilization rates on the evolutionary path, we set four groups of values from low to high: $K=130,145,150,160$. We show the simulation results in Figure 9.

We can learn from Figure 9 that the SME’s loan utilization ability is related to whether it can obtain renewed loans. When the capital utilization rate of the SME is low ($K=130,145$), despite the high motivation of government participation, there is still a high risk aversion on the part of the bank and guarantee agency, ultimately leading to the inability of SMEs to obtain loans, which verifies conclusion 3. As capital utilization rises, the SME will also accelerate and stabilize the “compliance” strategy, while the bank and guarantee agency will be more willing to provide services to SMEs. This also proves that green innovation will alleviate SMEs’ financing constraints. The government and banks should jointly strengthen the post-loan supervision to ensure that loans can be fully used for green technological upgrades and help SMEs to improve their credit ratings.

5.5. The Impacts of Interest Rate and Guarantee Fee Rate on the Evolution

The “whitelist” mechanism helps banks and guarantee agencies to screen out creditworthy SMEs, and banks and guarantee agencies can reasonably adjust the interest rate and guarantee fee rate. To analyze the impacts of different interest rates and guarantee fee rates on the evolutionary, we set four groups of values under the condition of low capital utilization of SMEs ($K=145$): $r_1=0.06,r_2=0.025,{\alpha }_1=w=0.01$; $r_1=0.055,r_2=0.02,{\alpha }_1=w=0.01$; $r_1=0.05,r_2=0.01,{\alpha }_1=w=0.01$; and $r_1=0.045,r_2=0.01,{\alpha }_1=w=0$. As SMEs on the “whitelist” have relatively high credit ratings, we increase the probability of compliance of SMEs, so we set $m=0.5$. We show the simulation results in Figure 10.

It can be seen from Figure 10 that after we increase the probability of SMEs’ compliance, gradually increasing the interest rate and guarantee fee rate did not have a significant impact on the evolution rate of SMEs under the scenario of low capital utilization ($K=145$). Compared to the simulation results in Figure 9, the bank is significantly more motivated to cooperate with the guarantee agency and is no longer fully risk-averse to SMEs, validating conclusion 3. In addition, when interest rates fall to a certain level ($r_1=0.045$), the bank will still choose to lend to SMEs on the “whitelist,” even if the government removes the interest rate subsidy. Regardless of the length of the evolution, when the guaranteed fee rate falls to 1%, removing the government guarantee fee subsidy will significantly reduce the probability that the guarantee agency chooses to “guarantee.” Therefore, the government should provide appropriate premium subsidies to guarantee providers in the early stages of SMEs’ green innovation. In summary, we can conclude that it is necessary to build a “whitelist” mechanism for financing guarantees, and screening high-quality SMEs can effectively prevent and control credit risk losses.

6. Conclusions

By constructing a quadratic evolutionary game model, we analyze the specific path of policy financing guarantees to support SMEs to develop green technology, and numerically simulate the main factors affecting the stability of the game system. Our research findings and relevant policy recommendations are as follows:

(1) Considering the risk-sharing model, we find that the traditional risk-sharing model involving only banks and guarantee agencies cannot improve the enthusiasm of guarantors, and the participation of the government and re-guarantee agencies is the key to alleviating the financing difficulties of SMEs. Taking the risk-sharing model currently operating in several Chinese provinces as an example, under the condition that banks and guarantee agencies bear 20% and 40% of the risk ratio, respectively, in the short term, reducing the proportion of risk taken by the government moderately will increase the government’s willingness to participate in the policy financing guarantee system. In the long run, the government should lower the entry threshold for the guarantee industry, encourage private capital to join the guarantee system, and expand the business scale of guarantee agencies. This approach can reduce fiscal expenditure while reasonably sharing the risk pressure of the re-guarantee agencies, ultimately promoting the stable operation of the overall policy financial guarantee system.

(2) In the early stage of GTI for SMEs, compared with only implementing guarantee fee subsidies, only implementing interest subsidies can improve the probability of SMEs repaying loans on time. However, when the guaranteed fee rate falls to a certain level, if the government cancels the guaranteed fee subsidy, it will significantly dampen the enthusiasm of guarantee institutions. Therefore, in the initial stage of GTI for SMEs, the government should formulate corresponding policy incentive mechanisms, provide appropriate subsidies to SMEs and guarantee agencies, and improve the green financial service system based on market mechanisms. Banks should innovate to create green financial services, reasonably adjust green credit issuance and interest rates, increase credit support for green technologies and green projects, and effectively strengthen the construction of internal risk-assessment mechanisms. Guarantee agencies and re-guarantee agencies should enhance cooperation with banks, insurance companies, and the government and accelerate the construction of a multi-guarantee system that supports the development of green technologies.

(3) When improving the probability of SME compliance slightly ($m=0.4\to m=0.5$), even with higher lending rates and guarantee fee rates, the probability of SMEs compliance does not decrease significantly, which also reveals the importance of screening high-quality SMEs in preventing and controlling credit risks. Banks, guarantee agencies, and the government should accelerate the improvement of credit evaluation systems for SMEs. Using big data, blockchain, and other financial technology tools to develop digital finance, integrate financial data, tax data, due diligence data, and other multi-dimensional data to carry out credit rating for SMEs will enable a “whitelist” of financing guarantees to be built and enhance credit for high-quality enterprises.

(4) For SMEs with low credit ratings ($m=0.4$), the degree of GTI is a key factor affecting whether they can obtain subsequent financial support. SMEs should make full and effective use of loan funds, strengthen cooperation with scientific research institutions and tertiary institutions, and make more profound efforts in clean production, new material research, and resource recycling according to their strengths. SMEs must establish a long-term green business philosophy, produce green products trusted by consumers, and always aim for GTI. Moreover, the government and financial institutions should increase supervision of SMEs, improve risk identification in all aspects of lending, severely punish defaulting enterprises and weaken post-loan risks.

This study has certain limitations: we do not consider the impact of the capital and the magnification of guarantee agencies in the game system to simplify the calculation. We only study the full guarantee mode without an in-depth discussion of other guarantee modes. We will conduct more profound research combined with actual policy financing guarantee cases.