Research on Stochastic Evolution Game of Green Technology Innovation Alliance of Government, Industry, University, and Research with Fuzzy Income

Abstract

At present, the high complexity of the environment, the uncertainty of income, and the choice of strategies have attracted extensive attention from all walks of life who are committed to studying the game of collaborative innovation between government and industry–university–research. Based on this, first of all, with the help of stochastic evolutionary game theory and fuzzy theory, this paper constructs a multi-party stochastic evolutionary game model of green technology innovation about the government guidelines and the joint promotion of industry, universities, and research institutes. Secondly, it discusses the evolution law of behavior strategies of each game subject and the main factors to maintain the alliance’s stability under fuzzy income. The numerical simulation results show the following: (1) Reputation gains have a significant positive correlation with the evolution stability of alliance behavior, and the incorporation of reputation gains or losses will effectively maintain the cooperation stability of the alliance. (2) Under the influence of product greenness, government subsidies, and long-term benefits, it will promote the pace consistency of cooperative decision-making between industry, universities, and research institutes, and accelerate the evolution of alliances. (3) The enterprise’s ability and the research party’s ability will restrict each other. When one party’s ability is low, its willingness to choose a cooperation strategy may be slightly low due to technology spillover and other reasons. When the two parties’ abilities match, their behavior strategies will increase their willingness to cooperate with their abilities. Compared with the traditional evolutionary game, this study fully considers the uncertainty of the environment and provides theoretical support and practical guidance for the high-quality development strategy of the industry–university–research green technology innovation alliance.

1. Introduction

The cooperation mode of government, industry, university, and research (GIUR), guided by the government, led by research and research institutions, and participated in by enterprises, has been a concrete embodiment of the development level and improvement speed of China’s scientific and technological innovation capacity for many years. How to use modern science and technology to analyze and solve the difficult problem of the GIUR career development at this stage is very important. Therefore, the application of green innovation technology and the combination of reputation benefits to improve the willingness of all parties to cooperate has undoubtedly opened up a new path for the development of new GIUR.

As a widely accepted procedure to solve problems in engineering, economics, psychology, biology, business, and politics, game theory is widely used in various fields [1]. However, practice has shown that classical game theory is too idealistic and has certain theoretical flaws. In 1973, Smith and Price [2] proposed Evolutionary Stable Strategies (ESS) to study the dynamics of natural selection based on practical situations, which opened the way to the study of evolutionary game theory. Although the existing literature draws on evolutionary game theory, aiming at the stability of the IUR alliance and green technology innovation [3,4,5], the multi-party evolutionary game model under different scenarios is constructed and studied. All these have laid the ideological foundation for this study. However, in past research, the government, as an important participant in the game, was rarely considered to enter the game model, which resulted in the research gap on government participation and its impact on the system.

Under the realistic background of high uncertainty, the existing research often only analyzes a certain influencing factor or a single game subject, largely ignores many factors and reputation interests, and deeply studies reputation interests’ influence on each participant’s decision-making in green technology innovation alliance evolution under an uncertain environment and the main factors that undermine the steady state of the alliance, to further improve the game model, reduce government costs, improve the benefits of each game subject and make the alliance cooperation more stable on the basis of previous research, and consolidate the national green technology innovation and the development of the green economy. At the same time, it also lays a solid theoretical foundation for the follow-up study on the stability of multi-party green technology innovation alliances and provides more reliable decision-making reference value for all players.

In a word, there are the paper’s main works: (1) Considering the uncertainty of the real world, the random interference system is introduced into the strategic research of the alliance, and a multi-party stochastic evolution game model of green technology innovation policy in the GIUR is constructed. Using the stochastic evolution game theory, the stability conditions of the three-party strategy in the random interference environment are solved. (2) Based on the fuzzy number theory, the deterministic expectation function is uncertain, that is, it has a fuzzy expected income, which further promotes the study of the GIUR game under an uncertain environment. (3) In the GIUR alliance game, the factors such as reputation income and product greenness are emphatically considered, and the main factors’ influence on the green technology innovation alliance’s stability is specifically analyzed through numerical simulation.

The subsequent sections of this paper are structured as follows. Section 2 provides an overview of the literature. Section 3 introduces the correlation theory of fuzzy numbers, analyzes the game problem of the GIUR league, and constructs the game model accordingly. Section 4 presents a stability analysis of the constructed game model. Section 5 is a simulation analysis designed to examine the impact of key parameters on evolutionary outcomes and trajectories. Finally, we summarize the analysis results, propose management insights, and propose ideas and directions for future development.

2. Literature Review

2.1. Game Model of GIUR

In the late 20th century, scholars Etzkowitz and Leydesdorff introduced the government as a party to the evolutionary game model and put forward the theory of the interactive relationship among the government, industry, and universities, thus explaining a new relationship among the three parties in the evolution of economic alliances—”triple helix” [6]. Some scholars have found that scientific research projects with high diffusion and spillover are generally inseparable from the support of the government, such as projects to improve the green production model of industry [7,8,9]. Later, Chen and others [10] added the government as the main body of the game to the study of the green technology innovation alliance and included the hypothesis of green degree variables, which filled the gap that the existing literature lacked in attention to the role of government as the main body and the study of green technology innovation in the government, industry, university, and research (GIUR). However, they have not considered the influence of reputation gains in a heterogeneous society on game players’ decision-making in the GIUR alliance of green technology innovation.

2.2. The Stochastic Evolutionary Game Model

Evolutionary game dynamics include not only deterministic evolutionary game dynamics of infinite populations but also stochastic evolutionary game dynamics of finite populations. Deterministic evolutionary game dynamics study mixed uniform infinite populations, while the replica kinetic equations are used to describe the evolution of the strategy [11,12,13]. But due to the complexity of the realistic environment, participating in the subject of limited rational, the overall number of finite, stochastic evolution game models, that depicted game subject strategy evolution process, more appropriate strong instability of the objective world, and containing the stochastic dynamic process of the random evolution game became the main directions of the game. In the study of stochastic evolution games, experts and scholars combine stochastic analysis theory [14,15] and game theory [16,17] to introduce Gaussian white noise, and the purpose of doing so is to better portray the overall random disturbance. The research mainly explores the related theories such as stochastic evolutionary game [18] and stochastic differential equation [19] and applies them to production and life [14,20]. In 2015, Wallace combined game theory with economic application and gave the related theory and application of stochastic evolutionary game dynamics in economic development [21]. The stability of strategic alliance, the strategic choice of innovation subject and its optimal high subsidy boundary, and the variable sensitivity analysis of the model have also become hot research applications [14,22].

2.3. A Game Model with Fuzzy Numbers

The participants’ strategy set and their payoff expectations are known precisely in evolutionary game theory. In general, due to the objective environment complexity and artificial uncertainty, the participants’ strategy sets and game benefits cannot be determined. At this time, fuzzy set theory appeared, which undoubtedly provided a new favorable direction and tool for solving these problems. Zadeh [23] first put forward the fuzzy set theory, which can effectively solve the uncertainty problem by manipulating mathematical terms. Nowadays, fuzzy mathematics has become a great weapon for dealing with knowledge problems in fields such as control, decision-making, engineering, and biomedicine. After that, the proposal of fuzzy coalition and fuzzy cooperative games also laid a solid foundation for the later research on two-player zero-sum games [24,25,26]. Some scholars have studied the two-person zero-sum game from the perspective of fuzzy strategy and fuzzy payoff [26,27] and obtained the game’s approximate optimal mixed strategy in a fuzzy environment using the Monte Carlo method [28]. With the deepening of the research on fuzzy games, its research has been extended to fuzzy cooperative games [29,30]. Not only are the problems of the double-matrix game [31,32] and cooperative game [33] with fuzzy payment analyzed and solved but also the game dynamics of the Moran process evolution with interval returns and fuzzy returns and its application [34] are studied. In addition, it is undoubtedly a good method to combine fuzzy mathematics with evolutionary games and explore fuzzy economic problems in reality by copying dynamic equations [35,36,37]. As a result, the promotion and application of fuzzy games should be highly concerned, because it has better applicability.

Up to now, the existing research literature was usually based on fuzzy mathematics or using replication dynamic equations to explore the infinite population’s evolutionary game but lacked research on the finite population‘s stochastic evolutionary game. Based on this, this paper studies the GIUR stochastic evolution game dynamics with fuzzy payoffs, comprehensively considering five variables: the degree of random interference, fuzzy reputation payoffs, product greenness, and the capabilities of industry and university research parties (UR), and gives the fuzzy game expectation matrix of this model, constructs the GIUR multi-party model, generalizes and proves the stability discriminant lemma of fuzzy stochastic games and the stable equilibrium solution of the model, and finally uses the numerical simulation results to investigate the sensitivity of each parameter to the alliance evolution process.

3. Theoretical Analysis and Model Construction

3.1. The Basic Knowledge of Fuzzy Numbers

Definition 1.

Fuzzy set is a collection of objects with attributes described by fuzzy concepts. Set $A$ as a fuzzy set, $A=\left\{\left(x_1,{\mu }_A\left(x_1\right)\right),\left(x_2,{\mu }_A\left(x_2\right)\right),\dots ,\left(x_k,{\mu }_A\left(x_k\right)\right)\right\}$, ${\mu }_A\left(x\right)$ is the membership degree of an element $x$ to a fuzzy set $A$, which maps each element $x$ in $x$ to a real number in an interval $\mathit{\left[}\mathit{0}\mathit{,}\mathit{1}\mathit{\right]}$.

Definition 2.

For a fuzzy variable  $\tilde{p}=\left(l,m,r\right)$, if its membership function is  ${\mu }_p\left(x\right):R\to \mathit{\left[}\mathit{0}\mathit{,}\mathit{1}\mathit{\right]}$, i.e.,

$${\mu }_p\left(x\right)=\left\{\begin{array}{c}{\displaystyle \frac{x}{m-l}}-{\displaystyle \frac{l}{m-l}},x\in \left[1,m\right]\\ {\displaystyle \frac{u}{r-m}}-{\displaystyle \frac{x}{r-m}},x\in \left[m,r\right]\\ 0,other\end{array}\right.$$

(1)

where $x\in R,l\le m\le r$, $l$ and $r$ are lower bound and upper bound, respectively. Where $x=m$, ${\mu }_p\left(m\right)=1$. In particular, as $l=m=r$, $p$ degenerated into a real number $m$, making $m=\left(m,m,m\right)$. Then, $\tilde{p}=\left(l,m,r\right)$ is called the triangular fuzzy number. The corresponding membership function is shown in Figure 1.

Definition 3.

The sum of any two triangular fuzzy numbers ${\tilde{p}}_1=\left(l_1,m_1,r_1\right)$ and ${\tilde{p}}_2=\left(l_2,m_2,r_2\right)$, and the operation rules are as follows [38]:

Addition: ${\tilde{p}}_1\oplus {\tilde{p}}_2=\left(l_1+l_2,m_1+m_2,r_1+r_2\right)$.

Subtraction: ${\tilde{p}}_1\Theta {\tilde{p}}_2=\left(l_1-r_2,m_1-m_2,r_1-l_2\right)$.

Multiplication: ${\tilde{p}}_1\otimes {\tilde{p}}_2=\left(l_1\ast l_2,m_1\ast m_2,r_1\ast r_2\right)$.

Number multiplication: $\epsilon \otimes {\tilde{p}}_1=\left(\epsilon \ast l_1,\epsilon \ast m_1,\epsilon \ast r_1\right),\epsilon >0,\epsilon \in R$.

Division: ${\displaystyle \frac{{\tilde{p}}_1}{{\tilde{p}}_2}}=\left({\displaystyle \frac{l_1}{r_2}},{\displaystyle \frac{m_1}{m_2}},{\displaystyle \frac{r_1}{l_2}}\right),l_i,m_i,r_i>0,i=1,2$.

Among them, ⊕, Θ, ⊗, respectively, represent the addition, subtraction, and multiplication operations of fuzzy numbers.

Definition 4.

According to the comparison rule of triangular fuzzy numbers [39], the possibility of the ${\tilde{p}}_1>{\tilde{p}}_2$ of any two numbers ${\tilde{p}}_1=\left(l_1,m_1,r_1\right)$ and ${\tilde{p}}_2=\left(l_2,m_2,r_2\right)$ is

$$V\left({\tilde{p}}_1\ge {\tilde{p}}_2\right)=\left\{\begin{array}{c}0,r_1\le l_2\\ {\displaystyle \frac{r_1-l_2}{\left(m_2-m_1\right)+\left(r_1-l_2\right)}},m_1<m_{2}and{r}_1>l_2\\ 1,m_1\ge m_2\end{array}\right.$$

Definition 5.

Let any $k$ number of triangular fuzzy numbers be set ${\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_k$, and according to the comparison rule of triangular fuzzy numbers [39], the possibility of ${\tilde{p}}_i\ge {\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_k$ is

$$V\left({\tilde{p}}_i\ge {\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_k\right)=\mathit{min}\{V\left({\tilde{p}}_i\ge {\tilde{p}}_1\right),V\left({\tilde{p}}_i\ge {\tilde{p}}_2\right),\dots ,V\left({\tilde{p}}_i\ge {\tilde{p}}_k\right)\},i=1,2,\dots ,k$$

Definition 6.

Optimal pure strategy solution based on possibility [22]. $\tilde{G}=\left\{N,S_i,p_{ij}\right\}$is a triangular fuzzy matrix, in which $N$ is an all players’ set, $S_i$ is a strategies’ set of player $i$, and $p_{ij}$ is the benefit function of strategies $j$ adopted by player $i$. The situation $\left\{p_i^{\ast },q_i^{\ast }\right\}$ is an optimal pure strategy solution the possibility degree $V_{i^{\ast }{j}^{\ast }}$, which satisfies the necessary and sufficient conditions $v_1=\underset{1\le i\le m}{max}\underset{1\le j\le n}{min}{\gamma }_{ij}=\underset{1\le j\le n}{min}\underset{1\le i\le m}{max}{\gamma }_{ij}=v_2$, and needs to be satisfied $V_{i^{\ast }{j}^{\ast }}=V\left(p_{i^{\ast }{j}^{\ast }}\ge p_{i{j}^{\ast }},i=1,2,\dots ,m\right)\otimes V\left(p_{i^{\ast }{j}^{\ast }}\ge p_{i^{\ast }j},j=1,2,\dots ,n\right)>0$.

Definition 7

($\lambda$-cut set [25]). Let $\tilde{A}$ be a fuzzy set on the universe $X$, $\lambda \in \left[0,1\right]$, remember: ${\tilde{A}}_{\lambda }=\left\{x\in X\left|{\mu }_{\lambda }\left(x\right)\ge \lambda \right.\right\}$, ${\tilde{A}}_{\lambda }$ is called the $\lambda$-cut set of $\tilde{A}$, and $\lambda$ is the confidence level or threshold. In particular, the kernel ${\tilde{A}}_{1,}$ called fuzzy set $\tilde{A}$ when $\lambda =1$ is called, remember as $Ker\tilde{A}$.

If $\tilde{A}$ is set as a fuzzy subset on the universe, take $\lambda \in \left[0,1\right]$ at will, remember ${\tilde{A}}_{\lambda }=\left\{x\left|{\mu }_{\lambda }\left(x\right)\ge \lambda \right.\right\}$, then, ${\tilde{A}}_{\lambda }$ is the level set of $\tilde{A}$. Therefore, ${\tilde{A}}_{\lambda }$ can be expressed as ${\tilde{A}}_{\lambda }=\left[\underset{¯}{{\tilde{A}}_{\lambda }},\overline{{\tilde{A}}_{\lambda }}\right]$, and $\underset{¯}{{\tilde{A}}_{\lambda }}$ and $\overline{{\tilde{A}}_{\lambda }}$ are the left and right boundaries of ${\tilde{A}}_{\lambda },$ respectively.

For any $\lambda \in \left[0,1\right]$, $\underset{¯}{{\tilde{A}}_{\lambda }}$ and $\overline{{\tilde{A}}_{\lambda }}$ of the $\lambda$ level sets of triangular fuzzy number $\tilde{A}=\left(l,m,r\right)$ can be expressed as $\underset{¯}{{\tilde{A}}_{\lambda }}=l+\left(m-l\right)\lambda$, $\overline{{\tilde{A}}_{\lambda }}=r-\left(r-m\right)\lambda ,$ respectively.

Definition 8

 ([40]). Set interval number $\left[a,b\right]$ and $\left[c,d\right]$, among $a<b,c<d$, then,

(1) $\left[a,b\right]+\left[c,d\right]=\left[a+c,b+d\right]$.

(2) $-\left[a,b\right]=\left[-b,-a\right]$.

(3) $\left[a,b\right]-\left[c,d\right]=\left[a-d,b-c\right]$.

(4) If $ab>0$, then $\left[a,b{]}^{-1}=\right[b^{-1},a^{-1}]$.

(5) $\left[a,b\right]\times \left[c,d\right]=[\mathit{min}\{ac,ad,bc,bd\},\mathit{max}\{ac,ad,bc,bd\}]$.

(6) If $cd>0$, then ${\displaystyle \frac{\left[a,b\right]}{\left[c,d\right]}}=[\mathit{min}\{{\displaystyle \frac{a}{c}},{\displaystyle \frac{a}{d}},{\displaystyle \frac{b}{c}},{\displaystyle \frac{b}{d}}\},\mathit{max}\{{\displaystyle \frac{a}{c}},{\displaystyle \frac{a}{d}},{\displaystyle \frac{b}{c}},{\displaystyle \frac{b}{d}}\}]$.

(7) If $k>0$, then $k\times \left[a,b\right]=\left[ka,kb\right]$; if $k<0$, then $k\times \left[a,b\right]=\left[kb,ka\right]$.

Definition 9

 ([41]). Let any two interval numbers be $a=\left[a^{-},a^{+}\right]$ and $b=\left[b^{-},b^{+}\right]$, and give the comparison rules of interval numbers as follows:

$$\left\{\begin{array}{c}a<biff{a}^{+}<b^{-}\\ a\subseteq biff{a}^{-}\ge b^{-}and{a}^{+}\le b^{+}\end{array}\right.$$

It is worth noting that this definition does not apply to the number of intervals where two intervals overlap.

All of the above definitions and descriptions of fuzzy numbers are shown in

Table 1. Definition of fuzzy number.

Define the Number	Define the Name	Mathematical Expressions	Ilustrate
Definition 1	Fuzzy sets	$A=\left\{\left(x_1,{\mu }_A\left(x_1\right)\right),\left(x_2,{\mu }_A\left(x_2\right)\right),\dots ,\left(x_k,{\mu }_A\left(x_k\right)\right)\right\}$	${\mu }_A\left(x\right)$ is the membership degree of an element $x$ to a fuzzy set $A$
Definition 2	Trigonometric fuzzy number	${\mu }_p\left(x\right)=\left\{\begin{array}{c}{\displaystyle \frac{x}{m-l}}-{\displaystyle \frac{l}{m-l}},x\in \left[1,m\right]\\ {\displaystyle \frac{u}{r-m}}-{\displaystyle \frac{x}{r-m}},x\in \left[m,r\right]\\ 0,other\end{array}\right.$	$x\in R,l\le m\le r$, $l$ and $r$ are lower bound and upper bound, respectively, where $x=m$, ${\mu }_p\left(m\right)=1$. In particular, as $l=m=r$, $p$ degenerated into a real number $m$, making $m=\left(m,m,m\right)$. Then, $\tilde{p}=\left(l,m,r\right)$ is called thetriangular fuzzy number.
Definition 3	Rules for the operation of trigonometric fuzzy numbers	Addition: ${\tilde{p}}_1\oplus {\tilde{p}}_2=\left(l_1+l_2,m_1+m_2,r_1+r_2\right)$. Subtraction: ${\tilde{p}}_1\Theta {\tilde{p}}_2=\left(l_1-r_2,m_1-m_2,r_1-l_2\right)$. Multiplication: ${\tilde{p}}_1\otimes {\tilde{p}}_2=\left(l_1\ast l_2,m_1\ast m_2,r_1\ast r_2\right)$. Number multiplication: $\epsilon \otimes {\tilde{p}}_1=\left(\epsilon \ast l_1,\epsilon \ast m_1,\epsilon \ast r_1\right),\epsilon >0,\epsilon \in R$. Division: ${\displaystyle \frac{{\tilde{p}}_1}{{\tilde{p}}_2}}=\left({\displaystyle \frac{l_1}{r_2}},{\displaystyle \frac{m_1}{m_2}},{\displaystyle \frac{r_1}{l_2}}\right),l_i,m_i,r_i>0,i=1,2$.	Arithmetic rules based on interval endpoints.
Definition 4	Trigonometric fuzzy number comparison rules	$V\left({\tilde{p}}_1\ge {\tilde{p}}_2\right)=\left\{\begin{array}{c}0,r_1\le l_2\\ {\displaystyle \frac{r_1-l_2}{\left(m_2-m_1\right)+\left(r_1-l_2\right)}},m_1<m_{2}and{r}_1>l_2\\ 1,m_1\ge m_2\end{array}\right.$	The possibility of the ${\tilde{p}}_1>{\tilde{p}}_2$ of any two numbers.
Definition 5	Multiple triangular fuzzy numbers are compared to the rule	$V\left({\tilde{p}}_i\ge {\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_k\right)=\mathit{min}\{V\left({\tilde{p}}_i\ge {\tilde{p}}_1\right),$ $V\left({\tilde{p}}_i\ge {\tilde{p}}_2\right),\dots ,V\left({\tilde{p}}_i\ge {\tilde{p}}_k\right)\},i=1,2,\dots ,k$	The possibility of ${\tilde{p}}_i\ge {\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_k$.
Definition 6	Optimal strategy solution	$v_1=\underset{1\le i\le m}{max}\underset{1\le j\le n}{min}{\gamma }_{ij}=\underset{1\le j\le n}{min}\underset{1\le i\le m}{max}{\gamma }_{ij}=v_2$, and needs to be satisfied $V_{i^{\ast }{j}^{\ast }}=V\left(p_{i^{\ast }{j}^{\ast }}\ge p_{i{j}^{\ast }},i=1,2,\dots ,m\right)\otimes V\left(p_{i^{\ast }{j}^{\ast }}\ge p_{i^{\ast }j},j=1,2,\dots ,n\right)>0$.	The optimal pure strategy solution of the probability is sufficient and necessary.
Definition 7	$\lambda$-cut set	${\tilde{A}}_{\lambda }=\left\{x\in X\left|{\mu }_{\lambda }\left(x\right)\ge \lambda \right.\right\}$	${\tilde{A}}_{\lambda }$ is called the $\lambda$-cut set of $\tilde{A}$.
Definition 8	Rules for the operation of interval number rules	(1) $\left[a,b\right]+\left[c,d\right]=\left[a+c,b+d\right]$. (2) $-\left[a,b\right]=\left[-b,-a\right]$. (3) $\left[a,b\right]-\left[c,d\right]=\left[a-d,b-c\right]$. (4) If $ab>0$, then $\left[a,b{]}^{-1}=\right[b^{-1},a^{-1}]$. (5) $\left[a,b\right]\times \left[c,d\right]=\left[\mathit{min}\right\{ac,ad,bc,bd\},\mathit{max}\{ac,ad$ $,bc,bd\}]$. (6) If $cd>0$, then ${\displaystyle \frac{\left[a,b\right]}{\left[c,d\right]}}=[\mathit{min}\{{\displaystyle \frac{a}{c}},{\displaystyle \frac{a}{d}},{\displaystyle \frac{b}{c}},{\displaystyle \frac{b}{d}}\},\mathit{max}\{{\displaystyle \frac{a}{c}},{\displaystyle \frac{a}{d}}$ $,{\displaystyle \frac{b}{c}},{\displaystyle \frac{b}{d}}\}]$. (7) If $k>0$, then $k\times \left[a,b\right]=\left[ka,kb\right]$; if $k<0$, then $k\times \left[a,b\right]=\left[kb,ka\right]$.	Including addition, negation, subtraction, reciprocal, multiplication, division, and number multiplication operations of interval numbers.
Definition 9	Interval number comparison rules	$\left\{\begin{array}{c}a<biff{a}^{+}<b^{-}\\ a\subseteq biff{a}^{-}\ge b^{-}and{a}^{+}\le b^{+}\end{array}\right.$,	This definition does not apply to the number of intervals where two intervals overlap.

.

3.2. Problem Description and Model Construction

3.2.1. Description and Analysis of the Game Problem of the GIUR Alliance with Fuzzy Numbers

At present, some experts and scholars have found that government intervention can make up for market failure, and it can improve the complicated internal and external environment induced by technical complexity or public opinion through laws, policies, and other means [8,9], further improve the willingness of players to cooperate, maintain the alliance’s long-term stability, and stimulate market activity. According to the stakeholder theory [42] and thw related literature variables [10,14,43], the stakeholders of this alliance are the government, industry, and the UR. Therefore, the following are the model’s basic assumptions.

Assumption 1.

The game’s participants include government, industry, and the UR, all of whom are bounded rationality.

Assumption 2.

The government’s strategy is (supervision, no supervision). The total cost of the government’s choice of supervision and non-supervision are ${\tilde{C}}_1$ and ${\tilde{C}}_4$, respectively, and the gains are ${\tilde{S}}_1$ and ${\tilde{S}}_4$, respectively, among which, ${\tilde{S}}_1>{\tilde{S}}_4>0$.

Assumption 3.

The government’s incentive to industry is ${\tilde{K}}_1$, and the incentive to the UR is ${\tilde{K}}_2$.

Assumption 4.

The strategy of the industry is (cooperation, betrayal in the middle). The total cost of industry’s choice of cooperation and midway betrayal are  ${\tilde{C}}_2$ and ${\tilde{C}}_5,$ respectively, and the gains are ${\tilde{S}}_2$ and ${\tilde{S}}_5,$ respectively, among which ${\tilde{S}}_2>{\tilde{S}}_5>0$. In the process of cooperation, the technical knowledge transfer ability of industry is $\tilde{\alpha }$, and the research and development ability is $\tilde{q}$, $\tilde{\alpha },\tilde{q}\in \left(0,1\right)$.

Assumption 5.

The strategy of the UR is (active research and development, betrayal in the middle). The total cost of choosing active R&D and betrayal in the middle are ${\tilde{C}}_3$ and ${\tilde{C}}_6,$ respectively,and the gains are ${\tilde{S}}_3$ and ${\tilde{S}}_6,$ respectively, where ${\tilde{S}}_3>{\tilde{S}}_6>0$. In the process of cooperation, the ability of technical knowledge expansion is $\tilde{\beta }$, and research and development promotion is $\tilde{p}$, $\tilde{\alpha },\tilde{q}\in \left(0,1\right)$.

Assumption 6.

Under the supervision of the government, the party who chooses to betray halfway should compensate $\tilde{P},$ the betrayed party. When the UR chooses to betray in the middle, the value of the knowledge, technology, and ideas he learned from the beginning to the middle is ${\tilde{L}}_1$. When an enterprise chooses to betray in the middle, the value of the semi-finished products or inspiration obtained during the period from the beginning to the middle is ${\tilde{L}}_2$.

Assumption 7.

The total value added of green innovation brought by the choice of continuing cooperation between non-governmental players is $\Delta \tilde{\pi }$. Assuming that industry’s proportion in the value added of green innovation is $\tilde{\theta }$, the proportion of the UR is $1-\tilde{\theta }$, which, $\tilde{\theta }\in \left(0,1\right)$. $\tilde{\epsilon }$ is the greenness of products produced jointly. Here, the proportion of value added is directly proportional to the proportion of input.

Assumption 8.

About reputation gains: when the government chooses to supervise, it gains ${\tilde{R}}_1$ benefit, otherwise it loses income ${\tilde{R}}_4$. When an enterprise chooses to cooperate or betray halfway, it gains ${\tilde{R}}_2$ and loses profits ${\tilde{R}}_5,$ respectively. When the UR chooses active R&D and betrays midway, it gains ${\tilde{R}}_3$ and loses income ${\tilde{R}}_6$ respectively.

Assumption 9.

For the government, the probability of choosing supervision and non-supervision are $\tilde{x}\left(0<\tilde{x}<1\right)$and $1-\tilde{x},$ respectively. For industry, the probability of choosing cooperation and betrayal in the middle are $\tilde{y}\left(0<\tilde{y}<1\right)$ and $1-\tilde{y}$ respectively. For the UR, the probability that it chooses active R&D and midway betrayal are $\tilde{z}\left(0<\tilde{z}<1\right)$ and $1-\tilde{z},$ respectively.

According to the aforementioned parameter hypotheses, the payoff matrix of the GIUR evolutionary game is created as shown in

Table 2. Payoff matrix.

Strategy Selection				Academia and Research Parties
				Actively Research and Develop (z)	Betray in the Middle (1 − z)
Government	Supervise ($\tilde{x}$)	Companies	Cooperate ($\tilde{y}$)	${\tilde{S}}_1+{\tilde{R}}_1-{\tilde{K}}_1-{\tilde{K}}_2-{\tilde{C}}_1$  ${\tilde{S}}_2+{\tilde{R}}_2+{\tilde{K}}_1+\tilde{\epsilon }\tilde{\theta }\Delta \tilde{\pi }\left(1+\tilde{\alpha }\tilde{q}\right)-{\tilde{C}}_2$  ${\tilde{S}}_3+{\tilde{R}}_3+{\tilde{K}}_2+\tilde{\epsilon }\left(1-\tilde{\theta }\right)\Delta \tilde{\pi }\left(1+\tilde{\beta }\tilde{p}\right)-{\tilde{C}}_3$	${\tilde{S}}_1+{\tilde{R}}_1-{\tilde{K}}_1-{\tilde{C}}_1$  ${\tilde{S}}_2+{\tilde{R}}_2+{\tilde{K}}_1+\tilde{P}-{\tilde{C}}_2$  ${\tilde{S}}_6+{\tilde{L}}_2\tilde{\beta }\tilde{p}-{\tilde{R}}_6-{\tilde{C}}_6-\tilde{P}$
			Midway betray ($1-\tilde{y}$)	${\tilde{S}}_1+{\tilde{R}}_1-{\tilde{K}}_2-{\tilde{C}}_1$  ${\tilde{S}}_5+{\tilde{L}}_1\tilde{\alpha }\tilde{q}-{\tilde{R}}_5-\tilde{P}-{\tilde{C}}_5$  ${\tilde{S}}_3+{\tilde{R}}_3+{\tilde{K}}_2+\tilde{P}-{\tilde{C}}_3$	${\tilde{S}}_1+{\tilde{R}}_1-{\tilde{C}}_1$   ${\tilde{S}}_5+{\tilde{L}}_1\tilde{\alpha }\tilde{q}-{\tilde{R}}_5-{\tilde{C}}_5$   ${\tilde{S}}_6+{\tilde{L}}_2\tilde{\beta }\tilde{p}-{\tilde{R}}_6-{\tilde{C}}_6$
	Not supervise ($1-\tilde{x}$)	Companies	Cooperate ($\tilde{y}$)	${\tilde{S}}_4-{\tilde{R}}_4-{\tilde{C}}_4$  ${\tilde{S}}_2+{\tilde{R}}_2+\tilde{\epsilon }\tilde{\theta }\Delta \tilde{\pi }\left(1+\tilde{\alpha }\tilde{q}\right)-{\tilde{C}}_2$  ${\tilde{S}}_3+{\tilde{R}}_3+\tilde{\epsilon }\left(1-\tilde{\theta }\right)\Delta \tilde{\pi }\left(1+\tilde{\beta }\tilde{p}\right)-{\tilde{C}}_3$	${\tilde{S}}_4-{\tilde{R}}_4-{\tilde{C}}_4$  ${\tilde{S}}_2+{\tilde{R}}_2-{\tilde{C}}_2$  ${\tilde{S}}_6+{\tilde{L}}_2\tilde{\beta }\tilde{p}-{\tilde{R}}_6-{\tilde{C}}_6$
			Midway betray ($1-\tilde{y}$)	${\tilde{S}}_4-{\tilde{R}}_4-{\tilde{C}}_4$  ${\tilde{S}}_5+{\tilde{L}}_1\tilde{\alpha }\tilde{q}-{\tilde{R}}_5-{\tilde{C}}_5$  ${\tilde{S}}_3+{\tilde{R}}_3-{\tilde{C}}_3$	${\tilde{S}}_4-{\tilde{R}}_4-{\tilde{C}}_4$  ${\tilde{S}}_5+{\tilde{L}}_1\tilde{\alpha }\tilde{q}-{\tilde{R}}_5-{\tilde{C}}_5$  ${\tilde{S}}_6+{\tilde{L}}_2\tilde{\beta }\tilde{p}-{\tilde{R}}_6-{\tilde{C}}_6$

Note: All the above parameters are greater than or equal to 0 and are triangular fuzzy numbers.

.

3.2.2. The Evolution Game Model of the GIUR Considering Reputation Gains

For convenience, this paper uses $\tilde{x}\left(t\right)$ to represent the proportion of partners, and the following description uses $\tilde{x}$ instead of $\tilde{x}\left(t\right)$. Therefore, the partner’s replication dynamic equation is expressed as

$$d\tilde{x}=\tilde{x}\left({\tilde{f}}_1-\tilde{\Phi }\right)dt$$

(2)

Suppose that ${\tilde{f}}_1$ and ${\tilde{f}}_2$ are the average fitness of all collaborators and traitors, respectively, determined by the game’s payoff matrix. The overall average fitness is $\tilde{\Phi }=\tilde{x}{\tilde{f}}_1+\left(1-\tilde{x}\right){\tilde{f}}_2$. Therefore, in the evolutionary game with two strategies, the explicit expression of the replication dynamic equation is [16,44,45]

$$d\tilde{x}=\tilde{x}\left({\tilde{f}}_1-\left(\tilde{x}{\tilde{f}}_1+\left(1-\tilde{x}\right){\tilde{f}}_2\right)\right)dt=\tilde{x}\left(1-\tilde{x}\right)\left({\tilde{f}}_1-{\tilde{f}}_2\right)dt.$$

(3)

According to the payoff matrix, the replication dynamic equations for the three parties’ selection strategies are obtained as follows:

Assuming that the payoffs of the government’s choice of “supervision” and “non-supervision” are ${\tilde{U}}_1$ and ${\tilde{U}}_2$, respectively, then the government’s replication dynamic equation is obtained

$$d\tilde{x}=\tilde{x}\left(1-\tilde{x}\right)\left({\tilde{U}}_1-{\tilde{U}}_2\right)dt=\tilde{x}\left(1-\tilde{x}\right)\left(-{\tilde{K}}_1\tilde{y}-{\tilde{K}}_2\tilde{z}+{\tilde{S}}_1-{\tilde{S}}_4+{\tilde{R}}_1+{\tilde{R}}_4+{\tilde{C}}_4-{\tilde{C}}_1\right)dt$$

(4)

Assuming that the payoffs of the enterprise’s choice of “cooperation” and “midway betrayal” are ${\tilde{V}}_1$ and ${\tilde{V}}_2$, respectively, then the enterprise’s replication dynamic equation is obtained

$$\begin{array}{c}d\tilde{y}=\tilde{y}\left(1-\tilde{y}\right)\left({\tilde{V}}_1-{\tilde{V}}_2\right)dt=\tilde{y}\left(1-\tilde{y}\right)(\tilde{\epsilon }\tilde{\theta }\Delta \tilde{\pi }\left(1+\tilde{\alpha }\tilde{q}\right)\tilde{z}-{\tilde{L}}_1\tilde{\alpha }\tilde{q}+({\tilde{K}}_1+\\ \tilde{P})\tilde{x}+{\tilde{S}}_2-{\tilde{S}}_5+{\tilde{R}}_2+{\tilde{R}}_5-{\tilde{C}}_2+{\tilde{C}}_5)dt\end{array}$$

(5)

Assuming that the payoffs of the UR’s choice of “active R&D” and “midway betrayal” are ${\tilde{W}}_1$ and ${\tilde{W}}_2$, respectively, then the UR’s replication dynamic equation is obtained

$$\begin{array}{c}d\tilde{z}=\tilde{z}\left(1-\tilde{z}\right)\left({\tilde{W}}_1-{\tilde{W}}_2\right)dt==\tilde{z}\left(1-\tilde{z}\right)(\tilde{\epsilon }\left(1-\tilde{\theta }\right)\Delta \tilde{\pi }\left(1+\tilde{\beta }\tilde{p}\right)\tilde{y}-{\tilde{L}}_2\tilde{\beta }\tilde{p}+\\ \left({\tilde{K}}_2+\tilde{P}\right)\tilde{x}+{\tilde{S}}_3-{\tilde{S}}_6+{\tilde{R}}_3+{\tilde{R}}_6-{\tilde{C}}_3+{\tilde{C}}_6)dt\end{array}$$

(6)

3.2.3. The Stochastic Evolution Game Model of the GIUR Alliance Considering Reputation Gains

The real world is a dynamic organism, and its high uncertainty will affect the strategy choice of alliance members to a certain extent, and all individuals in the system will be disturbed by noise. Therefore, many scholars have devoted themselves to the study of stochastic evolutionary games [46,47,48,49]. Yang et al. [50] established the bidding behavior game about power generation industry with the help of stochastic evolutionary game theory and stochastic differential equations knowledge. They suggest that more effective strategies have greater certainty about population evolutionary direction and the evolutionary rate of equilibrium point. Zhu et al. [15] explored the green credit mechanism model based on stochastic evolutionary game theory. The results show that although the reward and punishment mechanism is helpful to green credit and production, increasing the reward is not conducive to the government’s ability to perform its regulatory duties. Feng et al. [51] used the Gram–Schmidt orthogonalization process and Ito formula to derive the equation of the stochastic replicator, which was used for the dynamics and evolutionary stability of the stochastic replicator. To investigate the effect of data sharing on model stability, Dong et al. [43] established a three-party stochastic evolutionary game model and introduced Gaussian noise to portray random disturbance. Here, with the help of stochastic analysis theory, this paper adds Gaussian white noise, establishes the corresponding stochastic evolutionary game model, and then analyzes the substantial impact of various factors on the GIUR system stability. Therefore, the stochastic replication dynamic equation of government, industry, and the UR is obtained [14]:

$$\begin{array}{c}d\tilde{x}\left(t\right)=\tilde{x}\left(t\right)\left(1-\tilde{x}\left(t\right)\right)(-{\tilde{K}}_1\tilde{y}-{\tilde{K}}_2\tilde{z}+{\tilde{S}}_1-{\tilde{S}}_4+{\tilde{R}}_1+{\tilde{R}}_4+{\tilde{C}}_4-\\ {\tilde{C}}_1)dt+\sigma \tilde{x}\left(t\right)d\omega \left(t\right)\end{array}$$

(7)

$$\begin{array}{c}d\tilde{y}\left(t\right)=\tilde{y}\left(t\right)\left(1-\tilde{y}\left(t\right)\right)[\tilde{\epsilon }\tilde{\theta }\Delta \tilde{\pi }\left(1+\tilde{\alpha }\tilde{q}\right)\tilde{z}-{\tilde{L}}_1\tilde{\alpha }\tilde{q}+\left({\tilde{K}}_1+\tilde{P}\right)\tilde{x}+{\tilde{S}}_2-\\ {\tilde{S}}_5+{\tilde{R}}_2+{\tilde{R}}_5-{\tilde{C}}_2+{\tilde{C}}_5]dt+\sigma \tilde{y}\left(t\right)d\omega \left(t\right)\end{array}$$

(8)

$$\begin{array}{c}d\tilde{z}\left(t\right)=\tilde{z}\left(t\right)\left(1-\tilde{z}\left(t\right)\right)[\tilde{\epsilon }\left(1-\tilde{\theta }\right)\Delta \tilde{\pi }\left(1+\tilde{\beta }\tilde{p}\right)\tilde{y}-{\tilde{L}}_2\tilde{\beta }\tilde{p}+\left({\tilde{K}}_2+\tilde{P}\right)\tilde{x}+\\ {\tilde{S}}_3-{\tilde{S}}_6+{\tilde{R}}_3+{\tilde{R}}_6-{\tilde{C}}_3+{\tilde{C}}_6]dt+\sigma \tilde{z}\left(t\right)d\omega \left(t\right)\end{array}$$

(9)

where $\sigma$ is a random disturbance term, $\sigma =0$ indicates that the model has no random perturbation factor, and $\sigma =1$ indicates that the model has a random perturbation factor. $N\left(0,\sqrt{h}\right)\omega \left(t\right)$ is a one-dimensional standard Brownian motion, $d\omega \left(t\right)$ stands for Gaussian white noise, and when $t>0$, the step size $h>0$, and its increment $\Delta \omega \left(t\right)=\omega \left(t+h\right)-\omega \left(t\right)$ obeys the normal distribution.

Equations (7)–(9) are one-dimensional $It\widehat{o}$ stochastic differential equations with Gaussian random interference. Generally speaking, the explicit solutions of most nonlinear $It\widehat{o}$ stochastic differential equations are difficult to find. However, under certain conditions, the solutions’ existence and uniqueness of the equation can be obtained, and solutions’ basic characteristics can also be studied further.

Let the initial moment of the game subject be $t=0$, there is $\tilde{x}\left(0\right)=\left[0,0\right],\tilde{y}\left(0\right)=\left[0,0\right],\tilde{z}\left(0\right)=\left[0,0\right]$, and because $d\omega \left(t\right)\left|t\right.=0={\omega }^{\prime }\left(t\right)dt\left|t=0\right.=0$, then the equation has at least zero solution, that is, when $\sigma =0$, the system will continue to maintain this state, so the zero solution must be one of the equation’s equilibrium solutions.

4. Model Stability Analysis

4.1. Analysis of Strategic Nash Equilibrium Solution of the GIUR League Game with Triangular Fuzzy Numbers

(1) Pure strategy Nash equilibrium solution based on possibility.

According to the definition, we can calculate:

$$\begin{array}{c}V\left({\tilde{a}}_{11}\ge {\tilde{a}}_{i1},i=1,2,3,4\right)=\mathit{min}\{V\left({\tilde{a}}_{11}\ge {\tilde{a}}_{11}\right),V\left({\tilde{a}}_{11}\ge {\tilde{a}}_{21}\right),V({\tilde{a}}_{11}\ge \\ {\tilde{a}}_{31}),V\left({\tilde{a}}_{11}\ge {\tilde{a}}_{41}\right)\}\end{array}$$

(10)

$$\begin{array}{c}V\left({\tilde{a}}_{12}\ge {\tilde{a}}_{i2},i=1,2,3,4\right)=\mathit{min}\{V\left({\tilde{a}}_{12}\ge {\tilde{a}}_{12}\right),V\left({\tilde{a}}_{12}\ge {\tilde{a}}_{22}\right),V({\tilde{a}}_{12}\ge \\ {\tilde{a}}_{33}),V\left({\tilde{a}}_{12}\ge {\tilde{a}}_{42}\right)\}\end{array}$$

(11)

In the same way:

$$V\left({\tilde{a}}_{21}\ge {\tilde{a}}_{i1},i=1,2,3,4\right),V\left({\tilde{a}}_{31}\ge {\tilde{a}}_{i1},i=1,2,3,4\right),V\left({\tilde{a}}_{41}\ge {\tilde{a}}_{i1},i=1,2,3,4\right),$$

$$V\left({\tilde{a}}_{22}\ge {\tilde{a}}_{i2},i=1,2,3,4\right),V\left({\tilde{a}}_{32}\ge {\tilde{a}}_{i2},i=1,2,3,4\right),V\left({\tilde{a}}_{42}\ge {\tilde{a}}_{i2},i=1,2,3,4\right).$$

According to the above definition,

$$V_{11}=V\left({\tilde{a}}_{11}\ge {\tilde{a}}_{i1},i=1,2,3,4\right)\ast V\left({\tilde{b}}_{11}\ge {\tilde{b}}_{j1},j=1,2,3,4\right)\ast V\left({\tilde{c}}_{11}\ge {\tilde{c}}_{1k},k=1,2\right),$$

(12)

$$V_{12}=V\left({\tilde{a}}_{12}\ge {\tilde{a}}_{i2},i=1,2,3,4\right)\ast V\left({\tilde{b}}_{12}\ge {\tilde{b}}_{j2},j=1,2,3,4\right)\ast V\left({\tilde{c}}_{12}\ge {\tilde{c}}_{1k},k=1,2\right),$$

(13)

In the same way: $V_{21}$, $V_{22}$, $V_{31}$, $V_{32}$, $V_{41}$, $V_{42}$.

According to Definition 5, when $V_{ij}>0,i=1,2,3,4;j=1,2,3,4$, the game’s optimal pure strategy can be obtained.

Mixed strategy Nash equilibrium solution:

Based on the payoff matrix (see Table 1), if the alliance is not optimal pure strategy solution, then, according to the Nash equilibrium existence theorem I [26], a Nash equilibrium solution existing for any finite game at least, then the system will at least have a mixed strategy equilibrium solution.

Then, the average expected returns of the government, industry, and the UR are $E_1\left(\tilde{p},\tilde{q},\tilde{f}\right)$, $E_2\left(\tilde{p},\tilde{q},\tilde{f}\right)$, and $E_3\left(\tilde{p},\tilde{q},\tilde{f}\right)$, respectively.

By differentiating the above utility functions, the first-order conditions for the optimization of the strategies of the government, industry, and the UR are obtained, respectively:

$${\displaystyle \frac{\partial E_1\left(\tilde{p},\tilde{q},\tilde{f}\right)}{\partial \tilde{p}}}=0,{\displaystyle \frac{\partial E_2\left(\tilde{p},\tilde{q},\tilde{f}\right)}{\partial \tilde{q}}}=0,{\displaystyle \frac{\partial E_3\left(\tilde{p},\tilde{q},\tilde{f}\right)}{\partial \tilde{f}}}=0.$$

There is a mixed strategy $\left({\tilde{p}}^{\ast },{\tilde{q}}^{\ast },{\tilde{f}}^{\ast }\right)$.

4.2. Analysis of Nash Equilibrium Solution of Stochastic Evolutionary Game Model of the GIUR Alliance Game

In reality, the alliance is bound to be disturbed by many environmental factors, thus affecting the alliance’s long-term stability. In view of this, as far as the GIUR alliance stability is concerned, the addition of random factors is worth considering. When solving the equilibrium solution, we must ensure its existence and stability. The following Lemma 1 [52,53,54] gives the stability judgment theorem of the stochastic evolution game model’s equilibrium solution, from which the stability judgment theorem of the fuzzy stochastic game model’s equilibrium solution can be tried to prove.

Lemma 1.

Let fuzzy stochastic differential equation $\tilde{x}=\tilde{x}\left(t\right)=\left[\underset{\_}{x}\left(t\right),\overline{x}\left(t\right)\right],t\ge 0$ satisfy the solution of initial value problem of $It\widehat{o}$ differential equation:

$$d\tilde{x}\left(t\right)=d\left[\underset{\_}{x}\left(t\right),\overline{x}\left(t\right)\right]=\left[\underset{\_}{f}\left(t,\underset{\_}{x}\left(t\right)\right),\overline{f}\left(t,\overline{x}\left(t\right)\right)\right]dt+\left[\underset{\_}{g}\left(t,\underset{\_}{x}\left(t\right)\right),\overline{g}\left(t,\overline{x}\left(t\right)\right)\right]d\omega \left(t\right),$$

$$\forall t\ge 0,\tilde{x}\left(t_0\right)={\tilde{x}}_0=\left[{\underset{\_}{x}}_0,{\overline{x}}_0\right]$$

The existence of continuously differentiable function $V\left(t,\tilde{x}\right)$ and normal numbers $c_1,c_2$ makes $c_1{\left|\tilde{x}\right|}^p<V\left(t,\tilde{x}\right)<c_2{\left|\tilde{x}\right|}^p$ hold.

Assuming a normal number $\gamma$ exists and makes $LV\left(t,\tilde{x}\right)\le -\gamma V\left(t,\tilde{x}\right),t\ge 0$, then the zero solution of Equation (7) is p-order exponentially stable and holds $E{\left|\tilde{x}\left(t,{\tilde{x}}_0\right)\right|}^p\le \left({\displaystyle \frac{c_2}{c_1}}\right){\left|{\tilde{x}}_0\right|}^p{e}^{-\gamma t},t\ge 0$.

Assuming a normal number exists and makes $LV\left(t,\tilde{x}\right)\ge \gamma V\left(t,\tilde{x}\right),t\ge 0$, then the zero solution of Equation (7) is p-order exponentially unstable, and $E{\left|\tilde{x}\left(t,{\tilde{x}}_0\right)\right|}^p\ge \left({\displaystyle \frac{c_1}{c_2}}\right){\left|{\tilde{x}}_0\right|}^p{e}^{\gamma t},t\ge 0$ is established, where $LV\left(t,\tilde{x}\right)=V_t\left(t,\tilde{x}\right)+V_x\left(t,\tilde{x}\right)\tilde{f}\left(t,\tilde{x}\right)+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\tilde{g}}^2\left(t,\tilde{x}\right)V_{xx}\left(t,\tilde{x}\right)$.

Proof. 

Let $V\left(t,\tilde{x}\right)=\tilde{x}=\left[\underset{\_}{x},\overline{x}\right]$, $V\left(t,\tilde{y}\right)=\tilde{y}=\left[\underset{\_}{y},\overline{y}\right]$, $V\left(t,\tilde{z}\right)=\tilde{z}=\left[\underset{\_}{z},\overline{z}\right]$, $\tilde{x}\in \left[0,1\right]$, $\tilde{y}\in \left[0,1\right]$, $\tilde{z}\in \left[0,1\right]$, $c_1=c_2=1$, and $p=1$, then $\left|\tilde{x}\right|\le V\left(t,\tilde{x}\right)\le \left|\tilde{x}\right|$ is clearly established. Let $LV\left(t,\tilde{x}\right)=\tilde{f}\left(t,\tilde{x}\right)=\left[\underset{\_}{f}\left(t,\underset{\_}{x}\right),\overline{f}\left(t,\overline{x}\right)\right]$, assuming that there is a constant k, $-kV\left(t,\tilde{x}\right)=\left\{\begin{array}{c}\left[-k\overline{x},-k\underset{\_}{x}\right],k>0\\ \left[k\underset{\_}{x},k\overline{x}\right],k\le 0\end{array}\right.$ from Definition 8 and Definition 9. When $\underset{\_}{f}\left(t,\underset{\_}{x}\right)\ge -k\overline{x}$ and $\overline{f}\left(t,\overline{x}\right)\ge -k\underset{\_}{x}$, $\left[\underset{\_}{f}\left(t,\underset{\_}{x}\right),\overline{f}\left(t,\overline{x}\right)\right]\subseteq \left[-k\overline{x},-k\underset{\_}{x}\right]$ is valid; or when $\underset{\_}{f}\left(t,\underset{\_}{x}\right)\ge k\underset{\_}{x}$ and $\overline{f}\left(t,\overline{x}\right)\ge k\overline{x}$, $\left[\underset{\_}{f}\left(t,\underset{\_}{x}\right),\overline{f}\left(t,\overline{x}\right)\right]\subseteq \left[k\underset{\_}{x},k\overline{x}\right]$ is valid, then $LV\left(t,\tilde{x}\right)\le -kV\left(t,\tilde{x}\right),t\ge 0$ is valid and $E\left|\tilde{x}\left(t,{\tilde{x}}_0\right)\right|\le \left|{\tilde{x}}_0\right|e^{-\gamma t},t\ge 0$. And this point, $k\in \left[-{\displaystyle \frac{\underset{\_}{f}\left(t,\underset{\_}{x}\right)}{\overline{x}}},-{\displaystyle \frac{\overline{f}\left(t,\overline{x}\right)}{\underset{\_}{x}}}\right]$ or $k\le {\displaystyle \frac{\underset{\_}{f}\left(t,\underset{\_}{x}\right)}{\underset{\_}{x}}}andk\ge {\displaystyle \frac{\overline{f}\left(t,\overline{x}\right)}{\overline{x}}}$, is not valid, excluding the latter. Therefore, the constant k, condition holds.□

For the stochastic replication dynamic Equations (7)–(9), let $V\left(t,\tilde{x}\right)=\tilde{x}=\left[\underset{\_}{x},\overline{x}\right],V\left(t,\tilde{y}\right)=\tilde{y}=\left[\underset{\_}{y},\overline{y}\right],V\left(t,\tilde{z}\right)=\tilde{z}=\left[\underset{\_}{z},\overline{z}\right],\tilde{x}\in \left[0,1\right],\tilde{y}\in \left[0,1\right],\tilde{z}\in \left[0,1\right],c_1=c_2=1,p=1,\gamma =1$ and $LV\left(t,\tilde{x}\right)=\tilde{f}\left(t,\tilde{x}\right),LV\left(t,\tilde{y}\right)=\tilde{f}\left(t,\tilde{y}\right),LV\left(t,\tilde{z}\right)=\tilde{f}\left(t,\tilde{z}\right)$, then the conditions for making the zero solution of Equations (7)–(9) p-order exponentially stable are as follows:

$$\left\{\begin{array}{c}\tilde{x}\left(-{\tilde{K}}_1\tilde{y}-{\tilde{K}}_2\tilde{z}+{\tilde{S}}_1-{\tilde{S}}_4+{\tilde{R}}_1+{\tilde{R}}_4-{\tilde{C}}_1+{\tilde{C}}_4\right)\le -\tilde{x};\\ \tilde{y}\left[\tilde{\epsilon }\tilde{\theta }\Delta \tilde{\pi }\left(1+\tilde{\alpha }\tilde{q}\right)\tilde{z}-{\tilde{L}}_1\tilde{\alpha }\tilde{q}+\left({\tilde{K}}_1+\tilde{P}\right)\tilde{x}+{\tilde{S}}_2-{\tilde{S}}_5+{\tilde{R}}_2+{\tilde{R}}_5-{\tilde{C}}_2+{\tilde{C}}_5\right]\le -\tilde{y};\\ \tilde{z}\left[\tilde{\epsilon }\left(1-\tilde{\theta }\right)\Delta \tilde{\pi }\left(1+\tilde{\beta }\tilde{p}\right)\tilde{y}-{\tilde{L}}_2\tilde{\beta }\tilde{p}+\left({\tilde{K}}_2+\tilde{P}\right)\tilde{x}+{\tilde{S}}_3-{\tilde{S}}_6+{\tilde{R}}_3+{\tilde{R}}_6-{\tilde{C}}_3+{\tilde{C}}_6\right]\le -\tilde{z}.\end{array}\right.$$

(14)

Since $\tilde{x}\in \left[0,1\right],\tilde{y}\in \left[0,1\right],\tilde{z}\in \left[0,1\right]$, the conditions that satisfy the above equation are

$$\left\{\begin{array}{c}{\tilde{K}}_1+{\tilde{K}}_2-{\tilde{S}}_1+{\tilde{S}}_4-{\tilde{R}}_1-{\tilde{R}}_4+{\tilde{C}}_1-{\tilde{C}}_4\ge 1;\\ {\tilde{L}}_1\tilde{\alpha }\tilde{q}-{\tilde{S}}_2+{\tilde{S}}_5-{\tilde{R}}_2-{\tilde{R}}_5+{\tilde{C}}_2-{\tilde{C}}_5\ge 1;\\ {\tilde{L}}_2\tilde{\beta }\tilde{p}-{\tilde{S}}_3+{\tilde{S}}_6-{\tilde{R}}_3-{\tilde{R}}_6+{\tilde{C}}_3-{\tilde{C}}_6\ge 1\end{array}\right..$$

(15)

If the zero solution of Equations (7)–(9) are p-order exponentially unstable, the conditions are as follows:

$$\left\{\begin{array}{c}\tilde{x}\left(-K_1\tilde{y}-{\tilde{K}}_2\tilde{z}+{\tilde{S}}_1-{\tilde{S}}_4+{\tilde{R}}_1+{\tilde{R}}_4-{\tilde{C}}_1+{\tilde{C}}_4\right)\ge \tilde{x};\\ \tilde{y}\left[\tilde{\epsilon }\tilde{\theta }\Delta \tilde{\pi }\left(1+\tilde{\alpha }\tilde{q}\right)\tilde{z}-{\tilde{L}}_1\tilde{\alpha }\tilde{q}+\left({\tilde{K}}_1+\tilde{P}\right)\tilde{x}+{\tilde{S}}_2-{\tilde{S}}_5+{\tilde{R}}_2+{\tilde{R}}_5-{\tilde{C}}_2+{\tilde{C}}_5\right]\ge \tilde{y};\\ \tilde{z}\left[\tilde{\epsilon }\left(1-\tilde{\theta }\right)\Delta \tilde{\pi }\left(1+\tilde{\beta }\tilde{p}\right)\tilde{y}-{\tilde{L}}_2\tilde{\beta }\tilde{p}+\left({\tilde{K}}_2+\tilde{P}\right)\tilde{x}+{\tilde{S}}_3-{\tilde{S}}_6+{\tilde{R}}_3+{\tilde{R}}_6-{\tilde{C}}_3+{\tilde{C}}_6\right]\ge \tilde{z}.\end{array}\right.$$

(16)

Since $\tilde{x}\in \left[0,1\right],\tilde{y}\in \left[0,1\right],\tilde{z}\in \left[0,1\right]$, on the basis of Formula (16), if the zero solution moment of Formulas (7)–(9) are p-order exponentially stable, the conditions are as follows:

$$\left\{\begin{array}{c}{\tilde{S}}_1-{\tilde{S}}_4+{\tilde{R}}_1+{\tilde{R}}_4-{\tilde{C}}_1+{\tilde{C}}_4\le 1;\\ \tilde{\epsilon }\tilde{\theta }\Delta \tilde{\pi }\left(1+\tilde{\alpha }\tilde{q}\right)+{\tilde{K}}_1+\tilde{P}-{\tilde{L}}_1\tilde{\alpha }\tilde{q}+{\tilde{S}}_2-{\tilde{S}}_5+{\tilde{R}}_2+{\tilde{R}}_5-{\tilde{C}}_2+{\tilde{C}}_5\le 1;\\ \tilde{\epsilon }\left(1-\tilde{\theta }\right)\Delta \tilde{\pi }\left(1+\tilde{\beta }\tilde{p}\right)+{\tilde{K}}_2+\tilde{P}-{\tilde{L}}_2\tilde{\beta }\tilde{p}+{\tilde{S}}_3-{\tilde{S}}_6+{\tilde{R}}_3+{\tilde{R}}_6-{\tilde{C}}_3+{\tilde{C}}_6\le 1\end{array}\right..$$

(17)

To sum up, when the equations satisfy all conditions, the zero-order solutions of Equations (7)–(9) are p-order exponentially stable. This game’s only stable strategy is (supervision, cooperation, active research, and development).

5. Numerical Simulation

5.1. Stochastic Taylor Expansion of Replicated Dynamic Equations

Since the replicated dynamic equation is a nonlinear $It\widehat{o}$ stochastic differential equation, its analysis cannot be solved directly, it is solved numerically using the stochastic Taylor expansion method.

For $It\widehat{o}$ stochastic differential equations of type

$$d\tilde{x}\left(t\right)=d\left[\underset{\_}{x}\left(t\right),\overline{x}\left(t\right)\right]=\left[\underset{\_}{f}\left(t,\underset{\_}{x}\left(t\right)\right),\overline{f}\left(t,\overline{x}\left(t\right)\right)\right]dt+\left[\underset{\_}{g}\left(t,\underset{\_}{x}\left(t\right)\right),\overline{g}\left(t,\overline{x}\left(t\right)\right)\right]d\omega \left(t\right),$$

$$\forall 0\le t\le T.$$

where $t\in \left[t_0,T\right],\tilde{x}\left(t_0\right)={\tilde{x}}_0,{\tilde{x}}_0\in R$, and $\omega \left(t\right)$ represents one-dimensional standard Brownian motion. If $h={\displaystyle \frac{\left(T-t_0\right)}{N}},t_n=t_0+nh$, there is its stochastic Taylor expansion:

$$\begin{array}{c}\tilde{x}\left(t_{n+1}\right)=\tilde{x}\left(t_n\right)+h\tilde{f}\left(\tilde{x}\left(t_n\right)\right)+\Delta {\omega }_n\tilde{g}\left(\tilde{x}\left(t_n\right)\right)+{\displaystyle \frac{1}{2}}[\left(\Delta {\omega }_n{)}^2-h\right]\tilde{g}\left(\tilde{x}\left(t_n\right)\right){\tilde{g}}^{\prime }\left(\tilde{x}\left(t_n\right)\right)\\ +{\displaystyle \frac{1}{2}}{h}^2[\tilde{f}\left(\tilde{x}\left(t_n\right)\right)\tilde{f}\prime \tilde{x}\left(t_n\right))]+{\displaystyle \frac{1}{2}}{\tilde{g}}^2\left[\left(\tilde{x}\left(t_n\right)\right)\tilde{f}″\left(\tilde{x}\left(t_n\right)\right)\right]+R\end{array}$$

(18)

where $R$ is a residual term.

In practical application, thw Euler method and Milstein method are generally used to simulate the model [55]. The Euler method and Milstein high-order numerical method for solving equations are obtained by intercepting some terms of stochastic Taylor expansion. In fact, the essence of the Milstein higher-order method is the stochastic Taylor expansion truncated to the first-order expression after discarding the residual term. So, the improved Equation (18) is

$$\tilde{x}\left(t_{n+1}\right)=\tilde{x}\left(t_n\right)+h\tilde{f}\left(\tilde{x}\left(t_n\right)\right)+\Delta {\omega }_n\tilde{g}\left(\tilde{x}\left(t_n\right)\right)+{\displaystyle \frac{1}{2}}[\left(\Delta {\omega }_n{)}^2-h\right]\tilde{g}\left(\tilde{x}\left(t_n\right)\right)\tilde{g}\prime \left(\tilde{x}\left(t_n\right)\right)$$

(19)

Therefore, by solving Equations (7)–(9) with the Milstein higher-order method, we can obtain

$$\begin{array}{c}\tilde{x}\left(t_{n+1}\right)=\tilde{x}\left(t_n\right)+h\tilde{x}\left(t\right)\left(1-\tilde{x}\left(t\right)\right)\left(-{\tilde{K}}_1\tilde{y}-{\tilde{K}}_2\tilde{z}+{\tilde{S}}_1-{\tilde{S}}_4+{\tilde{R}}_1+{\tilde{R}}_4+{\tilde{C}}_4-{\tilde{C}}_1\right)+\\ \Delta {\omega }_n\sigma \tilde{x}\left(t_n\right)+{\displaystyle \frac{1}{2}}[\left(\Delta {\omega }_n{)}^2-h\right]{\sigma }^2\tilde{x}\left(t_n\right).\end{array}$$

(20)

$$\begin{array}{c}\tilde{y}\left(t_{n+1}\right)=\tilde{y}\left(t_{n+1}\right)+h\tilde{y}\left(t_n\right)\left(1-\tilde{y}\left(t_n\right)\right)[\tilde{\epsilon }\tilde{\theta }\Delta \tilde{\pi }\left(1+\tilde{\alpha }\tilde{q}\right)\tilde{z}-{\tilde{L}}_1\tilde{\alpha }\tilde{q}+\left({\tilde{K}}_1+\tilde{P}\right)\tilde{x}+{\tilde{S}}_2-\\ {\tilde{S}}_5+{\tilde{R}}_2+{\tilde{R}}_5-{\tilde{C}}_2+{\tilde{C}}_5]+\Delta {\omega }_n\sigma \tilde{y}\left(t_n\right)+{\displaystyle \frac{1}{2}}[\left(\Delta {\omega }_n{)}^2-h\right]{\sigma }^2\tilde{y}\left(t_n\right).\end{array}$$

(21)

$$\begin{array}{c}\tilde{z}\left(t_{n+1}\right)=\tilde{z}\left(t_n\right)+h\tilde{z}\left(t_n\right)\left(1-\tilde{z}\left(t_n\right)\right)[\tilde{\epsilon }\left(1-\tilde{\theta }\right)\Delta \tilde{\pi }\left(1+\tilde{\beta }\tilde{p}\right)\tilde{y}-{\tilde{L}}_2\tilde{\beta }\tilde{p}+\left({\tilde{K}}_2+\tilde{P}\right)\tilde{x}+\\ {\tilde{S}}_3-{\tilde{S}}_6+{\tilde{R}}_3+{\tilde{R}}_6-{\tilde{C}}_3+{\tilde{C}}_6]+\Delta {\omega }_n\sigma \tilde{z}\left(t_n\right)+{\displaystyle \frac{1}{2}}[\left(\Delta {\omega }_n{)}^2-h\right]{\sigma }^2\tilde{z}\left(t_n\right).\end{array}$$

(22)

To analyze the influence of random interference intensity $\sigma$, product greenness $\tilde{\epsilon }$, reputation benefits ${\tilde{R}}_i$, government incentives (${\tilde{K}}_1$, ${\tilde{K}}_2$), technical knowledge transfer ability $\tilde{\alpha }$ and R&D ability $\tilde{q}$ of industry, technical knowledge expansion ability $\tilde{\beta }$ and R&D promotion ability $\tilde{p}$ of the UR on the stochastic evolution process of the GIUR, this part uses Matlab2019a to simulate, analyze, and demonstrate relevant parameters’ influence on the game players’ strategy selection, and discuss the relationship between reputation benefits and other factors, and the specific analysis results will be presented in Section 5.2. According to the specific definition of different parameters in this paper, under the condition of ensuring the stability of the zero solution p-order index of the stochastic evolution game model, it is assumed that after the initial value of the relevant variable parameters affecting the stability of the alliance is (the unit is CNY one million) ${\tilde{C}}_1=\left[8,9,10\right]$, ${\tilde{C}}_2=\left[10,11,12\right]$, ${\tilde{C}}_3=\left[9,10,11\right]$, ${\tilde{C}}_4=\left[1,1,2\right]$, ${\tilde{C}}_5=\left[3,3,4\right]$, ${\tilde{C}}_6=\left[2,2,3\right]$, ${\tilde{S}}_1=\left[11,12,13\right]$, ${\tilde{S}}_2=\left[12,13,14\right]$, ${\tilde{S}}_3=\left[12,13,14\right]$, ${\tilde{S}}_4=\left[20,20,21\right]$, ${\tilde{S}}_5=\left[28,29,30\right]$, ${\tilde{S}}_6=\left[30,31,32\right]$, $\tilde{P}=\left[6,6,7\right]$, ${\tilde{L}}_1=\left[8,10,12\right]$, ${\tilde{L}}_2=\left[7,9,11\right]$, $\Delta \tilde{\pi }=\left[10.5,11.5,12.5\right]$, $\tilde{\theta }=\left[0.3,0.5,0.8\right]$, $\tilde{\epsilon }=\left[0.4,0.7,0.9\right]$, ${\tilde{K}}_1=\left[11,11,12\right]$, ${\tilde{K}}_2=\left[12,12,13\right]$, $\tilde{\alpha }=\left[0.0,0.3,0.6\right]$, $\tilde{\beta }=\left[0.0,0.3,0.6\right]$, $\tilde{q}=\left[0.0,0.3,0.5\right]$, and $\tilde{p}=\left[0.1,0.3,0.4\right]$. Generally, the government pays more attention to reputation income, followed by the UR, and industry pay more attention to economic income, that is, ${\tilde{R}}_1=\left[13,14,15\right]$, ${\tilde{R}}_2=\left[4,5,6\right]$, ${\tilde{R}}_3=\left[6,6,7\right]$, ${\tilde{R}}_4=\left[8,8,9\right]$, ${\tilde{R}}_5=\left[2,2,3\right]$, and ${\tilde{R}}_6=\left[2,3,3\right]$. The initial state of evolution is $\tilde{x}=\left[0.3,0.5,0.7\right]$, $\tilde{y}=\left[0.3,0.5,0.7\right]$, and $\tilde{z}=\left[0.3,0.5,0.7\right]$, and the step size $h=0.01$.

First of all, from the relevant definitions and Definition 6, we obtain $V_{11}=0$, $V_{12}=0$, $V_{21}=0$, $V_{22}\approx 0.98507>0$, $V_{31}=0$, $V_{32}=0$, $V_{41}=0$, $V_{42}=0$. It can be obtained that (supervision, midway betrayal, midway betrayal) is the game’s optimal pure strategy solution based on the possibility of about 0.98507.

5.2. Variable Sensitivity Analysis

From the pure fuzzy game income of this model, the optimal pure strategy solution is that the government chooses the supervision strategy, while both industry and the UR choose the midway betrayal strategy. However, we often hope that all the players will develop in the direction of win–win cooperation in order to obtain higher returns or better social effects, thus promoting economic development. Then, if the unstable international situation and environment and long-term social effects are taken into account, what changes will their strategies have? Next, we will analyze the sensitivity of the variables in this dynamic model. Definition 7 is used to denazify the variables of the fuzzy triangular numbers by performing $\lambda$-horizontal truncation to form an interval fuzzy number with only maxima and minima to simplify the evolutionary process. Here $\lambda =0.5$ is taken. At this time, $\tilde{x}=\left[x_1,x_2\right]$, $\tilde{y}=\left[y_1,y_2\right]$, and $\tilde{z}=\left[z_1,z_2\right]$ represent the possibility of strategy selection of government, industry, and the UR under fuzzy interval numbers.

5.2.1. Random Interference Intensity

This part mainly explores how the random interference intensity affects the behavior strategy of game players. Based on Milstein’s higher-order method, only the random interference intensity $\sigma$ ($\sigma =0,0.1,0.3,0.5,1$) in Equations (7)–(9) are changed. The strategy evolution process of government, industry, and the UR is simulated.

From Figure 2, the smaller the interference intensity, the smaller the evolution fluctuation, and the faster the evolution stability. When the interference intensity is extremely small $\sigma =0.1$, the government, industry, and the UR gradually evolve to a stable state. With the increase in interference intensity $\sigma$, after $\sigma \ge 0.3$, the players tend not to participate in the cooperative strategy because of the influence of interference factors. This proves that uncertain factors greatly affect the strategic choice of game players. Government departments should strictly control environmental factors, reduce the interference of the external environment, and create a green cooperation platform. The following evolution processes all default to $\sigma =0.1$ for ensuring the model’s validity and accuracy.

5.2.2. Influence of Reputation Gains on the Evolution Process

This part mainly explores whether reputation gains will affect the stability of the alliance. Only the reputation benefits ${\tilde{R}}_i,i=1,2,\dots ,6$ of different strategies of each game subject are changed, and the three parties’ strategy evolution processes are simulated.

From Figure 3, we analyze the evolution of reputation gains ${\tilde{R}}_1,{\tilde{R}}_2,{\tilde{R}}_3$ and ${\tilde{R}}_4,{\tilde{R}}_5,{\tilde{R}}_6$ under different scenarios for values when the government, industry, and the UR choose to cooperate and not cooperate, respectively. We find that no matter which strategy the three parties choose, their willingness to cooperate will become stronger with the increase in reputation and income. When the reputation gains are low, they tend to wait and see, gradually reduce the willingness to cooperate, and finally choose the non-cooperation strategy. From Figure 3a–f, we can find that the government is actually more concerned about the reputation gains, and compared with the other two parties, the reputation gains it has gained or been punished for are larger, so its evolution trend has always been positively correlated with the reputation gains. We also find that when the reputation gains or penalties increase, industry’s willingness to cooperate is stronger than that of the other two parties, and their strategic choices are more polarized. Too high or too low reputation gain or reputation loss will affect the strategy choice, which will lead to the alliance’s stability, thus leading all parties to choose opportunistic behavior strategies.

5.2.3. Influence of Product Greenness on Evolution Process

This part mainly explores the influence of product greenness on the stability of the GIUR alliance. With other parameters are unchanged, changing the value of product greenness $\tilde{\epsilon }$ simulates the evolution trend of government, industry, and the UR.

From Figure 4a–c, in the alliance’s process, the change in product greenness makes it difficult to shake the government’s strategic choice. It is observed in this part that when the lower green degree of products changes, although the strategic choice of industry and the UR will eventually evolve to a stable state, industry and the UR will not choose cooperative strategies at the beginning because of early investment, technology spillover, and harvest risk, and with the increase in iterations, that is, considering the long-term profitability of the alliance, they will gradually tend to choose cooperative strategies. In addition, increasing the product greenness will accelerate the players’ behavior strategy to the stable strategy, and the higher the product greenness, the faster the polarization of the fuzzy income evolution. In addition, this paper also finds a feature that by increasing green degree and iteration times, the industry’s cooperation intention is always higher than that of the UR, and the choice of cooperation strategies between them is more and more consistent. This shows that product greenness can change the stability of the alliance to a certain extent, and an appropriate product greenness is conducive to maintaining the stability of the green innovation alliance.

5.2.4. The Influence of the Capabilities of Industry and the UR on the Evolution Process

This part mainly explores the influence of the enterprise’s own technological knowledge transfer ability $\tilde{\alpha }$ and R&D ability $\tilde{q}$ on its strategic choice, as well as the influence of the UR’s own technological knowledge expansion ability $\tilde{\beta }$ and R&D promotion ability $\tilde{p}$ on its strategic choice. Only by changing the values of their own related capabilities, the strategy evolution processes of simulating the three parties are obtained.

From Figure 5a–f, the changes in the capabilities of industry and the UR will not affect the government’s evolution. In the green technology innovation alliance’s evolution process, the industry’s own ability and the UR’s own ability will restrict each other. When the two sides’ own abilities are low, in fact, the parties’ willingness to cooperate is relatively strong. Although, with the improvement in their own ability, the other side may gradually prefer to choose the halfway betrayal strategy due to problems such as technology spillover or integrity. However, for themselves, the improvement of their own ability will make them more willing to cooperate, because they have more room for development. Moreover, comparing Figure 5a and Figure 5d, this paper also finds that when one party’s own ability is low, its willingness to choose a cooperation strategy is slightly lower. This shows that the ability of both sides will induce the change in behavior strategies of all gamers. In fact, when both parties are more capable, there is a great possibility that the problem of domination of the benefits of technological knowledge will arise, which will make the capable parties choose to betray in the middle of the process and seek better low-cost development instead. Therefore, it is particularly important to match the ability of each party between industry and the UR.

6. Conclusions and Discussion

6.1. Discussion

GIUR collaborative innovation has always been one of the national important development directions by the attention of the scholars and its research results, but the existing research is mostly about the game process based on a definite strategy set, pay function, or revenue function game model, without considering the real environment of ambiguity and randomness; this is not in line with reality. Because these variables in a complex environment are not necessarily of a certain value, they may be an interval value or a triangle value, or a trapezoidal shape value, this is uncertain, so only studying a general definite game model may not allow us to greatly describe the system game process with uncertainty. In addition, reducing environmental uncertainty, providing government policy advantages, improving reputation and product greenness, and other factors not only meet the needs of human society for characteristic supply chain, manufacturing industry, medicine, and other aspects but also open up the mass consumption market of green products, thus realizing the bilateral sustainable development of environment and economy [52,53,54,55].

Therefore, in this paper, considering the green development background of reputation gains, adding fuzzy numbers and Gaussian white noise, a stochastic evolutionary game model of the GIUR with fuzzy numbers is constructed. Through a given numerical simulation, the effectiveness of the model is verified, as well as the influence of uncertain factors, reputation gains, product greenness, and the scientific research capabilities of industry and the UR on the stability of the model. It is found in this paper that the higher the reputation income and product greenness, the more positively it will affect the stability of the system. However, when industry and the UR have very strong capabilities, we should pay attention to whether there will be technical knowledge spillover problems, which will destroy the stability of the alliance model. This requires the government to improve the reputation mechanism and incorporate factors such as the capabilities of industry and the UR into more game models to solve social life problems in more fields.

In fact, many factors affect the development of the GIUR. Based on some documents, this paper only studies and discusses the issue of the green GIUR under the influence of reputation gains in an uncertain environment and has not considered all the factors. Nowadays, as one of the important factors in a country’s development, green innovation technology’s frontier development mainly focuses on new energy, architecture, biology, and industry. In particular, energy conversion is cleaner, more convenient, and more sustainable [56]. The comprehensive construction of the model from important factors such as reputation and income provides many new ideas for solving the multiple interests’ coordinated development. Due to the influence of some realistic factors, this paper only studies and discusses the game problem of the GIUR under fuzzy income on the basis of much of the literature and does not consider all the factors. The government’s policy on a good market environment for innovative technologies and the search for the best reputation benefits, the influence of the partnership between industry and decision-makers of the UR, the propaganda and promotion ability of the media, the income distribution of green innovative technologies, and the level and intensity of their intellectual property protection [57] will be the factors that our model will consider in detail in the next step. Recently, based on much of the literature, we found that consumers are also a very important subject in the innovation alliance, so we consider including consumers as an important game subject in the game and construction of a GIUR–consumer four-party game. We will also consider more realistic factors, fully analyze the stability of the GIUR innovation alliance in an uncertain environment, further improve the model, and provide strong and effective theoretical support and better strategic guidance for the green and stable development of the world innovation alliance.

6.2. Conclusions

Compared with the general evolution of random games, this paper introduces the fuzzy number into the game model, builds the fuzzy income matrix, using triangular fuzzy number and its related fuzzy set theory analysis under the government of the different strategy combination income, and uses the existence of Nash equilibrium theorem and related theory analysis of the game strategy Nash equilibrium, thus illustrating the rationality of the system subject decision. In addition, this paper uses numerical simulation to verify the influence of multiple factors on the dynamic system of GIUR and provides the corresponding theoretical basis for decision-makers.

Specifically, this paper considers the degree of stochastic interference, reputation, product greenness and enterprise and the learning research ability of five key variables, using the fuzzy theory and stochastic differential equation theory to verify the stability of the game model and the key variables in the evolution process, and discusses the synergistic relationship between them. These corrections are expected to better depict the high uncertainty of real society and verify the main influencing factors of the game subject strategy. We find that in the evolution process of the GIUR green technology innovation alliance, random interference factors are negatively correlated with the stability of the alliance. The random interference intensity will interfere with the speed at which alliance members evolve into stable strategies (strict supervision, cooperation, active research, and development). Reputation income and product greenness are positively related to the stability of the alliance. The stronger the reputation income, the stronger the willingness of the three parties to maintain the alliance’s stability. The improvement of product greenness will increase the value added of green innovation brought by the IUR cooperation and lower the probability of choosing opportunism, but we must combine the capabilities of both sides. However, reputation gains or product greenness beyond a reasonable range will cause the players to weigh the cost and the gains, no matter whether it is excessively high or low, which will lead the players to choose (unsupervised, betrayed halfway, betrayed halfway) strategies. In addition, from the perspective of their own capabilities, the improvement of their own capabilities plays a significant role in their own development, but for the other party, it may reduce the willingness to cooperate because of the overflow of technical knowledge. Therefore, if the alliance is to be sustained and stable, all factors of all parties should restrict each other and be in a relatively good state.

Based on this, there are some ideas in this study. Firstly, improve the external environment. The government should continue to introduce guiding policies for the UR, formulate reward and punishment mechanisms suitable for the market, create a good market environment, and improve the willingness of the UR to cooperate so as to attract more R&D capital investment. Secondly, reputation gains and product greenness are rationalized. Combined with the national conditions, the government draws up a reasonable reputation income, conducts a real-time investigation on the capabilities and honest operation of industry and the UR, encourages industry or the UR with strong capabilities to form strong alliances, strengthens the transformation of technological knowledge achievements, smooths the media channels, increases the publicity of new green technology products, mobilizes the enthusiasm of many members to participate in the alliance, enhances international influence, and accelerates the development of national green technology.