Knowledge Transfer within Enterprises from the Perspective of Innovation Quality Management: A Decision Analysis Based on the Stackelberg Game

Abstract

It is of great significance to study the effectiveness of knowledge transfer between the new and the veteran employees within enterprises for promoting sustainable innovation from the perspective of innovation quality management. However, few studies have examined the impact of innovation quality on the effectiveness of knowledge transfer between the new and veteran employees. In addition, knowledge of how reward and punishment incentives affect the effectiveness of knowledge transfer in innovation quality management remains insufficient. Since the amount of knowledge transfer is an important aspect of the effectiveness of knowledge transfer, this paper constructs a Stackelberg game model with an innovation-quality-oriented threshold of the knowledge transfer amount and investigates the amount of knowledge transfer between new and veteran employees in the collaborative innovation of research and development (R&D) projects within enterprises. A case study was used to reveal that the innovation-quality-oriented threshold for the knowledge transfer amount can effectively promote the amount of knowledge transfer between the new and the veteran employees in collaborative innovation. Moreover, reward is more effective than punishment in promoting the amount of knowledge transfer to exceed the innovation-quality-oriented threshold. This study enriches the theories of knowledge transfer games under quality management. By virtue of end-to-end project management strategies, modern multimedia technologies, and reward incentives this study can be used to conduct quality control during project execution, to promote knowledge retention in R&D projects, the innovation quality of projects, and the achievement of the Sustainable Development Goals (SDGs). The research methodology employed in this paper was limited to a case study, and the data utilized are not empirical data.

1. Introduction

In the current era of the knowledge economy, knowledge has become the core support of enterprise competitiveness and the basis of strategic organizational design [1]. Continuously updating and accumulating organizational knowledge has played an irreplaceable role in the survival and maintenance of market competitiveness for enterprises [2]. The knowledge of internal enterprises is mainly concentrated in the research and development (R&D) and design processes relevant to the organizational activities of employees such as R&D, design, production, and sales [3]. A key factor in organizational success lies in the transfer and sharing of knowledge among employees [4]. Therefore, research on knowledge transfer among internal employees holds significant value and importance for an organization.

By effectively retaining the knowledge of existing R&D employees and fully absorbing the new knowledge brought by new R&D employees, the organization can achieve continuous updating and accumulation of knowledge [5]. The knowledge transfer between the new and the veteran employees within an enterprise includes technical aspects, team culture, project experience, and more [6,7,8,9]. As rational economic agents, both the new and the veteran employees may have an interest in maximizing the benefits of knowledge transfer during the process. This indicates that they may engage in strategic gaming [10,11,12] to optimize their own knowledge transfer benefits during knowledge transfer interactions. In addition, quality management [13] and the incentives of reward and punishment [14] play crucial roles in promoting knowledge transfer. However, there is a lack of research on how reward and punishment incentives in the context of innovation quality management affect the effectiveness of knowledge transfer between the new and the veteran employees.

This paper makes the following two contributions to the literature. First, this paper introduces innovation quality targets for the amount of knowledge transfer, which enables the innovation quality targets to be evaluated justifiably through a quantified knowledge transfer amount. Second, this paper establishes a Stackelberg game model for decision-making on the amount of knowledge transfer between the new and the veteran employees, and delves into the different roles of reward and punishment incentives in promoting the effectiveness of knowledge transfer. However, these contributions were made based on the fact that the research methodology employed in this paper was limited to a case study, and the data utilized are not empirical data.

The remaining sections of this paper are organized as follows. Section 2 provides a literature review. Section 3 elaborates on the research hypotheses and model establishment. Section 4 analyzes the Stackelberg model, and the model results are analyzed and discussed. Section 5 presents a numerical analysis. Section 6 provides the conclusions.

2. Literature Review

Knowledge transfer plays a crucial role in promoting the effective utilization and innovative development of knowledge [15,16,17]. The following provides a literature review relevant to the research topic in this paper.

2.1. Tacit Knowledge Transfer

Based on the degree of difficulty in knowledge transfer, knowledge can be divided into explicit knowledge and tacit knowledge [18]. Explicit knowledge refers to knowledge that can be explicitly expressed and communicated, and its transfer is relatively easy [19]. Tacit knowledge is often closely related to an individual’s experience, intuition, know-how, skills, etc., and is difficult for outsiders to directly observe and understand. Tacit knowledge requires indirect transmission through methods such as practice, demonstration, and interaction, and its transfer is relatively more difficult [20,21], and is crucial for maintaining the sustainable development of an enterprise [22]. Successful knowledge transfer relies heavily on the effective transfer of tacit knowledge [23]. In tacit knowledge transfer, complex working relationships and behaviors are not only interconnected and interdependent, but also interact, inspire, complement, and constrain each other, jointly constructing a dynamic and vibrant knowledge ecological network [24]. It is worth mentioning that knowledge learning occurs not only in the knowledge reception process, but also in the knowledge output process [25]. During the reciprocal and cyclical transformation mechanism between tacit and explicit knowledge [26], tacit knowledge is continuously externalized, while explicit knowledge is gradually internalized. Currently, the research on tacit knowledge transfer focuses on dynamic processes and influencing factors.

2.2. Intergenerational Knowledge Transfer

Knowledge transfer between generations of employees is more conducive to promoting innovative behaviors among younger employees compared to knowledge transfer among peers [27]. Younger employees can not only establish connections and engage in in-depth exchanges with older employees, but also share different types of knowledge and experiences to collaboratively complete work tasks and achieve shared goals [28]. References [29,30] studied the promoting effect of organizational culture, communication methods, and technology on intergenerational knowledge transfer. According to reference [31], modern technological tools such as social media and online learning platforms provide employees of different generations with more opportunities for communication and learning. Successful intergenerational knowledge transfer requires comprehensive consideration of factors at multiple levels [32,33]. The knowledge transfer between engineering experts and apprentices is also a crucial aspect of intergenerational knowledge transfer. According to reference [34], trust, support from senior management, and employees’ intrinsic motivation can effectively increase the success of knowledge collaboration between engineering experts and apprentices, thereby promoting innovation and development within the entire team. At present, the research on intergenerational knowledge transfer is characterized by depth, systematization, interdisciplinarity, and practical application.

2.3. Effectiveness of Knowledge Transfer

The purpose of knowledge transfer is to achieve the sharing and utilization of knowledge, thereby promoting the occurrence of innovation. However, merely focusing on whether the act of knowledge transfer occurs is insufficient to adequately respond to the dynamic changes in the external market and the diverse needs of different entities [35]. The effectiveness of knowledge transfer is a key indicator for measuring the quality of knowledge transfer, reflecting the degree to which the objectives of knowledge transfer are achieved. References [35,36,37,38,39,40,41] interpreted and discussed the effectiveness of knowledge transfer from different perspectives. Reference [36] points out that the effectiveness of knowledge transfer is reflected in the breadth of the transferring parties and the amount of knowledge transferred. The amount of knowledge transferred is determined by the depth and breadth of the knowledge. Meanwhile, the breadth of knowledge embodies the diversified characteristics of knowledge [37]. In the study of the effectiveness of IT outsourcing knowledge transfer, reference [38] also adopts the amount of knowledge transfer as a measure of its effect. Effective feedback from recipients on whether the knowledge has been understood not only helps the sender adjust their teaching strategy but also ensures the accurate transmission and efficient absorption of knowledge. In addition, the interaction [42,43,44] as well as the implicit and complex nature of knowledge [45] has vital effects on the effectiveness of knowledge transfer. Incentives also have a significant impact on the effectiveness of knowledge transfer. Economic incentives [46], reciprocity and avoidance of punishment [47], organizational support [48], and a positive atmosphere [49,50] can significantly improve the effectiveness of knowledge transfer. However, there are still some challenges and shortcomings in deepening the research content and improving the research methods on the effectiveness of knowledge transfer.

2.4. Impact of Knowledge Transfer on Innovation Performance

The transfer and sharing of knowledge among employees within an organization can promote individual innovative behaviors [27], have a positive effect on innovation performance [51], and are deeply embedded in the increasingly complex collaborative innovation networks among organizations [52]. The mediating role of knowledge transfer among automation, employee creativity, and innovation performance [53] also indicates that knowledge transfer has a significant impact on both employee creativity and innovation performance. Importantly, knowledge transfer can ultimately promote the overall improvement of the enterprise’s innovation quality [54]. There is a significant positive relationship between total quality management and organizational performance, and knowledge transfer and innovative capability play a mediating role between the two [55]. A study conducted in reference [56] on the role of knowledge transfer and organizational innovation in quality and environmental management found that there is a significant positive correlation between quality management and knowledge transfer. Influencing factors, processes and mechanisms of knowledge transfer, and performance evaluation and measurement are extensively covered in the innovation performance of knowledge transfer.

2.5. Games of Knowledge Transfer

Game theory helps analyze the behaviors of participants in knowledge transfer. Reference [11] established a static game model for tacit knowledge transfer, and found that reward and punishment incentives can promote the occurrence of knowledge transfer. References [10,12,57,58,59] investigated the impact of knowledge transfer costs on knowledge transfer strategies and introduced complex networks into the game of knowledge transfer. Through different perspectives, they explored the interactions and dynamic evolution processes among individuals in complex systems. Furthermore, references [38,60] utilized Stackelberg game theory to analyze the effectiveness of knowledge transfer from the perspective of maximizing the benefits to the knowledge transfer subjects. Games of knowledge transfer reveal how the knowledge transfer participants formulate knowledge transfer strategies based on their own interest demands, and how they interact and compete with other participants. At present, games of knowledge transfer have been widely concerned and studied.

The literature review shows that the effective transfer of tacit knowledge plays an irreplaceable role in enhancing employee capabilities, driving innovation, and strengthening the competitiveness of enterprises. Hence, organizations need to actively promote and encourage internal knowledge sharing behavior [61]. When both the new and the veteran employees are willing to transfer knowledge to each other, their frequent interactions and close collaboration can enhance knowledge transfer behavior at the organizational level [9]. However, relatively few studies have focused on the new and veteran employees as the main players in the knowledge transfer game; in particular, research on decision-making regarding the effectiveness of knowledge transfer between the new and veteran employees seems to be lacking. In addition, few studies have delved into the impact of innovation quality on the effectiveness of knowledge transfer between the new and veteran employees. Furthermore, how reward and punishment incentives affect the effectiveness of the knowledge transfer within the framework of innovation quality management remains poorly defined in current knowledge transfer game research and requires further in-depth exploration.

3. Research Hypotheses and Model Establishment

3.1. Research Hypotheses

In this study, the veteran employees of a company are the older ones who have years of R&D experience within the enterprise, while the new employees are the younger ones who have just joined the company. During the process of knowledge transfer, due to differences in the accumulation of project experience, knowledge matching, and the degree of integration into the corporate culture, the new and the veteran employees naturally play different roles. The veteran employees have a high degree of alignment with the enterprise’s R&D needs, extensive experience in R&D projects, and a deep understanding of the internal operations and incentive mechanisms of the enterprise. They take responsibility for integrating and creating new knowledge, ensuring its alignment with the enterprise’s R&D needs. They shoulder the crucial task of guiding the new employees to internalize this knowledge. Conversely, the new employees lack a deep understanding of the corporate culture but are full of enthusiasm and potential for R&D work, and are eager to enhance their professional skills and integrate more with the team through learning from the veteran employees, and ultimately contribute their value to the R&D work. Based on the above, the following basic hypotheses are presented:

Hypothesis 1. 

In the knowledge transfer between the new and veteran employees, the veteran employees play a leading role, and the new employees play a subordinate role.

Hypothesis 2. 

In the cooperative innovation of R&D projects, enterprises also attach great importance to the leadership role of the veteran employees in the knowledge transfer, so they actively strengthen the leadership position of the veteran employees. This leads to an information advantage of the veteran employees over the new employees in the process of their knowledge transfer.

Hypothesis 3. 

In the knowledge transfer between the new and veteran employees, both the new and the veteran employees are rational economic parties, and they strive to maximize the benefits of knowledge transfer. Moreover, due to the information advantage of the veteran employees, they are able to accurately grasp the decision-making models, behaviors, and responses of the new employees.

Quality management plays a crucial role in promoting knowledge transfer. Especially in the control target system of R&D projects, clear quality goals can motivate the enthusiasm of both the new and the veteran employees, encourage them to share crucial knowledge and experience, thus strengthening team collaboration and enhancing the quality of R&D project outcomes [13]. This indicates that sufficient and appropriate knowledge transfer is a key factor in achieving these innovation quality targets. Based on the above, the following basic hypothesis is presented:

Hypothesis 4. 

To support the depth and breadth of innovation in R&D projects and to ensure the innovation quality of these projects, the amount of knowledge transfer between the new and the veteran employees needs to reach a certain level.

Thresholds for knowledge transfer amount may to some extent reflect the innovation quality objectives of R&D projects. These thresholds, which are closely linked to the specific innovative needs of the R&D projects, represent the required minimum amount of knowledge transfer between the new and the veteran employees during the collaborative innovation process. These thresholds not only serve as the minimum quality assurance for collaborative innovation in R&D projects, but also demonstrate a positive relationship between the amount of knowledge transfer of employees and the innovation quality of R&D projects. When the amount of knowledge transfer reaches or exceeds the threshold, it will be conducive to ensuring the innovation quality of R&D projects, promoting the absorption and dissemination of new knowledge, and driving the sustainable development of the enterprise. Based on the above, the following basic hypothesis is presented:

Hypothesis 5. 

In the knowledge transfer between the new and veteran employees, in order to discourage opportunistic behaviors such as insufficient knowledge transfer and non-cooperation in the collaborative innovation of R&D projects, the enterprise adopts the following punishment incentive function for the innovation-quality-oriented threshold for the knowledge transfer amount:

$$f(x_i,x_{\theta },\theta )=\theta \max (0,x_{\theta }-x_i)$$

(1)

where $x_{\theta }$ represents the innovation-quality-oriented threshold for the knowledge transfer amount, which is related to the specific innovation needs of the R&D project; $\theta$ represents the punishment factor corresponding to the loss of the innovation benefit of the R&D project averaged on the threshold of the knowledge transfer amount; and $x_i$ represents the amount of knowledge transfer of employees.

Drawing on the previous research on knowledge transfer [10,12,57], the following basic hypothesis is presented:

Hypothesis 6. 

The knowledge transfer revenue in collaborative innovation between the new and the veteran employees in R&D projects are influenced by the following factors:

(1)

Knowledge transfer amount $x_F/x_L$: Drawing on reference [38], this paper measures the amount of knowledge transfer based on the market value of knowledge. Specifically, knowledge with a value of CNY 1 is defined as a unit of knowledge amount. In order to reflect the actual situation of knowledge transfer, let us assume that the maximum amount of knowledge that can be transferred by the new and the veteran employees in the collaborative innovation of R&D projects is represented by $T_F$ and $T_L$, respectively. The actual amount of knowledge transferred by the new and the veteran employees in the collaborative innovation of R&D projects is represented by $x_F$ and $x_L$, respectively, satisfying $0\le x_F\le T_F$ and $0\le x_L\le T_L$.

(2)

Direct absorption coefficient of knowledge ${\sigma }_F/{\sigma }_L$: The ability of the new and the veteran employees to directly absorb knowledge in the collaborative innovation of R&D projects is represented by ${\sigma }_F$ and ${\sigma }_L$, respectively. ${\sigma }_F{x}_L$ and ${\sigma }_L{x}_F$ represent the amount of knowledge that the new employees and the veteran employees directly absorb from each other, respectively.

(3)

Knowledge synergy coefficient ${\eta }_F/{\eta }_L$: In the process of collaborative innovation in R&D projects, the ability of the new and the veteran employees to synergistically create new knowledge through knowledge transfer is represented by ${\eta }_F$ and ${\eta }_L$, respectively. ${\eta }_F{x}_L^m{x}_F^n$ and ${\eta }_L{x}_L^m{x}_F^n$ represent the amount of new knowledge synergistically created by the new and the veteran employees during this process, where $m$ and $n$ represent the elasticity coefficients of the knowledge transfer amount for the new and the veteran employees, satisfying the conditions $m>0$, $n>0$, and $m+n=1$.

(4)

Cross-organizational value co-creation benefit rate ${\lambda }_F/{\lambda }_L$: The benefit rates earned by the new and the veteran employees to innovate knowledge with external alliance members are represented by ${\lambda }_F$ and ${\lambda }_L$, respectively. ${\lambda }_F({\sigma }_F{x}_L+{\eta }_F{x}_L^m{x}_F^n)$ and ${\lambda }_L({\sigma }_L{x}_F+{\eta }_L{x}_L^m{x}_F^n)$, respectively, signify the cross-organizational value co-creation knowledge benefit that the new and the veteran employees obtain by using the incremental amount of knowledge they acquire within the enterprise.

(5)

Knowledge transfer cost coefficient ${\epsilon }_F/{\epsilon }_L$: The cost of transferring a unit of knowledge between the new and the veteran employees is represented by ${\epsilon }_F$ and ${\epsilon }_L$; ${\epsilon }_F{x}_F$ and ${\epsilon }_L{x}_L$ represent the costs paid by the new and the veteran employees when the amount of knowledge transferred is $x_F$ and $x_L$.

(6)

Knowledge transfer reward coefficient $w$: The reward earned by the new and the veteran employees for transferring a unit of knowledge is represented by $w$; ${wx}_F$ and ${wx}_L$ represent the reward benefits earned by the new and the veteran employees when the amount of knowledge transferred is $x_F$ and $x_L$, respectively.

(7)

Innovation-quality-oriented threshold for the knowledge transfer amount $x_{\theta }$: The innovation-quality-oriented threshold for the knowledge transfer amount must be less than the knowledge transfer capabilities of the new and the veteran employees, implying that $0<x_{\theta }<min(T_L,T_F)$.

(8)

Punishment factor $\theta$: A steeper punishment factor, $\theta >max\left(({\lambda }_F+1){\eta }_{F}n{(T_L/x_{\theta })}^m,({\lambda }_L+1){\eta }_{L}m{(T_F/x_{\theta })}^n\right)$, is assumed to discourage opportunistic behaviors such as insufficient knowledge transfer and non-cooperation during knowledge transfer.

3.2. Stackelberg Game Model

Merely focusing on whether the act of knowledge transfer occurs is insufficient to adequately respond to the dynamic changes in knowledge transfer between the new and the veteran employees. Therefore, it is necessary to explore the adequacy and effectiveness of knowledge transfer by the knowledge transfer amount.

In order to explore the adequacy and effectiveness of knowledge transfer between the new and the veteran employees in the context of innovation quality management, this paper takes the amount of knowledge transfer as the decision variable for knowledge transfer. Based on Hypotheses 1, 2, and 3, the dynamic process of the knowledge transfer game between the new and the veteran employees can be described as follows: First, the veteran employees first determine the optimal knowledge transfer amount to the new employees based on the principle of maximizing their own benefit, taking into account the enterprise’s incentive mechanisms as well as their own knowledge transfer parameters. Then, while observing the amount of knowledge transfer determined by the veteran employees, the new employees determine the optimal knowledge transfer amount to the veteran employees based on the principle of maximizing their own benefit.

In Stackelberg games, players are explicitly divided into two roles: the leader and the follower. The core of the game lies in the fact that the leader makes the first decision, while the follower, after observing the leader’s decision, formulates his own response. This model embodies the sequentiality of decision-making and informational asymmetry, where the leader can anticipate the optimal response of the follower when formulating their strategy, whereas the follower can only formulate their strategy based on the observed decision of the leader [62]. Although the Stackelberg game is a model of non-cooperative game theory, it also emphasizes the mutual influence and dependence between the leader and the follower. Through rational strategy formulation and dynamic adjustment, it can promote both cooperation and competition between the leader and the follower, thereby promoting the healthy development of the whole industry.

Given the distinctive characteristics of the clear leader–follower relationships, the significant information asymmetries, and the goal of maximizing knowledge transfer benefits through dynamic interactions, the Stackelberg game model becomes an ideal choice for analyzing the above knowledge transfer game, while other game models are less suitable in comparison. Hence, based on the above hypotheses, this paper develops a Stackelberg game-based decision model to determine the optimal knowledge transfer amount in R&D projects between the new and the veteran employees. The purpose of the usage of the Stackelberg game-based decision model is to explore the effectiveness of knowledge transfer with the knowledge transfer amount, thereby promoting effective knowledge transfer between the new and the veteran employees in R&D projects. Based on Hypothesis 6, the Stackelberg game-based decision model considers the knowledge transfer benefit, which consists of the following five components:

(1)

Benefit of the sum of direct knowledge absorption and the knowledge synergy: This component comes from the collaborations between the new and the veteran employees within the enterprise, and it represents the incremental amount of knowledge that the new and the veteran employees acquire within the enterprise. It is expressed as ${\sigma }_F{x}_L+{\eta }_F{x}_L^m{x}_F^n$ for the new employees, and ${\sigma }_L{x}_F+{\eta }_L{x}_L^m{x}_F^n$ for the veteran employees.

(2)

Benefit of the cross-organizational value co-creation: This component comes from the collaborations between employees within the enterprise and external collaborators [10]. In Hypothesis 6, it is the product of the cross-organizational value co-creation benefit rate and the incremental amount of knowledge acquired within the enterprise, expressed as ${\lambda }_F({\sigma }_F{x}_L+{\eta }_F{x}_L^m{x}_F^n)$ for the new employees, and ${\lambda }_L({\sigma }_L{x}_F+{\eta }_L{x}_L^m{x}_F^n)$ for the veteran employees.

(3)

Cost of the knowledge transfer: This component is expressed as ${\epsilon }_F{x}_F$ for the new employees, and ${\epsilon }_L{x}_L$ for the veteran employees.

(4)

Reward benefit of knowledge transfer: This component is expressed as $w{x}_F$ for the new employees, and ${wx}_L$ for the veteran employees.

(5)

Punishment of the innovation-quality-oriented threshold for the knowledge transfer amount: This component is expressed as $\theta \max \left(0,x_{\theta }-x_F\right)$ for the new employees, and $\theta \max \left(0,x_{\theta }-x_L\right)$ for the veteran employees.

Based on the components and constraints in Hypothesis 6, the Stackelberg game-based decision model is shown in Equations (2) and (3) with the constraints of C1, C2, C3, C4, and C5.

$$U_L=({\lambda }_L+1)({\sigma }_L{x}_F+{\eta }_L{x}_L^m{x}_F^n)-{\epsilon }_L{x}_L+w{x}_L-\theta \max \left(0,x_{\theta }-x_L\right)$$

(2)

$$U_F=({\lambda }_F+1)({\sigma }_F{x}_L+{\eta }_F{x}_L^m{x}_F^n)-{\epsilon }_F{x}_F+w{x}_F-\theta \max \left(0,x_{\theta }-x_F\right)$$

(3)

s.t.

$$\begin{array}{c}C1: 0\le x_F\le T_F\\ C2: 0\le x_L\le T_L\\ C3: m+n=1,m>0,n>0\\ C4: 0<x_{\theta }<\min (T_L,T_F)\\ C5: \theta >\max \left(({\lambda }_F+1){\eta }_{F}n{\left(T_L/x_{\theta }\right)}^m,({\lambda }_L+1){\eta }_{L}m{\left(T_F/x_{\theta }\right)}^n\right)\end{array}$$

Equation (2) is the knowledge transfer benefit function $U_L$ of the veteran employees who occupy the leader position. Equation (3) is the knowledge transfer benefit function $U_F$ of the new employees who act in the follower role. This model fills the gap in the research on the knowledge transfer game from the perspective of innovation quality management by introducing an innovation-quality-oriented threshold for the knowledge transfer amount ($x_{\theta }$).

4. Analysis and Discussion of Stackelberg Game Model

4.1. Equilibrium Strategy for Knowledge Transfer

Backward induction analysis is often used to solve the equilibrium strategies of a Stackelberg game. For derivations in backward induction analysis, Equations (2) and (3) in the Stackelberg game model are equivalently transformed into the following piecewise Functions (4) and (5):

$$U_L=\left\{\begin{array}{c}({\lambda }_L+1)({\sigma }_L{x}_F+{\eta }_L{x}_L^m{x}_F^n)-{\epsilon }_L{x}_L+w{x}_L,{ x}_L\ge x_{\theta }\\ \left({\lambda }_L+1\right)\left({\sigma }_L{x}_F+{\eta }_L{x}_L^m{x}_F^n\right)-{\epsilon }_L{x}_L+w{x}_L-\theta \left(x_{\theta }-x_L\right), x_L<x_{\theta }\end{array}\right.$$

(4)

$$U_F=\left\{\begin{array}{c}({\lambda }_F+1)({\sigma }_F{x}_L+{\eta }_F{x}_L^m{x}_F^n)-{\epsilon }_F{x}_F+w{x}_F, x_F\ge x_{\theta }\\ ({\lambda }_F+1)({\sigma }_F{x}_L+{\eta }_F{x}_L^m{x}_F^n)-{\epsilon }_F{x}_F+w{x}_F-\theta (x_{\theta }-x_F), x_F<x_{\theta }\end{array}\right.$$

(5)

Because $\underset{x_L\to x_{\theta }^{+}}{\lim }{U}_L=\underset{x_L\to x_{\theta }^{-}}{\lim }{U}_L=U_{L,x\theta }$ and $\underset{x_F\to x_{\theta }^{+}}{\lim }{U}_F=\underset{x_F\to x_{\theta }^{-}}{\lim }{U}_F=U_{F,x\theta }$, both $U_L$ and $U_F$ are continuous at $x_{\theta }$.

For backward induction analysis, we take the first-order derivative of $U_F$ with respect to $x_F$. Due to the piecewise function form of $U_F$, the first-order derivative of $U_F$ with respect to $x_F$ is expressed as Equations (6a) and (6b):

$${\displaystyle \frac{\partial U_F}{\partial x_F}}=\left({\lambda }_F+1\right){\eta }_{F}n{\left(x_L/x_F\right)}^m-{\epsilon }_F+w, x_F\ge x_{\theta }$$

(6a)

$${\displaystyle \frac{\partial U_F}{\partial x_F}}=\left({\lambda }_F+1\right){\eta }_{F}n{\left(x_L/x_F\right)}^m-{\epsilon }_F+w+\theta , x_F<x_{\theta }$$

(6b)

The second-order derivative of $U_F$ with respect to $x_F$ is expressed as Equation (7):

$${\displaystyle \frac{{\partial }^2{U}_F}{\partial {\left(x_F\right)}^2}}=-({\lambda }_F+1){\eta }_{F}mn{\left(x_L\right)}^m/{\left(x_F\right)}^{m+1}$$

(7)

As seen from Equation (7), when $x_F>0$ and $x_L>0$, there exists an optimal knowledge transfer amount for the new employees in the process of knowledge transfer, which can maximize the new employees’ knowledge transfer benefit.

For convenience, let $A={\displaystyle \frac{({\lambda }_F+1){\eta }_{F}n}{{\epsilon }_F-w}}$, $B={\displaystyle \frac{({\lambda }_F+1){\eta }_{F}n}{{\epsilon }_F-w-\theta }}$, $C={\displaystyle \frac{({\lambda }_L+1){\eta }_{L}m}{{\epsilon }_L-w}}$, and $D={\displaystyle \frac{({\lambda }_L+1){\eta }_{L}m}{{\epsilon }_L-w-\theta }}$.

Theorem 1. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$$\begin{gathered} w+({\lambda }_F+1){\eta }_{F}n{\left(T_L/T_F\right)}^m<{\epsilon }_F<w+({\lambda }_F+1){\eta }_{F}n{\left(T_L/x_{\theta }\right)}^m, \\ {\epsilon }_L<w+({\lambda }_L+1)({\sigma }_L{A}^{1/m}+{\eta }_L{A}^{n/m}) \end{gathered}$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=T_L{A}^{1/m}$, $x_L^{*}=T_L$, with $x_{\theta }<x_F^{*}<T_F$.

Theorem 2. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$$\begin{gathered} w+({\lambda }_F+1){\eta }_{F}n{\left(x_{\theta }/T_F\right)}^m<{\epsilon }_F<w+({\lambda }_F+1){\eta }_{F}n, \\ w+({\lambda }_L+1)({\sigma }_L{A}^{1/m}+{\eta }_L{A}^{n/m})<{\epsilon }_L<w+\theta +({\lambda }_L+1)({\sigma }_L{A}^{1/m}+{\eta }_L{A}^{n/m}) \end{gathered}$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=x_{\theta }{A}^{1/m}$, $x_L^{*}=x_{\theta }$, with $x_{\theta }<x_F^{*}<T_F$.

In backward induction, the optimal response of the new employee is first determined, then the optimal knowledge transfer amount of the veteran employee is determined, and finally the optimal knowledge transfer amount of the new employee is determined. In addition, to help understand the proofs, the following sketches are provided in Figure 1.

Proof of Theorems 1 and 2. 

Backward induction is used in the proof. The detailed proof is shown as follows.

When ${\epsilon }_F$ satisfies the condition of Theorem 1, it can be derived that Equation (6b) is greater than zero, indicating that $U_F$ is monotonically increasing with respect to $x_F$ when $x_F<x_{\theta }$. In addition, $U_F$ is continuous at $x_F=x_{\theta }$. To help understand, a sketch of $U_F$, Equation (4), is shown in Figure 1a. Therefore, the optimal response of the new employee can be obtained by setting Equation (6a) equal to zero, i.e., the optimal response of the new employee is $x_F=x_L{A}^{1/m}$, satisfying $x_F\ge x_{\theta }$.

Using the optimal response of the new employee, the veteran employee can obtain the following knowledge transfer benefit Equation (8) and its first-order derivatives with respect to $x_L$, i.e., Equations (9a) and (9b):

$$U_L=\left\{\begin{array}{c}({\lambda }_L+1)({\sigma }_L{x}_L{A}^{1/m}+{\eta }_L{x}_L{A}^{n/m})-{\epsilon }_L{x}_L+w{x}_L, { x}_L\ge x_{\theta }\\ ({\lambda }_L+1)({\sigma }_L{x}_L{A}^{1/m}+{\eta }_L{x}_L{A}^{n/m})-{\epsilon }_L{x}_L+w{x}_L-\theta (x_{\theta }-x_L), x_L<x_{\theta }\end{array}\right.$$

(8)

$${\displaystyle \frac{\partial U_L}{\partial x_L}}=({\lambda }_L+1)({\sigma }_L{A}^{1/m}+{\eta }_L{A}^{n/m})-{\epsilon }_L+w, x_L\ge x_{\theta }$$

(9a)

$${\displaystyle \frac{\partial U_L}{\partial x_L}}=\left({\lambda }_L+1\right)\left({\sigma }_L{A}^{1/m}+{\eta }_L{A}^{n/m}\right)-{\epsilon }_L+w+\theta , x_L<x_{\theta }$$

(9b)

When ${\epsilon }_L$ satisfies the condition of Theorem 1, Equations (9a) and (9b) are greater than zero, which indicates that the knowledge transfer benefit for the veteran employee, Equation (8), is monotonically increasing with respect to $x_L$, and that the optimal knowledge transfer amount of the veteran employee is $x_L^{*}=T_L$. To help understand, a sketch of $U_L$, Equation (8), is shown in Figure 1b. By substituting $x_L^{*}=T_L$ into the optimal response of the new employee $A^{1/m}$, the optimal knowledge transfer amount of the new employee is $x_F^{*}=T_L{A}^{1/m}$, satisfying $x_{\theta }<x_F^{*}<T_F$ when ${\epsilon }_F$ satisfies the condition of Theorem 1. Hence, Theorem 1 holds.

Similar to the proof of Theorem 1, when ${\epsilon }_F$ satisfies the condition of Theorem 2, the optimal response of the new employee is $x_F=x_L{A}^{1/m}$, satisfying $x_F\ge x_{\theta }$, the knowledge transfer benefit for the veteran employee is Equation (8), and its first-order derivatives with respect to $x_L$ are Equations (9a) and (9b). When ${\epsilon }_L$ satisfies the condition in Theorem 2, Equation (9a) is greater than zero while Equation (9b) is smaller than zero. According to the first derivative test for extremes, $x_{\theta }$ is the maximum point for the knowledge transfer benefit $U_L$, Equation (8), i.e., $x_L^{*}=x_{\theta }$. To help understand, a sketch of $U_L$, Equation (8), is shown in Figure 1c. By substituting $x_L^{*}=x_{\theta }$ into the optimal response of the new employee, the optimal knowledge transfer amount of the new employee is $x_F^{*}=x_{\theta }{A}^{1/m}$, satisfying $x_{\theta }<x_F^{*}<T_F$. Hence, Theorem 2 holds. The proof ends here.  □

Theorem 3. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$${\epsilon }_F>w+\theta +({\lambda }_F+1){\eta }_{F}n{\left(T_L/x_{\theta }\right)}^m,{\epsilon }_L<w+({\lambda }_L+1)({\sigma }_L{B}^{1/m}+{\eta }_L{B}^{n/m})$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=T_L{B}^{1/m}$, $x_L^{*}=T_L$, with $0<x_F^{*}<x_{\theta }$.

Theorem 4. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$$\begin{gathered} {\epsilon }_F>w+\theta +({\lambda }_F+1){\eta }_{F}n, \\ w+({\lambda }_L+1)({\sigma }_L{B}^{1/m}+{\eta }_L{B}^{n/m})<{\epsilon }_L<w+\theta +({\lambda }_L+1)({\sigma }_L{B}^{1/m}+{\eta }_L{B}^{n/m}) \end{gathered}$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=x_{\theta }{B}^{1/m}$, $x_L^{*}=x_{\theta }$, with $0<x_F^{*}<x_{\theta }$.

Proof of Theorems 3 and 4. 

Backward induction is used in the proof. The detailed proof is shown as follows.

When ${\epsilon }_F$ satisfies the condition of Theorem 3, the optimal response of the new employee is $x_F=x_L{B}^{1/m}$, satisfying $x_F<x_{\theta }$ by setting Equation (6b) equal to zero, while $x_F=x_L{A}^{1/m}$ does not satisfy $x_F\ge x_{\theta }$ by setting Equation (6a) equal to zero. This is because when $x_F=x_L{A}^{1/m}$ satisfies $x_F\ge x_{\theta }$, ${\epsilon }_F$ will contradict the condition in Theorem 3.

Using the optimal response of the new employee, the veteran employee can obtain the following knowledge transfer benefit Equation (10) and its first-order derivatives with respect to $x_L$, i.e., (11a) and (11b):

$$U_L=\left\{\begin{array}{c}({\lambda }_L+1)({\sigma }_L{x}_L{B}^{1/m}+{\eta }_L{x}_L{B}^{n/m})-{\epsilon }_L{x}_L+w{x}_L, { x}_L\ge x_{\theta }\\ ({\lambda }_L+1)({\sigma }_L{x}_L{B}^{1/m}+{\eta }_L{x}_L{B}^{n/m})-{\epsilon }_L{x}_L+w{x}_L-\theta (x_{\theta }-x_L), x_L<x_{\theta }\end{array}\right.$$

(10)

$${\displaystyle \frac{\partial U_L}{\partial x_L}}=({\lambda }_L+1)({\sigma }_L{B}^{1/m}+{\eta }_L{B}^{n/m})-{\epsilon }_L+w, x_L\ge x_{\theta }$$

(11a)

$${\displaystyle \frac{\partial U_L}{\partial x_L}}=\left({\lambda }_L+1\right)\left({\sigma }_L{B}^{1/m}+{\eta }_L{B}^{n/m}\right)-{\epsilon }_L+w+\theta , x_L<x_{\theta }$$

(11b)

When ${\epsilon }_F$ satisfies the condition of Theorem 3, Equations (11a) and (11b) are greater than zero, which indicates that the knowledge transfer benefit for the veteran employee, Equation (10), is monotonically increasing with respect to $x_L$, and that the optimal knowledge transfer amount of the veteran employee is $x_L^{*}=T_L$. This situation is similar to the sketch in Figure 1b. By substituting $x_L^{*}=T_L$ into the optimal response of the new employee, the optimal knowledge transfer amount of the new employee is $x_F^{*}=T_L{B}^{1/m}$, satisfying $x_L\ge x_{\theta }$ when ${\epsilon }_L$ satisfies the condition of Theorem 3. Hence, Theorem 3 holds.

Similar with the proof of Theorem 3, when ${\epsilon }_F$ satisfies the condition of Theorem 4, the optimal response of the new employee is $x_F=x_L{B}^{1/m}$, satisfying $x_F<x_{\theta }$, the knowledge transfer benefit for the veteran employee is Equation (10), and its first-order derivatives with respect to $x_L$ are Equations (11a) and (11b). When ${\epsilon }_L$ satisfies the condition of Theorem 4, Equation (11b) is greater than zero while Equation (11a) is smaller than zero. According to the first derivative test for extremes, $x_{\theta }$ is the maximum point for the knowledge transfer benefit $U_L$, Equation (10), i.e., $x_L^{*}=x_{\theta }$. This situation is similar to the sketch in Figure 1c. By substituting $x_L^{*}=x_{\theta }$ into the optimal response of the new employee, the optimal knowledge transfer amount of the new employee is $x_F^{*}=x_{\theta }{B}^{1/m}$, satisfying 0$<x_F^{*}<x_{\theta }$. Hence, Theorem 4 holds. The proof ends here.  □

Theorem 5. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$${\epsilon }_F<w, {\epsilon }_L<w+({\lambda }_L+1){\eta }_{L}m{\left(T_F/T_L\right)}^n$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=T_F$, $x_L^{*}=T_L$.

Theorem 6. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$${\epsilon }_F<w, w+({\lambda }_L+1){\eta }_{L}m{\left(T_F/T_L\right)}^n<{\epsilon }_L<w+({\lambda }_L+1){\eta }_{L}m{\left(T_F/x_{\theta }\right)}^n$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=T_F$, $x_L^{*}=T_F{C}^{1/n}$, with $x_{\theta }<x_L^{*}<T_L$.

Theorem 7. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$${\epsilon }_F<w, {\epsilon }_L>w+\theta +({\lambda }_L+1){\eta }_{L}m{\left(T_F/x_{\theta }\right)}^n$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=T_F$, $x_L^{*}=T_F{D}^{1/n}$, with $0<x_L^{*}<x_{\theta }$.

Theorem 8. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$${\epsilon }_F<w, w+({\lambda }_L+1){\eta }_{L}m{\left(T_F/x_{\theta }\right)}^n<{\epsilon }_L<w+\theta +({\lambda }_L+1){\eta }_{L}m{\left(T_F/x_{\theta }\right)}^n$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=T_F$, $x_L^{*}=x_{\theta }$.

Proof of Theorems 5, 6, 7, and 8. 

Backward induction is used in the proof. The detailed proof is shown as follows.

When ${\epsilon }_F$ satisfies the condition of Theorem 5, Equations (6a) and (6b) are greater than zero, which indicates that $U_F$ is monotonically increasing with respect to $x_F$, and that the optimal response of the new employee is $x_F^{*}=T_F$. Using the optimal response of the new employee, the veteran employee can obtain the following knowledge transfer benefit Equation (12) and its first-order derivatives with respect to $x_L$, i.e., (13a) and (13b):

$$U_L=\left\{\begin{array}{c}({\lambda }_L+1)({\sigma }_L{T}_F+{\eta }_L{x}_L^m{T}_F^n)-{\epsilon }_L{x}_L+w{x}_L, { x}_L\ge x_{\theta }\\ ({\lambda }_L+1)({\sigma }_L{T}_F+{\eta }_L{x}_L^m{T}_F^n)-{\epsilon }_L{x}_L+w{x}_L-\theta (x_{\theta }-x_L), x_L<x_{\theta }\end{array}\right.$$

(12)

$${\displaystyle \frac{\partial U_L}{\partial x_L}}=({\lambda }_L+1){\eta }_{L}m{\left(T_F/x_L\right)}^n-{\epsilon }_L+w, x_L\ge x_{\theta }$$

(13a)

$${\displaystyle \frac{\partial U_L}{\partial x_L}}=\left({\lambda }_L+1\right){\eta }_{L}m{\left(T_F/x_L\right)}^n-{\epsilon }_L+w+\theta , x_L<x_{\theta }$$

(13b)

When ${\epsilon }_L$ satisfies the condition of Theorem 5, Equations (13a) and (13b) are greater than zero, which indicates that $U_L$, Equation (12), is a monotonically increasing function with respect to $x_L$ and that the optimal knowledge transfer amount of the veteran employee is $x_L^{*}=T_L$. Hence, Theorem 5 holds.

Similar to the proof of Theorem 5, when ${\epsilon }_F$ satisfies the condition of Theorem 6, the optimal response of the new employee is $x_F^{*}=T_F$, the knowledge transfer benefit for the veteran employee is Equation (12), and its first-order derivatives with respect to $x_L$ are Equations (13a) and (13b). When ${\epsilon }_L$ satisfies the condition of Theorem 6, it can be derived that Equation (13b) is greater than zero, indicating that the knowledge transfer benefit for the veteran employee, Equation (12), is monotonically increasing with respect to $x_L$ when $x_L<x_{\theta }$. In addition, Equation (12) is continuous at $x_L=x_{\theta }.$ To help understand, a sketch of $U_L$, Equation (12), is shown in Figure 1d. Therefore, the optimal knowledge transfer amount of the veteran employee can be obtained by setting Equation (13a) equal to zero, i.e., the optimal knowledge transfer amount of the veteran employee is $x_L^{*}=T_F{C}^{1/n}$ satisfying $x_{\theta }<x_L^{*}<T_L$. Hence, Theorem 6 holds.

Similar to the proof of Theorem 5, when ${\epsilon }_F$ satisfies the condition of Theorem 7, the optimal response of the new employee is $x_F^{*}=T_F$ and the knowledge transfer benefit for the veteran employee is $x_L^{*}=T_F{D}^{1/n}$, satisfying $x_L^{*}<x_{\theta }$ by setting Equation (13b) equal to zero, while $x_L^{*}=T_F{C}^{1/n}$ does not satisfy $x_L^{*}\ge x_{\theta }$ by setting Equation (13a) equal to zero. This is because when $x_L=T_F{C}^{1/n}$ satisfies $x_L\ge x_{\theta }$, ${\epsilon }_L$ will contradict $D^{1/n}$. Hence, Theorem 7 holds.

Similar to the proof of Theorem 5, when ${\epsilon }_F$ satisfies the condition of Theorem 8, the optimal response of the new employee is $x_F^{*}=T_F$, the knowledge transfer benefit for the veteran employee is Equation (12), and its first-order derivatives with respect to $x_L$ are Equations (13a) and (13b). When ${\epsilon }_L$ satisfies the condition of Theorem 8, Equation (13b) is greater than zero while Equation (13a) is smaller than zero. According to the first derivative test for extremes, $x_{\theta }$ is the maximum point for the knowledge transfer benefit $U_L$, Equation (12), i.e., $x_L^{*}=x_{\theta }$. To help understand, a sketch of $U_L$, Equation (12), is shown in Figure 1e. Hence, Theorem 8 holds. The proof ends here.  □

Theorem 9. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$$w+({\lambda }_F+1){\eta }_{F}n{\left(T_L/x_{\theta }\right)}^m<{\epsilon }_F<w+\theta , {\epsilon }_L<w+({\lambda }_L+1){\eta }_{L}m{\left(x_{\theta }/T_L\right)}^n$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=x_{\theta }$, $x_L^{*}=T_L$.

Theorem 10. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions, 

$$\begin{gathered} w+({\lambda }_F+1){\eta }_{F}n{\left(T_L/x_{\theta }\right)}^m<{\epsilon }_F<w+\theta , \\ w+({\lambda }_L+1){\eta }_{L}m{\left(x_{\theta }/T_L\right)}^n<{\epsilon }_L<w+({\lambda }_L+1){\eta }_{L}m \end{gathered}$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=x_{\theta }$, $x_L^{*}=x_{\theta }{C}^{1/n}$, with $x_{\theta }<x_L^{*}<T_L$.

Theorem 11. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions, 

$$w+({\lambda }_F+1){\eta }_{F}n{\left(T_L/x_{\theta }\right)}^m<{\epsilon }_F<w+\theta , {\epsilon }_L>w+\theta +({\lambda }_L+1){\eta }_{L}m$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=x_{\theta }$, $x_L^{*}=x_{\theta }{D}^{1/n}$, with $0<x_L^{*}<x_{\theta }$.

Theorem 12. 

When the knowledge transfer cost coefficients of the new and the veteran employees, i.e., ${\epsilon }_F$ and ${\epsilon }_L$, satisfy the following conditions,

$$w+({\lambda }_F+1){\eta }_{F}n{\left(T_L/x_{\theta }\right)}^m<{\epsilon }_F<w+\theta ,{ \epsilon }_L>w+\theta +({\lambda }_L+1){\eta }_{L}m$$

the equilibrium strategy for knowledge transfer between the new and the veteran employees under the Stackelberg game is $x_F^{*}=x_{\theta }$, $x_L^{*}=x_{\theta }$.

Proof of Theorems 9, 10, 11, and 12. 

Backward induction is employed in the proof. The detailed proof is shown as follows.

When ${\epsilon }_F$ satisfies the condition of Theorem 9, Equation (6b) is greater than 0 while Equation (6a) is smaller than 0, which indicates that the optimal response of the new employee is $x_F^{*}=x_{\theta }$ according to the first derivative test for extremes. To help understand, a sketch of $U_F$, Equation (4), is shown in Figure 1f.

Using the optimal response of the new employee, the veteran employee can obtain the following knowledge transfer benefit Equation (14) and its first-order derivatives with respect to $x_L$, i.e., (15a) and (15b):

$$U_L=\left\{\begin{array}{c}({\lambda }_L+1)({\sigma }_L{x}_{\theta }+{\eta }_L{x}_L^m{x}_{\theta }^n)-{\epsilon }_L{x}_L+w{x}_L, { x}_L\ge x_{\theta }\\ ({\lambda }_L+1)({\sigma }_L{x}_{\theta }+{\eta }_L{x}_L^m{x}_{\theta }^n)-{\epsilon }_L{x}_L+w{x}_L-\theta (x_{\theta }-x_L), x_L<x_{\theta }\end{array}\right.$$

(14)

$${\displaystyle \frac{\partial U_L}{\partial x_L}}=({\lambda }_L+1){\eta }_{L}m{\left(x_{\theta }/x_L\right)}^n-{\epsilon }_L+w, x_L\ge x_{\theta }$$

(15a)

$${\displaystyle \frac{\partial U_L}{\partial x_L}}=\left({\lambda }_L+1\right){\eta }_{L}m{\left(x_{\theta }/x_L\right)}^n-{\epsilon }_L+w+\theta , x_L<x_{\theta }$$

(15b)

When ${\epsilon }_L$ satisfies the condition of Theorem 9, Equations (15a) and (15b) are greater than zero, which indicates that $U_L$, Equation (14), is monotonically increasing with respect to $x_L$, and that the optimal knowledge transfer amount of the veteran employees is $x_L^{*}=T_L$. Hence, Theorem 9 holds.

Similar to the proof of Theorem 9, when ${\epsilon }_F$ satisfies the condition of Theorem 10, the optimal response of the new employee is $x_F^{*}=x_{\theta }$, the knowledge transfer benefit for the veteran employee is Equation (14), and its first-order derivatives with respect to $x_L$ are Equations (15a) and (15b). When ${\epsilon }_L$ satisfies the condition of Theorem 10, it can be derived that Equation (15b) is greater than zero, indicating that the knowledge transfer benefit for the veteran employee, Equation (14), is monotonically increasing with respect to $x_L$ when $x_L<x_{\theta }$. In addition, Equation (14) is continuous at $x_L=x_{\theta }.$ This situation is similar to the sketch in Figure 1d. Therefore, the optimal knowledge transfer amount of the veteran employee can be obtained by setting Equation (15a) equal to zero, i.e., the optimal knowledge transfer amount of the veteran employee is $x_L^{*}=x_{\theta }{C}^{1/n}$, satisfying $x_{\theta }<x_L^{*}<T_L$. Hence, Theorem 10 holds.

Similar to the proof of Theorem 9, when ${\epsilon }_F$ satisfies the condition of Theorem 11, the optimal response of the new employee is $x_F^{*}=x_{\theta }$, the knowledge transfer benefit for the veteran employee is Equation (14), and its first-order derivatives with respect to $x_L$ are Equations (15a) and (15b). When ${\epsilon }_L$ satisfies the condition of Theorem 11, the optimal knowledge transfer amount of the veteran employee is $x_L^{*}=x_{\theta }{D}^{1/n}$ satisfying $x_L^{*}<x_{\theta }$ by setting Equation (15b) equal to zero, while $x_L^{*}=x_{\theta }{C}^{1/n}$ does not satisfy $x_L^{*}\ge x_{\theta }$ by setting Equation (15a) equal to zero. This is because when $x_L=x_{\theta }{C}^{1/n}$ satisfies $x_L\ge x_{\theta }$, ${\epsilon }_L$ will contradict the condition of Theorem 11. Hence, Theorem 11 holds.

Similar to the proof of Theorem 9, when ${\epsilon }_F$ satisfies the condition of Theorem 12, the optimal response of the new employee is $x_F^{*}=x_{\theta }$, the knowledge transfer benefit for the veteran employee is Equation (14), and its first-order derivatives with respect to $x_L$ are Equations (15a) and (15b). When ${\epsilon }_L$ satisfies the condition of Theorem 12, Equation (15b) is greater than zero while Equation (15a) is smaller than zero. According to the first derivative test for extremes, $x_{\theta }$ is the maximum point for the knowledge transfer benefit $U_L$, Equation (14), i.e., $x_L^{*}=x_{\theta }$. This situation is similar to the sketch in Figure 1c. Hence, Theorem 12 holds. The proof ends here.  □

4.2. Analysis and Discussion for Equilibrium Results

Based on Theorems 1–12, the following propositions can be drawn:

Proposition 1. 

When the knowledge transfer cost coefficient of either the new employees or the veteran employees is greater than the sum of the reward coefficient and twice the punishment factor, the amount of knowledge transfer of that party will fail to reach the innovation-quality-oriented threshold for the amount of knowledge transfer.

It can be proved as follows. Based on the constraints C4 and C5, we can derive that $\theta >\max \left(({\lambda }_F+1){\eta }_{F}n, ({\lambda }_L+1){\eta }_{L}m\right)$. Combining this relationship with Theorems 3, 4, 7, and 11, we can conclude that when the knowledge transfer cost coefficient of either the new employees or the veteran employees exceeds the sum of the reward coefficient and is twice the punishment factor, the amount of knowledge transfer of that party will be less than the innovation-quality-oriented threshold for the amount of knowledge transfer, i.e., $x_{\theta }$.

The innovation-quality-oriented threshold for the amount of knowledge transfer is an important criterion to measure the success of R&D projects. In the collaborative innovation of R&D projects, this threshold is the minimum amount of knowledge transfer that both the new and the veteran employees need to achieve. Once this threshold is met, it signifies sufficient interaction and communication between the new and the veteran employees in the R&D project, enabling the project to achieve the quality standards set by the enterprise in terms of both the depth and breadth of innovation.

However, when either the new employees or the veteran employees fail to reach the threshold of knowledge transfer oriented by innovation quality, the depth and diversity of knowledge incorporated into the R&D project will be limited. As a result, it becomes difficult for the R&D project to achieve the desired level of innovation in both depth and breadth. Proposition 1 clearly points out the critical influence of knowledge transfer cost on the amount of knowledge transfer.

Proposition 2. 

When the amount of knowledge transfer of either the new or the veteran employees is greater than 0 but less than the threshold of knowledge transfer oriented by innovation quality, the amount of knowledge transfer is positively correlated with the knowledge transfer reward coefficient and the punishment factor.

It can be proved as follows. According to Theorem 3 and Theorem 4, when the veteran employees’ knowledge transfer amount is greater than 0 but less than the threshold oriented by innovation quality, their knowledge transfer amount is denoted as $x_F^{*}=T_L{B}^{1/m}$ in Theorem 3 and $x_F^{*}=x_{\theta }{B}^{1/m}$ in Theorem 4 with $0<x_F^{*}<x_{\theta }$, where $B={\rho }_F({\lambda }_F+1){\eta }_{F}n/({\epsilon }_F-w-\theta )$. According to Theorem 7 and Theorem 11, when the new employees’ knowledge transfer amount is greater than 0 but less than the threshold oriented by innovation quality, their knowledge transfer amount is denoted as $x_L^{*}=T_F{D}^{1/n}$ in Theorem 7 and $x_L^{*}=x_{\theta }{D}^{1/n}$ in Theorem 11 with $0<x_L^{*}<x_{\theta }$, where $D={\rho }_L({\lambda }_L+1){\eta }_{L}m/({\epsilon }_L-w-\theta )$. From this, it can be inferred that when the knowledge transfer amount of either the new or the veteran employees is greater than 0 but less than the threshold oriented by innovation quality, the knowledge transfer amount is related to the knowledge transfer reward coefficient and the punishment factor.

Taking the first-order derivatives of $B^{1/m}$ with respect to the knowledge transfer reward coefficient ($w$) and the punishment factor ($\theta$), we have the following relationships:

$$sign\left({\displaystyle \frac{\partial B^{1/m}}{\partial w}}\right)=sign\left({\displaystyle \frac{B^{n/m}({\lambda }_F+1){\eta }_{F}n}{m{({\epsilon }_F-w-\theta )}^2}}\right)>0, sign\left({\displaystyle \frac{\partial B^{1/m}}{\partial \theta }}\right)=sign\left({\displaystyle \frac{B^{n/m}({\lambda }_F+1){\eta }_{F}n}{m{({\epsilon }_F-w-\theta )}^2}}\right)>0.$$

Taking the first-order derivatives of $D^{1/n}$ with respect to the knowledge transfer reward coefficient ($w$) and the punishment factor ($\theta$), we have the following relationships:

$$sign\left({\displaystyle \frac{\partial D^{1/n}}{\partial w}}\right)=sign\left({\displaystyle \frac{D^{m/n}({\lambda }_L+1){\eta }_{L}m}{n{({\epsilon }_L-w-\theta )}^2}}\right)>0, sign\left({\displaystyle \frac{\partial D^{1/n}}{\partial \theta }}\right)=sign\left({\displaystyle \frac{D^{m/n}({\lambda }_L+1){\eta }_{L}m}{n{({\epsilon }_L-w-\theta )}^2}}\right)>0.$$

Proposition 3. 

When the knowledge transfer amount of either the new or the veteran employees is greater than the innovation-quality-oriented threshold for the knowledge transfer amount but less than their maximum knowledge transfer amount, their knowledge transfer amount is positively correlated with the knowledge transfer reward coefficient. However, it is not related to the punishment factor.

It can be proved as follows. According to Theorem 1 and Theorem 2, when the knowledge transfer amount of the new employees is greater than the threshold for the knowledge transfer amount but less than their maximum knowledge transfer amount, their knowledge transfer amount is $x_F^{*}=T_L{A}^{1/m}$ in Theorem 1 and $x_F^{*}=x_{\theta }{A}^{1/m}$ in Theorem 2 with $x_{\theta }<x_F^{*}<T_F$, where $A=({\lambda }_F+1){\eta }_{F}n/({\epsilon }_F-w)$. According to Theorem 6 and Theorem 10, when the knowledge transfer amount of the veteran employees is greater than the threshold for the knowledge transfer amount but less than their maximum knowledge transfer amount, their knowledge transfer amounts is $x_L^{*}=T_F{C}^{1/n}$ in Theorem 6 and $x_L^{*}=x_{\theta }{C}^{1/n}$ in Theorem 10 with $x_{\theta }<x_L^{*}<T_L$, where $C=({\lambda }_L+1){\eta }_{L}m/({\epsilon }_L-w)$. From this, it can be inferred that when the knowledge transfer amount of either the new employees or the veteran employees is larger than the threshold oriented by innovation quality but less than the maximum amount of knowledge transfer, the knowledge transfer amount is related to the knowledge transfer reward coefficient, but it is not related to the punishment factor.

Taking the first-order derivatives of $A^{1/m}$ and the first-order derivatives of $C^{1/n}$ with respect to the knowledge transfer reward coefficient ($w$), we have the following relationships:

$$sign\left({\displaystyle \frac{\partial A^{1/m}}{\partial w}}\right)=sign\left({\displaystyle \frac{A^{n/m}({\lambda }_F+1){\eta }_{F}n}{m{({\epsilon }_F-w)}^2}}\right)>0, sign\left({\displaystyle \frac{\partial C^{1/n}}{\partial w}}\right)=sign\left({\displaystyle \frac{C^{m/n}({\lambda }_L+1){\eta }_{L}m}{n{({\epsilon }_L-w)}^2}}\right)>0$$

It is worth noting that once the amount of knowledge transferred by the employees exceeds the threshold oriented by innovation quality, the role of the punishment factor in further incentivizing the employees to increase the amount of knowledge transfer becomes less significant. This indicates that while punishment measures play a positive role in ensuring the basic quality of R&D projects, their scope of influence is limited. In contrast, reward incentives always play a significant role in promoting knowledge transfer, effectively stimulating employees’ willingness to transfer knowledge and increasing the amount of knowledge transfer.

By combining Proposition 2 and Proposition 3, it can be seen that within the framework of innovation quality management, punishment measures and reward incentives play distinct yet equally crucial roles in promoting knowledge transfer. Punishment measures play a crucial role in ensuring the fundamental level of R&D quality. Nevertheless, when it comes to further motivating employees to increase their knowledge transfer amount and achieve more thorough and effective knowledge transfer, the effectiveness of punishment measures is relatively limited. In contrast, reward incentives demonstrate a more powerful motivating effect. In addition to motivating employees to exceed the knowledge transfer threshold, reward incentives can encourage them to achieve maximum knowledge transfer, thereby conferring significant competitive advantages and sustainable development momentum upon the enterprise. This further emphasizes the importance of employing reward and punishment strategies in innovation quality management in an appropriate manner, as well as the necessity of leveraging reward mechanisms to stimulate employees’ intrinsic motivation and maximize the effectiveness of knowledge transfer.

5. Numerical Analysis

5.1. Case Study

Case studies are often used in theoretical studies to better explain the situation through numerical analysis. However, the case selected should fit the research scenario of the paper, and serve a specific set of parameters for the research scenario. Considering that the framework of the knowledge transfer benefits in this paper is a new development based on knowledge transfer within e-commerce enterprises [10], we choose an e-commerce enterprise as a case in this paper. In addition, e-commerce enterprises often communicate and interact with a wider range of external stakeholders including customers, suppliers, and business partners [10] than other categories of companies in R&D activities, which can better fit the framework of the knowledge transfer benefits in this paper. As is known, JD.com is one of the typical representatives of e-commerce enterprises. Therefore, in this section, JD.com is taken as the scenario to conduct a detailed numerical analysis for decision-making on the knowledge transfer amount.

JD.com attaches great importance to the quality of R&D cooperation and innovation in the field of artificial intelligence and Internet technology. In order to enhance R&D competitiveness, JD.com established R&D cooperation alliances with Tencent in early 2015, and shared R&D resources in terms of knowledge, experience, and technology through project cooperation, aiming to achieve mutual benefit and win–win results [63]. JD.com’s project management office (PMO) implements an end-to-end project management strategy including quality management, collaboration management, and schedule management [64]. JD.com’s PMO directs project managers to conduct quality control during project execution, thereby improving the quantity and quality of project completion [65].

In simulations, we assume that there is a R&D project team in JD.com with a veteran employee as the project R&D manager and the new employees as the ordinary R&D project members, and that the R&D project team is in charge of three small R&D projects I, II, and III for improving resource utilization, focusing on the business as the core, being guided by measurable value, aiming to improve efficiency, and adhering to the concept of win–win cooperation [64,65]. In addition, we believe that JD.com’s PMO can direct the R&D project manager to set different innovative quality management goals for different projects based on the business, strategic positioning, customer requirements, and working hours of the projects, thus requiring different knowledge transfer thresholds for different projects. Therefore, we assume that the knowledge transfer amount thresholds for R&D projects I, II, and III are set to $x_{\theta }=10, 60, 150$, respectively. The simulations consider the Stackelberg game for the knowledge transfer amount between the veteran R&D manager and the new employee.

Parameters can be set based on the R&D project team of JD.com as follows:

First, since JD.com has accumulated sufficient experience in R&D cooperation with other companies, the parameters, including the cross-organizational value co-creation benefit coefficient, the knowledge synergy coefficient, and the direct knowledge absorption coefficient of the employees of JD.com’s R&D project team, should be greater than zero. In general, the cross-organizational value co-creation benefit coefficient, the knowledge synergy coefficient, and the direct knowledge absorption coefficient, as well as the maximum amount of knowledge transfer, are relatively fixed in a period of time. With reference to previous literature [10,12,57] and different characteristics of the veteran R&D manager and the new employee, the direct knowledge absorption coefficient is set to ${\sigma }_L=0.2,{\sigma }_F=0.6$, the cross-organizational value co-creation benefit coefficient is set to ${\lambda }_L\in \{0.2, 0.5\}, {\lambda }_F\in \{0.1, 0.5\}$, and the knowledge synergy coefficient is set to ${\eta }_L\in \{0.3, 0.5\}, {\eta }_F\in \{0.1, 0.5\}$ for the veteran R&D manager and the new employee, respectively. In addition, the maximal knowledge transfer amount of the veteran R&D manager and the new employee is set at 200 to meet the requirements of the R&D projects I, II, and III.

Second, because of the different knowledge transfer tasks and innovation quality thresholds faced by the veteran R&D manager and the new employee, the knowledge transfer cost coefficient remains a range of variation, as do both the reward coefficient and the punishment factor. Following previous literature [10,12,57], the reward coefficient $w$ is varied from 0.2 to 0.5, while the punishment factor $\theta$ is varied from 0.5 to 2.0. In addition, to fully investigate the knowledge transfer behavior, the knowledge transfer cost coefficient for the veteran R&D manager and the new employee (${\epsilon }_L/{\epsilon }_F$) is varied from 0.1 to 5.0.

Since the simulation results for R&D projects I, II, and III are similar in Section 5.2, Section 5.3, Section 5.4, Section 5.5 and Section 5.6, only R&D project II is used as an example in Section 5.2, Section 5.3, Section 5.4, Section 5.5 and Section 5.6 to save space. However, to compare the simulation results for R&D projects I, II, and III, Section 5.7 includes examples of R&D projects I, II, and III.

5.2. Impact of a Relatively Large Knowledge Transfer Cost Coefficient on the Knowledge Transfer Amount

For when the knowledge transfer cost coefficient is relatively large, the simulation results of the knowledge transfer amount between the new employee and the veteran R&D manager with different parameters are illustrated in Figure 2, where the initial parameters are set as $w=0.2, \theta =0.5$, and ${\lambda }_F=0.1,{ \eta }_F=0.1$ for Figure 2a, and ${\lambda }_L=0.2, {\eta }_L=0.3$ for Figure 2b. Results in Figure 2 show that, when the knowledge transfer cost coefficients of the new employee and the veteran R&D manager satisfy the conditions in Theorems 3, 4, 7, and 11, their amount of knowledge transfer will be below the threshold for the knowledge transfer amount ($x_{\theta }=60$). In addition, as the e parameters $w, \theta , {\lambda }_F, {\lambda }_L, {\eta }_F, {\eta }_L$ increase, the amount of the knowledge transfer increases to some extent. However, as the knowledge transfer cost coefficient increases, the amount of knowledge transfer between the new employee and the veteran R&D manager decreases gradually. It is evident from Figure 2 that, when the knowledge transfer cost coefficient of either the new employee or the veteran R&D manager exceeds the sum of the reward coefficient and is twice the punishment factor, the amount of knowledge transfer of that party will fail to reach the threshold required for innovation quality. Moreover, the higher the knowledge transfer cost coefficient is, the more the amount of knowledge transfer approaches 0, which validates Proposition 1.

5.3. Effects of the Reward and the Punishment on the Knowledge Transfer Amount When the Knowledge Transfer Amount Fails to Reach the Threshold

For when the amount of knowledge transfer fails to reach the threshold, the simulation results of the relationships between the amount of knowledge transfer and the reward coefficient as well as the punishment factor are shown in Figure 3 and Figure 4. The set of initial parameters ${\lambda }_F, {\lambda }_L, {\eta }_F, {\eta }_L$ for Figure 3 and Figure 4 is same as with Figure 2, and the knowledge transfer cost coefficient is fixed as 1.0 for Figure 3. Results in the figures illustrate that when the amount of knowledge transfer of either the new employee or the veteran R&D manager fails to reach the threshold, the amount of knowledge transfer of that party will increase as the reward coefficient ($w$) and the punishment factor ($\theta$) increase. At the same time, increasing the cross-organizational value co-creation benefit coefficient (${\lambda }_F/{\lambda }_L$) and the knowledge synergy coefficient (${\eta }_F/{\eta }_L$) can also increase the amount of knowledge transfer to some extent. These results validate Proposition 2. This indicates that the knowledge transfer amount can be brought close to the threshold by increasing the reward coefficient and the punishment factor. In addition, the quality of collaborative innovation in R&D projects can be ensured by increasing the reward coefficient and the punishment factor.

5.4. Effect of Punishment Factor on Maintaining the Threshold of Knowledge Transfer Amount

The impact of the knowledge transfer cost coefficient on the amount of knowledge transfer from the veteran R&D manager under different punishment factors of 0.5, 1.0, 1.5, and 2.0 is illustrated in Figure 5. Results in Figure 5 demonstrate that, as the knowledge transfer cost coefficient increases, the amount of knowledge transfer will gradually decrease. However, when the knowledge transfer amount drops to the threshold level required for innovation quality, it will remain at that threshold level within a certain range of knowledge transfer cost coefficients. Moreover, the larger the punishment factor is, the wider the range of knowledge transfer cost coefficients that maintain the threshold level of knowledge transfer is. This indicates that through punishment measures, the amount of knowledge transfer between the new employee and the veteran R&D manager can be maintained at the required threshold level in order to achieve innovation quality goals. Moreover, increasing the punishment factor can effectively deter opportunistic behaviors such as insufficient knowledge transfer. At the same time, it also implies that punishment measures can only ensure that the amount of knowledge transfer between the new employee and the veteran R&D manager meets the minimum level required for achieving innovation quality goals. From this, when the amount of knowledge transfer of the new employee and the veteran R&D manager is at the threshold level, increasing the reward coefficient can further improve the quality of project cooperation innovation. On the contrary, punishment measures have little effect on further improving the quality of project cooperation innovation even if the punishment factor is increased.

5.5. Effects of the Reward on the Knowledge Transfer Amount When the Knowledge Transfer Amount Exceeds Threshold

For when the knowledge transfer amount exceeds the threshold, the simulation results of the relationship between the reward coefficient and the knowledge transfer amount are shown in Figure 6. Also, the set of initial parameters ${\lambda }_F, {\lambda }_L, {\eta }_F, {\eta }_L$ for Figure 6 is same as with Figure 2.

The results in Figure 6 show that the knowledge transfer amount increases with the increase in the knowledge transfer reward coefficient. In addition, increasing the cross-organizational value co-creation benefit coefficient (${\lambda }_F/{\lambda }_L$) and the knowledge synergy coefficient (${\eta }_F/{\eta }_L$) can further increase the amount of knowledge transfer. This indicates that when the knowledge transfer amount of either the new employee or the veteran R&D manager exceeds the threshold, it is positively correlated with the knowledge transfer reward coefficient. This verifies Proposition 3.

5.6. Influence of the Knowledge Transfer Amount of the New Employee on the Veteran Employee

Since the maximal knowledge transfer amount is not fixed but changing in the following simulation, different variations of the knowledge transfer cost coefficient are considered. The simulation results of the influence of the knowledge transfer amount of the new employee on the veteran R&D manager are shown in Figure 7, where $U$ represents the uniform distribution. The results in the figures show that there exists an influence of the knowledge transfer amount of the new employee on the veteran R&D manager, even when different variations of the knowledge transfer cost coefficient are considered. By comparing Figure 7a,b, it can be seen that the knowledge transfer cost coefficients in Figure 7a are smaller than those in Figure 7b, which also indicates that a relatively small knowledge transfer cost coefficient easily leads to a relatively large amount of knowledge transfer. However, in practice, such an influence of the knowledge transfer amount of the new employee on the veteran R&D manager may exist over a long period of time, because each R&D project requires a cycle of R&D, and the maximal knowledge transfer amount cannot be changed frequently in a relatively short period of time.

5.7. Impact of Different Thresholds on the Amount of Knowledge Transfer

The impact of the thresholds for the knowledge transfer amount of R&D projects I, II, and III on the amount of knowledge transfer is analyzed in the following simulation, and the simulation results are shown in Figure 8. Results in Figure 8 show that different thresholds of the knowledge transfer amount produce different curves of the knowledge transfer amount, and higher thresholds result in higher levels of curves of the knowledge transfer amount. This phenomenon is directly caused by the enterprise’s innovation quality management practices in R&D projects, as the innovation quality objectives often require a certain depth and breadth of innovation. To support the depth and breadth of innovation, the amount of knowledge transfer from the new employee and the veteran R&D manager needs to reach a certain level. This means that in the collaborative R&D innovation of a project, the new employee and the veteran R&D manager need to engage in frequent interactions and knowledge transfer to acquire sufficient knowledge related to the R&D project, in order to support their innovative thinking and practice in the collaborative R&D process. Only when the amount of knowledge transferred is sufficient can the new employee and the veteran R&D manager have more options and references, thus increasing the chances of successful innovation in collaborative R&D and ensuring the quality of innovation.

In addition, results in Figure 8 also indicate that R&D projects with higher thresholds of the knowledge transfer amount tend to require a lower knowledge transfer cost coefficient to ensure the quality of collaborative innovation in the R&D projects. One possible reason for the above phenomenon can be explained as follows. An excessively high threshold for the knowledge transfer amount may involve more tacit knowledge that is difficult to explicate and encode in the process of knowledge transfer, which easily leads to an increase in the cost of knowledge transfer. Only when the total cost of knowledge transfer through incentive compensation is reduced and the willingness of employees to transfer tacit knowledge is improved can employees carry out cooperative innovation in R&D projects requiring higher thresholds of the knowledge transfer amount.

5.8. Application of the Stackelberg Game Model

Note that the Stackelberg game model should be applied to the knowledge transfer scenario where the veteran employees are in a dominant position while the new employees are in a subordinate position. In practical applications, the PMO should first utilize end-to-end project management strategies and take a comprehensive view of the business, strategic positioning, customer requirements, and workload to reasonably determine the innovation quality thresholds for the knowledge transfer amount. Another key issue in practical applications is how to determine the knowledge transfer amount that occurs between the new and the veteran employees during their interactions. To address this issue, our recommendation is to develop an integrated solution based on modern multimedia technology. By using a high-definition video recording system, we aim to comprehensively capture and document interaction scenarios between the new and the veteran employees. This will not only provide a solid technical basis for the PMO to scientifically evaluate the amount of knowledge transferred and direct project managers to conduct quality control during project execution, but also significantly promote deeper learning outcomes and knowledge retention among employees in R&D projects through the review of video content. Finally, more attention should be paid to reward incentives to compensate for knowledge transfer costs and to promote the amount of knowledge transfer and the innovation quality of projects.

It is also important to note that the Stackelberg game model is limited by the cost of knowledge transfer as a linear function of the amount of knowledge transferred. Otherwise, the equilibrium strategies obtained in this paper are no longer valid. Given the above limitation, the Stackelberg game model should be applied to a scenario where advanced information transfer technologies with higher levels of automation, such as digital platforms, online communication tools, multimedia, etc., are used to improve the efficiency of knowledge transfer and reduce the cost of knowledge transfer during the knowledge transfer between the new and veteran employees.

In addition, the Stackelberg game model of knowledge transfer between the new and the veteran employees is vital for the Sustainable Development Goals (SDGs) [66], and it can be applied to promote the achievement of the SDGs. First, under the guidance of the model, enterprises can effectively promote the effect of knowledge transfer among employees, reduce trial and error and resource consumption, and ensure the development quality of new products and technologies, thereby promoting economic growth and reducing resource waste and environmental pollution. Second, the model results show that enterprises should be encouraged to adopt rewards to promote knowledge transfer, which can not only improve the innovation quality, but also help to cultivate a more inclusive and sustainable work environment. Third, with the help of the model, e-commerce enterprises can cooperate better with suppliers to jointly promote green supply chain management. By sharing environmental standards and promoting green procurement policies, they can reduce the carbon footprint of the entire supply chain. All of this not only promotes durable, inclusive, and sustainable economic growth and decent work, contributing to SDG 8 [66], but also encourages responsible consumption and production practices, in line with SDG 12 [66], ultimately reducing resource waste and pollution.

6. Conclusions

Based on the innovation-quality-oriented thresholds for the amount of knowledge transfer, this paper established a Stackelberg game model for decision-making on the amount of knowledge transfer between the new and the veteran employees in R&D projects. Using the decision model, this paper explored the amount of knowledge transfer in cooperative innovation between the new and the veteran employees from the perspective of innovation quality management. It should be noted that the research methodology employed in this paper was limited to a case study, and that the data utilized are not empirical data. In the case study, JD.com is selected as the case and serves a specific set of parameters. Based on the case study and the specific set of parameters, the following conclusions are drawn:

First, the knowledge transfer cost coefficient has an important influence on the amount of knowledge transferred between the new and veteran employees. If the knowledge transfer cost coefficient is smaller than the reward coefficient, the amount of knowledge transfer can be maximized. If the knowledge transfer cost coefficient is larger than the sum of the reward coefficient and twice the punishment factor, the amount of knowledge transfer may not reach the innovation-quality-oriented knowledge transfer amount threshold.

Second, the punishment factor can motivate employees to meet the requirement of the basic innovation quality standard for the knowledge transfer amount in the R&D project, which is mainly based on its effects deterring opportunistic behaviors such as insufficient knowledge transfer. However, its role in further increasing the knowledge transfer to exceed the innovation-quality-oriented knowledge transfer amount threshold is relatively limited even if the punishment factor is increased. Conversely, the reward coefficient can not only motivate employees to exceed the innovation-quality-oriented threshold but also encourage them to achieve the maximum of knowledge transfer to improve innovation quality of the project R&D.

Finally, different innovation-quality-oriented thresholds of the knowledge transfer amount produce different curves of the knowledge transfer amount, and higher thresholds result in higher levels of curves of the knowledge transfer amount.

In addition, it is noted that the conclusions were made based on the analysis of the equilibrium strategies of the Stackelberg game model, which is limited by the cost of knowledge transfer as a linear function of the amount of knowledge transferred. However, this work contributes to the achievement of SDG 8 (decent work and economic growth) and SDG 12 (responsible consumption and production).