Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 5551625 | https://doi.org/10.1155/2021/5551625

Fang Song, Xiwu Hu, Junfeng Li, Michael Allen Watson, "Altruistic Punishment, Status Conflict, and Knowledge Sharing in the Workplace: An Evolutionary Game Model", Discrete Dynamics in Nature and Society, vol. 2021, Article ID 5551625, 24 pages, 2021. https://doi.org/10.1155/2021/5551625

Altruistic Punishment, Status Conflict, and Knowledge Sharing in the Workplace: An Evolutionary Game Model

Academic Editor: Polinpapilinho Katina
Received12 Jan 2021
Revised01 Mar 2021
Accepted08 Mar 2021
Published12 Apr 2021

Abstract

Second- and third-party altruistic punishments can both improve knowledge sharing, but it is debatable which does so better. We thus concurrently consider the influence of status conflict advantage and disadvantage on knowledge sharing. We also analyze knowledge sharing by introducing an evolutionary game model and conducting simulations, and reveal the following: (1) the appropriate choice for an organization that is increasing its knowledge contribution is third-party punishment; (2) the status conflict advantage and disadvantage play important roles in knowledge sharing. When the status conflict advantage is greater than the disadvantage, the two players eventually share knowledge, and vice versa. Interestingly, when the status conflict advantage is equal to the disadvantage, the player with the stronger sharing inclination has a powerful impact on the player who would rather hoard knowledge, thus ultimately leading him/her to share. (3) Guanxi, knowledge capacity, and synergy revenue have positive effects on knowledge sharing, while the knowledge sharing cost has a negative influence on them. Moreover, the stable equilibrium result is affected by the initial system state. In sum, our results not only promote the understanding of altruistic punishment and status conflict regarding knowledge sharing but also provide practical implications for both organizations and leaders.

1. Introduction

Knowledge sharing refers to employees sharing their work-related experiences, expertise, know-how, and contextual information with their peers through informal and formal interactions within or across teams or work units [1]. Knowledge sharing has thus received considerable attention, due to its vital role in the development of organizations’ competitive advantages [2]. However, in real life, individuals cannot share their knowledge with others at all times because of knowledge ownership and the need for the protection and pursuit of their own interests. That is, promoting knowledge sharing is a difficult task, as hoarding and guardedly considering knowledge are natural human tendencies, meaning we cannot expect individuals to simply share their ideas and insights because it is the right thing to do [3]. In short, individuals will not share knowledge if they think they can obtain more benefits from hoarding instead of sharing [2]. Consequently, effective incentives can improve knowledge sharing behaviors in the workplace, with extant studies showing that knowledge sharing cannot be forced or mandated but can instead be encouraged and facilitated [4, 5]. The threat of punishment can effectively induce individuals to share their knowledge, since negative incentives or punishment can be used to reduce free-rider pay-off [6].

Altruistic punishment has thus received considerable attention in academia, as it challenges the basic “homo-economicus” model of traditional economics. Psychological, economic, and anthropological research shows that subjects have strong tendencies to behave prosocially and are motivated by the personal desires to punish antisocial behaviors at a cost to themselves [7]. That is, not only those whose interests are affected will punish violators (second-party punishment) but also third parties whose interests have not been harmed will also do so (third-party punishment). Such punishment (both second- and third-parties) can effectively maintain social norms and is also called altruistic punishment. As effective incentive measures, second- and third-party punishments can essentially enhance employees' knowledge sharing behaviors. However, there is no comparative study on the promotion effects of second- and third-party punishments on knowledge sharing. Therefore, we aim to tackle this line of research in the present study.

Individuals’ pursuit of status leads to status conflicts, which are attempts to defend or elevate one’s own relative status [8]. Prior research on the relationship between status conflict and knowledge sharing shows that knowledge sharing is a behavior that has both benefits and risks. On the one hand, the need for status drives employees to generously share tacit knowledge and special expertise to obtain social recognition and status, as conferred by supervisory appraisal; thus, this can help them achieve sustainable status conflict advantage over other colleagues [912]. On the other hand, knowledge sharing can threaten contributors’ power and status by reducing their value to the organization and damaging their position, because their expertise and special know-how constitute the source of their current status and employment [11, 13]. Therefore, knowledge sharing can lead to status conflict disadvantage for contributors. Although existing research can help us understand the relationship between status conflict and knowledge sharing, there is currently no study on the relationship between status conflict and knowledge sharing behaviors that considers status conflict advantage and disadvantage concurrently. We aim to explore this line of research in this study.

To this end, we analyze knowledge sharing behaviors by introducing an evolutionary game model with the following incentive mechanisms: second-party punishment, third-party punishment, and mixed second-and-third-party punishment, to determine which punishment mechanism best promotes knowledge sharing within an organization. The model is then used to simultaneously consider status conflict advantage and disadvantage. The stable equilibrium is then obtained by solving the replicator dynamic equations and quantifying the universal dilemma strength. The following issues are investigated by using Matlab for a programming simulation: (1) which among second- and third-party punishments is better for improving knowledge sharing and what happens if we adopt the mixed incentive punishment mechanism; (2) how do status conflict advantage and disadvantage influence knowledge sharing behaviors; and (3) how can we provide practical implications for organizations and leaders to encourage knowledge owners to contribute their knowledge? Finally, the practical implications for organizations and leaders are discussed.

The remainder of this paper is organized as follows. Section 2 provides the theoretical foundation by introducing the concepts of altruistic punishment, second- and third-party punishments, status conflict, guanxi, and evolutionary game. Section 3 constructs the evolutionary game model by defining variables and the pay-off functions matrix and investigates the evolutionary stable strategy of the model. Section 4 presents the simulation analysis and the results. Section 5 draws conclusions and presents management implications, as well as the research limitations and future research directions.

2. Relevant Literature

Prior studies have hitherto contributed to the better understanding of knowledge sharing; however, there are some questions requiring further investigation. For example, as different types of altruistic punishment, second- and third-party punishments can both improve knowledge sharing to some extent; nevertheless, which one promotes knowledge contributions better remains debatable. Additionally, sharing knowledge leads to simultaneous individual status conflict advantage and disadvantage, but there are currently no studies analyzing concurrently the knowledge sharing for both sides of the status conflict. Furthermore, most previous works regarding knowledge sharing are empirical and have several limitations. Therefore, it is necessary to investigate knowledge sharing using different research methods. Specifically, we analyze knowledge sharing by introducing an evolutionary game model with the following main factors: second- and third-party punishments and status conflict advantage and disadvantage. The model is then used to simultaneously consider guanxi, professional and general knowledge capacity, and professional and general absorptive capacity. Finally, we conducted a simulation using Matlab.

2.1. Altruistic Punishment and Knowledge Sharing

Punishment can effectively reduce free riding. However, punishment is a public good that benefits everyone and, similar to other public goods, is vulnerable to free riding. Those who benefit without paying punishment costs gain more than those who punish. Punishment is thus a topic of wide interest to scholars in the form of altruistic punishment [14], which describes the sacrifice of self-interest to punish the violations of social norms, such as fairness or reciprocity [15, 16]. This behavior is often modeled using the ultimatum game (UG) [1719], which illustrates that a sizeable number of people from a wide variety of backgrounds are willing to punish others at a cost to themselves to prevent unfair outcomes or sanction unfair behavior [20]. Among the representative theories used to explain this phenomenon are the fairness preference, strong reciprocity, social preference, social norm activation, emotion theories. Because these punishments can maintain and consolidate social norms, they are called altruistic punishment [20].

The important impact of altruistic punishment is that it challenges the basic hypothesis in traditional economics on the existence of an economically rational agent with two characteristics: it is selfish and pursues the maximization of self-interest instinctively [21]. However, the studies on altruistic punishment indicate that individuals’ behaviors can also be irrational and willing to maintain social norms even it means they need to sacrifice their own interests. Such altruistic punishment helps sustain cooperation in society and is present across a highly diverse range of human populations and under various patterns [22, 23].

2.2. Second- and Third-Party Punishments and Knowledge Sharing

Altruistic punishment includes second- and third-party punishments, both being widely used to punish noncooperators to preserve social norms. Second-party punishment, which is performed by punishers who are directly affected by those whom they punish [24], has attracted the attention of scholars. “An eye for an eye and a tooth for a tooth” is a typical example here. Nevertheless, a growing literature stream switched focus to another non-negligible type of punishment, the one by third parties who are independent of or are only indirectly affected by those whom they punish [25, 26]. Third-party punishment is a pure altruistic behavior. For instance, Fehr and Gachter study the enforcement mechanisms behind social norms, indicating that a large percentage of subjects are willing to enforce distribution and cooperation norms, although they incur costs and reap no economic benefits from their sanctions nor have they been directly harmed by the norm violation [25]. A stranger defending the weak against the strong is a good example. In short, third-party punishment is an important tool to maintain social norms in society.

In most studies, second- and third-party punishments are discussed together and are generally regarded as altruistic punishment types. However, despite both being altruistic behaviors, there are some differences between them. Some studies focus on the mechanisms of second- and third-party punishments and the impacts of anger on these punishment types show that the anger about personal harm, which is a trigger of second-party punishment, is a distinct emotion from the empathic anger due to witnessing injustice or harm to someone else, which is a trigger of third-party punishment [25, 27]. That is, although both triggered emotions represent anger, they have different elicitors, thus causing different punishment. Bendor et al. identify that second-party punishment is easily affected by personal or situational factors; in other words, third-party punishment is more stable than second-party punishment [28]. Leibbrandt et al. prove that many third parties do not punish in an impartial or normative manner [26]. However, only few studies have investigated the effects of second- and third-party punishments on cooperation using experimental methods. For example, Zhou et al. find that third parties punish more frequently, severely, and less antisocially, resulting in a higher contribution level than that driven by second-party punishment [24].

While these studies inform the relationship between altruistic punishment and cooperative behaviors, there is currently no comparative study of the promotion effects of second- and third-party punishments on knowledge sharing that utilizes an evolutionary game model and an ensuing simulation. We thus analyze knowledge sharing behaviors by introducing an evolutionary game model with second-, third-, and mixed second-and-third-party punishment mechanisms to determine which one improves knowledge sharing best within an organization. From the ensuing Matlab programming simulation, we can better understand the effects of altruistic punishment on knowledge sharing and the differences between second- and third-party punishments.

2.3. Status Conflict and Knowledge Sharing

The employees in an organization have the need to pursue not only tangible resources, such as money, but also intangible resources, such as status [29]. However, status is pervasive in human behavior, both in the work environment and in everyday life [30], and the desire for status is a fundamental human motive. Status is defined as the respect and admiration, and voluntary deference individuals are afforded by others, thus being distinct from related constructs such as power, financial success, and social belongingness [31]. People’s subjective well-being, self-esteem, and mental and physical health depend on the status level they are accorded by others, the importance of status being observed across individuals who differed in culture, gender, age, and personality, supporting the universality of the status motive [29, 31]. During the process of an individual's status pursuit, status conflicts arise when individuals attempts to protect or enhance their relative status.

As a sustainable competitive advantage, knowledge—especially professional knowledge—can enable employees to obtain status and power [32]. Prior research indicates that employees will use knowledge sharing as an interpersonal strategy to gain recognition and respect, thus improving their social prestige [33]. Additionally, individuals can show superiority compared to peers and expert authorities by sharing knowledge to boost their reputation and status [3335]. The research on knowledge sharing from the perspective of status conflict motivation also proves that the individuals who are promotion-focused show a higher willingness to take risks, use this approach as a behavioral strategy, and are more likely to explore the advantages of novel behavior directions [36]. In short, they are more inclined to share knowledge with others. Nevertheless, knowledge sharing not only brings employees advantages in a status conflict but has also a disadvantage. Namely, without the control of knowledge, especially expertise, knowledge holders lose the competitive advantage from knowledge, which threatens directly their power and status in the organization [11, 37]. Previous studies show that knowledge sharing may lead individuals to lose unique value and affects their competitive advantage, thus threatening their status [13, 38, 39]. Therefore, sharing knowledge brings both benefits and losses to individuals. Generally, knowledge owners gain respect and prestige by sharing knowledge, which has a positive impact on status conflicts; however, knowledge contributors also lose the control of their expertise, which has a negative effect on status conflicts. In this paper, we consider concurrently the positive and negative influences on status conflict and denote them as status conflict advantages and disadvantages, respectively.

2.4. Guanxi and Knowledge Sharing

Guanxi is a stable and enduring feature of China’s national culture, as well as a source of security and protection in an unpredictable society [40]. Guanxi is today among the most important, talked about, and studied phenomena in China, lying at the heart of China’s social order, its economic structure, and changing institutional landscape. It is considered important in almost every area of life from politics to business and from officialdom to street life and as informal connection, thus being also essential to gaining approval for or access to just about everything in China [4143]. In other words, a key aspect of Chinese culture is being based on guanxi [44], which is different from the concepts of interpersonal relationships or social networks in western society. The Chinese phrase “guanxi” consists of two characters: “guan” means a gate or a hurdle and “xi” refers to a tie, a relationship, or a connection. Therefore, guanxi literally means “pass the gate and get connected” [41]. In summary, guanxi refers to interpersonal relationships or connections and can be applied not only to kinship and friendship relationships but also to social connections, such as dyadic relationships [45, 46].

Prior works on guanxi have contributed to the improved understanding of guanxi in China. For instance, Fei published his book “From the Soil: The Foundation of Chinese Society” [44] in 1940s after 10 years of field study, pointing out the differences between the U.S. society, which is organized by voluntary associations, and the Chinese society, which is organized by concentric guanxi circles. The Chinese are self-centered in communicating with others, extending from the family (the core) to relatives, friends, and so on. People who are farther away from the center of the circle are more alienated. The core Chinese values are the differentiated attitudes towards parents, children, siblings, kinsmen, friends, and so on, which Fei calls “differentiated modes of association” [44]. In the Chinese context, individual rights are not universalistic but guanxi-specific and particularistic—that is, no tie, no obligation, and no rights [41, 44]. Jacobs, a western scholar, was the first one to study relationships in the context of China in Taiwan. He uses the English transliteration “kuan-hsi” to express relationships and argues that the Chinese have long suggested that particularistic ties, or the so-called kuan-hsi, play an important role in politics [45]. Chiao [47] uses the Chinese Pinyin “guanxi” to refer to relationships in China, marking the uniqueness of relationships in the Chinese culture compared with western culture. One of the most representative studies on guanxi is Huang’s “Human feelings and face: A game of Chinese power” [46], whose investigation of guanxi and “face” deepens the understanding of guanxi in China. Based on this prior study, Yang [48] categorizes guanxi in China into three groups: between family members; between familiar people, such as neighbors, friends, and colleagues; and between strangers or mere acquaintances. These three relationship categories have completely different social and psychological meanings to the involved parties and are governed by different sets of interpersonal rules.

A guanxi strategy can be defined as the development and use of personal networks by managers to gain status [40]. Knowledge sharing is a type of interpersonal interaction behavior among individuals. That is, individuals can obtain status under a guanxi strategy by means of sharing knowledge. Therefore, according to the theory of the “differentiated modes of association,” knowledge owners make decisions on whether or how much to share their knowledge with others based on guanxi, defined as the location of a concentric circle from the center. Hence, we introduce the variable guanxi in our model to represent the Chinese cultural content.

2.5. Applying Evolutionary Game Theory in Analyzing Knowledge Sharing

The basic assumption in classical game theory is that agents are “completely rational,” which leads to limitations in analyzing practical problems, because, in real life, agents are bonded rational instead of completely rational. Evolutionary game theory uses the basic assumption of individuals’ bonded rationality and combines game theoretic analysis with dynamic evolution. It considers the equilibrium when individuals with bounded rationality seek optimization over time, namely, it depicts the evolution of a game over time. Therefore, evolutionary game theory is a useful tool for discussing the behaviors of individuals with bonded rationality and a refinement of the Nash equilibrium, thus extending and expanding the classic game theory.

While previous works regarding knowledge sharing have helped us better understand knowledge sharing behaviors, most of them are empirical. Therefore, these studies have several limitations, such as the use of variables based on a questionnaire completed by a single source over one time period [12]. As such, more research on knowledge sharing using different research methods is needed. Some scholars have investigated knowledge sharing from the perspective of social dilemmas in realistic organizations. The frameworks presented in these studies come from sociological research on cooperation and inform the sociopsychological processes governing exchanges among employees [2]. For example, Lin studies the relationship between knowledge sharing and indirect reciprocity with computational experiment [49]. These studies broaden the scope of the investigation on knowledge sharing, but further explorations are still required.

Nowadays, game theory has become one of the most common methods used to solve social dilemmas [50], along with evolutionary game theory, whose assumptions are based on learning ability and bounded rationality instead of complete rationality [51]. In evolutionary game theory, agents need to improve their strategies by imitating, copying, and learning constantly, after which the game eventually reaches stability, which is also known as the “evolutionary stable strategy” (ESS). Employees’ knowledge sharing behaviors within an organization can be regarded as infinite dynamic repeated games, and their strategies may lead to conflicts of interests among relevant individuals. Meanwhile, free riding is a strategy that can bring individuals maximum benefits. Under these circumstances, evolutionary game theory can be used to explore knowledge sharing behaviors.

Few studies have hitherto discussed knowledge sharing between organizations by applying evolutionary game theory. For instance, Li et al. construct an evolutionary game model to analyze the dynamic evolution process of a firm’s knowledge sharing behavior in a setting of supply chain networks [52], while Wang et al. investigate the behavioral characteristics of interfirm knowledge sharing in innovation clusters [53]. As there is currently no comparative study on the promotion effects of second- and third-party punishments on knowledge sharing that uses evolutionary game theory and an ensuing simulation, we aim to tackle this approach.

3. Model

3.1. Variables

To better describe the influence of knowledge on knowledge sharing, we classify knowledge into two types—general and professional knowledge—according to their complexity, flowability, and natural and social attributes. General knowledge refers to the open or universal knowledge in the organization, including rules and regulations, values, and culture, while professional knowledge refers to knowledge related to the product, technology, market, and operation of an organization. That is, professional knowledge rather refers to personal knowledge, experience, skills, or insight. These two types of knowledge are valued differently within organizations [54]. When individuals perceive the knowledge they possess as a valuable commodity, knowledge sharing becomes a process mediated by the decisions of what knowledge to share, when to share, and who to share it with [55]. Therefore, the type of knowledge affects the knowledge sharing behavior. General knowledge is easy to be shared, and employees can obtain it with a low cost; conversely, professional knowledge is difficult to share, since it is more valuable.

To simplify our model, we assume that the players in the evolutionary game are two employees from one organization. This game can be considered as an infinite dynamic repeated game, where each player has two options in selecting his/her strategy as either KS (knowledge sharing) or NKS (no knowledge sharing). It is impossible for either of these two players to fully understand the other one's utility function, which makes the situation an incomplete information game based on bounded rationality. Each player knows his/her own circumstances, including knowledge capacity, guanxi, status, and the gain and loss from sharing knowledge. Therefore, the decision on whether to share knowledge depends on the judgement of his/her expected revenue. Our model includes a number of variables, defined as follows.

3.1.1. , , , : Knowledge Capacity

It denotes the maximal quantity of sharable knowledge for the two players. The knowledge is classified into two categories: professional and general knowledge. Hence, and refer to the maximal quantity of sharable professional and general knowledge, respectively, for the first player, and and refer to the maximal quantity of sharable professional and general knowledge, respectively, for the second player. Therefore, the sums of the sharable knowledge for these two players are and , respectively.

3.1.2. , : Guanxi Index

It refers to one player’s perception of guanxi with the other player. Each player decides how to interact with the other one based on the theory of the “differentiated mode of association.” That is, a closer guanxi leads to increases responsibility and less attention paid to payback, which translates as a higher willingness to share knowledge. Under the influence of guanxi, the actual sums of sharable knowledge for the two players are and .

We concurrently consider the positive and negative influences of status conflict when players choose to share knowledge, which are described as status conflict advantage and disadvantage, respectively. The variables are as follows.

3.1.3. , : Status Conflict Advantage Coefficient from Sharing Professional Knowledge

This is the effect of status conflict advantage when a player shares professional knowledge. Individuals can boost their reputation by sharing professional knowledge, which has a positive effect on status conflict. The status conflict advantage of these two players from sharing professional knowledge is reflected in the status conflict advantage coefficient and the quantities of professional knowledge they actually share, respectively, denoted as and .

3.1.4. , : Status Conflict Advantage Coefficient from Sharing General Knowledge

This is the effect of the status conflict advantage of a player when he/she shares general knowledge. As an interpersonal strategy, sharing general knowledge can also bring status conflict advantage to the individual. The status conflict advantages of the two players from sharing general knowledge are reflected in the status conflict advantage coefficient and the quantities of general knowledge they actually share, respectively, denoted as and . Additionally, individuals can obtain a higher advantage from sharing professional knowledge compared to sharing general knowledge because professional knowledge is more valuable and they can better demonstrate their professional and technical skills. Therefore, we assume that , .

3.1.5. , : Status Conflict Disadvantage Coefficient from Sharing Professional Knowledge

This is the effect variable of the status conflict disadvantage of a player when he/she shares professional knowledge. Sharing general knowledge does not bring individuals a status conflict disadvantage because general knowledge is easy to share. Conversely, sharing professional knowledge means losing the control of expertise for an individual, which then threatens directly his/her power and status. The status conflict disadvantage from sharing professional knowledge for these two players is reflected in the status conflict disadvantage coefficient and the quantities of professional knowledge they actually share, respectively, denoted as and .

3.1.6. , : Professional Knowledge Absorptive Capacity

This is the effect variable of the professional knowledge that a player can understand and absorb. In our model, we define the direct excess benefit as the benefit of the free rider from absorbing professional knowledge when he/she chooses NKS and the other player chooses KS. The direct excess benefit of the free rider is influenced by his/her professional knowledge absorptive capacity and the quantity of professional knowledge the other player actually shares and can be expressed as and for the two players, respectively.

3.1.7. , : General Knowledge Absorptive Capacity

This is the effect variable of the general knowledge that a player can understand and absorb. In our model, we define the direct benefit as the benefit of the free rider from absorbing general knowledge when he/she chooses NKS and the other player chooses KS. The direct benefit of the free rider is related to his/her general knowledge absorption capacity and the quantity of general knowledge the other player actually shares and can be expressed as and for the two players, respectively. Additionally, professional knowledge is more difficult to be absorbed compared to general knowledge because of its characteristics. Hence, we assume that , .

3.1.8. , : Synergy Revenue Coefficient of KS

It refers to the additional benefit generated from knowledge sharing. When both players share knowledge, they may create additional new knowledge due to knowledge fusion and, consequently, reciprocal benefits [52]. The synergy of the two players is affected by the sums of the professional and general knowledge they actually share, which can be expressed as and , respectively., : knowledge sharing cost. Based on transaction cost economics, KS incurs costs, including the opportunity, technical, and time costs that a sharer needs to pay.: second-party punishment cost. Under second-party punishment, the punisher, as a player in the game, needs to pay the cost of punishing the free rider.: penalty of the second-party punishment. Under second-party punishment, the free rider needs to pay the penalty of choosing NKS.: penalty of the third-party punishment. Under third-party punishment, the free rider needs to pay for the penalty of choosing NKS.

The relevant parameters under the different punishment mechanisms are shown in Table 1.


ParametersPunishment mechanism
Second-party punishmentThird-party punishmentMixed second-and-third-party punishments


3.2. Pay-Off Function Matrix

The pay-off functions for both players can be discussed under the following four scenarios.(1)Both player1 and player2 choose KS. At this moment, the benefits and costs of both players include synergy revenue, status conflict advantage from sharing professional and general knowledge, status conflict disadvantage from sharing professional knowledge, and knowledge sharing cost. The pay-off functions of player1 and player2 are and , respectively.(2)Player1 chooses KS and player2 chooses NKS. There can be three situations under this scenario, described as follows.Situation 1: Second-party punishment is adopted. At this moment, player2 is punished by player1. Player1 needs to pay for the second-party punishment cost, , while player2 needs to pay for the penalty of second-party punishment, , since he/she chooses to be a free rider.Situation 2: Third-party punishment is adopted. At this moment, player2 is punished by the independent third party. Player2 needs to pay the penalty of third-party punishment, , since he/she chooses to be a free rider.Situation 3: Mixed second-and-third-party punishment is adopted. At this moment, player2 is simultaneously punished by player1 and the independent third party. Player1 needs to pay the second-party punishment cost, , while player2 needs to pay the penalty of second-and-third-party punishment, , since he/she chooses to be a free rider.Based on these three situations, the benefits and costs of player1 include the status conflict advantage from sharing professional and general knowledge, status conflict disadvantage from sharing professional knowledge, knowledge sharing cost, and second-party punishment cost. The pay-off function of player1 is expressed as . The benefits and costs of player2 include direct excess benefit, which is the benefit of the free rider from absorbing shared professional knowledge; direct benefit, which is the benefit of the free rider from absorbing shared general knowledge; and the penalty, which is from the second- or/and third-party punishment. The pay-off function of player2 is .(3)Player1 chooses NKS, and player2 chooses KS. Similar to scenario (2), the benefits and costs of player2 include the status conflict advantage from sharing professional and general knowledge, status conflict disadvantage from sharing professional knowledge, knowledge sharing cost, and second-party punishment cost. The pay-off function of player2 is expressed as . The benefits and costs of player1 include direct excess benefit, direct benefit, and penalty, which are from the second- or/and third-party punishment. The pay-off function of player1 is .(4)Both player1 and player2 choose NKS. At this moment, both players have no benefits, since they do not share knowledge. Moreover, the second-party punishment is not implemented; thus, no one needs to pay for the penalty of second-party punishment. However, third-party punishment is present and both players need to pay for its penalty, , since they both break the rule of cooperation. In other words, both players’ pay-off is under this scenario.

Overall, the pay-off function matrices under those four scenarios are summarized in Table 2.


Player 2
KSNKS

Player 1KS
NKS

3.3. Evolutionary Stable Strategy
3.3.1. Solving Evolutionary Stable Strategy by Applying Replicator Dynamic Equation

Based on the evolutionary game, we further assume that the initial probability of player1 choosing KS is ; thus, the initial probability of choosing NKS would be . Similarly, the initial probability of player2 choosing KS is , meaning the initial probability of choosing NKS would be . Moreover, and are functions of time.

The expected revenue of player1 when he/she chooses KS is defined as per the following equation:

The expected revenue of player1 when he/she chooses NKS is defined in the following equation:

The expected average revenue of player1 is defined as per the following equation:

Based on symmetry, the expected revenues of player2 when he/she chooses KS and NKS, and the expected average revenue are, respectively, expressed as per the following equations:

Then, we apply the replicator dynamic equation for player1 and player2 as

To solve the stable strategy in equations (7) and (8), we need to obtain the critical value.

If , we get , or , or

If , we get , or , or

Then, it can be concluded that there are five evolutionary equilibrium points during the game: (0, 0), (0, 1), (1, 0), (1, 1), .

Subsequently, the characteristics of the evolutionary equilibrium points can be analyzed based on the Jacobian matrix of the dynamic system [56]. According to equations (7) and (8), the Jacobian matrix can be expressed as

The value of the Jacobian matrix isThe trace of the Jacobian matrix can be expressed as

Next, the value and trace of the Jacobian matrix at each evolutionary equilibrium point are calculated and the results shown in Table 3.

Afterwards, the evolutionary equilibrium points can be discussed, we define polynomials ①–⑥ as follows.Polynomial ①:  Polynomial ②:  Polynomial ③:  Polynomial ④:  Polynomial ⑤:  Polynomial ⑥:


Evolutionary equilibrium point

(0, 0)

(0, 1)

(1, 0)

(1, 1)