Effects of conservative-militant defection strategies on the evolution of cooperation in social dilemma

Abstract

Competition in the international arena and business realm offers avenues for individual growth and advancement. Individuals using different means of competition can obtain unequal rewards. This paper claims that when no consensus is reached in business activities, defectors will choose conservative or militant defection strategies during market competition. Conservative defectors, who are in a comparatively weak position, need to pay the costs brought by market share losses. However, their personal abilities cannot be ignored, which prompts them to bravely choose the conservative defection strategy. This brings rewards to conservative defectors. Militant defectors, typically in stronger positions, also receive greater rewards. Research results establish an evolutionary game model of three strategies, the cooperation strategy, the conservative defection strategy, and the militant defection strategy. After the system is stable, this model displays two stable states. Through numerical simulation, it can be found that the personal abilities of conservative defectors play a decisive role in promoting cooperation. However, the market share losses of conservative defectors have periodical impacts on cooperation. Moreover, the threats of militant defectors to cooperation should be comprehensively considered in combination with the personal abilities of conservative defectors.

Introduction

Cooperation and defection are decisions that human beings must face to survive^1,2,3. The law of the “survival of the fittest” states that individuals are self-interested and hope to maximize their personal interests. As a result, cooperation is at a disadvantage in defection^4,5. However, cooperation plays a key role in the field of public tax management⁶, public transportation management⁷and infectious disease control⁸. Therefore, it is particularly important to explore strategies to promote cooperation during the game.

Evolutionary Game Theory provides a general framework to address such issues^9,10,11. As scholars have gradually cast their eyes on game models, they have uncovered and refined numerous examples. Examples are prisoner’s dilemma game⁵, Rock-Scissor-Paper game¹², snowdrift game⁵, stag hunt game^13,14, and so on. Among them, prisoner’s dilemma game has been widely applied in addressing the issue of cooperation and defection¹⁵. Research on game mechanisms in cooperative behavior, scholars usually make improvements based on the two strategies of cooperation and defection, such as the prisoner’s dilemma game with an exit mechanism¹⁶, an aspiration mechanism^17,18, a reputation mechanism^19,20and a memory mechanism^21,22. In addition, time delay also has potential effects on cooperative actions²³. Further investigations consider the impacts of rules update^24,25,26and spatial game^27,28,29on promoting cooperation, including Fermi update rule³⁰and Moran update rule³¹, etc. As scholars delve deeper into the study of evolutionary game theory, they have found that the emerging cyclic dominance among competing strategies becoming a very well-known research path^32,33. Szolnoki et al. introduced strategy-dependent learning activity for specific strategies in systems where cyclic dominance emerges³⁴. When different interest subjects are divided, they adopt different strategies³⁵due to varying personalities and demands between them as well as limitations to their abilities³⁶. For example, individuals take drastic actions to gain more rewards for themselves, while businesses may use tactics to suppress competitors and maximize orders; governments, on the other hand, may opt for a conservative approach to monitor developments of the situation. In the process of a game between these subjects, both conservative and militant game behaviors are fully represented.

In business activities where benefits are not clearly distributed, games between employers and employees are very common. A typical example is the Dong Yuhui incident of live-streaming sales in the Douyin platform. Dong Yuhui was a live-streaming sales anchor of East Buy Holding Limited. The company needed to train more anchors to maintain sustainable development. Therefore, Dong Yuhui’s individual prevailing trend was suppressed to some extent during the company’s operations. At this time, East Buy Holding Limited took drastic actions to deny Dong Yuhui’s individual contributions (militant defection strategy), while Dong Yuhui chose to wait and see the development of the situation silently (conservative defection strategy). As time went on, Dong Yuhui’s individual prevailing trend was reflected, as most netizens supported Dong Yuhui who remained silent. In the end, forced by Dong Yuhui’s personal influence, East Buy Holding Limited chose to cooperate with Dong Yuhui again (cooperation strategy).

Conservative and militant strategies are reasonable choices for two types of individuals. In this paper, the defection strategy is subdivided into conservative and militant defection strategies. When individuals in a population play prisoner’s dilemma game in twos, conservative defectors will lose a certain market share in a short time. However, when playing the game with militant defectors, conservative defectors will win the initiative with their personal abilities or influences, which is reflected as increased rewards in the payoff matrix. When militant defectors play the game with cooperators, the former will win the compromise of cooperators and their rewards will increase. Through a mathematical model of three strategies, with conservative and militant defection strategies, this paper investigates the impact of conservative and militant defection strategies on the evolution of cooperation in a well-mixed population.

This paper is organized as follows. Section “Model and Theoretical Analysis” introduces the evolutionary game model with cooperation strategy, conservative defection strategy and militant defection strategy and gives the theoretical analysis for the well-mixed population. Section “Results” analyzes the evolution results by numerical simulation. Finally, the main conclusion are given in Section “Conclusion”.

Model and theoretical analysis

Model

Cooperation (C) and Defection (D) are two basic strategies in the prisoner’s dilemma. When two players cooperate, they receive rewards R. Conversely, if they both defect, then they face punishment P. In addition, when a player adopts the defection strategy while the other chooses cooperation, the defector obtains the temptation of defection T, while the cooperator obtains the loser’s payoff S. The rewards satisfy $T>R>P>S$ and $2R>T+S$. At this time, the defection strategy is the only Nash equilibrium strategy^37,38.

Defectors usually show conservative and militant characteristics due to different environments, personalities, and options. For individuals who choose the conservative defection strategy, personal abilities play a decisive role in promoting cooperation. For militant defectors, their tough attitude can usually force other defectors to compromise. This study explores three types of strategies that exist in a population: the cooperation strategy (C), the conservative defection strategy (CD), and the militant defection strategy (MD). Cooperation strategy often mean that both sides can compromise with each other to achieve the same goal. Conservative defection strategy is often adopted by employees in business behavior, who observe the direction of the situation to determine their own development. And militant defection strategy is often adopted by the dominant party with decision-making power, usually the employers. During the game, conservative defectors will disappear in the market. As time goes on, when they play the game with any individual in the population, there will be market share loss $a(0 \le a \le 1)$ while conservative defectors are disappearing. However, their personal abilities will bring rewards $\beta (0 \le \beta \le 1)$ to conservative defectors. Because militant defectors are in a stronger position, their rewards will increase $\delta (0 \le \delta \le 1)$ when they play the game with cooperators. If $R=1$,  $S=P=0$, and  $T=b\in (1,2]$, then the payoff matrix can be presented as:

$$\begin{aligned} \begin{aligned} \begin{array}{l} \quad \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C\quad \;\;\;\;CD\;\;\;MD\\ \begin{array}{*{20}{c}} C\\ {CD}\\ {MD} \end{array}\left( {\begin{array}{*{20}{c}} 1& 0& { - \delta }\\ {b - a}& { - a}& {\beta - a}\\ {b + \delta }& { - \beta }& 0 \end{array}} \right) . \end{array} \end{aligned} \end{aligned}$$

(1)

Consider an infinite well-mixed population to play the above game. Assume that x,   y and $z=1-x-y$ are the fraction of cooperators, conservative defectors and militant defectors, respectively.  Individual can interact with any other individuals in the group, and the average payoff of these three strategies are:

$$\begin{aligned} \begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{P_C}}& = & {x - \delta z,}\\ {{P_{CD}}}& = & {(b - a)x - ay + (\beta - a)z,}\\ {{P_{MD}}}& = & {(b + \delta )x - \beta y.} \end{array}} \right. \end{aligned} \end{aligned}$$

(2)

The replication equations can be calculated by $P_C$,  $P_{CD}$, and  $P_{MD}$ in Eq. (2).

$$\begin{aligned} \begin{aligned} \left\{ {\begin{array}{*{20}{l}} {\dot{x}}& = & {\frac{{dx}}{{dt}}}& = & {x({P_C} - {\bar{P}}),}\\ {\dot{y}}& = & {\frac{{dy}}{{dt}}}& = & {y({P_{CD}} - {\bar{P}}),}\\ {\dot{z}}& = & {\frac{{dz}}{{dt}}}& = & {z({P_{MD}} - {\bar{P}}).} \end{array}} \right. \end{aligned} \end{aligned}$$

(3)

Where the expected payoff of the whole population is ${{\bar{P}}}=x{P_C}+y{P_{CD}}+z{P_{MD}}$.

Theoretical analysis

The system (3) has six realistic equilibria: $E_1=(0,0,1)$, $E_2=(1,0,0)$, $E_3=(0,1,0)$, $E_4=({\frac{{a}}{{b-1}}},{\frac{{b-a-1}}{{b-1}}},{\frac{{2b-2a-2}}{{b-a}}})$, $E_5=(-{\frac{{\delta }}{{b-1}}}, 0, {\frac{{b-\delta -1}}{{b-1}}})$, $E_6=(x^*,y^*,z^*)$. Where

$$\begin{aligned}\begin{array}{*{20}{l}} {{x^*}}& \!\!\!\!=\!\!\!\! & {\frac{{\beta - a}}{{\beta + \delta }},}\\ {{y^*}}& \!\!\!\!=\!\!\!\! & {\frac{{{\delta ^2} + \beta \delta + a - \beta - ab + b\beta }}{{{{(\beta + \delta )}^2}}},}\\ {{z^*}}& \!\!\!\!=\!\!\!\! & {\frac{{(a - b + 1)\beta + (\delta + b - 1)a}}{{{{(\beta + \delta )}^2}}}.} \end{array}\end{aligned}$$

Due to $z=1-x-y$, the three strategy game model of conservative-militant defection strategy in system (3) can be transformed into:

$$\begin{aligned} \begin{aligned} \begin{array}{*{20}{l}} {\dot{x}}& = & {f(x,y)}& = & {(b - 1){x^3} + (\delta - b + 1){x^2} + \left[ {(a + \delta )y - \delta } \right] x,}\\ {\dot{y}}& = & {g(x,y)}& = & {(a - \beta ){y^2} + \left[ {(b - 1){x^2} + \beta (1 - x) - a} \right] y.} \end{array} \end{aligned} \end{aligned}$$

(4)

The Jacobian matrix of the differential equation system (4) can be obtained by calculating the first partial derivative as:

$$\begin{aligned} \begin{aligned} J = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial f(x,y)}}{{\partial x}}}& {\frac{{\partial f(x,y)}}{{\partial y}}}\\ {\frac{{\partial g(x,y)}}{{\partial x}}}& {\frac{{\partial g(x,y)}}{{\partial y}}} \end{array}} \right] . \end{aligned} \end{aligned}$$

(5)

Where

${\frac{{\partial f(x,y)}}{{\partial x}}} = {(3b - 3){x^2} + (2 - 2b + 2\delta )x + (a + \delta )y - \delta},$  

${\frac{{\partial f(x,y)}}{{\partial y}}}= {(a + \delta )x},$

$\ {\frac{{\partial g(x,y)}}{{\partial x}}} = {2(b-1)xy - \beta y},$

$\ {\frac{{\partial g(x,y)}}{{\partial y}}}= {(b - 1){x^2} + (2a - 2\beta )y - \beta x + \beta - a.}$

Stability analysis can be carried out by determining the determinant and the trace of Jacobian matrix (5) as following:

• For point $E_1=(0,0,1)$, $\vert {J}\vert =-\delta (\beta -a)$ and $trJ=\beta -a-\delta$.

  • When $\beta >a+\delta$, indicating that the equilibrium point $E_1$ is unstable.

  • When $\beta <a$, indicating that the equilibrium point $E_1$ is stable. As shown in Figure 1(a).

• For point $E_2=(1,0,0)$, $\vert {J}\vert =(b+\delta -1)(b-a-1)$ and $trJ=2b+\delta -a-2$. When $b-a-1>0$, $\vert {J}\vert <0$. Due to $b+\delta -1>0$, therefore the equilibrium point $E_2$ is unstable.

• For point $E_3=(0,1,0)$, $\vert {J}\vert =a(a-\beta )$ and $trJ=2a-\beta$. Due to $\lambda _1=a>0$, therefore the equilibrium point $E_3$ is unstable.

• For point $E_4=({\frac{{a}}{{b-1}}},{\frac{{b-a-1}}{{b-1}}},{\frac{{2b-2a-2}}{{b-a}}})$, $\vert {J}\vert =\frac{ak_1({\beta }k_1-ak_1+ak_2+2abk_2)}{(b-1)^2}$ and $trJ=\frac{ak_2+{\beta }k_1}{b-1}$, where $k_1=a-b+1$ and $k_2=a+\delta$.

  • When $a+1>b$, indicating that the equilibrium point $E_4$ is unstable.

  • Due to the mutual exclusion of $a+1<b$ and $\beta +a^2+a\delta +2ba^2+2ab\delta <ak_1$, it is not possible to satisfy $\vert {J}\vert >0$ and $trJ<0$, so the equilibrium point $E_4$ is unstable.

• For point $E_5=(-{\frac{{\delta }}{{b-1}}}, 0, {\frac{{b-\delta -1}}{{b-1}}})$, $\vert {J}\vert =\frac{(\beta -a){\delta }b^2+(\beta -a)b\delta ^2+(b-1)\delta ^2+(2b-1)a\delta +\delta (b\delta ^2+a\delta +\delta ^3-b\beta )}{b-1}$ and $trJ=\frac{b\delta +2\delta ^2+b\beta -ab+a}{b-1}$. Due to the mutual exclusion of $ab>b\delta +2\delta ^2+b\beta +a$ and $\beta >a$, it is not possible to satisfy $\vert {J}\vert >0$ and $trJ<0$, so the equilibrium point $E_5$ is unstable.

• For point $E_6=(x^*,y^*,z^*)$, $\vert {J}\vert ={\sigma _1}{\sigma _2}-{\sigma _3}{\sigma _4}$, $trJ={\sigma _1}+{\sigma _2}$. Where $\sigma _1=\phi _1+\phi _2$, $\sigma _2=\phi _3+\phi _4$.

  $$\begin{aligned}\begin{array}{*{20}{l}} {{\phi _1}}& \!\!\!\!=\!\!\!\! & {\frac{2ba^2-5ab\beta +3b\beta ^2-2a^2+5a\beta -3\beta ^2+a\delta ^2+a{\beta }{\delta }+\delta ^3+{\beta }{\delta ^2}+a\delta -{\beta }{\delta }-ab\delta +b{\beta }{\delta }}{{(\beta + \delta )^2 }},}\\ {{\phi _2}}& \!\!\!\!=\!\!\!\! & {\frac{{(2b-2-2\delta )(a-\beta )-\delta (\beta +\delta ) }}{{{{(\beta + \delta )}}}},}\\ {{\phi _3}}& \!\!\!\!=\!\!\!\! & {\frac{{(b+1)(a-\beta )^2+(2a-2\beta )(\delta ^2+{\beta }{\delta }+a-\beta -ab+b\beta }}{{{{(\beta + \delta )}^2}}},}\\ {{\phi _4}}& \!\!\!\!=\!\!\!\! & {\frac{{(2\beta +\delta )(\beta -a)}}{{{{(\beta + \delta )}}}},}\\ {{\sigma _3}}& \!\!\!\!=\!\!\!\! & {\frac{{-(a+\delta )(a-\beta )}}{{{{(\beta + \delta )}}}},}\\ {{\sigma _4}}& \!\!\!\!=\!\!\!\! & {\frac{{(2b-2)(a-\beta )(\delta ^2+{\beta }{\delta }+a-\beta -ab+b\beta )-(\beta (\beta +\delta )}}{{{{(\beta + \delta )}}}}.} \end{array} \end{aligned}$$

  When $(\phi _1+\phi _2)(\phi _3+\phi _4)-{\sigma _3}{\sigma _4}>0$ and $\phi _1+\phi _2+\phi _3+\phi _4<0$, indicating that the equilibrium point $E_6$ is stable. As shown in Figure 1(b).

Fig. 1

Simplex plot of evolutionary dynamics. The streamline represents the gradient selected under the replication equation.

The results of equilibrium point stability determination are shown in Table 1.

Table 1 Results of equilibrium point stability determination.

Results

Assume that the initial proportions of cooperators, conservative defectors, and militant defectors are 1/3. Firstly, when $b=1.3,$  $\delta =0.5,$  $\beta =0.9$, the proportions of the three groups after the system is stable are $\rho _C,$  $\rho _{CD},$ and  $\rho _{MD}$, respectively. The evolution results of the three proportions on a and the market share loss of conservative defectors are given below. Figure 2 shows that when the market share loss of conservative defectors is $a \in (0.16,0.26)$, the proportion of cooperators reaches maximum. When the market share loss of conservative defectors increases, the proportion of cooperators decreases rapidly; the proportion of conservative defectors first decreases slowly and then decreases rapidly. In contrast, the proportion of militant defectors increases rapidly.

Fig. 2

The fraction of cooperators(C),conservative-defectors(CD) and militant-defectors(MD) dependent on the variation of a. The other parameters are $b=1.3,$  $\delta =0.5,$  $\beta =0.9$.

Fig. 3

The fraction of cooperators(C),conservative-defectors(CD) and militant-defectors(MD) variation over time. The parameters are $b=1.3,$  $\delta =0.5,$  $\beta =0.9$ and a$\in \{0.1, 0.16, 0.2, 0.26, 0.3, 0.5, 0.7, 0.9\}$.

In Figure 3, assuming $b=1.3,$  $\delta =0.5$, and  $\beta =0.9$, then $a \in \{0.1,0.16,0.2,0.26,0.3,0.5,0.7,0.9\}$. The horizontal scale represents the progression of time, and the vertical scale represents the proportion of people who choose cooperation strategy (C), conservative defection strategy (CD) and militant defection strategy (MD). As shown, when the conservative loss is $a \in [0,0.16]$, there is no militant defector in the system, while conservative defectors are at an advantage. When the conservative loss is $a \in [0.16,0.26]$, the proportion of cooperators temporarily reaches maximum, which is the easiest way to promote cooperation. When the conservative loss is $a \in [0.26,1]$, the proportion of militant defectors begins to increase. Finally, there is no cooperator in the system. This reveals that when conservative defectors lose little and the reward is far greater than the loss, they can obtain great rewards if they seize the opportunity in competition. With the passage of time, conservative defectors gradually lose their competence in the market. Then, cooperators and defectors will experience a thaw when it is easiest for them to reach multi-party cooperation. When the market share of conservative defectors is occupied rapidly, militant defectors have more competitive advantages. Therefore, to reach cooperation, it is necessary to occupy the market share at any time.

Next, when $b=1.3, ~\delta =0.5, ~a=0.2$, the proportions of the three groups after the system is stable are $\rho _C,~ \rho _{CD},~ \rho _{MD}$, respectively. The evolutionary results of the three proportions on $\beta$ and the rewards brought by the personal abilities of conservative defectors are given below. Figure 4 shows that when the personal abilities of conservative defectors bring a reward larger than 0.84, a cooperation is gradually reached. Notably, it becomes evident that when conservative defectors have sufficiently strong personal abilities, militant defectors fail to gain any competitive advantage.

Fig. 4

The fraction of cooperators(C),conservative-defectors(CD) and militant-defectors(MD) dependent on the variation of $\beta$. The other parameters are $b=1.3,~\delta =0.5,~a=0.2$.

Fig. 5

The fraction of cooperators(C),conservative-defectors(CD) and militant-defectors(MD) variation over time. The parameters are $b=1.3,~\delta =0.5,~a=0.2$ and  $\beta \in \{0.1,0.3,0.5,0.6,0.8,0.84,0.89,0.95,1\}$.

In Figure 5, assuming $b=1.3, ~\delta =0.5, ~a=0.2$, then $\beta \in \{0.1,0.3,0.5,0.6,0.8,0.84,0.89,0.95,1\}$. The horizontal scale represents the progression of time, and the vertical scale represents the proportion of people who choose cooperation strategy (C), conservative defection strategy (CD) and militant defection strategy (MD). The personal abilities of conservative defectors play a decisive role in both personal development and promoting cooperation. Figure5(a) shows a scenario where conservative defectors have extremely weak personal abilities, resulting in there being only militant defectors in the system. When conservative defectors slightly show their personal abilities, the proportion of militant defectors immediately begins to decrease sharply, as shown in Figure 5(b)-(e). When $\beta \in [0.84,1]$, the people who choosing cooperation strategy as having a certain advantage over choosing the other two strategies, as shown in Figure 5(f)-(i). Thus, when conservative defectors have certain personal abilities, it is actually beneficial to their personal development. Militant defectors should attach importance to the personal abilities of conservative defectors. At the same time, the ultimate goal of conservative defectors is to promote cooperation. Therefore, a cooperation is reached when conservative defectors have extremely strong personal abilities.

Finally, when $b=1.3,~\beta =0.9, ~a=0.2$, the proportions of the three groups after the system is stable are $\rho _C, ~\rho _{CD}, ~\rho _{MD}$, respectively. The evolution results of the proportions on $\delta$ and the strong rewards of militant defectors are given below. Figure 6 shows that when $\delta$ is smaller than 0.56, the proportion of cooperators is the largest. When $\delta$ is greater than 0.56, the proportion of conservative defectors is the largest. In both scenarios, militant defectors account for the smallest proportion.

Fig. 6

The fraction of cooperators(C),conservative-defectors(CD) and militant-defectors(MD) dependent on the variation of $\delta$. The other parameters are $b=1.3,~\beta =0.9,~a=0.2$.

Fig. 7

The fraction of cooperators(C),conservative-defectors(CD) and militant-defectors(MD) variation over time. The parameters are $b=1.3,~\beta =0.9,~a=0.2$ and  $\delta \in \{0.1,0.3,0.5,0.56,0.7,0.9\}$.

In Figure 7, assuming $b=1.3, ~\beta =0.9, ~a=0.2$, then $\delta \in \{0.1,0.3,0.5,0.56,0.7,0.9\}$. The horizontal scale represents the progression of time, and the vertical scale represents the proportion of people who choose cooperation strategy (C), conservative defection strategy (CD) and militant defection strategy (MD). As shown in Figure 7(a)-(c), when the strong reward of militant defectors $\delta$ is smaller than 0.56, cooperators and conservative defectors do not have any panic about the strength of militant defectors. As a result, the people who choosing cooperation strategy as having a certain advantage over choosing the other two strategies. With the passage of time, when militant defectors begin to show their strength, they fail to have a great impact on cooperators and conservative defectors. In Figure 7(d)-(f), it can be seen that the strength shown by militant defectors fail to scare conservative defectors; however, it breaks the original cooperation. Therefore, militant defectors fail to play a decisive role in cooperation.

The analysis above indicates that the market share loss of conservative defectors when they disappear and the impact of their personal activities on cooperation are non-monotonic. Clearly, the strong rewards of militant defectors are unfavorable to cooperation. However, the evolution results of indicators describing conservative and militant defectors on the system are not yet clear. Therefore, it is necessary to further explore the common impact of $a-\delta$ and $\beta -\delta$ on the system.

Fig. 8

Heat maps of $\rho _C$(cooperators), $\rho _{CD}$(conservative-defectors) and $\rho _{MD}$(militant-defectors) with respect to $a-\delta$.

Assume $b=1.1,1.2,1.3$ and $\beta =0.9$. The heat maps of the proportions of cooperators, conservative defectors, and militant defectors on a and $\delta$ are shown in Figure 8. The horizontal scale represents the range of values for $\delta$, and the vertical scale represents the range of values for a. Figure 8(a),(d),(g) shows that the proportion of cooperators decreases with the increase of strong rewards of militant defectors and also decreases with the increase of market share loss of conservative defectors. This indicates that cooperation is unfavorable when the market share of conservative defectors is eroded sharply and militant defectors show strong threats. As shown in Figure 8(b),(e),(h), when the threats of militant defectors are not very strong, the market share loss of conservative defectors contributes to the increase of their proportion. This indicates that if militant defectors are not strongly suppressive, it is more beneficial to the development of conservative defectors. This can also be seen in Figure 8(c),(f),(i). The proportion of militant defectors is positively correlated with their strength.

Assume $b=1.1,1.2,1.3$ and  $a=0.2$. The heat maps of the proportions of cooperators, conservative defectors, and militant defectors on $\beta$ and $\delta$ are shown in Figure 9. The horizontal scale represents the range of values for $\delta$, and the vertical scale represents the range of values for $\beta$. Figure 9(a),(d),(g) show that when the strong rewards of militant defectors are low, personal abilities seem to promote cooperation. This also applies to conservative defectors, as shown in Figure 9(b),(e),(h). However, when strong rewards have enough temptation, the balance of this cooperation is gradually disrupted, as shown in Figure 9(c),(f),(i). Militant defectors cannot tolerate the gradual erosion of market share, thus beginning to suppress cooperators and conservative defectors with personal abilities. From this, it can be seen that as the strong rewards for militant defectors increases, the cooperation between the two parties eventually falls apart. Militant defectors are in a dominant position on both sides of the cooperation and have decision-making authority.

Fig. 9

Heat maps of $\rho _C$(cooperators), $\rho _{CD}$(conservative-defectors) and $\rho _{MD}$(militant-defectors) with respect to $\beta -\delta$.

Conclusion

In this paper, we take the “Dong Yuhui” incident of the famous anchor of East Buy Holding Limited Company as an example. Dong Yuhui’s influence in the live stream is similar to that of sports stars on the field. Their bargaining power in negotiations with employers is similar. If employers still choose to negotiate with these superstars in a tough manner, it may lead to a breakdown in negotiations, resulting in losses for the company. In order to maintain a balance in cooperation, employers can lower their own tough attitude and appropriately utilize employees’ personal abilities. Based on this, this paper attempts to resolve this social dilemma and try to find a compromise strategy that can benefit both employers and highly capable employees, allowing employers to have strong employees while ensuring that these capable employees are willing to work for them.

We extend the conventional defection strategy by introducing the conservative defection strategy and the militant defection strategy. In a well-mixed population, a mathematical model of three strategies, the cooperation strategy, the conservative defection strategy, and the militant defection strategy, is built. The evolutionary game model of the three strategies is established using the replication equation. The solution to the equilibrium point of the model is found, and the stability of the equilibrium point is judged. After the system is stable, this model displays two stable states: the stable state where all individuals choose the militant defection strategy and the stable state where the three strategies co-exist. Meanwhile, numerical stimulation was used to verify the stable states and evolution results.

The research results show that, in a well-mixed population, without considering strong reward of militant defectors, the personal abilities of conservative defectors play a decisive role in cooperation and are positively correlated with cooperation. When militant defectors demonstrate strong means, conservative defectors with too strong personal abilities can easily lead to the breakdown of cooperation between both sides. The strength of militant defectors can threaten cooperators and conservative defectors. When the employees benefits themselves too much and harms the employer’s rights, the employer will break the cooperation and exercise their own power. The market share loss of conservative defectors during their disappearance has periodical impacts on the cooperation strategy and defection strategies.

In future research, we will pay more attention to the impact of third-party internet users on the self-media sales model. And construct a model including radical posters, onlookers, and those trying to calm the situation the people who holding attitudes on the internet. Each group can choose between the militant defection strategy, the conservative defection strategy, and the cooperation strategy. We will establish a model to discuss the three-party three-strategy game model.