# Behavior Choice of Game Parties under the Interference of Cognition in the Game between Coal Miners and Supervisors

Authors: Jun Liu; Chen Wang; Xinhua Wang; Qingli Li
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In the game between coal miners and supervisors, the behavior choices of the game parties will be affected by cognitive factors. The analysis of the behavior choices of coal miners and supervisors under the influence of cognitive factors is helpful for the design of violation behavior control strategies. Firstly, a description method of subjective cognition based on the mathematical method of quantum theory is designed. Secondly, taking the subjective cognition of coal miners and supervisors as a random variable, a behavior evolution model with random variables is constructed. Thirdly, the impact of subjective cognition on the behavior choices of coal miners and supervisors is analyzed. Finally, the violation behavior control strategy is designed. It is found that when the violation probability decreases to a certain extent, the probability of supervision will change from the increase to decrease. When the probability of supervision decreases to a certain extent, the violation probability will change from the decrease to increase. Fluctuations in cognitive state can affect the change process of violation probability and supervision probability. The behavior control strategy designed according to the behavior evolution model can control the violation behavior in the situation of cognitive state fluctuation.


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## Body

## 1. Introduction

The violation behavior of coal miners is an important factor affecting safety production [1–4]. Cognition is a psychological factor affecting coal miners’ behavior [5, 6]. Cognition can be affected by some factors, such as mental pressure [7–9]. The relationship between pressure and cognitive competence is an inverted U-shaped curve [7]. In the game between coal miners and supervisors, the game parties’ cognition usually has bias. For example, coal miners may underestimate the risk of violation behavior and overestimate its utility. In China, the education level of coal miners is generally low, and their cognitive ability is limited. Under the influence of production pressure, life pressure, and other stochastic factors, their cognition of the utility of violation behavior tends to fluctuate. To analyze the impact of cognition on the game parties’ behavior choice in the game between coal miners and supervisors, we will design a description method of subjective cognition based on quantum theory and build a behavior evolution model considering the influence of subjective cognition based on evolutionary game theory.According to the behavioral decision theory, decision-makers are not entirely rational. Most people are also not completely selfish [10, 11]. The human brain can conduct complicated calculations and and have a complicated memory. However, biological and social evolutions equip us with a set of mental tools. We always use this set of tools to make decisions [12]. As a mental factor, cognition is in close relation to decision-making [13]. To better describe the impact of cognition on decision-making, some authors have introduced quantum theory to the decision-making field and made many achievements [14–24]. Research on cognitive psychology indicates that quantum probability can describe the decision-making of human beings in an elegant framework [14]. This provides another method for computing probabilities without falling into the restrictions that classical probability has regarded to model cognitive systems and decision-making [15–18]. The quantum-like decision-making model proposed by Moreira and Wichert can recognize the optimal decision in their belief space [19]. Bohr thinks that the mental process of people is similar to the course of quantum physics in many aspects and that quantum theory can be used as a mathematical tool for solving problems involving the thinking process of human beings [20]. The mathematical methods in quantum theory can explain humans’ cognitive processes, including judgment, decision-making, concept-making, inference, memory, and perception [21]. Busemeyer and Trueblood put forward six reasons for applying quantum theory to human cognition [22]. Vaio thinks that the mathematical methods in quantum theory can explain the behavior of violating logic and help in the establishment of mental and cognitive models [23]. The above study shows that it is feasible to describe the cognitive state of the game parties by making use of the mathematical methods in quantum theory.In terms of the game problem between the stakeholders in the production system of a coal mine, Lu and Wang conducted relevant research on the fractional supervision game model [24], the multiple evolution game model [25], the stability of the evolution game system [26], and the evolution dynamics model based on the delay and impulsive differential equation [27]. Furthermore, some authors have analyzed the dynamic process of a multiplayer game in a coal mine safety supervision system [28–30]. Some authors have also analyzed the symmetry of game benefits between managers and coal miners [31]. Moreover, they have also analyzed the game relationships between the stakeholders in the coal mine safety production system [32, 33]. These researches all provide a reference for the analysis of the behavior of game parties.The traditional game theory is built on the assumption that the game parties are completely rational [34]. It does not consider the irrational decision-making behaviors caused by some mental factors such as cognition. Compared with traditional game theory, the evolution game theory deems that the game parties will learn in the game and seek a relatively good strategy by trial and error instead of modelling people into superrational game parties [35–37]. The evolution game theory can be used to analyze the evolution process of the decision-making behavior of game parties [38–40]. For example, based on the evolution game theory, Shia et al. analyzed the dynamic evolutionary process of evacuation decision-making for different decision-makers in a wildfire event [41], and Du et al. analyzed the evolution process of stakeholders’ decision-making behaviors in construction and demolition waste management [42]. We will use evolutionary game theory to establish the behavior evolution model and then use the behavior evolution model to analyze the behavior evolution process of game parties in the game between coal miners and supervisors under the fluctuation of cognitive state and finally design a behavior control strategy.


## 2. Behavior Evolution Model considering the Interference of Cognition

### 2.1. Model Assumption

The relevant assumptions in the game between coal miners and supervisors are as follows.Coal miners have two behaviors for selection, operating in violation of the rules and operating according to the rules, which are labelled asA1 and A2, respectively. The probability of choosing A1 is x1 and the probability of choosing A2 is x2, where x1+x2=1. Supervisors have two behaviors for selection, supervising and not supervising, which can be labelled as B1 and B2, respectively. The probability of choosing B1 is y1 and the probability of choosing B2 is y2, where y1+y2=1.Coal miners have an incomplete cognition ofy1 and y2. Supervisors have an incomplete cognition of x1 and x2. Coal miners and supervisors can make behavior choices according to their subjective judgment.Operations in violation of rules can save time. Coal miners can obtainu0 if they operate according to the rules. Meanwhile, if they operate in violation of rules, they can obtain u0+I, where I is the utility of saving time. If supervisors find that coal miners operate in violation of the rules, coal miners will suffer a corresponding punishment f. If supervisors find that coal miners operate according to the rules, coal miners will obtain the award b. The enterprise loss that the operation in violation of the rules leads to will be l. If supervisors supervise, this loss can be avoided. The supervisory cost is c0. The game payment matrix can be shown as in Table 1.Table 1
Game payment matrix between coal miners and supervisors.
-B1B2A1uA11=u0+I−f,uB11=−c0uA12=u0+I,uB21=−lA2uA21=u0+b,uB12=−c0uA22=u0,uB22=0


### 2.2. Description of Cognitive State

In quantum theory, the quantum state can be used to represent the state of the quantum system [43, 44]. Labelled as ψ, the quantum state is the vector representation of Hn, which is the n-dimensional Hilbert Space. ⋅ is the symbol of Dirac [43]. It can be assumed that there is a quantum state ψ∈Hn, ψ=1, and the general definition is shown as follows:(1)ψ=∑j=1ncjj,where j is an n-dimensional column vector. The jth line is 1 and the other lines are 0. j is the basic state of the system. As a plural, cj is called the probability amplitude of j. cj2=pj, cj=pjeiθj, pj is the probability that the basic state j appears, eiθj=cosθj+isinθj, and θj is the argument. When n=2, 1=10, 2=01, c1=p1eiθ1=p1cosθ1+ip1sinθ1, c2=p2eiθ2=p2cosθ2+ip2sinθ2, and ψ=c1c2. The relationship between 1, 2, and ψ is shown in Figure 1.Figure 1
The relationship between1, 2, and ψ.In quantum theory,⋅ and ⋅ are called bra and ket, respectively [43]. ⋅ is the conjugate transpose of ⋅. For example, when n=2, ψ=c1∗c2∗, where c1∗=p1cosθ1−ip1sinθ1 and c2∗=p2cosθ2−ip2sinθ2. c1∗ and c2∗ are the complex conjugates of c1 and c2. c1c1∗=p1 and c2c2∗=p2.ψψ is called a density operator. ψψ=c1c2c1∗c2∗=c1c1∗c1c2∗c2c1∗c2c2∗=p1c1c2∗c2c1∗p2. 11=1010=1000, 12=1001=0100, 21=0110=0010, and 22=0101=0001. If D=u1u2u3u4, trD=u1+u4 is called the trace of the matrix D.According to the Dirac–Von Neumann axiom set [44], the observable quantity D of the quantum system is a self-adjoint operator of Hn. That is to say that D∗=D. D∗ is the conjugate transpose of D. In the quantum state ψ, the average value of D is D=ψDψ=trDψψ=trDρ. ρ=ψψ is the density operator. trDρ is the trace of the matrix Dρ.According to the type of expression of quantum state, Asano et al. described the cognitive process of decision-makers and Basieva et al. described the belief state of decision-makers [45, 46]. According to these studies, a coal miners’ cognitive state about whether supervisors supervise at t is shown as follows:(2)Atm=y1eiθm1t1m+y2eiθm2t2m,where 1m=10 and 2m=01 represent states where the supervisors supervise and do not supervise, respectively. y1eiθm1t2=y1 and y2eiθm2t2=y2.The supervisors’ cognitive state about whether coal miners operate in violation of the rules att is shown as follows:(3)Btc=x1eiφc1t1c+x2eiφc2t2c,where 1c=10 and 2c=01 represent states where the coal miners operate in violation of the rules and operate according to the rules, respectively.


### 2.3. Subjective Cognition on the Utility of Behavior

We draw lessons from the quantum-like decision-making models proposed by Asano et al. and Basieva et al. [45, 46] and then construct the operator UA1 which can be used to compare A1 with A0. A0 means that no action is taken. uA01 and uA02, which are the utilities of A0 in the supervisory and nonsupervisory states, are both 0. The definition of UA1 is shown as follows:(4)UA1=12DA0⟶A1−DA1⟶A0,where DA0⟶A1=∑k=12∑l=12uA1k−uA0lei∅lA0⟶kA1kmlm and DA1⟶A0=∑l=12∑k=12uA0l−uA1kei∅kA1⟶lA0lmkm. kmlm which is the operator of state transition represents that the state turns to km from lm. uA1k−uA0l represents that when lm turns to km, the coal miner will obtain uA1k while losing the chance to obtain uA0l in the meantime. uA1k−uA0l is called relative utility. DA0⟶A1 is the utility that A1 is relative to A0 and DA1⟶A0 is the utility that A0 is relative to A1. ∅lA0⟶kA1 is the argument. ∅lA0⟶kA1=−∅kA1⟶lA0 represents that uA1k−uA0l and uA0l−uA1k are a utility comparison with an opposite direction. When l=k, ∅lA0⟶kA1=0 represents the utility comparison in the same basic state. When l≠k, ∅lA0⟶kA1=π/2 represents the utility comparison in different basic states.At timet, the average value of UA1 is UA1t=trUA1∗ρAt=1/2trDA0⟶A1∗ρAt−DA1⟶A0∗ρAt. The density operator ρAt=AtmAtm is shown as follows:(5)ρAt=y1y1eiθm1t−θm1ty1y2eiθm1t−θm2ty2y1eiθm2t−θm1ty2y2eiθm2t−θm2t.The matrix forms ofDA0⟶A1 and DA1⟶A0 are shown as follows:(6)DA0⟶A1=uA11−uA01ei∅1A0⟶1A1uA11−uA02ei∅2A0⟶1A1uA12−uA01ei∅1A0⟶2A1uA12−uA02ei∅2A0⟶2A1,DA1⟶A0=uA01−uA11ei∅1A1⟶1A0uA01−uA12ei∅2A1⟶1A0uA02−uA11ei∅1A1⟶2A0uA02−uA12ei∅2A1⟶2A0,andUA1t is shown as follows:(7)UA1t=∑k=12∑l=12uA1k−uA0l∗ykylcosθlA0⟶kA1t,where θlA0⟶kA1t=∅lA0⟶kA1+θmkt−θmlt; uA0l=0. UA1t can be also expressed as(8)UA1t=∑k=12pk′tuA1k,where pk′t=∑l=12ylcosθlA0⟶kA1tyk. p1′t and p2′t are shown as follows: (9)p1′t=y1cosθ1A0⟶1A1t+y1y2cosθ2A0⟶1A1t,(10)p2′t=y2cosθ2A0⟶2A1t+y2y1cosθ1A0⟶2A1t,where p1′t and p2′t can be seen as the subjective judgment of coal mine workers on whether supervisors try hard to supervise. UA1t is the utility perception of A1.In (9) and (10), θ1A0⟶1A1t=∅1A0⟶1A1+θm1t−θm1t=0, θ2A0⟶2A1t=∅2A0⟶2A1+θm2t−θm2t=0, θ2A0⟶1A1t=∅2A0⟶1A1+θm1t−θm2t=π/2+θm1t−θm2t, and θ1A0⟶2A1t=∅1A0⟶2A1+θm2t−θm1t=π/2+θm2t−θm1t.cosθ1A0⟶1A1t=cosθ2A0⟶2A1t=0 and cosθ2A0⟶1A1t=−cosθ1A0⟶2A1t. Equations (9) and (10) can be transformed as(11)p1′t=y1+y1y2cosθAt,p2′t=y2−y2y1cosθAt,where θAt=θ2A0⟶1A1t, p1′t+p2′t=1, and cosθAt is a random variable. When cosθAt=0, p1′t=y1 and p2′t=y2. This represents that the coal miners’ cognition of y1 and y2 has no bias. When cosθAt>0, p1′t>y1, and p2′t<y2, this represents that coal miners overrate y1 and underestimate y2. When cosθAt<0, p1′t<y1, and p2′t>y2, this represents that coal miners underestimate y1 and overrate y2.According to the fluctuation incosθAt in a certain scope, we can describe the fluctuation of the game parties’ cognitive state under the effects of different factors. Furthermore, we can introduce the parameter σ and then use σ∗cosθAt to adjust the fluctuation range of the cognitive state. σ is a real number that is larger than 0. It can be assumed that cosθAt is a stochastic process of the following form:(12)ξt=cosωt+U,where ω is a real number and U is a random variable. For ensuring 0≤pk′t≤1, the following updating is made for pk′t:(13)pkt=0,pk′t<0,1,pk′t>1,pk′t,others.We lety1=0.8, y2=0.2, and σ=0.1. When U obeys the uniform distribution on 0,2π, p1t and p2t under the influence of ξt are shown in Figure 2. p1t is coal miners’ subjective cognition on y1. Under the influence of random factors, coal miners will have different judgments on y1 at different times. Sometimes they overestimate y1 and sometimes underestimate y1. Overestimation and underestimation are uncertain. Processes like this are usually described by stochastic processes [47, 48]. Figure 2 is an example of using the stochastic process to describe the fluctuation of the cognitive state. As can be seen from Figure 2, ξt can describe the uncertain process in which coal miners sometimes overestimate y1 and sometimes underestimate y1.Figure 2
Coal miners’ judgments ony1 and y2under the influence of random factors.After updating, the coal miners’ subjective cognition on the utility ofA1 is(14)UA1t=∑k=12pktuA1k.WhencosθAt>0, p1t≥y1, p2t≤y2, and UA1t=u0+I−fp1t+u0+Ip2t≤u0+I−fy1+u0+Iy2. This represents the state where the coal miners underestimate the utility of A1.WhencosθAt<0, p1t≤y1, p2t≥y2, and UA1t=u0+I−fp1t+u0+Ip2t≥u0+I−fy1+u0+Iy2. This represents the state where the coal miners overrate the utility of A1.Asano et al. regardcosθAt as the decision-maker’s estimate of risk [45]. Basieva et al. think that cosθAt is an interference factor that affects the subjective judgment of decision-makers, such as risk [46]. Thus, UA1t includes the influence of risk factors.


### 2.4. Subjective Cognition on the Utility of Behavior

Coal miners’ subjective cognition on the utility ofA2 is UA2t=∑k=12pk″tuA2k, pk″t=∑l=12ylcosθA′tyk, and θA′t=θlA0⟶kA2t. When l=k, ∅lA0⟶kA2=∅lA0⟶kA1=0. When l≠k, ∅lA0⟶kA2=∅lA0⟶kA1=π/2. Thus, θA′t=∅lA0⟶kA2+θmkt−θmlt=θlA0⟶kA1t=θAt, cosθA′t=cosθAt, and pk″t=pk′t. We use the following principle to judge the behavioral preference of coal miners:(15)A1≻A2,DAt>0,A2≼A1,DAt≤0,where DAt=UA1t−UA2t. UA1t is the coal miners’ subjective cognition on the utility of A1. UA2t is coal miners’ subjective cognition on the utility of A2. When coal miners think that the utility of A1 is greater than the utility of A2, they will prefer A1.According to the above method, we can construct the operatorDBt used to compare B1 with B2:(16)DBt=UB1t−UB2t,where(17)UB1t=∑k=12qktuB1k,UB2t=∑k=12qktuB2k,qkt=0,qk′t<0,1,qk′t>1qk′tOthers.,q1′t=x1+x1x2cosθBt,q2′t=x2−x2x1cosθBt,when DBt>0 and B1≻B2.


### 2.5. Behavior Evolution Model

The replicator dynamics equation proposed by Taylor and Jonker is one of the most important models in evolutionary game theory [49, 50]. It can be used to predict the evolution of population behavior [50]. Suppose that there are two groups, each with two different strategies. The replicator dynamics equation is shown as follows:(18)dxdt=xu1−u¯=x1−xu1−u2,dydt=yv1−v¯=y1−yv1−v2,where u1 and u2 are the utility of individuals in the first group when they choose strategy 1 and strategy 2, respectively. u¯ is the average utility of the first group. x is the proportion of choosing strategy 1. v1 and v2 are the utility of individuals in the second group when they choose strategy 3 and strategy 4, respectively. v¯ is the average utility of the second group. y is the proportion of those choosing strategy 3.We assume that the game between coal miners and supervisors is a repeated game with a random pair for the big group consisting of members whose learning speeds are relatively slow. We assume that coal miners and supervisors choose behaviors according to their subjective judgment. Their subjective cognition on the utility of each behavior isUA1t=uA11p1t+uA12p2t, UA2t=uA21p1t+uA22p2t, UB1t=uB11q1t+uB12q2t, and UB2t=uB21q1t+uB22q2t. According to the replication dynamic equation, the following behavior evolution model can be established:(19)dxdt=x1−xDAt,dydt=y1−yDBt,where(20)x=x1,y=y1,DAt=UA1t−UA2t,DBt=UB1t−UB2t,p1t=0,y+y1−ycosθAt<0,1,y+y1−ycosθAt>1,y+y1−ycosθAt,Others,q1t=0,x+x1−xcosθBt<0,1,x+x1−xcosθBt>1,x+x1−xcosθBt,Others.


## 2.1. Model Assumption

The relevant assumptions in the game between coal miners and supervisors are as follows.Coal miners have two behaviors for selection, operating in violation of the rules and operating according to the rules, which are labelled asA1 and A2, respectively. The probability of choosing A1 is x1 and the probability of choosing A2 is x2, where x1+x2=1. Supervisors have two behaviors for selection, supervising and not supervising, which can be labelled as B1 and B2, respectively. The probability of choosing B1 is y1 and the probability of choosing B2 is y2, where y1+y2=1.Coal miners have an incomplete cognition ofy1 and y2. Supervisors have an incomplete cognition of x1 and x2. Coal miners and supervisors can make behavior choices according to their subjective judgment.Operations in violation of rules can save time. Coal miners can obtainu0 if they operate according to the rules. Meanwhile, if they operate in violation of rules, they can obtain u0+I, where I is the utility of saving time. If supervisors find that coal miners operate in violation of the rules, coal miners will suffer a corresponding punishment f. If supervisors find that coal miners operate according to the rules, coal miners will obtain the award b. The enterprise loss that the operation in violation of the rules leads to will be l. If supervisors supervise, this loss can be avoided. The supervisory cost is c0. The game payment matrix can be shown as in Table 1.Table 1
Game payment matrix between coal miners and supervisors.
-B1B2A1uA11=u0+I−f,uB11=−c0uA12=u0+I,uB21=−lA2uA21=u0+b,uB12=−c0uA22=u0,uB22=0


## 2.2. Description of Cognitive State

In quantum theory, the quantum state can be used to represent the state of the quantum system [43, 44]. Labelled as ψ, the quantum state is the vector representation of Hn, which is the n-dimensional Hilbert Space. ⋅ is the symbol of Dirac [43]. It can be assumed that there is a quantum state ψ∈Hn, ψ=1, and the general definition is shown as follows:(1)ψ=∑j=1ncjj,where j is an n-dimensional column vector. The jth line is 1 and the other lines are 0. j is the basic state of the system. As a plural, cj is called the probability amplitude of j. cj2=pj, cj=pjeiθj, pj is the probability that the basic state j appears, eiθj=cosθj+isinθj, and θj is the argument. When n=2, 1=10, 2=01, c1=p1eiθ1=p1cosθ1+ip1sinθ1, c2=p2eiθ2=p2cosθ2+ip2sinθ2, and ψ=c1c2. The relationship between 1, 2, and ψ is shown in Figure 1.Figure 1
The relationship between1, 2, and ψ.In quantum theory,⋅ and ⋅ are called bra and ket, respectively [43]. ⋅ is the conjugate transpose of ⋅. For example, when n=2, ψ=c1∗c2∗, where c1∗=p1cosθ1−ip1sinθ1 and c2∗=p2cosθ2−ip2sinθ2. c1∗ and c2∗ are the complex conjugates of c1 and c2. c1c1∗=p1 and c2c2∗=p2.ψψ is called a density operator. ψψ=c1c2c1∗c2∗=c1c1∗c1c2∗c2c1∗c2c2∗=p1c1c2∗c2c1∗p2. 11=1010=1000, 12=1001=0100, 21=0110=0010, and 22=0101=0001. If D=u1u2u3u4, trD=u1+u4 is called the trace of the matrix D.According to the Dirac–Von Neumann axiom set [44], the observable quantity D of the quantum system is a self-adjoint operator of Hn. That is to say that D∗=D. D∗ is the conjugate transpose of D. In the quantum state ψ, the average value of D is D=ψDψ=trDψψ=trDρ. ρ=ψψ is the density operator. trDρ is the trace of the matrix Dρ.According to the type of expression of quantum state, Asano et al. described the cognitive process of decision-makers and Basieva et al. described the belief state of decision-makers [45, 46]. According to these studies, a coal miners’ cognitive state about whether supervisors supervise at t is shown as follows:(2)Atm=y1eiθm1t1m+y2eiθm2t2m,where 1m=10 and 2m=01 represent states where the supervisors supervise and do not supervise, respectively. y1eiθm1t2=y1 and y2eiθm2t2=y2.The supervisors’ cognitive state about whether coal miners operate in violation of the rules att is shown as follows:(3)Btc=x1eiφc1t1c+x2eiφc2t2c,where 1c=10 and 2c=01 represent states where the coal miners operate in violation of the rules and operate according to the rules, respectively.


## 2.3. Subjective Cognition on the Utility of Behavior

We draw lessons from the quantum-like decision-making models proposed by Asano et al. and Basieva et al. [45, 46] and then construct the operator UA1 which can be used to compare A1 with A0. A0 means that no action is taken. uA01 and uA02, which are the utilities of A0 in the supervisory and nonsupervisory states, are both 0. The definition of UA1 is shown as follows:(4)UA1=12DA0⟶A1−DA1⟶A0,where DA0⟶A1=∑k=12∑l=12uA1k−uA0lei∅lA0⟶kA1kmlm and DA1⟶A0=∑l=12∑k=12uA0l−uA1kei∅kA1⟶lA0lmkm. kmlm which is the operator of state transition represents that the state turns to km from lm. uA1k−uA0l represents that when lm turns to km, the coal miner will obtain uA1k while losing the chance to obtain uA0l in the meantime. uA1k−uA0l is called relative utility. DA0⟶A1 is the utility that A1 is relative to A0 and DA1⟶A0 is the utility that A0 is relative to A1. ∅lA0⟶kA1 is the argument. ∅lA0⟶kA1=−∅kA1⟶lA0 represents that uA1k−uA0l and uA0l−uA1k are a utility comparison with an opposite direction. When l=k, ∅lA0⟶kA1=0 represents the utility comparison in the same basic state. When l≠k, ∅lA0⟶kA1=π/2 represents the utility comparison in different basic states.At timet, the average value of UA1 is UA1t=trUA1∗ρAt=1/2trDA0⟶A1∗ρAt−DA1⟶A0∗ρAt. The density operator ρAt=AtmAtm is shown as follows:(5)ρAt=y1y1eiθm1t−θm1ty1y2eiθm1t−θm2ty2y1eiθm2t−θm1ty2y2eiθm2t−θm2t.The matrix forms ofDA0⟶A1 and DA1⟶A0 are shown as follows:(6)DA0⟶A1=uA11−uA01ei∅1A0⟶1A1uA11−uA02ei∅2A0⟶1A1uA12−uA01ei∅1A0⟶2A1uA12−uA02ei∅2A0⟶2A1,DA1⟶A0=uA01−uA11ei∅1A1⟶1A0uA01−uA12ei∅2A1⟶1A0uA02−uA11ei∅1A1⟶2A0uA02−uA12ei∅2A1⟶2A0,andUA1t is shown as follows:(7)UA1t=∑k=12∑l=12uA1k−uA0l∗ykylcosθlA0⟶kA1t,where θlA0⟶kA1t=∅lA0⟶kA1+θmkt−θmlt; uA0l=0. UA1t can be also expressed as(8)UA1t=∑k=12pk′tuA1k,where pk′t=∑l=12ylcosθlA0⟶kA1tyk. p1′t and p2′t are shown as follows: (9)p1′t=y1cosθ1A0⟶1A1t+y1y2cosθ2A0⟶1A1t,(10)p2′t=y2cosθ2A0⟶2A1t+y2y1cosθ1A0⟶2A1t,where p1′t and p2′t can be seen as the subjective judgment of coal mine workers on whether supervisors try hard to supervise. UA1t is the utility perception of A1.In (9) and (10), θ1A0⟶1A1t=∅1A0⟶1A1+θm1t−θm1t=0, θ2A0⟶2A1t=∅2A0⟶2A1+θm2t−θm2t=0, θ2A0⟶1A1t=∅2A0⟶1A1+θm1t−θm2t=π/2+θm1t−θm2t, and θ1A0⟶2A1t=∅1A0⟶2A1+θm2t−θm1t=π/2+θm2t−θm1t.cosθ1A0⟶1A1t=cosθ2A0⟶2A1t=0 and cosθ2A0⟶1A1t=−cosθ1A0⟶2A1t. Equations (9) and (10) can be transformed as(11)p1′t=y1+y1y2cosθAt,p2′t=y2−y2y1cosθAt,where θAt=θ2A0⟶1A1t, p1′t+p2′t=1, and cosθAt is a random variable. When cosθAt=0, p1′t=y1 and p2′t=y2. This represents that the coal miners’ cognition of y1 and y2 has no bias. When cosθAt>0, p1′t>y1, and p2′t<y2, this represents that coal miners overrate y1 and underestimate y2. When cosθAt<0, p1′t<y1, and p2′t>y2, this represents that coal miners underestimate y1 and overrate y2.According to the fluctuation incosθAt in a certain scope, we can describe the fluctuation of the game parties’ cognitive state under the effects of different factors. Furthermore, we can introduce the parameter σ and then use σ∗cosθAt to adjust the fluctuation range of the cognitive state. σ is a real number that is larger than 0. It can be assumed that cosθAt is a stochastic process of the following form:(12)ξt=cosωt+U,where ω is a real number and U is a random variable. For ensuring 0≤pk′t≤1, the following updating is made for pk′t:(13)pkt=0,pk′t<0,1,pk′t>1,pk′t,others.We lety1=0.8, y2=0.2, and σ=0.1. When U obeys the uniform distribution on 0,2π, p1t and p2t under the influence of ξt are shown in Figure 2. p1t is coal miners’ subjective cognition on y1. Under the influence of random factors, coal miners will have different judgments on y1 at different times. Sometimes they overestimate y1 and sometimes underestimate y1. Overestimation and underestimation are uncertain. Processes like this are usually described by stochastic processes [47, 48]. Figure 2 is an example of using the stochastic process to describe the fluctuation of the cognitive state. As can be seen from Figure 2, ξt can describe the uncertain process in which coal miners sometimes overestimate y1 and sometimes underestimate y1.Figure 2
Coal miners’ judgments ony1 and y2under the influence of random factors.After updating, the coal miners’ subjective cognition on the utility ofA1 is(14)UA1t=∑k=12pktuA1k.WhencosθAt>0, p1t≥y1, p2t≤y2, and UA1t=u0+I−fp1t+u0+Ip2t≤u0+I−fy1+u0+Iy2. This represents the state where the coal miners underestimate the utility of A1.WhencosθAt<0, p1t≤y1, p2t≥y2, and UA1t=u0+I−fp1t+u0+Ip2t≥u0+I−fy1+u0+Iy2. This represents the state where the coal miners overrate the utility of A1.Asano et al. regardcosθAt as the decision-maker’s estimate of risk [45]. Basieva et al. think that cosθAt is an interference factor that affects the subjective judgment of decision-makers, such as risk [46]. Thus, UA1t includes the influence of risk factors.


## 2.4. Subjective Cognition on the Utility of Behavior

Coal miners’ subjective cognition on the utility ofA2 is UA2t=∑k=12pk″tuA2k, pk″t=∑l=12ylcosθA′tyk, and θA′t=θlA0⟶kA2t. When l=k, ∅lA0⟶kA2=∅lA0⟶kA1=0. When l≠k, ∅lA0⟶kA2=∅lA0⟶kA1=π/2. Thus, θA′t=∅lA0⟶kA2+θmkt−θmlt=θlA0⟶kA1t=θAt, cosθA′t=cosθAt, and pk″t=pk′t. We use the following principle to judge the behavioral preference of coal miners:(15)A1≻A2,DAt>0,A2≼A1,DAt≤0,where DAt=UA1t−UA2t. UA1t is the coal miners’ subjective cognition on the utility of A1. UA2t is coal miners’ subjective cognition on the utility of A2. When coal miners think that the utility of A1 is greater than the utility of A2, they will prefer A1.According to the above method, we can construct the operatorDBt used to compare B1 with B2:(16)DBt=UB1t−UB2t,where(17)UB1t=∑k=12qktuB1k,UB2t=∑k=12qktuB2k,qkt=0,qk′t<0,1,qk′t>1qk′tOthers.,q1′t=x1+x1x2cosθBt,q2′t=x2−x2x1cosθBt,when DBt>0 and B1≻B2.


## 2.5. Behavior Evolution Model

The replicator dynamics equation proposed by Taylor and Jonker is one of the most important models in evolutionary game theory [49, 50]. It can be used to predict the evolution of population behavior [50]. Suppose that there are two groups, each with two different strategies. The replicator dynamics equation is shown as follows:(18)dxdt=xu1−u¯=x1−xu1−u2,dydt=yv1−v¯=y1−yv1−v2,where u1 and u2 are the utility of individuals in the first group when they choose strategy 1 and strategy 2, respectively. u¯ is the average utility of the first group. x is the proportion of choosing strategy 1. v1 and v2 are the utility of individuals in the second group when they choose strategy 3 and strategy 4, respectively. v¯ is the average utility of the second group. y is the proportion of those choosing strategy 3.We assume that the game between coal miners and supervisors is a repeated game with a random pair for the big group consisting of members whose learning speeds are relatively slow. We assume that coal miners and supervisors choose behaviors according to their subjective judgment. Their subjective cognition on the utility of each behavior isUA1t=uA11p1t+uA12p2t, UA2t=uA21p1t+uA22p2t, UB1t=uB11q1t+uB12q2t, and UB2t=uB21q1t+uB22q2t. According to the replication dynamic equation, the following behavior evolution model can be established:(19)dxdt=x1−xDAt,dydt=y1−yDBt,where(20)x=x1,y=y1,DAt=UA1t−UA2t,DBt=UB1t−UB2t,p1t=0,y+y1−ycosθAt<0,1,y+y1−ycosθAt>1,y+y1−ycosθAt,Others,q1t=0,x+x1−xcosθBt<0,1,x+x1−xcosθBt>1,x+x1−xcosθBt,Others.


## 3. Behavior Choice of Game Parties

### 3.1. Behavior Choice at Timet

The relevant assumptions in the game between coal miners and supervisors are as follows:Takingu0=7, I=3, f=6, b=1, c0=3, and l=5 as an example, we analyze the behavior choice of coal miners and supervisors in different situations. Substituting the above parameters into DAt and DBt, we can conclude DAt=−4p1t+31−p1t and DBt=2q1t−31−q1t. When DAt>0, coal miners will prefer the operation in violation of the rules. When DBt>0, supervisors will prefer supervision. With the change in θAt, DAt is shown in Figure 3(a). With the change in θBt, DBt is shown in Figure 3(b).Figure 3
The change process of preference of coal miners and supervisors. (a) The change process of coal miners’ preference with the change inθAt. (b) The change process of supervisors’ preference with the change in θBt.
(a)(b)According to Figure3(a), whether coal miners choose to operate in violation of rules depends on the probability of supervising and cosθAt. When cosθAt<0, the coal miners will overrate the utility of the operation in violation of the rules. When θAt remains constant, the smaller the probability of supervising, the more likely it is that coal miners will operate in violation of the rules. When y remains constant, coal miners will be more likely to operate in violation of the rules if they overrate the utility of the operation in violation of the rules seriously. According to Figure 3(b), when x and cosθBt are relatively large, supervisors will be more likely to supervise.


### 3.2. Behavior Choice with the Change int

The behavior evolution process is discussed in the two following situations. The first situation is complete cognition. In the condition of complete cognition,cosθAt and cosθBt are 0. The behavior evolution process is shown in Figure 4. x is the probability of choosing A1, which is also the violation probability. y is the probability of choosing B1, which is also the supervision probability.Figure 4
The evolution process in the condition of complete cognition. (a) The evolution process when the initial state isx0=0.5 andy0=0.3. (b) The evolution process when the initial state isx0=0.2 and y0=0.2.
(a)(b)According to Figure4, different original states will have different evolution processes. The evolution process shown in Figure 4 is not stable. As can be seen from Figure 4, an increase in the violation probability will increase the supervision probability. When the probability of supervision increases to a certain extent, the violation probability will change from the increase to decrease. When the violation probability decreases to a certain extent, the probability of supervision will decrease. When the probability of supervision decreases to a certain extent, the violation probability will change from the decrease to increase. In Figure 4(a), when t=15, x=0.77 and y=0.42. At this time, x will change from the increase to decrease. When t=16, x=0.58 and y=0.58. At this time, y will change from the increase to decrease. When t=17.4, x=0.42 and y=0.42. At this time, x will change from the decrease to increase.The second situation is fluctuation in the game parties’ cognitive state. It can be assumed thatcosθAt fluctuates in fA0,fA1 and cosθBt fluctuates infB0,fB1. We use cosθAt=fA0+fA1−fA0ξt and cosθBt=fB0+fB1−fB0ξt to describe the fluctuation in cosθAt and cosθBt in fA0,fA1 andfB0,fB1, respectively, where ξt=cosωt+U, and U obeys the uniform distribution of 0,2π. Through specific examples, we adopt the emulation mode to analyze the impact of the fluctuation in the game parties’ cognitive state on the evolution process and the stability of the balance points. If x∗,y∗ satisfies x∗1−x∗DAt=0 and y∗1−y∗DBt=0, x∗,y∗ is the balance point of (19). During the analysis, we use σ∗cosθAt to analyze the behavior evolution process after the amplification of the fluctuation.We takext=0=0.5 andyt=0=0.3 as the original state of the evolution system and then analyze the impact of the fluctuation in the game parties’ cognitive state on the behavior evolution process. When fA0<0<fA1 and fB0<0<fB1, the behavior evolution process is shown in Figure 5, where σ=2. According to Figure 5, if the fluctuation scope of cosθAt and cosθBt is bigger, the fluctuation scope of x and y will be bigger. If the fluctuation scope ofcosθAt and cosθBt is smaller, the behavior evolution process will be more similar to the behavior evolution process in the condition of complete cognition.Figure 5
The evolution processes when game parties’ cognitive state fluctuates in different ranges. (a) The behavior evolution process when the fluctuation range isfA0,fA1=−1,1 and fB0,fB1=−1,1. (b) The behavior evolution process when the fluctuation range is fA0,fA1=−0.7,0.7 and fB0,fB1=−0.7,0.7. (c) The behavior evolution process when the fluctuation range is fA0,fA1=−0.4,0.4 and fB0,fB1=−0.4,0.4. (d) The behavior evolution process when the fluctuation range is fA0,fA1=−0.1,0.1 and fB0,fB1=−0.1,0.1.
(a)(b)(c)(d)WhenfA1<0, fB1<0 or fA0>0, fB0>0, the behavior evolution process is shown in Figure 6. In Figure 6, x0,y0 is the behavior evolution process in the condition of complete cognition. Figure 6(a) is the behavior evolution process when cosθAt<0 and cosθBt<0. Figure 6(b) is the behavior evolution process when cosθAt>0 and cosθBt>0. cosθAt<0 represents that coal miners overrate the utility of A1. According to Figure 6(a), coal miners are more likely to operate in violation of the rules when they overrate the utility of A1. According to Figure 6(b), coal miners are more likely to operate according to the rules when they underestimate the utility of A1. The fluctuation in the game parties’ cognitive state can affect the process of behavior evolution.Figure 6
Comparison of evolution processes under complete cognition and cognitive state fluctuation. (a) The evolution process under complete cognition and the evolution process whenfA0,fA1=−0.8,−0.6 and fB0,fB1=−0.8,−0.6. (b) The evolution process under complete cognition and the evolution process whenfA0,fA1=0.6,0.8 and fB0,fB1=0.6,0.8.
(a)(b)


## 3.1. Behavior Choice at Timet

The relevant assumptions in the game between coal miners and supervisors are as follows:Takingu0=7, I=3, f=6, b=1, c0=3, and l=5 as an example, we analyze the behavior choice of coal miners and supervisors in different situations. Substituting the above parameters into DAt and DBt, we can conclude DAt=−4p1t+31−p1t and DBt=2q1t−31−q1t. When DAt>0, coal miners will prefer the operation in violation of the rules. When DBt>0, supervisors will prefer supervision. With the change in θAt, DAt is shown in Figure 3(a). With the change in θBt, DBt is shown in Figure 3(b).Figure 3
The change process of preference of coal miners and supervisors. (a) The change process of coal miners’ preference with the change inθAt. (b) The change process of supervisors’ preference with the change in θBt.
(a)(b)According to Figure3(a), whether coal miners choose to operate in violation of rules depends on the probability of supervising and cosθAt. When cosθAt<0, the coal miners will overrate the utility of the operation in violation of the rules. When θAt remains constant, the smaller the probability of supervising, the more likely it is that coal miners will operate in violation of the rules. When y remains constant, coal miners will be more likely to operate in violation of the rules if they overrate the utility of the operation in violation of the rules seriously. According to Figure 3(b), when x and cosθBt are relatively large, supervisors will be more likely to supervise.


## 3.2. Behavior Choice with the Change int

The behavior evolution process is discussed in the two following situations. The first situation is complete cognition. In the condition of complete cognition,cosθAt and cosθBt are 0. The behavior evolution process is shown in Figure 4. x is the probability of choosing A1, which is also the violation probability. y is the probability of choosing B1, which is also the supervision probability.Figure 4
The evolution process in the condition of complete cognition. (a) The evolution process when the initial state isx0=0.5 andy0=0.3. (b) The evolution process when the initial state isx0=0.2 and y0=0.2.
(a)(b)According to Figure4, different original states will have different evolution processes. The evolution process shown in Figure 4 is not stable. As can be seen from Figure 4, an increase in the violation probability will increase the supervision probability. When the probability of supervision increases to a certain extent, the violation probability will change from the increase to decrease. When the violation probability decreases to a certain extent, the probability of supervision will decrease. When the probability of supervision decreases to a certain extent, the violation probability will change from the decrease to increase. In Figure 4(a), when t=15, x=0.77 and y=0.42. At this time, x will change from the increase to decrease. When t=16, x=0.58 and y=0.58. At this time, y will change from the increase to decrease. When t=17.4, x=0.42 and y=0.42. At this time, x will change from the decrease to increase.The second situation is fluctuation in the game parties’ cognitive state. It can be assumed thatcosθAt fluctuates in fA0,fA1 and cosθBt fluctuates infB0,fB1. We use cosθAt=fA0+fA1−fA0ξt and cosθBt=fB0+fB1−fB0ξt to describe the fluctuation in cosθAt and cosθBt in fA0,fA1 andfB0,fB1, respectively, where ξt=cosωt+U, and U obeys the uniform distribution of 0,2π. Through specific examples, we adopt the emulation mode to analyze the impact of the fluctuation in the game parties’ cognitive state on the evolution process and the stability of the balance points. If x∗,y∗ satisfies x∗1−x∗DAt=0 and y∗1−y∗DBt=0, x∗,y∗ is the balance point of (19). During the analysis, we use σ∗cosθAt to analyze the behavior evolution process after the amplification of the fluctuation.We takext=0=0.5 andyt=0=0.3 as the original state of the evolution system and then analyze the impact of the fluctuation in the game parties’ cognitive state on the behavior evolution process. When fA0<0<fA1 and fB0<0<fB1, the behavior evolution process is shown in Figure 5, where σ=2. According to Figure 5, if the fluctuation scope of cosθAt and cosθBt is bigger, the fluctuation scope of x and y will be bigger. If the fluctuation scope ofcosθAt and cosθBt is smaller, the behavior evolution process will be more similar to the behavior evolution process in the condition of complete cognition.Figure 5
The evolution processes when game parties’ cognitive state fluctuates in different ranges. (a) The behavior evolution process when the fluctuation range isfA0,fA1=−1,1 and fB0,fB1=−1,1. (b) The behavior evolution process when the fluctuation range is fA0,fA1=−0.7,0.7 and fB0,fB1=−0.7,0.7. (c) The behavior evolution process when the fluctuation range is fA0,fA1=−0.4,0.4 and fB0,fB1=−0.4,0.4. (d) The behavior evolution process when the fluctuation range is fA0,fA1=−0.1,0.1 and fB0,fB1=−0.1,0.1.
(a)(b)(c)(d)WhenfA1<0, fB1<0 or fA0>0, fB0>0, the behavior evolution process is shown in Figure 6. In Figure 6, x0,y0 is the behavior evolution process in the condition of complete cognition. Figure 6(a) is the behavior evolution process when cosθAt<0 and cosθBt<0. Figure 6(b) is the behavior evolution process when cosθAt>0 and cosθBt>0. cosθAt<0 represents that coal miners overrate the utility of A1. According to Figure 6(a), coal miners are more likely to operate in violation of the rules when they overrate the utility of A1. According to Figure 6(b), coal miners are more likely to operate according to the rules when they underestimate the utility of A1. The fluctuation in the game parties’ cognitive state can affect the process of behavior evolution.Figure 6
Comparison of evolution processes under complete cognition and cognitive state fluctuation. (a) The evolution process under complete cognition and the evolution process whenfA0,fA1=−0.8,−0.6 and fB0,fB1=−0.8,−0.6. (b) The evolution process under complete cognition and the evolution process whenfA0,fA1=0.6,0.8 and fB0,fB1=0.6,0.8.
(a)(b)


## 4. Behavior Control Strategy

For controlling the operation in violation of the rules, it is necessary to let0,0 or 0,1 become the stable point of the evolution system. We first assume that coal miners and supervisors can achieve complete cognition and then use the Jacobi matrix to analyze the stability of the balance points of the evolution system [50]. Finally, we analyze whether the fluctuation of game parties’ cognitive state affects the control effect of the control strategy. Label dx/dt and dy/dt as fx,y and x,y, respectively. The Jacobi matrix is shown as follows: (21)J=fx′x,yfy′x,ygx′x,ygy′x,y,where(22)fx′x,y=1−2xuA11−uA21y+uA12−uA221−y,fy′x,y=x1−xuA11−uA21−uA12−uA22,gy′x,y=1−2yuB11−uB21x+uB12−uB221−x,gx′x,y=y1−yuB11−uB21−uB12−uB22,for the balance point x0,y0, when detJ=fx′x0,y0gy′x0,y0−fy′x0,y0gx′x0,y0>0, trJ=fx′x0,y0+gy′x0,y0<0, x0,y0 is a stable point of evolution.The condition where0,0 is a stable point has uA12−uA22uB12−uB22>0 and uA12−uA22+uB12−uB22<0. The condition where 0,1 is a stable point has uA11−uA21uB12−uB22<0 and uA11−uA21−uB12−uB22<0.BecauseuA12−uA22uB12−uB22=−Ic0<0, 0,0 is an unstable point. When I−f−b>0, uA11−uA21−uB12−uB22=I−f−b+c0>0. When I−f−b<0, uA11−uA21uB12−uB22=I−f−b−c0>0. 0,1 is also an unstable point. Therefore,0,0 or 0,1 cannot become a stable point only by “punishing coal miners for operating in violation of rules” and “awarding coal miners for operating according to the rules.”To make0,0 or 0,1 become a stable point, we introduce the “punishing supervisors for not supervising.” The combined strategy consisting of “punishing coal miners for operating in violation of rules,” “ awarding coal miners for operating according to the rules,” and “punishing supervisors for not supervising” is adopted to control the operation in violation of the rules. It can be assumed that the punishment that the supervisor will suffer if he does not supervise is f'. uB12−uB22=f′−c0.Whenf′−c0>0, uA12−uA22+uB12−uB22=I+f′−c0>0. When f′−c0<0, uA12−uA22uB12−uB22=If′−c0<0. Hence, the introduction of the strategy of “punishing supervisors for not supervising” cannot make 0,0 become a stable point. When uA11−uA21uB12−uB22=I−f−bf′−c0<0 and uA11−uA21−uB12−uB22=I−f−b−f′−c0<0, 0,1 is a stable point. At this point, the operation in violation of the rules on the part of the coal miner can be controlled. For this case, I−f−b=−4<0 and c0=3. Thus, when f′>3, 0,1 is a stable point.According to the impact of the fluctuation in the game parties’ cognitive state on behavior evolution, coal miners are more likely to choose to operate in violation of the rules when they underestimatey and the supervisors underestimate x. We let fA0,fA1=−1,−0.6 and fB0,fB1=−1,−0.6 and use σ∗cosθAt to amplify the coal miners’ underestimated degree of y and use σ∗cosθBt to amplify the supervisors’ underestimated degree of x. When f′<c0, the behavior evolution process is shown as in Figure 7(a). When f′>c0 and σ have different values, the behavior evolution process is shown in Figures 7(b)–7(d). In Figure 6, x0,y0 is the evolution process in the condition of complete cognition.Figure 7
The evolution processes under different levels of punishment and cognitive fluctuations. (a) The behavior evolution process whenf′=2.5 and σ=1. (b) The behavior evolution process when f′=3.5 and σ=1. (c) The behavior evolution process when f′=3.5 and σ=2. (d) The behavior evolution process when f′=3.5 and σ=10.
(a)(b)(c)(d)According to Figure7, the fluctuation in the game parties’ cognitive state can affect the time when the final effect of the control strategy is achieved. By using σ∗cosθAt and σ∗cosθBt to let the cognitive state have a fluctuation in different scopes, the system will finally stabilize in the point of 0,1. This indicates that cosθAt and cosθBt cannot affect the stability of 0,1. The fluctuation in the game parties’ cognitive state does not affect the eventual effect of the control strategy, but some fluctuations will extend the time when the final effect is achieved.


## 5. Discussion and Conclusions

The traditional game model is based on the assumption that both sides of the game are completely rational, which cannot reflect the subjective cognition of game parties on the utility of behavior. In the game between coal miners and supervisors, their cognitive ability is limited. Under the influence of random factors, the cognition of coal miners on the utility of the operation in violation of the rules is biased, sometimes overestimating its utility and sometimes underestimating its utility. When coal miners think that the utility of the operation in violation of the rules is higher than that of operations according to the rules, they will prefer the operation in violation of the rules. The subjective cognition and preference of behaviors can be described by (14) and (15). Actually, the cognition and preference of coal miners to the operation in violation of the rules are not fixed. Under the influence of random factors, the cognitive state and behavior preference of coal miners have stochastic volatility. (14) and (15) can describe this stochastic volatility. The description method of subjective cognition can be applied to the behavior analysis of other occupational groups.In the game process of coal miners and supervisors, the game parties will continue to learn. Through learning, their cognition will change. The change in cognition will affect their behavior choice. Based on the evolutionary game theory, the behavior evolution model, which is shown in (18), can describe the behavior evolution process of coal miners and supervisors. Compared with previous evolutionary game models, (18) embeds a stochastic process. This provides a reference for the analysis of the behavior evolution process under the interference of stochastic factors. The probability of supervisors’ choice of supervision and the cognition of coal miners on the probability of supervision will affect the cognition of coal miners on the utility of operation in violation of the rules. When the probability of supervision is low and the coal miners overrate the utility of the operation in violation of the rules, the coal miners will tend to choose the operation in violation of the rules, as is shown in Figure 3(a).The behaviors of coal miners and supervisors influence each other. When the fluctuation in the cognitive state is not considered and the probability of operation in violation of the rules is low, supervisors will tend to choose not to supervise. This will reduce the probability of supervision. The decrease in the probability of supervision will make coal miners tend to choose the operation in violation of the rules. The increase in the probability of operation in violation of the rules will make supervisors tend to choose supervision. The behavior relationship between coal miners and supervisors is shown in Figure4. When considering the fluctuation of cognitive state, the behavior evolution process of coal miners and supervisors will become complex, as is shown in Figure 5.Different coal miners have different cognitions. No matter how the cognitive state fluctuates, some coal miners may always overestimate the utility of the operation in violation of the rules. Compared with the coal miners without cognitive bias, they are more inclined to choose the operation in violation of the rules. This also makes supervisors more inclined to choose supervision. The behavior evolution process in this case is shown in Figure6(a). Some coal miners may always underestimate the effectiveness of the operation in violation of the rules. Compared with the coal miners without cognitive bias, they prefer to operate according to the rules. This makes supervisors more inclined to choose not to supervise. The behavior evolution process in this case is shown in Figure 6(b).It can be seen from Figures4–6 that managing the behavior of the coal miners only cannot completely control the operation in violation of the rules. It is also necessary to manage the behavior of supervisors. The combined strategy consisting of “punishing coal miners for operating in violation of rules,” “awarding coal miners for operating according to the rules,” and “punishing supervisors for not supervising” can make coal miners choose the operation according to rules and supervisors choose supervision. The fluctuation of cognitive states of coal miners and supervisors will not affect the final effect of this strategy.The description method of subjective cognition can describe the cognitive state, where coal miners and supervisors overestimate or underestimate the utility of behaviors under the influence of random factors. The behavior evolution model with random variables can describe the dynamic evolution process of the behavior choice of coal miners and supervisors. The behavior evolution model can be used to analyze the behavior relationship between coal miners and supervisors under the conditions of random fluctuations in cognitive state. In coal mine production, the behavior of coal miners is also affected by the behavior of their colleagues and production commanders. In the future, we will establish a behavior evolution model that includes multiple game players.



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