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Research ArticleStability of the Evolutionary Game System
andControl Strategies of Behavior Instability in CoalMine Safety
Management
Xinhua Wang ,1 Rongwu Lu ,2 Hao Yu ,1 and Dan Li 1
1College of Economics and Management, Shandong University of
Science and Technology, Qingdao 266590, China2College of
Mathematics and Systems Science, Shandong University of Science and
Technology, Qingdao 266590, China
Correspondence should be addressed to Rongwu Lu;
[email protected]
Received 26 September 2018; Revised 29 December 2018; Accepted
16 January 2019; Published 3 February 2019
Guest Editor: Baltazar Aguirre-Hernandez
Copyright © 2019 Xinhua Wang et al. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
In this paper, we try to find the right controlmethod for the
game behavior instability in coalmine safety management.Through
theanalysis and comparison of the system stabilitywith inflexible
andflexible costs and penalties, it can be concluded that the
dynamicalgame systemwith flexible costs (incentive rewards) and
flexible penalty mechanism can significantly reduce the dynamics of
unsafebehaviors in coal mine safety supervision. A combined
mechanism of incentive rewards and flexible penalty is put forward
toimprove the stability of the dynamical system and control the
instability of behaviors effectively. The results of model
simulationshow that the combinedmechanism has very good property
and can optimize and control the instability of behaviors and
strategiesof the interested parties. Based on the theoretical
conclusions, some control strategies and policy advice are proposed
for theimprovement of the system and measures of safety management
for government departments.
1. Introduction
The frequent occurrence of coal mine safety accidents
alwaysmakes the safety production situation in China become
thefocus of attention at home and abroad. Management chaos,illegal
operations, poor policy implementation, and regu-latory dislocation
have long become the prominent factorsaffecting the safety of coal
mine production. The traditionalsafety management is mainly based
on experience-orientedadministrative regulations and legislations.
However, due tothe differences of region, staff and technical
level, and soon, coal mine safety management has great complexity.
Theeffectiveness of some rules and regulations is temporary
andlimited in the implementation process.
In order to ensure that the safety legislation is
closelyintegrated with management practice, many scholars
havedeeply analyzed the influence factors of coal mine
safetysupervision and studied the relationship between the
occur-rences of unsafe behaviors and the interest distribution of
coalmine safety [1–6].
Through the previous studies on the game of safetysupervision,
it is found that the dynamical system of safetymanagement game
model hardly has the stability and self-control generally, and the
game behaviors of the playersalways have repeated volatility [7,
8]. The volatility of thebehaviors may sometimes provide wrong
information andcause unrealistic or even error related decisions
andmeasuresof coal mine safety management, which will seriously
affectthe efficiency of safety management.
In the previous studies of evolutionary game model incoalmine
safetymanagement, we have obtained the followingreplicator dynamics
[9, 10]:̇𝑥1 = 𝑥1 (1 − 𝑥1) [(𝑥2 + (1 − 𝑥2) 𝑥3) 𝑓1 − 𝑐1] ;̇𝑥2 = 𝑥2 (1
− 𝑥2) [(1 − 𝑥1) (𝑓1 + 𝑥3𝑓2) − 𝑐2] ;̇𝑥3 = 𝑥3 (1 − 𝑥3) [(1 − 𝑥1) (1 −
𝑥2) (𝑓1 + 𝑓2) − 𝑐3] (1)where 𝑐𝑖(𝑖 = 1, 2, 3) is the safety cost
(input) of coal miners(denoted as player I), safety inspection
groups inside the coal
HindawiComplexityVolume 2019, Article ID 6987427, 14
pageshttps://doi.org/10.1155/2019/6987427
http://orcid.org/0000-0003-3395-5615http://orcid.org/0000-0002-2281-7076http://orcid.org/0000-0002-8553-4997http://orcid.org/0000-0001-8410-4867https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/6987427
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2 Complexity
01
0.2
0.4
1
0.6
0.8
0.8
0.5 0.6
1
0.40.20 0 x1x2
x3
０0０1
０2
０3
０0(0.5, 0.3, 0.2)０1(0.7, 0.5,
0.3)０2(0.3,0.5,0.7)０3(0.5,0.7,0.9)
Phase diagram of tripartite game with different initial
values
(a)
0 50 100 150 200 250 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x1x2x3
Time series diagram in general with initial value ０1(0.7 0.5
0.3)
(b)
Figure 1: Phase graph of dynamical system (1) and time series
diagram of players’ strategy selection probabilities. Values
ofmodel parameters:(𝑐1, 𝑐2, 𝑐3, 𝑓1, 𝑓2) = (0.5, 0.2, 0.3, 1,
1.2).mine (player II), and safety regulatory departments of
localgovernment (player III). 𝑓𝑖(𝑖 = 1, 2) is the penalty
index(fine) when player I or player II has made mistakes
(unsafebehaviors) and is checked out, respectively.
According to the actual situation of China’s coal mineproduction
and coal mine safety regulation, we assumethat the parameters
described above satisfy the followingconditions:
(i) 𝑐2 ≤ 𝑐3 ≤ 𝑐1 ≤ 1;(ii) 𝑓𝑖 ≥ 1, 𝑖 = 1, 2.Phase diagram
analysis and time series diagram are
commonly used methods to study the stability of
nonlineardynamical systems. Phase diagram analysis method has
thecharacteristic of vivid intuition, which represents all
possiblestates of the dynamic system. Phase diagram analysis
greatlypromotes the wide application of differential
dynamicalsystem in many disciplines [11–15]. Time series graph
depictsthe process of function changing with time variable 𝑡,
whichcan simply and clearly describe the evolution process,
chaos,and synchronization of differential dynamical systems
[16–19]. In addition to the eigenvalue method, a large part ofmodel
analysis in this study uses these two methods.
The gamemodel contains toomany variables and param-eters, it is
difficult to obtain the model’s analytical solution[10]. In this
paper, we mainly use numerical simulationmethod to analyze the
evolutionary process. So, it is veryimportant to properly set the
initial values of parametersand variables. Coal mine safety
management involves var-ious types of coal production enterprises
with differenttechnological development levels in different
regions, andthe coal production process and safety management
statusof these enterprises themselves have great differences.
Thecorresponding parameters and initial values of variables
will
not be the same and changeless. Supposing that the
relevantmethod is applied in a specific coal production enterprise
or aspecific mining area, the model parameters and initial valuesof
variables should be deduced and estimated accordingto the actual
situation of coal production. In this way, themodel analysis will
be more pertinent and the correspondingconclusions will be more
accurate. In this paper, we merelyset the values of model
parameters and the initial value of thevariables theoretically
reasonably but do not make detailedand specific statistical
reasoning.
In order to reflect the dependence of the stability ofdynamic
system on the initial value, we choose severaldifferent initial
values of variables (each player’s strategyselection probabilities
of 𝑥1, 𝑥2, and 𝑥3) in the system.For example, we set the value of
(𝑥1(0), 𝑥2(0), 𝑥3(0)) to𝑃0(0.5, 0.3, 0.2), 𝑃1(0.7, 0.5, 0.3),
𝑃2(0.3, 0.5, 0.7), 𝑃3(0.5, 0.7),and 𝑃3(0.5, 0.7, 0.9),
respectively, in Figure 1(a).
Through the stability analysis of the dynamical system (1)(see
Figure 1, where Figure 1(a) is phase graph of dynamicalsystem and
Figure 1(b) demonstrates the time series of play-ers’ strategy
selection probabilities), we can find that thereare repeated
fluctuations in the game process of multipartysafety management. It
agrees with the actual situation ofsafety management to a certain
extent. If the coal miners donot adopt a safety production strategy
(or the proportion ofunsafe strategies in the production worker
group is larger)at the beginning, the regulators will increase the
level ofsafety supervision. With the increase of supervision
intensity,coal miners will take safe production behaviors (or
increasethe proportion of safety behavior choice) or even
completelydo not choose unsafe behavior. Then coal production
safetyconditions will improve, and then regulators will
reducesupervision, and then the unsafe behaviors will bemore
likelyto happen. Over time, coal mine safety will always be in
theprocess of cyclical fluctuations. This phenomenon appears
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Complexity 3
not only in coal mine safety management but also in
otherprocesses of production safety management [20–23].
The game model of safety regulation does not haveglobal
stability under general conditions [10]. Most of theexisting
studies proposed the system and measures for safetymanagement from
the points of view of the policy makers,according to the specific
situation (such as types of accidentsor illegal behaviors) [24–28].
In this way, even if the relevantpolicies and measures have played
a certain role in a shortperiod, in a short period, the long-term
effectiveness can notbe guaranteed because of the lack of control
(inhibition oravoidance) of behavior volatility of the interested
groups.Therepeated fluctuations and volatility (instability) often
providethe decision-makers with wrong information, which will
leadto unrealistic or even error decisions and measures of coalmine
safety management, and seriously affect the efficiencyof coal mine
safety management.
In order to definitely improve the efficiency of
safetyman-agement, some control methods are needed for
restrainingthe instability of players’s dynamical behavior and
strate-gies in the process of coal mine safety supervision. So,
inthis paper, we proceed from the relationship between themodel
parameters (or variables) and explore the functionarymechanism of
those model parameters (or variables) onthe stability of game
behaviors. As the design mechanismof model parameters and variables
(i.e., the formulationof incentive and penalty measures in the
process of safetysupervision) is found out, then we could put
forward thecontrol methods and control strategies for the
behaviorinstability. First, the stability of the dynamical system
of thegame model is studied in the cases of the generally
inflexiblecost subsidies and penalties index. Second, the stability
ofthe dynamical system is considered under the flexible
costfunctions and flexible penalty functions. We believe that
therelevant dynamic control methods and strategies will providea
theoretical basis for the regulation and control of coal minesafety
management system and measures.
The rest of this paper is organized as follows. In Section 2,we
analyze and compare the system stability of evolutionarygame models
in four cases: inflexible cost subsidies, flexiblecosts (incentive
rewards), inflexible penalties, and flexiblepenalties. In Section
3, we carry out further theoreticalanalysis under the
combinedmechanismof incentive rewardsand flexible penalties. Then,
in Section 4, we put forwardsome control strategies to suppress the
behavior instability insafety management system. Our paper is
finally concluded inSection 5.
2. Analysis on Stability of the DynamicalGame System
2.1. Stability of the Dynamical Game System with InflexibleCost
Subsidies. The inflexible cost subsidy mentioned in thisarticle
refers to those policy oriented fixed subsidies for theworkers
which are undifferentiated and averagely shared,such as the
underground allowances and the post allowances,etc. The increase of
inflexible subsidies is equivalent to whenthe reduction of the
safety production costs of coal miners,
or the safety inspection cost of safety inspection team, or
thesafety supervision cost of the government department (𝑐𝑖, 𝑖 =1,
2, 3) becomes smaller. Therefore, based on the generalsituation as
shown in Figure 1, we decrease the values of𝑐1, 𝑐2, and 𝑐3,
respectively, or synchronously to comparativelyanalyze the
stability of dynamical system of the game model.
Here, Figures 2(a), 2(b), and 2(c) are time series graphs
ofstrategy selection probabilities (𝑥1, 𝑥2, 𝑥3) when 𝑐1
decreasesgradually. Similarly, we get Figures 2(d), 2(e), and 2(f)
when𝑐2 decreases gradually and Figures 2(g), 2(g), and 2(i) when𝑐3
decreases gradually. Figures 3(a), 3(b), and 3(c) show thetime
series of strategy selection probabilities when 𝑐1, 𝑐2, and𝑐3
decreases synchronously.
As can be seen from Figures 2 and 3, the volatilityof the game
system has been reduced when the value of𝑐1, 𝑐2, or 𝑐3 becomes
smaller. However, the stability of thegame system has not changed
substantially. Under normalcircumstances (see Figures 2(a), 2(d),
and 2(g)), 𝑥1 still hasno asymptotic stability (controllability).
Only under someextreme conditions such as 𝑐1 = 0, 𝑐2 < 0, or 𝑐3
< 0which are difficult to reach in practice, will 𝑥1
(probability ofcoal miners selecting safe production behaviors)
quickly andsteadily approach 1 (see Figures 2(c), 2(f), 2(i), and
3(c)).
So, on the premise of guaranteeing the economic benefitsof coal
enterprises, it is necessary to suitably raise the policysubsidies
and the welfare level of the employees, which willrelevantly reduce
the safety input cost of the interested groupsand promote them to
increase investment in production.Then the production safety of
coal enterprises and safetymanagement work can be run at a higher
level.
2.2. Stability of the Dynamical Game System with Flexible
Cost(Incentive Rewards). In Section 2.1, we found that generalcost
reductions do not completely bring down the volatilityof the
dynamical system of game models. Salary is the sumof various forms
of remuneration or return obtained by theemployees of the
enterprise for providing labor or labor totheir work units,
including job salary, performance salary,bonus (month award,
quarterly award, annual award, etc.),allowance, labor bonus,
welfare, and so on. Salary manage-ment is the key to reflect the
vital interests and social valuesof employees, and also it is the
internal power of enterpriseoperation [29, 30], so that the
incentive salary system mustadhere to the principle of pay
according to performance.The implementation of performance
compensation should bebased on a scientific performance evaluation
system. Oth-erwise, the fairness of performance compensation will
notreach the purpose of motivating employees. In recent
years,various enterprises (including various types of enterprises
inthe coal industry) paymore andmore thewages of
employees,quarterly awards, year-end awards, and other variable
incometo the total income of workers.The discretionary variable
part(performance salary, bonus, etc.) plays a very important rolein
stimulating theworking enthusiasm in coal enterprises andimproving
safety efficiency.
Therefore, performance salary and bonus and some otherincentive
rewards should be taken into account of the players’costs of safety
production. Those who perform better will getmore rewards, and
their safety cost inputs will be reduced
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4 Complexity
0 50 100 150 200 250 300
Time series diagram when ＝1 =0.2
Ｒ1
Ｒ2Ｒ3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a) 𝑐1 = 0.2
0 50 100 150 200 250 300
Time series diagram when ＝1 =0.05
Ｒ1
Ｒ2Ｒ3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) 𝑐1 = 0.05
0 50 100 150 200 250 300
Time series diagram when ＝1 =0
Ｒ1
Ｒ2Ｒ3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(c) 𝑐1 = 0
0 50 100 150 200 250 300
Time series diagram when ＝2 =0
Ｒ1
Ｒ2Ｒ3
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(d) 𝑐2 = 0.1
0 50 100 150 200 250 300
Time series diagram when ＝2 =0
Ｒ1
Ｒ2Ｒ3
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(e) 𝑐2 = 0
0 50 100 150 200 250 300
Time series diagram when ＝2 =-0.1
Ｒ1
Ｒ2Ｒ3
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(f) 𝑐2 = −0.1
0 50 100 150 200 250 300
Time series diagram when ＝3 =0.1
Ｒ1
Ｒ2Ｒ3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(g) 𝑐3 = 0.1
0 50 100 150 200 250 300
Time series diagram when ＝3 =0
Ｒ1
Ｒ2Ｒ3
−0.2
0
0.2
0.4
0.6
0.8
1
(h) 𝑐3 = 0
0 50 100 150 200 250 300
Time series diagram when ＝3 =-0.1
Ｒ1
Ｒ2Ｒ3
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(i) 𝑐3 = −0.1
Figure 2: Time series graph of players’ strategy selection
probabilities when 𝑐1, 𝑐2, 𝑐3 have smaller values separately.
Initial values of variables:(𝑥1, 𝑥2, 𝑥3)|𝑡=0 = 𝑃1(0.7, 0.5,
0.3).accordingly. So we construct some flexible cost functions,such
as 𝑐𝑖(𝑥𝑖) = 𝑐𝑖(1−𝑥𝑖), 𝑐𝑖(𝑥𝑖) = 𝑐𝑖(1−𝑥2𝑖 ), 𝑐𝑖(𝑥𝑖) = 𝑐𝑖(1−√𝑥𝑖),to
analyze the stability of dynamical system with
incentiverewards.
If 𝑐𝑖(𝑥𝑖) = 𝑐𝑖(1−𝑥𝑖), 𝑖 = 1, 2, 3, are taken, income functionsof
the three players in the game model are
𝜋1 = −𝑥1 (1 − 𝑥1) 𝑐1 − (1 − 𝑥1) (𝑥2 + 𝑥3 − 𝑥2𝑥3) 𝑓1;𝜋2 = −𝑥2 (1
− 𝑥2) 𝑐2+ (1 − 𝑥1) [𝑥2𝑓1 − (1 − 𝑥2) 𝑥3𝑓2] ;
𝜋3 = −𝑥3 (1 − 𝑥3) 𝑐3 + (1 − 𝑥1) (1 − 𝑥2) 𝑥3 (𝑓1 + 𝑓2) .(2)
The replicator dynamics corresponding to the gamemodel are as
follows:
̇𝑥1 = 𝑥1 (1 − 𝑥1) [𝑥1𝑐1 + (𝑥2 + (1 − 𝑥2) 𝑥3) 𝑓1] ;̇𝑥2 = 𝑥2 (1 −
𝑥2) [𝑥2𝑐2 + (1 − 𝑥1) (𝑓1 + 𝑥3𝑓2)] ;̇𝑥3 = 𝑥3 (1 − 𝑥3) [𝑥3𝑐3 + (1 −
𝑥1) (1 − 𝑥2) (𝑓1 + 𝑓2)] .(3)
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Complexity 5
0 50 100 150 200 250 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ｒ1Ｒ2Ｒ3
＝1=0.2,＝2=0.1,＝3=0.15
(a)
0 50 100 150 200 250 3000.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Ｒ1Ｒ2Ｒ3
＝1=0,＝2=0,＝3=0
(b)
0 50 100 150 200 250 3000.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Ｒ1Ｒ2Ｒ3
＝1=0, ＝2=-0.1, ＝3=-0.1
(c)
Figure 3: Time series graph of players’ strategy selection
probability when 𝑐1, 𝑐2, 𝑐3 become smaller simultaneously. (𝑥1, 𝑥2,
𝑥3)|𝑡=0 =𝑃1(0.7, 0.5, 0.3).The Jacobian matrix of the dynamical
system (3) is
𝐽 = 𝜕 ( ̇𝑥1, ̇𝑥2, ̇𝑥3)𝜕 (𝑥1, 𝑥2, 𝑥3) = (𝑙1 𝑥1 (1 − 𝑥1) (1 − 𝑥3)
𝑓1 𝑥1 (1 − 𝑥1) (1 − 𝑥2) 𝑓1𝑥2 (1 − 𝑥2) (𝑓1 + 𝑓2𝑥3) 𝑙2 𝑥2 (1 − 𝑥1) (1
− 𝑥2) 𝑓2−𝑥3 (1 − 𝑥3) (1 − 𝑥2) (𝑓1 + 𝑓2) −𝑥3 (1 − 𝑥3) (1 − 𝑥1) (𝑓1 +
𝑓2) 𝑙3 ) (4)
where𝑙1 = (1 − 2𝑥1) (𝑥2 + 𝑥3 − 𝑥2𝑥3) 𝑓1 + (2𝑥1 − 3𝑥21) 𝑐1;𝑙2 =
(1 − 2𝑥2) (1 − 𝑥1) (𝑓1 + 𝑥3𝑓2) + (2𝑥2 − 3𝑥22) 𝑐2;𝑙3 = (1 − 2𝑥3) (1
− 𝑥1) (1 − 𝑥2) (𝑓1 + 𝑓2)+ (2𝑥3 − 3𝑥23) 𝑐3.(5)
Let ̇𝑥1 = ̇𝑥2 = ̇𝑥3 = 0, and we get eight pure
strategyequilibrium points of system (3), 𝑋𝑖(𝑖 = 0, 1, . . . ,
8):(0, 0, 0) , (1, 0, 0) , (0, 1, 0) , (0, 0, 1) , (1, 1, 0) , (1,
0, 1) ,(0, 1, 1) , (1, 1, 1) . (6)According to Formula (4), all of
the Jacobian matrices at theequilibriumpoints are diagonalmatrices
(see Table 1). And allof the eigenvalues are not larger than zero
at the four pointswith 𝑥1 = 1 (i.e., coal miners will choose safety
productionbehaviors), and the first eigenvalues of them are
negative.
According to the eigenvalue judgment method of evolu-tionary
game strategy (ESS) [31, 32], the ESS of coal minersis 𝑥1 = 1,
which means they will surely choose the safetyproduction strategy
in the case of this function form offlexible costs.
As can be shown in Figures 4 and 5, dynamical system(3) goes
into evolutionary stable state despite different initialvalues.
Especially for some reasonable initial state values,
theevolutionary process of the game system is very anticipatedand
preferable (see Figures 4(a) and 4(b)).
Therefore, if the incentive salary system is establishedin coal
enterprise, the proportion of unsafe behaviors willgreatly be
reduced and coal mine safety will be maintainedin a stable and
controllable state. An incentive salary systemmeans that the salary
should be linked to the performanceof coal workers with the
efficiency of safety production andthe frequency of unsafe
behavior, and the variable part of coalenterpriseworkers’
performancewages, bonuses, welfare, andso on should be increased in
their personal incomes.
2.3. Stability of the Dynamical Game System with
InflexiblePenalties. As can be seen in Figure 6, the volatility of
thedynamical system would be reduced when value of 𝑓1 or
𝑓2increases. In contrast to Figures 6(a), 6(b), and 6(c), we
findthat 𝑥1 approaches 1 and 𝑥2 increases with smaller
amplitudeswhen 𝑓1 becomes larger. That is, the increase in the
intensityof player I’s penalty can make coal workers increase
thewillingness to choose the behaviors of safe production, andthe
enterprise safety inspection department is also willing toincrease
the intensity of safety inspection.
By contrasting Figure 6(d) with Figure 6(e), we alsofind that 𝑥1
approaches 1 with smaller amplitudes when 𝑓2becomes larger.
However, the volatilities of 𝑥2 and 𝑥3 haveno significant reduction
when 𝑓1 or 𝑓2 becomes larger (evenwhen𝑓1 and𝑓2 become large
synchronously; see Figure 6(f)),while the values of 𝑥2 and 𝑥3 have
the trends of increasing.In addition, the increase in the intensity
of player II’s penaltymay increase the workload and tension in
safety regulation.
So the conclusion is that although the behavior instabilitywill
decrease with the increase of the values of 𝑓1 and
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6 Complexity
0 50 100 150 200 250 300Ｒ1Ｒ2Ｒ3
00.10.20.30.40.50.60.70.80.9
1＝Ｃ(ＲＣ)=＝Ｃ(1-ＲＣ), i=1,2,3; initial value=０1(0.7,0.5,0.3)
(a)
0 50 100 150 200 250 300Ｒ1Ｒ2Ｒ3
00.10.20.30.40.50.60.70.80.9
1＝Ｃ(ＲＣ)=＝Ｃ(1-ＲＣ), i=1,2,3; initial value=０0 (0.5,0.3,0.2)
(b)
0 50 100 150 200 250 300Ｒ1Ｒ2Ｒ3
−0.2
0
0.2
0.4
0.6
0.8
1
1.2＝Ｃ(ＲＣ)=＝Ｃ(1-ＲＣ), i=1,2,3; initial value=０2(0.3,0.5,0.7)
(c)
0 50 100 150 200 250 300Ｒ1Ｒ2Ｒ3
−0.2
0
0.2
0.4
0.6
0.8
1
1.2＝Ｃ(ＲＣ)=＝Ｃ(1-ＲＣ), i=1,2,3; initial value=０3(0.5,0.7,0.9)
(d)
Figure 4: Time series graph of players’ strategy selection
probability when costs become flexible. 𝑐𝑖(𝑥𝑖) = 𝑐𝑖(1 − 𝑥𝑖), 𝑖 = 1,
2, 3.
01
0.2
0.4
1
0.6
0.8
0.8
0.5 0.6
1
0.40.20 0
０1
０0
０2
０3
x1x2
x3
Flexible costs: ＝Ｃ(ＲＣ)=＝Ｃ(1-ＲＣ), i=1,2,3.
０0(0.5, 0.3, 0.2)０1(0.7, 0.5,
0.3)０2(0.3,0.5,0.7)０3(0.5,0.7,0.9)
Figure 5: Phase diagram of the tripartite game system with
incentive rewards.
-
Complexity 7
Table 1: Jacobian Matrix of Equilibrium Points with Dynamical
costs.
Equilibrium points (0,0,0) (0,1,0) (0,0,1) (0,1,1)J diag(0, 𝑓1,
𝑓1 + 𝑓2) diag(𝑓1, −𝑐2 − 𝑓1, 0) diag(𝑓1, 𝑓1 + 𝑓2, −𝑐3 − 𝑓1 − 𝑓2)
diag(𝑓1, −𝑐2 − 𝑓1 − 𝑓2, −𝑐3)Equilibrium points (1,0,0) (1,1,0)
(1,0,1) (1,1,1)J diag(−𝑐1, 0, 0) diag(−𝑐1, −𝑐2, 0) diag(−𝑐1, 0,
−𝑐3) diag(−𝑐1 − 𝑓1, −𝑐2, −𝑐3)
00.10.20.30.40.50.60.70.80.9
1
50 100 150 200 250 3000Ｒ1Ｒ2Ｒ3
＠1=2; ＠2=1.2
(a) 𝑓1 = 2, 𝑓2 = 1.2
00.10.20.30.40.50.60.70.80.9
1
50 100 150 200 250 3000Ｒ1Ｒ2Ｒ3
＠1=4; ＠2=1.2
(b) 𝑓1 = 4, 𝑓2 = 1.2
00.10.20.30.40.50.60.70.80.9
1
50 100 150 200 250 3000Ｒ1Ｒ2Ｒ3
＠1=8; ＠2=1.2
(c) 𝑓1 = 8, 𝑓2 = 1.2
00.10.20.30.40.50.60.70.80.9
1
50 100 150 200 250 3000Ｒ1Ｒ2Ｒ3
＠1=1; ＠2=4
(d) 𝑓1 = 1, 𝑓2 = 4
00.10.20.30.40.50.60.70.80.9
1
50 100 150 200 250 3000Ｒ1Ｒ2Ｒ3
＠1=1; ＠2=8
(e) 𝑓1 = 1, 𝑓2 = 8
00.10.20.30.40.50.60.70.80.9
1
50 100 150 200 250 3000Ｒ1Ｒ2Ｒ3
＠1=8; ＠2=8
(f) Both 𝑓1 and 𝑓2 are large
Figure 6: Time series graph of players’ strategy selection
probability when𝑓1 or𝑓2 has larger values. Initial value of
variables (𝑥1, 𝑥2, 𝑥3)|𝑡=0 =𝑃1(0.7, 0.5, 0.3).𝑓2, there is no
essential change of the volatility of thedynamical system. That is,
the ordinary increase of penaltyindex can not completely eliminate
the behavior instabilityof game models. So, on the one hand, it is
necessary tomake proper punishment for unsafe behaviors strictly
inthe coal mine safety management. On the other hand,
too,excessively severe penaltymeasuresmay lead to the instabilityof
the working state of the supervised objects, which maybring
negative effects on the efficiency of coal mine
safetysupervision.
2.4. Stability of the Dynamical Game System with
FlexiblePenalties. According to the conclusion of Section 2.3, the
sta-bility of players’ selections has not been completely
controlledas the values of 𝑓1, 𝑓2 become larger simply. Therefore,
weconsider some flexible penalty functions 𝑓𝑖(𝑥𝑖) to substitute
for general penalty index 𝑓𝑖 and analyze the influence
offlexible penalty mechanism on system stability.
The design of the flexible penalty function in this part isbased
on the following considerations: for the players who donot often
make mistakes, the penalty for accidental mistakesis relatively
light; for the players who often make mistakes(repeatedly taught
not to change), the more frequent themistakes are, the larger the
corresponding penalty will be.Therefore, the flexible penalty
functions 𝑓𝑖(𝑥𝑖)mainly discussthree functional forms to analyze and
be compared with eachother: 𝑀𝑖(1 − 𝑥𝑖),𝑀𝑖(𝑥−1𝑖 − 1),𝑀𝑖(𝑥−2𝑖 − 1), 𝑖
= 1, 2. Setthe values of function parameters 𝑀1 = 1 and 𝑀2 =
1.2(separately equivalent to the preceding parameter values of𝑓1
and 𝑓2), and numerical simulation results indicate thatflexible
penalty mechanism can significantly improve thestability of
dynamical systems of evolutionary game models
-
8 Complexity
00
0.20.4
10.60.8
0.60.80.4
0.21 0
0.5
1Flexible penalties: ＠Ｃ(ＲＣ)=＠Ｃ(1/ＲＣ-1), i=1,2.
０0 ０1
０2 ０3
０0(0.5, 0.3, 0.2)０1(0.7, 0.5,
0.3)０2(0.3,0.5,0.7)０3(0.5,0.7,0.9)
x1x2
x3
(a) 𝑓𝑖(𝑥𝑖) = 𝑓𝑖(𝑥−1𝑖 − 1), 𝑖 = 1, 2
0 10
0.2
0.80.2
0.4
0.60.4
0.6
0.4
0.8
0.6
1
0.20.8 01
０0
０1
０2
０3
x1 x2
x3
Flexible penalties: ＠Ｃ(ＲＣ)=＠Ｃ(1/Ｒ2
Ｃ-1), i=1,2.
０0(0.5, 0.3, 0.2)０1(0.7, 0.5,
0.3)０2(0.3,0.5,0.7)０3(0.5,0.7,0.9)
(b) 𝑓𝑖(𝑥𝑖) = 𝑓𝑖(𝑥−2𝑖 − 1), 𝑖 = 1, 2
Figure 7: Phase diagram of dynamical system of tripartite game
with flexible penalty strategy.
(see Figures 7, 8, and 9). In particular the latter two types
offlexible penalty function can effectively control the
evolution-ary stability of game model, which can make the
probabilityof players’ strategy selection 𝑥1, 𝑥2, and 𝑥3 stabilize
quickly.In addition, the latter two types can theoretically avoid
theunwanted situations like any equilibrium point 𝑥𝑖 equals
zero(where 𝑥𝑖 appears on the denominator).
The comparison between Figures 7(a) and 7(b) demon-strates that
the last function form 𝑀𝑖(𝑥−2𝑖 − 1) is even betterone because 𝑥1
can quickly and steadily tends to 1. Supposethat 𝑓𝑖(𝑥𝑖) = 𝑓𝑖(𝑥−2𝑖 −
1), 𝑖 = 1, 2, and then we get the incomefunction of the three
players in the game model as follows:𝜋1 = −𝑥1𝑐1 − (1 − 𝑥1) (𝑥2 + 𝑥3
− 𝑥2𝑥3) 𝑓1 (𝑥−21 − 1) ;𝜋2 = −𝑥2𝑐2 + (1 − 𝑥1)⋅ [𝑥2𝑓1 (𝑥−21 − 1) − (1
− 𝑥2) 𝑥3𝑓2 (𝑥−22 − 1)] ;𝜋3 = −𝑥3𝑐3 + (1 − 𝑥1) (1 − 𝑥2)⋅ 𝑥3 (𝑓1
(𝑥−21 − 1) + 𝑓2 (𝑥−22 − 1)) .
(7)
Replicator dynamic equations corresponding to gamemodelsare ̇𝑥1
= 𝑥1 (1 − 𝑥1) [−𝑐1 + (𝑥2 + 𝑥3 − 𝑥2𝑥3) 𝑓1 (𝑥−21 − 1)] ;̇𝑥2 = 𝑥2 (1 −
𝑥2) [−𝑐2+ (1 − 𝑥1) (𝑓1 (𝑥−21 − 1) + 𝑥3𝑓2 (𝑥−22 − 1))] ;̇𝑥3 = 𝑥3 (1
− 𝑥3) [−𝑐3+ (1 − 𝑥1) (1 − 𝑥2) (𝑓1 (𝑥−21 − 1) + 𝑓2 (𝑥−22 − 1))]
.
(8)
The Jacobian matrix of the dynamical system of the gamemodel is
𝐽 = 𝜕 ( ̇𝑥1, ̇𝑥2, ̇𝑥3)𝜕 (𝑥1, 𝑥2, 𝑥3) = (𝑗𝑖𝑗)3×3 (9)
where𝑗11 = (1 − 2𝑥1) [(𝑥2 + 𝑥3 − 𝑥2𝑥3) 𝑓1 (𝑥−21 − 1) − 𝑐1]− 2 (1
− 𝑥1) (𝑥2 + 𝑥3 − 𝑥2𝑥3) 𝑓1𝑥21 ;𝑗12 = 𝑥1 (1 − 𝑥1) (1 − 𝑥3) 𝑓1 (𝑥−21 −
1) ;𝑗13 = 𝑥1 (1 − 𝑥1) (1 − 𝑥2) 𝑓1 (𝑥−21 − 1) ;𝑗21 = 𝑥2 (1 − 𝑥2)(−𝑓1
(𝑥−21 − 1) − 2 (1 − 𝑥1) 𝑓1𝑥31+ 𝑓2 (𝑥−22 − 1) 𝑥3) ;𝑗22 = (1 − 2𝑥2)⋅
[(1 − 𝑥1) (𝑓1 (𝑥−21 − 1) + 𝑥3𝑓2 (𝑥−22 − 1)) − 𝑐2]− 2 (1 − 𝑥1) (1 −
𝑥2) 𝑥2𝑥3𝑓2𝑥32 ;𝑗23 = 𝑥2 (1 − 𝑥1) (1 − 𝑥2) 𝑓2 (𝑥−22 − 1) ;𝑗31 = −𝑥3
(1 − 𝑥3) (1 − 𝑥2) [−𝑓1 (𝑥−21 − 1)− 2 (1 − 𝑥1) 𝑓1𝑥31 + 𝑓2 (𝑥−22 −
1)] ;𝑗32 = −𝑥3 (1 − 𝑥3) (1 − 𝑥1) [−𝑓2 (𝑥−22 − 1)− 2 (1 − 𝑥2) 𝑓2𝑥32
+ 𝑓1 (𝑥−21 − 1)] ;𝑗33 = (1 − 2𝑥3)⋅ [(1 − 𝑥1) (1 − 𝑥2) (𝑓1 (𝑥−21 −
1) + 𝑓2 (𝑥−22 − 1))− 𝑐3] .
(10)
-
Complexity 9
Flexible penalties: ＠Ｃ(ＲＣ)=＠Ｃ(1-ＲＣ), i=1,2.
Ｒ1
Ｒ2
Ｒ3
50 100 150 200 250 30000
0.10.20.30.40.50.60.70.80.9
1
(a) 𝑓𝑖(𝑥𝑖) = 𝑓𝑖(1 − 𝑥𝑖), 𝑖 = 1, 2
Ｒ1
Ｒ2
Ｒ3
Flexible penalties: ＠Ｃ(ＲＣ)=＠Ｃ(1/ＲＣ-1), i=1,2.
50 100 150 200 250 30000
0.10.20.30.40.50.60.70.80.9
1
(b) 𝑓𝑖(𝑥𝑖) = 𝑓𝑖(𝑥−1𝑖 − 1), 𝑖 = 1, 2
Ｒ1
Ｒ2
Ｒ3
Flexible penalties: ＠Ｃ(ＲＣ)=＠Ｃ(1/Ｒ2
Ｃ-1), i=1,2.
50 100 150 200 250 30000
0.10.20.30.40.50.60.70.80.9
1
(c) 𝑓𝑖(𝑥𝑖) = 𝑓𝑖(𝑥−2𝑖 − 1), 𝑖 = 1, 2
Figure 8: Time series graph of dynamical system under flexible
penalty mechanism. Initial value is 𝑃0(0.5, 0.3, 0.2).Let ̇𝑥1 = ̇𝑥2
= ̇𝑥3 = 0, it is found that there are
only two pure policy equilibrium points in the system, 𝑋0 =(1,
1, 0);𝑋1 = (1, 1, 1), and there may be a mixed policyequilibrium
point 𝑋2 satisfying the following conditions:
(𝑥2 + 𝑥3 − 𝑥2𝑥3) 𝑓1 (𝑥−21 − 1) = 𝑐1;(1 − 𝑥1) [𝑓1 (𝑥−21 − 1) +
𝑥3𝑓2 (𝑥−22 − 1)] = 𝑐2;(1 − 𝑥1) (1 − 𝑥2) [𝑓1 (𝑥−21 − 1) + 𝑓2 (𝑥−22 −
1)] = 𝑐3.(11)
Through calculating, we get 𝐽(𝑋0) = diag(𝑐1, 𝑐2, −𝑐3); 𝐽(𝑋1)
=diag(𝑐1, 𝑐2, 𝑐3). The dynamical systems do not have evolu-tionary
stability in the two equilibrium fields because bothJacobian
matrices at the two pure strategy equilibrium pointshave positive
eigenvalues. In addition, Formula (11) showsthat the solution of 𝑥1
must be less than 1 even if theequilibrium point of mixed strategy
𝑋2 exists.
So it can be concluded from the numerical simulationsthat
flexible penalty mechanism can significantly reduce thedynamics of
unsafe behaviors in coal mine safety super-vision. Once the
penalties can be closely correlated with
the frequency of their mistakes, the players would liketo
consistently choose the safe behaviors (or improve theefficiency of
safety supervision) with great probability. Thenthe controllability
of safety supervision system and theperformance efficiency of
safety management measures canbe guaranteed. However, a shortcoming
of flexible penaltiesis that although 𝑥1 is an ESS, it does not
regularly converge to1 (see Figure 9(c)).
3. Stability of the Dynamical Game Systemunder Combined
Mechanism of IncentiveRewards and Flexible Penalties
Through the previous analysis in Sections 2.2 and 2.4, wefind
that incentive rewards (flexible costs) can raise players’selection
probabilities of safe behaviors and flexible penaltiescan bring
system stability. Considering the actual situation ofcoal
production, we certainly prefer to have an ESS of 𝑥1 = 1in the
dynamical safety management system. So we proposea combined
mechanism of incentive rewards and flexiblepenalties and then
analyze the stability of the evolutionarydynamical system on this
basis.
-
10 Complexity
Ｒ1
Ｒ2
Ｒ3
Flexible penalties: ＠Ｃ(ＲＣ)=＠Ｃ(1-ＲＣ), i=1,2.
50 100 150 200 250 30000
0.10.20.30.40.50.60.70.80.9
1
(a) 𝑓𝑖(𝑥𝑖) = 𝑓𝑖(1 − 𝑥𝑖), 𝑖 = 1, 2
Ｒ1
Ｒ2
Ｒ3
Flexible penalties: ＠Ｃ(ＲＣ)=＠Ｃ(1/ＲＣ-1), i=1,2.
50 100 150 200 250 30000
0.10.20.30.40.50.60.70.80.9
1
(b) 𝑓𝑖(𝑥𝑖) = 𝑓𝑖(𝑥−1𝑖 − 1), 𝑖 = 1, 2
Ｒ1
Ｒ2
Ｒ3
Flexible penalties: ＠Ｃ(ＲＣ)=＠Ｃ(1/Ｒ2
Ｃ-1), i=1,2.
50 100 150 200 250 30000
0.10.20.30.40.50.60.70.80.9
1
(c) 𝑓𝑖(𝑥𝑖) = 𝑓𝑖(𝑥−2𝑖 − 1), 𝑖 = 1, 2
Figure 9: Time series graph of dynamical system under flexible
penalty mechanism. Initial value is 𝑃1(0.7, 0.5, 0.3).Two cases are
chosen as an example to be analyzed and
compared in this part. One case is all players unified
incentiverewards and flexible penalties; i.e., 𝑐𝑖(𝑥𝑖) = 𝑐𝑖(1 − 𝑥𝑖),
𝑖 = 1, 2,and𝑓𝑖(𝑥𝑖) = 𝑓𝑖(𝑥−2𝑖 −1), 𝑖 = 1, 2, 3.Theother case is that
playerI (coal miners) is subjected to incentive reward
mechanism,and player II is subjected to flexible penalty
mechanism,and player III is not given any incentive reward and
flexiblepenalty (i.e., 𝑐1(𝑥1) = 𝑐1(1−𝑥1), 𝑓2(𝑥2) = 𝑓2(𝑥−22 −1), and
therest of the parameters remain unchanged). As shown in thephase
diagrams (see Figure 10), both results of the two casesof combined
mechanisms are very satisfying (𝑥1 = 1 is ESS).
The evolutionary processes of more cases of combinedmechanisms
are demonstrated by the time series graphs (seeFigure 11). Through
mutual comparison among them, wecan find that some of them bring
more satisfying results(see Figures 11(a) and 11(b)). Although
starting from differentinitial values, the dynamical system always
tends to be ideallystable in very short time (player I chooses safe
productionstrategy with 𝑥1 = 1 and the values of 𝑥2 and 𝑥3 arealso
stable in some reasonable values). Therefore, we canconclude that
this combined mechanism of incentive rewardand flexible penalties
can optimize and control the instabilityof the behavior strategy
selection in the safety supervision
model. This conclusion is undoubtedly a highlight of thisstudy.
These combined mechanisms may be the key measureto solve the
behavior fluctuations in the safety managementsystem.
4. Control Strategies for theImprovement of the System
andMeasures of Safety Management
At present, the formulation of safety management policiesand
measures is mostly based on the temporary measurestaken according
to the occurrence of accidents [1, 33]. Thepolicies are adjusted
with the occurrence of accidents. Therelevant supervision measures
are often characterized byaccident types and lack of comprehensive
analysis of safetysystem, which is one of the important reasons for
frequentaccidents. There are many factors that affect the
gamebehavior and game strategy of each interested group. Thestudy
of the regulatory mechanism of influencing factors isan important
guarantee for the rationality of the formulationof management
rules. Only when the management rules canregulate the bad game
behavior of the players and the gamerules can the management rules
be effective.
-
Complexity 11
01
0.2
0.4
1
0.6
0.8
0.8
0.5 0.6
1
0.40.20 0
０0
０1
０2
０3
x1x2
x3
Flexible costs and penalties:
＝Ｃ(ＲＣ)=＝Ｃ(1-ＲＣ),i=1,2,3;＠Ｃ(ＲＣ)=＠Ｃ(1/Ｒ2
Ｃ-1),i=1,2.
０0(0.5, 0.3, 0.2)０1(0.7, 0.5,
0.3)０2(0.3,0.5,0.7)０3(0.5,0.7,0.9)
(a)
01
0.2
0.4
1
0.6
0.8
0.8
0.5 0.6
1
0.40.20 0
０0
０1
０2
０3
x1x2
x3
Flexible ＝1 and ＠2: ＝1(Ｒ1)=＝1(1-Ｒ1); ＠2(Ｒ2)=＠2(1/Ｒ2
2-1)
０0(0.5, 0.3, 0.2)０1(0.7, 0.5,
0.3)０2(0.3,0.5,0.7)０3(0.5,0.7,0.9)
(b)
Figure 10: Phase diagrams of dynamical system of tripartite game
under combined mechanisms of incentive reward and flexible
penalty.
According to the researches of the coal mine safetysupervision
model in Sections 2 and 3, we finally put forwardthe following
regulation and control strategies of coal minesafety management
system and supervision measures.
(1) Restraint Measures in Safety Supervision System of CoalMine
Safety Management. In Section 2.1, it is concluded thatraising
policy subsidies and the welfare level of the employeeswould
relevantly reduce the safety input cost and make theproduction
safety of coal enterprises and safety managementwork run at a
higher level. So the governmental super-vision department should
formulate some policy-based orindustry-specific constraints for
safety supervision behaviors,includingmandatory provisions on the
supervision strategiesof internal and external safety supervision
and inspectiondepartments. In order to better improve the overall
produc-tion safety level and safety management level, the
majorityof front-line production workers in coal enterprises can
becalled upon to exercise self-restraint on their own
safetyproduction activities spontaneously (strictly disciplined
andsupervised by each other). It is more important especiallyin
large and medium-sized state-owned coal enterprises,where the
quality of coal mine workers is very good and thetechnology of
safety production is more advanced.
It should be noted that the so-called subsidies andemployee
benefits are not necessarily limited to the economiclevel. For
ordinary employees, especially the coal minersworking in the
production line, enterprises should increaseinvestment in regular
safety knowledge training and safetyskills drills to improve the
safety quality and safety awarenessof employees and inhibit the
occurrence of unsafe behavior.For enterprise safety managers and
government personnel incharge of safety supervision departments, it
is necessary tocarry out regular or irregular study of safety
skills and safetymanagement knowledge in coal mine safety, so as to
improvethe safety knowledge and safety management level of
safetymanagement personnel and ensure the implementation ofsafety
supervision policies and measures.
(2) Energetically Promoting the Incentive Salary System inthe
Coal Industry. In Section 2.2, we found that incentivesalary system
can greatly reduce the proportion of unsafebehaviors and impels
coal mine safety to be stable and con-trollable. Incentive reward
system will reform the traditionalunreasonable wage and salary
distribution in coal enterprisesand significantly increase wage
distribution ratio of safety-related posts and production links,
such as high technicalrequirements, high production safety
intensity, and heavymanagement responsibility. Then the staff and
workers ofcoal enterprises can be guided to spontaneously
strengthenthe improvement of safety production technology and
theexertion of safety production capacity.
There are still some backward phenomena in the
salarydistribution system of nowadays’ coal enterprises in
China,such as the serious tendency of internal income
distributionequalitarianism and unreasonable internal income
distri-bution relationship, which seriously affect the efficiency
ofsafety supervision in coal enterprises and the level of
safetymanagement [34]. Safety production is the most importanttask
in the coal enterprises. So we should vigorously promotethe
incentive salary system in the coal industry, especially inlarge
and medium-sized state-owned coal enterprises. Post-performance pay
system, year-end accident-free incentives,and other ways should be
adopted properly to improve thecorrelation between the income of
workers and their safetyproduction efficiency and safety
behavior.
(3) Establishing the Flexible Penalty Mechanism of
UnsafeBehavior in Coal Mine Safety System. It can be seen from
theresearch results in Sections 2.4 and 3 that the flexible
penaltymechanism based on the frequency of unsafe behavior
canbetter reflect the management mode of tight integration insafety
management. More closely related to the occurrenceof coal mine
safety behaviors, the flexible penalty mecha-nism can better
guarantee the stability of coal mine safetymanagement.
-
12 Complexity
Ｒ1
Ｒ2
Ｒ3
＠1, ＠2 and ＝1 are flexible
50 100 150 200 250 30000
0.10.20.30.40.50.60.70.80.9
1
(a)
Ｒ1
Ｒ2
Ｒ3
＠1, ＠2 and ＝1, ＝2 are flexible
300100 150 200 2505000
0.10.20.30.40.50.60.70.80.9
1
(b)
Ｒ1
Ｒ2
Ｒ3
＠1, ＠2 and ＝1, ＝2, ＝3 are flexible
50 100 150 200 250 30000
0.10.20.30.40.50.60.70.80.9
1
(c)
Ｒ1
Ｒ2
Ｒ3
＠1 and ＝1 are flexible
00.10.20.30.40.50.60.70.80.9
1
50 100 150 200 250 3000
(d)
Ｒ1
Ｒ2
Ｒ3
＠1 and ＝1, ＝2 are flexible
50 100 150 200 250 30000
0.10.20.30.40.50.60.70.80.9
1
(e)
Ｒ1
Ｒ2
Ｒ3
＠1 and ＝1, ＝2, ＝3 are flexible
00.10.20.30.40.50.60.70.80.9
1
50 100 150 200 250 3000
(f)
Figure 11: Time series graphs of dynamical system under combined
mechanisms of incentive reward and flexible penalty. Initial value
is𝑃1(0.7, 0.5, 0.3).In coal mine safety management, the penalty
measures
against unsafe behaviors of coal miners are very important.The
difficulty and inadequacy of flexible cost mechanism areto have
certain requirements for the operation profitability ofcoal mine
enterprises. Actually, for many slow-growing smalland medium-sized
coal mine enterprises, they may prefer tomotivate employees through
punitive measures.
5. Conclusions
To control players’ behavior instability in coal mine
safetymanagement, this study starts from the stability analysisof
the dynamical game system under the four cases ofinflexible costs
(static subsidies), flexible costs (incentiverewards), inflexible
penalties, and flexible penalties. Then, we
-
Complexity 13
discuss the stability of the dynamical system under
combinedmechanism of cost and penalty and find out which methodis
more effective to control the instability of dynamicalsystem.
Through the comparative analysis of inflexible andflexible rewards
and penalties mechanism, it can be foundthat if the salary and
penalty mechanism of coal enterprisescan be closely related to
their safety production behaviors(or safety supervision behaviors),
the stability of the gamedynamical system of coal enterprises’
safety supervisioncan be enhanced. We propose a combined mechanism
ofincentive reward and flexible penalties for the improvementof the
safety management system to restrain the emergenceand spread of
unsafe behaviors in the coal mine safety system.Some results of
this research provide a theoretical basis formore reasonable and
more effective policies of coal minesafety management.
It is worth noting that it is necessary to carry outthe modern
financial management of coal enterprises. Thecoal industry should
vigorously improve the performancewages, quarterly awards, year-end
bonuses, welfare, and othervariable income in the proportion of
employees’salaries andstrive to build and improve the incentive
salary system forcoal enterprise employees. And, more importantly,
moreclose connection should be established between the system
ofstaff safety performance evaluation and the system of rewardsand
punishment.
In the future, the corresponding tuning mechanismfor the
parameters will be investigated in different safetymanagement
systems. In order to improve the accuracy ofmodel parameters, we
should investigate and analyze theactual situation of coal mine
safety management by meansof questionnaire survey and field visit,
so as to improve thepracticability of the model method in this
paper. Besides,some optimal techniques should also be considered
forimproving control performance of the dynamical system,such as
extremum seeking control (ESC) method [35, 36].We will continue our
efforts to study the optimization of thedynamical system
stability.
Data Availability
The data used to support the findings of this study areincluded
within the article.
Conflicts of Interest
The authors declare that there are no conflicts of
interestregarding the publication of this paper.
Acknowledgments
This work is supported by National Natural Science Founda-tion
of China (Grant no. 51574157): Research on MultilateralGame and
Control Strategies in Coal Mine Safety System.
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