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Game theory model of corporate governance: Fitting real market environment

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This paper uses two game models to fit the market subject behavior in corporategovernance from the dynamic and static dimensions. Further expanding the maingovernance subjects and adding variables with practical significance, suchas "bribery" and "social welfare," into the game theory model, some conclusions withpractical significance are found in this paper. For example, the factors such as thepossibility of checking out manager's violation, the punishment of the irregularities, and the supervision cost of the supervisory board, have an essential impact onthebehavior of corporate governance. In the static game model, this paper finds that intersubjective games and restrictions prevent “the worst result” fromoccurring. Besides, social welfare even can be improved under certain conditions. Inthemulti-coexistence game model, the equilibrium point of the static game model is uncertain, which makes the social welfare situation uncertain. The conclusions of thegame models can be used for reference to improve the corporate governancemechanism. JEL classification: G30; M10; D21; D81
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Game theory model of corporate governance: Fitting
real market environment
Yunxuan Zhu ( sunnyzyxxyz@stu.pku.edu.cn )
Peking University
Research Article
Keywords: corporate governance, dynamic game model, static game model
Posted Date: July 20th, 2023
DOI: https://doi.org/10.21203/rs.3.rs-2401729/v3
License: This work is licensed under a Creative Commons Attribution 4.0 International License. 
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Additional Declarations: No competing interests reported.
Game theory model of corporate governance:
Fitting real market environment1
Yunxuan Zhu*2
Guanghua School of Management, Peking University, China
Abstract
This paper uses two game models to fit the market subject behavior in corporate
governance from the dynamic and static dimensions. Further expanding the main
governance subjects and adding variables with practical significance, such as
"bribery" and "social welfare," into the game theory model, some conclusions with
practical significance are found in this paper. For example, the factors such as the
possibility of checking out manager's violation, the punishment of the irregularities,
and the supervision cost of the supervisory board, have an essential impact on the
behavior of corporate governance. In the static game model, this paper finds that
intersubjective games and restrictions prevent “the worst result” from occurring.
Besides, social welfare even can be improved under certain conditions. In the
multi-coexistence game model, the equilibrium point of the static game model is
uncertain, which makes the social welfare situation uncertain. The conclusions of the
game models can be used for reference to improve the corporate governance
mechanism.
Key words: corporate governance; dynamic game model; static game model
JEL classification: G30; M10; D21; D81
1The author sincerely thanks Professor Haotian Xiang for his suggestions and modifications to this article and
Professor Lei Ming's guidance on the game model-solving process. Both professors have checked every detail of
this paper. Of course, all possible errors are the responsibility of the authors.
2Contact information:sunnyzyxxyz@stu.pku.edu.cn
1 Introduction
The principal-agent problem is the core problem of corporate governance. The
main problem in the study of corporate governance is how to protect the rights and
interests of fund providers(Shleifer&Vishny,1996). The literature on corporate
governance tends to focus on "incentive" and "restriction," and the institutional
arrangements are discussed or designed based on that. Researchers often only pay
attention to controlling shareholders and discuss how to solve the principal-agent
problem between them. With the development of modern enterprise management
theory, Researchers generally believe that it is more practical to expand the central
bodies of corporate governance(Holderness,2003). Li and Dong (2003) first
constructed a tripartite dynamic game model to deal with corruption in corporate
governance and put forward countermeasures according to the equilibrium solution.
Dyck and Zingales (2006)believe that social media and the public should also be paid
attention to in the promotion of corporate governance, and the media can effectively
solve corporate governance problems by influencing corporate reputation. Dai
(2011)believes that adverse reports from the media can effectively curb the financial
restatement of listed companies in China. At the same time, the external market will
play an essential complementary role in improving the corporate governance
mechanism (Yi et al., 2010). However, little literature has considered internal and
external corporate governance subjects, and the interaction between central bodies
will undoubtedly lead to corruption (Rodriguez et al.,2006). Without a supervision
mechanism, individuals and enterprises tend to use illegal means to achieve success
(Merton, 1938). In addition, as an important social indicator to reflect the quality of
life, social welfare needs to be paid attention to all the time in the progress of
enterprise reform and development (Zhen,2013). However, these indicators with
practical significance are rarely introduced into the study of corporate governance.
Establishing a theoretical model based on the actual market environment has
important practical significance for promoting the practice and reform of corporate
governance.
Much literature has documented that corporate governance is primarily studied in
mechanism design. Ross and Laffont (1993)have made outstanding contributions to
the establishment of an "incentive" mechanism of corporate governance by
introducing "private information," such as "moral hazard" and "adverse selection,"
into the model. The literature in the 1970s is mainly based on a macro-perspective,
which usually put forward a mechanism to solve the principal-agent problem. In the
1980s, quantitative methods are widely used to analyze the principal-agent problem.
For example, Spence and Zeckhauser (1988)built a mathematical analysis framework
to quantify the principal-agent problem. Based on their research, Mirrlees has
dramatically contributed to the information theory and incentive theory. The optimal
policy design Mirrlees adopted is a classical non-symmetry information Economics
method. Holmstrom has further developed Mirrlees’ work and perfected the
principal-agent model based on Mirrlees. Since then, many scholars have further
considered the principal-agent problem from various aspects. For example, Harris,
Antel, and Green et al.have analyzed the principal-agent problem from executive
incentive, enterprise incentive, internal control, and budget constraint. Although the
former perspective can grasp the core problem of corporate governance, it needs a
more accurate quantitative analysis. From the perspective of "quantitative analysis,"
the mathematical method has been applied successfully. However, the model is often
too "delicate" and relies on a series of strict assumptions, which makes the
applicability of the conclusion questionable. As a result, Cabrera Garcia, Tobias Hiller,
and Kasiri et al.began to use a variety of game models to analyze the principal-agent
problems in corporate governance, such as information hiding, skimming problems,
and so on. The game theory model makes the model environment closer to the real
market environment by relaxing assumptions. It is worth noting that the
"principal-agent" problem is extremely complex and also includes signal transfer
theory, incentive theory, common agency theory, and so on. The game theory model
cannot be considered optimal in all circumstances, and different methods should be
adopted in different situations.
The main innovation of this paper lies in further expanding the subject of corporate
governance and introducing the external environment, such as the reaction of the
public, into the model. In addition, introducing "bribery," "social welfare," and other
variables into the model makes the assumptions more consistent with the actual
market, making the conclusions somewhat explanatory.3
The paper is organized as follows: Section 2 describes the dynamic game model,
and Section 3 is the static game model. The summary and conclusion are presented in
Section 4.
2 Dynamic game model
In this paper, we use the classic framework of corporate finance theory and
theoretically assume that the relationship between shareholders and managers is a
pure "principal-agent" relationship.4This paper divides regulators into internal
regulators (supervisory board) and external regulators (supervision department) and
further introduces the significant factor "bribery" into the model. The assumptions
about "bribery" are as follows. First, only when the managers are motivated to
infringe on shareholders’ interests can they bribe the supervisory board. Second, if the
supervisory board accepts bribes from the manager, the board will not punish the
manager or report the manager's illegal behaviors to the external supervision
department.5Third, external supervision departments are the ones that do not take
bribes.
In this paper, "social welfare" refers to the change in social welfare in a broad sense.
The change in social welfare is mainly affected by the media and public response. The
response of the media and the public mainly depends on the behavior of the
3It is worth noting that the game theory model constructed in this paper still depends on widely
accepted assumptions of human behavior. However, the corresponding assumptions are more in
line with the real situation. In addition, irrational behaviors and overlapping behaviors between the
subjects also exist in the real market environment. In this paper, we give normative explanations
for these situations.
4This indicates that managers are not shareholders and shareholders do not carry out daily
company management.
5If the supervisory board does not accept the managers’ bribes, the supervisory board will
naturally report the manager's illegal behaviors to the regulatory authorities.
supervision departments. If the public expresses "satisfaction" with the behavior of
the supervision departments, the level of social welfare will rise; if the public
expresses "dissatisfaction" with the behavior of the supervision departments, the level
of social welfare will fall. Under the positive response, the overall social welfare level
is improved (for example, the market is more orderly), and the supervision department
also can obtain a part of social welfare (for example, the credibility of the supervision
department is improved); under the negative response (for example, the public thinks
that the behavior results of the supervision department contribute to unhealthy
tendencies of the society), the overall social welfare growth is zero or even reduced
(this paper assumes that the reduced value is -S). The supervision department cannot
get a part of social welfare.
Table
Setting and description of the dynamic game model
Firstly, we use the dynamic game model to describe the players’ behavior in
corporate governance. In the dynamic game model, there is a sequence of players'
actions. The latter can observe the former's action and then decides how he should act.
We assume that the company’s managers act first, and then the shareholders act.
When the shareholders' interests are violated, they first appeal to the company’s
supervisory board. The supervisory board reports the violations to the external
supervision department, which announces the investigation results to the public. The
public makes positive or negative reactions to the investigation results.6
Symbol
setting
description

Benefits obtained by managers from
infringing shareholders' interests

>
0
The benefits paid by the manager to
bribe the supervisory board
1. If the managers do not intend to infringe on the
interests of shareholders, they will not bribe the
supervisory board
2. If the supervisory board accepts bribes, it will
not punish the managers nor report to the external
supervision department
6This is a typical sequential game model.
3. External supervision department cannot be
bribed
1
Costs for shareholders to appeal to
the supervisory board
2>
1
2
Costs for shareholders to appeal to
the supervision department
1
Regulatory costs of the supervisory
board
1> 0
2
Regulatory costs of the external
supervision department
> 0
1
Coefficient greater than 1
2>
1
, The punishment of the supervision
department is greater than that of the supervisory
board.
2
Coefficient greater than 1
1
Coefficient between 0 and 1
The benefit-sharing ratio between the supervisory
board and shareholders
2
Coefficient between 0 and 1
The benefit-sharing ratio between the supervision
department and shareholders
Coefficient between 0 and 1
The welfare-sharing ratio between the supervision
department and the public
Social welfare
Social welfare in a broad sense. For example, the
improvement of social credibility and the
improvement of the market environment
1
The possibility of the managers
infringing the interests of the
shareholders
0<
<1
2
Possibility of Shareholders’ Appeal
3
The possibility for the supervisory
board to report the managers illegal
behaviors
4
The possibility for the supervision
department to check out the illegal
behaviors of managers
5
Possibility of positive public
response
Figure 1 shows the game tree of the dynamic game model.
Manager
Infringement:P1;Compliance:(1-P1)
Shareholder
Appeal: P2; Give up: (1-P2)
Supervisory board
Report: P3
Hide: (1-P3)
Supervision department
Check out: P4
Fail: (1-P4)
The public
Positive: P5
Negative: (1-P5)
1 2 3 4 5 6 7 ….. Conditions ….. 26 27 28 29 30 31 32
Fig1.Game tree based on the dynamic game model
Figure 1 shows the sequence of sequential games. We can also show the
information conveyed in the game tree as a table. In this way, we get Table 2.
Table
Game order
According to figure1, the order of the players can also be given in Table 2.
Manager
Infringement
Appeal
Supervisory
board
Report
Supervision
department
Check
out
The
public
Positive
1
Negative
2
Fail
Positive
3
Negative
4
Hide
Check
out
Positive
5
Negative
6
Fail
Positive
7
Negative
8
Give
up
Report
Check
out
Positive
9
Negative
10
Fail
Positive
11
Negative
12
Hide
Check
Positive
13
out
Negative
14
Fail
Positive
15
Negative
16
Compliance
Appeal
Report
Check
out
Positive
17
Negative
18
Fail
Positive
19
Negative
20
Hide
Check
out
Positive
21
Negative
22
Fail
Positive
23
Negative
24
Give
up
Report
Check
out
Positive
25
Negative
26
Fail
Positive
27
Negative
28
Hide
Check
out
Positive
29
Negative
30
Fail
Positive
31
Negative
32
According to Table 1, table 2, figure 1, table 3, and Table 4, we can finally show
the income matrix of all parties in the dynamic game model.
Table
Income matrix of shareholders and operators
According to our model settings and assumptions, the income matrix of
shareholders and managers can be shown in Table 3.
The income matrix of Shareholders
The income matrix of Managers
Conditions
Income
Income
1-2

1
2+ 1
1
1

+ 1
2
2


1

2

3-4

1
2+ 1
1
1


1

5-6

1
2+ 1
2
2


2

7-8

1
2

9-10

+ 1
1
1

+ 1
2
2


1

2

11-12

+ 1
1
1


1

13-14

+ 1
2
2


2

15-16


17-24
1
2
0
25-32
0
0
Table
Income matrix of the Supervisory board, Supervision department, and the
Public
According to the same rules, the income matrix of the supervisory board,
supervision department, and the public can be shown in Table 4.
The income matrix of the Public
The income matrix of the Supervision
department
The income matrix of the Supervisory
board
Conditions
Income
Income
Income
1
1
2
2

+

2
1
1
B
1
2
0
2
2

2
1
1

1
3
0
2
1
1
B
1
4
-S
2
1
1
B
1
5
1
2
2

+

2
1
6
0
2
2

2
1
7
0
2
1
8
-S
2
1
9
1
2
2

+

2
1
1
B
1
10
0
2
2

2
1
1
B
1
11
0
2
1
1
B
1
12
0
2
1
1
B
1
13
1
2
2

+

2
14
0
2
2

2
15
0
2
16
0
2
17-32
0
2
1
The equilibrium solution of the dynamic game model is obtained by the “backward
induction” approach. First, according to the maximization of the expected return, the
optimal solution of the last actor is obtained and brought into the equation of the
penultimate actor. The equilibrium solutions of the dynamic game model are formed
by continuing this process. Therefore, we first seek the optimal solution of "the
public":
5
=
1
2
3
4
51
1
2
31
41
5
+
1
21
3
4
51
1
21
31
41
5
+
11
2
3
4
51
+
11
21
3
4
51
When we achieve the expected revenue maximization, the first-order derivative
equals zero. Therefore, we can further get the following formula:


2
3
41
+
2
31
4+
21
3
41
+
21
31
4+ 1
2
3
41
+ 1
21
3
41
= 0
By simplification, we can get the following:
2
=
4
1

4
|
1
|(1)
Eq. (1) represents that the possibility of shareholders' appeal is closely related to
the possibility of the supervision department checking out the managers’ violations.
The more likely the supervision department finds out about the managers’ violation,
the more likely the shareholders will appeal; otherwise, the less likely they are.
Next, we seek the optimal solution of the "supervision department":

4

4
=
1
2
3
2+
1
2
2
1
2
3
2+
1
3
2
1
2
3
2
+
11
21
3
2
1
1
2
3
2
+ 1
1
2
3
2
1
1
21
3
2
+ 1
1
21
3
2
1
11
2
3
2
+ 1
11
2
3
2
1
11
21
3
2
+ 1
11
21
3
2
+
1
5
2
2
+

2
+
1
1
5
2
2
2= 0
By simplification, we can get the following:
1
5

+
1
2
2
= 0
Take absolute values on both sides, and we can get:
5=

2
2

 =
2
2

 (2)7
Eq. (2) represents that, since is an exogenous variable,  is a constant of
social welfare enjoyed by the supervision department. Therefore, the positive or
negative public response to the supervision department’s behavior largely depends on
the supervision department’s punishments for the managers' violations (2)and the
benefit-sharing ratio affected by the performance of their duties (2). If the
supervision department inflicts strong punishments on the managers (that means the
value of 2is large) or the supervision department gets big profits from punishing
managers (that means the value of 2is large), then the public is more likely to make
a positive response, and vice versa.
In the same way, we seek the optimal solution of the "supervisory board":

3

3
=
1
2
4
5+
1
2
41
5+
1
21
4
5
+
1
21
41
5
1
1
1
1
2
4
5+
1
2
41
5+
1
21
4
5
+
1
21
41
5
1
+
11
2
4+
11
21
4
1
1
1
+
11
2
4+
11
21
4

1
1
2
4+ 1
1
21
4
1
1
2
4
1
1
21
4+ 1
11
2
4
+ 1
11
21
4
1
11
2
4
1
11
21
4
1= 0
7In the case of
1
0, we can get:
5
=
2
2

 . Since the value of probability pis greater
than 0, the absolute value is taken on both sides, and formula (2) can be obtained.
By simplification, we can get the following:
1
1
1
1+
2
1
= 0
And we can further get the following:
2=
+
1

1
1

1
(3)
According to Eq.(1), combined with Eq.(3), we can further obtain:
4=
+
1

1
1
1

1 +
+
1

1
1

(4)
If we examine Eq.(4) closely, we can find that (
+
1
1
1
)is the
common part of the numerator and denominator. This part of the fraction has the
relationship of the same increase and decrease. Similarly, the absolute amount of
increase and decrease is also the same. Therefore, the key factor of the
4
value is
1
1
.
1
is the benefit-sharing ratio between the supervision department
and the public. This paper assumes that in repeated games, the benefit-sharing
coefficient between the public and the supervision department tends to be stable,
so
1
tends to be a constant in essence. We can see that the possibility that the
supervision department checks out the managers’ violations directly depends on the
supervisory cost (
2
)of the supervisory board. This conclusion is different from
previous literature. Most of the previous literature believed that the possibility of
checking out the violations by the supervision department directly depends on the
supervisory cost of the supervision department (
2
). However, in the sequential
game model, the supervisory board is equivalent to the "upstream" source of
regulation, while the supervision department is equivalent to the "downstream"
tributary of regulation. The higher the supervisory cost of the supervisory board, the
more likely the supervisory board is not to carry out the supervision. Therefore, the
supervisory board has fewer incentives to report managers’ violations to the
supervision department, and the supervision department is less likely to find out the
managers’ violations (the lower the
4
value). From this model, we can see that the
motivation for the supervisory board to supervise will directly determine the
efficiency of the supervision department. Suppose the board exists in name only and
does not conduct substantive supervision in the real market. In that case, the
possibility that the supervision department checks out the managers’ infringements
will be greatly reduced. At the same time, Eq.(4) implies a conclusion that the
external supervision department represents "social justice and fairness." The
probability of the violations checked out by the supervision department is not related
to the supervisory cost of the supervision department itself, which shows that the
supervision department is to maintain "market justice" in a great sense rather than just
a trade-off between cost and benefit.
Next, we seek the optimal solution of "shareholders":

2

2
=
11
3
4
1
2+
11
31
4
1
2
+
1
3
4
1
2+
1
31
4
1
2
+ 1
1
1
2= 0
By simplification, we can get the following:
1
4
1
3
4+
1
1
4
1
3+
1
3
4+
1
3
4+
1
3
1
3
4+ 1
1
1
2= 0
And then we can finally get:
1=
2= 0
(5)
Eq.(5) tells us a somewhat subtle conclusion that in the dynamic game process, the
appeal cost is 0. Obviously, in the real world, the cost of appeal is as inevitable as the
cost of investigation and may be very high. Then this result (
1=
2= 0
) may
show that: in repeated games, the appeal cost is insignificant compared with the
interests infringed by the managers. Suppose the shareholders waive their rights to
appeal in repeated games because of the appeal cost. In that case, the managers will
repeatedly infringe on the interests of shareholders and ultimately make the appeal
cost appear insignificant compared with all the infringed interests. The game model
indicates that if the repeated violations of the managers are inevitable, then at the
beginning of the game, as long as the managers violate the interests of the
shareholders, shareholders will certainly choose to appeal.
To be specific:
First, at the equilibrium point where the appeal cost is zero, as long as the managers
violate the interests of shareholders, then shareholders will certainly appeal. (C1)
Second, according to the income matrix of managers, it can be clearly known that
the equilibrium point of managers is "Compliance with laws and regulations" when
shareholders will certainly appeal (
1
0). (C2)
Third, according to the hypothesis of this paper, without infringing the interests of
shareholders, the managers will not bribe the supervisory board, so the "bribe" is zero.
( 0) (C3)
At last, we seek the optimal solution of "managers":

1

1
=
1
4
3+
3
4
+
4
3
4
2
+
3
3
4
1
+
3
4
1
2
= 0
By simplification, we can get the following:
+

3
4
2
3
1
= 0
And then we can get the following:
3=
4
2

+

1

Combined with Eq.(4), we can further obtain:
3=
(
+
1

1
1

)
2

1

1 +
+
1

1
1


1

+


1

(6)
According to Eq.(6), the actions of the supervisory board are influenced by many
factors. To simplify the analysis, we can see that (
+
1
1
1
)is the
common part of the numerator and denominator. This part of the fraction has the same
increase and decrease relationship. And the absolute amount of increase and decrease
is also the same. We assume that the supervisory cost
1
of the supervisory board
is fixed and
1
is a constant.8Therefore,
1
1
can also be regarded as a
constant. For Eq.(6), the core influencing factors are
2
,( ), and
(
1
).
2
can be understood as the disciplinary strength of the supervision
department;( )and (
1
)is the trade-off between bribery and
infringement of interests. More specifically, ( )is the trade-off between
"bribery" and "infringement of interests" by managers; (
1
)is the
trade-off between "bribery" and "fine" by the supervisory board. The greater the
disciplinary strength of the supervision department (the greater the value of
2
),
the greater the possibility that the supervisory board will report violations to the
supervision department and less likely to cover up for the "managers."
( )shows the decision-making process of the managers: when the bribe
given by the managers to the supervisory board ()is less than the profit obtained by
the managers()from infringing the interests of shareholders (namely:
BJ<
0
), the probability of the supervisory board reporting violations (
3
)will
decrease; but when the bribe given by the managers to the supervisory board ()is
8The reasons have been given in the previous paragraphs.
greater than the profit obtained by the managers()from infringing the interests
of shareholders(namely: >
0
), the probability of the supervisory board
reporting violations (
3
)will increase. This shows that when "self-interest
maximization" is not the leading aim of managers but "self-protection" is the leading
aim (that is, the cost of bribery exceeds the income obtained), the supervisory board
will "refuse to harbor" the managers out of the awareness of "excessive risks." (
1
)shows the decision-making process of the supervisory board, supervisory
board has to make tough decisions between the "bribe" of managers and the "fine"
obtained from conscientious performance of duties. If the "bribe"()becomes bigger
and the fine (
1
)becomes smaller, the denominator of the fraction becomes
bigger, and
3
becomes smaller, which means that the possibility of the supervisory
board reporting violations will be reduced; otherwise, the possibility will be increased.
This indicates that the supervisory board will make a trade-off between two kinds of
interest and choose a relatively larger one.9
Table
Equilibrium solutions of the dynamic game model
From the above calculations and combined with C1, C2, and C3, we can get the
dynamic game model’s equilibrium solutions, as shown in Table 5.
Possibility
Equilibrium point
1
0
2
1
1
1

1
3
(
1
1
1

)
2

1
1+
1
1
1


1

+

1

4
1
1
1

1
1 +
1
1
1

5
2
2


9This conclusion does not conflict with the previous conclusion, and it shows the
decision-making process of the supervisory board under "normal" circumstances.
Therefore, the conclusions of the dynamic game model are as follows:
First, the possibility of shareholders' appeal is closely related to the possibility of
the supervision department checking out the managers’ violations.
Second, punishments on the managers and the profits from punishing managers can
greatly affect the public’s response.
Third, the possibility that the supervision department checks out the managers’
violations directly depends on the supervisory cost of the supervisory board and not
the supervision department itself.
Fourth, if the managers’ repeated violations are inevitable, shareholders will
certainly choose to appeal at the beginning of the game.
Fifth, the greater the disciplinary strength of the supervision department, the less
likely for the supervisory board to cover up for the "managers."
Sixth, the supervisory board will "refuse to harbor" the managers because of the
awareness of "excessive risks." However, the supervisory board will make a trade-off
between two kinds of interests under normal circumstances.
3 Static game model
Next, this paper constructs a static game model to further explore corporate
governance. This static game model is based on a "macro" perspective. The reader
may be confused because the static game model is set up quite differently from the
dynamic game model, making it difficult to understand the connection between the
two models. In fact, to some extent, macroeconomic behavior is the "aggregation" of
microeconomic behavior, but macroeconomic behavior also shows different
characteristics from microeconomic behavior. The dynamic game model in this paper
describes the behaviors of different corporate governance bodies at the micro level
and tries to find the decision-making mechanism and rules of micro corporate
governance behaviors. The static game model established in this paper describes the
rules and mechanisms of corporate governance behaviors at the macro level (similar
to the aggregation of micro corporate governance behaviors). Therefore, this static
model improves the whole theoretical model of the corporate governance game in this
paper. The static model is also an essential piece of the puzzle.
We assume that there are only three market participant types: infringers, defenders,
and regulators. Managers are divided into "infringers," who will infringe upon the
interests of shareholders, and "defenders," who will try their best to serve the interests
of shareholders. This paper assumes that if the manager does not infringe on the
interests of shareholders, the "compliance costs" faced by the manager are
(potential loss); if the manager's income multiplier is , then  is used to
represent the total income obtained by the manager from the shareholders (the total
income is expressed as a multiple of the "compliance costs"). If the cost of
supervision is and the penalty multiplier is (
> 1
), the penalty cost of the
infringer monitored by the regulator is . Assume that the initial participants in the
market are infringers and defenders, and the total number of infringers and defenders
is . In the presence of regulators, the number of regulators is , the total number of
the three market participants is (
+
), and regulators are in the same market
environment as defenders and bear compliance costs().
Table
Setting and description of the static game model
The main variables and settings of the model are given in Table 6.
Symbol
description
Expected revenue of infringers
Expected revenue of defenders
Expected revenue of regulators
Expected returns of the whole society
Total number of infringers and defenders
Number of regulators
The cost of supervision
penalty multiplier,
> 1
compliance costs
Income multiplier,
> 1
Total number of market participants (
+
)
It can be seen from the model that the expected return of infringers, defenders, and
regulators can be given by the following formula:

 
󰀦

 
󰀦

 
󰀦
When all participants are defenders (without regulators), the market's overall return
(or social welfare) reaches the maximum, and the overall expected return will achieve
the Pareto optimality:
 

󰀦
When there are infringers in the market (without regulators), the equilibrium point
of the whole society lies in the position where the individual return is zero (because
  , then 
 󰀦 
 
󰀦).Therefore, the whole society
will be full of infringers, and social groups will fall into predicaments:

That is to say, when there is an infringer (without a regulator) in society, the
optimal decision of the individual is to become an infringer as well. In this case, the
overall welfare of the society is 0. Further, it is assumed that the market has two types
of subjects: infringers and defenders. The number of infringers is   
          
, representing the total number of market subjects. Furthermore, infringers and
defenders constitute binary random variables. Assuming that the regulator has a
unique position, it is responsible for supervising two types of subjects (that means the
regulator is a sampler of the market). We assume that the regulator takes samples at
one time, the number of          
       of a sample containing a specific
number of infringers and defenders is:
  


From this, we can get the income matrix of infringers and defenders.
Infringement
Compliance
Being Discovered

  
󰀦

 
󰀦

 
󰀦

 
󰀦
Not being discovered
And the profit matrix of regulators.
Infringement
Compliance
Find out

  
󰀦

 
󰀦

 
󰀦

 
󰀦
Fail
From the above income matrix, it can be seen that for market participants
(infringers and defenders) and regulators, the expected return of individuals in a
single sampling can be given below:
=




󰀦 




󰀦 

 󰀦



 
󰀦 


 󰀦

 
󰀦


 
󰀦(7)




󰀦  





󰀦 

 
󰀦 



 󰀦  (8)




󰀦 




󰀦 

 󰀦 



 󰀦


 󰀦 



 
󰀦(9)
From Eq.(7), Eq.(8), and Eq.(9), we can see that the total expected return of the
three parties is:




󰀦



󰀦



󰀦




󰀦


󰀦



󰀦 



󰀦




󰀦



󰀦


󰀦



󰀦 



󰀦 



󰀦 

 
󰀦   (10)
From Eq. (10), the total expected return of the society can be further obtained:

  
󰀦
When there are only defenders in society:

 
󰀦
When there are only infringers in society:

 
󰀦
When there are only regulators in society:

 
󰀦
Eq.(10) describes a society in which there are both infringers, defenders, and
regulators. The following is a further analysis of the differences between the overall
expected return of the society when the society is composed of the above three parties
and the expected return when there only exists a single market subject:
. When   , we can get:
 
.When     , we can get:
δ
γ  δ
γ
.When γ γ  δ γ δ, we can get:
δ
γ
.When    , we can get:
 
As      stands for "compliance costs,"
Inequality .doesn't hold, and we exclude this possibility. It can be seen from the
above that in the case of tripartite coexistence, the overall expected return of the
society may be in three intervals. In the model of tripartite coexistence, due to the
constraints and restrictions among the three parties, the overall return of the society
can be higher than the "absolutely selfish" world (the whole society will be full of
infringers, and social groups will fall into predicaments,  ), and can also be
lower than the "Pareto optimality" that the society should have achieved. Therefore,
even in the theoretical model, many possibilities exist for social welfare.
When      
    
         

        
           
           
 
            
            
      
    
           
             
            
           
           
      


  erall expected return of the society is higher than
that when only defenders exist, and this improved result is achieved when three
parties coexist in the model rather than when only one party exists. Therefore, it can
be concluded that the common restrictions of the three parties can realize the
maximization of social welfare under certain conditions. Moreover, the introduction
of regulators and defenders avoids the "worst result" (the social welfare is zero).
However, the equilibrium point of the game model is uncertain.
4 Conclusion
This paper uses two game models to fit the market subject behavior in corporate
governance from a dynamic and static dimension. Adding variables with practical
significance, such as "bribery" and "social welfare," into the game theory model can
better fit the real market environment. In the part of the dynamic game model, this
paper draws six conclusions.10 In the part of the static game model, the main
conclusions are as follows:
First, in the model of tripartite coexistence, the restrictions among the different
parties avoid the possibility of "the worst result." Under certain conditions, social
welfare can even be improved.
Second, in the multi-coexistence game model, the equilibrium point of the static
game model is uncertain, which makes the social welfare situation uncertain.
The relevant conclusions have corresponding implications for improving corporate
governance, optimizing system design, and external supervision. As a symbol of
social justice, the supervision departments should strengthen the punishment of the
behaviors of infringing the interests of shareholders while actively performing their
duties and create a just and free market environment; As a part of the company, the
supervisory board should try its best to reduce the cost of daily supervision and carry
out regular supervision; Shareholders, as relative bodies in the enterprises, should
10 The dynamic game model is an "open"model, and different perspectives may draw different
conclusions.
appeal in time when they are infringed by managers so as to reduce the possibility of
repeated violations. At the same time, we can improve the motivation of social
supervision by reducing regulatory costs. It is worth noting that it is difficult to
determine the actual regulatory costs, income multiplier, and other variables in the
real market, and there are differences in culture and development levels between
different regions and countries. Therefore, objectively, different equilibrium points
and levels of social welfare can be expected. This paper does not draw a general
conclusion but, based on fitting the real market, concludes with certain universality.
In conclusion, based on the market environment, we find that the intensity of
stimulation and the maintenance of fair and free competition are indispensable steps
for establishing an efficient corporate governance mechanism.
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