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Game theory model of corporate governance: Fitting real market environment

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Abstract

In this paper, I use two game models to fit the market subject behavior in corporate governance from dynamic dimension and static dimension, respectively. By 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 to check out manager's violation, the punishment to the irregularities, the supervision cost of the supervisory board, have an important impact on the behavior of the corporate governance. In the static game model, this paper finds that intersubjective games and restrictions avoid “the worst result” to occur. 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 two game models can be used for reference to further improve the corporate governance mechanism. JEL classification: G30; M10; D21; D81
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
DOI: https://doi.org/10.21203/rs.3.rs-2401729/v2
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
November 20,2022
Abstract
In this paper, I use two game models to fit the market subject behavior in corporate
governance from dynamic dimension and static dimension, respectively. By 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 to check out manager's violation, the punishment to the
irregularities, the supervision cost of the supervisory board, have an important impact
on the behavior of the corporate governance. In the static game model, this paper
finds that intersubjective games and restrictions avoid “the worst result” to occur.
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
two game models can be used for reference to further improve the corporate
governance mechanism.
Key words: corporate governance; dynamic game model; static game model
JEL classification: G30; M10; D21; D81
1 Introduction
The principal-agent problem is the core problem of corporate governance. The
1I sincerely thank Professor Haotian Xiang for his suggestions and modifications to this article, as well as
Professor Lei Ming's guidance on the game model solving process. Both professors have checked every detail of
this paper. Of course, all possible mistakes are mine.
2Contact information:sunnyzyxxyz@stu.pku.edu.cn
main problem in the study of corporate governance is how to protect the rights and
interests of fund providers(Shleifer and Vishny,1996). The literature of corporate
governance tend 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 main
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)believes 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 negative 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 important complementary role in improving the corporate
governance mechanism (Yi et al., 2010). However, little literature has taken both
internal and external corporate governance subjects into consideration, and the
interaction between main bodies will certainly lead to corruption (Rodriguez et
al.,2006). In the absence of 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. How to establish theoretical model based on the real
market environment has important practical significance for promoting the practice
and reform of corporate governance.
Has been documented in a lot of literature, corporate governance is primarily
studied in the aspect of mechanism design. Ross and Laffont (1993)have made
outstanding contributions to the establishment of "incentive" mechanism of corporate
governance through introducing "private information", such as "moral hazard" and
"adverse selection", into the model. The literature in the 1970s is mostly 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)build a
mathematical analysis framework to quantify the principal-agent problem. Based on
their research, Mirrlees has made great contribution to the information theory and
incentive theory. The optimal policy design adopted by Mirrlees is a classical method
of the non-symmetry information Economics. 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 the aspects of executive incentive, enterprise incentive, internal control
and budget constraint. Although the former perspective can grasp the core problem of
corporate governance, it lacks accurate quantitative analysis. From the perspective of
"quantitative analysis", the mathematical method has been applied successfully, yet
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 problem 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, which also includes signal
transfer theory, incentive theory, common agency theory and so on. The game theory
model cannot be simply considered as the optimal model 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, by introducing "bribery", "social welfare" and
other variables into the model, the assumptions are more consistent with the real
market, making the conclusions more explanatory to some extent.3
The paper is organized as follows:section 2 describes the dynamic game model,
section 3 is the static game model. Conclusions 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 have the motivation to
infringe on the interests of shareholders can they bribe the supervisory board. Second,
if the supervisory board accepts the 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
don't take bribes.
In this paper, "social welfare" refers to the change of social welfare in a broad sense.
The change of 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
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
3It is worth noting that the game theory model constructed in this paper still depends on widely
accepted assumptions of human behavior, but the corresponding assumptions are more in line with
the real situation. In addition, irrational behaviors and the overlapping of behaviors between the
subjects also exist in the real market environment. In this paper, I give normative explanations for
these situations.
4This indicates that managers are not shareholders and shareholders do not carry out daily
management of the company.
5If the supervisory board does not accept the bribes from the managers, then the supervisory
board will naturally report the manager's illegal behaviors to the regulatory authorities.
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) and supervision department cannot
get a part of social welfare.
Table
Setting and description of dynamic game model
Firstly, I use 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. I assume
that the managers of the company act first, and then the shareholders act. When the
shareholders' interests are violated, they first appeal to the supervisory board of the
company. The supervisory board reports the violations to the external supervision
department, and the external supervision department 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 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
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
supervision department
1
Regulatory costs of the supervisory
board
1
>0
2
Regulatory costs of the external
2
>0
6This is a typical sequential game model.
supervision department
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
Benefit-sharing ratio between supervisory
board and shareholders
2
Coefficient between 0 and 1
Benefit-sharing ratio between supervision
department and shareholders
Coefficient between 0 and 1
Welfare-sharing ratio between supervision
department and the public
Social welfare
Social welfare in a broad sense.for example,
the improvement of social credibility and the
improvement of 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.
Fig1.Game tree based on dynamic game model
Figure 1 shows the sequence of sequential games. We can also show the
information conveyed in the game tree in the form of 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
out
Positive
13
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 and figure 1,table 3 and table 4 respectively, 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 supervisory board,supervision department and the public
According to the same rules, the income matrix of supervisory board, supervision
department and the public can be shown in table 4.
The income matrix of the public
The income matrix of supervision department
The income matrix of 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 dynamic game model is obtained by “backward
induction” approach. First, according to the maximization of expected return, the
optimal solution of the last actor is obtained and brought into the equation of the
penultimate actor. By continuing this process, the equilibrium solutions of the
dynamic game model are formed. Therefore, we first seek the optimal solution of "the
public":
=
S
S
+
S
S
+
S+
S
When we achieve the expected revenue maximization, the first-order derivative
equals to zero. Therefore, we can further get the following formula:


+
+
+
+
+
By simplification,we can get:

(1)
The Eq. (1) represents that the possibility of shareholders' appeal is closely related
to the possibility of supervision department checking out the violations of managers.
The more likely the supervision department is to find out the managers’ violation, the
more likely the shareholders are to appeal; otherwise, the less likely they are.
Next, we seek the optimal solution of "supervision department", by simplification,
we can get:


Take absolute values on both sides, and we can get (please refer to Appendix (1)
for the specific calculation process):




 (2)7
The 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 behavior of the supervision department depends to a
large extent on the punishments of the supervision department against the managers'
violations (2) and 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 "supervisory board", by
simplification, we can get (please refer to Appendix (2) for the specific calculation
process):

And we can further get:



(3)
According to Eq.(1), combined with Eq.(3), we can further obtain:



 

(4)
If we examine Eq.(4) closely, we can find that (
+
1
1
1
)is the
common part of numerator and denominator. This part of the fraction has the
relationship of the same increase and decrease, and the absolute amount of increase
and decrease is also the same. Therefore, the key factor of
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
tends to be a
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.
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
supervisory board. This conclusion is different from previous literatures. Most of
previous literatures 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 violations of the managers (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. If the board exists in
name only and does not carry out substantive supervision in the real market, then 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", by simplification, we can get
(please refer to Appendix (3) for the specific calculation process):
And then we can finally get:
(5)
Eq.(5) tells us a somewhat subtle conclusion that in the process of dynamic game,
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 cost of appeal is insignificant compared with the
interests infringed by the managers. If the shareholders waive their rights to appeal
because of the appeal cost, in repeated games, 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", by simplification, we can get
(please refer to Appendix (4) for the specific calculation process):
 


And then we can get:
 

Combined with Eq.(4),we can further obtain:




 







(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 numerator and denominator. This part of the fraction has the
relationship of the same increase and decrease, 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
)are 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: H 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 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 the "self-interest maximization"is not the leading aim of managers,
8The reasons have been given in the previous paragraphs.
but "self-protection" as 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 dynamic game model
From the above calculations and combined with C1,C2 and C3, we can get the
equilibrium solutions of dynamic game model, as is 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


Therefore, the conclusions of dynamic game model are as follows:
First, the possibility of shareholders' appeal is closely related to the possibility of
supervision department checking out the violations of managers.
Second, punishments on the managers and the profits from punishing managers can
9This conclusion does not conflict with the previous conclusion, and it shows the
decision-making process of the supervisory board under "normal" circumstances.
greatly affect the response of the public.
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 repeated violations of the managers are inevitable, then 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 out of the
awareness of "excessive risks", but 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. We assume that there are only three types of market participants,
including 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 also bear the compliance costs ().
Table
Setting and description of 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
=1
+

>
=1
+

).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 the 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 there are two types of
subjects in the market: infringers and defenders. The number of infringers is , the
number of defenders is , and the total number of infringers and defenders is
(
=
+
), which represents 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 to supervise two types of subjects
(that means, the regulator is sampler of the market). We assume that the regulator
takes samples at one time, the number of in the sample is , and the number
of (equals to )in the sample is ( ), then the probability 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 single
sampling can be given blow:







 
󰀦󰀦


󰀦






󰀦

  󰀦

  󰀦


 󰀦(7)








󰀦󰀦





󰀦

 
󰀦(8)







 
󰀦󰀦





󰀦




󰀦

 
󰀦


 󰀦(9)
From Eq.(7), Eq.(8) ,Eq.(9), we can see that the total expected return of the three
parties are:



 󰀦





 󰀦󰀦



 󰀦


󰀦



󰀦


󰀦



󰀦


 󰀦


󰀦



󰀦



󰀦



󰀦



󰀦(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
3

2
>
, we can get:
.When 
> 3

2
>
 , we can get:
.When 
> 3

2
>
 , we can get:
.When 
> 3

2
, we can get:
As
> 1
, we can further get:
< 0
.Since 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,
= 0
), and can also be
lower than the "Pareto optimality" that the society should have achieved. Therefore,
even in the theoretical model, there also exist many possibilities of social welfare.
When
3

2
>
 , we can get:
>
2(

1)
. Since stands for "income
multiplier" and
> 1
, thus,
2(
1)
can be regarded as a proportional coefficient
(similarly,
1
2

1
can also be regarded as a proportional coefficient). Therefore, the
key to the problem lies in the comparison between compliance costs ( ) and
regulatory costs ( ). It can be seen that when the regulatory costs (multiplied by
1
2(

1)
) of the society are less than the compliance costs, the regulatory authority has
the motivation to supervise, and the overall expected return of the society is the
largest; when the regulatory costs (multiplied by
1
2

1
) of the society are greater than
the compliance costs, the society will lack the motivation to supervise. At this time,
the overall expected return of the society is lower than the Pareto optimality, but the
overall expected return is still greater than zero. This shows that in the model of
tripartite coexistence, the restrictions among the three parties avoid the possibility of
"the worst result", and under certain conditions, social welfare can even been
improved.
When
2(

1) >
>
2

1
, the overall 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 occurrence of "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 dynamic dimension and static dimension respectively. By adding
variables with practical significance, such as "bribery" and "social welfare", into the
game theory model, the model can better fit the real market environment. In the part
of dynamic game model, this paper draws six conclusions.10 In the part of 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", and under certain conditions, social
welfare can even been 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 to 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
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 the regulatory costs. It is worth noting that it is difficult to
determine the actual regulatory costs, income multiplier and other variables in real
market, and there are differences in culture and development level between different
regions and countries. Therefore, objectively, different equilibrium points and
10 The dynamic game model is an "open"model, and different conclusions may be drawn from
different perspectives.
different levels of social welfare can be expected. This paper does not draw a general
conclusion, but on the premise of fitting the real market, draws a conclusion 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 an
indispensable step for the establishment of an efficient corporate governance
mechanism.
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Appendix
1) The optimal solution of "supervision department" is given by:


+
+

+
+
+
+
+
+
 
+

2) The optimal solution of "supervisory board" is given by:





3) The optimal solution of "shareholders" is given by:


4) The optimal solution of "managers" is given by:










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