An evolutionary game analysis of the prefabricated building supply side under dynamic government reward and punishment mechanisms

Abstract

From the standpoint of the supply side of prefabricated buildings, an evolutionary game model between the government and developers is constructed under the static and dynamic reward and punishment mechanisms in order to promote the development of prefabricated buildings and realize the transformation and upgrading of the construction industry. The simulation study of the example is conducted, and the impact of policy cost, subsidy upper limit, and tax upper limit on the system’s evolutionary stability strategy is examined. The findings demonstrate that when the government chooses static subsidy and static tax policies, there is no evolutionary stable strategy in the game system. The flaws of the static reward and punishment system can be adequately compensated for by the dynamic subsidy and dynamic tax policy, leading to an evolutionary stable state. The likelihood of developers producing prefabricated structures is negatively connected with policy cost and subsidy upper limits and favorably correlated with tax upper limits, according to the dynamic reward and punishment mechanism. The findings of the study have significant implications for government policy that will encourage the widespread use of prefabricated structures in China.

Keywords

Prefabricated building dynamic reward and punishment mechanism evolutionary game theory numerical simulation

Introduction

The building sector has steadily grown in importance to the national economy as urbanization has been accelerated. Prefabricated structures have emerged as the only option for modernizing and transforming China’s construction sector as a result of pressure from energy conservation, environmental protection, efficiency, and other factors.¹ The state has gradually developed support programs to promote the development of prefabricated structures in this environment. In 2016, the “Several Opinions of the CPC Central Committee and the State Council on Further Strengthening the Management of Urban Planning and Construction” clearly stated that “strive to use ten years or so to make prefabricated buildings account for 30% of new buildings.” At present, China’s prefabricated buildings have made great progress. By 2019, the scale of China’s prefabricated building market has climbed to 42.75% of the global proportion, but the proportion of prefabricated buildings in the new construction area is only about 10 %. The difference between the percentage of new construction and the 30% percentage is still very large. The “14th Five-Year Plan” objective still faces several obstacles. For instance, the low economic scale benefit, the high cost of construction, the poor construction technology, the low level of industrialization, and the low consumer perception of prefabricated buildings limit the developers’ excitement for the development of prefabricated structures. As a result, it is critical that the government set up a robust incentive system to encourage the orderly and seamless development of prefabricated buildings.

At present, domestic and foreign scholars have done a lot of research on how the government formulates policies to promote the development of prefabricated buildings. Fan and Wu² analyzed the impact of different incentive strategies on the level of high-rise green building development from a cost-benefit perspective using AHP and system dynamics model comparisons, and the results of the study showed that transaction costs are a determining factor in developing high-rise green buildings and that they should be included in the cost-benefit analysis in order to improve the design of the incentive mechanism. Based on the externality theory, Yu et al.³ constructed a bargaining game model between the government, developers and consumers and studied its externality benefit distribution mechanism so as to promote the development of the assembly building industry. He et al.⁴ constructed an evolutionary game model between the government and component producers, and analyzed the factors that weakened component producers’ production at different stages of the development of the assembly building market and the adjustment of the government’s incentive strategy. Jiang and Wang et al.^5,6 analyzed the interaction and evolutionary stabilization strategies of the government and developers through evolutionary game theory, explored the influencing factors of developers’ behavioral choices, and proposed corresponding incentive strategies based on the findings. Wang et al.⁷ used the policy tool perspective to establish an indicator evaluation system for assembly building incentive policies, constructed an evaluation model using gray correlation analysis, and proposed that governments at all levels should continuously improve incentive policies, pay attention to the balance of incentive policies at all levels of the industrial chain, and increase policy support for assembly building managers and operators.

Yang et al.⁸ revealed the changes in the game strategies of government groups and investment groups in encouraging green retrofits and implementing green retrofits through evolutionary game analysis. Yang et al.⁹ designed the government compensation mechanism for prefabricated buildings from two aspects of supervision and incentive. Adetayo et al.¹⁰ established a hierarchical structure diagram based on ISM and found that preferential policies such as subsidies, rewards, and tax reductions are important driving forces for the supply of prefabricated housing. Chen¹¹ established a “government-developer” prefabricated building revenue loss allocation model from the perspective of revenue loss to provide developers with corresponding economic compensation policies. El-Abidi et al.¹² quantitatively analyzed the incentive policies of prefabricated buildings in various parts of China and formulated different incentive strategies for different development stages and different beneficiaries. Meng and Strobel et al.^13,14 sorted out the government's fiscal and tax incentive policies in the housing industrialization and put forward the idea of formulating incentive policies from multiple perspectives. Wang et al.¹⁵ used the Mitt-Horn policy implementation model to analyze the implementation of China’s current prefabricated building policy and put forward the corresponding policy improvement path. At the same time, evolutionary game theory has been widely used in the development of prefabricated buildings. Li et al.¹⁶ constructed an evolutionary game among the three groups of design units, manufacturers, and construction enterprises under government intervention and provided policy incentive strategies for the coordinated development of the industrial chain. Fan and Hui¹⁷ constructed an evolutionary game model of local government incentive policies and innovation strategies of prefabricated construction enterprises, and analyzed the effective incentive strategies of the government. Dou et al.^18,19 pill took the interests of the government and developers as the starting point, used game theory to explore the behavior strategies of both parties in the development of prefabricated buildings, and elaborated on the mechanism of government incentive policies.

To sum up, from the perspective of existing research, most of them are government incentive policy research, and there are few studies on government punishment mechanism. There is a dearth of study on the mechanisms through which the government and construction units interact, and the majority of research subjects are construction units and construction units. Despite the literature²⁰ on evolutionary game analysis between the government and developers, there are still some limitations: most studies assume that the policy intensity is fixed, and there is little research on the government’s dynamic reward and punishment mechanism, which has been widely used in the development of green buildings,²¹ enterprise carbon emission reduction,²² promotion of electric vehicles,²³ lead battery production,²⁴ etc., to show that it is more effective. This paper establishes a supply-side evolutionary game model of the government and developers and discusses the system under the government’s static and dynamic reward and punishment mechanism on the basis of existing research, presuming that both the government and developers are limited rational persons and taking into account the interests of all parties. The evolutionary stability strategy and its mechanism of action on the choice of developer behavior, focusing on the analysis of the role of subsidy amount, tax punishment, and policy cost in the effective incentive of the government and the choice of developer strategy under the dynamic reward and punishment mechanism, and providing a theoretical basis for policy formulation and the development of prefabricated buildings.

Evolutionary game modeling

Model description

The development of prefabricated buildings involves multiple participants such as the government, consumers, developers, designers, construction parties, supervisors, and prefabricated parts manufacturers.^25–27 The pre-investment of prefabricated buildings is high. As an important participant on the supply side, the financing ability and development decision of developers are the key factors for their development, which determines the development direction of the prefabricated building market. Due to the existence of the externality of prefabricated buildings, the spontaneous adjustment of the market is difficult to achieve the effective allocation of resources. It needs the intervention and regulation of the government to guide the participants through administrative means and policy formulation, and drive the steady development of the prefabricated building market. Evolutionary stabilization strategy refers to the fact that in the process of the game, due to the limited rationality of both parties, it is impossible to find the most optimal strategy and the optimal equilibrium point at the very beginning of the game. Therefore, the players need to keep learning in the process of the game, and they will gradually correct their mistakes and keep imitating and improving their own and other people’s most favorable strategies in the past. After a period of imitating and correcting mistakes, all the players will converge to a certain stable strategy. Therefore, this paper chooses the government and developers as the game subject, based on the limited rationality of the two subjects, the establishment of the government and the developers of the evolution of the game model, the two behaviors interact with each other and through repeated learning to achieve the evolution of a stable strategy.

Fundamental assumption

Assumption 1: The government and the developer are selected as the main players of the game, assuming that they are all limited rational persons and the rationality of the two sides of the game is not equal, and both of them repeatedly play the game under the condition of asymmetric information, and constantly seek for the optimal strategy.

Assumption 2: the government’s strategy set: {incentive and restraint policies, formal policies}, the probability of the government choosing incentive and restraint policies is x, and the probability of choosing formal policies is 1−x, xÎ[0,1]. The strategy set of developers is: {develop prefabricated buildings, develop traditional buildings}, in which prefabricated buildings are not distinguished by assembly rate, the probability of developers choosing to develop prefabricated buildings is y, and the probability of choosing to develop traditional buildings is 1−y, yÎ[0,1].

Profit and loss parameter setting

In order to ensure that the setting and interpretation of the profit and loss parameters of the game parties are objective and reasonable, and reflect the difference between the prefabricated building and the traditional cast-in-place building, this paper combines the research of Hu et al. to sort out the selected indicators and parameters. For the game subjects of the government and developers, a total of 11 parameters were set up, as shown in Table 1.

Table 1.

Parameterization and interpretation of gains and losses on both sides of the game.

Game subject	Parameter setting and interpretation
Government	i: When the government adopts incentive and restraint policies, the amount of subsidies for selecting prefabricated building developers (economic subsidies, tax incentives, credit support, etc.)
	c₁: The cost of policy research, formulation, and publicity when the government adopts incentive and restraint policies
	c₂: When developers choose traditional buildings, the government pays the cost of governance (to solve the traditional construction method of high energy consumption, high pollution cost of manpower, and material resources)
	p₁: When developers choose traditional buildings, the government gains (land revenue and related taxes)
	p₂: When developers choose prefabricated buildings, the government gains (land revenue, related taxes, government credibility, resource conservation, environmental benefits, etc.)
Developer	c₃: Developers need to pay the cost when choosing traditional buildings
	c₄: Incremental cost of prefabricated buildings for developers (such as incremental cost of prefabricated building construction, technology research and development cost, and marketing cost)
	s: When the government adopts incentive and restraint policies, developers are punished for failing to meet the standards when developing traditional buildings (such as carbon tax, and energy tax)
	p₃: The direct economic benefits of developers' choice of traditional buildings
	p₄: The direct economic benefits of developers choice of prefabricated buildings
	p₅: Indirect benefits (corporate image, brand value, etc.) when developers choose prefabricated buildings

Note: The above parameters are greater than 0.

Game payoff matrix construction

According to the hypothesis and explanation of the appeal model, the game payment matrix between the government and the developer can be constructed as shown in Table 2.

Table 2.

Payment matrix for the government-developer game.

Both sides of the game		Developer
Both sides of the game		Development of prefabricated buildings(y)	Development of traditional architecture(1-y)
Government	Incentive and restraint policies(x)	(p₂-i-c₁, p₄+p₅+i-c₃-c₄)	(p₁+s-c₁-c₂, p₃-c₃-s)
Government	Formal policy(1-x)	(p₂, p₄+p₅-c₃-c₄)	(p₁-c₂, p₃-c₃)

According to the principle of evolutionary game, the expected revenue and the average government revenue of the government’s choice of incentive and restraint policies and formal policies are U_G1, U_G2, and U_G, respectively, and the following formula can be obtained:

\begin{array}{l} U_{G 1} = y (p_{2} - i - c_{1}) + (1 - y) (p_{1} + s - c_{1} - c_{2}) \\ U_{G 2} = y p_{2} + (1 - y) (p_{1} - c_{2}) \\ U_{G} = x U_{G 1} + (1 - x) U_{G 2} \end{array}

(1)

The expected revenue and average revenue of developers to develop prefabricated buildings and traditional buildings are U_D1, U_D2, and U_D, respectively, and the following formula can be obtained:

\begin{array}{l} U_{D 1} = x (p_{4} + p_{5} + i - c_{3} - c_{4}) + (1 - x) (p_{4} + p_{5} - c_{3} - c_{4}) \\ U_{D 2} = x (p_{3} - c_{3} - s) + (1 - x) (p_{3} - c_{3}) \\ U_{D} = y U_{D 1} + (1 - y) U_{D 2} \end{array}

(2)

Game analysis of assembly building supply side evolution under static reward and punishment mechanisms

Static incentives and penalties refer to the fact that the incentives and penalties set by the government department are fixed regardless of the strategy chosen by the developer and do not change according to the developer’s choice of strategy.

Replication dynamic equation analysis

The replication dynamic equation of the government’s choice to implement incentive and restraint policies:

F (x) = \frac{d x}{d t} = x (U_{G 1} - U_{G}) = x (1 - x) [- y (i + s) + s - c_{1}]

(3)

Developers choose to develop the replication dynamic equation of prefabricated buildings:

F (y) = \frac{d y}{d t} = y (U_{D 1} - U_{D}) = y (1 - y) [x (i + s) + p_{4} + p_{5} - c_{4} - p_{3})]

(4)

From the above formula, the two-dimensional dynamic system equation can be obtained:

{\begin{cases} \frac{d x}{d t} = x (1 - x) [- y (i + s) + s - c_{1}] \\ \frac{d y}{d t} = y (1 - y) [x (i + s) + p_{4} + p_{5} - c_{4} - p_{3}] \end{cases}

(5)

If we want to ensure the government’s reward and punishment mechanism to play its due effect, we need to ensure that the total benefit after subsidies to prefabricated building developers is greater than the total benefit after taxation on the development of traditional buildings, and the government’s taxation should meet the governance needs, that is, $p_{4} + p_{5} - c_{4} + i > p_{3} - s$ , $p_{1} + s - c_{1} - c_{2} > 0$ and $i > 0$ , $s > 0$ . The stability analysis of the system will be carried out on this premise.

Let F(x) = 0, F(y) = 0, five evolution equilibrium points can be obtained: (0,0), (0,1), (1,1), (1,0), ( $x^{*}$ , $y^{*}$ ), if and only if $0 < \frac{p_{3} + c_{4} - p_{4} - p_{5}}{i + s} < 1$ , $0 < \frac{s - c_{1}}{i + s} < 1$ , $x^{*} = \frac{p_{3} + c_{4} - p_{4} - p_{5}}{i + s}$ , $y^{*} = \frac{s - c_{1}}{i + s}$ are the equilibrium points of the system.

Equilibrium point stability analysis

According to the viewpoint proposed by Fredman, the stability of the equilibrium point of the evolutionary system can be analyzed by the local stability of the Jacobian matrix. When the determinant of the equilibrium point det(J) > 0 and tr(J) < 0, it indicates that the system tends to be locally asymptotically stable during the dynamic evolution process. This point is regarded as the local evolutionary stability strategy (ESS) of the system.²⁰ The system Jacobian matrix is as follows:

J = [\begin{array}{l} \frac{\partial F (x)}{\partial (x)} & \frac{\partial F (x)}{\partial F (y)} \\ \frac{\partial F (y)}{\partial (x)} & \frac{\partial F (y)}{\partial (y)} \end{array}] = [\begin{array}{l} (1 - 2 x) [- y (i + s) + s - c_{1}] & - x (1 - x) (i + s) \\ y (1 - y) (i + s) & (1 - 2 y) [x (i + s) + p_{4} + p_{5} - c_{4} - p_{3}] \end{array}]

When

0 < \frac{p_{3} + c_{4} - p_{4} - p_{5}}{i + s} < 1

0 < \frac{s - c_{1}}{i + s} < 1

, in order to facilitate analysis, let

a = p_{4} + p_{5} - c_{4}

, which means the net income obtained by developers when using prefabricated buildings relative to traditional buildings;

b = p_{1} - c_{2}

, which means the government’s income when developers develop traditional buildings. The stability analysis results of each equilibrium point are shown in Table 3.

Table 3.

Evolutionary stabilization strategies for systems with static reward and punishment mechanisms.

Point of equilibrium	det(J)	Sign	tr(J)	Sign	Result
(0,0)	(s−c₁) (a−p₃)	—	s-c₁+a−p₃	Inconclusive	Saddle point
(0,1)	(i+c₁) (a−p₃)	—	p₃-a-i-c₁	Inconclusive	Saddle point
(1,1)	(i+c₁) ( p₃−a-i−s)	—	c₁-s+p₃−a	Inconclusive	Saddle point
(1,0)	(c₁−s) (i + s+a−p₃)	—	c₁+i+a−p₃	Inconclusive	Saddle point
( $x^{}$ , $y^{}$ )	xy(1−x) (1−y)(i + s)²	+	0	0	Center point

It can be seen from Table 3 that (0,0), (0,1), (1,0), and (1,1) are not stable. The characteristic root corresponding to the equilibrium point ( $x^{*}$ , $y^{*}$ ) is a pure imaginary number. According to the relevant theoretical research, this point only has a stable limit cycle and does not have asymptotic stability, that is, the trajectory of the system evolution is a closed-loop curve around the center point, which cannot be automatically stabilized to the equilibrium point.²¹

From the above analysis, it can be seen that the government and developers do not have evolutionary stable strategies under the static reward and punishment mechanism, and the change of any variable will have a greater impact on the strategic choice of both sides of the game. When the initial state changes, the system evolves to different equilibrium points to achieve the evolutionary stability strategy of each case. The following analyzes the evolutionary stability strategy of the system under different initial states:

(1) When s-c₁<0, and p₃>a, the evolutionary stable strategy is (0,0);

(2) when s-c₁>0 and p₃<a, the evolutionary stable strategy is (0,1);

(3) when s-c₁<0 and p₃<a, the evolutionary stable strategy is (0,1);

(4) when s-c₁>0 and p₃<a + i + s, there is no evolutionary stable strategy;

(5) when s-c₁>0 and p₃>a + i + s(p₃>a), the evolutionary stable strategy is (1,0).

It can be seen from (1) that when the government’s tax on the development of traditional building developers is less than the cost of policy research, formulation, and promotion when adopting incentive and restraint policies, and the income from the development of traditional buildings is greater than the net income obtained by prefabricated buildings, the evolution trend of the government and developers is {a mere formality policy, the development of traditional buildings}, which usually occurs in the early stage of the development of prefabricated buildings.

It can be seen from (2) and (3) that when the net income from the development of prefabricated buildings is greater than the income from the development of traditional buildings, no matter how the government’s tax on traditional buildings and the cost of promoting prefabricated buildings change, the evolution trend of the government and developers tends to {incentive and restraint policies, development of prefabricated buildings}. Developers are driven by interests to actively develop prefabricated buildings. At this time, the government can dilute the reward and punishment mechanism and gradually improve the supporting security policies.

It can be seen from (4) that when the government’s tax on the development of traditional building developers is greater than the cost of formulating and promoting incentive and restraint policies, and the profit of developing prefabricated buildings is gradually increasing but still less than the profit of developing traditional buildings, the two sides do not have an evolutionary stability strategy, and the two sides of the game try each other. This is a critical period for the development of prefabricated buildings, and the government must attach great importance to it.

It can be seen from (5) that when the government’s tax on the development of traditional building developers is greater than the cost of formulating and promoting incentive and restraint policies, and the benefits of developing traditional buildings are greater than the benefits of developing prefabricated buildings, the evolution trend of the government and developers is {incentive and restraint policies, developing traditional buildings}.This situation shows that although the government is improving the tax policy to encourage the development of prefabricated buildings, the profits of developing traditional buildings are still considerable, and developers still tend to develop traditional buildings. At this time, the government should actively adjust and increase subsidies to stimulate the enthusiasm of developers to develop prefabricated buildings.

Game analysis of assembly building supply side evolution under dynamic reward and punishment mechanisms

In the previous model, the government’s reward and punishment strength is a fixed value, which is called a static reward and punishment mechanism. Under the dynamic reward and punishment mechanism, the government’s reward and punishment to developers is related to the developer’s strategic choice. When the probability of developers choosing to develop traditional buildings is higher, the government’s tax upper limit can be appropriately raised, thus exerting a deterrent effect. On the contrary, the tax upper limit can be reduced, so as to avoid the failure of government regulation and unnecessary fiscal expenditure due to the inflexibility of the static reward and punishment mechanism. The government’s subsidy to prefabricated building developers is $i (y) = (1 - y) \bar{i}$ , and $\bar{i}$ indicates the upper limit of the subsidy amount. It means that when the probability of developers choosing traditional (1−y) is higher, the government’s subsidy to developers choosing prefabricated buildings is greater. In addition, the government’s tax on traditional building developers is $s (y) = (1 - y) \bar{s}$ and $\bar{s}$ represents the upper limit of the tax amount, which means that the higher the probability of developers choosing the traditional (1−y), the more tax penalties they will be subject to. The evolutionary game dynamic system between the government and developers under the dynamic reward and punishment mechanism of the government is reconstructed as follows.

The replication dynamic equation is as follows:

\begin{array}{l} F (x) = \frac{d x}{d t} = x (1 - x) [- y (i (y) + s (y)) + s (y) - c_{1}] = x (1 - x) [- y (1 - y) (\bar{i} + \bar{s}) + (1 - y) \bar{s} - c_{1}] \\ F (y) = \frac{d y}{d t} = y (1 - y) [x ((i (y) + s (y)) + a - p_{3}] = y (1 - y) [x (1 - y) (\bar{i} + \bar{s}) + a - p_{3}] \end{array}

Let F(x) = 0, F(y) = 0, five evolution equilibrium points can be obtained: (0,0), (0,1), (1,1), (1,0), ( $x_{1}^{*}$ , $y_{1}^{*}$ ), where if and only if $0 < \frac{2 (p_{3} - a)}{\bar{i} + \sqrt{{\bar{i}}^{2} + 4 c_{1} (\bar{i} + \bar{s})}} < 1, 0 < \frac{\bar{i} + 2 \bar{s} - \sqrt{{\bar{i}}^{2} + 4 c_{1} (\bar{i} + \bar{s})}}{2 (\bar{i} + \bar{s})} < 1$ , $x_{1}^{*} = \frac{2 (p_{3} - a)}{\bar{i} + \sqrt{{\bar{i}}^{2} + 4 c_{1} (\bar{i} + \bar{s})}}, y_{1}^{*} = \frac{\bar{i} + 2 \bar{s} - \sqrt{{\bar{i}}^{2} + 4 c_{1} (\bar{i} + \bar{s})}}{2 (\bar{i} + \bar{s})}$ are the equilibrium points of the system, and the Jacobian matrix is as follows:

J^{'} = [\begin{array}{l} (1 - 2 x) [- y (1 - y) (\bar{i} + \bar{s}) + (1 - y) \bar{s} - c_{1}] & x (1 - x) [(2 y - 1) (\bar{i} + \bar{s}) - \bar{s}] \\ y {(1 - y)}^{2} (\bar{i} + \bar{s}) & (1 - 2 y) (a - p 3) + x (1 - y) (\bar{i} + \bar{s}) (1 - 3 y) \end{array}]

Similarly, the evolutionary stability strategy of the system under dynamic subsidies and dynamic taxation is shown in Table 4.

Table 4.

Evolutionary stabilization strategies for systems with dynamic reward and punishment mechanisms.

Point of equilibrium	det(J)	Sign	tr(J)	Sign	Result
(0,0)	$(\bar{s} - c_{1}) (a - p_{3})$	—	$\bar{s} - c_{1} + a - p_{3}$	Inconclusive	Saddle point
(0,1)	$- c_{1} (p_{3} - a)$	—	$p_{3} - a - c_{1}$	Inconclusive	Saddle point
(1,1)	$c_{1} (p_{3} - a)$	+	$p_{3} - a + c_{1}$	+	Saddle point
(1,0)	$(c_{1} - \bar{s}) (a - p_{3} + \bar{i} + \bar{s})$	—	$c_{1} - \bar{s} + a - p_{3} + \bar{i} + \bar{s}$	Inconclusive	Saddle point
( $x^{}$ , $y^{}$ )	+	+	—	—	Stable point

It can be seen from the above table that the system equilibrium points (0, 0), (0, 1), (1, 1), and (1, 0) are not stable. The characteristic root of the Jacobian matrix corresponding to the equilibrium point ( $x_{1}^{*}$ , $y_{1}^{*}$ ) is a pair of characteristic composite roots with negative real parts. According to Taylor’s research, ( $x_{1}^{*}$ , $y_{1}^{*}$ ) is a stable point, which has progressive stability. The evolutionary trend of both parties is a spiral curve that tends to this point, which indicates that there is an evolutionary stable strategy between the government and the developer under dynamic subsidies and dynamic tax policies. When the probability of the government choosing incentive and restraint policies is $x_{1}^{*}$ , the probability of the developer choosing to develop prefabricated buildings is $y_{1}^{*}$ .

According to the asymptotic stability point ( $x_{1}^{*}$ , $y_{1}^{*}$ ), the influence of dynamic reward and punishment mechanism and related parameter changes on the strategy choice of government and developers is analyzed. When the government implements the incentive and restraint policy, the influence of policy cost c₁, government subsidy upper limit $\bar{i}$ , and government tax upper limit $\bar{s}$ on the strategy choice of the game subject is as follows:

(1) $\frac{\partial x_{1}^{*}}{\partial c_{1}} < 0, \frac{\partial x_{1}^{*}}{\partial \bar{i}} < 0, \frac{\partial x_{1}^{*}}{\partial \bar{s}} < 0$

(2) $\frac{\partial y_{1}^{*}}{\partial c_{1}} < 0, \frac{\partial y_{1}^{*}}{\partial \bar{i}} < 0, \frac{\partial y_{1}^{*}}{\partial \bar{s}} > 0$

It can be seen from (1) that $x_{1}^{*}$ is the decreasing function of $c_{1}$ , $\bar{i}$ , and $\bar{s}$ . That is, the probability of the government adopting incentive and restraint policies is inversely proportional to the cost of government incentive policies, the upper limit of subsidies, and the upper limit of taxes.

It can be seen from (2) that $y_{1}^{*}$ is the decreasing function of $c_{1}$ , $\bar{i}$ , and the increasing function of $\bar{s}$ . That is, the probability of developers developing prefabricated buildings is inversely proportional to the cost of government incentive policies and the upper limit of subsidies, and is proportional to the upper limit of taxation.

The above results show that the higher the cost of government investment in the formulation and promotion of incentive and restraint policies, the lower the probability of policy implementation, and the lower the willingness of developers to develop prefabricated buildings. When the government’s tax on developers of traditional buildings is increased, the probability of developers choosing to develop prefabricated buildings will increase, and the government will appropriately reduce the policy cost. When the government’s subsidy for developers who choose prefabricated buildings increases, in the short term, developers will rely too much on financial subsidies to choose to develop prefabricated buildings, resulting in increased financial burden, and the government will reduce the promotion cost of prefabricated buildings. Over time, the probability that developers choose to develop prefabricated buildings will decrease. Therefore, a reasonable amount of tax and subsidy is particularly important for the promotion of prefabricated buildings.

Calculus analysis

Under the premise of satisfying the model requirements $a + i > p_{3} - s$ , and $i > 0$ , $s > 0$ , the initial parameters of the system are assigned according to the actual situation, and the values are simulated and analyzed by MATLAB. The initial parameter values were set as follows: $c_{1} = 0.7$ , $a = 1.8$ , $b = 2$ , $p_{3} = 3$ , $\bar{i} = 1$ , $\bar{s} = 1.2$ (Unit: Millions). The simulation start time is 0, and the end time is 50 (Unit: Month).

Firstly, based on the setting of the initial parameter values, the evolution process of the system under the government’s static reward and punishment mechanism and the dynamic reward and punishment mechanism is analyzed, respectively.

From Figure 1, it can be seen that under the static reward and punishment mechanism, the evolution path of the government and developers is a closed-loop curve around the equilibrium point, that is, under the static reward and punishment mechanism, the system does not have an evolutionary stability strategy. From Figure 2, it can be seen from that under the dynamic reward and punishment mechanism, the behavior evolution trajectory of the government and developers is constantly approaching around the equilibrium point, spiraling convergence, and finally stabilizing at the equilibrium point, indicating that the system has an evolutionary stability strategy under the dynamic reward and punishment mechanism. Consistent with the previous analysis, the appeal simulation results show that the government’s adoption of dynamic taxation and dynamic subsidy systems can effectively make up for the shortcomings of static taxation and static subsidy policies.

Figure 1.

Evolutionary path diagram under static reward and punishment mechanism.

Figure 2.

Evolutionary path diagram under dynamic reward and punishment mechanism.

Secondly, under the dynamic reward and punishment mechanism, the influence of each parameter change on the decision probability of the government and the developer is further discussed. The initial decision probability is (0.2, 0.2):

(1) The impact of the change of policy cost c₁ on the probability of strategy selection of government and developers. Under the condition that other parameters remain unchanged and meet the conditions of $a + i > p_{3} - s$ , $b + s - c_{1}$ , the value of c₁ is set to 0.7, 1.2, and 0.4, and the stability strategy changes of the government and developers under different conditions are discussed, respectively, as shown in Figures 3 and 4. With the increase of policy cost, the probability of government choosing incentive and restraint policies is reduced, which leads to the probability of developers choosing prefabricated buildings is also reduced, which verifies that in the dynamic reward and punishment mechanism x and y are the subtraction functions of c₁. From a practical point of view, the increase in policy costs makes the financial burden heavier, and the probability of the government implementing incentive and restraint policies is reduced. When the policy cost reaches a high value, the government will stop implementing incentive and restraint policies. From the perspective of developers, the increase in policy costs may be transferred to the construction costs of developers, resulting in an increase in the indirect costs of developers developing prefabricated buildings, reducing profit margins, thereby reducing the enthusiasm of developers to develop prefabricated buildings.

(2) The influence of the change of the upper limit value $\bar{i}$ of government subsidy on the probability of strategy selection of government and developers. Under the condition that other parameters remain unchanged, the value of $\bar{i}$ is set to 1, 1.5, and 2, respectively. The behavior strategies of the government and developers are shown in Figures 5 and 6. With the increase of the upper limit of the government subsidy amount, when the stability strategy is reached, the probability of the government choosing the incentive and restraint policy is reduced, and the probability of the developer choosing the prefabricated building is also reduced, which verifies that x and y are the decreasing functions of $\bar{i}$ under the dynamic reward and punishment mechanism. As shown in Figure 6, in a short period of time, increasing the upper limit of government subsidies can increase the probability of developers choosing prefabricated buildings. However, as time goes on, when developers reach a stable strategy, the probability of increasing government subsidies for developers to choose prefabricated buildings decreases. This is because the long-term dependence of developers on financial subsidies will bring huge financial pressure to the government, limit the development of other activities, and thus reduce the government’s incentive willingness. The willingness of developers to develop prefabricated buildings will also be reduced, so blindly increasing the amount of subsidies will not increase the probability of developers applying prefabricated buildings.

(3) The influence of the change of the upper limit $\bar{s}$ of the tax amount on the probability of the government and the developer’s strategic choice. Under the condition that other parameters remain unchanged, the value of $\bar{s}$ is set to 1.2, 1.7, and 2.2, respectively. The behavior strategies of the government and developers are shown in Figures 7 and 8. With the increase of the upper limit of the government’s tax quota, when the stable strategy is reached, the probability of the government’s choice of incentive and restraint policies is reduced, and the probability of developers choice of prefabricated buildings is increased, which verifies that under the dynamic reward and punishment mechanism, x is a decreasing function of $\bar{s}$ and y is an increasing function of $\bar{s}$ . From a practical point of view, in the early stage of construction projects, when the tax upper limit increases, the cost of developers to develop traditional buildings increases, thereby reducing profits, so developers will spontaneously turn to the development of prefabricated construction projects. With the increase in the probability of developers developing prefabricated buildings, the government will re-learn new strategies to reduce policy costs, thereby reducing the probability of implementing incentive and restraint policies. This shows that the government should also strengthen the tax management system when formulating incentive policies.

Figure 3.

Impact of policy cost on government strategy choice.

Figure 4.

Impact of policy cost on developers’ strategy choice.

Figure 5.

Impact of subsidy amount on government strategy choice.

Figure 6.

Impact of subsidy amount on developers’ strategy choice.

Figure 7.

Impact of tax quota on government strategy choice.

Figure 8.

Impact of tax quota developers’ strategy selection.

Conclusion and suggestions

This paper mainly analyzes the evolution paths and equilibrium points of the government and developers under the condition of limited rationality under the static and dynamic reward and punishment mechanisms of assembled buildings, and discusses the impacts of the policy cost, the upper limit value of subsidy amount, and the upper limit value of tax amount on the evolutionary stabilization strategies of the two sides of the game. The following conclusions are drawn from the study:

(1) When the government adopts a static reward and punishment mechanism, that is, the financial subsidies and taxes on developers are fixed. Under the condition of $p_{4} + p_{5} - c_{4} + i > p_{3} - s$ , $p_{1} + s - c_{1} - c_{2} > 0$ , the government and developers tend to pursue their own interests, and the evolution path of both sides is cyclical around the equilibrium point, which cannot reach a stable state.

(2) When the government adopts a dynamic reward and punishment mechanism, that is, the government dynamically adjusts financial subsidies and tax quotas according to the probability of developers developing prefabricated buildings. Similarly, under the condition of $p_{4} + p_{5} - c_{4} + i > p_{3} - s$ , $p_{1} + s - c_{1} - c_{2} > 0$ , it can effectively solve the problem that there is no evolutionary strategy under the static reward and punishment mechanism. This shows that the dynamic reward and punishment mechanism is more in line with the needs of the market. The government makes policy adjustments according to the developer’s behavior strategy, and the game process between the two sides will develop in an ideal direction.

(3) Under the dynamic reward and punishment mechanism, as an important starting point for economic and policy regulation, the government’s behavior strategy is affected by many factors. The probability of implementing incentive and restraint policies is inversely proportional to the policy cost, the upper limit of subsidy amount, and the upper limit of tax amount. As the main body of the supply side of prefabricated buildings, the behavior choice of developers is related to the implementation factors of the government. The probability of developing prefabricated buildings is inversely proportional to the policy cost and the upper limit of the subsidy amount, and is proportional to the upper limit of the tax amount.

Through the above conclusions, this paper puts forward the following suggestions for the formulation of government policies and the development of assembly:

(1) Dynamic reward and punishment mechanisms are put in place. The government’s system of rewards and penalties has to be modified to reflect developers’ preferred conduct. Prefabricated structures now have a lower net income than traditional constructions. The government uses a fixed system of rewards and penalties, and the actions of developers are unpredictable. A steady evolution plan cannot be reached by the two parties. This combination of policies is counterproductive for the market and will surely raise the financial burden. In order to maximize the incentive for developers to create prefabricated buildings, the government should closely monitor the behavior of developers through supervision and other methods, and dynamically adjust the rewards and punishments according to the completion of the phased goals. When the net income of developing prefabricated buildings exceeds that of traditional buildings, developers will be driven by interests to actively develop prefabricated buildings. At that time, the government can weaken the incentive and restraint policies and give full play to the market vitality.

(2) Reduce policy costs. The behavior approach of developers will be impacted by the change in policy cost. The expense of policy study, formulation, and advertising will be passed on to developers as a result of the rise in policy costs, which will decrease the likelihood that prefabricated structures will be developed. As a result, the government should increase the effectiveness of oversight, implement the use of policy costs precisely, cut back on wasteful spending, and lower policy costs. In addition, the government may streamline the approval process, save time and money, and perform its duties more effectively by opening a green channel for prefabricated building developers.

(3) Rationalize the amount of subsidies. Blindly raising the upper limit value of financial subsidies will not enhance the probability of developers to develop assembled buildings, but on the contrary, it will reduce the enthusiasm of developers, increase the financial pressure of the government, and impede the development of construction industrialization. Therefore, the amount of subsidies must be scientifically measured, in response to changes in the market, according to the dynamic behavior of developers to choose to adjust the subsidy strategy in a timely manner. In addition to the need for certain subsidies to developers should also be in the administrative costs, urban construction support costs, approval process, quality assurance deposit, etc., to reduce the development cost of assembly building, forming a good policy dividend, and developers should also seize the policy dividend, the introduction of advanced technology, training professionals, take the initiative to develop the assembly building, and assume the corresponding social responsibility.

(4) Improve the tax system. From the previous analysis, it can be seen that raising the upper limit value of the tax amount to develop traditional building developers can play an effective role in incentivizing the development and application of assembled buildings, so the government should strengthen the tax management system, starting from multiple perspectives, such as tax policy, land policy and fiscal policy, and use tax as a kind of auxiliary incentives to form a dynamic and effective disciplinary mechanism.

(5) Increase publicity. The growth of the scale of prefabricated buildings will ultimately be paid by consumers. At this stage, consumers’ insufficient understanding of prefabricated buildings, coupled with their high prices, leads to insufficient demand for prefabricated buildings. The government should increase publicity to guide consumers to purchase prefabricated buildings. Specifically, it can increase the construction of pilot cities and pilot projects for prefabricated buildings, continuously open up the market, and enable consumers to have a better understanding of prefabricated buildings through publicity channels. Appropriately give consumers purchase subsidies, loan interest rate concessions, etc., increase their willingness to purchase, and increase the demand side of prefabricated buildings to drive the development of the supply side of prefabricated buildings.

However, this paper is mainly from the supply side perspective, respectively, explored the static and dynamic reward and punishment mechanism under the government and developers of the evolution of the stabilization strategy, while the assembly building development process also involves the design unit, the construction unit, and many other stakeholders, due to space limitations, this paper has not yet ventured into the study of other stakeholders. Therefore, in the subsequent research, the behavior of other subjects can be included in the scope of analysis, and the multi-party game model can be constructed to further deepen the research.

Statements and declarations

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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10.

Adetayo

Innocent