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The Three-party Evolutionary Game Under "Re-familization" 
Family Policies 

Shutong Qian* 

Nankai University, Tianjin, China 

*Corresponding author: 1912229@mail.nankai.edu.cn 

Abstract. With the development of society and the change of fertility culture, the fertility rate is 
gradually decreasing around the world, which leads to a lot of problems, so the Chinese government 
is seeking effective family policies to improve fertility levels. Based on the evolutionary game theory, 
the article defines the three participants in the evolutionary process of fertility choice and their payoff 
functions, then constructs a three-party evolutionary game model among individuals, firms and local 
governments, and finally analyzes the strategic choices of each party under the government-led 
family policies. The findings suggest that economic policies that directly give cash incentives and tax 
benefits are more effective in increasing fertility rate and creating a fertility-friendly social 
environment. Thus, the author suggests that the government should mainly use economic policies 
to promote the increase of fertility level and solve the social problem of low fertility rate. 

Keywords: Re-familization, Fertility Choice, Family Policy; Evolutionary Game Theory. 

1. Introduction 

With the development of society and the change of fertility culture, the fertility rate has gradually 

decreased around the world, which is often below the replacement level of 2.1, and the population 

size is on the trend of shrinking. The low fertility rate brings multiple problems such as population 

aging, labor shortage and lack of economic vitality, and has received wide attention from academic 

circles. 

China has achieved low fertility level for nearly 30 years and now enters a very low fertility stage. 

To encourage childbirth, the Chinese government has loosened fertility restrictions, but those policies 

didn’t work well. In that situation, an NPC deputy said in an interview that the length of maternity 

leave for women should be extended to three years while liberalizing the three-child birth policy and 

granting birth incentives. Both policies he mentioned are typical "re-familization" family policies. 

The term "re-familization" refers to the idea that families should bear the costs of childbirth, rather 

than sharing them with other welfare producers (such as the government, the market and society). 

Recent changes in policies suggest that the Chinese government prefers to adopt "re-familization" 

family policies in the short term to increase fertility rate. Thus, this paper focuses on the differences 

in the effects of different "re-familization" family policies. The article examines the choices of 

individuals, firms, and local governments in fertility issues through an evolutionary game approach, 

then analyzes the interests of each party and the equilibrium of the game, and finally makes 

suggestions about family policy choices for the Chinese government. 

2. Literature Review 

2.1 Factors that Influencing Fertility Behavior 

The factors that influencing fertility choices have been analyzed comprehensively in the academic 

community and can be summarized into four main categories: Political, economic, social, and 

demographic factors. 

Political factors have the most direct and targeted influence on fertility behavior, which include 

fertility policies, family policies, and welfare policies. Fertility policy refers to guidelines set by the 

national government to regulate the fertility behavior of people in reproductive age, and China's 

family planning policy is a typical fertility policy. Strictly enforced quantitative fertility restrictions 



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and high fines for over-births have a significant effect on fertility reduction, but the relaxation of 

fertility restrictions does not lead to a sustained increase in fertility [1-4]. Family policies can be 

divided into two types: "De-familization" family policies and "re-familization" family policies. "De-

familization" family policies refer to service policies (providing childcare services). "Re-

familization" family policies refer to economic policies (providing fertility bonuses and tax 

preferences) and time policies (extending maternity leave and paternity leave). Researches have 

generally shown that "de-familization" family policies are more conducive to increasing fertility rate, 

but they require a large investment in creating childcare service system, which is difficult to achieve 

in a short period of time in a country with a large population like China [5]. The social welfare and 

security system may affect individuals' measure of economic benefits from childbearing. The better 

public education, public health care, and retirement security are, the lower people's expectations of 

economic benefits from their children they have, and thus their willingness to have children decrease 

[6,7]. 

Economic factors affect people's fertility behavior mainly by influencing the cost of childbearing 

for families, and can be divided into society production methods, income levels, and the cost of 

childbearing. Increased female labor force participation and delayed retirement due to industrialized 

production methods have a negative impact on fertility levels [8-10]. The income level has a 

significant effect on fertility behavior, but the direction of its influence is somewhat controversial 

among researchers. While an increase in income may increase the ability to raise children, it also 

increases the opportunity cost of childbearing, which may lead to a reduction in fertility behavior 

[11,12]. In addition to the income per capita, the distribution of income and the stability of income 

sources can also influence people’s fertility choices [13-15]. Fertility costs include both direct and 

indirect costs: Direct costs are the money spent from the mother's pregnancy to the child's self-

sufficiency, including living costs, education costs, and medical costs. Indirect costs are the time and 

effort spent in raising a child, and the opportunity cost of income decline and potential promotion 

opportunity lost. Their increase has a significant inhibitory effect on fertility rate [16-18]. 

Social factors tend to act on fertility behavior by influencing the mental gains people derive from 

their children, including cultural customs and education levels. Gender-specific preferences, the 

importance attached to reproductive behavior in different cultures, and the social status of women can 

influence people's reproductive choices [11,19]. As the education level of people in childbearing ages 

increases, their participation in the workforce, marriage and childbearing are delayed, and both 

fertility intentions and fertility rates decrease [20]. 

Demographic factors are intermediate variables affecting fertility, covering factors such as marital 

status, age at childbearing, family structure, and the "low fertility trap". Since this aspect is not 

covered in this paper, it will not be discussed here. 

2.2 Researches on Fertility Issues from the Game Theory Perspective 

By combing through the literature related to fertility issues, it is found that most scholars currently 

use quantitative analysis to examine the effects of different factors on fertility. For example, Zhiqiang 

and Zaiping analyzed the uneven sharing of the fertility net cost among the government, firms, and 

families through a labor supply and demand model [21]. Zilian used a time-series and panel data 

vector autoregressive model to analyze the mutual shock effect between income and fertility [22]. 

Minja and Hyungseok used the period-child progressive total fertility rate to describe marriage and 

childbearing behavior, and compared the effects of factors such as women's education level and the 

development of childcare services on fertility through regression analysis [23]. 

Only a small amount of literature using game theory approach to study fertility issues. There is 

still some room for academic development in this area. In previous studies, Yuyin, Hanmin and 

Kainan have provided a comprehensive overview of the characteristics and application scope of the 

evolutionary game theory in their articles [24,25]. The objects of evolutionary games are often groups 

of people who change their strategies over time. The evolutionary game theory follows the 

assumption of finite rationality and assumes that participants do not possess full knowledge of the 



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game structure and rules, so they cannot adopt the optimal strategy at the beginning, but they would 

constantly modify and improve their behavior in the course of time, imitating the successful strategies. 

Therefore, the evolutionary game model can better reflect the complexity and diversity of participants, 

better simulate the behavior of people in the real society and provide a theoretical basis for policy 

making. These characteristics of evolutionary game theory make it very applicable to the study of 

fertility issues. When making a fertility decision, individuals are often not fully aware of their firm’s 

attitudes toward childbirth and possible policies in the future. In that case, people would rely heavily 

on the choices made by the group in the past. Over time, some random factors emerge to influence 

people's fertility decisions, which is very close to the assumptions of the evolutionary game theory. 

Therefore, this paper chooses to apply the research method of evolutionary game to examine the 

fertility issues, to analyze the influence of different factors on the participants and the final 

evolutionary stabilization strategy, which is also the innovation of this paper. 

3. Fertility Behavior Evolutionary Game Model 

3.1 Definition of Game Participants 

In the fertility behavior evolutionary game model, there are three types of participants: Individuals, 

firms and local governments. It is assumed that all three parties involved in the game are not perfectly 

rational. Under this condition, each party is unable to find their own optimal strategy at the beginning, 

but they would adjust their strategies according to their experience. 

Individuals are the decision makers and actual bearers of fertility costs. For individuals, the costs 

and benefits of childbearing are the main factors that influencing their reproductive choices. The costs 

of childbearing can be divided into direct and indirect costs: Direct costs include living costs, 

educational costs, medical costs, which are affected by the price level. Indirect costs include the time 

and effort spent on raising children, as well as the possible loss of job income and promotion 

opportunities due to childbearing. The benefits of childbirth can be divided into economic benefits 

and spiritual benefits: Economic benefits include the income that children bring to the family when 

they become adults, the insurance for retirement and the ability to resist risks, which are influenced 

by the production method and the economic condition of the family. Spiritual benefits include the 

sense of completeness and satisfaction from childbirth, as well as an increase in the couple's 

relationship with each other, and relief of social pressure, which are influenced by cultural customs 

and the education level. Individuals are influenced by many factors when making fertility choices and 

have different perceptions of the fertility costs and benefits, and their opinions of having a child 

change over time, so individuals are assumed to be imperfectly rational. 

Firms are the fertility cost co-sharers. In the short term, childbirth only imposes multiple costs on 

the firm, including the efficiency loss due to interruption of employees' work and adjustment of 

employees' work schedules, and the expense of recruiting new employees or paying overtime to other 

employees to share the work. In the long term, childbirth brings multiple benefits to the business, 

including future workforce and potential consumers. Firms may only focus on short-term costs, or 

take long-term benefits into consideration. Firms’ different perceptions towards fertility make their 

choices somewhat unstable, so firms are assumed to be imperfectly rational. 

Local governments are the fertility policy makers and supervisors. In the short run, family policies 

require a large amount of funding and human resources to implement and monitor; different family 

policies may have different effects on the government's reputation. In the long run, fertility will bring 

many social benefits, such as accumulating future labor productivity, boosting domestic demand, 

passing on national culture, and ensuring human reproduction; at the same time, fertility will bring 

future tax revenue to the government and increase government revenue. A short-sighted government 

will only consider the short-term costs and benefits, while a forward-thinking government will take 

potential benefits and social gains into account. Different government attitudes may affect their policy 

choices and there is some instability, so local governments are assumed to be imperfectly rational. 



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3.2 Model Hypothesis and Parameters 

Hypothesis 1: Individuals are participant 1, firms are participants 2, and local governments are 

participants 3. All three parties have imperfect rationality, and the strategy choice evolves gradually 

over time to their optimal strategy. 

Hypothesis 2: Individuals have a strategy set of having children and not having children. 

Individuals choose to have children at a rate of 𝑥 and choose not to have children at a rate of (1 −
𝑥). 

Hypothesis 3: Firms have a strategy set of being maternity friendly and being maternity 

discriminatory. Being maternity friendly means that the company treats pregnant employees normally, 

arranges appropriate work contents for them, allows them to take maternity leave, and gives them the 

wages and subsidies they deserve. Being maternity discriminatory means that firms discriminate 

against pregnant employees by transferring them to other jobs, reducing their salaries, or even 

dismissing them. Firms choose to be maternity friendly at a rate of 𝑦 and choose to be maternity 

discriminatory at a rate of (1 − 𝑦). 

Hypothesis 4: Local governments have a strategy set of economic policy and time policy. 

Economic policy means that local governments offer direct childbirth bonuses and childcare subsidies 

for families and give tax incentives to individuals who have children and fertility-friendly firms. Time 

policy means that local governments extend statutory maternity leave for women and paternity leave 

for men. Local governments choose the economic policy at a rate of 𝑧 and the time policy at a rate 

of (1 − 𝑧). 

Other parameters are set as shown in Table 1, and all parameters are positive. 

 

Table 1. Parameter descriptions 
Parameter Description 

Fertility cost for individual 𝐶1 
The direct and indirect costs when an individual chooses to have 

children. 

Cost of maternity leave 

extension for individual 𝐶2 

Employment difficulties and promotion difficulties for individuals due to 

extended maternity leave. 

Additional fertility cost for 

individual 𝐶3 

Wages and job opportunities lost for childbearers when they are in a 

fertility discriminatory firm. 

Fertility cost for company 𝐶4 

Fertility friendly firms incurs efficiency losses, overtime payments and 

expenses from recruiting new employees when their employees choose 

to have children. 

Cost of maternity leave 

extension for company 𝐶5 

The increase of firm costs due to the extension of statutory maternity 

leave. 

Fertility revenue for individual 

𝑅1 
The financial and spiritual benefits to the person who have children. 

Revenue of maternity leave 

extension for individual 𝑅2 

The decline in childcare costs for individual due to the extension of 

maternity leave. 

Fertility bonus 𝐵 
Under the economic policy, local governments give fertility bonuses and 

childcare allowances to childbearers. 

Tax preference for individual 

𝑇1 
Tax benefits to individuals who have children under economic policies. 

Tax preference for company 

𝑇2 
Tax benefits to fertility-friendly firms under economic policies. 

Penalty 𝑃 Penalties when a firm discriminates pregnant employees. 

Reputation enhancement 𝑆1 The increase of firm reputation when a firm is fertility friendly. 

Reputation decline 𝑆2 The decline of firm reputation when a firm is fertility discriminatory. 

Credibility enhancement 𝐺1 
The increase in government credibility when a local government choose 

economic policies. 

Credibility decline 𝐺2 
The decline in government credibility when a local government choose 

time policies. 



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3.3 Local Government-firm-individual Game Model Construction 

According to the parameters set in the previous section, the payoff matrix shown in Table 2 is 

constructed. 
 

Table 2. Payoff matrix of individuals, firms and local governments 

 Local government: economic policy (𝑧) 
Local government: time policy (1 −

𝑧) 

 
Firm: maternity 

friendly (𝑦) 

Firm: maternity 

discriminatory (1 −
𝑦) 

Firm: 

maternity 

friendly (𝑦) 

Firm: maternity 

discriminator (1 −
𝑦) 

Individual 

have 

children 

(𝑥) 

(−𝐶1 + 𝑅1 +
𝐵 + 𝑇1, −𝐶4 +
𝑇2 + 𝑆1, −𝐵 −
𝑇1 − 𝑇2 + 𝐺1) 

(−𝐶1 − 𝐶3 + 𝑅1 +
𝐵 + 𝑇1 + 𝑃, −𝑃 −
𝑆2, −𝐵 − 𝑇1 + 𝐺1) 

(−𝐶1 − 𝐶2 +
𝑅1 + 𝑅2, 

−𝐶4 − 𝐶5 +
𝑆1, −𝐺2) 

(−𝐶1 − 𝐶2 − 𝐶3 +
𝑅1 + 𝑅2 + 𝑃, 

−𝑃 − 𝑆2, −𝐺2) 

not have 

children 

(1 − 𝑥) 

(0, 𝑇2 + 𝑆1, 

−𝑇2 + 𝐺1) 
(0, −𝑆2, 𝐺1) 

(−𝐶2, 𝑆1, 

−𝐺2) 
(−𝐶2, −𝑆2, −𝐺2) 

 

Here are the expected benefits 𝑈𝑥1 for individuals choosing to have children, 𝑈𝑥2 for not having 

children, and the replication dynamic equation 𝐹: 

 

    (1) 

 

𝑈𝑥2 = 𝑦(1 − 𝑧)(−𝐶2) + (1 − 𝑦)(1 − 𝑧)(−𝐶2)                      (2) 

 

          (3) 

 

Here are the expected benefits 𝑈𝑦1 for maternity friendly firms, 𝑈𝑦2 for maternity discriminatory 

firms, and the replication dynamic equation 𝐺: 

 

𝑈𝑦1 = 𝑥𝑧(−𝐶4 + 𝑇2 + 𝑆1) + (1 − 𝑥)𝑧(𝑇2 + 𝑆1) + 𝑥(1 − 𝑧)(−𝐶4 − 𝐶5 + 𝑆1) + (1 − 𝑥)(1 − 𝑧)𝑆1  (4) 

 

𝑈𝑦2 = 𝑥𝑧(−𝑃 − 𝑆2) + (1 − 𝑥)𝑧(−𝑆2) + x(1 − z)(−𝑃 − 𝑆2) + (1 − x)(1 − z)(−𝑆2)    (5) 

 

𝐺 =
𝑑𝑦

𝑑𝑡
= 𝑦(1 − 𝑦)(𝑈𝑦1 − 𝑈𝑦2) = 𝑦(1 − 𝑦)[(𝑆1 + 𝑆2) − (𝐶4 + 𝐶5 − 𝑃)𝑥 + 𝑇2𝑧 + 𝐶5𝑥𝑧]   (6) 

 

Here are the expected benefits 𝑈𝑧1 for local governments implementing economic policy, 𝑈𝑧2 

for implementing time policy, and the replication dynamic equation 𝐻: 

 
𝑈𝑧1 = 𝑥𝑦(−𝐵 − 𝑇1 − 𝑇2 + 𝐺1) + 𝑥(1 − 𝑦)(−𝐵 − 𝑇1 + 𝐺1) + (1 − 𝑥)𝑦(−𝑇2 + 𝐺1) + (1 − 𝑥)(1 − 𝑦)𝐺1  (7) 

 

𝑈𝑧2 = 𝑥𝑦(−𝐺2) + 𝑥(1 − 𝑦)(−𝐺2) + (1 − 𝑥)𝑦(−𝐺2) + (1 − 𝑥)(1 − 𝑦)(−𝐺2)       (8) 

 

𝐻 =
𝑑𝑧

𝑑𝑡
= 𝑧(1 − 𝑧)(𝑈𝑧1 − 𝑈𝑧2) = 𝑧(1 − 𝑧)[(𝐺1 + 𝐺2) − (𝐵 + 𝑇1)𝑥 − 𝑇2𝑦]        (9) 



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4. Model Analysis 

4.1 Stability Analysis of Individual Strategy Choice 

Let 𝐹(𝑥) = 0 , getting 𝑥 = 0 , 𝑥 = 1 , and (−𝐶1 − 𝐶3 + 𝑅1 + 𝑅2 + 𝑃) + (𝐶3 − 𝑃)𝑦 + (𝐵 +
𝑇1 − 𝑅2)𝑧 = 0. If (−𝐶1 − 𝐶3 + 𝑅1 + 𝑅2 + 𝑃) + (𝐶3 − 𝑃)𝑦 + (𝐵 + 𝑇1 − 𝑅2)𝑧 = 0, then 𝐹(𝑥) ≡ 0, 

which means that regardless of the proportion of individuals choosing to have or not to have children, 

their strategies do not change over time. If (−𝐶1 − 𝐶3 + 𝑅1 + 𝑅2 + 𝑃) + (𝐶3 − 𝑃)𝑦 + (𝐵 + 𝑇1 −
𝑅2)𝑧 ≠ 0, letting 𝐹(𝑥) = 0, 𝑥 = 0 and 𝑥 = 1 are the two stable points of 𝑥. This means that 

individuals will be stable in their strategy as long as there are no mutating factors changing the 

strategy choice. 

Set 𝑎 = 𝐶3 − 𝑃 , and 𝑎  represents the gap between the additional fertility cost and the 

compensation. Since 𝐶3 is often greater than 𝑃, 𝑎 > 0 is set here. The effect of the individuals’ 

strategy choice rate on their stable equilibrium strategy is:  

 
𝜕𝐹(𝑥)

𝜕𝑥
= (1 − 2𝑥)[(−𝐶1 + 𝑅1 + 𝑅2 − 𝑎) + 𝑎𝑦 + (𝐵 + 𝑇1 − 𝑅2)𝑧]            (10) 

 

Letting 𝜆𝑦′ =
(𝐶1−𝑅1−𝑅2+𝑎)+(𝑅2−𝐵−𝑇1)𝑧

𝑎
, when 𝑦 > 𝜆𝑦′, 

𝜕𝐹(𝑥)

𝜕𝑥
|
𝑥=1

< 0, the individual evolutionary 

stabilization strategy is 𝑥∗ = 1, which means that individuals tend to have children; when 𝑦 < 𝜆𝑦′, 
𝜕𝐹(𝑥)

𝜕𝑥
|
𝑥=0

< 0 , the individual evolutionary stabilization strategy is 𝑥∗ = 0 , which means that 

individuals tend to not have children. Letting 𝜆𝑧′ =
(−𝐶1+𝑅1+𝑅2−𝑎)+𝑎𝑦

𝑅2−𝐵−𝑇1
, the value of (𝑅2 − 𝐵 − 𝑇1) 

would hugely affect the individual evolutionary stability strategy, and the relationship between 𝑥 and 

𝑧 is not significant. 

From the analysis above, it is clear that individuals are influenced by the choice of firms but is not 

significantly related to the choice of local governments. 𝑥 increases as 𝑦 rises. When firms tend to 

be maternity friendly, individuals tend to have children. In addition, whether individuals choose to 

have children is related to the fertility cost 𝐶1, the fertility revenue 𝑅1, the revenue of maternity leave 

extension 𝑅2 and the gap between the additional fertility cost and the compensation 𝑎, as well as the 

fertility bonus 𝐵 and the tax preference 𝑇1 . When 𝐶1  is smaller, and 𝑅1 , 𝑅2 , 𝐵, 𝑇1  and 𝑎 are 

bigger, 𝑦 > 𝜆𝑦′ exists, which means that individuals tend to have children.  

4.2 Stability Analysis of Firm Strategy Choice 

Let 𝐺(𝑦) = 0 , getting 𝑦 = 0, 𝑦 = 1, and (𝑆1 + 𝑆2) − (𝐶4 + 𝐶5 − 𝑃)𝑥 + 𝑇2𝑧 + 𝐶5𝑥𝑧 = 0. If 

(𝑆1 + 𝑆2) − (𝐶4 + 𝐶5 − 𝑃)𝑥 + 𝑇2𝑧 + 𝐶5𝑥𝑧 = 0, then 𝐺(𝑦) ≡ 0, which means that regardless of the 

proportion of firms choosing to be maternity friendly or maternity discriminatory, their strategies do 

not change over time. If (𝑆1 + 𝑆2) − (𝐶4 + 𝐶5 − 𝑃)𝑥 + 𝑇2𝑧 + 𝐶5𝑥𝑧 ≠ 0, letting 𝐺(𝑦) = 0, 𝑦 = 0 

and 𝑦 = 1 are the two stable points of 𝑦. This means that firms will be stable in their strategy as 

long as there are no mutating factors changing the strategy choice. 

Since 𝐶4 is often greater than 𝑃, 𝐶4 − 𝑃 > 0 is set here. The effect of the firms’ strategy choice 

rate on their stable equilibrium strategy is:  

 
𝜕𝐺(𝑦)

𝜕𝑦
= (1 − 2𝑦)[(𝑆1 + 𝑆2) − (𝐶4 + 𝐶5 − 𝑃)𝑥 + 𝑇2𝑧 + 𝐶5𝑥𝑧]             (11) 

 

Letting 𝜆𝑥′ =
(𝑆1+𝑆2)+𝑇2𝑧

(𝐶4+𝐶5−𝑃)−𝐶5𝑧
, when 𝑥 > 𝜆𝑥′ , 

𝜕𝐺(𝑦)

𝜕𝑦
|
𝑦=0

< 0, the firm evolutionary stabilization 

strategy is 𝑦∗ = 0, which means that firms tend to be maternity discriminatory; when 𝑥 < 𝜆𝑥′ , 
𝜕𝐺(𝑦)

𝜕𝑦
|
𝑦=1

< 0, the firm evolutionary stabilization strategy is 𝑦∗ = 1, which means that firms tend to 



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be maternity friendly. Similarly, letting 𝜆𝑧′ =
(𝐶4+𝐶5−𝑃)𝑥−(𝑆1+𝑆2)

𝑇2+𝐶5𝑥
, when 𝑧 > 𝜆𝑧′, 

𝜕𝐺(𝑦)

𝜕𝑦
|
𝑦=1

< 0, the 

firm evolutionary stabilization strategy is 𝑦∗ = 1 , which means that firms tend to be maternity 

friendly; when 𝑧 < 𝜆𝑧′, 
𝜕𝐺(𝑦)

𝜕𝑦
|
𝑦=0

< 0, the firm evolutionary stabilization strategy is 𝑦∗ = 0, which 

means that firms tend to be maternity discriminatory.  

From the analysis above, it is clear that firms are influenced by the choices of individuals and local 

governments. 𝑦 increases as 𝑥 decreases and 𝑧 rises. When individuals tend not to have children 

and local governments tend to choose economic policy, firms tend to be maternity friendly. In 

addition, whether firms choose to be maternity friendly is related to the fertility cost 𝐶4, the cost of 

maternity leave extension 𝐶5, the penalty 𝑃 and the tax preference 𝑇2, as well as the reputation 

enhancement 𝑆1 and the reputation decline 𝑆2. When 𝐶4 and 𝐶5 are smaller, and 𝑃, 𝑇2 and (𝑆1 +
𝑆2) are bigger, 𝑥 < 𝜆𝑥′ exists, which means that firms tend to be maternity friendly.  

4.3 Stability Analysis of Local Government Strategy Choice 

Let 𝐻(𝑧) = 0, getting 𝑧 = 0, 𝑧 = 1, and (𝐺1 + 𝐺2) − (𝐵 + 𝑇1)𝑥 − 𝑇2𝑦 = 0. If (𝐺1 + 𝐺2) −
(𝐵 + 𝑇1)𝑥 − 𝑇2𝑦 = 0 , then 𝐻(𝑧) ≡ 0 , which means that regardless of the proportion of local 

governments choosing economic policy or time policy, their strategies do not change over time. If 

(𝐺1 + 𝐺2) − (𝐵 + 𝑇1)𝑥 − 𝑇2𝑦 ≠ 0, letting 𝐻(𝑧) = 0, 𝑧 = 0 and 𝑧 = 1 are the two stable points 

of 𝑧. This means that local governments will be stable in their strategy as long as there are no 

mutating factors changing the strategy choice. 

The effect of the local governments’ strategy choice rate on their stable equilibrium strategy is:  

 
𝜕𝐻(𝑧)

𝜕𝑧
= (1 − 2𝑧)[(𝐺1 + 𝐺2) − (𝐵 + 𝑇1)𝑥 − 𝑇2𝑦]                  (12) 

 

Letting 𝜆𝑥′ =
(𝐺1+𝐺2)−𝑇2𝑦

𝐵+𝑇1
, when 𝑥 > 𝜆𝑥′ , 

𝜕𝐻(𝑧)

𝜕𝑧
|
𝑧=0

< 0 , the local government evolutionary 

stabilization strategy is 𝑧∗ = 0, which means that local governments tend to choose time policy; 

when 𝑥 < 𝜆𝑥′ , 
𝜕𝐻(𝑧)

𝜕𝑧
|
𝑧=1

< 0, the local government evolutionary stabilization strategy is 𝑧∗ = 1, 

which means that local governments tend to choose economic policy. Similarly, letting 𝜆𝑦′ =
(𝐺1+𝐺2)−(𝐵+𝑇1)𝑥

𝑇2
, when 𝑦 > 𝜆𝑦′ , 

𝜕𝐻(𝑧)

𝜕𝑧
|
𝑧=0

< 0 , the local government evolutionary stabilization 

strategy is 𝑧∗ = 0, which means that local governments tend to choose time policy; when 𝑦 < 𝜆𝑦′, 
𝜕𝐻(𝑧)

𝜕𝑧
|
𝑧=1

< 0, the local government evolutionary stabilization strategy is 𝑧∗ = 1, which means that 

local governments tend to choose economic policy.  

From the analysis above, it is clear that local governments are influenced by the choices of 

individuals and firms. 𝑧 increases as 𝑥 and 𝑦 decreases. When individuals tend not to have children 

and firms tend to be maternity discriminatory, local governments tend to choose economic policy. In 

addition, whether local governments choose economic policy is related to the fertility bonus 𝐵, the 

tax preferences 𝑇1 and 𝑇2, as well as the credibility enhancement 𝐺1 and the credibility decline 𝐺2. 

When 𝐵, 𝑇1 and 𝑇2 are smaller, and (𝐺1 + 𝐺2) is bigger, 𝑥 < 𝜆𝑥′ exists, which means that local 

governments tend to choose economic policy.  

4.4 Evolutionary Equilibrium of Fertility Behavior System 

The equilibrium point derived from a single replicated dynamic equation is not necessarily an 

evolutionary stable strategy for the system. Therefore, the stability of the evolutionary equilibrium 

point of three parties will be further analyzed in this paper. Here is the Jacobi matrix obtained from 

the dynamic replication equations of the three parties: 
 



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269 

𝐽 = [
𝐽11 𝐽12 𝐽13

𝐽21 𝐽22 𝐽23

𝐽31 𝐽32 𝐽33

] =

[
 
 
 
 
𝜕𝐹

𝜕𝑥

𝜕𝐹

𝜕𝑦

𝜕𝐹

𝜕𝑧

𝜕𝐺

𝜕𝑥

𝜕𝐺

𝜕𝑦

𝜕𝐺

𝜕𝑧

𝜕𝐻

𝜕𝑥

𝜕𝐻

𝜕𝑦

𝜕𝐻

𝜕𝑧]
 
 
 
 

                       (13) 

 

𝐽11 =
𝜕𝐹

𝜕𝑥
= (1 − 2𝑥)[(−𝐶1 − 𝐶3 + 𝑅1 + 𝑅2 + 𝑃) + (𝐶3 − 𝑃)𝑦 + (𝐵 + 𝑇1 − 𝑅2)𝑧]       (14) 

 

𝐽12 =
𝜕𝐹

𝜕𝑦
= 𝑥(1 − 𝑥)(𝐶3 − 𝑃)                          (15) 

 

𝐽13 =
𝜕𝐹

𝜕𝑧
= 𝑥(1 − 𝑥)(𝐵 + 𝑇1 − 𝑅2)                        (16) 

 

𝐽21 =
𝜕𝐺

𝜕𝑥
= 𝑦(1 − 𝑦)[−𝐶4 − 𝐶5 + 𝑃 + 𝐶5𝑧]                     (17) 

 

𝐽22 =
𝜕𝐺

𝜕𝑦
= (1 − 2𝑦)[(𝑆1 + 𝑆2) − (𝐶4 + 𝐶5 − 𝑃)𝑥 + 𝑇2𝑧 + 𝐶5𝑥𝑧]             (18) 

 

𝐽23 =
𝜕𝐺

𝜕𝑧
= 𝑦(1 − 𝑦)(𝑇2 + 𝐶5𝑥)                         (19) 

 

𝐽31 =
𝜕𝐻

𝜕𝑥
= 𝑧(1 − 𝑧)(−𝐵 − 𝑇1)                         (20) 

 

𝐽32 =
𝜕𝐻

𝜕𝑦
= 𝑧(1 − 𝑧)(−𝑇2)                            (21) 

 

𝐽33 =
𝜕𝐻

𝜕𝑧
= (1 − 2𝑧)[(𝐺1 + 𝐺2) − (𝐵 + 𝑇1)𝑥 − 𝑇2𝑦]                 (22) 

 

From the previous analysis, it is clear that the equilibrium points of the system are: 𝐸1(0, 0, 0), 

𝐸2(1, 0, 0), 𝐸3(0, 1, 0), 𝐸4(0, 0, 1), 𝐸5(1, 1, 0), 𝐸6(1, 0, 1), 𝐸7(0, 1, 1), and 𝐸8(1, 1, 1). Since this 

paper focuses on explaining how to create a fertility-friendly social environment and promote fertility 

rate through appropriate family policies, the stability of 𝐸5(1, 1, 0)  and 𝐸8(1, 1, 1)  is mainly 

analyzed here. 

Firstly, here is the situation at 𝐸5(1, 1, 0): 

 

𝐽 = [

𝐶1 − 𝑅1 − 𝑅2 0 0
0 −(𝑆1 + 𝑆2 − 𝐶4 − 𝐶5 + 𝑃) 0
0 0 𝐺1 + 𝐺2 − 𝐵 − 𝑇1 − 𝑇2

]          (23) 

 

Only when 𝐶1 < 𝑅1 + 𝑅2, 𝑆1 + 𝑆2 > 𝐶4 + 𝐶5 − 𝑃, and 𝐺1 + 𝐺2 < 𝐵 + 𝑇1 + 𝑇2 hold at the same 

time, 𝐸5(1, 1, 0) is the evolutionary stabilization strategy of the system. Individuals choose to have 

children, when the fertility cost 𝐶1 is less than the sum of the fertility revenue 𝑅1 and the revenue 

of maternity leave extension 𝑅2. Firms choose to be maternity friendly, when the gap in the impact 

of a firm's choice on reputation (𝑆1 + 𝑆2) is greater than the increase in cost it brings (𝐶4 + 𝐶5 − 𝑃). 

Local governments choose time policy, when the gap in the impact of a local government’s choice 

on credibility (𝐺1 + 𝐺2) is less than the increase in cost it brings (𝐵 + 𝑇1 + 𝑇2). 

Then, here is the situation at 𝐸8(1, 1, 1): 

 

𝐽 = [

𝐶1 − 𝑅1 − 𝐵 − 𝑇1 0 0
0 −(𝑆1 + 𝑆2 − 𝐶4 + 𝑃 + 𝑇2) 0
0 0 −(𝐺1 + 𝐺2 − 𝐵 − 𝑇1 − 𝑇2)

]     (24) 



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Only when 𝐶1 < 𝑅1 + 𝐵 + 𝑇1, 𝑆1 + 𝑆2 + 𝑇2 > 𝐶4 − 𝑃, and 𝐺1 + 𝐺2 > 𝐵 + 𝑇1 + 𝑇2 hold at the 

same time, 𝐸8(1, 1, 1) is the evolutionary stabilization strategy of the system. Individuals choose to 

have children, when the fertility cost 𝐶1 is less than the sum of the fertility revenue 𝑅1 and the 

economic compensation (𝐵 + 𝑇1). Firms choose to be maternity friendly, when the sum of the gap in 

the impact of a firm's choice on reputation (𝑆1 + 𝑆2) and the tax preference 𝑇2 is greater than the 

increase in cost it brings (𝐶4 − 𝑃). Local governments choose economic policy, when the gap in the 

impact of a local government’s choice on credibility (𝐺1 + 𝐺2) is greater than the increase in cost it 

brings (𝐵 + 𝑇1 + 𝑇2). 

5. Conclusion 

In the evolution of fertility behavior, there are many factors affecting participants’ decisions, and 

any participant changing its own choice will change the costs and benefits of others, thus affecting 

their strategic choices. Therefore, when choosing family policies, local governments need to consider 

all the factors, so as to guide firms to have fertility-friendly attitudes and promote individuals to have 

children, and thus gradually increase the fertility rate in China. 

From the replication dynamic equation of individuals, it is clear that the proportion of local 

governments’ choice does not have a significant effect on individuals' fertility choice. Local 

governments can only change people's fertility choice by affect their maternity leave extension gain 

𝑅2, fertility reward B and fertility tax benefit 𝑇1. The larger 𝑅2, 𝐵 and 𝑇1 are, the higher proportion 

of individuals choosing to have children would be. Among these parameters, 𝐵  and 𝑇1  can be 

directly decided by local governments. 𝑅2 is affected by the price level, and it is not possible for 

local governments to change the price level in China through a single family policy. Thus, economic 

policies are more feasible than time policies. Therefore, this paper suggests that the government 

choose economic policies, or a combination of family policies based on economic policies to increase 

the benefits individuals receive from childbirth and reduce the costs of childbirth by granting fertility 

bonuses, childcare subsidies, and providing tax benefits to childbearers, so as to promote the increase 

of fertility. 

From the replication dynamic equation of firms, it is clear that economic policies are more helpful 

for firms to choose fertility-friendly attitudes and create a fertility-friendly social environment. 

Meanwhile, the larger reputation gap (𝑆1 + 𝑆2), penalty 𝑃, and tax preference 𝑇2 are, the higher 

proportion of firms choosing to be fertility friendly would be. Among these parameters, 𝑇2 can be 

directly determined by local governments. 𝑃 is related to the probability of employees choosing labor 

arbitration, and the probability is related to workers' knowledge of labor law and local governments’ 

efforts to popularize the law. The greater probability of labor arbitration is, the greater the penalties 

would be. (𝑆1 + 𝑆2) is related to social concern about fertility issues and women's rights, and efforts 

to promote corporate culture. Therefore, this paper suggests that local governments, while choosing 

economic policies, should strengthen the popularization of labor laws, increase workers' awareness 

of protecting their legal rights, and make employees dare to engage in labor arbitration, so that the 

cost of maternity discriminatory firms increases. Local governments can also advocate a maternity-

friendly concept in society and offer recognition to maternity-friendly firms, so that making (𝑆1 + 𝑆2) 

increases, thus increasing the benefits of fertility-friendly firms. 

From the replication dynamic equation of local governments, it is clear that when individuals tend 

not to have children, firms tend to be fertility discriminatory, local governments tend to choose 

economic policies. Such an analysis is also consistent with the current low fertility rate and negative 

social perceptions of fertility in China, suggesting that local governments should choose economic 

policies. 

From the Jacobi matrix and the analysis above, it is clear that the equilibrium point 𝐸8(1, 1, 1) is 

the evolutionary stabilization strategy of the replication dynamic system, only when 𝐶1 < 𝑅1 + 𝐵 +
𝑇1 , 𝑆1 + 𝑆2 + 𝑇2 > 𝐶4 − 𝑃 , and 𝐺1 + 𝐺2 > 𝐵 + 𝑇1 + 𝑇2  hold at the same time. At that point, 

individuals choose to have children, firms choose to be fertility friendly, and local governments 



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choose economic policies. Meanwhile, local governments should take a longer-term view, and take 

social benefits of fertility and future financial resources into account, as well as calculate the benefits 

in a comprehensive way, so that the system eventually converges to 𝐸8(1, 1, 1) and achieves the goal 

of higher fertility rate and a fertility-friendly society. 

In the model analysis part, this paper didn’t distinguish the gender of individuals, but generally 

grouping them into one category, ignoring the gender differences in the fertility costs and their effects 

on fertility decisions, which is a shortcoming of this paper. In future studies, scholars can further 

refine the groups of individuals to make their research more accurate. 

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