# Stochastic Evolutionary Game in Food Takeaway Management with Improved Replication Dynamic Equations in Postepidemic Era

Authors: Ying Zhang; Haojie Zhu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The rapid growth of the food takeaway industry during COVID-19 raises a claim to its quality improvement, especially the hygiene condition. In the takeaway chain, the food takeaway platforms, the restaurants, and the distributors are responsible for food hygiene. Accordingly, this study employs an evolutionary game to study the takeaway management among them. It constructs their traditional replication dynamic equations after parameter design. Then, it improves the equations by introducing the incentive relationships and interference factors. Through the equations, it analyzes and simulates the stochastic evolution process of the three parties, exploring the influence of some parameters’ numerical changes. After that, it concludes that the proportional expenditure coefficient for illegal rectification, the incentive coefficient, the interference intensity coefficient, and the step size can impact the efficiency, the results, or the fluctuations in the evolution, obtaining methods to theoretically control the evolution. Based on that, it puts forward targeted suggestions for constructing the contract mechanism of the takeaway, which provides a reference for the management of food takeaways practically in the postepidemic era.


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## Body

## 1. Introduction

During COVID-19, many restaurants develop takeaway services to increase their revenue [1]. According to iiMedia Research, by 2020, 91.6% of catering enterprises have developed takeaway businesses. From 2011 to 2020, the number of online takeaway users in China has reached 456 million, and the market scale has reached 664.62 billion RMB.However, many users are worried about food hygiene in its supply chain. Some scholars suggested that the hygienic conditions of online takeaway are worse than those of other food producers [2]. Since the virus can be transmitted through water, aerosols, and contact [3], users are concerned about the infection through food, packaging, and contact with delivery personnel [4]. In fact, due to the particularity of online takeaway food, it is more difficult to supervise, which increases the risk of hygiene problems. Besides, it may distribute contaminated food to large geographical areas quickly, which poses a threat to public health [5]. Therefore, it is necessary to strengthen its quality management to eliminate virus infection in postepidemic era.Hygiene risks in the takeaway are normally in food production, packaging, and delivery. In terms of production, some scholars have obtained potential etiological agents as food materials through investigations [6]. However, it is even more important to formulate safety criteria for takeaway food producers due to a lack of uniform standards [7]. Therefore, some scholars proposed to improve the laws, regulations, and industry standards to construct a comprehensive food safety system [8]. Afterward, many food regulatory agencies in different countries published such rules that contribute to the standardized and precise management of takeaway food hygiene [9]. Concerning packaging, Olaimat et al. stated that contact with food packaging or containers contaminated with SARS-CoV-2 can spread the virus to the mouth, nose, and eyes [10]. Given these problems, Tian analyzed the shortcomings of current takeaway packaging, putting forward a packaging design that achieves “zero contact” and “safe sealing” [11]. Through questionnaire analysis, Jiang et al. put forward a takeaway packaging program that takes into account food hygiene and packaging costs [12]. About delivery, since catering takeaway platforms are widely connected to third-party logistics and crowd-sourcing logistics, it is difficult to monitor food hygiene issues in meal collection and distribution [13]. Accordingly, it is essential to maintain the hygiene habits of the distributors [14]. Chandrasekhar et al. also claimed that the most typical takeaway problem was not only the lack of hygiene in the kitchen but also in the delivery stages [15]. In this regard, many scholars used modern technologies to ensure delivery safety. For example, Wu et al. designed a robot for food distribution in college, which achieves “zero-touch” meal delivery [16]. In terms of the whole-process management of takeaway food, Zhang et al. provided specific suggestions for food enterprises, distributors, and government administration in food production, delivery, and supervision [17]. Bai et al. put forward guidance in food supervision and sterilization [18]. Besides, scholars like Iftekhar built the traceability system in the takeaway food supply chain [19, 20].As a powerful method to study decision-making, game theory considers the behavior of individuals and their profits in a game and studies their optimization strategies. Scholars frequently employ it to understand and predict the players’ strategies. Furthermore, evolutionary game theory is a dynamic method to study decision-making. Compared with classical game theory that requires rational players, in an evolutionary game, the individuals are semirational, which means their decisions are not always optimal. Afterward, their strategies can be inherited by their descendants. Its main principle is that individuals with higher payoffs tend to replicate more frequently, so their strategy will spread in the population [21]. Accordingly, it is suitable for exploring the players’ strategy. For instance, Wang and Liu used it to analyze the evolution between dairy enterprises and government to strengthen dairy management through supervision [22]. As for takeaway management, Ye and Sun built and analyzed a three-party static game model among the government, takeaway platforms, and takeaway restaurants in the takeaway chain, concluding that the punishment helps to maintain a safe environment [23]. Hui et al. analyzed the game between takeaway restaurants and takeaway platforms, suggesting that the platforms must severely punish unqualified restaurants and reduce the supervision costs to ensure food quality [24]. Similarly, Lv and Huo constructed an evolutionary game model of takeaway restaurants and platforms, finding that the strategic choices of them mainly depend on the government’s supervision [25]. However, most related researches are deficient in their models and analysis. Initially, most of them pay attention to food production and its supervision, regardless of food packaging, distribution, and other factors that also play vital roles in the takeaway process and affect food hygiene. Moreover, they usually neglect the incentive relationships between different strategies. The two basic concepts of evolutionary game theory are evolutionary stability strategy and replication dynamic equation, in which the former emphasizes the mutation in evolution, and the latter emphasizes the choice in evolution [26]. Scholars widely use the latter for research, ignoring the interdependence and mutual incentives between strategies in the same group, which is also a disadvantage in current studies of takeaway management. Eventually, they generally suppose without the influence of complex external interference.Accordingly, this study employs the evolutionary game to study the dynamic evolution mechanism of the tripartite game among platform parties, restaurants, and distributors in the takeaway process, providing theoretical references for strengthening the quality management of takeaways. Compared with previous research, its innovations are as follows:(1)
It involves takeaway platforms, takeaway restaurants, and takeaway distributors as the main groups in the game, which puts the main groups that influence takeaway hygiene in the process into consideration, including food production, packaging, distribution, and supervision.(2)
It improves the replication dynamic equation in the evolutionary game by introducing incentive factors, which consider the incentive relationships between different strategies in practical decision-making.(3)
It introduces Gaussian white noise into the improved replication dynamic equation to simulate the influence of interference factors in reality on decision-making, which is more suitable for the actual scene.The rest of this article is organized as follows:The second part makes assumptions about the revenue and expenditure of the platform, the restaurant, and the distributor and establishes a revenue matrix. The third part firstly establishes the three-party replication dynamic equation and improves it, then introduces random interference into the equation, and analyzes its stability. The fourth part conducts numerical simulation through assignment, analyzing the influence of different factors on the evolution process and results. The fifth part summarizes the research results of this research and proposes the corresponding cooperation mechanism suggestions.


## 2. Model Assumptions and Analysis

According to the first part, hygiene safety problems typically happen during food production and packaging by the restaurants, and food collection and delivery by the riders (or distributors). In terms of production, normally, it involves food materials, material storage, and the food cooking process. The study presents every part in the takeaway chain and its risks are shown in Figure1.Figure 1
The risks in the takeaway food chain. (a) Food materials, (b) food storage, (c) meal cooking, (d) food packaging, (e) food delivery, (f) takeaway restaurants, and (g) takeaway distributors.According to previous research, some players seek higher earnings by providing illegal work, which cuts their costs. However, it can lead to hygiene risks when the restaurants use sleazy materials in food production and packaging, or the distributors do not protect the food and its packaging from virus contamination. Fortunately, takeaway platforms can restrain their illegal work through strict surveillance and punishment.Based on that, this study makes the following assumptions to build the evolutionary game model.Hypothesis 1.
The players in the three-party game model consist of takeaway platforms, the restaurants, and the distributors. They have two work strategies, respectively. From the aspect of the platforms, the strategic space is supervision and nonsupervision, and the proportional distribution of the two strategies isx,1−x. From the perspective of the restaurants, the strategic space is normal production and packaging, and illegal production and packaging, and the proportional distribution is y,1−y. For the distributors, the strategic space is normal delivery and illegal delivery, and the proportional distribution is z,1−z. Among them, x,y,z∈0,1. In the game, the three groups are bounded rational. They continuously choose and alter their strategies to seek higher profits in the game.Hypothesis 2.
The basic income of takeaway platforms, the restaurants, and the distributors areE1, E2, and E3, respectively, and remain unchanged. However, their final profit changes with different strategy combinations that lead to different costs and financial punishments.Hypothesis 3.
For takeaway platforms, the cost of supervision isC1. If they do not take such measures, it will lead to potential hygiene risks and cause loss D. On the contrary, if they take regular supervision and timely rectification, the loss can be avoided. For the restaurants, the cost of normal production and packaging is C2 and the cost of illegal work is C3. When the platforms find illegal work, they are required to rectify and impose a fine. The quality of the rectified work is the same as under normal working conditions, and the cost of rectification is recorded as θ times the cost of illegal production and packaging, that is, θC3, θ＞1, and the fine is F. In the same way, the cost of the distributors during normal work is C4, and the cost of illegal work is C5. When the platforms find illegal work, they are also required to rectify and impose a fine. The rectification cost is as θ times the cost of the illegal delivery, namely, θC5, and the fine is F.Hypothesis 4.
When a hygiene problem happens, the platforms are the first to be held accountable. However, the problem is caused by the joint decisions of the three parties, so both the restaurants and the distributors should also be jointly and severally liable and bear the corresponding losses. Therefore, this article assumes that the platforms will transfer the loss of hygiene problems to the restaurants and the distributors with a special transmission coefficientk.
Table1 lists the parameters designed in the game.
According to the above assumptions, Table2 presents the tripartite income matrix.Table 1
Parameter design.
ParametersMeaningsE1The basic income of takeaway platformsE2The basic income of the restaurantsE3The basic income of the distributorsC1The cost of supervision by takeaway platformsC2The cost of normal production packaging by restaurantsC3The cost of illegal production and packaging by restaurantsC4The cost of normal delivery by distributorsC5The cost of illegal delivery by distributorsθThe expenditure proportion coefficient for rectification,θ>1FThe fines for illegal workDThe losses caused by hygiene incidentskThe transmission coefficient of lossesxThe rate that the platforms choose the supervisionyThe rate that the restaurants choose normal workzThe rate that the distributors choose normal workTable 2
The tripartite income matrix.
DistributorsTakeaway platformsSupervisionxNonsupervision1−xRestaurantsNormal production packagingyNormal deliveryzE2−C2E2−C2E3−C4E3−C4E1−C1E1Illegal delivery1−zE2−C2E2−C2−kDE3−θC5−FE3−C5−kDE1−C1+FE1−DIllegal production and packaging1−yNormal deliveryzE2−θC3−FE2−C3−kDE3−C4E3−C4−kDE1−C1+FE1−DIllegal delivery1−zE2−θC3−FE2−C3−kDE3−θC5−FE3−C5−kDE1−C1+2FE1−D


## 3. Model Construction and Solution

### 3.1. The Replication Dynamic Equations of the Three Parties

From the perspective of takeaway platforms, the profit when they choose the supervision is recorded asU11, and that when they choose not to supervise is recorded as U12. Based on that, their average expected profit is noted as U1¯, and the replication dynamic equation is Fx. The result is as follows:(1)U11=yzE1−C1+y1−zE1−C1+F+1−yzE1−C1+F+1−y1−zE1−C1+2F,(2)U12=yzE1+y1−zE1−D+1−yzE1−D+1−y1−zE1−D,(3)U1¯=xU11+1−xU12,(4)Fx=xU11−U1¯=xx−1C1+Dyz−D+Fy+Fz−2F.Concerning the restaurants, the profit when they choose normal production and packaging is recorded asU21, and that when they choose illegal work is recorded as U22. Based on that, their average expected return is noted as U2¯, and the replication dynamic equation is Fy. The result is as follows:(5)U21=xzE2−C2+x1−zE2−C2+1−xzE2−C2+1−x1−zE2−C2−kD,(6)U22=xzE2−θC3−F+x1−zE2−θC3−F+1−xzE2−C3−kD+1−x1−zE2−C3−kD,(7)U2¯=yU21+1−yU22,(8)Fy=yU21−U2¯=yy−1C2−C3xθ+C3x−C3+Dkxz−Dkz−Fx.From the aspect of the distributors, the income when they choose normal distribution is recorded asU31, and that when selecting illegal distribution is recorded as U32. Based on that, their average expected income is noted as U3¯ and the replication dynamic equation is Fz. The result is as follows:(9)U31=xyE3−C4+x1−yE3−C4+1−xyE3−C4+1−x1−yE3−C5−kD,(10)U32=xyE3−θC5−F+x1−yE3−θC5−F+1−xyE3−C5−kD+1−x1−yE3−C5−kD,(11)U3¯=zU31+1−zU32,(12)Fz=zU31−U3¯=zz−1−C4xy+C4x+C4y+C5xy−C5xθ−C5y+Dkxy−Dky−Fx.


### 3.2. The Improved Dynamic Equations of the Three Parties

Assuming that at timet, the strategy sets of the platforms, the restaurants, and the distributors are P1,P2, B1,B2, and D1,D2, respectively. From the aspect of the platforms, there are p1t of them choose a supervisory strategy (denoted as P1), the proportion of which to the total number of the platforms is xt. On this basis, their average expected profit at time t is U11.Forxt, the computation function is as follows:(13)xt=p1t∑i=12pit.With time elapsing, the number of platforms that adopt this strategy will change, and the dynamic change speedp1′t is proportional to the number of people p1t who choose the strategy and its expected profit U11.Briefly, the computation function ofp1t is as follows:(14)p1′t=μ1p1tU11.In formula (14), μ1μ1>0 is the influence factor of strategy P1 (the supervision strategy of takeaway platforms). In the game, there is a mutual influence between different strategies. That is, the strategy with faster diffusion speed is more influential. The larger μ1 is, the more influential role P1 plays in decision-making. On the contrary, the weaker it is, the less influential P1 is.Taking the derivation of formula (13), the study improves the replication dynamic equation of the platforms’ strategy selection based on formula (14):(15)x′t=p1′t∑i=12pit−p1t∑i=12pi′t∑i=12pit2=p1t∑i=12pitp1′tp1t−∑i=12pi′t∑i=12pit=xtμ1p1tU11p1t−∑i=12μipitU1it∑i=12pit=μ1xt1−xtU11−μ2μ1U12.In formula (15), μ2μ2>0 is the influence factor of strategy P2 (the nonsupervision strategy). Similarly, it deduces the replication dynamics equations under all three groups:(16)x′t=μ1xt1−xtU11−μ2μ1U12,y′t=σ1yt1−ytU21−σ2σ1U22,z′t=φ1zt1−ztU31−φ2φ1U32.Now, it defines an incentive coefficientl=μ2/μ1. In the game, it means the incentive effect of the platforms’ nonsupervision strategy P2 on the supervision strategy P1. When l>1, it means that P2 restrains P1; when l<1, it means that P1 restrains P2; when l=1, it means that there is no incentive relationship between them. In the same way, m=σ2/σ1 is defined to indicate the incentive relationship of the restaurants’ illegal production and packaging strategy B2 to normal production and packaging strategy B1 in the game. Finally, n=φ2/φ1 is defined to indicate the incentive relationship of the distributors’ illegal distribution strategy D2 to the normal distribution strategy D1 in the game. Substituting them into equation set (16), respectively, it obtains the following equation set:(17)x′t=μ1xt1−xtU11−lU12,y′t=σ1yt1−ytU21−mU22,z′t=φ1zt1−ztU31−nU32.Whenl=m=n=1 and μ1=σ1=φ1=1, it obtains their traditional replication dynamic equations:(18)x′t=xt1−xtU11−U12,y′t=yt1−ytU21−U22,z′t=zt1−ztU31−U32.


### 3.3. The Stochastic Evolutionary Game Model of the Three Parties

In the game of takeaway platforms, the restaurants, and the distributors, various random factors are disturbing them, which make a difference to the stability of the evolutionary game system. To better simulate the random interference caused by uncertain factors in the game, based on the theory of stochastic analysis, this study introduces Gaussian white noise into the game process and rewrites the equations as follows:(19)dxt=μ1xt1−xtU11−lU12dt+πxt1−xtdωt,dyt=σ1yt1−ytU21−mU22dt+πyt1−ytdωt,dzt=φ1zt1−ztU31−nU32dt+πφ1zt1−ztdωt.In the equations,ωt is the standard one-dimensional Brown motion process, and the increment Δωt=ωt+h−ωt obeys the normal distribution N0,h. πxt1−xtdωt plays the role of the random interference, where π represents the random interference intensity, and xt1−xt reaches the maximum only when xt=1−xt, which is in line with the actual situation.Then, this section employs the SDE (stochastic differential equation) judgment theorem to analyze the stability of the equations [27].The theorem is as given below.A stochastic differential equation is given as follows:(20)dxt=ft,xtdt+gt,xtdωt,xt0=x0.Suppose there is a functionVt,x and positive constants c1 and c2, it exists c1xp≤Vt,x≤c2xp, t≥0. In this condition, if there is a positive constant ε that makes LVt,x≤−εVt,x, t≥0, then the equation’s zero solution pth moment is exponentially stable and there is Ext,t0p<c2/c1x0pe−εt, t≥0.Based on the above theorem, this research judges the stability of the stochastic differential equations (19).LetVt,x=x, Vt,y=y, and Vt,z=z, in which x,y,z∈0,1, and then let c1=c2=1, p=1, ε=1. Therefore, there are LVt,x=ft,x, LVt,y=ft,y, and LVt,z=ft,z.Becausex,y,z∈0,1, this study analyzes the stability based on the following equations:(21)LVt,x=ft,x=μ1xU11−lU12,LVt,y=ft,y=σ1yU21−mU22,LVt,z=ft,z=φ1zU31−nU32.It substitutesU11, U12, U21, U22, U31, and U32 into the above equations and obtains the following equations:(22)LVt,x=μ1x−C1−Dlyz+Dl−Fy−Fz+2F−E1l+E1,(23)LVt,y=σ1y−C2+C3mxθ−C3mx+C3m−Dkmx+Dkm−Dkxz+Dkx+Dkz−Dk+Fmx−E2m+E2,(24)LVt,z=φ1zC4xy−C4x−C4y+C5nxθ−C5nx+C5n−C5xy+C5x+C5y−C5−Dknx+Dkn−Dkxy+Dkx+Dky−Dk+Fnx−E3n+E3.According to the theorem, if the zero solutionpth moment of equations (19) is exponentially stable, then it needs to meet the following equations:(25)μ1x−C1−Dlyz+Dl−Fy−Fz+2F−E1l+E1≤−x,(26)σ1y−C2+C3mxθ−C3mx+C3m−Dkmx+Dkm−Dkxz+Dkx+Dkz−Dk+Fmx−E2m+E2≤−y,(27)φ1zC4xy−C4x−C4y+C5nxθ−C5nx+C5n−C5xy+C5x+C5y−C5−Dknx+Dkn−Dkxy+Dkx+Dky−Dk+Fnx−E3n+E3≤−z.Since the stochastic improved replication dynamic equations in this study are all nonlinear Itȏ’s SDEs (stochastic differential equations) whose solutions cannot be obtained directly, this study employs stochastic Taylor expansion to solve them numerically.Itȏ’s stochastic differential equation is given as follows:(28)dxt=ft,xtdt+gt,xtdωt.In the above equation,t0=0 and t∈0,T. The time interval 0,T is divided into N parts that consist of t0,t1, t1,t2,..., tN−1,tN, in which 0=t0<t1<t2<…<tN=T, the step size h=T/N, and the node tn=nhn=1,2,3,…,N. Besides, xt0=x0, yt0=y0, and zt0=z0, in which x0,y0,z0∈R.This research uses the explicit forward Euler method to expand formula (28), obtaining the following equation:(29)xn+1=xn+hfxn+Δωngxn.It expands equation (19) with this method, reaching the following recursion formulas:(30)xn+1=xn+μ1xn1−xn−C1−Dlynzn+Dl−Fyn−Fzn+2F−E1l+E1h+Δωnπxn1−xn,(31)yn+1=yn+σ1yn1−yn−C2+C3mxnθ−C3mxn+C3m−Dkmxn+Dkm−Dkxnzn+Dkxn+Dkzn−Dk+Fmxn−E2m+E2h+Xωnπyn1−yn,(32)zn+1=zn+φ1zn1−znC4xnyn−C4xn−C4yn+C5nxnθ−C5nxn+C5n−C5xnyn+C5xn+C5yn−C5−Dknxn+Dkn−Dkxnyn+Dkxn+Dkyn−Dk+Fnxn−E3n+E3h+Δωnπzn1−zn.


## 3.1. The Replication Dynamic Equations of the Three Parties

From the perspective of takeaway platforms, the profit when they choose the supervision is recorded asU11, and that when they choose not to supervise is recorded as U12. Based on that, their average expected profit is noted as U1¯, and the replication dynamic equation is Fx. The result is as follows:(1)U11=yzE1−C1+y1−zE1−C1+F+1−yzE1−C1+F+1−y1−zE1−C1+2F,(2)U12=yzE1+y1−zE1−D+1−yzE1−D+1−y1−zE1−D,(3)U1¯=xU11+1−xU12,(4)Fx=xU11−U1¯=xx−1C1+Dyz−D+Fy+Fz−2F.Concerning the restaurants, the profit when they choose normal production and packaging is recorded asU21, and that when they choose illegal work is recorded as U22. Based on that, their average expected return is noted as U2¯, and the replication dynamic equation is Fy. The result is as follows:(5)U21=xzE2−C2+x1−zE2−C2+1−xzE2−C2+1−x1−zE2−C2−kD,(6)U22=xzE2−θC3−F+x1−zE2−θC3−F+1−xzE2−C3−kD+1−x1−zE2−C3−kD,(7)U2¯=yU21+1−yU22,(8)Fy=yU21−U2¯=yy−1C2−C3xθ+C3x−C3+Dkxz−Dkz−Fx.From the aspect of the distributors, the income when they choose normal distribution is recorded asU31, and that when selecting illegal distribution is recorded as U32. Based on that, their average expected income is noted as U3¯ and the replication dynamic equation is Fz. The result is as follows:(9)U31=xyE3−C4+x1−yE3−C4+1−xyE3−C4+1−x1−yE3−C5−kD,(10)U32=xyE3−θC5−F+x1−yE3−θC5−F+1−xyE3−C5−kD+1−x1−yE3−C5−kD,(11)U3¯=zU31+1−zU32,(12)Fz=zU31−U3¯=zz−1−C4xy+C4x+C4y+C5xy−C5xθ−C5y+Dkxy−Dky−Fx.


## 3.2. The Improved Dynamic Equations of the Three Parties

Assuming that at timet, the strategy sets of the platforms, the restaurants, and the distributors are P1,P2, B1,B2, and D1,D2, respectively. From the aspect of the platforms, there are p1t of them choose a supervisory strategy (denoted as P1), the proportion of which to the total number of the platforms is xt. On this basis, their average expected profit at time t is U11.Forxt, the computation function is as follows:(13)xt=p1t∑i=12pit.With time elapsing, the number of platforms that adopt this strategy will change, and the dynamic change speedp1′t is proportional to the number of people p1t who choose the strategy and its expected profit U11.Briefly, the computation function ofp1t is as follows:(14)p1′t=μ1p1tU11.In formula (14), μ1μ1>0 is the influence factor of strategy P1 (the supervision strategy of takeaway platforms). In the game, there is a mutual influence between different strategies. That is, the strategy with faster diffusion speed is more influential. The larger μ1 is, the more influential role P1 plays in decision-making. On the contrary, the weaker it is, the less influential P1 is.Taking the derivation of formula (13), the study improves the replication dynamic equation of the platforms’ strategy selection based on formula (14):(15)x′t=p1′t∑i=12pit−p1t∑i=12pi′t∑i=12pit2=p1t∑i=12pitp1′tp1t−∑i=12pi′t∑i=12pit=xtμ1p1tU11p1t−∑i=12μipitU1it∑i=12pit=μ1xt1−xtU11−μ2μ1U12.In formula (15), μ2μ2>0 is the influence factor of strategy P2 (the nonsupervision strategy). Similarly, it deduces the replication dynamics equations under all three groups:(16)x′t=μ1xt1−xtU11−μ2μ1U12,y′t=σ1yt1−ytU21−σ2σ1U22,z′t=φ1zt1−ztU31−φ2φ1U32.Now, it defines an incentive coefficientl=μ2/μ1. In the game, it means the incentive effect of the platforms’ nonsupervision strategy P2 on the supervision strategy P1. When l>1, it means that P2 restrains P1; when l<1, it means that P1 restrains P2; when l=1, it means that there is no incentive relationship between them. In the same way, m=σ2/σ1 is defined to indicate the incentive relationship of the restaurants’ illegal production and packaging strategy B2 to normal production and packaging strategy B1 in the game. Finally, n=φ2/φ1 is defined to indicate the incentive relationship of the distributors’ illegal distribution strategy D2 to the normal distribution strategy D1 in the game. Substituting them into equation set (16), respectively, it obtains the following equation set:(17)x′t=μ1xt1−xtU11−lU12,y′t=σ1yt1−ytU21−mU22,z′t=φ1zt1−ztU31−nU32.Whenl=m=n=1 and μ1=σ1=φ1=1, it obtains their traditional replication dynamic equations:(18)x′t=xt1−xtU11−U12,y′t=yt1−ytU21−U22,z′t=zt1−ztU31−U32.


## 3.3. The Stochastic Evolutionary Game Model of the Three Parties

In the game of takeaway platforms, the restaurants, and the distributors, various random factors are disturbing them, which make a difference to the stability of the evolutionary game system. To better simulate the random interference caused by uncertain factors in the game, based on the theory of stochastic analysis, this study introduces Gaussian white noise into the game process and rewrites the equations as follows:(19)dxt=μ1xt1−xtU11−lU12dt+πxt1−xtdωt,dyt=σ1yt1−ytU21−mU22dt+πyt1−ytdωt,dzt=φ1zt1−ztU31−nU32dt+πφ1zt1−ztdωt.In the equations,ωt is the standard one-dimensional Brown motion process, and the increment Δωt=ωt+h−ωt obeys the normal distribution N0,h. πxt1−xtdωt plays the role of the random interference, where π represents the random interference intensity, and xt1−xt reaches the maximum only when xt=1−xt, which is in line with the actual situation.Then, this section employs the SDE (stochastic differential equation) judgment theorem to analyze the stability of the equations [27].The theorem is as given below.A stochastic differential equation is given as follows:(20)dxt=ft,xtdt+gt,xtdωt,xt0=x0.Suppose there is a functionVt,x and positive constants c1 and c2, it exists c1xp≤Vt,x≤c2xp, t≥0. In this condition, if there is a positive constant ε that makes LVt,x≤−εVt,x, t≥0, then the equation’s zero solution pth moment is exponentially stable and there is Ext,t0p<c2/c1x0pe−εt, t≥0.Based on the above theorem, this research judges the stability of the stochastic differential equations (19).LetVt,x=x, Vt,y=y, and Vt,z=z, in which x,y,z∈0,1, and then let c1=c2=1, p=1, ε=1. Therefore, there are LVt,x=ft,x, LVt,y=ft,y, and LVt,z=ft,z.Becausex,y,z∈0,1, this study analyzes the stability based on the following equations:(21)LVt,x=ft,x=μ1xU11−lU12,LVt,y=ft,y=σ1yU21−mU22,LVt,z=ft,z=φ1zU31−nU32.It substitutesU11, U12, U21, U22, U31, and U32 into the above equations and obtains the following equations:(22)LVt,x=μ1x−C1−Dlyz+Dl−Fy−Fz+2F−E1l+E1,(23)LVt,y=σ1y−C2+C3mxθ−C3mx+C3m−Dkmx+Dkm−Dkxz+Dkx+Dkz−Dk+Fmx−E2m+E2,(24)LVt,z=φ1zC4xy−C4x−C4y+C5nxθ−C5nx+C5n−C5xy+C5x+C5y−C5−Dknx+Dkn−Dkxy+Dkx+Dky−Dk+Fnx−E3n+E3.According to the theorem, if the zero solutionpth moment of equations (19) is exponentially stable, then it needs to meet the following equations:(25)μ1x−C1−Dlyz+Dl−Fy−Fz+2F−E1l+E1≤−x,(26)σ1y−C2+C3mxθ−C3mx+C3m−Dkmx+Dkm−Dkxz+Dkx+Dkz−Dk+Fmx−E2m+E2≤−y,(27)φ1zC4xy−C4x−C4y+C5nxθ−C5nx+C5n−C5xy+C5x+C5y−C5−Dknx+Dkn−Dkxy+Dkx+Dky−Dk+Fnx−E3n+E3≤−z.Since the stochastic improved replication dynamic equations in this study are all nonlinear Itȏ’s SDEs (stochastic differential equations) whose solutions cannot be obtained directly, this study employs stochastic Taylor expansion to solve them numerically.Itȏ’s stochastic differential equation is given as follows:(28)dxt=ft,xtdt+gt,xtdωt.In the above equation,t0=0 and t∈0,T. The time interval 0,T is divided into N parts that consist of t0,t1, t1,t2,..., tN−1,tN, in which 0=t0<t1<t2<…<tN=T, the step size h=T/N, and the node tn=nhn=1,2,3,…,N. Besides, xt0=x0, yt0=y0, and zt0=z0, in which x0,y0,z0∈R.This research uses the explicit forward Euler method to expand formula (28), obtaining the following equation:(29)xn+1=xn+hfxn+Δωngxn.It expands equation (19) with this method, reaching the following recursion formulas:(30)xn+1=xn+μ1xn1−xn−C1−Dlynzn+Dl−Fyn−Fzn+2F−E1l+E1h+Δωnπxn1−xn,(31)yn+1=yn+σ1yn1−yn−C2+C3mxnθ−C3mxn+C3m−Dkmxn+Dkm−Dkxnzn+Dkxn+Dkzn−Dk+Fmxn−E2m+E2h+Xωnπyn1−yn,(32)zn+1=zn+φ1zn1−znC4xnyn−C4xn−C4yn+C5nxnθ−C5nxn+C5n−C5xnyn+C5xn+C5yn−C5−Dknxn+Dkn−Dkxnyn+Dkxn+Dkyn−Dk+Fnxn−E3n+E3h+Δωnπzn1−zn.


## 4. Simulation and Analysis

To intuitively demonstrate the evolution process of takeaway platforms, takeaway restaurants, and takeaway distributors, this study assigns the parameters and calculates the results. To simplify the calculation, based on Chinese salary policy and some parameters set in previous research [23], this study assumes that C1=20, C2=20, C3=10, C4=20, C5=10, E1=25, E2=25, E3=25, D=20, F=5, k=0.8, and μ1=σ1=φ1=1. It supposes that x0=0.5, y0=0.5, and z0=0.5, which means that at the beginning, the three groups are all in a neutral position in decision-making and are most likely to be influenced by external interference due to limited experience. Besides, it sets T=3 and N=150, thus the step length h=0.02, which is small enough to ensure the accuracy of the simulation. By adjusting the value of the rectification expenditure proportion coefficient θ, the incentive coefficients l, m, and n, the stochastic interference intensity π, and the step size h, it explores their influences on the evolution and the theoretical method to improve the work quality of the players, accordingly ensuring food safety in the takeaway chain.


### 4.1. The Impact of the Rectification Expenditure Proportion Coefficient

This study hypothesizesθ=1.2, 1.4, 1.6, and 1.8 respectively in the experiment, which means the expenditure for rectification grows during the experiments. Based on that, it simulates the evolutionary game process of the three groups, as shown in Figure 2.Figure 2
The impact ofθ on evolution. (a) The platforms’ evolution, (b) the restaurants’ evolution, and (c) the distributors’ evolution.
(a)(b)(c)Whenθ=1.2 and x,y,z converges at 1,0,0, which means that the set of ESSs (evolutionary stable strategies) for platforms, restaurants, and distributors is supervision, illegal production and packaging, and illegal distribution). In spite of the platforms’ supervision, the other two sides work illegally, which not only provides unqualified food but also makes the expense of supervision useless. As a result, the study tries improving this by increasing the expenditure for rectification. When θ=1.4, x and y still converge at 1 and 0, respectively. For z, its final value in the given time is very close to 0. It means that in the end, the platforms still choose supervision, the restaurants still provide illegal work, and some of the distributors do not provide illegal work any longer, though the rate of them is pretty low. As θ continues growing, when θ=1.6 or 1.8, x,y,z converges at 0,1,1, which means that the set of ESSs of the three groups becomes nonsupervision, normal production and packaging, and normal distribution. In conclusion, concerning the restaurants and the distributors, they tend to provide normal work when facing higher rectification expenditure. In this sense, it suggests that a higher rectification expenditure proportion coefficient helps to restrain their illegal work and improve their work quality, accordingly maintaining food safety. From the aspect of the platforms, although the rectification expenditure proportion coefficient does not influence their decision-making directly, according to equation (30), it impacts them indirectly by influencing those of the restaurants and the distributors. As a result, it finds that when θ=1.6 or 1.8, x stops growing and starts falling in the time interval of 0.5,1. In other words, some platforms ignore their responsibility, choosing nonsupervision when the restaurants and the distributors tend to provide normal work, which can be harmful to takeaway food chain management.Besides, through the comparison between the evolution whenθ=1.2 and 1.4, it finds that y and z decline more slowly and turbulently as θ grows; that is, the efficiency of the evolution decreases. It means that the restaurants and the distributors are more erratic in decision-making when facing higher rectification expenditure. In addition, based on the comparison when θ=1.6 and 1.8, and the sets of ESSs for the restaurants y and the distributors z are both 1, it finds that as θ grows, it requires less time to reach the stable results; that is, the efficiency of the evolution increases. Therefore, it concludes that a higher rectification expenditure proportion coefficient is negative to their illegal work and positive to normal work. Similarly, it discovers that the coefficient also restrains the platforms’ enthusiasm for inspection, which suggests that it is not a good idea to improve the work of the restaurants and the distributors by just increasing the rectification expenditure proportion coefficient.


### 4.2. The Impact of the Incentive Coefficients

Based on the initial values, this study keepsθ=1.2, adjusting the values of the incentive coefficients l, m, and n and exploring their influence on the three groups’ evolution. To get rid of the duplicate discussion, it only selectively analyzes the numerical simulation in this part.Whenl=m=n=1, there are no incentive relationships between the two strategies of the three sides, respectively, which is the same as a traditional stochastic differential equation. In this condition, the set of ESSs in the game is 1,0,0, which means the restaurants and the distributors provide low-quality work despite the platforms’ supervision. This is the worst condition in the game, for even the platforms' contribution to inspection cannot improve the work of the other two parties. As a result, the study attempts to adjust the incentive coefficients for the restaurants and the distributors to improve their work.It keepsl=1 and adjusts m and n, making m<1 and n<1. In these circumstances, for the restaurants, the strategy of normal production and packaging restrains the other strategy; for the distributors, the strategy of normal delivery can also restrain the other strategy. Figure 3 shows the evolution under the assumption.Figure 3
The impact ofm and n on the evolution. (a) The platforms’ evolution, (b) the restaurants’ evolution, and (c) the distributors’ evolution.
(a)(b)(c)The experiment separately supposes thatm=n=0.8, m=n=0.5, and m=n=0.2 and conducts the simulation. The results show that when m=n=0.8, although the set of the ESSs of the three sides is still 1,0,0, the evolution efficiencies of the restaurants and the distributors both diminish compared with the condition when m=n=1. When m=n=0.5 or 0.2, x,y,z converges at 0,1,1, which means the set of ESSs becomes nonsupervision, normal production and packaging, and normal distribution as time goes by. In addition, the comparison between these two conditions finds that when the restaurants and the distributors tend to provide normal work, smaller m and n make their evolution more efficient. However, the platforms also change their strategies from supervision to nonsupervision when m and n are smaller. When platforms ignore their responsibility for supervision, it can cause managerial problems in the takeaway chain, which further harms the public’s trust in them and impedes their development. Briefly, the decrease of the incentive coefficients makes the platforms negative in supervision and makes the restaurants and the distributors positive in food production, packaging, and delivery. Therefore, the adjustment of m and n cannot make all the three groups work responsibly.Next, it tries adjusting the incentive coefficients for the platforms to improve their work. It keepsm=n=0.2 and adjusts the value of l, making l<1, which means that the platforms’ supervision strategy restrains the nonsupervision strategy. It separately assumes that l=1, 0.8, 0.5, and 0.2 and conducts the simulation based on that, reaching the results, as shown in Figure 4.Figure 4
The impact ofl on evolution. (a) The platforms’ evolution, (b) the restaurants’ evolution, and (c) the distributors’ evolution.
(a)(b)(c)As the figure suggests, in the beginning,l=1 and x,y,z converges at 0,1,1. When l=0.8 and 0.5 separately, although the ESSs of x are still 0, the evolutionary efficiency is a little lower. When l=0.2, x converges at 1 and x,y,z converges at 1,1,1. In this condition, the set of ESSs is supervision, normal production and packaging, and normal distribution, which is the best situation in the game. The changes of l also impact the evolution of the restaurants and the distributors. Specifically, due to the inhibiting effect, the platforms’ supervision strategy plays on the nonsupervision strategy, the restaurants and the distributors react a little more quickly in the time interval of 0,1.5, and it is more obvious in the evolution of the restaurants. In conclusion, the platforms’ impetus for supervision enhances the work quality of themselves and convinces the other two groups to work legally in the game. Accordingly, the takeaway platforms must keep positive in the inspection.In brief, the incentive coefficients can improve the work quality of all three groups in the takeaway chain when the standardized work restrains the illegal one.


### 4.3. The Impact of the Stochastic Factors

This study simulates the evolution process of the restaurants in different levels of the stochastic interference intensityπ. Based on previous assumptions about the parameters, it hypothesizes π=0, 0.5, and 1, 2 respectively and simulates the evolution process. Due to the increase of π, the evolution becomes more turbulent and can lead to different results even under the same parameter assignments. To simplify the analysis, this study only chooses one condition to explain, which is shown in Figure 5.Figure 5
The impact ofπ on evolution. (a) The platforms’ evolution, (b) the restaurants’ evolution, and (c) the distributors’ evolution.
(a)(b)(c)As Figure5 shows, when π=0, that is, the game runs without stochastic interference, x,y,z converges at 1,0,0, and all their evolutionary paths are smooth. When π=0.5 and 1, respectively, it causes fluctuations in their evolution. Additionally, it fluctuates more apparently with higher π, which means that the stochastic interference makes the players dither about their strategies. As π continually grows, it leads to even severer fluctuations, and x,y,z converges at 0,1,1 when π=2. In this condition, the restaurants and the distributors choose normal work in the end. In this sense, interference seems to be more beneficial to improve hygiene than the interference-free condition.However, this is just an accidental event. In the experiment, whenπ reaches a high value, x, y, and z can evolve to different results each time, even other parameters stay unchanged. The study conducts 100 experiments under the same parameter assignments, discovering that the set of the ESSs can be 1,0,0 or 0,1,1. Through statistics, it obtains their distributions, as shown in Figure 6.Figure 6
The distribution of different ESSs sets.According to the figure, whenπ=0.5, only 2% of the experiments reach 0,1,1; when π=1, only 17% of the experiments reach 0,1,1; when π=2, only 23% of the experiments reach 0,1,1. It means that stronger interference can increase the probability of 0,1,1. Although its frequency increases with higher π, in most experiments, the ESSs still keep 1,0,0.In brief, the stochastic interference strength impacts the evolution process and results by causing a fluctuation in the game. Although sometimes it brings good results, the probability is low. Apart from that, the fluctuation in their decision-making can disturb the market order. As a result, it is necessary to avoid external interference and deal with it timely.


### 4.4. The Impact of the Step Size

Since stochastic differential equations cannot be directly numerically solved, this study expands the equations obtained in Section3.3, transforming the continuous system into a discrete system for analysis. However, the stability may be different in some cases. Specifically, for the discrete system, when the step size grows, its accuracy becomes worse, and accordingly the stability of the original continuous system may be lost. In order to test the stability of the discrete system, this section takes the case in Section 4.1 as an example to simulate and analyze the evolution process of the platforms’ decision-making by adjusting the step size.According toh=T/N, the step size is decided by T and N. Specifically, it grows when T increases or N decreases. Therefore, this study verifies the stability of the discrete system under the above two conditions respectively. Also, considering that random disturbance may lead to different evolution results, this study selects the typical result for analysis after a number of experiments.On the one hand, based on the parameters in Section4.1, it hypothesizes T=3, 6, and 12, respectively, under which h=0.02, 0.04, and 0.08, respectively. In this case, it simulates the evolutionary game process of takeaway platforms, as shown in Figure 7.Figure 7
The impact ofT on takeaway platforms’ evolution. (a) T=3, (b) T=6, and (c) T=12.
(a)(b)(c)On the other hand, it hypothesizesN=150, 60, and 20, respectively, under which h=0.02, 0.05, and 0.15, respectively. In this case, it simulates the evolutionary game process of takeaway platforms, as shown in Figure 8.Figure 8
The impact ofN on takeaway platforms’ evolution. (a) N=150, (b) N=60, and (c) N=20.
(a)(b)(c)The simulation results show that although the increase of the step size affects the evolution process of takeaway platforms to a certain extent, it is not easy to directly change the ESS analyzed in this study. Besides, the impact ofθ on the evolution speed referred to in Section 4.1 still holds. Therefore, the discrete system is of sufficient stability to support the conclusion of the study.However, the evolution does mutates more frequently ash grows. Accordingly, in experiments, it is necessary to make h small enough in simulation so as to ensure the results as close as possible to the continuous system. Apart from that, in reality, the government should encourage the development of takeaway industry, providing the game players with more frequent decision-making opportunities.


## 4.1. The Impact of the Rectification Expenditure Proportion Coefficient

This study hypothesizesθ=1.2, 1.4, 1.6, and 1.8 respectively in the experiment, which means the expenditure for rectification grows during the experiments. Based on that, it simulates the evolutionary game process of the three groups, as shown in Figure 2.Figure 2
The impact ofθ on evolution. (a) The platforms’ evolution, (b) the restaurants’ evolution, and (c) the distributors’ evolution.
(a)(b)(c)Whenθ=1.2 and x,y,z converges at 1,0,0, which means that the set of ESSs (evolutionary stable strategies) for platforms, restaurants, and distributors is supervision, illegal production and packaging, and illegal distribution). In spite of the platforms’ supervision, the other two sides work illegally, which not only provides unqualified food but also makes the expense of supervision useless. As a result, the study tries improving this by increasing the expenditure for rectification. When θ=1.4, x and y still converge at 1 and 0, respectively. For z, its final value in the given time is very close to 0. It means that in the end, the platforms still choose supervision, the restaurants still provide illegal work, and some of the distributors do not provide illegal work any longer, though the rate of them is pretty low. As θ continues growing, when θ=1.6 or 1.8, x,y,z converges at 0,1,1, which means that the set of ESSs of the three groups becomes nonsupervision, normal production and packaging, and normal distribution. In conclusion, concerning the restaurants and the distributors, they tend to provide normal work when facing higher rectification expenditure. In this sense, it suggests that a higher rectification expenditure proportion coefficient helps to restrain their illegal work and improve their work quality, accordingly maintaining food safety. From the aspect of the platforms, although the rectification expenditure proportion coefficient does not influence their decision-making directly, according to equation (30), it impacts them indirectly by influencing those of the restaurants and the distributors. As a result, it finds that when θ=1.6 or 1.8, x stops growing and starts falling in the time interval of 0.5,1. In other words, some platforms ignore their responsibility, choosing nonsupervision when the restaurants and the distributors tend to provide normal work, which can be harmful to takeaway food chain management.Besides, through the comparison between the evolution whenθ=1.2 and 1.4, it finds that y and z decline more slowly and turbulently as θ grows; that is, the efficiency of the evolution decreases. It means that the restaurants and the distributors are more erratic in decision-making when facing higher rectification expenditure. In addition, based on the comparison when θ=1.6 and 1.8, and the sets of ESSs for the restaurants y and the distributors z are both 1, it finds that as θ grows, it requires less time to reach the stable results; that is, the efficiency of the evolution increases. Therefore, it concludes that a higher rectification expenditure proportion coefficient is negative to their illegal work and positive to normal work. Similarly, it discovers that the coefficient also restrains the platforms’ enthusiasm for inspection, which suggests that it is not a good idea to improve the work of the restaurants and the distributors by just increasing the rectification expenditure proportion coefficient.


## 4.2. The Impact of the Incentive Coefficients

Based on the initial values, this study keepsθ=1.2, adjusting the values of the incentive coefficients l, m, and n and exploring their influence on the three groups’ evolution. To get rid of the duplicate discussion, it only selectively analyzes the numerical simulation in this part.Whenl=m=n=1, there are no incentive relationships between the two strategies of the three sides, respectively, which is the same as a traditional stochastic differential equation. In this condition, the set of ESSs in the game is 1,0,0, which means the restaurants and the distributors provide low-quality work despite the platforms’ supervision. This is the worst condition in the game, for even the platforms' contribution to inspection cannot improve the work of the other two parties. As a result, the study attempts to adjust the incentive coefficients for the restaurants and the distributors to improve their work.It keepsl=1 and adjusts m and n, making m<1 and n<1. In these circumstances, for the restaurants, the strategy of normal production and packaging restrains the other strategy; for the distributors, the strategy of normal delivery can also restrain the other strategy. Figure 3 shows the evolution under the assumption.Figure 3
The impact ofm and n on the evolution. (a) The platforms’ evolution, (b) the restaurants’ evolution, and (c) the distributors’ evolution.
(a)(b)(c)The experiment separately supposes thatm=n=0.8, m=n=0.5, and m=n=0.2 and conducts the simulation. The results show that when m=n=0.8, although the set of the ESSs of the three sides is still 1,0,0, the evolution efficiencies of the restaurants and the distributors both diminish compared with the condition when m=n=1. When m=n=0.5 or 0.2, x,y,z converges at 0,1,1, which means the set of ESSs becomes nonsupervision, normal production and packaging, and normal distribution as time goes by. In addition, the comparison between these two conditions finds that when the restaurants and the distributors tend to provide normal work, smaller m and n make their evolution more efficient. However, the platforms also change their strategies from supervision to nonsupervision when m and n are smaller. When platforms ignore their responsibility for supervision, it can cause managerial problems in the takeaway chain, which further harms the public’s trust in them and impedes their development. Briefly, the decrease of the incentive coefficients makes the platforms negative in supervision and makes the restaurants and the distributors positive in food production, packaging, and delivery. Therefore, the adjustment of m and n cannot make all the three groups work responsibly.Next, it tries adjusting the incentive coefficients for the platforms to improve their work. It keepsm=n=0.2 and adjusts the value of l, making l<1, which means that the platforms’ supervision strategy restrains the nonsupervision strategy. It separately assumes that l=1, 0.8, 0.5, and 0.2 and conducts the simulation based on that, reaching the results, as shown in Figure 4.Figure 4
The impact ofl on evolution. (a) The platforms’ evolution, (b) the restaurants’ evolution, and (c) the distributors’ evolution.
(a)(b)(c)As the figure suggests, in the beginning,l=1 and x,y,z converges at 0,1,1. When l=0.8 and 0.5 separately, although the ESSs of x are still 0, the evolutionary efficiency is a little lower. When l=0.2, x converges at 1 and x,y,z converges at 1,1,1. In this condition, the set of ESSs is supervision, normal production and packaging, and normal distribution, which is the best situation in the game. The changes of l also impact the evolution of the restaurants and the distributors. Specifically, due to the inhibiting effect, the platforms’ supervision strategy plays on the nonsupervision strategy, the restaurants and the distributors react a little more quickly in the time interval of 0,1.5, and it is more obvious in the evolution of the restaurants. In conclusion, the platforms’ impetus for supervision enhances the work quality of themselves and convinces the other two groups to work legally in the game. Accordingly, the takeaway platforms must keep positive in the inspection.In brief, the incentive coefficients can improve the work quality of all three groups in the takeaway chain when the standardized work restrains the illegal one.


## 4.3. The Impact of the Stochastic Factors

This study simulates the evolution process of the restaurants in different levels of the stochastic interference intensityπ. Based on previous assumptions about the parameters, it hypothesizes π=0, 0.5, and 1, 2 respectively and simulates the evolution process. Due to the increase of π, the evolution becomes more turbulent and can lead to different results even under the same parameter assignments. To simplify the analysis, this study only chooses one condition to explain, which is shown in Figure 5.Figure 5
The impact ofπ on evolution. (a) The platforms’ evolution, (b) the restaurants’ evolution, and (c) the distributors’ evolution.
(a)(b)(c)As Figure5 shows, when π=0, that is, the game runs without stochastic interference, x,y,z converges at 1,0,0, and all their evolutionary paths are smooth. When π=0.5 and 1, respectively, it causes fluctuations in their evolution. Additionally, it fluctuates more apparently with higher π, which means that the stochastic interference makes the players dither about their strategies. As π continually grows, it leads to even severer fluctuations, and x,y,z converges at 0,1,1 when π=2. In this condition, the restaurants and the distributors choose normal work in the end. In this sense, interference seems to be more beneficial to improve hygiene than the interference-free condition.However, this is just an accidental event. In the experiment, whenπ reaches a high value, x, y, and z can evolve to different results each time, even other parameters stay unchanged. The study conducts 100 experiments under the same parameter assignments, discovering that the set of the ESSs can be 1,0,0 or 0,1,1. Through statistics, it obtains their distributions, as shown in Figure 6.Figure 6
The distribution of different ESSs sets.According to the figure, whenπ=0.5, only 2% of the experiments reach 0,1,1; when π=1, only 17% of the experiments reach 0,1,1; when π=2, only 23% of the experiments reach 0,1,1. It means that stronger interference can increase the probability of 0,1,1. Although its frequency increases with higher π, in most experiments, the ESSs still keep 1,0,0.In brief, the stochastic interference strength impacts the evolution process and results by causing a fluctuation in the game. Although sometimes it brings good results, the probability is low. Apart from that, the fluctuation in their decision-making can disturb the market order. As a result, it is necessary to avoid external interference and deal with it timely.


## 4.4. The Impact of the Step Size

Since stochastic differential equations cannot be directly numerically solved, this study expands the equations obtained in Section3.3, transforming the continuous system into a discrete system for analysis. However, the stability may be different in some cases. Specifically, for the discrete system, when the step size grows, its accuracy becomes worse, and accordingly the stability of the original continuous system may be lost. In order to test the stability of the discrete system, this section takes the case in Section 4.1 as an example to simulate and analyze the evolution process of the platforms’ decision-making by adjusting the step size.According toh=T/N, the step size is decided by T and N. Specifically, it grows when T increases or N decreases. Therefore, this study verifies the stability of the discrete system under the above two conditions respectively. Also, considering that random disturbance may lead to different evolution results, this study selects the typical result for analysis after a number of experiments.On the one hand, based on the parameters in Section4.1, it hypothesizes T=3, 6, and 12, respectively, under which h=0.02, 0.04, and 0.08, respectively. In this case, it simulates the evolutionary game process of takeaway platforms, as shown in Figure 7.Figure 7
The impact ofT on takeaway platforms’ evolution. (a) T=3, (b) T=6, and (c) T=12.
(a)(b)(c)On the other hand, it hypothesizesN=150, 60, and 20, respectively, under which h=0.02, 0.05, and 0.15, respectively. In this case, it simulates the evolutionary game process of takeaway platforms, as shown in Figure 8.Figure 8
The impact ofN on takeaway platforms’ evolution. (a) N=150, (b) N=60, and (c) N=20.
(a)(b)(c)The simulation results show that although the increase of the step size affects the evolution process of takeaway platforms to a certain extent, it is not easy to directly change the ESS analyzed in this study. Besides, the impact ofθ on the evolution speed referred to in Section 4.1 still holds. Therefore, the discrete system is of sufficient stability to support the conclusion of the study.However, the evolution does mutates more frequently ash grows. Accordingly, in experiments, it is necessary to make h small enough in simulation so as to ensure the results as close as possible to the continuous system. Apart from that, in reality, the government should encourage the development of takeaway industry, providing the game players with more frequent decision-making opportunities.


## 5. Conclusion

Aiming at the improvement of takeaway management, this study takes the three groups of takeaway platforms, restaurants, and distributors as the main players, establishing a tripartite evolutionary game model. It introduces incentive coefficients in decision-making, which improves the traditional replication dynamic equation. Besides, it employs Gaussian white noise to simulate the interference factors, in reality, establishing Itȏstochastic differential equations. Based on that, it analyzes the influence of some factors on the evolution through numerical simulation after the expansion of the equations. It suggests that parameters, such as the rectification expenditure proportion coefficient, the incentive coefficient between the two strategies of any one group of players, the random interference intensity, and the step size in decision-making, affect the evolution speed and result. On this basis, this study puts forward some suggestions on improving the hygiene management of takeaways:(1)
Control the value of the expenditure proportion coefficient for rectification. Simulation experiments come to a conclusion that an appropriate increase in this coefficient can effectively improve the work quality of the restaurants and the distributors. Therefore, takeaway platforms can appropriately increase their value when formulating contracts, which adds to the cost of illegal work. Nonetheless, they ought to analyze specific problems in detail and avoid a blind increase. Meanwhile, government departments should also standardize the set of related contracts and the distribution of benefits to improve the quality of the work of the restaurants and the distributors and simultaneously guarantee their legitimate income to maintain long-term and stable cooperation in the takeaway chain.(2)
Make the normal work strategy restrain the illegal work strategy. Simulation results show that when the normal work strategy inhibits the illegal work strategy, the work quality of the three parties in the takeaway process can be effectively improved; otherwise, it may decrease. Therefore, takeaway platforms can strengthen restaurants’ and distributors’ professionalism and social responsibility by strengthening vocational education and raising barriers to entry. Moreover, government departments can strengthen the platforms’ and the public’s awareness of supervision through publicity, education, and so on to suppress the illegal work of the three parties.(3)
Deal with external interference. Experiments suggest that external random factors may interfere with the game’s stability and ESSs, which harm decision-making. Therefore, the platform should take fine management into practice, involving a standardized takeaway process and work quality in every part of the process, obligatory vocational education for its partners before they come to work, and regular supervision during their work to reduce random factors in all parts from food production to delivery. Besides, it ought to technically develop risk prediction and emergency response capabilities to reduce the loss caused by random factors. From the aspect of government departments, they should cooperate with the public forces to improve the supervision of takeaway platforms, restaurants, and the distributors to ensure their work quality.In addition, there are some limitations to this research. First of all, the platforms, restaurants, and the distributors in the takeaway process are in a more complicated system, in which more factors affect their strategic choices. This research is not comprehensive enough to consider all those factors in the tripartite game and will employ more detailed parameter settings in line with real scenes in the future. Besides, due to the limitations of data collection and the idealized parameters allocation, this article only conducts numerical simulations through assignment, and future experiments will combine actual cases for more in-depth research.



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*Source: 9058007-2022-07-06.xml*