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Evolutionary Game Analysis of the Dissemination of False Information by Multiple Parties after Major Emergencies

Evolutionary Game Analysis of the Dissemination of False Information by Multiple Parties after... 1. IntroductionSince 2019, with the outbreak of COVID‐19, and ongoing events such as the China‐India border dispute and the Indonesia air crash, major emergencies have had a great impact on public security and social stability and prosperity. Since major emergencies have a relatively strong influence, spread widely, and command high levels of public attention, they can quickly become key topics in general public discussion. Internet users are anonymous, interact with each other, and are located worldwide, which means that major emergencies not only influence the real world but also have a secondary effect on the Internet. For example, after the outbreak of COVID‐19, various online clusters were formed and members of those online clusters were eager to obtain specific information and any updates about the progress of the pandemic. However, the information has a hysteresis effect so that details could not be published instantly, leaving space for people with negative intentions. Therefore, any false information related to COVID‐19 released by such people could influence Internet users who took part in related discussions. If no interference or control is introduced, Internet users can easily experience negative feelings and panic due to making incorrect judgments, and such panic can lead to secondary impacts, which can intensify social instability and increase pressure on governments.Based on the above analysis, this study constructs an evolutionary game model describing how information is disseminated among regulatory institutions, opinion leaders, and Internet users in response to major emergencies. Of these three major parties, regulatory institutions need to decide whether to spend additional time and energy on supervising online information platforms (i.e., choosing strict or loose regulatory strategies), opinion leaders need to decide what kind of information they release (i.e., releasing positive information or false information), and Internet users need to decide whether to adopt or reject information released by opinion leaders. Based on relevant knowledge of evolutionary game theory, the model is solved and evolutionary stability strategies of different parties are analyzed, and then Matlab2017b is used to numerically simulate evolutionary trends under different strategy combinations and varying game parameters.This study answers these two key questions:(1)What are the possible strategies of regulatory institutions, opinion leaders, and Internet users?(2)What is the optimal strategy for a stable society? How can we guide each party to choose their optimal strategy?This review aims to study different behavior of the participants after major emergencies by constructing a tripartite evolutionary game model among regulatory institutions, opinion leaders, and Internet users, which is quite innovative among other studies. In detail, it proceeds as follows: first, Internet users usually place some trust in opinion leaders, but such trust will not be absolute. Therefore, Internet users’ psychological identity with the opinion leaders has been defined as an important parameter and is included in this model construction creatively; second, by solving this model, game strategies promoting social stability and development are concluded, which further provide theoretical basis and decision‐making references for regulatory institutions to deal with online‐cluster behavior after major emergencies.2. Literature ReviewThere is currently no clear definition of false information and rumors in the academic literature. When studying transmission mechanisms of such information, it is generally accepted that the concepts of false information and rumors are consistent and that both represent inauthentic information generated and disseminated via certain media. This paper therefore does not distinguish between false information and rumors and refers to them collectively as false information for convenience.Since the birth of human society, false information has been part of the development of human civilization. Research on false information has attracted great attention from the academic community in various different fields. Some scholars have focused on the dissemination process of false information. Askarizadeh et al. [1] believed that factors such as the public’s recognition of information, social anxiety, and richness of the content would affect the speed of information dissemination. Askarizadeh and Ladani [2] proposed a soft rumor control model to avoid the dissemination of false information by raising public awareness of the false information. Prollochs et al. found that different emotions of different parties cause the scale and duration of false information dissemination to vary [3]. Myilsamy et al. used relevant theories from physics to study the dissemination process of false information [4]. Other scholars have studied the behavior of different parties separately in the information dissemination process. Schuetz et al. divided the behavior of Internet users after receiving false information into three categories: ignoring, accepting but not disseminating, and disseminating for a second time [5]. Ai et al. concluded that the anxiety level of information recipients increases with the spread of false information [6]. Wang et al. considered that during the spread of information, those who disseminate information can release either positive or negative information, and they thereby constructed a model of different strategy choices made by different parties [7].Some scholars carried out research by focusing on how people receive information. Cai et al. revised and optimized ASTRAEA protocols, which enhances the possibility of participants receiving true information and adds a new channel of information acquisition [8]. Martins and Pinto et al. believed that people are easier to acquire true information rather than false one during social activities offline compared with online [9]. Jia et al. held the view that with the development of Internet, people are more likely to receive information via social network platforms that are based on Internet such as smart phone, tablet, and laptop [10]. Zhang et al. thought public transport including subway, bus, and taxi is a major offline information exchange channel for citizens living in Beijing, China [11]. By conducting a questionnaire survey of 963 participants in a first‐tier international city, Yao et al. found that people are able to acquire information via both officially and independently operated media accounts; however, they have a higher reliance on accounts operated officially [12].Similarly, research has been performed on the behavior of opinion leaders in the process of false information dissemination. Opinion leaders have a large influence on the information dissemination process [13]. Some scholars [14–16] have developed a broad definition of opinion leaders that they are able to influence and mold other people’s opinions deeply. Liu et al. concluded that if an opinion leader followed by an individual chose a new behavior, then this individual would do the same and change their original behavior [17]. Zhao et al. found that the influence level of opinion leaders is affected by the degree of trust they receive from their followers [18].On the basis of species evolution theory in the field of biology, Smith et al. proposed important concepts in evolutionary game theory [19]. This has been widely used by the academic community in research on the decision‐making behavior of bounded rational groups. Based on the evolutionary game model, Johari et al. analyzed manufacturers’ pricing strategies [20]. Similarly, An et al. studied innovative behavior between financial and regulatory institutions [21]. Taking the COVID‐19 pandemic as an example, Xu et al. constructed an evolutionary game model of different parties to describe their decision‐making behavior relating to epidemic prevention and control after the occurrence of public health emergencies [22]. Ji et al. also used evolutionary game theory to investigate the behavior and choices of local governments and automakers relating to new energy vehicle subsidy policies [23].In summary, according to classic game model requirements, the different parties should be completely rational and have full access to information about each other [24]. However, the parties in this study are bounded rational, so the classic game model is not suitable here. Conversely, the parties in evolutionary game models can be bounded rational and access to information among different parties does not need to be fully open. Therefore, this study chooses regulatory institutions, opinion leaders, and Internet users as the different parties and constructs a model based on evolutionary game theory.3. Basic Assumptions and Model Construction3.1. Major ParticipantsAfter the outbreak of a major emergency, most of the attention of ordinary Internet users is on issues related to the emergency. Different users will discuss their own interests and concerns, so online clusters focusing on different issues related to the major emergency will be formed. The major participants of such online clusters are opinion leaders with authoritative voices, ordinary Internet users, and regulatory institutions with the right to supervise and regulate, and these three are designated as the parties in the game in this study. Regulatory institutions are defined as functional departments that are able to supervise and regulate the whole Internet, while imposing punishment on disseminators of false information. Opinion leaders are defined as people whose statements or behavior are influential and can shape others’ opinions, and ordinary Internet users are defined as people who are able to exchange information via Internet platforms.Starting with regulatory institutions, false online information can start to appear after major emergencies. If no supervision or regulation is imposed on its production and spread, it is possible for the false information to go viral, leading to negative impacts and even triggering mass events, causing emergent incidents and secondary impacts. Therefore, regulatory institutions could increase the resources allocated to supervision and regulation, thereby regulating information dissemination platforms more strictly, so as to reduce the probability of producing false information (hereinafter referred to as “strict regulation”); similarly, regulation institutions can also reduce the resources allocated to supervision and regulation of information dissemination platforms, and direct more attention to dealing with the secondary effects of major emergencies (hereinafter referred to as “loose regulation”). Opinion leaders could guide people in a positive direction: after investigating and collecting evidence of key Internet discussion topics, they could release positive information, attract and maintain Internet users’ attention and raise their own profile (hereinafter referred to as “positive information”). Opinion leaders could also guide people in a negative direction, that is, by faking information or starting rumors to release false information, so as to attract Internet users’ attention and raise their profile that way (hereinafter referred to as “false information”). Internet users could adopt information from opinion leaders (hereinafter referred to as “adoption”), or reject that information (hereinafter referred to as “rejection”).All three game parties are bounded rational and capable of learning. When information is not complete, they cannot instantly judge whether a decision optimizes their own interests. However, as each party is capable of learning, they can gradually find a strategy that mostly optimizes their own interests as their learning progresses. The game strategies described above can be categorized as (strict regulation, positive information, adoption), (strict regulation, positive information, rejection), (strict regulation, false information, adoption), (strict regulation, false information, rejection), (loose regulation, positive information, adoption), (loose regulation, positive information, rejection), (loose regulation, false information, adoption), and (loose regulation, false information, rejection).3.2. Basic Assumptions and DefinitionsThe assumptions and definitions made in the tripartite evolutionary game model of regulatory institutions, opinion leaders, and Internet users are as follows:(1)The probability of regulatory institutions choosing strict regulation is x(0 ≤ x ≤ 1), so the probability of regulatory institutions choosing loose regulation is 1 − x; the probability of opinion leaders releasing positive information is y(0 ≤ y ≤ 1), so the probability of opinion leaders releasing false information is 1 − y; the probability of Internet users adopting is z(0 ≤ z ≤ 1), so the probability of rejecting is 1 − z. The values of x, y, and z vary with time t.(2)The psychological identification of Internet users with opinion leaders is λ(0 < λ < 1), which represents how Internet users selectively adopt information released by opinion leaders. This paper assumes that the psychological identification of Internet users with opinion leaders is a trust relationship formed after following their comments over a long period. When the quantity is close to 0, there is no trust between Internet users and opinion leaders, and when it is close to 1, Internet users have complete trust in opinion leaders. Due to the existence of psychological identity, Internet users would trust information released by opinion leaders selectively, and it is more in line with reality. Therefore, the psychological identification of Internet users with opinion leaders is set as a vital parameter in the construction of this model. (Note: if the value of Internet users’ psychological identification with opinion leaders equals 0, it means opinion leaders make unrepresentative comments; therefore, such comments would not be accepted by Internet users; if the value equals 1, it means Internet users trust comments from opinion leaders blindly, which does not occur in reality.)(3)If regulatory institutions choose strict regulation instead of loose regulation, the additional cost is ΔC. However, strict regulation of information platforms will produce some income Sr. When regulatory institutions choose loose regulation and Internet users reject any comments from opinion leaders, the lack of important information about major emergencies will cause panic among Internet users and a resulting cost Pr. When Internet users adopt false information from opinion leaders, regulatory institutions will experience losses (e.g., decreases in government credibility and loss from public panic and mass events) due to incorrect judgments and their subsequent impact; when Internet users adopt positive information, this will prompt a boost in social stability Ir. When regulatory institutions impose strict regulation, false information from opinion leaders has an impact Ps on social stability; when regulatory institutions impose loose regulation, the impact on social stability is Pl.(4)When opinion leaders release positive information, the cost of investigation and evidence collection is Cl, while the reward under strict regulation from regulatory institutions is I; the comment is Mp; when opinion leaders release false information, no costs are incurred, and the comment is Mn; since false information in the context of major emergencies is very likely to have greater influence on the Internet, which may lead to online mass events with negative impacts and even cause secondary impacts of major emergencies, regulatory institutions have a probability of π1 (strict regulation) and π2 (loose regulation) that they will impose punishment Lr on opinion leaders disseminating false information.(5)As false information is more likely to stimulate attention and discussion among Internet users, it brings additional income ΔIl to opinion leaders when they release false information that is adopted. Opinion leaders who publish positive information and those who publish false information will lose attention when their comments are not adopted by the public. When comments are not adopted by Internet users, the attention lost by opinion leaders disseminating positive information is Lp, and the attention lost by opinion leaders disseminating false information is Ln.(6)By focusing attention on major emergencies, Internet users obtain a sense of participation Ip and satisfaction In, after adopting positive information from opinion leaders. Similarly, Internet users incur time and energy costs Ca.(7)When Internet users choose to adopt information released by opinion leaders, the cost incurred in judging the information is (1‐λ)Cn. When Internet users adopt positive information, they internalize a number of opinion leaders’ comments λMp (i.e., the sense of safety); when Internet users adopt negative information, the negative income is λMn (i.e., the sense of panic; as the comments are false information, this is included in the game model calculation as –λMn).(8)All the parameters mentioned above are greater than zero, and 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1, 0 ≤ π2 ≤ π1 ≤ 1, Ln > Lp, 0 < λ < 1, In > Ip, Pl > Ps.The definitions of the main parameter symbols are summarized in Table 1.1TableParameters and symbols.ParameterDefinitionλInternet users’ psychological identification with opinion leaders.ΔCAdditional regulatory costs paid by regulators when they choose strict regulation.SrIncome of regulatory institutions when they choose strict regulation (e.g., approval from Internet users, improvement of self‐image, etc.).PrLoss incurred by regulatory institutions when they loosen regulation and Internet users reject comments released by opinion leaders (e.g., increasing panic felt by Internet users due to missing information and its impact on social stability).PLoss incurred by regulatory institutions when Internet users adopt false information and their subsequent incorrect judgement has negative impact (e.g., decrease in government credibility, losses due to public panic, and mass events).PsImpact of negative information released by opinion leaders when regulatory institutions choose strict regulation.PlImpact of negative information released by opinion leaders when regulatory institutions choose loose regulation.IrIncrease of social stability when Internet users adopt positive information.π1Probability that regulatory institutions punish opinion leaders disseminating false information under strict regulation.π2Probability that regulatory institutions punish opinion leaders disseminating false information under loose regulation.LrPunishment imposed by regulatory institutions on opinion leaders disseminating false information.ClCosts of investigation and evidence collection paid by opinion leaders disseminating positive information.IReward for opinion leaders disseminating positive information under strict regulation (e.g., regulatory institutions’ recognition of opinion leaders).MpComments from opinion leaders disseminating positive information.MnComments from opinion leaders disseminating negative information.ΔIlAdditional income gained by opinion leaders when Internet users adopt false information (e.g., attention, visitor traffic).LpLoss of attention when Internet users reject positive information released by opinion leaders.LnLoss of attention when Internet users reject false information released by opinion leaders.SnIncome of Internet users when strict regulation of information platforms brings benefits (e.g., generally a greater dissemination of correct information).CaCosts of time and energy of Internet users when they adopt comments from opinion leaders.PnPanic caused by a general lack of information when Internet users reject comments under loose regulation.IPThe sense of participation and satisfaction when Internet users adopt positive information released by opinion leaders.InThe sense of participation and satisfaction when internet users adopt negative information released by opinion leaders.CnCosts incurred by Internet users in independently verifying the accuracy of information released by opinion leaders.xProbability that regulatory institutions choose strict regulation.YProbability that opinion leaders choose to release positive information.ZProbability that Internet users adopt information released by opinion leaders.3.3. Model ConstructionBased on the assumptions proposed above, it is concluded that when the game strategy is (strict regulation, positive information, adoption), the income of regulatory institutions is −ΔC + Ir + Sr, the income of opinion leaders is I − Cl, and the income of Internet users is Ip + λMp + Sn − (1 − λ)Cn − Ca. When the game strategy is (strict regulation, positive information, rejection), the income of regulatory institutions is −ΔC + Sr, the income of opinion leaders is I − Cl − Lp, and the income of Internet users is Sn. When the game strategy is (strict regulation, false information, adoption), the income of regulatory institutions is −ΔC − P − Ps + Sr, the income of opinion leaders is −π1Lr + ΔIl, and the income of Internet users is In − λMn + Sn − (1 − λ)Cn − Ca. When the game strategy is (strict regulation, false information, rejection), the income of regulatory institutions is −ΔC + Sr − Ps, the income of opinion leaders is −π1Lr − Ln, and the income of Internet users is Sn. When the game strategy is (loose regulation, positive information, adoption), the income of regulatory institutions is Ir, the income of opinion leaders is –Cl, and the income of Internet users is IP + λMp−(1 − λ)Cn − Ca. When the game strategy is (loose regulation, positive information, rejection), the income of regulatory institutions is −Pr, the income of opinion leaders is −Cl − Lp, and the income of Internet users is −Pn. When the game strategy is (loose regulation, false information, adoption), the income of regulatory institutions is −P − Pl, the income of opinion leaders is −π2Lr + ΔIl, and the income of Internet users is In − λMn−(1 − λ)Cn − Ca. When the game strategy is (loose regulation, false information, rejection), the income of regulatory institutions is −Pr − Pl, the income of opinion leaders is −π2Lr − Ln, and the income of Internet users is −Pn. The income of each game participant for each combination of strategies is shown in Table 2.2TableThe tripartite game income matrix for regulatory institutions, opinion leaders, and Internet users.Regulatory institutions and opinion leadersInternet usersAdoptionRejectionRegulatory institutionsStrict regulationOpinion leadersPositive information−ΔC + Ir + Sr−ΔC + SrI − ClI − Cl − LpIp + λMp + Sn−(1 − λ)Cn − CaSnFalse information−ΔC − P − Ps + Sr−ΔC + Sr − Ps−π1Lr + ΔIl−π1Lr − LnIn − λMn + Sn−(1 − λ)Cn − CaSnLoose regulationOpinion leadersPositive informationIr−Pr−Cl−Cl − LpIP + λMp−(1 − λ)Cn − Ca−PnFalse information−P − Pl−Pr − Pl−π2Lr + ΔIl−π2Lr − LnIn − λMn−(1 − λ)Cn − Ca−PnBased on the income matrix shown in Table 2, denoting Ur1 as the expected income of regulatory institutions when they impose strict regulation of information dissemination platforms, Ur2 as the expected income of regulatory institutions when they impose loose regulation of information dissemination platforms, and Ur¯ as the expected overall annual income of regulatory institutions, then the following equations are obtained for Ur1,Ur2, and Ur¯:1Ur1=yz−ΔC+Ir+Sr+y1−z−ΔC+Sr+1−yz−ΔC−Ps−P+Sr+1−y1−z−ΔC+Sr−Ps=yzIr+P−zP+yPs−ΔC−Ps+SrUr2=yzIr+1−yz−P−Pl+y1−z−Pr+1−y1−z−P−Pr=yzIr+P+zPr−P+yPl−Pr−PlUr¯=xUr1+1−xUr2=xyPs−Pl−xzPr+yzIr+P+xPr−Ps+Pl+Sr−ΔC+yPl+zPr−P−Pr−Pl.Therefore, the replicator dynamic equation for regulatory institutions with strict regulation is 2Fx=dxdt=xUr1−Ur¯=x1−xUr1−Ur2=x1−xPl+Pr−Ps+Sr−ΔC−yPl−Ps−zPr.Similarly, denoting Ul1 as the expected income of opinion leaders disseminating positive information, Ul2 as the expected income of opinion leaders disseminating negative information, and Ul¯ as the expected overall annual income of opinion leaders, then the following equations are obtained for Ul1, Ul2, and Ul¯:3Ul1=xz−Cl+I+x1−z−Cl+I−Lp+1−xz−Cl+1−x1−z−Cl−Lp=xI−Lp−Cl+zLpUl2=xz−π1Lr+ΔIl+x1−z−π1Lr−Ln+1−xz−π2Lr+ΔIl+1−x1−z−π2Lr−Ln=−xπ1−π2Lr+zLn+ΔIl−Ln−π2LrUl¯=yUl1+1−yUl2=xyI+π1−π2Lr+yz−Ln+Lp−ΔIl−xπ1−π2Lr+yLn−Cl−Lp+π2Lr+zLn+ΔIl−Ln−π2Lr.Therefore, the replicator dynamic equation for opinion leaders disseminating positive information is4Fy=dydt=yUl1−Ul¯=y1−yUl1−Ul2=y1−y Ln−Cl−Lp+π2Lr+xLrπ1−π2+xI−zLn−Lp+ΔIl.Denoting Un1 as the expected income of Internet users when they adopt information released by opinion leaders, Un2 as the expected income of Internet users when they reject information released by opinion leaders, and Un¯ as the expected overall annual income of Internet users, then the following equations are obtained for Un1, Un2, and Un¯:5Un1=xyλMp+Ip+Sn−1−λCn−Ca+x1−yIn−λMn+Sn−1−λCn−Ca+1−xyIp+λMp−1−λCn−Ca+1−x1−yIn−λMn−1−λCn−Ca=xSn+y−In+Ip+λMn+Mp+In−1−λCn−Ca−λMnUn2=xySn+x1−ySn+1−xy−Pn+1−x1−y−Pn=−Pn+xPn+SnUn¯=zUn1+1−zUn2=−xzPn+yzIp−In+λMn+Mp+xPn+Sn +zIn−Ca−Cn+Pn+λCn−Mn−Pn.Therefore, the replicator dynamic equation for Internet users when they adopt information released by opinion leaders is 6Fz=dzdt=zUn1−Un¯=z1−zUn1−Un2=z1−zIn−Ca−Cn+Pn+λCn−Mn+yIp−In+λMn+λMp−xPn.4. Systematic Stability AnalysisBy combining the replicator dynamic, and (2) (4), and (6), we obtain the following replicator dynamic system of regulatory institutions, opinion leaders, and Internet users:7Fx=x1−xPl+Pr−Ps+Sr−ΔC−yPl−Ps−zPrFy=y1−yLn−Cl−Lp+π2Lr+xLrπ1−π2+xI−zLn−Lp+ΔIlFz=z1−zIn−Ca−Cn+Pn+λCn−Mn+yIp−In+λMn+λMp−xPn.Setting the value of (7) as 0, we solve and obtain the partial stable equilibrium points of this system: E1(0,0,0), E2(0,0,1), E3(0,1,0), E4(0,1,1), E5(1,0,0), E6(1,0,1), E7(1,1,0), E8(1,1,1), and E9(x∗,y∗,z∗). As partial equilibrium points are not always evolutionary equilibrium points of a system, a stability analysis should be carried out. According to Ritzberger and Weibull [25], for a tripartite evolutionary game, asymptotical stability analysis only needs to be applied to the system's pure strategy Nash equilibrium points, so point E9(x∗,y∗,z∗) is not considered. Based on the method of Friedman [26], and applying partial stability of the Jacobian matrix to perform an asymptotical stability analysis on the above eight pure strategy Nash equilibrium points, the system’s Jacobian matrix is as follows:8J=∂Fx∂x,∂Fx∂y,∂Fx∂z∂Fy∂x,∂Fy∂y,∂Fy∂z∂Fz∂x,∂Fz∂y,∂Fz∂z=J11J12J13J21J22J23J31J32J33,where9J11=12−xPl+Pr−Ps+Sr−ΔC+yPs−Pl−zPrJ12=xx−1Pl−PsJ13=xx−1PrJ21=y1−yI+Lrπ1−π2J22=12−yLn−Cl−Lp+π2Lr+xLrπ1−π2+xI−zLn−Lp+ΔIlJ23=yy−1Ln−Lp+ΔIlJ31=zz−1PnJ32=z1−zIp−In+λMn+λMpJ33=12−zIn−Ca−Cn+Pn+λCn−Mn+yIp−In+λMn+λMp−xPn.By inputting the eight pure strategy Nash equilibrium points into the Jacobian matrix (8), eigenvalues for different equilibrium states are obtained as shown in Table 3.3TableSystem eigenvalues under different equilibrium points.No.Equilibrium pointsEigenvaluesλ1λ2λ3(1)E1 (0,0,0)Pl + Pr + Sr − ΔC − PsLn − Cl − Lp + π2LrIn − Cn − Ca + Pn + λ(Cn − Mn)(2)E2 (0,0,1)Pl + Sr − ΔC − Psπ2Lr − Cl − ΔIl−[In − Cn − Ca + Pn + λ(Cn − Mn)](3)E3 (0,1,0)Pr + Sr − ΔC−(Ln − Cl −Lp + π2Lr)Ip − Cn − Ca + Pn + λ(Cn + Mp)(4)E4 (0,1,1)Sr − ΔC−(π2Lr − Cl − ΔIl)−[Ip − Cn − Ca + Pn + λ(Cn + Mp)](5)E5 (1,0,0)−(Pl + Pr + Sr − ΔC − Ps)I − Cl + Ln − Lp + π1LrIn − Cn − Ca + λ(Cn − Mn)(6)E6 (1,0,1)−(Pl + Sr − ΔC − Ps)I − Cl + π1Lr − ΔIl−[In − Cn − Ca + λ(Cn − Mn)](7)E7 (1,1,0)−(Pr + Sr − ΔC)−(I − Cl + Ln − Lp + π1Lr)Ip − Cn − Ca + λ(Cn + Mp)(8)E8 (1,1,1)ΔC − Sr−(I − Cl + π1Lr − ΔIl)−[Ip − Cn − Ca + λ(Cn + Mp)]According to the first method of Liapunov [27], when any partial stable equilibrium point is input into the Jacobian matrix, if all the eigenvalues of the matrix are negative, then the point is asymptotically stable and the system will tend to a stable state, meaning that game strategy is an evolutionary game strategy; if at least one eigenvalue of the Jacobian matrix is not negative, then that point is an unstable point, and the system will be unstable as well. Equilibrium point E1(0,0,0) is used as an example here to investigate the stability of the system.When Pr + Sr + Pl < ΔC + Ps, Ln + π2Lr < Cl + Lp, and In + Pn < Ca + λMn+(1 − λ)Cn, the eigenvalues of the Jacobian matrix at point E1(0,0,0) are all negative, so E1(0,0,0) is an evolutionary stable point and the system will tend to a stable state, so the strategy combining loose regulation, false information, and rejection is an evolutionary stability strategy under these conditions. This means that, in the end, regulatory institutions would choose loose regulation, opinion leaders would release false information, and Internet users would reject information released by opinion leaders.Similarly, the conditions for the seven other pure strategy Nash equilibrium points to be evolutionary stable points are shown in Table 4.4TableConditions for system to be asymptotical stability at each equilibrium point.No.Equilibrium pointConditions for system to be asymptotical stability(1)E1 (0,0,0)Pr + Sr + Pl < ΔC + Ps, Ln + π2Lr < Cl + Lp, In + Pn < Ca + λMn+(1 − λ)Cn(2)E2 (0,0,1)Pl + Sr< ΔC + Ps, π2Lr < Cl + ΔIl, In + Pn > Ca + λMn+(1 − λ)Cn(3)E3 (0,1,0)Pr + Sr< ΔC, Ln + π2Lr > Cl + Lp, Ip + Pn + λMp < Ca+(1 − λ)Cn(4)E4 (0,1,1)Sr< ΔC, π2Lr > Cl + ΔIl, Ip + Pn + λMp > Ca + (1 − λ)Cn(5)E5 (1,0,0)Pr + Sr + Pl > ΔC + Ps, I + Ln + π1Lr < Cl + Lp, In < Ca + λMn + (1 − λ)Cn(6)E6 (1,0,1)Pl + Sr> ΔC + Ps, I + π1Lr < Cl + ΔIl, In > Ca + λMn+(1 − λ)Cn(7)E7 (1,1,0)Pr + Sr> ΔC, I + Ln + π1Lr > Cl + Lp, Ip + λMp < Ca+(1 − λ)Cn(8)E8 (1,1,1)ΔC < Sr, I + π1Lr > Cl + ΔIl, Ip + λMp > Ca+(1 − λ)Cn5. Numerical Simulation AnalysisTo effectively verify the asymptotical stability of the equilibrium points discussed above, Matlab2017b is applied to perform a numerical simulation analysis of regulatory institutions, opinion leaders, and Internet users.Considering the real world, when major emergencies occur, secondary impacts on the public should be avoided so as to maintain social stability. Therefore, the more favorable equilibrium points are E1(0,0,0), E4(0,1,1), E5(1,0,0), and E8(1,1,1). Among these, the strategy that minimizes the burden on regulatory institutions and maximizes their benefits is E4(0,1,1).Therefore, this study only analyses the above four equilibrium points (E1, E4, E5, E8). We investigate in detail (x, y, z) = (0.2, 0.4, 0.6) as the initial condition for the probabilities that regulatory institutions choose strict regulation, opinion leaders release positive information, and Internet users adopt information released by opinion leaders. For clarity, in Figures 1–4, the probabilities that regulatory institutions choose strict regulation during a strategy selection process are shown as blue lines, the probabilities that opinion leaders choose to release positive information during a strategy selection process are shown as green lines, and the probabilities that Internet users choose to adopt information during a strategy selection process are shown as red lines.1Figure(a)(b)Tripartite evolutionary game trending figure stable at point E1 (0,0,0) between “regulatory institutions, opinion leaders and Internet users.” (a) Strategy selection process. (b) 3D simulation diagram of the system.2Figure(a)(b)Tripartite evolutionary game trending figure stable at point E4 (0,1,1) between “regulatory institutions, opinion leaders and Internet users.” (a) Strategy selection process. (b) 3D simulation diagram of the system.3Figure(a)(b)Tripartite evolutionary game trending figure stable at point E5 (1,0,0) between “regulatory institutions, opinion leaders and Internet users.” (a) Strategy selection process. (b) 3D simulation diagram of the system.4Figure(a)(b)Tripartite evolutionary game trending figure stable at point E8 (1,1,1) between “regulatory institutions, opinion leaders and Internet users”. (a) Strategy selection process. (b) 3D simulation diagram of the system.5.1. Equilibrium Point E1(0,0,0) Is Asymptotically StableFrom Table 4, it is found that the necessary conditions for E1(0,0,0) to be asymptotically stable are Pr + Sr + Pl < ΔC + Ps, Ln + π2Lr < Cl + Lp, and In + Pn < Ca + λMn+(1 − λ)Cn. Assigning values to each parameter and performing the numerical simulation, the tripartite evolutionary game trending figure stable at point E1(0,0,0) between “regulatory institutions, opinion leaders, and Internet users” is obtained, as in Figure 1. (Set ΔC = 6, Sr = 1, Pr = 3, P = 4, Ps = 3, Pl = 4, Ir = 2, π1 = 0.6, π2 = 0.4, Lr = 3, Cl = 3, I = 2, Mp = 6, Mn = 8, ΔIl = 1, Lp = 2, Ln = 3, Sn = 2, Ca = 2, Pn = 3, IP = 5, In = 7, Cn = 10, λ = 0.6.)As shown in Figure 1(a), under the condition of Pr + Sr + Pl < ΔC + Ps, Ln + π2Lr < Cl + Lp, and In + Pn < Ca + λMn+(1 − λ)Cn, when, as time evolves, the strategy choice probabilities for regulatory institutions, opinion leaders, and Internet users all converge to 0. To further prove that the result of Figure 1(a) is a certain event, set the value of x, y, z between the section of 0.05 to 1, respectively, with 0.1 defined as its interval, and then carry out a three‐dimension simulation. With the result of Figure 1(b), it is found that the three‐dimensional dynamical system will converge at E1(0,0,0) and finally reach its stable status regardless of the initial value of x, y, and z, which means the evolutionary game equilibrium result is the strategy combination of loose regulation, negative information, and rejection. Above theoretical analysis is verified by Figures 1(a) and 1(b).It is concluded that when all three conditions are matched at the same time: (a) the sum of the income gained by regulatory institutions applying strict regulation, the difference in impact of negative information under strict regulation and loose regulation imposed by regulatory institutions, and the cost to regulatory institutions when Internet users reject any information released by opinion leaders under loose regulation and become panicked is smaller than the additional costs paid by regulatory institutions to apply strict regulation; (b) the sum of the loss of attention borne by opinion leaders when their false information is rejected by Internet users and the punishment for opinion leaders disseminating false information, imposed by regulatory institutions under loose regulation, is smaller than the investigation costs of opinion leaders when they release positive information; (c) the sum of the sense of participation and satisfaction gained by Internet users when they adopt false information released by opinion leaders and the panic caused by a lack of information when information released by opinion leaders is rejected by Internet users under loose regulation, is smaller than the sum of the information gained by Internet users when they adopt false information, the costs of time and energy in adopting information, and the costs of judging the accuracy of information released by opinion leaders.After the major emergencies under these conditions, regulatory institutions would tend toward looser regulations on information dissemination platforms, opinion leaders would tend toward releasing false information, and Internet users would tend toward rejecting any information from opinion leaders.5.2. Equilibrium Point E4 (0,1,1) Is Asymptotically StableFrom Table 4, it is seen that the conditions for E4(0,1,1) being asymptotically stable are that Sr< ΔC, π2Lr> Cl + ΔIl, and Ip + Pn + λMp > Ca+(1 − λ)Cn. Assigning values to each parameter and performing the numerical simulation, the tripartite evolutionary game trending figure stable at point E4(0,1,1) between “regulatory institutions, opinion leaders, and Internet users” can be obtained as in Figure 2. (Set C = 10, Sr = 6, Pr = 3, P = 4, Ps = 3, Pl = 4, Ir = 2, π1 = 0.6, π2 = 0.4, Lr = 11, Cl = 3, I = 2, Mp = 6, Mn = 8, ΔIl = 1, Lp = 2, Ln = 3, Sn = 2, Ca = 2, Pn = 3, IP = 5, In = 7, Cn = 10, λ = 0.6.)As shown in Figure 2(a), under the condition of Sr< ΔC, π2Lr> Cl + ΔIl, and Ip + Pn + λMp > Ca+(1 − λ)Cn, when, as time evolves, the strategy choice probability for regulatory institutions will gradually converge to 0, while the strategy choice probability for opinion leaders and Internet users will gradually converge to 1. To further prove that the result of Figure 2(a) is a certain event, set the value of x, y, z between the section of 0.05 to 1, respectively, with 0.1 defined as its interval, and then carry out a three‐dimension simulation. With the result of Figure 2(b), it is found that the three‐dimensional dynamical system will converge at E4(0,1,1) and finally reach a stable status, which means the evolutionary game equilibrium result is the strategy combination of loose regulation, positive information, and adoption. Above theoretical analysis is verified by Figures 2(a) and 2(b).It is concluded that when all three conditions are matched at the same time: (a) the income of regulatory institutions applying strict regulation is smaller than the additional costs of regulation; (b) the sum of investigation costs of opinion leaders disseminating positive information and the additional costs of opinion leaders when they release false information and having them adopted by Internet users, is smaller than the punishment borne by opinion leaders when they release false information under loose regulation imposed by regulatory institutions; (c) the sum of the sense of participation and satisfaction gained by Internet users when they adopt positive information from opinion leaders, the panic caused by a lack of information when information released by opinion leaders is rejected by Internet users under loose regulation, and the information gained by Internet users when they adopt positive information, is greater than the sum of time and energy costs borne by Internet users when they adopt information released by opinion leaders and the costs of judging the accuracy of information released by opinion leaders.After major emergencies under these conditions, regulatory institutions would tend toward looser regulation of information dissemination platforms, opinion leaders would tend toward releasing positive information, and Internet users would tend toward adopting information from opinion leaders.5.3. Equilibrium Point E5(1,0,0) Is Asymptotically StableFrom Table 4, it is seen that the conditions for E5(1,0,0) being asymptotically stable are Pr + Sr + Pl > ΔC + Ps, I + Ln + π1Lr < Cl + Lp, and In < Ca + λMn + (1 − λ)Cn. Assigning values to each parameter and performing the numerical simulation, the tripartite evolutionary game trending figure stable at point E5(1,0,0) between “regulatory institutions, opinion leaders, and Internet users” is obtained, as in Figure 3. (Set ΔC = 6, Sr = 10, Pr = 3, P = 4, Ps = 3, Pl = 4, Ir = 2, π1 = 0.6, π2 = 0.4, Lr = 5, Cl = 7, I = 2, Mp = 6, Mn = 8, ΔIl = 1, Lp = 2, Ln = 3, Sn = 2, Ca = 2, Pn = 3, IP = 5, In = 7, Cn = 10, λ = 0.6.)As shown in Figure 3(a), under the condition of Pr + Sr + Pl > ΔC + Ps, I + Ln + π1Lr < Cl + Lp, and In < Ca + λMn+(1 − λ)Cn, when, as time evolves, the strategy choice probability for regulatory institutions will gradually converge to 1 and the strategy choice probabilities for opinion leaders and Internet users will gradually converge to 0. To further prove that the result of Figure 3(a) is a certain event, set the value of x, y, z between the section of 0.05 to 1, respectively, with 0.1 defined as its interval, and then carry out a three‐dimension simulation. With the result of Figure 3(b), it is found that the three‐dimensional dynamical system will converge at E5 (1,0,0), and finally reach a stable state, which means the evolutionary game equilibrium result is the strategy combination of strict regulation, negative information, and rejection. Above theoretical analysis is verified by Figures 3(a) and 3(b).It is concluded that when all three conditions are matched at the same time: (a) the sum of the income of regulatory institutions applying strict regulation, the difference in impact caused by negative information under strict regulation and loose regulation imposed by regulatory institutions, and the loss to regulatory institutions caused by public panic when they choose loose regulation and Internet users reject any information released by opinion leaders, is greater than the additional costs to regulatory institutions in applying strict regulation; (b) the sum of the reward gained by opinion leaders when they release positive information under strict regulation, the additional loss of attention borne by opinion leaders when they release false information that is rejected by Internet users, and the punishment borne by opinion leaders disseminating false information under strict regulation, is smaller than the investigation costs to opinion leaders in releasing positive information; (c) the sum of the information gained by Internet users when they adopt false information, the costs of energy and time borne by Internet users in adopting information, and the costs of judging the accuracy of information released by opinion leaders, is greater than the sense of participation and satisfaction gained by Internet users when they adopt false information released by opinion leaders.After major emergencies under these conditions, regulatory institutions would tend toward tighter regulations of information dissemination platforms, opinion leaders would tend toward releasing false information, and Internet users would tend toward rejecting any information from opinion leaders.5.4. Equilibrium Point E8(1,1,1) Is Asymptotically StableFrom Table 4, it is seen that the conditions for E8(1,1,1) to be asymptotically stable are that ΔC < Sr, I + π1Lr > Cl + ΔIl, and Ip + λMp > Ca+(1 − λ)Cn. Assigning values to each parameter and performing the numerical simulation, the tripartite evolutionary game trending figure stable at point E8(1,1,1) between “regulatory institutions, opinion leaders, and Internet users” is obtained, as in Figure 4. (Set ΔC = 6, Sr = 10, Pr = 3, P = 4, Ps = 3, Pl = 4, Ir = 2, π1 = 0.6, π2 = 0.4, Lr = 5, Cl = 3, I = 2, Mp = 6, Mn = 8, ΔIl = 1, Lp = 2, Ln = 3, Sn = 2, Ca = 2, Pn = 3, IP = 5, In = 7, Cn = 10, λ = 0.6.)As shown in Figure 4(a), under the condition of ΔC < Sr, I + π1Lr > Cl + ΔIl, and Ip + λMp > Ca+(1 − λ)Cn, when, as time evolves, the strategy choice probabilities for regulatory institutions, opinion leaders, and Internet users will gradually converge to 1. To further prove that the result of Figure 4(a) is a certain event, set the value of x, y, z between the section of 0.05 to 1, respectively, with 0.1 defined as its interval, and then carry out a three‐dimension simulation. With the result of Figure 4(b), it is found that the three‐dimensional dynamical system converges at E8 (1,1,1) and finally reaches a stable state, which means the evolutionary game equilibrium result is the strategy combination of strict regulation, positive information, and adoption. Above theoretical analysis is verified by Figures 4(a) and 4(b).It is concluded that when all three conditions are matched at the same time: (a) the additional income of regulatory institutions applying strict regulation is greater than their additional regulatory costs; (b) the sum of the reward gained by opinion leaders when they release positive information under strict regulation and the punishment borne by opinion leaders disseminating false information under strict regulation, is greater than the investigation costs of opinion leaders when they release positive information and the additional income gained by opinion leaders when Internet users adopt their false information; (c) the sum of the sense of participation and satisfaction gained by Internet users when they adopt positive information and the information gained by Internet users when they adopt positive information, is greater than the sum of the time and energy costs borne by Internet users when they adopt information released by opinion leaders and the costs of judging the accuracy of information released by opinion leaders.After major emergencies under these conditions, regulatory institutions would tend toward tighter regulations of information dissemination platforms, opinion leaders would tend toward releasing positive information, and Internet users would tend toward adopting information from opinion leaders.Based on the above simulation analysis result, four game strategy points of E1, E5, E8, E8 are able to avoid a second‐time influence on the public caused by false information dissemination after major emergencies; however, all of them have some shortage except E4. The advantage of game strategy point E1 (0,0,0) is that no extra work should be done by regulatory institutions, so work pressure decreases, but the downside is that opinion leaders could release false information with no interference. The advantage of game strategy point E5 (1,0,0) is that there will be no public panic as Internet users reject opinion leaders’ false information; however, extra costs should be paid by regulatory institutions. The advantage of game strategy point E8 (1,1,1) is that the society will run smoothly due to Internet users’ acceptance of positive information released by opinion leaders, but once again, extra costs should be paid by regulatory institutions.6. ConclusionThe Internet is currently experiencing rapid development, such that a sustained release of false information triggered by a major emergency can impose a level of harm and panic on the public just as great as the impact of the emergency itself. Because classical game theory cannot describe the behavior of parties with bounded rationality who do not fully share information with each other, this study adopts evolutionary game theory, with the innovative approach of adding Internet users’ psychological identification with opinion leaders as an important parameter, to research the behavior of false information transmission among different participants after major emergencies. This is done by constructing a tripartite evolutionary game model between regulatory institutions, opinion leaders, and Internet users. Based on the solutions of models and numerical simulation, the main conclusions and insights are as follows:First, considering the real world, when major emergencies happen, secondary impacts should be avoided so as to maintain social stability. Strategy choices favorable for social stability require that Internet users adopt positive information released by opinion leaders or reject false information, and adoption of positive information is the more favorable of the two. Four strategy choices meet this condition. From the perspective of regulatory institutions, to relieve pressure, the best solution is to maintain the regulation level of information dissemination platforms and not to add additional regulatory costs. They should rely on opinion leaders to release positive information and Internet users to adopt positive information.Second, for the public and regulatory institutions, the probability of choosing the most optimal strategy is positively correlated with the punishment imposed by regulatory institutions on opinion leaders disseminating false information, the reward provided by regulatory institutions for opinion leaders disseminating positive information, the sense of participation and satisfaction gained by Internet users when they adopt positive information, the content richness of authentic information released by opinion leaders, and Internet users’ psychological identification with opinion leaders. It is negatively correlated with the costs of investigation and evidence collection borne by opinion leaders when they release positive information, the additional income of opinion leaders when they release false information and have it adopted, the time and energy costs of Internet users when they adopt information from opinion leaders, and the costs of judging the accuracy of information independently.Therefore, by taking approaches such as increasing the punishment for opinion leaders who release false information, increasing the reward for opinion leaders who release positive information, improving progress in investigation of major emergencies and publishing authentic information, and improving the information collection channels for opinion leaders, regulatory institutions can help opinion leaders and Internet users make better choices. Opinion leaders can attract Internet users’ attention by increasing the extent to which Internet users psychologically identify with them and enriching their authentic content in future information releases so as to help Internet users adopt positive information from them.Last but not least, this study outperforms other studies with classic game theory as the research method because bounded rational people capable of learning are chosen as game parties. All participants from all three groups are able to learn from and simulate others and thus dynamically adjusting their game strategies and maximize their own interests on the basis of others’ choices. Evolutionary game theory fully satisfies the research aim of studying three groups of game participants. However, there remain some limitations in this study. People are affected by emotions, which increase the complexity of their behavior choices in different situations, so model construction in this study is biased due to not taking this into consideration. Moreover, the model described here applies only to the influence of false information on game participants after major emergencies. In the future, we expect that a model able to analyze the reaction of game participants to false more generally will be constructed and will improve upon our model’s accuracy and also broaden its applicability to a wider range of circumstances.Data AvailabilityAll the data used to support the findings of this study are included within the article.Conflicts of InterestThe authors declare that there are no conflicts of interest regarding the publication of this study.AcknowledgmentsThis research was funded by Natural Science Foundation of China (no. 71771112) and Project of Liaoning Provincial Federation Social Science Circles of China (no. 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Evolutionary Game Analysis of the Dissemination of False Information by Multiple Parties after Major Emergencies

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Abstract

1. IntroductionSince 2019, with the outbreak of COVID‐19, and ongoing events such as the China‐India border dispute and the Indonesia air crash, major emergencies have had a great impact on public security and social stability and prosperity. Since major emergencies have a relatively strong influence, spread widely, and command high levels of public attention, they can quickly become key topics in general public discussion. Internet users are anonymous, interact with each other, and are located worldwide, which means that major emergencies not only influence the real world but also have a secondary effect on the Internet. For example, after the outbreak of COVID‐19, various online clusters were formed and members of those online clusters were eager to obtain specific information and any updates about the progress of the pandemic. However, the information has a hysteresis effect so that details could not be published instantly, leaving space for people with negative intentions. Therefore, any false information related to COVID‐19 released by such people could influence Internet users who took part in related discussions. If no interference or control is introduced, Internet users can easily experience negative feelings and panic due to making incorrect judgments, and such panic can lead to secondary impacts, which can intensify social instability and increase pressure on governments.Based on the above analysis, this study constructs an evolutionary game model describing how information is disseminated among regulatory institutions, opinion leaders, and Internet users in response to major emergencies. Of these three major parties, regulatory institutions need to decide whether to spend additional time and energy on supervising online information platforms (i.e., choosing strict or loose regulatory strategies), opinion leaders need to decide what kind of information they release (i.e., releasing positive information or false information), and Internet users need to decide whether to adopt or reject information released by opinion leaders. Based on relevant knowledge of evolutionary game theory, the model is solved and evolutionary stability strategies of different parties are analyzed, and then Matlab2017b is used to numerically simulate evolutionary trends under different strategy combinations and varying game parameters.This study answers these two key questions:(1)What are the possible strategies of regulatory institutions, opinion leaders, and Internet users?(2)What is the optimal strategy for a stable society? How can we guide each party to choose their optimal strategy?This review aims to study different behavior of the participants after major emergencies by constructing a tripartite evolutionary game model among regulatory institutions, opinion leaders, and Internet users, which is quite innovative among other studies. In detail, it proceeds as follows: first, Internet users usually place some trust in opinion leaders, but such trust will not be absolute. Therefore, Internet users’ psychological identity with the opinion leaders has been defined as an important parameter and is included in this model construction creatively; second, by solving this model, game strategies promoting social stability and development are concluded, which further provide theoretical basis and decision‐making references for regulatory institutions to deal with online‐cluster behavior after major emergencies.2. Literature ReviewThere is currently no clear definition of false information and rumors in the academic literature. When studying transmission mechanisms of such information, it is generally accepted that the concepts of false information and rumors are consistent and that both represent inauthentic information generated and disseminated via certain media. This paper therefore does not distinguish between false information and rumors and refers to them collectively as false information for convenience.Since the birth of human society, false information has been part of the development of human civilization. Research on false information has attracted great attention from the academic community in various different fields. Some scholars have focused on the dissemination process of false information. Askarizadeh et al. [1] believed that factors such as the public’s recognition of information, social anxiety, and richness of the content would affect the speed of information dissemination. Askarizadeh and Ladani [2] proposed a soft rumor control model to avoid the dissemination of false information by raising public awareness of the false information. Prollochs et al. found that different emotions of different parties cause the scale and duration of false information dissemination to vary [3]. Myilsamy et al. used relevant theories from physics to study the dissemination process of false information [4]. Other scholars have studied the behavior of different parties separately in the information dissemination process. Schuetz et al. divided the behavior of Internet users after receiving false information into three categories: ignoring, accepting but not disseminating, and disseminating for a second time [5]. Ai et al. concluded that the anxiety level of information recipients increases with the spread of false information [6]. Wang et al. considered that during the spread of information, those who disseminate information can release either positive or negative information, and they thereby constructed a model of different strategy choices made by different parties [7].Some scholars carried out research by focusing on how people receive information. Cai et al. revised and optimized ASTRAEA protocols, which enhances the possibility of participants receiving true information and adds a new channel of information acquisition [8]. Martins and Pinto et al. believed that people are easier to acquire true information rather than false one during social activities offline compared with online [9]. Jia et al. held the view that with the development of Internet, people are more likely to receive information via social network platforms that are based on Internet such as smart phone, tablet, and laptop [10]. Zhang et al. thought public transport including subway, bus, and taxi is a major offline information exchange channel for citizens living in Beijing, China [11]. By conducting a questionnaire survey of 963 participants in a first‐tier international city, Yao et al. found that people are able to acquire information via both officially and independently operated media accounts; however, they have a higher reliance on accounts operated officially [12].Similarly, research has been performed on the behavior of opinion leaders in the process of false information dissemination. Opinion leaders have a large influence on the information dissemination process [13]. Some scholars [14–16] have developed a broad definition of opinion leaders that they are able to influence and mold other people’s opinions deeply. Liu et al. concluded that if an opinion leader followed by an individual chose a new behavior, then this individual would do the same and change their original behavior [17]. Zhao et al. found that the influence level of opinion leaders is affected by the degree of trust they receive from their followers [18].On the basis of species evolution theory in the field of biology, Smith et al. proposed important concepts in evolutionary game theory [19]. This has been widely used by the academic community in research on the decision‐making behavior of bounded rational groups. Based on the evolutionary game model, Johari et al. analyzed manufacturers’ pricing strategies [20]. Similarly, An et al. studied innovative behavior between financial and regulatory institutions [21]. Taking the COVID‐19 pandemic as an example, Xu et al. constructed an evolutionary game model of different parties to describe their decision‐making behavior relating to epidemic prevention and control after the occurrence of public health emergencies [22]. Ji et al. also used evolutionary game theory to investigate the behavior and choices of local governments and automakers relating to new energy vehicle subsidy policies [23].In summary, according to classic game model requirements, the different parties should be completely rational and have full access to information about each other [24]. However, the parties in this study are bounded rational, so the classic game model is not suitable here. Conversely, the parties in evolutionary game models can be bounded rational and access to information among different parties does not need to be fully open. Therefore, this study chooses regulatory institutions, opinion leaders, and Internet users as the different parties and constructs a model based on evolutionary game theory.3. Basic Assumptions and Model Construction3.1. Major ParticipantsAfter the outbreak of a major emergency, most of the attention of ordinary Internet users is on issues related to the emergency. Different users will discuss their own interests and concerns, so online clusters focusing on different issues related to the major emergency will be formed. The major participants of such online clusters are opinion leaders with authoritative voices, ordinary Internet users, and regulatory institutions with the right to supervise and regulate, and these three are designated as the parties in the game in this study. Regulatory institutions are defined as functional departments that are able to supervise and regulate the whole Internet, while imposing punishment on disseminators of false information. Opinion leaders are defined as people whose statements or behavior are influential and can shape others’ opinions, and ordinary Internet users are defined as people who are able to exchange information via Internet platforms.Starting with regulatory institutions, false online information can start to appear after major emergencies. If no supervision or regulation is imposed on its production and spread, it is possible for the false information to go viral, leading to negative impacts and even triggering mass events, causing emergent incidents and secondary impacts. Therefore, regulatory institutions could increase the resources allocated to supervision and regulation, thereby regulating information dissemination platforms more strictly, so as to reduce the probability of producing false information (hereinafter referred to as “strict regulation”); similarly, regulation institutions can also reduce the resources allocated to supervision and regulation of information dissemination platforms, and direct more attention to dealing with the secondary effects of major emergencies (hereinafter referred to as “loose regulation”). Opinion leaders could guide people in a positive direction: after investigating and collecting evidence of key Internet discussion topics, they could release positive information, attract and maintain Internet users’ attention and raise their own profile (hereinafter referred to as “positive information”). Opinion leaders could also guide people in a negative direction, that is, by faking information or starting rumors to release false information, so as to attract Internet users’ attention and raise their profile that way (hereinafter referred to as “false information”). Internet users could adopt information from opinion leaders (hereinafter referred to as “adoption”), or reject that information (hereinafter referred to as “rejection”).All three game parties are bounded rational and capable of learning. When information is not complete, they cannot instantly judge whether a decision optimizes their own interests. However, as each party is capable of learning, they can gradually find a strategy that mostly optimizes their own interests as their learning progresses. The game strategies described above can be categorized as (strict regulation, positive information, adoption), (strict regulation, positive information, rejection), (strict regulation, false information, adoption), (strict regulation, false information, rejection), (loose regulation, positive information, adoption), (loose regulation, positive information, rejection), (loose regulation, false information, adoption), and (loose regulation, false information, rejection).3.2. Basic Assumptions and DefinitionsThe assumptions and definitions made in the tripartite evolutionary game model of regulatory institutions, opinion leaders, and Internet users are as follows:(1)The probability of regulatory institutions choosing strict regulation is x(0 ≤ x ≤ 1), so the probability of regulatory institutions choosing loose regulation is 1 − x; the probability of opinion leaders releasing positive information is y(0 ≤ y ≤ 1), so the probability of opinion leaders releasing false information is 1 − y; the probability of Internet users adopting is z(0 ≤ z ≤ 1), so the probability of rejecting is 1 − z. The values of x, y, and z vary with time t.(2)The psychological identification of Internet users with opinion leaders is λ(0 < λ < 1), which represents how Internet users selectively adopt information released by opinion leaders. This paper assumes that the psychological identification of Internet users with opinion leaders is a trust relationship formed after following their comments over a long period. When the quantity is close to 0, there is no trust between Internet users and opinion leaders, and when it is close to 1, Internet users have complete trust in opinion leaders. Due to the existence of psychological identity, Internet users would trust information released by opinion leaders selectively, and it is more in line with reality. Therefore, the psychological identification of Internet users with opinion leaders is set as a vital parameter in the construction of this model. (Note: if the value of Internet users’ psychological identification with opinion leaders equals 0, it means opinion leaders make unrepresentative comments; therefore, such comments would not be accepted by Internet users; if the value equals 1, it means Internet users trust comments from opinion leaders blindly, which does not occur in reality.)(3)If regulatory institutions choose strict regulation instead of loose regulation, the additional cost is ΔC. However, strict regulation of information platforms will produce some income Sr. When regulatory institutions choose loose regulation and Internet users reject any comments from opinion leaders, the lack of important information about major emergencies will cause panic among Internet users and a resulting cost Pr. When Internet users adopt false information from opinion leaders, regulatory institutions will experience losses (e.g., decreases in government credibility and loss from public panic and mass events) due to incorrect judgments and their subsequent impact; when Internet users adopt positive information, this will prompt a boost in social stability Ir. When regulatory institutions impose strict regulation, false information from opinion leaders has an impact Ps on social stability; when regulatory institutions impose loose regulation, the impact on social stability is Pl.(4)When opinion leaders release positive information, the cost of investigation and evidence collection is Cl, while the reward under strict regulation from regulatory institutions is I; the comment is Mp; when opinion leaders release false information, no costs are incurred, and the comment is Mn; since false information in the context of major emergencies is very likely to have greater influence on the Internet, which may lead to online mass events with negative impacts and even cause secondary impacts of major emergencies, regulatory institutions have a probability of π1 (strict regulation) and π2 (loose regulation) that they will impose punishment Lr on opinion leaders disseminating false information.(5)As false information is more likely to stimulate attention and discussion among Internet users, it brings additional income ΔIl to opinion leaders when they release false information that is adopted. Opinion leaders who publish positive information and those who publish false information will lose attention when their comments are not adopted by the public. When comments are not adopted by Internet users, the attention lost by opinion leaders disseminating positive information is Lp, and the attention lost by opinion leaders disseminating false information is Ln.(6)By focusing attention on major emergencies, Internet users obtain a sense of participation Ip and satisfaction In, after adopting positive information from opinion leaders. Similarly, Internet users incur time and energy costs Ca.(7)When Internet users choose to adopt information released by opinion leaders, the cost incurred in judging the information is (1‐λ)Cn. When Internet users adopt positive information, they internalize a number of opinion leaders’ comments λMp (i.e., the sense of safety); when Internet users adopt negative information, the negative income is λMn (i.e., the sense of panic; as the comments are false information, this is included in the game model calculation as –λMn).(8)All the parameters mentioned above are greater than zero, and 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1, 0 ≤ π2 ≤ π1 ≤ 1, Ln > Lp, 0 < λ < 1, In > Ip, Pl > Ps.The definitions of the main parameter symbols are summarized in Table 1.1TableParameters and symbols.ParameterDefinitionλInternet users’ psychological identification with opinion leaders.ΔCAdditional regulatory costs paid by regulators when they choose strict regulation.SrIncome of regulatory institutions when they choose strict regulation (e.g., approval from Internet users, improvement of self‐image, etc.).PrLoss incurred by regulatory institutions when they loosen regulation and Internet users reject comments released by opinion leaders (e.g., increasing panic felt by Internet users due to missing information and its impact on social stability).PLoss incurred by regulatory institutions when Internet users adopt false information and their subsequent incorrect judgement has negative impact (e.g., decrease in government credibility, losses due to public panic, and mass events).PsImpact of negative information released by opinion leaders when regulatory institutions choose strict regulation.PlImpact of negative information released by opinion leaders when regulatory institutions choose loose regulation.IrIncrease of social stability when Internet users adopt positive information.π1Probability that regulatory institutions punish opinion leaders disseminating false information under strict regulation.π2Probability that regulatory institutions punish opinion leaders disseminating false information under loose regulation.LrPunishment imposed by regulatory institutions on opinion leaders disseminating false information.ClCosts of investigation and evidence collection paid by opinion leaders disseminating positive information.IReward for opinion leaders disseminating positive information under strict regulation (e.g., regulatory institutions’ recognition of opinion leaders).MpComments from opinion leaders disseminating positive information.MnComments from opinion leaders disseminating negative information.ΔIlAdditional income gained by opinion leaders when Internet users adopt false information (e.g., attention, visitor traffic).LpLoss of attention when Internet users reject positive information released by opinion leaders.LnLoss of attention when Internet users reject false information released by opinion leaders.SnIncome of Internet users when strict regulation of information platforms brings benefits (e.g., generally a greater dissemination of correct information).CaCosts of time and energy of Internet users when they adopt comments from opinion leaders.PnPanic caused by a general lack of information when Internet users reject comments under loose regulation.IPThe sense of participation and satisfaction when Internet users adopt positive information released by opinion leaders.InThe sense of participation and satisfaction when internet users adopt negative information released by opinion leaders.CnCosts incurred by Internet users in independently verifying the accuracy of information released by opinion leaders.xProbability that regulatory institutions choose strict regulation.YProbability that opinion leaders choose to release positive information.ZProbability that Internet users adopt information released by opinion leaders.3.3. Model ConstructionBased on the assumptions proposed above, it is concluded that when the game strategy is (strict regulation, positive information, adoption), the income of regulatory institutions is −ΔC + Ir + Sr, the income of opinion leaders is I − Cl, and the income of Internet users is Ip + λMp + Sn − (1 − λ)Cn − Ca. When the game strategy is (strict regulation, positive information, rejection), the income of regulatory institutions is −ΔC + Sr, the income of opinion leaders is I − Cl − Lp, and the income of Internet users is Sn. When the game strategy is (strict regulation, false information, adoption), the income of regulatory institutions is −ΔC − P − Ps + Sr, the income of opinion leaders is −π1Lr + ΔIl, and the income of Internet users is In − λMn + Sn − (1 − λ)Cn − Ca. When the game strategy is (strict regulation, false information, rejection), the income of regulatory institutions is −ΔC + Sr − Ps, the income of opinion leaders is −π1Lr − Ln, and the income of Internet users is Sn. When the game strategy is (loose regulation, positive information, adoption), the income of regulatory institutions is Ir, the income of opinion leaders is –Cl, and the income of Internet users is IP + λMp−(1 − λ)Cn − Ca. When the game strategy is (loose regulation, positive information, rejection), the income of regulatory institutions is −Pr, the income of opinion leaders is −Cl − Lp, and the income of Internet users is −Pn. When the game strategy is (loose regulation, false information, adoption), the income of regulatory institutions is −P − Pl, the income of opinion leaders is −π2Lr + ΔIl, and the income of Internet users is In − λMn−(1 − λ)Cn − Ca. When the game strategy is (loose regulation, false information, rejection), the income of regulatory institutions is −Pr − Pl, the income of opinion leaders is −π2Lr − Ln, and the income of Internet users is −Pn. The income of each game participant for each combination of strategies is shown in Table 2.2TableThe tripartite game income matrix for regulatory institutions, opinion leaders, and Internet users.Regulatory institutions and opinion leadersInternet usersAdoptionRejectionRegulatory institutionsStrict regulationOpinion leadersPositive information−ΔC + Ir + Sr−ΔC + SrI − ClI − Cl − LpIp + λMp + Sn−(1 − λ)Cn − CaSnFalse information−ΔC − P − Ps + Sr−ΔC + Sr − Ps−π1Lr + ΔIl−π1Lr − LnIn − λMn + Sn−(1 − λ)Cn − CaSnLoose regulationOpinion leadersPositive informationIr−Pr−Cl−Cl − LpIP + λMp−(1 − λ)Cn − Ca−PnFalse information−P − Pl−Pr − Pl−π2Lr + ΔIl−π2Lr − LnIn − λMn−(1 − λ)Cn − Ca−PnBased on the income matrix shown in Table 2, denoting Ur1 as the expected income of regulatory institutions when they impose strict regulation of information dissemination platforms, Ur2 as the expected income of regulatory institutions when they impose loose regulation of information dissemination platforms, and Ur¯ as the expected overall annual income of regulatory institutions, then the following equations are obtained for Ur1,Ur2, and Ur¯:1Ur1=yz−ΔC+Ir+Sr+y1−z−ΔC+Sr+1−yz−ΔC−Ps−P+Sr+1−y1−z−ΔC+Sr−Ps=yzIr+P−zP+yPs−ΔC−Ps+SrUr2=yzIr+1−yz−P−Pl+y1−z−Pr+1−y1−z−P−Pr=yzIr+P+zPr−P+yPl−Pr−PlUr¯=xUr1+1−xUr2=xyPs−Pl−xzPr+yzIr+P+xPr−Ps+Pl+Sr−ΔC+yPl+zPr−P−Pr−Pl.Therefore, the replicator dynamic equation for regulatory institutions with strict regulation is 2Fx=dxdt=xUr1−Ur¯=x1−xUr1−Ur2=x1−xPl+Pr−Ps+Sr−ΔC−yPl−Ps−zPr.Similarly, denoting Ul1 as the expected income of opinion leaders disseminating positive information, Ul2 as the expected income of opinion leaders disseminating negative information, and Ul¯ as the expected overall annual income of opinion leaders, then the following equations are obtained for Ul1, Ul2, and Ul¯:3Ul1=xz−Cl+I+x1−z−Cl+I−Lp+1−xz−Cl+1−x1−z−Cl−Lp=xI−Lp−Cl+zLpUl2=xz−π1Lr+ΔIl+x1−z−π1Lr−Ln+1−xz−π2Lr+ΔIl+1−x1−z−π2Lr−Ln=−xπ1−π2Lr+zLn+ΔIl−Ln−π2LrUl¯=yUl1+1−yUl2=xyI+π1−π2Lr+yz−Ln+Lp−ΔIl−xπ1−π2Lr+yLn−Cl−Lp+π2Lr+zLn+ΔIl−Ln−π2Lr.Therefore, the replicator dynamic equation for opinion leaders disseminating positive information is4Fy=dydt=yUl1−Ul¯=y1−yUl1−Ul2=y1−y Ln−Cl−Lp+π2Lr+xLrπ1−π2+xI−zLn−Lp+ΔIl.Denoting Un1 as the expected income of Internet users when they adopt information released by opinion leaders, Un2 as the expected income of Internet users when they reject information released by opinion leaders, and Un¯ as the expected overall annual income of Internet users, then the following equations are obtained for Un1, Un2, and Un¯:5Un1=xyλMp+Ip+Sn−1−λCn−Ca+x1−yIn−λMn+Sn−1−λCn−Ca+1−xyIp+λMp−1−λCn−Ca+1−x1−yIn−λMn−1−λCn−Ca=xSn+y−In+Ip+λMn+Mp+In−1−λCn−Ca−λMnUn2=xySn+x1−ySn+1−xy−Pn+1−x1−y−Pn=−Pn+xPn+SnUn¯=zUn1+1−zUn2=−xzPn+yzIp−In+λMn+Mp+xPn+Sn +zIn−Ca−Cn+Pn+λCn−Mn−Pn.Therefore, the replicator dynamic equation for Internet users when they adopt information released by opinion leaders is 6Fz=dzdt=zUn1−Un¯=z1−zUn1−Un2=z1−zIn−Ca−Cn+Pn+λCn−Mn+yIp−In+λMn+λMp−xPn.4. Systematic Stability AnalysisBy combining the replicator dynamic, and (2) (4), and (6), we obtain the following replicator dynamic system of regulatory institutions, opinion leaders, and Internet users:7Fx=x1−xPl+Pr−Ps+Sr−ΔC−yPl−Ps−zPrFy=y1−yLn−Cl−Lp+π2Lr+xLrπ1−π2+xI−zLn−Lp+ΔIlFz=z1−zIn−Ca−Cn+Pn+λCn−Mn+yIp−In+λMn+λMp−xPn.Setting the value of (7) as 0, we solve and obtain the partial stable equilibrium points of this system: E1(0,0,0), E2(0,0,1), E3(0,1,0), E4(0,1,1), E5(1,0,0), E6(1,0,1), E7(1,1,0), E8(1,1,1), and E9(x∗,y∗,z∗). As partial equilibrium points are not always evolutionary equilibrium points of a system, a stability analysis should be carried out. According to Ritzberger and Weibull [25], for a tripartite evolutionary game, asymptotical stability analysis only needs to be applied to the system's pure strategy Nash equilibrium points, so point E9(x∗,y∗,z∗) is not considered. Based on the method of Friedman [26], and applying partial stability of the Jacobian matrix to perform an asymptotical stability analysis on the above eight pure strategy Nash equilibrium points, the system’s Jacobian matrix is as follows:8J=∂Fx∂x,∂Fx∂y,∂Fx∂z∂Fy∂x,∂Fy∂y,∂Fy∂z∂Fz∂x,∂Fz∂y,∂Fz∂z=J11J12J13J21J22J23J31J32J33,where9J11=12−xPl+Pr−Ps+Sr−ΔC+yPs−Pl−zPrJ12=xx−1Pl−PsJ13=xx−1PrJ21=y1−yI+Lrπ1−π2J22=12−yLn−Cl−Lp+π2Lr+xLrπ1−π2+xI−zLn−Lp+ΔIlJ23=yy−1Ln−Lp+ΔIlJ31=zz−1PnJ32=z1−zIp−In+λMn+λMpJ33=12−zIn−Ca−Cn+Pn+λCn−Mn+yIp−In+λMn+λMp−xPn.By inputting the eight pure strategy Nash equilibrium points into the Jacobian matrix (8), eigenvalues for different equilibrium states are obtained as shown in Table 3.3TableSystem eigenvalues under different equilibrium points.No.Equilibrium pointsEigenvaluesλ1λ2λ3(1)E1 (0,0,0)Pl + Pr + Sr − ΔC − PsLn − Cl − Lp + π2LrIn − Cn − Ca + Pn + λ(Cn − Mn)(2)E2 (0,0,1)Pl + Sr − ΔC − Psπ2Lr − Cl − ΔIl−[In − Cn − Ca + Pn + λ(Cn − Mn)](3)E3 (0,1,0)Pr + Sr − ΔC−(Ln − Cl −Lp + π2Lr)Ip − Cn − Ca + Pn + λ(Cn + Mp)(4)E4 (0,1,1)Sr − ΔC−(π2Lr − Cl − ΔIl)−[Ip − Cn − Ca + Pn + λ(Cn + Mp)](5)E5 (1,0,0)−(Pl + Pr + Sr − ΔC − Ps)I − Cl + Ln − Lp + π1LrIn − Cn − Ca + λ(Cn − Mn)(6)E6 (1,0,1)−(Pl + Sr − ΔC − Ps)I − Cl + π1Lr − ΔIl−[In − Cn − Ca + λ(Cn − Mn)](7)E7 (1,1,0)−(Pr + Sr − ΔC)−(I − Cl + Ln − Lp + π1Lr)Ip − Cn − Ca + λ(Cn + Mp)(8)E8 (1,1,1)ΔC − Sr−(I − Cl + π1Lr − ΔIl)−[Ip − Cn − Ca + λ(Cn + Mp)]According to the first method of Liapunov [27], when any partial stable equilibrium point is input into the Jacobian matrix, if all the eigenvalues of the matrix are negative, then the point is asymptotically stable and the system will tend to a stable state, meaning that game strategy is an evolutionary game strategy; if at least one eigenvalue of the Jacobian matrix is not negative, then that point is an unstable point, and the system will be unstable as well. Equilibrium point E1(0,0,0) is used as an example here to investigate the stability of the system.When Pr + Sr + Pl < ΔC + Ps, Ln + π2Lr < Cl + Lp, and In + Pn < Ca + λMn+(1 − λ)Cn, the eigenvalues of the Jacobian matrix at point E1(0,0,0) are all negative, so E1(0,0,0) is an evolutionary stable point and the system will tend to a stable state, so the strategy combining loose regulation, false information, and rejection is an evolutionary stability strategy under these conditions. This means that, in the end, regulatory institutions would choose loose regulation, opinion leaders would release false information, and Internet users would reject information released by opinion leaders.Similarly, the conditions for the seven other pure strategy Nash equilibrium points to be evolutionary stable points are shown in Table 4.4TableConditions for system to be asymptotical stability at each equilibrium point.No.Equilibrium pointConditions for system to be asymptotical stability(1)E1 (0,0,0)Pr + Sr + Pl < ΔC + Ps, Ln + π2Lr < Cl + Lp, In + Pn < Ca + λMn+(1 − λ)Cn(2)E2 (0,0,1)Pl + Sr< ΔC + Ps, π2Lr < Cl + ΔIl, In + Pn > Ca + λMn+(1 − λ)Cn(3)E3 (0,1,0)Pr + Sr< ΔC, Ln + π2Lr > Cl + Lp, Ip + Pn + λMp < Ca+(1 − λ)Cn(4)E4 (0,1,1)Sr< ΔC, π2Lr > Cl + ΔIl, Ip + Pn + λMp > Ca + (1 − λ)Cn(5)E5 (1,0,0)Pr + Sr + Pl > ΔC + Ps, I + Ln + π1Lr < Cl + Lp, In < Ca + λMn + (1 − λ)Cn(6)E6 (1,0,1)Pl + Sr> ΔC + Ps, I + π1Lr < Cl + ΔIl, In > Ca + λMn+(1 − λ)Cn(7)E7 (1,1,0)Pr + Sr> ΔC, I + Ln + π1Lr > Cl + Lp, Ip + λMp < Ca+(1 − λ)Cn(8)E8 (1,1,1)ΔC < Sr, I + π1Lr > Cl + ΔIl, Ip + λMp > Ca+(1 − λ)Cn5. Numerical Simulation AnalysisTo effectively verify the asymptotical stability of the equilibrium points discussed above, Matlab2017b is applied to perform a numerical simulation analysis of regulatory institutions, opinion leaders, and Internet users.Considering the real world, when major emergencies occur, secondary impacts on the public should be avoided so as to maintain social stability. Therefore, the more favorable equilibrium points are E1(0,0,0), E4(0,1,1), E5(1,0,0), and E8(1,1,1). Among these, the strategy that minimizes the burden on regulatory institutions and maximizes their benefits is E4(0,1,1).Therefore, this study only analyses the above four equilibrium points (E1, E4, E5, E8). We investigate in detail (x, y, z) = (0.2, 0.4, 0.6) as the initial condition for the probabilities that regulatory institutions choose strict regulation, opinion leaders release positive information, and Internet users adopt information released by opinion leaders. For clarity, in Figures 1–4, the probabilities that regulatory institutions choose strict regulation during a strategy selection process are shown as blue lines, the probabilities that opinion leaders choose to release positive information during a strategy selection process are shown as green lines, and the probabilities that Internet users choose to adopt information during a strategy selection process are shown as red lines.1Figure(a)(b)Tripartite evolutionary game trending figure stable at point E1 (0,0,0) between “regulatory institutions, opinion leaders and Internet users.” (a) Strategy selection process. (b) 3D simulation diagram of the system.2Figure(a)(b)Tripartite evolutionary game trending figure stable at point E4 (0,1,1) between “regulatory institutions, opinion leaders and Internet users.” (a) Strategy selection process. (b) 3D simulation diagram of the system.3Figure(a)(b)Tripartite evolutionary game trending figure stable at point E5 (1,0,0) between “regulatory institutions, opinion leaders and Internet users.” (a) Strategy selection process. (b) 3D simulation diagram of the system.4Figure(a)(b)Tripartite evolutionary game trending figure stable at point E8 (1,1,1) between “regulatory institutions, opinion leaders and Internet users”. (a) Strategy selection process. (b) 3D simulation diagram of the system.5.1. Equilibrium Point E1(0,0,0) Is Asymptotically StableFrom Table 4, it is found that the necessary conditions for E1(0,0,0) to be asymptotically stable are Pr + Sr + Pl < ΔC + Ps, Ln + π2Lr < Cl + Lp, and In + Pn < Ca + λMn+(1 − λ)Cn. Assigning values to each parameter and performing the numerical simulation, the tripartite evolutionary game trending figure stable at point E1(0,0,0) between “regulatory institutions, opinion leaders, and Internet users” is obtained, as in Figure 1. (Set ΔC = 6, Sr = 1, Pr = 3, P = 4, Ps = 3, Pl = 4, Ir = 2, π1 = 0.6, π2 = 0.4, Lr = 3, Cl = 3, I = 2, Mp = 6, Mn = 8, ΔIl = 1, Lp = 2, Ln = 3, Sn = 2, Ca = 2, Pn = 3, IP = 5, In = 7, Cn = 10, λ = 0.6.)As shown in Figure 1(a), under the condition of Pr + Sr + Pl < ΔC + Ps, Ln + π2Lr < Cl + Lp, and In + Pn < Ca + λMn+(1 − λ)Cn, when, as time evolves, the strategy choice probabilities for regulatory institutions, opinion leaders, and Internet users all converge to 0. To further prove that the result of Figure 1(a) is a certain event, set the value of x, y, z between the section of 0.05 to 1, respectively, with 0.1 defined as its interval, and then carry out a three‐dimension simulation. With the result of Figure 1(b), it is found that the three‐dimensional dynamical system will converge at E1(0,0,0) and finally reach its stable status regardless of the initial value of x, y, and z, which means the evolutionary game equilibrium result is the strategy combination of loose regulation, negative information, and rejection. Above theoretical analysis is verified by Figures 1(a) and 1(b).It is concluded that when all three conditions are matched at the same time: (a) the sum of the income gained by regulatory institutions applying strict regulation, the difference in impact of negative information under strict regulation and loose regulation imposed by regulatory institutions, and the cost to regulatory institutions when Internet users reject any information released by opinion leaders under loose regulation and become panicked is smaller than the additional costs paid by regulatory institutions to apply strict regulation; (b) the sum of the loss of attention borne by opinion leaders when their false information is rejected by Internet users and the punishment for opinion leaders disseminating false information, imposed by regulatory institutions under loose regulation, is smaller than the investigation costs of opinion leaders when they release positive information; (c) the sum of the sense of participation and satisfaction gained by Internet users when they adopt false information released by opinion leaders and the panic caused by a lack of information when information released by opinion leaders is rejected by Internet users under loose regulation, is smaller than the sum of the information gained by Internet users when they adopt false information, the costs of time and energy in adopting information, and the costs of judging the accuracy of information released by opinion leaders.After the major emergencies under these conditions, regulatory institutions would tend toward looser regulations on information dissemination platforms, opinion leaders would tend toward releasing false information, and Internet users would tend toward rejecting any information from opinion leaders.5.2. Equilibrium Point E4 (0,1,1) Is Asymptotically StableFrom Table 4, it is seen that the conditions for E4(0,1,1) being asymptotically stable are that Sr< ΔC, π2Lr> Cl + ΔIl, and Ip + Pn + λMp > Ca+(1 − λ)Cn. Assigning values to each parameter and performing the numerical simulation, the tripartite evolutionary game trending figure stable at point E4(0,1,1) between “regulatory institutions, opinion leaders, and Internet users” can be obtained as in Figure 2. (Set C = 10, Sr = 6, Pr = 3, P = 4, Ps = 3, Pl = 4, Ir = 2, π1 = 0.6, π2 = 0.4, Lr = 11, Cl = 3, I = 2, Mp = 6, Mn = 8, ΔIl = 1, Lp = 2, Ln = 3, Sn = 2, Ca = 2, Pn = 3, IP = 5, In = 7, Cn = 10, λ = 0.6.)As shown in Figure 2(a), under the condition of Sr< ΔC, π2Lr> Cl + ΔIl, and Ip + Pn + λMp > Ca+(1 − λ)Cn, when, as time evolves, the strategy choice probability for regulatory institutions will gradually converge to 0, while the strategy choice probability for opinion leaders and Internet users will gradually converge to 1. To further prove that the result of Figure 2(a) is a certain event, set the value of x, y, z between the section of 0.05 to 1, respectively, with 0.1 defined as its interval, and then carry out a three‐dimension simulation. With the result of Figure 2(b), it is found that the three‐dimensional dynamical system will converge at E4(0,1,1) and finally reach a stable status, which means the evolutionary game equilibrium result is the strategy combination of loose regulation, positive information, and adoption. Above theoretical analysis is verified by Figures 2(a) and 2(b).It is concluded that when all three conditions are matched at the same time: (a) the income of regulatory institutions applying strict regulation is smaller than the additional costs of regulation; (b) the sum of investigation costs of opinion leaders disseminating positive information and the additional costs of opinion leaders when they release false information and having them adopted by Internet users, is smaller than the punishment borne by opinion leaders when they release false information under loose regulation imposed by regulatory institutions; (c) the sum of the sense of participation and satisfaction gained by Internet users when they adopt positive information from opinion leaders, the panic caused by a lack of information when information released by opinion leaders is rejected by Internet users under loose regulation, and the information gained by Internet users when they adopt positive information, is greater than the sum of time and energy costs borne by Internet users when they adopt information released by opinion leaders and the costs of judging the accuracy of information released by opinion leaders.After major emergencies under these conditions, regulatory institutions would tend toward looser regulation of information dissemination platforms, opinion leaders would tend toward releasing positive information, and Internet users would tend toward adopting information from opinion leaders.5.3. Equilibrium Point E5(1,0,0) Is Asymptotically StableFrom Table 4, it is seen that the conditions for E5(1,0,0) being asymptotically stable are Pr + Sr + Pl > ΔC + Ps, I + Ln + π1Lr < Cl + Lp, and In < Ca + λMn + (1 − λ)Cn. Assigning values to each parameter and performing the numerical simulation, the tripartite evolutionary game trending figure stable at point E5(1,0,0) between “regulatory institutions, opinion leaders, and Internet users” is obtained, as in Figure 3. (Set ΔC = 6, Sr = 10, Pr = 3, P = 4, Ps = 3, Pl = 4, Ir = 2, π1 = 0.6, π2 = 0.4, Lr = 5, Cl = 7, I = 2, Mp = 6, Mn = 8, ΔIl = 1, Lp = 2, Ln = 3, Sn = 2, Ca = 2, Pn = 3, IP = 5, In = 7, Cn = 10, λ = 0.6.)As shown in Figure 3(a), under the condition of Pr + Sr + Pl > ΔC + Ps, I + Ln + π1Lr < Cl + Lp, and In < Ca + λMn+(1 − λ)Cn, when, as time evolves, the strategy choice probability for regulatory institutions will gradually converge to 1 and the strategy choice probabilities for opinion leaders and Internet users will gradually converge to 0. To further prove that the result of Figure 3(a) is a certain event, set the value of x, y, z between the section of 0.05 to 1, respectively, with 0.1 defined as its interval, and then carry out a three‐dimension simulation. With the result of Figure 3(b), it is found that the three‐dimensional dynamical system will converge at E5 (1,0,0), and finally reach a stable state, which means the evolutionary game equilibrium result is the strategy combination of strict regulation, negative information, and rejection. Above theoretical analysis is verified by Figures 3(a) and 3(b).It is concluded that when all three conditions are matched at the same time: (a) the sum of the income of regulatory institutions applying strict regulation, the difference in impact caused by negative information under strict regulation and loose regulation imposed by regulatory institutions, and the loss to regulatory institutions caused by public panic when they choose loose regulation and Internet users reject any information released by opinion leaders, is greater than the additional costs to regulatory institutions in applying strict regulation; (b) the sum of the reward gained by opinion leaders when they release positive information under strict regulation, the additional loss of attention borne by opinion leaders when they release false information that is rejected by Internet users, and the punishment borne by opinion leaders disseminating false information under strict regulation, is smaller than the investigation costs to opinion leaders in releasing positive information; (c) the sum of the information gained by Internet users when they adopt false information, the costs of energy and time borne by Internet users in adopting information, and the costs of judging the accuracy of information released by opinion leaders, is greater than the sense of participation and satisfaction gained by Internet users when they adopt false information released by opinion leaders.After major emergencies under these conditions, regulatory institutions would tend toward tighter regulations of information dissemination platforms, opinion leaders would tend toward releasing false information, and Internet users would tend toward rejecting any information from opinion leaders.5.4. Equilibrium Point E8(1,1,1) Is Asymptotically StableFrom Table 4, it is seen that the conditions for E8(1,1,1) to be asymptotically stable are that ΔC < Sr, I + π1Lr > Cl + ΔIl, and Ip + λMp > Ca+(1 − λ)Cn. Assigning values to each parameter and performing the numerical simulation, the tripartite evolutionary game trending figure stable at point E8(1,1,1) between “regulatory institutions, opinion leaders, and Internet users” is obtained, as in Figure 4. (Set ΔC = 6, Sr = 10, Pr = 3, P = 4, Ps = 3, Pl = 4, Ir = 2, π1 = 0.6, π2 = 0.4, Lr = 5, Cl = 3, I = 2, Mp = 6, Mn = 8, ΔIl = 1, Lp = 2, Ln = 3, Sn = 2, Ca = 2, Pn = 3, IP = 5, In = 7, Cn = 10, λ = 0.6.)As shown in Figure 4(a), under the condition of ΔC < Sr, I + π1Lr > Cl + ΔIl, and Ip + λMp > Ca+(1 − λ)Cn, when, as time evolves, the strategy choice probabilities for regulatory institutions, opinion leaders, and Internet users will gradually converge to 1. To further prove that the result of Figure 4(a) is a certain event, set the value of x, y, z between the section of 0.05 to 1, respectively, with 0.1 defined as its interval, and then carry out a three‐dimension simulation. With the result of Figure 4(b), it is found that the three‐dimensional dynamical system converges at E8 (1,1,1) and finally reaches a stable state, which means the evolutionary game equilibrium result is the strategy combination of strict regulation, positive information, and adoption. Above theoretical analysis is verified by Figures 4(a) and 4(b).It is concluded that when all three conditions are matched at the same time: (a) the additional income of regulatory institutions applying strict regulation is greater than their additional regulatory costs; (b) the sum of the reward gained by opinion leaders when they release positive information under strict regulation and the punishment borne by opinion leaders disseminating false information under strict regulation, is greater than the investigation costs of opinion leaders when they release positive information and the additional income gained by opinion leaders when Internet users adopt their false information; (c) the sum of the sense of participation and satisfaction gained by Internet users when they adopt positive information and the information gained by Internet users when they adopt positive information, is greater than the sum of the time and energy costs borne by Internet users when they adopt information released by opinion leaders and the costs of judging the accuracy of information released by opinion leaders.After major emergencies under these conditions, regulatory institutions would tend toward tighter regulations of information dissemination platforms, opinion leaders would tend toward releasing positive information, and Internet users would tend toward adopting information from opinion leaders.Based on the above simulation analysis result, four game strategy points of E1, E5, E8, E8 are able to avoid a second‐time influence on the public caused by false information dissemination after major emergencies; however, all of them have some shortage except E4. The advantage of game strategy point E1 (0,0,0) is that no extra work should be done by regulatory institutions, so work pressure decreases, but the downside is that opinion leaders could release false information with no interference. The advantage of game strategy point E5 (1,0,0) is that there will be no public panic as Internet users reject opinion leaders’ false information; however, extra costs should be paid by regulatory institutions. The advantage of game strategy point E8 (1,1,1) is that the society will run smoothly due to Internet users’ acceptance of positive information released by opinion leaders, but once again, extra costs should be paid by regulatory institutions.6. ConclusionThe Internet is currently experiencing rapid development, such that a sustained release of false information triggered by a major emergency can impose a level of harm and panic on the public just as great as the impact of the emergency itself. Because classical game theory cannot describe the behavior of parties with bounded rationality who do not fully share information with each other, this study adopts evolutionary game theory, with the innovative approach of adding Internet users’ psychological identification with opinion leaders as an important parameter, to research the behavior of false information transmission among different participants after major emergencies. This is done by constructing a tripartite evolutionary game model between regulatory institutions, opinion leaders, and Internet users. Based on the solutions of models and numerical simulation, the main conclusions and insights are as follows:First, considering the real world, when major emergencies happen, secondary impacts should be avoided so as to maintain social stability. Strategy choices favorable for social stability require that Internet users adopt positive information released by opinion leaders or reject false information, and adoption of positive information is the more favorable of the two. Four strategy choices meet this condition. From the perspective of regulatory institutions, to relieve pressure, the best solution is to maintain the regulation level of information dissemination platforms and not to add additional regulatory costs. They should rely on opinion leaders to release positive information and Internet users to adopt positive information.Second, for the public and regulatory institutions, the probability of choosing the most optimal strategy is positively correlated with the punishment imposed by regulatory institutions on opinion leaders disseminating false information, the reward provided by regulatory institutions for opinion leaders disseminating positive information, the sense of participation and satisfaction gained by Internet users when they adopt positive information, the content richness of authentic information released by opinion leaders, and Internet users’ psychological identification with opinion leaders. It is negatively correlated with the costs of investigation and evidence collection borne by opinion leaders when they release positive information, the additional income of opinion leaders when they release false information and have it adopted, the time and energy costs of Internet users when they adopt information from opinion leaders, and the costs of judging the accuracy of information independently.Therefore, by taking approaches such as increasing the punishment for opinion leaders who release false information, increasing the reward for opinion leaders who release positive information, improving progress in investigation of major emergencies and publishing authentic information, and improving the information collection channels for opinion leaders, regulatory institutions can help opinion leaders and Internet users make better choices. Opinion leaders can attract Internet users’ attention by increasing the extent to which Internet users psychologically identify with them and enriching their authentic content in future information releases so as to help Internet users adopt positive information from them.Last but not least, this study outperforms other studies with classic game theory as the research method because bounded rational people capable of learning are chosen as game parties. All participants from all three groups are able to learn from and simulate others and thus dynamically adjusting their game strategies and maximize their own interests on the basis of others’ choices. Evolutionary game theory fully satisfies the research aim of studying three groups of game participants. However, there remain some limitations in this study. People are affected by emotions, which increase the complexity of their behavior choices in different situations, so model construction in this study is biased due to not taking this into consideration. Moreover, the model described here applies only to the influence of false information on game participants after major emergencies. In the future, we expect that a model able to analyze the reaction of game participants to false more generally will be constructed and will improve upon our model’s accuracy and also broaden its applicability to a wider range of circumstances.Data AvailabilityAll the data used to support the findings of this study are included within the article.Conflicts of InterestThe authors declare that there are no conflicts of interest regarding the publication of this study.AcknowledgmentsThis research was funded by Natural Science Foundation of China (no. 71771112) and Project of Liaoning Provincial Federation Social Science Circles of China (no. 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Journal

Complexity – Wiley

Published: Jan 1, 2022

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