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<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Public Health</journal-id>
<journal-title>Frontiers in Public Health</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Public Health</abbrev-journal-title>
<issn pub-type="epub">2296-2565</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
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<article-meta>
<article-id pub-id-type="doi">10.3389/fpubh.2024.1373747</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Public Health</subject>
<subj-group>
<subject>Original Research</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>Behavioral decision-making of government, agricultural product producers, and consumers on agricultural product quality and safety regulation in a digital environment</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name><surname>Huo</surname> <given-names>Hong</given-names></name>
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</contrib>
<contrib contrib-type="author" corresp="yes">
<name><surname>Liu</surname> <given-names>Xiangyu</given-names></name>
<xref ref-type="corresp" rid="c001"><sup>&#x002A;</sup></xref>
<uri xlink:href="https://loop.frontiersin.org/people/2634702/overview"/>
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<aff><institution>Management School, Harbin University of Commerce</institution>, <addr-line>Harbin</addr-line>, <country>China</country></aff>
<author-notes>
<fn fn-type="edited-by" id="fn0001"><p>Edited by: Ibrahim A. Elshaer, King Faisal University, Saudi Arabia</p></fn>
<fn fn-type="edited-by" id="fn0002"><p>Reviewed by: Diego Nocetti, University of Tarapac&#x00E1;, Chile</p><p>The Anh Han, Teesside University, United Kingdom</p></fn>
<corresp id="c001">&#x002A;Correspondence: Xiangyu Liu, <email>liuxiangyu0515@163.com</email></corresp>
</author-notes>
<pub-date pub-type="epub">
<day>02</day>
<month>04</month>
<year>2024</year>
</pub-date>
<pub-date pub-type="collection">
<year>2024</year>
</pub-date>
<volume>12</volume>
<elocation-id>1373747</elocation-id>
<history>
<date date-type="received">
<day>20</day>
<month>01</month>
<year>2024</year>
</date>
<date date-type="accepted">
<day>18</day>
<month>03</month>
<year>2024</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#x00A9; 2024 Huo and Liu.</copyright-statement>
<copyright-year>2024</copyright-year>
<copyright-holder>Huo and Liu</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/">
<p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p>
</license>
</permissions>
<abstract>
<p>The quality and safety of agricultural products are related to people&#x2019;s lives and health, economic development, and social stability, and have always been a hot issue of concern to the government and society. The rapid development of digital traceability technology in the digital environment has brought new opportunities for the supervision of agricultural product quality and safety, but the frequent occurrence of agricultural product safety incidents in recent years has exposed many problems such as the lack of governmental supervision, unstandardized production process of enterprises, and weak consumer awareness. To improve the cooperation efficiency of stakeholders and ensure the quality and safety of agricultural products, this paper proposes a dynamic model based on evolutionary game theory. The model incorporates the government, agricultural product producers, and farmers, and evaluates the stability and effectiveness of the system under different circumstances. The results of the study show that there are multiple evolutionary stabilization strategies in the tripartite evolutionary game model of agricultural product quality and safety supervision, and there are corresponding evolutionary stabilization conditions. There are several factors affecting the stability of the system, the most important of which are government regulation, severe penalties for agricultural product producers, and incentives. When these factors reach a certain threshold, the stakeholder cooperation mechanism can establish an evolutionarily stable strategy. This study contributes to the understanding of the operational mechanism of stakeholder cooperation in agricultural product quality and safety regulation in the digital environment and provides decision support and policy recommendations for stakeholders to promote the sustainable development and optimization of agricultural product quality and safety regulation.</p>
</abstract>
<kwd-group>
<kwd>agricultural product quality and safety</kwd>
<kwd>digital traceability system</kwd>
<kwd>quality and safety supervision</kwd>
<kwd>game evolution</kwd>
<kwd>public health</kwd>
</kwd-group>
<counts>
<fig-count count="4"/>
<table-count count="4"/>
<equation-count count="21"/>
<ref-count count="56"/>
<page-count count="10"/>
<word-count count="8666"/>
</counts>
<custom-meta-wrap>
<custom-meta>
<meta-name>section-at-acceptance</meta-name>
<meta-value>Public Health Policy</meta-value>
</custom-meta>
</custom-meta-wrap>
</article-meta>
</front>
<body>
<sec sec-type="intro" id="sec1">
<label>1</label>
<title>Introduction</title>
<p>The quality and safety of agricultural products are the foundation of food quality and safety, which is related to people&#x2019;s health and life safety, and has a bearing on farmers&#x2019; income increase and the high-quality development of agriculture and rural areas (<xref ref-type="bibr" rid="ref1">1</xref>, <xref ref-type="bibr" rid="ref2">2</xref>). Ensuring the quality and safety of agricultural products has become an important challenge as the demand for agricultural products increases year after year with the growing population and accelerated urbanization (<xref ref-type="bibr" rid="ref3">3</xref>, <xref ref-type="bibr" rid="ref4">4</xref>). With the continuous progress of science and technology and the popularization of information technology, digital technology has brought new opportunities for supervising agricultural quality and safety (<xref ref-type="bibr" rid="ref5">5</xref>), such as real-time monitoring and collection of agricultural product quality data, and agricultural product quality traceability based on blockchain. However, there have been frequent incidents of quality and safety of all kinds of agricultural products (<xref ref-type="bibr" rid="ref6">6</xref>, <xref ref-type="bibr" rid="ref7">7</xref>), such as the &#x201C;stale grain&#x201D; incident, the &#x201C;Sudan red incident&#x201D; (<xref ref-type="bibr" rid="ref8">8</xref>), the &#x201C;malachite green incident,&#x201D; the &#x201C;artificial honey&#x201D; incident in Wuhan, Hubei, and other places, and so on. These events have exposed that digital traceability technology has not been effectively used in the supervision of agricultural product quality and safety, and how to apply digital technology to the supervision of agricultural product quality and safety has become an urgent problem to be solved (<xref ref-type="bibr" rid="ref9">9</xref>).</p>
<p>The process of monitoring the quality and safety of agricultural products involves the government, agricultural production enterprises, consumers, and other multi-interested parties (<xref ref-type="bibr" rid="ref10">10</xref>, <xref ref-type="bibr" rid="ref11">11</xref>). Scholars at china and abroad have conducted some studies on the roles played by various subjects in the supervision of agricultural product quality and safety. The government plays a leading role in the process of regulating the quality and safety of agricultural products and is involved in the complete industrial chain of agricultural products from field to table (<xref ref-type="bibr" rid="ref12">12</xref>, <xref ref-type="bibr" rid="ref13">13</xref>). Teng et al. (<xref ref-type="bibr" rid="ref14">14</xref>) argue that effective government regulation can promote farmers&#x2019; green production behavior. According to Bhatt, the regulation of agricultural product trading has gradually tended to be government-led (<xref ref-type="bibr" rid="ref15">15</xref>), with coordinated social supervision covering the media, consumers, the general public, and even farmers and enterprises. Agricultural production enterprises, as important providers of agricultural products, also play an important role in the management of agricultural product quality and safety (<xref ref-type="bibr" rid="ref16">16</xref>, <xref ref-type="bibr" rid="ref17">17</xref>). Enterprises, as a fit between business and information flows, have different advantages in food safety and quality control practices (<xref ref-type="bibr" rid="ref18">18</xref>). Lezoche et al. (<xref ref-type="bibr" rid="ref19">19</xref>) believe that the core enterprises of the agricultural supply chain have an important position in the agricultural supply chain, can influence other members of the supply chain to maintain a dynamic cooperation mechanism, and play an important role in quality and safety management (<xref ref-type="bibr" rid="ref20">20</xref>). Consumers, as direct stakeholders (<xref ref-type="bibr" rid="ref21">21</xref>), have become important participants in the supervision of agricultural product quality and safety (<xref ref-type="bibr" rid="ref22">22</xref>). Introducing consumer participation in governance and giving full play to the power of consumer groups can effectively alleviate the problem of insufficient regulatory resources and help eliminate regulatory blind spots, which is an inevitable choice for social co-governance of food safety (<xref ref-type="bibr" rid="ref23">23</xref>).</p>
<p>Traceability is vital in food quality and safety management (<xref ref-type="bibr" rid="ref24">24</xref>). With the deep integration of digital technologies such as the Internet of Things, big data, cloud computing, blockchain, and other digital technologies with intelligent agriculture (<xref ref-type="bibr" rid="ref2">2</xref>, <xref ref-type="bibr" rid="ref25">25</xref>), the digital agricultural product traceability system provides a new way of thinking for the supervision of agricultural product quality and safety (<xref ref-type="bibr" rid="ref26">26</xref>). Consumers are also more inclined to buy traceable produce (<xref ref-type="bibr" rid="ref21">21</xref>). Establishing the digital agricultural product traceability system requires the joint participation of the government, agricultural product producers, processors, inspection and certification organizations, consumers and other main bodies, which is indispensable.</p>
<p>An evolutionary game model is a mathematical model that uses the principles of evolution and the framework of game theory to study the interactions between individuals in a biological population (<xref ref-type="bibr" rid="ref27">27</xref>). The model aims to explore how the frequency of different types of individuals in a population changes over time and how this change is affected by interactions between individuals and environmental influences (<xref ref-type="bibr" rid="ref28">28</xref>, <xref ref-type="bibr" rid="ref29">29</xref>). Evolutionary game models are often used by scholars to discuss the interrelationships between the three parties in an interaction. For example, in environmental monitoring, Encarna&#x00E7;&#x00E3;o et al. (<xref ref-type="bibr" rid="ref30">30</xref>) develop a new framework based on evolutionary game theory, envisioning that the state, business and civil sectors are faced with the dilemma of deciding between maintaining the status quo or shifting to a new paradigm, and the results show that public intervention is essential for shifting to a new paradigm, and that synergies between the private and civil sector are an important step in supporting the paradigm shift. In Healthcare Investing, Alalawi et al. (<xref ref-type="bibr" rid="ref31">31</xref>) provide a theoretical and simulation analysis of healthcare business models involving Public Healthcare Providers, Private Healthcare Providers and Patients, contributing to the modeling of the healthcare economy by analyzing the dynamics of agents and the emergence of collaborative behaviors in the three populations. Bova et al. (<xref ref-type="bibr" rid="ref32">32</xref>) use evolutionary game models to explore the role governments can play in building regulatory markets for AI systems to prevent reckless behavior. So it can be concluded from previous studies that the evolutionary game model can help us analyze the conflict of interest and cooperation between different stakeholders (<xref ref-type="bibr" rid="ref33">33</xref>). In our research, the field of agricultural product quality and safety supervision involves multiple stakeholders (<xref ref-type="bibr" rid="ref34">34</xref>), and the evolutionary game model helps to reveal the complex game relationship among them. Simulating the strategy selection and decision-making process of each participant helps to optimize the regulatory system and improve the quality and safety of agricultural products (<xref ref-type="bibr" rid="ref35">35</xref>). In the regulation of agricultural product quality and safety, information asymmetry often leads to increased difficulty in regulation, and the dynamic process of information transfer and gaming can be better understood using evolutionary game models (<xref ref-type="bibr" rid="ref36">36</xref>). The evolutionary game model can predict the behavioral evolution paths of different players in different contexts (<xref ref-type="bibr" rid="ref37">37</xref>), which helps to formulate more effective regulatory strategies and countermeasures and improve regulatory effectiveness quality and safety levels.</p>
<p>Therefore, scholars have widely used game theory in the study of quality and safety control of agricultural products (<xref ref-type="bibr" rid="ref38">38</xref>, <xref ref-type="bibr" rid="ref39">39</xref>). Chen et.al (<xref ref-type="bibr" rid="ref40">40</xref>) analyzed input capacity constraints&#x2019; impact on food quality and quality regulation through game theory. Based on the evolutionary game theory, a game model between the government, farmers, and consumers was established, and the results showed that the government subsidy strength, to farmers, consumer trust coefficient, and willingness to pay the premium for carbon-labeled agricultural products were positively correlated with the adoption of low-carbon production behavior by farmers. Chen et al. (<xref ref-type="bibr" rid="ref41">41</xref>) introduced the social preference theory to construct an evolutionary game model among multiple subjects and studied how to guide the behavioral decisions of multiple subjects to be standardized and rationalized (<xref ref-type="bibr" rid="ref42">42</xref>). Teng et al. (<xref ref-type="bibr" rid="ref14">14</xref>) studied the evolutionary decision-making behavior of government, farmers, and consumers based on the perspective of agricultural product quality and safety. Ma et al. (<xref ref-type="bibr" rid="ref43">43</xref>) constructed a three-party evolutionary game model of consumers, government, and farmers in the context of COVID-19 prevention and control normalization. The results showed that the cost of government regulation, the evaluation of the government by consumers and pig farmers, the government&#x2019;s subsidies to pig farmers and consumers, and the proportion of stakeholder behaviors affect the formation of the three-party relationship.</p>
<p>Some scholars have used evolutionary game theory to model incentives and agreement compliance. For example, Li et al. (<xref ref-type="bibr" rid="ref44">44</xref>) established an evolutionary game model of the governance mechanism of the recycling industry and analyzed the impact of government punishment on the behavioral strategies of recycling firms. Sasaki et al. (<xref ref-type="bibr" rid="ref45">45</xref>) combine key aspects of characterizing different punishment mechanisms in an evolutionary game-theoretic perspective and introduce a strategy of simultaneous commitment to cooperation and peer punishment to investigate a new mechanism for maintaining social order. Starting from a game-theoretic model that captures hegemonic competition in the field using artificial intelligence techniques, Han et al. (<xref ref-type="bibr" rid="ref46">46</xref>) show how sanctions, when applied unconditionally to potentially unsafe behaviors, may produce socially undesired outcomes. With the help of evolutionary game theory, Ogbo et al. (<xref ref-type="bibr" rid="ref47">47</xref>) analyze how ex-ante commitment can enhance the coordination of parties in two-by-two and multi-party interactions when the outcome presents an asymmetric payoff structure. Barrett uses a simple game model to illustrate whether and how treaties and related institutions can change incentives to align national self-interests with collective interests (<xref ref-type="bibr" rid="ref48">48</xref>). Han, on the other hand, shows that evolutionary game theory provides a suitable tool for studying the evolution of cooperative behaviors in social dilemmas as they are governed by institutional incentives and prior commitments (<xref ref-type="bibr" rid="ref49">49</xref>). Therefore, this paper also adopts evolutionary game theory to discuss the dynamic reward and dynamic punishment mechanism under the participation of multiple actors in the supervision of agricultural product quality and safety.</p>
<p>Some scholars have also taken digitization into account. Wan (<xref ref-type="bibr" rid="ref50">50</xref>) applies big data technology to the governance of agricultural product quality and safety and utilizes extensive data methods to study the critical control points in the traceability process of agricultural products. Considering the altruistic reciprocity of supermarkets and the fairness concern of processors, respectively, Qin et al. (<xref ref-type="bibr" rid="ref51">51</xref>) placed the Stackelberg game model under the Corporate social responsibility of processors and investigated the effects of Corporate social responsibility, altruistic reciprocity, and fairness concern on the quality improvement of agricultural supply chains.</p>
<p>In summary, most of the existing studies are on traditional agricultural product quality and safety supervision, and some scholars have studied agricultural product quality and safety traceability and governance from the perspective of big data technology. Few scholars have applied evolutionary game models to agricultural quality and safety regulation in digital environments. Therefore, this paper adopts the evolutionary game method to study the behavioral strategies of the government, agricultural product producers, and consumers in the process of agricultural product quality and safety supervision under the digital environment, analyze the mutual influence of strategic choices between different subjects, and provide theoretical references for the construction of the digital agricultural product traceability system, to further improve the quality and safety of agricultural products, safeguard the rights and interests of consumers, and continuously maintain the stability of the social order.</p>
</sec>
<sec id="sec2">
<label>2</label>
<title>Model assumptions and construction</title>
<sec id="sec3">
<label>2.1</label>
<title>Model assumptions</title>
<p>In this paper, we choose three subjects: government, agricultural producers, and consumers, and put forward the following hypotheses:</p>
<list list-type="order">
<list-item><p>The government is Participant 1, the agricultural producer is Participant 2, and the consumer is Participant 3. All three participants are assumed to be finite-rational and aim to maximize profit.</p></list-item>
<list-item><p>The government&#x2019;s strategic choice is to &#x201C;Regulate&#x201D; or &#x201C;Not regulate.&#x201D; Assume that the probability that the government chooses to &#x201C;regulate&#x201D; is x (0&#x003C;x&#x003C;1), and the probability that it chooses &#x201C;Not regulate&#x201D; is 1-x. When the government chooses to &#x201C;Regulate,&#x201D; it will incur the cost of regulation, but it will also enhance the government&#x2019;s image and trust, and gain certain benefits; when the government chooses &#x201C;Not regulate,&#x201D; it will not incur the cost of regulation, but it will lower the government&#x2019;s image and trust, and incur the certain loss of trust.</p></list-item>
<list-item><p>The strategic choice of the agricultural producer is &#x201C;Build&#x201D; or &#x201C;Not build&#x201D;. Assume that the probability that an agricultural enterprise chooses to &#x201C;build&#x201D; is y (0&#x003C;y&#x003C;1), and the probability that it chooses to &#x201C;not build&#x201D; is 1-y. When an agricultural producer chooses to &#x201C;building,&#x201D; i.e., construct a digital traceability system for agricultural products, it will incur the cost of construction, but the government will provide incentives, and at the same time improve the reputation and trust of the enterprise; When an agricultural product manufacturer chooses &#x201C;not build&#x201D;, i.e., chooses not to build a digital traceability system for agricultural products, it will save the technical, human and financial costs of building a traceability system, but it will be penalized by the government accordingly, and at the same time, it will also reduce the reputation and trust of the enterprise.</p></list-item>
<list-item><p>The consumer&#x2019;s strategic choice is &#x201C;satisfied&#x201D; or &#x201C;dissatisfied&#x201D;. Assume that the probability that the consumer chooses &#x201C;satisfied&#x201D; is z (0&#x003C;z&#x003C;1), then the probability that the consumer chooses &#x201C;dissatisfied&#x201D; is 1-z. When consumers are &#x201C;satisfied&#x201D; with the products produced by agricultural product manufacturers and the operation of the digital agricultural product traceability system, they will receive corresponding economic, health and environmental benefits. When consumers are &#x201C;dissatisfied&#x201D; with the products produced by agricultural product manufacturers and the operation of the digital agricultural products traceability system, they will incur corresponding rights defense costs. In contrast, agricultural product manufacturers will incur corresponding losses.</p></list-item>
</list>
<p>Based on the above, the game process between the government, agricultural producers, and consumers is visualized by building a game tree, as shown in <xref ref-type="fig" rid="fig1">Figure 1</xref>.</p>
<fig position="float" id="fig1">
<label>Figure 1</label>
<caption>
<p>Gaming tree.</p>
</caption>
<graphic xlink:href="fpubh-12-1373747-g001.tif"/>
</fig>
</sec>
<sec id="sec4">
<label>2.2</label>
<title>Model construction</title>
<sec id="sec5">
<label>2.2.1</label>
<title>Parameter setting</title>
</sec>
<sec id="sec6">
<label>2.2.2</label>
<title>Payment matrix</title>
<p>According to the parameter settings in <xref ref-type="table" rid="tab1">Table 1</xref>, the payment matrix of the three parties of the game is constructed, as shown in <xref ref-type="table" rid="tab2">Table 2</xref>.</p>
<table-wrap position="float" id="tab1">
<label>Table 1</label>
<caption>
<p>Related parameter settings.</p>
</caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th align="left" valign="top">Parameter name</th>
<th align="left" valign="top">Parameter description</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M7"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle">Costs incurred by Governments choosing to &#x201C;Regulate&#x201D;.</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M8"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle">Gains from the Government&#x2019;s choice to &#x201C;Regulate&#x201D;.</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M9"><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle">Loss of trust resulting from the Government&#x2019;s choice to &#x201C;Not regulate&#x201D;.</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M10"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle">Costs incurred by agro-producing firms choosing to &#x201C;Build&#x201D;.</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M11"><mml:mi>A</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle">Government incentives for agricultural producers to choose &#x201C;Build&#x201D;.</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M12"><mml:mi>P</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle">Penalties for agricultural producers choosing not to construct.</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M13"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle">Benefits to consumers when they are &#x201C;Satisfied&#x201D; with the produce they purchase.</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M14"><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle">Costs incurred by consumers in purchasing agricultural products.</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M15"><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle">Costs of advocacy when consumers are &#x201C;Dissatisfied&#x201D;.</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M16"><mml:mi>L</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle">Losses incurred by agricultural producers when consumers opt for &#x201C;Dissatisfied&#x201D;.</td>
</tr>
</tbody>
</table>
</table-wrap>
<table-wrap position="float" id="tab2">
<label>Table 2</label>
<caption>
<p>Payment matrix for the three parties of the game.</p>
</caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th align="center" valign="top" colspan="3" rowspan="2">Strategic choice</th>
<th align="center" valign="top" colspan="2">Government</th>
</tr>
<tr>
<th align="left" valign="top">Regulate x</th>
<th align="left" valign="top">Not regulate 1-x</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left" valign="middle" rowspan="4">Agricultural producers</td>
<td align="left" valign="middle" rowspan="2">Build <inline-formula><mml:math id="M17"><mml:mi>y</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle">Satisfied <inline-formula><mml:math id="M18"><mml:mi>z</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M19"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>A</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M20"><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M21"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M22"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M23"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M24"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td>
</tr>
<tr>
<td align="left" valign="middle">Dissatisfied <inline-formula><mml:math id="M25"><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M26"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M27"><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M28"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M29"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M30"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M31"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula></td>
</tr>
<tr>
<td align="left" valign="middle" rowspan="2">Not build <inline-formula><mml:math id="M32"><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle">Satisfied <inline-formula><mml:math id="M33"><mml:mi>z</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M34"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M35"><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M36"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M37"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M38"><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M39"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td>
</tr>
<tr>
<td align="left" valign="middle">Dissatisfied <inline-formula><mml:math id="M40"><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M41"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M42"><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M43"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M44"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>, 0, <inline-formula><mml:math id="M45"><mml:mo>&#x2212;</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math></inline-formula></td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
</sec>
</sec>
<sec id="sec7">
<label>3</label>
<title>Model analysis</title>
<p>An evolutionarily stable strategy is a strategy in a population that maintains a high degree of adaptability in the face of different strategies and cannot be replaced by other strategies (<xref ref-type="bibr" rid="ref52">52</xref>). This strategy maintains its dominant position in the group through stability and non-invasiveness, and remains relatively stable during the evolutionary process. We search for more stable strategies by building game models and analyzing the evolution of strategies.</p>
<sec id="sec8">
<label>3.1</label>
<title>Government&#x2019;s evolutionarily stable strategy</title>
<p>The expected returns to government regulation and non-regulation<inline-formula><mml:math id="M46"><mml:msub><mml:mi>E</mml:mi><mml:mn>11</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M47"><mml:msub><mml:mi>E</mml:mi><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula>, and the average expected return <inline-formula><mml:math id="M48"><mml:msub><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula> are:</p>
<disp-formula id="E1"><mml:math id="M49"><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mi>E</mml:mi><mml:mn>11</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mi>z</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>z</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:mfenced></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mspace width="0.25em"/><mml:mo>=</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi><mml:mi>z</mml:mi><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mi>P</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E2"><mml:math id="M50"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mi>E</mml:mi><mml:mn>12</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mi>z</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>z</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E3"><mml:math id="M51"><mml:msub><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mn>11</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:msub><mml:mi>E</mml:mi><mml:mn>12</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mfenced open="[" close="]"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi><mml:mi>z</mml:mi><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mi>P</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:math></disp-formula>
<p>The replication dynamic equation for government strategy choice is:</p>
<disp-formula id="E4"><mml:math id="M52"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy="true">/</mml:mo><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mn>11</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mfenced open="[" close="]"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mi>z</mml:mi><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mi>P</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>When <inline-formula><mml:math id="M53"><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>Z</mml:mi><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M54"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mfrac><mml:mtext>&#x2261;</mml:mtext><mml:mn>0</mml:mn></mml:math></inline-formula>, It is in a steady state regardless of the value of x. If <inline-formula><mml:math id="M55"><mml:mi>y</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mfrac><mml:mrow><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>Z</mml:mi><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula>, it is in a steady state at <inline-formula><mml:math id="M56"><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> and <inline-formula><mml:math id="M57"><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>.</p>
<p>Perform a derivation of <inline-formula><mml:math id="M59"><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced><mml:mo>:</mml:mo><mml:mspace width="0.25em"/><mml:malignmark/><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>2</mml:mn><mml:mi>x</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mfenced close="]" open="["><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mi>z</mml:mi><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mi>P</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:math></inline-formula>.</p>
<p>When <inline-formula><mml:math id="M60"><mml:mi>y</mml:mi><mml:mo>&#x003E;</mml:mo><mml:mfrac><mml:mrow><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>Z</mml:mi><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M61"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M62"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M63"><mml:msup><mml:mi>x</mml:mi><mml:mo>&#x2217;</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> is an evolutionarily stable strategy. When <inline-formula><mml:math id="M64"><mml:mi>y</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mfrac><mml:mrow><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>Z</mml:mi><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M65"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M66"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M67"><mml:msup><mml:mi>x</mml:mi><mml:mo>&#x2217;</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>is an evolutionarily stable strategy.</p>
</sec>
<sec id="sec9">
<label>3.2</label>
<title>Evolutionary stabilization strategies for agricultural producers</title>
<p>The expected returns of agricultural producers<inline-formula><mml:math id="M68"><mml:msub><mml:mi>E</mml:mi><mml:mn>21</mml:mn></mml:msub></mml:math></inline-formula>, <inline-formula><mml:math id="M69"><mml:msub><mml:mi>E</mml:mi><mml:mn>22</mml:mn></mml:msub></mml:math></inline-formula>, and the average expected returns <inline-formula><mml:math id="M70"><mml:msub><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula> are:</p>
<disp-formula id="E5"><mml:math id="M71"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mi>E</mml:mi><mml:mn>21</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mi>z</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>z</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E6"><mml:math id="M72"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mi>E</mml:mi><mml:mn>22</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mi>z</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E7"><mml:math id="M73"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mn>21</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:msub><mml:mi>E</mml:mi><mml:mn>22</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>+</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="[" close="]"><mml:mrow><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>The replication dynamic equation for the strategy choice of agricultural producers is:</p>
<disp-formula id="E8"><mml:math id="M74"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>y</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mi mathvariant="italic">dy</mml:mi><mml:mo stretchy="true">/</mml:mo><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mn>11</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>y</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mfenced open="[" close="]"><mml:mrow><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>If <inline-formula><mml:math id="M75"><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>x</mml:mi><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M76"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>y</mml:mi></mml:mfenced></mml:mrow><mml:mi mathvariant="italic">dy</mml:mi></mml:mfrac><mml:mtext>&#x2261;</mml:mtext><mml:mn>0</mml:mn></mml:math></inline-formula>, it is stable regardless of the value of y. If <inline-formula><mml:math id="M77"><mml:mi>z</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mfrac><mml:mrow><mml:mi>x</mml:mi><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:math></inline-formula>, then it is in a steady state at y&#x2009;=&#x2009;0 and y&#x2009;=&#x2009;1.</p>
<p>Perform the derivation on <inline-formula><mml:math id="M78"><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>y</mml:mi></mml:mfenced></mml:math></inline-formula>:</p>
<disp-formula id="E9"><mml:math id="M79"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>y</mml:mi></mml:mfenced></mml:mrow><mml:mi mathvariant="italic">dy</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>2</mml:mn><mml:mi>x</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mfenced open="[" close="]"><mml:mrow><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:mfenced></mml:math></disp-formula>
<p>When <inline-formula><mml:math id="M80"><mml:mi>z</mml:mi><mml:mo>&#x003E;</mml:mo><mml:mfrac><mml:mrow><mml:mi>x</mml:mi><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:math></inline-formula>,<inline-formula><mml:math id="M81"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>y</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M82"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>y</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M83"><mml:msup><mml:mi>y</mml:mi><mml:mo>&#x2217;</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> is an evolutionarily stable strategy. When <inline-formula><mml:math id="M84"><mml:mi>z</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mfrac><mml:mrow><mml:mi>x</mml:mi><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M85"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>y</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M86"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>y</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M87"><mml:msup><mml:mi>y</mml:mi><mml:mo>&#x2217;</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> is an evolutionarily stable strategy.</p>
</sec>
<sec id="sec10">
<label>3.3</label>
<title>Evolutionary stabilization strategies for consumers</title>
<p>Consumer satisfaction and dissatisfaction expected returns <inline-formula><mml:math id="M88"><mml:msub><mml:mi>E</mml:mi><mml:mn>31</mml:mn></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math id="M89"><mml:msub><mml:mi>E</mml:mi><mml:mn>32</mml:mn></mml:msub></mml:math></inline-formula>, and average expected return <inline-formula><mml:math id="M90"><mml:msub><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover><mml:mn>3</mml:mn></mml:msub></mml:math></inline-formula> are:</p>
<disp-formula id="E10"><mml:math id="M91"><mml:msub><mml:mi>E</mml:mi><mml:mn>31</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mi>x</mml:mi><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></disp-formula>
<disp-formula id="E11"><mml:math id="M92"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mi>E</mml:mi><mml:mn>32</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E12"><mml:math id="M93"><mml:msub><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover><mml:mn>3</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>z</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mn>31</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:msub><mml:mi>E</mml:mi><mml:mn>32</mml:mn></mml:msub></mml:math></disp-formula>
<p>The replication dynamic equation for consumer strategy choice is:</p>
<disp-formula id="E13"><mml:math id="M94"><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>z</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mi>z</mml:mi><mml:mo stretchy="true">/</mml:mo><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>z</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mn>31</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi mathvariant="normal">z</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi mathvariant="normal">z</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mo stretchy="true">(</mml:mo><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mi mathvariant="normal">y</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mi mathvariant="normal">y</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">x</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">y</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mspace width="0.25em"/><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">y</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">x</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>If <inline-formula><mml:math id="M95"><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:math></inline-formula>,<inline-formula><mml:math id="M96"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>z</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:mfrac><mml:mtext>&#x2261;</mml:mtext><mml:mn>0</mml:mn></mml:math></inline-formula>, it is stable regardless of the value of z. If <inline-formula><mml:math id="M97"><mml:mi>y</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:math></inline-formula>, then it is in a steady state at z&#x2009;=&#x2009;0 and z&#x2009;=&#x2009;1.</p>
<p>Perform the derivation on <inline-formula><mml:math id="M98"><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>z</mml:mi></mml:mfenced></mml:math></inline-formula>:</p>
<disp-formula id="E14"><mml:math id="M99"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>z</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>2</mml:mn><mml:mi>z</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mi mathvariant="normal">y</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mi mathvariant="normal">y</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>+</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">x</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">y</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">y</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mi mathvariant="normal">y</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>&#x2212;</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">x</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi mathvariant="normal">x</mml:mi><mml:mi mathvariant="normal">y</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mfenced></mml:math></disp-formula>
<p>When <inline-formula><mml:math id="M100"><mml:mi>y</mml:mi><mml:mo>&#x003E;</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M101"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>z</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M102"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>z</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M103"><mml:msup><mml:mi>z</mml:mi><mml:mo>&#x2217;</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>is an evolutionarily stable strategy. When <inline-formula><mml:math id="M104"><mml:mi>y</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:mi>L</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M105"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>z</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M106"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>z</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true">|</mml:mo><mml:msub><mml:mrow><mml:mtext>&#x00A0;</mml:mtext></mml:mrow><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>&#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M107"><mml:msup><mml:mi>z</mml:mi><mml:mo>&#x2217;</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>is an evolutionarily stable strategy.</p>
</sec>
<sec id="sec11">
<label>3.4</label>
<title>Stability analysis</title>
<p>The system equilibrium point is <inline-formula><mml:math id="M108"><mml:msub><mml:mi>E</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>0</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula>, <inline-formula><mml:math id="M109"><mml:msub><mml:mi>E</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula>, <inline-formula><mml:math id="M110"><mml:msub><mml:mi>E</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>1</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula>, <inline-formula><mml:math id="M111"><mml:msub><mml:mi>E</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>0</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula>, <inline-formula><mml:math id="M112"><mml:msub><mml:mi>E</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula>, <inline-formula><mml:math id="M113"><mml:msub><mml:mi>E</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula>, <inline-formula><mml:math id="M114"><mml:msub><mml:mi>E</mml:mi><mml:mn>7</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>1</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula>, <inline-formula><mml:math id="M115"><mml:msub><mml:mi>E</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula> from <inline-formula><mml:math id="M116"><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>,<inline-formula><mml:math id="M117"><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>y</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>,<inline-formula><mml:math id="M118"><mml:mi>F</mml:mi><mml:mfenced open="(" close=")"><mml:mi>z</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>. The Jacobi matrix of the three-party evolutionary game <inline-formula><mml:math id="M119"><mml:mi>J</mml:mi><mml:mo>=</mml:mo><mml:mfenced open="[" close="]"><mml:mtable columnalign="center"><mml:mtr columnalign="center"><mml:mtd columnalign="center"><mml:msub><mml:mi>A</mml:mi><mml:mn>11</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mi>A</mml:mi><mml:mn>12</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mi>A</mml:mi><mml:mn>13</mml:mn></mml:msub></mml:mtd></mml:mtr><mml:mtr columnalign="center"><mml:mtd columnalign="center"><mml:msub><mml:mi>A</mml:mi><mml:mn>21</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mi>A</mml:mi><mml:mn>22</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mi>A</mml:mi><mml:mn>23</mml:mn></mml:msub></mml:mtd></mml:mtr><mml:mtr columnalign="center"><mml:mtd columnalign="center"><mml:msub><mml:mi>A</mml:mi><mml:mn>31</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mi>A</mml:mi><mml:mn>32</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign="center"><mml:msub><mml:mi>A</mml:mi><mml:mn>33</mml:mn></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mfenced></mml:math></inline-formula>.</p>
<disp-formula id="E15"><mml:math id="M120"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mi>A</mml:mi><mml:mn>11</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant="normal">x</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mi mathvariant="normal">z</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mo>+</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="0.25em"/><mml:mfenced open="(" close=")"><mml:mrow><mml:mi mathvariant="normal">x</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi mathvariant="normal">R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mi mathvariant="normal">z</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
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<p>Substituting the eight equilibrium points into the Jacobi matrix, respectively, the corresponding eigenvalues of each equilibrium point can be obtained, as shown in <xref ref-type="table" rid="tab3">Table 3</xref>. According to the evolutionary game theory, the equilibrium point that satisfies the Jacobi matrix when all the eigenvalues are negative is the evolutionary equilibrium point.</p>
<table-wrap position="float" id="tab3">
<label>Table 3</label>
<caption>
<p>Eigenvalues of the Jacobi matrix.</p>
</caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th align="left" valign="top">Balance point</th>
<th align="left" valign="top">Eigenvalue 1</th>
<th align="left" valign="top">Eigenvalue 2</th>
<th align="left" valign="top">Eigenvalue 3</th>
</tr>
</thead>
<tbody>
<tr>
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</tr>
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<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M139"><mml:msub><mml:mi>E</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>0</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M140"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M141"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M142"><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula></td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M143"><mml:msub><mml:mi>E</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M144"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M145"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M146"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula></td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M147"><mml:msub><mml:mi>E</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M148"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M149"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>P</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M150"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M151"><mml:msub><mml:mi>E</mml:mi><mml:mn>7</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>1</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M152"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M153"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>L</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M154"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula></td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M155"><mml:msub><mml:mi>E</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M156"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M157"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi></mml:math></inline-formula></td>
<td align="left" valign="middle"><inline-formula><mml:math id="M158"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula></td>
</tr>
</tbody>
</table>
</table-wrap>
<p>Scenario 1 When<inline-formula><mml:math id="M159"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, the trust loss of the government choosing not to regulate is smaller than the cost of choosing to regulate. The punishment of the government that the agricultural product producer chooses not to build the digital agricultural product traceability system is smaller than the cost of choosing to build it. The benefit the consumer gets when satisfied is more significant than the cost of defending his rights when dissatisfied. At this point, as can be seen from <xref ref-type="table" rid="tab4">Table 4</xref>, the eigenvalues of the Jacobian matrix corresponding to the equilibrium point <inline-formula><mml:math id="M160"><mml:msub><mml:mi>E</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula> are all less than 0. The equilibrium point <inline-formula><mml:math id="M161"><mml:msub><mml:mi>E</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula> is stable, and its corresponding evolutionary strategy is (Not regulate, Not build, Satisfied).</p>
<table-wrap position="float" id="tab4">
<label>Table 4</label>
<caption>
<p>Stability analysis of the equilibrium solution.</p>
</caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th align="left" valign="top">Balance point</th>
<th align="left" valign="top">Eigenvalue symbol</th>
<th align="left" valign="top">Stability</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M162"><mml:msub><mml:mi>E</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>0</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle">Having positive value</td>
<td align="left" valign="middle">Unsteady point</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M163"><mml:msub><mml:mi>E</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle">When <inline-formula><mml:math id="M164"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, all negative values</td>
<td align="left" valign="middle">ESS</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M165"><mml:msub><mml:mi>E</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>1</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle">Having positive value</td>
<td align="left" valign="middle">Unsteady point</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M166"><mml:msub><mml:mi>E</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>0</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle">Having positive value</td>
<td align="left" valign="middle">Unsteady point</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M167"><mml:msub><mml:mi>E</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>0</mml:mn><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle">Having positive value</td>
<td align="left" valign="middle">Unsteady point</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M168"><mml:msub><mml:mi>E</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle">When <inline-formula><mml:math id="M169"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> and <inline-formula><mml:math id="M170"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, all negative values</td>
<td align="left" valign="middle">ESS</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M171"><mml:msub><mml:mi>E</mml:mi><mml:mn>7</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>1</mml:mn><mml:mn>0</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle">Having positive value</td>
<td align="left" valign="middle">Unsteady point</td>
</tr>
<tr>
<td align="left" valign="middle"><inline-formula><mml:math id="M172"><mml:msub><mml:mi>E</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula></td>
<td align="left" valign="middle">When <inline-formula><mml:math id="M173"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> and <inline-formula><mml:math id="M174"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, all negative values</td>
<td align="left" valign="middle">ESS</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>Scenario 2 When <inline-formula><mml:math id="M175"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>and<inline-formula><mml:math id="M176"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, the cost to the government of choosing to regulate is less than the benefit of choosing to regulate. The penalties imposed on agricultural producers for choosing not to build a traceability system are greater than the sum of the benefits and incentives of choosing to build one. When satisfied, the benefits of food safety to consumers are more significant than the costs of defending their rights when dissatisfied. At this point, as can be seen from <xref ref-type="table" rid="tab4">Table 4</xref>, the eigenvalues of the Jacobian matrix corresponding to the equilibrium point <inline-formula><mml:math id="M177"><mml:msub><mml:mi>E</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula> are all less than 0. The equilibrium point <inline-formula><mml:math id="M178"><mml:msub><mml:mi>E</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula> is stable, and its corresponding evolutionary strategy is (Regulate, Not build, Satisfied).</p>
<p>Scenario 3 When <inline-formula><mml:math id="M179"><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> and <inline-formula><mml:math id="M180"><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, the cost to the government of choosing to regulate is less than the benefit it receives from choosing to regulate, and the cost to agricultural producers of constructing a digital traceability system is less than the penalties imposed by the government in the event of consumer dissatisfaction. At this point, as can be seen from <xref ref-type="table" rid="tab4">Table 4</xref>, the eigenvalues of the Jacobian matrix corresponding to the equilibrium point <inline-formula><mml:math id="M181"><mml:msub><mml:mi>E</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula> are all less than 0. The equilibrium point <inline-formula><mml:math id="M182"><mml:msub><mml:mi>E</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mfenced open="(" close=")" separators=",,"><mml:mn>1</mml:mn><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:mfenced></mml:math></inline-formula> is stable, and its corresponding evolutionary strategy is (Regulate, Build, Satisfied).</p>
</sec>
</sec>
<sec id="sec12">
<label>4</label>
<title>Simulation analysis</title>
<p>Scenario 1 <inline-formula><mml:math id="M183"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>. When <inline-formula><mml:math id="M184"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M185"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M186"><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M187"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M188"><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M189"><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M190"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M191"><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M192"><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M193"><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>. The results of the simulation experiment are shown in <xref ref-type="fig" rid="fig2">Figure 2</xref>. As can be seen from the figure, the probability that the government chooses to regulate keeps decreasing until it is 0; the probability that the agricultural producer chooses to construct keeps decreasing until it is 0; and the probability that the consumer is satisfied keeps increasing until it is 1. Therefore, the evolutionary equilibrium state tends to be (0,0,1). In this case, consumers choose not to file complaints due to the high cost of defending their rights, and the probability of government regulation will be very low, which makes agricultural producers not to build a digital traceability system.</p>
<fig position="float" id="fig2">
<label>Figure 2</label>
<caption>
<p>Evolutionary outcome of scenario 1.</p>
</caption>
<graphic xlink:href="fpubh-12-1373747-g002.tif"/>
</fig>
<p>Scenario 2 <inline-formula><mml:math id="M194"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> and <inline-formula><mml:math id="M195"><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>. When <inline-formula><mml:math id="M196"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M197"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>9</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M198"><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M199"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M200"><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M201"><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M202"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M203"><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M204"><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M205"><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>. The results of the simulation experiment are shown in <xref ref-type="fig" rid="fig3">Figure 3</xref>. From the figure, it can be seen that the probability that the government chooses to regulate keeps rising until it is 1; the probability that the agricultural producer chooses to construct keeps falling until it is 0; and the probability that the consumer is satisfied keeps rising until it is 1, so the evolutionary equilibrium state tends to (1, 0, 1). Compared with Case 1, the benefit <inline-formula><mml:math id="M206"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula> gained from the regulation of agricultural product producers increases, and the probability of their choosing to regulate increases, the cost <inline-formula><mml:math id="M207"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula> of agricultural product producers constructing a digital traceability system increases, and even though the government will penalize them, the cost of constructing a digital traceability system is higher than that of penalizing them, so that agricultural product producers will still tend to choose not to construct it. The consumer cannot obtain the quality traceability information of agricultural products they purchase, and the cost of consumers defending their rights is high, so the probability that the consumer chooses to file a complaint decreases.</p>
<fig position="float" id="fig3">
<label>Figure 3</label>
<caption>
<p>Evolutionary outcome of scenario 2.</p>
</caption>
<graphic xlink:href="fpubh-12-1373747-g003.tif"/>
</fig>
<p>Scenario 3 <inline-formula><mml:math id="M208"><mml:msub><mml:mi mathvariant="normal">C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> and <inline-formula><mml:math id="M209"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>P</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>. When <inline-formula><mml:math id="M210"><mml:msub><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M211"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>9</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M212"><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M213"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M214"><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M215"><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M216"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M217"><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M218"><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M219"><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>. The results of the simulation experiment are shown in <xref ref-type="fig" rid="fig4">Figure 4</xref>. From the figure, it can be seen that the probability of the government choosing to regulate has been rising until it is 1; the probability of the agricultural production enterprise choosing to build has been rising until it is 1; the probability of the consumer being satisfied has been rising until it is 1, so the evolutionary equilibrium state tends to (<xref ref-type="bibr" rid="ref1">1</xref>). Compared to Case 1, the gain <inline-formula><mml:math id="M220"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula> gained from government regulation increases, the probability of its regulation increases, the penalty <inline-formula><mml:math id="M221"><mml:mi>P</mml:mi></mml:math></inline-formula> for not constructing a digital traceability system for agricultural product producers increases, the probability of agricultural product producers constructing it increases, consumers can obtain traceability information about their products, agricultural product producers tend to produce high-quality products, and the probability of consumer satisfaction increases.</p>
<fig position="float" id="fig4">
<label>Figure 4</label>
<caption>
<p>Evolutionary outcome of scenario 3.</p>
</caption>
<graphic xlink:href="fpubh-12-1373747-g004.tif"/>
</fig>
</sec>
<sec id="sec13">
<label>5</label>
<title>Conclusion and recommendation</title>
<p>This paper constructs a three-party evolutionary game model among the government, agricultural product producers, and consumers in the digital environment, analyzes the evolutionary process of agricultural product quality and safety issues involving the government, agricultural product producers, and consumers, and studies the strategic choices of the government, agricultural product producers and consumers as well as their influencing factors through the analysis of arithmetic examples. It is found that there are multiple evolutionary stabilization strategies in the tripartite evolutionary game model of agricultural product quality and safety regulation. When the benefits gained from government regulation, the cost of constructing a digital traceability system for agricultural product producers, and the penalties imposed on agricultural product producers are constantly changing, there will be (Not regulate, Not build, Satisfied), (Regulate, Not build, Satisfied), and (Regulate, Build, Satisfied) in order.</p>
<p>From this, the following recommendations can be drawn: (1) The government should set appropriate penalties and impose severe penalties for agricultural producers who violate production standards and quality requirements to force agricultural producers to improve quality (<xref ref-type="bibr" rid="ref53">53</xref>). (2) Reduce the cost of enterprises to build a digital traceability system for agricultural products (<xref ref-type="bibr" rid="ref54">54</xref>). The government can provide relevant financial support and reduce or waive relevant taxes and fees, etc., to reduce the cost of building a digital agricultural product traceability system for agricultural product production enterprises and encourage more enterprises to join the ranks of digital management (3). Actively guide consumers&#x2019; understanding of the quality and safety traceability of agricultural products and teach them to inquire about product origin and quality information through the traceability system (<xref ref-type="bibr" rid="ref55">55</xref>, <xref ref-type="bibr" rid="ref56">56</xref>). Advocating that priority be given to agricultural products with digital quality and safety traceability marks to increase consumers&#x2019; confidence and sense of security in purchasing. Consumers should also actively report suspected agricultural product quality problems to government regulators and actively participate in supervision and public opinion monitoring to safeguard the quality and safety of agricultural products. This study also has some limitations. In the next step, we will expand the scope of the study to include third-party testing organizations and the public, and explore the characteristics and laws of multi-body behavioral decision-making in agricultural product quality and safety supervision. In addition, the game model we chose has a bias in describing the real decision-making environment, and we will also choose a more appropriate model to improve our study in the subsequent research.</p>
</sec>
<sec sec-type="data-availability" id="sec14">
<title>Data availability statement</title>
<p>The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.</p>
</sec>
<sec sec-type="author-contributions" id="sec15">
<title>Author contributions</title>
<p>HH: Conceptualization, Formal analysis, Resources, Validation, Writing &#x2013; original draft, Writing &#x2013; review &#x0026; editing. XL: Conceptualization, Data curation, Funding acquisition, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing &#x2013; original draft, Writing &#x2013; review &#x0026; editing.</p>
</sec>
</body>
<back>
<sec sec-type="funding-information" id="sec16">
<title>Funding</title>
<p>The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This research was supported by three funding: The National Social Science Found of China, Grant/Award Number: 23BJY191, 23BJY151; Heilongjiang Nature Science Foundation, Grant/Award Number: LH2022G014; Heilongjiang philosophy and Social Science Foundation, Grant/Award Number: 21GLC187, 22GLD356.</p>
</sec>
<sec sec-type="COI-statement" id="sec17">
<title>Conflict of interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec id="sec100" sec-type="disclaimer">
<title>Publisher&#x2019;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
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