Dynamic Evolution Game Strategy of Government, Power Grid, and Users in Electricity Market Demand-Side Management

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732615622"> <meta name="title" content="Dynamic Evolution Game Strategy of Government, Power Grid, and Users in Electricity Market Demand-Side Management"> <meta name="description" content="In the process of promoting demand-side management, the core stakeholder groups are government departments, power grid companies, and electricity users. Due to the different positions and conflicting interests of the three parties in the game, intense and complex battles will occur. This paper investigates a tripartite evolutionary game involving government, power grid companies, and electricity users in the context of demand-side management (DSM) and analyzes the dynamic interactions between government departments, power grid companies, and electricity users within the framework of DSM using evolutionary game theory. Using evolutionary game theory, we explore how incentives and strategic interactions among these three stakeholders evolve over time, affecting the stability of DSM policies. The model addresses the asymmetry in the decision-making process and examines the dynamic equilibrium outcomes under various scenarios. The results provide insights into the optimal design of incentive mechanisms to enhance DSM adoption. The findings offer practical recommendations to improve DSM policies, fostering balanced interests between government, grid companies, and users. This research contributes to a deeper understanding of strategic interactions in DSM, revealing how adaptive behaviors can enhance energy efficiency. It also underscores the importance of carefully designed incentive mechanisms in achieving long-term stability and cooperation among key stakeholders." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/mathematics-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="Dynamic Evolution Game Strategy of Government, Power Grid, and Users in Electricity Market Demand-Side Management"> <meta name="dc.creator" content="Xin Shen"> <meta name="dc.creator" content="Jianlin Tang"> <meta name="dc.creator" content="Yijing Zhang"> <meta name="dc.creator" content="Bin Qian"> <meta name="dc.creator" content="Jiahao Li"> <meta name="dc.creator" content="Mi Zhou"> <meta name="dc.creator" content="Yitao Zhao"> <meta name="dc.creator" content="Yujun Yin"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Mathematics 2024, Vol. 12, Page 3249"> <meta name="dc.date" content="2024-10-17"> <meta name ="dc.identifier" content="10.3390/math12203249"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="In the process of promoting demand-side management, the core stakeholder groups are government departments, power grid companies, and electricity users. Due to the different positions and conflicting interests of the three parties in the game, intense and complex battles will occur. This paper investigates a tripartite evolutionary game involving government, power grid companies, and electricity users in the context of demand-side management (DSM) and analyzes the dynamic interactions between government departments, power grid companies, and electricity users within the framework of DSM using evolutionary game theory. Using evolutionary game theory, we explore how incentives and strategic interactions among these three stakeholders evolve over time, affecting the stability of DSM policies. The model addresses the asymmetry in the decision-making process and examines the dynamic equilibrium outcomes under various scenarios. The results provide insights into the optimal design of incentive mechanisms to enhance DSM adoption. The findings offer practical recommendations to improve DSM policies, fostering balanced interests between government, grid companies, and users. This research contributes to a deeper understanding of strategic interactions in DSM, revealing how adaptive behaviors can enhance energy efficiency. It also underscores the importance of carefully designed incentive mechanisms in achieving long-term stability and cooperation among key stakeholders." > <meta name="dc.subject" content="power demand-side management" > <meta name="dc.subject" content="incentive mechanism" > <meta name="dc.subject" content="evolutionary game theory" > <meta name="dc.subject" content="evolutionary stable equilibrium" > <meta name="dc.subject" content="replicator dynamics equations" > <meta name ="prism.issn" content="2227-7390"> <meta name ="prism.publicationName" content="Mathematics"> <meta name ="prism.publicationDate" content="2024-10-17"> <meta name ="prism.volume" content="12"> <meta name ="prism.number" content="20"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="3249" > <meta name="citation_issn" content="2227-7390"> <meta name="citation_journal_title" content="Mathematics"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="Dynamic Evolution Game Strategy of Government, Power Grid, and Users in Electricity Market Demand-Side Management"> <meta name="citation_publication_date" content="2024/1"> <meta name="citation_online_date" content="2024/10/17"> <meta name="citation_volume" content="12"> <meta name="citation_issue" content="20"> <meta name="citation_firstpage" content="3249"> <meta name="citation_author" content="Shen, Xin"> <meta name="citation_author" content="Tang, Jianlin"> <meta name="citation_author" content="Zhang, Yijing"> <meta name="citation_author" content="Qian, Bin"> <meta name="citation_author" content="Li, Jiahao"> <meta name="citation_author" content="Zhou, Mi"> <meta name="citation_author" content="Zhao, Yitao"> <meta name="citation_author" content="Yin, Yujun"> <meta name="citation_doi" content="10.3390/math12203249"> <meta name="citation_id" content="mdpi-math12203249"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2227-7390/12/20/3249"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2227-7390/12/20/3249/pdf?version=1729157712"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2227-7390/12/20/3249/pdf?version=1729157712"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2227-7390/12/20/3249/pdf?version=1729157712"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2227-7390/12/20/3249/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2227-7390/12/20/3249/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2227-7390/12/20/3249/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2227-7390/12/20/3249/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2227-7390/12/20/3249/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2227-7390/12/20/3249/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/mathematics-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2227-7390/12/20/3249" /> <meta property="og:title" content="Dynamic Evolution Game Strategy of Government, Power Grid, and Users in Electricity Market Demand-Side Management" /> <meta property="og:description" content="In the process of promoting demand-side management, the core stakeholder groups are government departments, power grid companies, and electricity users. Due to the different positions and conflicting interests of the three parties in the game, intense and complex battles will occur. This paper investigates a tripartite evolutionary game involving government, power grid companies, and electricity users in the context of demand-side management (DSM) and analyzes the dynamic interactions between government departments, power grid companies, and electricity users within the framework of DSM using evolutionary game theory. Using evolutionary game theory, we explore how incentives and strategic interactions among these three stakeholders evolve over time, affecting the stability of DSM policies. The model addresses the asymmetry in the decision-making process and examines the dynamic equilibrium outcomes under various scenarios. The results provide insights into the optimal design of incentive mechanisms to enhance DSM adoption. The findings offer practical recommendations to improve DSM policies, fostering balanced interests between government, grid companies, and users. This research contributes to a deeper understanding of strategic interactions in DSM, revealing how adaptive behaviors can enhance energy efficiency. It also underscores the importance of carefully designed incentive mechanisms in achieving long-term stability and cooperation among key stakeholders." /> <meta property="og:image" content="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001-550.jpg?1729157782" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); 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data-bg="creme"> <i class="fa fa-file-text"></i> </div> </div> </div> </div> </div> </div> </div> <article ><div class='html-article-content'> <span itemprop="publisher" content="Multidisciplinary Digital Publishing Institute"></span><span itemprop="url" content="https://www.mdpi.com/2227-7390/12/20/3249"></span> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <h1 class="title hypothesis_container" itemprop="name"> Dynamic Evolution Game Strategy of Government, Power Grid, and Users in Electricity Market Demand-Side Management </h1> <div class="art-authors hypothesis_container"> by <span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13280537' data-options='is_hover:true, hover_timeout:5000'> Xin Shen</div><div id="profile-card-drop13280537" data-dropdown-content class="f-dropdown content 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href="https://scholar.google.com/scholar?q=Xin%20Shen" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup> 1</sup><span style="display: inline; margin-left: 5px;"></span><a class="toEncode emailCaptcha visibility-hidden" data-author-id="13280537" href="/cdn-cgi/l/email-protection#c1eea2afa5eca2a6a8eeadeea4aca0a8adecb1b3aeb5a4a2b5a8aeafe2f1f1f0a3f6f2f0f7f0a5f1a3f0a0f0a5f3a2f0f1f1f1f0f5f2f2f5f3f5f4f5f1f4a5f0f1f0a2f0a4"><sup><i class="fa fa-envelope-o"></i></sup></a>, </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13280538' data-options='is_hover:true, hover_timeout:5000'> Jianlin Tang</div><div id="profile-card-drop13280538" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img 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class="toEncode emailCaptcha visibility-hidden" data-author-id="13280538" href="/cdn-cgi/l/email-protection#96b9f5f8f2bbf5f1ffb9fab9f3fbf7fffabbe6e4f9e2f3f5e2fff9f8b5a6a6a6a5a0f7a6f4a6a2a6a0a6a5a6a2a7f3a6f4a6a2a6f2a5a3a6afa7afa6f2a4f7a3f4a3f5a3afa2a2a6afa6a3a6a1"><sup><i class="fa fa-envelope-o"></i></sup></a>, </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13280539' data-options='is_hover:true, hover_timeout:5000'> Yijing Zhang</div><div id="profile-card-drop13280539" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Yijing Zhang</span></div></div></div><div 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href="/cdn-cgi/l/email-protection#507f333e347d3337397f3c7f353d31393c7d20223f24353324393f3e73606061626731613261646134606361636160616361646134626561696069613463316432643364696564616961656167"><sup><i class="fa fa-envelope-o"></i></sup></a>, </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13280540' data-options='is_hover:true, hover_timeout:5000'> Bin Qian</div><div id="profile-card-drop13280540" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Bin Qian</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a 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href="/cdn-cgi/l/email-protection#1d327e7379307e7a7432713278707c7471306d6f7269787e697472733e2d2d2c2d2a242c2a2d2d2d7e2c2e2d7e2c2a2c2e2d7e2c2a2e242925297b297c282a2c7c2c2b2c29"><sup><i class="fa fa-envelope-o"></i></sup></a></span> </div> <div class="nrm"></div> <span style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-item"><sup>1</sup></div> <div class="affiliation-name ">Measurement Center, Yunnan Power Grid Co., Ltd., Kunming 650051, China</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>2</sup></div> <div class="affiliation-name ">Electric Power Research Institute, China Southern Power Grid Co., Ltd., Guangzhou 510530, China</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Author to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(20), 3249; <a href="https://doi.org/10.3390/math12203249">https://doi.org/10.3390/math12203249</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 2 September 2024</span> / <span style="display: inline-block">Revised: 23 September 2024</span> / <span style="display: inline-block">Accepted: 26 September 2024</span> / <span style="display: inline-block">Published: 17 October 2024</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/Artificial_Intelligence_Game_Theory ">Artificial Intelligence and Game Theory</a>)<br/> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-1500643" aria-controls="drop-supplementary-1500643" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-1500643" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2227-7390/12/20/3249/pdf?version=1729157712" data-name="Dynamic Evolution Game Strategy of Government, Power Grid, and Users in Electricity Market Demand-Side Management" data-journal="mathematics">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2227-7390/12/20/3249/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2227-7390/12/20/3249/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2227-7390/12/20/3249/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2227-7390/12/20/3249/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001.png?1729157780" title=" <strong>Figure 1</strong><br/> <p>Schematic diagram of symmetric and asymmetric evolutionary game structures [<a href="#B46-mathematics-12-03249" class="html-bibr">46</a>]. In the figure (<b>a</b>), it illustrates the structure of a symmetric evolutionary game. In figure (<b>b</b>), it demonstrates the structure of an asymmetric evolutionary game.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002a.png?1729157782" title=" <strong>Figure 2</strong><br/> <p>The simulation results of the 2P2S-AEG system. In figure (<b>a</b>), it illustrates the evolution trend of the proportion <span class="html-italic">x</span> of individuals selecting strategy S<sub>A1</sub> in group A over time <span class="html-italic">t</span>. In figure (<b>b</b>), it demonstrates the evolution trend of the proportion <span class="html-italic">y</span> of individuals selecting strategy S<sub>B1</sub> in group B over time <span class="html-italic">t</span>. In figure (<b>c</b>), it reveals the evolution trend of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over time <span class="html-italic">t</span> in the game system.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002b.png?1729157784" title=" <strong>Figure 2 Cont.</strong><br/> <p>The simulation results of the 2P2S-AEG system. In figure (<b>a</b>), it illustrates the evolution trend of the proportion <span class="html-italic">x</span> of individuals selecting strategy S<sub>A1</sub> in group A over time <span class="html-italic">t</span>. In figure (<b>b</b>), it demonstrates the evolution trend of the proportion <span class="html-italic">y</span> of individuals selecting strategy S<sub>B1</sub> in group B over time <span class="html-italic">t</span>. In figure (<b>c</b>), it reveals the evolution trend of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over time <span class="html-italic">t</span> in the game system.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g003.png?1729157786" title=" <strong>Figure 3</strong><br/> <p>The tripartite evolutionary game framework among the government, the power grid company, and power users in demand-side management (DSM). This figure illustrates the strategic interactions among the three parties, where each entity’s strategy selection impacts the others’ payoffs. The government incentivizes or refrains from incentivizing DSM policies, the grid company decides whether to implement these policies, and the power users choose to participate or opt out of DSM services. The figure reflects the dynamic evolution of these strategic choices over time, with the goal of achieving a stable evolutionary equilibrium among all participants.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004a.png?1729157788" title=" <strong>Figure 4</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (6, 1, 3, 7, 2, 4) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and −<span class="html-italic">d</span> &lt; <span class="html-italic">e</span> − <span class="html-italic">f</span> &lt; 0. In figure (<b>b</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (6, 1, 3, 4, 2, 7) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> + <span class="html-italic">d</span> &lt; 0. In figure (<b>c</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (3, 4, 1, 4, 2, 7) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> + <span class="html-italic">d</span> &lt; 0. In figure (<b>d</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (3, 4, 1, 7, 2, 4) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and −<span class="html-italic">d</span> &lt; <span class="html-italic">e</span> − <span class="html-italic">f</span> &lt; 0. In figure (<b>e</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (5, 2, 4, 3, 1, 6) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> &gt; 0. In figure (<b>f</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (7, 2, 4, 3, 6, 5) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> &gt; 0.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004b.png?1729157790" title=" <strong>Figure 4 Cont.</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (6, 1, 3, 7, 2, 4) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and −<span class="html-italic">d</span> &lt; <span class="html-italic">e</span> − <span class="html-italic">f</span> &lt; 0. In figure (<b>b</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (6, 1, 3, 4, 2, 7) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> + <span class="html-italic">d</span> &lt; 0. In figure (<b>c</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (3, 4, 1, 4, 2, 7) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> + <span class="html-italic">d</span> &lt; 0. In figure (<b>d</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (3, 4, 1, 7, 2, 4) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and −<span class="html-italic">d</span> &lt; <span class="html-italic">e</span> − <span class="html-italic">f</span> &lt; 0. In figure (<b>e</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (5, 2, 4, 3, 1, 6) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> &gt; 0. In figure (<b>f</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (7, 2, 4, 3, 6, 5) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> &gt; 0.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005a.png?1729157792" title=" <strong>Figure 5</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (7, 5, 1, 6, 2, 1, 4) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and −<span class="html-italic">d</span>′ &lt; <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>b</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (7, 4, 1, 3, 2, 5, 12) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and <span class="html-italic">d</span>′ + <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>c</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 5, 2, 3, 2, 1, 9) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and <span class="html-italic">d</span>′ + <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>d</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 5, 2, 7, 3, 1, 6) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and −<span class="html-italic">d</span>′ &lt; <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>e</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 1, 2, 7, 3, 4, 5) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &gt; 0. In figure (<b>f</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (3, 4, 1, 7, 2, 6, 5) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &gt; 0.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005b.png?1729157793" title=" <strong>Figure 5 Cont.</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (7, 5, 1, 6, 2, 1, 4) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and −<span class="html-italic">d</span>′ &lt; <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>b</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (7, 4, 1, 3, 2, 5, 12) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and <span class="html-italic">d</span>′ + <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>c</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 5, 2, 3, 2, 1, 9) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and <span class="html-italic">d</span>′ + <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>d</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 5, 2, 7, 3, 1, 6) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and −<span class="html-italic">d</span>′ &lt; <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>e</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 1, 2, 7, 3, 4, 5) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &gt; 0. In figure (<b>f</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (3, 4, 1, 7, 2, 6, 5) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &gt; 0.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g006.png?1729157795" title=" <strong>Figure 6</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid-power user three-group asymmetric evolutionary game system when taking 1000 times of repeated evolution game. Figure (<b>a</b>) shows the phase trajectory of (<span class="html-italic">x</span>, <span class="html-italic">y</span>, <span class="html-italic">z</span>), and Figure (<b>b</b>) demonstrates the phase trajectories of (<span class="html-italic">x</span>, <span class="html-italic">t</span>), (<span class="html-italic">y</span>, <span class="html-italic">t</span>) and (<span class="html-italic">z</span>, <span class="html-italic">t</span>).</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g007.png?1729157798" title=" <strong>Figure 7</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid-power user three-group asymmetric evolutionary game system when taking 2000 times of repeated evolution game. Figure (<b>a</b>) shows the phase trajectory of (<span class="html-italic">x</span>, <span class="html-italic">y</span>, <span class="html-italic">z</span>), and Figure (<b>b</b>) demonstrates the phase trajectories of (<span class="html-italic">x</span>, <span class="html-italic">t</span>), (<span class="html-italic">y</span>, <span class="html-italic">t</span>) and (<span class="html-italic">z</span>, <span class="html-italic">t</span>).</p> "> </a> </div> <a class="button button--color-inversed" href="/2227-7390/12/20/3249/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <section id="FeaturedApplication" type=""><h2 data-nested="1"> Featured Application</h2><div class="html-p"><b>This article mainly utilizes the advantages and characteristics of evolutionary game theory and, based on the ideas and methods of evolutionary game theory, describes the relationship between the government, power grid enterprises, and power users as a learning progressive evolution system, focusing on the evolution process of the relationship between various stakeholders and the influencing factors of evolutionary stability in the electricity market demand-side management. It provides a reasonable explanation for the spontaneous formation of interest equilibrium between government, power grid enterprises, and power users, as well as theoretical references and policy recommendations for government regulation of the demand-side electricity market.</b></div></section><section id="Abstract" type=""><h2 data-nested="1"> Abstract</h2><div class="html-p">In the process of promoting demand-side management, the core stakeholder groups are government departments, power grid companies, and electricity users. Due to the different positions and conflicting interests of the three parties in the game, intense and complex battles will occur. This paper investigates a tripartite evolutionary game involving government, power grid companies, and electricity users in the context of demand-side management (DSM) and analyzes the dynamic interactions between government departments, power grid companies, and electricity users within the framework of DSM using evolutionary game theory. Using evolutionary game theory, we explore how incentives and strategic interactions among these three stakeholders evolve over time, affecting the stability of DSM policies. The model addresses the asymmetry in the decision-making process and examines the dynamic equilibrium outcomes under various scenarios. The results provide insights into the optimal design of incentive mechanisms to enhance DSM adoption. The findings offer practical recommendations to improve DSM policies, fostering balanced interests between government, grid companies, and users. This research contributes to a deeper understanding of strategic interactions in DSM, revealing how adaptive behaviors can enhance energy efficiency. It also underscores the importance of carefully designed incentive mechanisms in achieving long-term stability and cooperation among key stakeholders.</div></section> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=power+demand-side+management">power demand-side management</a>; <a href="/search?q=incentive+mechanism">incentive mechanism</a>; <a href="/search?q=evolutionary+game+theory">evolutionary game theory</a>; <a href="/search?q=evolutionary+stable+equilibrium">evolutionary stable equilibrium</a>; <a href="/search?q=replicator+dynamics+equations">replicator dynamics equations</a></div> <div class="html-gwd-group"><div id="html-keywords-title">MSC:</div> 65M12</div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-mathematics-12-03249' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>In the context of global energy transitions, managing electricity demand has become a critical issue for ensuring energy efficiency and reducing environmental impact. Many countries, including China, are facing challenges related to rising energy consumption, inefficient energy use, and the growing need to integrate renewable energy sources into the grid. In particular, China is experiencing rapid economic growth, which has led to a sharp increase in electricity demand. This growth, coupled with the need to reduce carbon emissions, has placed immense pressure on the national grid. Demand-side management (DSM) plays a vital role in balancing electricity supply and demand, particularly as smart grid technologies and renewable energy sources become more integrated into power systems. However, a significant challenge lies in aligning the interests of the three core stakeholders: government, power grid companies, and electricity users. This misalignment can hinder the effective implementation of DSM strategies, reducing the potential benefits of energy savings and peak load reduction. Against the backdrop of ongoing energy transition in various countries, the proportion of electricity in terminal energy consumption continues to rise [<a href="#B1-mathematics-12-03249" class="html-bibr">1</a>].</div><div class='html-p'>Demand-side management (DSM) has emerged as a crucial strategy to address these challenges. DSM involves adjusting consumer demand for electricity through various methods such as financial incentives, real-time pricing, and the promotion of energy-efficient technologies. By implementing DSM, the government seeks to reduce energy wastage, lower peak loads, and promote the use of cleaner energy sources. However, the effective implementation of DSM requires the cooperation of three key stakeholders: the government, power grid companies, and electricity users. The government is tasked with designing DSM policies and providing the necessary incentives for grid companies and users to participate. Power grid companies, on the other hand, need to balance the costs of implementing DSM programs with the benefits of improved grid stability and operational efficiency. Finally, electricity users must decide whether to participate in DSM programs based on the potential cost savings and the convenience of doing so. The misalignment of interests between these stakeholders has made the implementation of DSM a complex challenge. As a result, the potential benefits of energy savings and peak load reduction may not be fully realized.</div><div class='html-p'>In addition, the emergence of smart grids has provided a powerful information exchange platform for various levels of the grid, promoting the development of aspects such as power generation, transmission and distribution processes, and scheduling, improving the stability of system operations, and facilitating the rational allocation of electric resources [<a href="#B2-mathematics-12-03249" class="html-bibr">2</a>,<a href="#B3-mathematics-12-03249" class="html-bibr">3</a>,<a href="#B4-mathematics-12-03249" class="html-bibr">4</a>]. For example, on the demand side, the increasing popularity of smart homes and electric vehicles has enhanced load responsiveness [<a href="#B5-mathematics-12-03249" class="html-bibr">5</a>]. Participants on the demand side are becoming more diverse, and by optimizing the balance between supply and demand sides, the benefits for both can be improved, promoting healthy economic development [<a href="#B6-mathematics-12-03249" class="html-bibr">6</a>]. Currently, the power supply and demand sides in the new power systems exhibit characteristics of overall balance, surplus supply in some regions, and tight supply during peak periods in certain areas [<a href="#B7-mathematics-12-03249" class="html-bibr">7</a>,<a href="#B8-mathematics-12-03249" class="html-bibr">8</a>]. In this situation, increasing user participation in electricity usage and incentivizing users to engage in demand-side management can effectively enhance electricity efficiency, promote supply-demand balance, and achieve energy conservation and emission reduction. Therefore, electricity demand-side management methods are increasingly being emphasized by the state and society.</div><div class='html-p'>Generally, DSM in power system is defined as follows [<a href="#B9-mathematics-12-03249" class="html-bibr">9</a>,<a href="#B10-mathematics-12-03249" class="html-bibr">10</a>,<a href="#B11-mathematics-12-03249" class="html-bibr">11</a>]: through effective management of electricity users and the use of reasonable and effective incentive mechanisms, it aims to enhance the enthusiasm of electricity users to participate in smart electricity distribution, thereby promoting the efficient allocation of electricity resources from the source. Currently, China’s demand-side management has achieved significant effectiveness, with a notable increase in electricity efficiency [<a href="#B12-mathematics-12-03249" class="html-bibr">12</a>,<a href="#B13-mathematics-12-03249" class="html-bibr">13</a>]. The “14th Five-Year Plan” indicates that the new focus of demand-side management is to formulate policies that align with the current electricity supply and demand situation [<a href="#B14-mathematics-12-03249" class="html-bibr">14</a>], promote the development of smart grids, facilitate the sale of clean energy, and advance structural reforms on the energy supply side. Since the power grid is responsible for both providing electricity and selling power, the costs and benefits during the DSM project process can vary. The enthusiasm of grid companies to implement projects can be influenced by project costs and benefits. If a grid company invests in a project but receives low returns, with the main benefits accruing to other participants, there is a possibility that the grid company may halt project construction. Similarly, if the benefits for electricity users are lower than their expenses, they may also choose to discontinue using related services.</div><div class='html-p'>In this study, we use evolutionary game theory to model the dynamic interactions between these stakeholders and explore how different DSM incentive schemes can promote cooperation and lead to a stable, long-term equilibrium that benefits all parties. To this end, this paper seeks to address the question: How can evolutionary game theory be applied to model and improve the interactions between these stakeholders in the context of DSM, leading to a stable, long-term cooperative equilibrium? The key to achieving this lies in understanding the complex dynamics between the stakeholders, who often have conflicting interests and varying incentives for participation in DSM programs. Governments aim to reduce energy consumption and promote sustainability; power grid companies are concerned with operational efficiency and cost, while users seek to minimize electricity costs and maintain convenience.</div><div class='html-p'>Despite the importance of DSM, there remains a gap in the literature regarding the application of evolutionary game theory to model these multi-stakeholder dynamics. Previous research has mainly focused on classical game theory approaches, which assume fully rational decision-making. However, real-world stakeholders often make decisions under bounded rationality, adjusting their strategies based on past experiences and evolving incentives. This study applies evolutionary game theory to better capture these dynamics, providing insights into how long-term cooperation between government, grid companies, and users can be fostered under different incentive mechanisms.</div><div class='html-p'>The specific question analyzed in this paper is: What evolutionary stable strategies (ESS) emerge from the interactions between government, power grid companies, and users under different DSM incentive schemes, and how do these strategies affect the overall efficiency of DSM implementation? Through the use of replicator dynamics and simulations, this research aims to identify stable equilibria that promote cooperation and improve DSM policy effectiveness. The findings will offer practical recommendations for policymakers and power grid companies to better design DSM programs that align the interests of all stakeholders, thereby enhancing energy efficiency and reducing peak demand.</div><div class='html-p'>To this end, for government departments, demand-side management of electricity can reduce energy loss, alleviate the tight electricity situation in cities, improve energy utilization efficiency, promote the rational allocation of electricity resources, reduce pollutant emissions, improve the ecological environment, drive the research and development of energy-saving equipment, enhance the quality of economic development in the electricity sector, and promote green economic development and the optimization of energy consumption structure. For grid companies, implementing demand-side management can lower peak load, improve the stability of power supply systems, ensure the safe and economical operation of the grid, and increase economic benefits. For electricity users, utilizing demand-side management services can reduce electricity bills and consumption, allowing corporate users to enhance their profit margins.</div><div class='html-p'>This paper uses evolutionary game theory to study the three-party game problem in demand-side management of the electricity market. Game theory, also known as strategic theory, originated in the field of economics [<a href="#B15-mathematics-12-03249" class="html-bibr">15</a>,<a href="#B16-mathematics-12-03249" class="html-bibr">16</a>,<a href="#B17-mathematics-12-03249" class="html-bibr">17</a>]. Game theory is used to analyze how multiple players, based on their level of rationality and the information they possess, assess the interests of all players, thereby making decisions that maximize the benefits for their own group. Game theory has been applied in the field of electrical engineering for many years, initially in the direction of the electricity market, and now has widespread applications in areas such as electricity markets [<a href="#B18-mathematics-12-03249" class="html-bibr">18</a>], power system planning [<a href="#B19-mathematics-12-03249" class="html-bibr">19</a>], coordination [<a href="#B20-mathematics-12-03249" class="html-bibr">20</a>], and scheduling [<a href="#B21-mathematics-12-03249" class="html-bibr">21</a>]. Classical game theory is based on the premise that participants have complete rationality and complete information, focusing on static equilibrium and comparative static equilibrium, which presents significant flaws when studying real-world issues. In contrast, evolutionary game theory (EGT) features bounded rationality and limited information, emphasizing dynamic equilibrium and positing that individual decisions are determined through dynamic processes such as learning, imitation, and communication within the group [<a href="#B22-mathematics-12-03249" class="html-bibr">22</a>,<a href="#B23-mathematics-12-03249" class="html-bibr">23</a>,<a href="#B24-mathematics-12-03249" class="html-bibr">24</a>,<a href="#B25-mathematics-12-03249" class="html-bibr">25</a>]. By studying the behavioral changes of a group through this dynamic process, game theory can predict future group behavior, making it more suitable for analyzing real-world problems.</div><div class='html-p'>Actually, in recent years, increasing attention has been given to improving energy utilization, particularly in the context of electricity, which is a critical resource in the global energy transition. However, traditional energy production and consumption methods, characterized by inefficiency and environmental degradation, pose significant challenges to sustainable development. Countries are striving to adopt cleaner and more efficient energy practices, with electricity becoming a central component of this transformation. As electricity’s share of total energy consumption rises, it becomes imperative to develop mechanisms that optimize its use, both from supply and demand perspectives. This is where demand-side management (DSM) becomes crucial.</div><div class='html-p'>The central research question addressed in this paper is how to effectively manage DSM through the dynamic interactions of key stakeholders: the government, power grid companies, and electricity users. Current literature identifies significant barriers to balancing these stakeholders’ competing interests, making DSM implementation a complex challenge. This study focuses on understanding the evolution of these interactions over time and identifying stable equilibrium strategies that can lead to sustainable DSM outcomes.</div><div class='html-p'>By leveraging evolutionary game theory, this paper provides a novel approach to analyzing the strategic behaviors of these stakeholders under various incentive mechanisms. Evolutionary game theory is particularly suited for this analysis because it accounts for the bounded rationality of players and allows for dynamic adjustments of strategies based on learning and adaptation. This study extends current research by applying a multi-agent framework that focuses on long-term stability and cooperation among stakeholders, ultimately offering practical solutions to improve DSM policy design and implementation.</div><div class='html-p'>Overall, this paper uses evolutionary game theory, focusing on incentive mechanisms, to analyze the structure of demand-side management in the electricity market. It studies the factors affecting system stability through modeling, providing analytical data for relevant electricity policies, and offering references to accelerate the development of demand-side management in the electricity sector. This paper investigates the dynamic interactions among government entities, power grid companies, and electricity users in the context of demand-side management (DSM) in the electricity market, utilizing evolutionary game theory as a core analytical framework. The research contributes new insights into how strategic behaviors evolve over time among these key stakeholders and offers concrete policy recommendations based on the analysis. Below, the main research contributions, key conclusions, and innovative aspects of the study are detailed.</div><dl class='html-order'><dt id=''>(1)</dt><dd><div class='html-p'><b>Thorough Analysis of Stakeholder Dynamics in DSM</b></div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'>Government-Grid-User Interaction: The paper develops a comprehensive evolutionary game model that captures the dynamic interactions between three key stakeholders in the DSM process: government bodies, power grid companies, and electricity users. These interactions are modeled as a multi-agent, multi-strategy game where each entity’s strategic choices influence the others.</div></li><li><div class='html-p'>Strategic Incentives: The research emphasizes how different incentives (e.g., subsidies, policy benefits, penalties) from the government can affect the behavior of grid companies and electricity users. By adjusting these incentives, the government can steer the entire system towards a more efficient and cooperative outcome in DSM.</div></li></ul></div></dd><dt id=''>(2)</dt><dd><div class='html-p'><b>Application of Evolutionary Game Theory to DSM</b></div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'>Comparison of Classical and Evolutionary Game Theory: The paper highlights the limitations of classical game theory, which assumes complete rationality and static equilibria, and contrasts it with evolutionary game theory, which better reflects real-world scenarios with bounded rationality and dynamic decision-making. This novel application of evolutionary game theory allows for a more realistic analysis of stakeholder behavior in DSM.</div></li><li><div class='html-p'>Replicator Dynamics and Stability Analysis: By utilizing replicator dynamics equations, the study models how the proportion of stakeholders adopting different strategies evolves over time. The Lyapunov stability method is applied to identify evolutionarily stable strategies (ESS), which offer insights into how long-term stability in the DSM process can be achieved under various game conditions.</div></li></ul></div></dd><dt id=''>(3)</dt><dd><div class='html-p'><b>Key Research Findings</b></div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'>Cooperation is Essential for DSM Success: The simulations reveal that cooperation between government, grid companies, and users is essential for the long-term success of DSM policies. When properly incentivized, grid companies are more likely to implement DSM projects, and users are more inclined to adopt energy-saving behaviors. The research demonstrates that achieving a balance of interests among all stakeholders can create a win-win scenario.</div></li><li><div class='html-p'>Incentive Mechanisms Drive Evolutionary Stability: One of the major conclusions is that carefully designed incentive mechanisms (such as government subsidies for DSM adoption or penalties for non-compliance) can significantly affect the system’s convergence towards a stable equilibrium. The study finds that when incentives align the interests of the grid and users with those of the government, stable, long-term cooperation is more likely.</div></li><li><div class='html-p'>Role of Government as a Dominant Player: The government is shown to play a critical role as the dominant player in the DSM game, able to influence the behavior of other stakeholders through policy interventions. However, the study also points out that without sufficient incentives, both grid companies and users may resist cooperation, leading to an unstable or inefficient system.</div></li></ul></div></dd><dt id=''>(4)</dt><dd><div class='html-p'><b>Innovative Contributions to the Field</b></div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'>First Integration of Evolutionary Game Theory into DSM Research: This study is among the first to apply evolutionary game theory to the problem of DSM in the electricity market, offering a new analytical tool for understanding the strategic behavior of multiple stakeholders over time. This approach goes beyond the limitations of static game models traditionally used in energy policy analysis.</div></li><li><div class='html-p'>Introduction of Asymmetric and Multi-Group Game Models: The paper incorporates both symmetric and asymmetric game models, reflecting the real-world complexity of information asymmetry and varying degrees of strategic advantage among stakeholders. This adds depth to the analysis by showing how different levels of information and influence affect the strategic decisions of the government, grid companies, and users.</div></li><li><div class='html-p'>Numerical Simulations for Policy Optimization: By conducting extensive numerical simulations, the study provides concrete examples of how different game scenarios play out under various policy conditions. These simulations offer a practical reference for policymakers, demonstrating which incentive structures are most effective in promoting DSM adoption and achieving energy efficiency goals.</div></li></ul></div></dd><dt id=''>(5)</dt><dd><div class='html-p'><b>Policy Implications and Future Directions</b></div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'>Practical Recommendations for DSM Policy: The findings provide actionable recommendations for improving DSM policies. For example, the study suggests that financial subsidies for energy-efficient equipment can significantly enhance user participation, especially for large industrial consumers. Similarly, for grid companies, regulatory targets, and financial incentives can encourage the implementation of DSM technologies.</div></li><li><div class='html-p'>Insights into Adaptive Strategies and Long-Term Sustainability: The research offers valuable insights into how adaptive strategies can promote long-term sustainability in energy management. The evolutionary game approach illustrates how stakeholders’ strategies evolve in response to changing incentives, highlighting the importance of continuous adjustment in policy design.</div></li><li><div class='html-p'>Further Applications of Evolutionary Game Theory in Energy Markets: The innovative use of evolutionary game theory in this paper sets the stage for its broader application in other areas of energy markets. This includes analyzing bidding strategies in wholesale electricity markets, cooperative investments in renewable energy, and dynamic pricing strategies for distributed energy resources.</div></li></ul></div></dd><dt id=''>(6)</dt><dd><div class='html-p'><b>Contributions to Broader Research in Game Theory and Energy Policy</b></div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'>Bridging the Gap between Theory and Practice: This paper contributes to the theoretical development of evolutionary game theory while simultaneously providing practical insights for energy policymakers. It effectively bridges the gap between abstract game theory models and real-world policy challenges in energy management, making a significant contribution to both fields.</div></li><li><div class='html-p'>Framework for Analyzing Complex Stakeholder Interactions: The study offers a robust framework for analyzing complex multi-stakeholder interactions in energy markets. It demonstrates how the behavior of key actors can be modeled and predicted over time, providing a valuable tool for future research in both electricity market design and other domains involving strategic decision-making under uncertainty.</div></li></ul></div></dd></dl><div class='html-p'>Furthermore, the summary of key innovations is elaborated as follows.</div><ul class='html-bullet'><li><div class='html-p'>Pioneering use of evolutionary game theory in analyzing demand-side management in electricity markets, providing new perspectives on stakeholder behavior.</div></li><li><div class='html-p'>Development of asymmetric and multi-group game models that account for information asymmetry and varying stakeholder influence, enhancing the realism of the analysis.</div></li><li><div class='html-p'>Comprehensive numerical simulations that demonstrate the practical effects of different incentive mechanisms on DSM adoption and stakeholder cooperation.</div></li><li><div class='html-p'>Actionable policy recommendations grounded in rigorous theoretical and empirical analysis, offering a roadmap for optimizing DSM policies in the energy sector.</div></li></ul><div class='html-p'>This research contributes both theoretically and practically to the fields of game theory and energy policy, offering new methodologies and insights that can be applied to broader contexts in energy market design and management.</div><div class='html-p'>The remaining part of this paper is organized as follows: <a href="#sec2-mathematics-12-03249" class="html-sec">Section 2</a> introduces the background of demand management in our country and the practical significance of promoting its development. It also summarizes the existing research results related to demand-side management in domestic and international electricity markets and evolutionary game theory at this stage, followed by a brief introduction to the structure and research approach of this paper. In <a href="#sec3-mathematics-12-03249" class="html-sec">Section 3</a>, this paper introduces the basic concepts of classical game theory and evolutionary game theory, compares the two, and then presents several important concepts in the evolutionary game theory discussed in this paper. It briefly describes how to assess the evolutionary stability of game models based on the Lyapunov method. Based on this, <a href="#sec4-mathematics-12-03249" class="html-sec">Section 4</a> first studies the two-population, two-strategy asymmetric evolutionary game (2P2S-AEG) under general conditions, establishes the corresponding game model, and also conducts simulation verification, laying a theoretical foundation for the subsequent research. Further, <a href="#sec5-mathematics-12-03249" class="html-sec">Section 5</a> first analyzes the game structure of the incentive mechanisms among government departments, grid companies, and electricity users in demand-side management of the electricity market. Next, based on the ideas and methods of evolutionary game theory, it builds the government-grid and government-user evolutionary game models and then uses MATLAB R2023b (v23.2.0.2428915) for dynamic simulation of the models, followed by an analysis of the simulation results. Finally, <a href="#sec6-mathematics-12-03249" class="html-sec">Section 6</a> is a summary of the research content of this paper.</div></section><section id='sec2-mathematics-12-03249' type=''><h2 data-nested='1'> 2. Literature Review</h2><section id='sec2dot1-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 2.1. Demand-Side Management in Electricity Market</h4><div class='html-p'>Since the reform and opening up, China’s economic strength has rapidly increased, and the electric power economy has developed well. Daily life, travel, and production activities are now difficult to separate from the use of electricity. Electricity holds a unique and irreplaceable important position as an energy source. In China, the electricity market is monopolized by the state. However, certain types of pollution are inevitably caused during electricity production activities. The concept of DSM was introduced in our country in the last century. As of now, significant achievements have been made in demand-side management in China. According to the annual announcement by the National Development and Reform Commission in 2019, the grid company has met national targets in demand-side management work, saving a large amount of electric energy. The continuous deepening of the electricity market reform has provided conditions for establishing a market-based incentive mechanism for DSM. Some provinces in China have begun pilot programs to implement a DSM market mechanism based on big data. In some areas, DSM platforms have been established to promote DSM projects using big data. During the summer peak electricity load period, these regions effectively reduced peak electricity load by about 10 million kW. The ongoing reform of the electricity market is a prerequisite for establishing a market-based incentive mechanism for DSM. Places like Zhejiang in China have already applied this mechanism in practical work, while Guangdong in China has proposed corresponding implementation plans and measures, and provinces like Jiangsu in China are conducting pilot projects. Some regions are actively expanding the scale of electricity user access to DSM platforms, leveraging the big data advantages of the platform to help enterprises optimize their electricity usage behavior. The development of smart grids has promoted the implementation of demand-side management, as smart grids can efficiently allocate and manage wide-area demand-side and supply-side electric energy resources, improve functional efficiency, and facilitate the integration of renewable energy into the grid [<a href="#B26-mathematics-12-03249" class="html-bibr">26</a>].</div><div class='html-p'>In recent years, both domestically and internationally, there has been a growing awareness of the benefits brought by DSM, prompting researchers to invest significant effort into studying its connotations and practical applications. For example, Zhou et al. [<a href="#B27-mathematics-12-03249" class="html-bibr">27</a>] comprehensively considered demand-side resources and supply-side resources, incorporating them into power planning to construct a comprehensive resource strategic planning model. The application of this model can meet social electricity demand while reducing the installed capacity on the supply side and lowering pollutant emissions. Wang et al. [<a href="#B28-mathematics-12-03249" class="html-bibr">28</a>], under the new circumstances of DSM system reform in China, approached the issue from the perspectives of partnerships and profit models, establishing an organizational framework for DSM and studying its future development models. By establishing relevant laws and incentive measures, optimizing the electricity pricing system, and supervising the implementation of DSM by the power grid, the development of DSM can be promoted. Zhou et al. [<a href="#B29-mathematics-12-03249" class="html-bibr">29</a>] identified three directions for the upgraded development of DSM: macro management of electricity consumption, micro-management of electricity consumption, and comprehensive electricity service upgrades. Jiang et al. [<a href="#B30-mathematics-12-03249" class="html-bibr">30</a>] constructed a regulatory model for DSM, analyzing the behavioral changes and strategic choices of government regulatory departments and power supply enterprises during the implementation of DSM, ultimately suggesting that regulatory departments could penalize power supply enterprises that do not implement DSM policies and improve the DSM system. Sun et al. [<a href="#B31-mathematics-12-03249" class="html-bibr">31</a>] built an interactive evaluation model for residential electricity consumption focused on demand-side management, centered around load optimization and user demand response, and verified the model’s effectiveness through experiments. By adopting reasonable reward and punishment measures, residents can be guided to improve their electricity consumption methods, thereby enhancing the implementation effectiveness of DSM. Based on this, Mishra et al. [<a href="#B32-mathematics-12-03249" class="html-bibr">32</a>] adopted the game theoretical method to investigate demand-side management, considering distributed energy storage and generation with discomfort as an objective. The results indicate that the proposed game method can be used to increase the average energy cost, and the proposed game model does not significantly impact the distributed dispatchable generation and energy storage profile. Ji et al. [<a href="#B33-mathematics-12-03249" class="html-bibr">33</a>] conducted a systematic review of the game-theoretic applications for decision-making behavior on the energy demand side, where the non-cooperative game, cooperative game, Stackelberg game, Bayesian game, and evolutionary game methods are reviewed on their applications on the demand side. The review shows that these game-theoretical methods can benefit both the grid and the users, for example, by reducing the peak-to-average ratios and energy costs of the users. Based on these game models, Belhaiza et al. [<a href="#B34-mathematics-12-03249" class="html-bibr">34</a>] proposed a game theoretic model for solving the issue of the multiperiodic smart grid demand-side management with shifted demand. In addition, Mishra and Parida [<a href="#B35-mathematics-12-03249" class="html-bibr">35</a>] proposed an aggregative game approach for capturing the interaction between users and utility, which indicates that the proposed pricing scheme is effective in terms of consumers’ energy bills and system peak reduction.</div><div class='html-p'>Overall, the demand-side management of the electricity market is still in a rapid development stage. Many experts have studied the importance of DSM in the electricity market, its operational methods, and the organizational framework of participants. They have elaborated on the positive role of DSM in the development of distribution systems and the optimization of electricity pricing mechanisms and analyzed the issues in areas such as government regulation of grid companies, incentivizing residents to use DSM services, implementing time-of-use pricing, and optimizing residential electricity consumption behavior. Some scholars have used game theory methods to construct corresponding models for analysis and verification and have proposed suggestions for the development of the DSM system.</div></section><section id='sec2dot2-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 2.2. Game-Theoretical Methods</h4><div class='html-p'>Game theory, as a powerful mathematical tool for studying competitive phenomena in research, has the following domestic and international research status in the electrical field. Mei et al. [<a href="#B36-mathematics-12-03249" class="html-bibr">36</a>] established a game planning model for hybrid power systems of solar energy, wind energy, and energy storage, analyzing the capacity optimization configuration of solar energy, wind energy, and energy storage under game strategies of complete non-cooperation, partial cooperation, and complete cooperation, and examined the stability of the Nash equilibrium solution when wind speed has uncertain characteristics. Voropai and Ivanova [<a href="#B37-mathematics-12-03249" class="html-bibr">37</a>] constructed a cooperative game model involving the government, power generation enterprises, and users in the context of power generation growth planning, pointing out that cooperation significantly enhances overall returns. Xie et al. [<a href="#B38-mathematics-12-03249" class="html-bibr">38</a>] established a tripartite game model involving regulatory agencies, power generation enterprises, and users, analyzing the evolutionary stable strategies of the system under static and dynamic reward and punishment mechanisms, proving that the three parties cannot achieve evolutionary stability under static mechanisms, while their strategies can reach stable equilibrium under dynamic mechanisms. Ekneligoda and Weaver [<a href="#B39-mathematics-12-03249" class="html-bibr">39</a>] studied the transient stability control problem of microgrids using dynamic games. Gao and Sheng [<a href="#B40-mathematics-12-03249" class="html-bibr">40</a>] applied game theory to analyze the spontaneous evolution process of bidding strategies among power generation enterprises, highlighting the importance of the government in formulating reasonable bidding rules and regulating the electricity market. Li et al. [<a href="#B41-mathematics-12-03249" class="html-bibr">41</a>] modeled the bidding strategies of power generation enterprises based on differences in their generation scales using evolutionary game theory, showing that the multi-group replicator dynamic game model can adequately reflect the dynamic process of bidding among power generation enterprises and their pricing strategies under different regulatory measures. Sun et al. [<a href="#B42-mathematics-12-03249" class="html-bibr">42</a>] analyzed the purchasing and selling strategies of electricity companies and the risk factors they face, establishing a two-tier game model between electricity companies and power users using evolutionary game theory. Xu [<a href="#B43-mathematics-12-03249" class="html-bibr">43</a>] established a game model in a hybrid power system where multiple power generation companies have various generation technologies, concluding that moderate subsidies for wind power can increase its utilization rate and enhance energy efficiency. Zhu et al. [<a href="#B44-mathematics-12-03249" class="html-bibr">44</a>] studied the DSM and control issues of a type of mesh-structured smart grid based on evolutionary game theory. Miorandi and Pellegrini [<a href="#B45-mathematics-12-03249" class="html-bibr">45</a>] researched a distributed control method that can be forcibly implemented by operators based on pricing strategies. Additionally, Cheng et al. [<a href="#B46-mathematics-12-03249" class="html-bibr">46</a>] used evolutionary game theory to analyze the behavioral decision-making issues faced by multi-group users participating in smart electricity allocation in the grid, providing theoretical references for the decision-making problems of non-fully rational participants in the smart grid field. Zhu and Gao [<a href="#B47-mathematics-12-03249" class="html-bibr">47</a>] proposed a dynamic analysis model of smart electricity user participation based on evolutionary game theory. Chai et al. [<a href="#B48-mathematics-12-03249" class="html-bibr">48</a>] proposed a two-tier game algorithm for the bidirectional dynamic interaction between electricity users and power companies under the condition that all electricity users participate in smart electricity allocation.</div><div class='html-p'>Overall, with the gradual opening of the electricity market, the types of decision-making entities are becoming increasingly complex, and competition is intensifying. Therefore, relevant scholars have chosen to use game theory to study various decision-making issues in the electricity market. The research groups include governments, power generators, corporate users, and residential users, with research directions covering the supply side, distribution side, and demand side, and research content including bidding strategies, sales strategies, and the long-term equilibrium characteristics of spontaneous evolutionary processes. Among them, evolutionary game theory has a clear advantage in analyzing the optimization decision-making problems of multiple interest groups in the electricity market due to its focus on long-term dynamic evolution and stability.</div></section></section><section id='sec3-mathematics-12-03249' type=''><h2 data-nested='1'> 3. Evolutionary Game-Theoretic Model in General Conditions</h2><div class='html-p'>This section introduces the fundamental concepts and construction methods of classical and evolutionary game theory, which are necessary for understanding the subsequent analysis of the three-party interaction between the government, power grid, and electricity users in demand-side management (DSM). For readers unfamiliar with evolutionary game theory, this section serves as an essential primer, providing the theoretical foundation required to grasp the complexities of the later sections. Furthermore, the construction methods discussed here directly inform the modeling and simulation processes that follow. By establishing these fundamentals, we ensure that the subsequent application of the theory to real-world scenarios is both comprehensible and well-supported. In the context of DSM, the replicator dynamics equations are used to model the frequency with which each group (government, power grid, and users) adjusts its strategy based on the payoffs received. The tripartite nature of the system adds complexity, as each group’s decision-making is influenced not only by their own benefits but also by the dynamic responses of the other two stakeholders.</div><section id='sec3dot1-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 3.1. Classical Game Theory</h4><div class='html-p'>Game theory refers to a theory and method that studies how multiple groups, based on the information they have about their opponents, assess the equilibrium stability of all strategy combinations and make decisions to maximize the interests of their own group. It is applicable for analyzing phenomena with complex, competitive relationships. The historical development is as follows: In 1928, scholar John von Neumann successfully proved the basic principles of game theory, marking its official birth. In 1994, von Neumann and Oskar Morgenstern jointly published “<span class='html-italic'>Theory of Games and Economic Behavior</span>”, which laid the foundation for the theoretical system of game theory [<a href="#B49-mathematics-12-03249" class="html-bibr">49</a>]. Subsequently, many scholars began to study game theory, leading to the gradual introduction of important theories such as non-cooperative games, repeated games, and evolutionary games [<a href="#B50-mathematics-12-03249" class="html-bibr">50</a>,<a href="#B51-mathematics-12-03249" class="html-bibr">51</a>,<a href="#B52-mathematics-12-03249" class="html-bibr">52</a>,<a href="#B53-mathematics-12-03249" class="html-bibr">53</a>]. The content of game theory has become increasingly refined and mature, providing a mathematical theory for the study and analysis of actual competitive phenomena, and has been widely applied in fields such as economics, ecology, management, and engineering technology.</div><div class='html-p'>The basic assumptions of classical game theory are summarized as follows. First, participants possess the trait of complete rationality, with the goal of maximizing their own interests; second, there is common knowledge, meaning that in the perception of all participants, they are completely rational and not influenced by the emotional aspects of humans; third, before the game officially begins, a fixed game situation and a game structure containing game information will be provided, and the players will engage in the game within this structure.</div><div class='html-p'>A complete game structure should at least include three basic elements: participants, strategy sets, and utility functions. Participants are the decision-making entities represented by <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> <mo>}</mo> </mrow> </semantics></math>, indicating a set of <span class='html-italic'>n</span> individuals involved in the game. The strategy set, also known as the strategy space, represents all possible combinations of strategies that the players can choose during the game, denoted as <math display='inline'><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mo>{</mo> <msub> <mi>S</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>S</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>S</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> </semantics></math> for the strategy sets of all players. The utility function is the function that measures the utility obtained by the players after the decisions are implemented, also referred to as the payoff function. The utility function can be used to measure the gains or losses that players can achieve in this game. The decision-maker’s utility vector is represented as <math display='inline'><semantics> <mrow> <mi>U</mi> <mo>=</mo> <mo>{</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> </semantics></math>. Generally, players will choose strategies that yield higher returns or lower costs. In general, a classic game theory can be represented as <math display='inline'><semantics> <mrow> <mi>G</mi> <mo>=</mo> <mo>{</mo> <mi>N</mi> <mo>;</mo> <mtext> </mtext> <mi>S</mi> <mo>;</mo> <mtext> </mtext> <mi>U</mi> <mo>}</mo> </mrow> </semantics></math>.</div><div class='html-p'>Classical game theory has the following shortcomings in practical use. First, in reality, it is generally impossible for game participants to achieve complete rationality when making decisions. Here, various complex factors such as participants’ backgrounds, knowledge reserves, psychological abilities, personal preferences, and behavioral habits all limit their rationality. Second, participants are generally unable to grasp all the information within the game environment. Due to differences in their own capabilities and positions, the amount of information each participant can obtain varies significantly, with those who possess more information having a greater advantage in the game. Additionally, in classical game theory, solving for Nash equilibrium is quite complex. The solution for Nash equilibrium refers to strategies that balance the interests of all participants, allowing each participant to be satisfied. However, classical game theory itself does not clearly define the formation process of Nash equilibrium, making it unsuitable for studying and analyzing multi-party game issues.</div><div class='html-p'>Game theory can generally be divided into three types [<a href="#B5-mathematics-12-03249" class="html-bibr">5</a>]: non-cooperative games, cooperative games, and evolutionary games. This paper only applies evolutionary game theory, and the following will briefly introduce its basic concepts.</div></section><section id='sec3dot2-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 3.2. Evolutionary Game Theory</h4><section id='sec3dot2dot1-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 3.2.1. Bounded Rationality Assumption</h4><div class='html-p'>Evolutionary game theory is based on Darwin’s theory of biological evolution and survival of the fittest and combines evolutionary thought with classical game theory to analyze the evolution of population behavior. It is based on the assumption that individual rationality is limited and takes into account phenomena such as learning, imitation, communication, and feedback among individuals within a group. Evolutionary game theory is generally considered to have been proposed by Maynard Smith in 1973 while studying biological evolution phenomena [<a href="#B44-mathematics-12-03249" class="html-bibr">44</a>]. The decision-making processes of humans and other organisms in nature when dealing with certain real-world problems, such as whether to cooperate or compete, have similarities. Therefore, following a research approach and methodology based on analogical reasoning, evolutionary game theory is well-suited for studying and analyzing human game behavior.</div><div class='html-p'>In contrast to the basic conditions of classical game theory, evolutionary game theory assumes that players possess only limited rationality. When mutations occur, participants adjust their strategies in a timely manner based on the experiences left by predecessors, their own experiences, and the current feedback, continuously repeating the game until reaching an equilibrium state. Classical game theory, on the other hand, posits that participants only need to play the game once to derive the optimal strategy. Compared to classical game theory, evolutionary game theory places greater emphasis on the dynamic evolutionary characteristics of strategies rather than the equilibrium nature of the strategies themselves [<a href="#B45-mathematics-12-03249" class="html-bibr">45</a>,<a href="#B46-mathematics-12-03249" class="html-bibr">46</a>,<a href="#B47-mathematics-12-03249" class="html-bibr">47</a>]. Evolutionary game theory takes into account the constantly changing game situations in reality, making it more aligned with real-world scenarios when analyzing actual game situations and participants’ behavioral strategies, resulting in conclusions that are more valuable for reference.</div><div class='html-p'>Next, this paper will introduce several important concepts: replicator dynamics (RD), multi-group evolutionarily stable strategies (MESSs), and asymptotically stable equilibrium points (ASEPs).</div></section><section id='sec3dot2dot2-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 3.2.2. Replicator Dynamics</h4><div class='html-p'>In order to more accurately model the dynamic process by which participants adjust their strategies, Taylor and Juck proposed the concept of RD. The theoretical basis is that individuals possess a dynamic process of imitation, learning, communication, and feedback, i.e., the behaviors of individuals influence each other. RD is used to model the dynamic adjustment process of strategies, revealing the evolutionary pattern of population numbers or proportions [<a href="#B48-mathematics-12-03249" class="html-bibr">48</a>]. Replicator dynamics considers that in multiple dynamic games of the population since individuals tend to choose strategies with higher payoffs and lower risks, the proportion of strategies with high payoffs being chosen after multiple rounds of the game is increasing. So, the replicator dynamics can be described by a dynamic differential equation for the probability or frequency of a particular strategy <math display='inline'><semantics> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> </mrow> </semantics></math> being chosen in a given population. This dynamic differential equation is the replicator dynamics equation and can be expressed as:<div class='html-disp-formula-info' id='FD1-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mfenced close="]" open="["> <mrow> <mi>E</mi> <mfenced> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> </mrow> </mfenced> <mo>−</mo> <msub> <mi>E</mi> <mrow> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> </mrow> </msub> <mfenced> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> </mrow> </mfenced> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> where the value of RD (i.e., <math display='inline'><semantics> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> </semantics></math>) is proportional to <math display='inline'><semantics> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </semantics></math>, denoted by the proportion of the population choosing the strategy, and also proportional to the magnitude by which its expected return <math display='inline'><semantics> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> exceeds the population mean return <math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mrow> <mi>av</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> [<a href="#B48-mathematics-12-03249" class="html-bibr">48</a>].</div><div class='html-p'>If the return of some individuals choose a pure strategy <math display='inline'><semantics> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> </mrow> </semantics></math> is higher than the average return of the population, the proportion of individuals choosing that strategy increases. If the proportion of individuals choosing a pure strategy <math display='inline'><semantics> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> </mrow> </semantics></math> remains constant over time after many rounds of the game, and the payoff function is time-independent, that strategy becomes a stable evolutionary point for the population.</div></section><section id='sec3dot2dot3-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 3.2.3. Multi-Group Evolutionarily Stable Strategies</h4><div class='html-p'>When a certain strategy is chosen by the majority of individuals in a group, a small number of individuals adopting a mutant strategy cannot invade that group. At this point, according to the mechanism of natural selection, this mutant strategy can only adapt to the system by changing itself, or it will be eliminated by the system without making any changes. This strategy effectively prevents the invasion of other mutant strategies, has higher stability than other strategies, and is an evolutionarily stable strategy (ESS).</div><div class='html-p'>Assuming in a game involving <span class='html-italic'>n</span> multiple groups, the strategy combination <math display='inline'><semantics> <mrow> <mi>X</mi> <mo>=</mo> <mo>{</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>}</mo> <mo>∈</mo> <mi>Ψ</mi> </mrow> </semantics></math> is a multi-group evolutionarily stable strategy (MESS) where <math display='inline'><semantics> <mi>Ψ</mi> </semantics></math> is the strategy space. For the mutation strategy <math display='inline'><semantics> <mrow> <mi>Y</mi> <mo>=</mo> <mo>{</mo> <msub> <mi>Y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mo>}</mo> <mo>∈</mo> <mi>Ψ</mi> </mrow> </semantics></math>, it is defined as follows. When <math display='inline'><semantics> <mrow> <mi>Y</mi> <mo>≠</mo> <mi>X</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mo>∃</mo> <mn>0</mn> <mo><</mo> <mi>w</mi> <mo><</mo> <mn>1</mn> </mrow> </semantics></math>, for <math display='inline'><semantics> <mrow> <mo>∀</mo> <mn>0</mn> <mo><</mo> <mi>ϖ</mi> <mo><</mo> <mi>w</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mo>∀</mo> <msup> <mi>Y</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>ϖ</mi> <mi>Y</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mi>X</mi> </mrow> </semantics></math> (where <math display='inline'><semantics> <mrow> <mo>∀</mo> <msup> <mi>Y</mi> <mo>′</mo> </msup> <mo>≠</mo> <mi>X</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mo>∀</mo> <msup> <mi>Y</mi> <mo>′</mo> </msup> <mo>≠</mo> <mi>Y</mi> </mrow> </semantics></math>), <math display='inline'><semantics> <mrow> <mo>∃</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> </mrow> </semantics></math>, such that the expected payoff of the <span class='html-italic'>i</span>-th group when selecting a strategy <math display='inline'><semantics> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>Y</mi> <mi>i</mi> </msub> </mrow> </semantics></math>, while other groups select the strategy combination <math display='inline'><semantics> <mrow> <msub> <mi>X</mi> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msup> <mi>Y</mi> <mo>′</mo> </msup> <mo>,</mo> <mo>∀</mo> <msub> <mi>X</mi> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, satisfies <div class='html-disp-formula-info' id='FD2-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>></mo> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div></div></section><section id='sec3dot2dot4-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 3.2.4. Asymptotically Stable Equilibrium Point</h4><div class='html-p'>Set as mixing strategies <math display='inline'><semantics> <mrow> <mi>M</mi> <mo>,</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> <mo>∈</mo> <mi>δ</mi> </mrow> </semantics></math>, if <math display='inline'><semantics> <mrow> <msup> <mi>M</mi> <mo>*</mo> </msup> </mrow> </semantics></math> is an evolutionary stable equilibrium strategy, then the following conditions must be satisfied for this mixed strategy <math display='inline'><semantics> <mrow> <msup> <mi>M</mi> <mo>*</mo> </msup> </mrow> </semantics></math>. The equilibrium condition is that there is <math display='inline'><semantics> <mrow> <mo>∀</mo> <mi>M</mi> <mo>∈</mo> <mi>δ</mi> </mrow> </semantics></math>, then <math display='inline'><semantics> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> <mo>,</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> <mo stretchy="false">)</mo> </mrow> </semantics></math>. The stabilization condition is that if <math display='inline'><semantics> <mrow> <mo>∀</mo> <mi>M</mi> <mo>≠</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> <mo>,</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> <mo stretchy="false">)</mo> </mrow> </semantics></math>, then we can obtain <math display='inline'><semantics> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> <mo>,</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>. If these two conditions are met, the group state <math display='inline'><semantics> <mrow> <msup> <mi>P</mi> <mo>*</mo> </msup> <mo>=</mo> <msup> <mi>M</mi> <mo>*</mo> </msup> </mrow> </semantics></math> will finally be an asymptotically stable equilibrium for the evolutionary game.</div></section><section id='sec3dot2dot5-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 3.2.5. Determination of Evolutionary Stability</h4><div class='html-p'>Lyapunov stability theory is generally used to distinguish whether the internal equilibrium point of a system is in an evolutionary steady state or an asymptotic smooth point. The corresponding Jacobi matrix and its internal equilibrium point can be obtained according to the replicator dynamic equations of evolutionary game theory.</div><div class='html-p'>If the system is at an internal equilibrium point where the real parts of all eigenvalues of the corresponding Jacobi matrix are negative, i.e., when <span class='html-italic'>Det</span>(<b><span class='html-italic'>J</span></b>) > 0 and <span class='html-italic'>Tr</span>(<b><span class='html-italic'>J</span></b>) < 0 are satisfied, the equilibrium point is an evolutionarily stable equilibrium point of the system, and the system can obtain an ESS at this equilibrium point and is in a strict Nash equilibrium state. This equilibrium is an evolutionarily unstable internal equilibrium if all real parts are positive, i.e., when <span class='html-italic'>Det</span>(<b><span class='html-italic'>J</span></b>) > 0 and <span class='html-italic'>Tr</span>(<b><span class='html-italic'>J</span></b>) > 0 are satisfied. The equilibrium is the saddle point or center of the system if all real parts are positive, negative, or zero. The saddle point is at the threshold between evolutionary stability and instability but is still an unstable equilibrium. In summary, if the asymptotically stable equilibrium point of a dynamic model of the replicators in the system is the equilibrium point of the pure strategy case of the system, then it must be an ESS of the system and in a strict Nash equilibrium.</div></section></section></section><section id='sec4-mathematics-12-03249' type=''><h2 data-nested='1'> 4. Evolutionary Game-Theoretic Model</h2><div class='html-p'>This section provides a detailed analysis of symmetric and asymmetric evolutionary games, with a focus on a typical two-population asymmetric evolutionary game scenario. This section is crucial because it not only demonstrates the practical application of the theoretical concepts introduced in <a href="#sec3-mathematics-12-03249" class="html-sec">Section 3</a> but also showcases the significant advantages of evolutionary game theory in handling real-world, multi-stakeholder scenarios. The theoretical analysis and numerical simulation in this section serve as an illustrative example that supports the real-world case study in the next section. By doing so, it provides readers with a complete understanding of how evolutionary game theory can be applied to complex interactions in the energy market, particularly in demand-side management (DSM). In the tripartite system, the government’s payoffs are a function of system stability, energy savings, and public welfare benefits, while the grid companies’ payoffs depend on operational efficiency, cost savings, and incentive payouts. The users’ decisions are driven by direct financial benefits and long-term energy savings. Each player’s payoffs are interdependent, reflecting the asymmetric nature of the game.</div><section id='sec4dot1-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 4.1. Symmetric and Asymmetric Game Situations</h4><div class='html-p'>In a symmetric game, all participants know each other’s preference information, and there is no discrepancy; at this time, there is only one replicator dynamic equation. In contrast, in asymmetric games, the participants’ information about each other’s preferences is asymmetric and only partially known [<a href="#B48-mathematics-12-03249" class="html-bibr">48</a>], with discrepancies, at which point there are two replicator dynamic equations for the game system. In addition, it is also possible to determine symmetric and asymmetric based on the payment parameters of the Jacobi matrix; if the payment parameters are symmetric, it is a symmetric evolutionary game, and vice versa, it is an asymmetric evolutionary game. According to the reference [<a href="#B46-mathematics-12-03249" class="html-bibr">46</a>], the schematic structure of symmetric evolutionary game and asymmetric evolutionary game in the general case are shown in <a href="#mathematics-12-03249-f001" class="html-fig">Figure 1</a>a and <a href="#mathematics-12-03249-f001" class="html-fig">Figure 1</a>b, respectively.</div></section><section id='sec4dot2-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 4.2. A Two-Group Asymmetric Evolutionary Game Model for the General Case</h4><section id='sec4dot2dot1-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 4.2.1. Theoretical Analysis</h4><div class='html-p'>In a general two-population, two-strategy asymmetric evolutionary game (2P2S-AEG), the two populations can be represented by population A and population B, respectively. It is assumed that there are only two pure strategies in the strategy set of each population. Based on this, the strategy set of population A is set as <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">A</mi> </msub> <mo>=</mo> <mfenced close="}" open="{"> <mrow> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">A</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">A</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> and the strategy set of population B is set as <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">B</mi> </msub> <mo>=</mo> <mfenced close="}" open="{"> <mrow> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">B</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">B</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>. Here, strategies <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">A</mi> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">A</mi> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </semantics></math> are a pair of opposite pure strategies. For example, if the strategy chosen by an individual in population A is a cooperative strategy, it is represented by strategy <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">A</mi> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </semantics></math>, and the strategy is non-cooperation as denoted by <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">A</mi> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </semantics></math>. Similarly, strategies <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </semantics></math> should also be set as a pair of opposite strategies. In the multi-round game, assume that the variable <span class='html-italic'>x</span> is the proportion of individuals in population A adopting strategy <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">A</mi> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </semantics></math>, and <span class='html-italic'>y</span> is the proportion of individuals in population B adopting strategy <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </semantics></math>. Accordingly, 1 − <span class='html-italic'>x</span> and 1 − <span class='html-italic'>y</span> are the proportions of individuals in population A adopting strategy <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">A</mi> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </semantics></math> and individuals in population B adopting strategy <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </semantics></math>, respectively, where <span class='html-italic'>x</span>, <span class='html-italic'>y</span> ∈ [0, 1]. Therefore, the payoff matrix of the general 2P2S-AEG can be expressed as <div class='html-disp-formula-info' id='FD3-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd columnalign="right"> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd> <mrow/> </mtd> <mtd> <mrow> <mtext> </mtext> <mover> <mover> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd> <mrow> <mi>y</mi> </mrow> </mtd> <mtd> <mrow> <mn>1</mn> <mo>−</mo> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">B</mi> <mn>1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">B</mi> <mn>2</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo stretchy="true">︷</mo> </mover> <mrow> <mi>Group</mi> <mtext> </mtext> <mi mathvariant="normal">B</mi> </mrow> </mover> </mrow> </mtd> </mtr> </mtable> <mtext> </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>Group</mi> <mtext> </mtext> <mi mathvariant="normal">A</mi> <mfenced close="" open="{"> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mrow> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">A</mi> <mn>1</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">A</mi> <mn>2</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mfenced close="]" open="["> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mtext> </mtext> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mtext> </mtext> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo>,</mo> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd> <mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <mtext> </mtext> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div> where in this formula, <span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>, <span class='html-italic'>g</span>, and <span class='html-italic'>h</span>, respectively, represent the payoff parameters of population A and population B under different combinations of game strategies. The parameters used in the simulation represent key costs and benefits associated with the implementation of DSM strategies. Specifically the following:</div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'><span class='html-italic'>a</span> denotes the initial cost to the government for implementing the incentive strategy SG1;</div></li><li><div class='html-p'><span class='html-italic'>b</span> represents the long-term financial and social gains to the government from incentivizing DSM;</div></li><li><div class='html-p'><span class='html-italic'>c</span> reflects the operational costs incurred by the power grid for adopting strategy SPG1;</div></li><li><div class='html-p'><span class='html-italic'>d</span> is the revenue benefit to the power grid from successfully implementing DSM policies;</div></li><li><div class='html-p'><span class='html-italic'>e</span> and <span class='html-italic'>f</span> represent the changes in societal gains and system performance for the government and power grid, respectively, due to DSM’s success.</div></li></ul></div><div class='html-p'>These parameters were selected to simulate real-world trade-offs in DSM adoption, allowing for the observation of both short-term instability and long-term equilibrium states. From this, the following replicator dynamics equation can be obtained as <div class='html-disp-formula-info' id='FD4-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mrow> <mo>=</mo> </mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>γ</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mn>4</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> where the variables in (4) are represented as <div class='html-disp-formula-info' id='FD5-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>c</mi> <mo>−</mo> <mi>g</mi> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>γ</mi> <mn>3</mn> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>γ</mi> <mn>4</mn> </msub> <mo>=</mo> <mi>f</mi> <mo>−</mo> <mi>h</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div></div><div class='html-p'>From Equation (5), the corresponding Jacobian matrix <b><span class='html-italic'>J</span></b> can be obtained as <div class='html-disp-formula-info' id='FD6-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="bold-italic">J</mi> <mo>=</mo> <mfenced close="]" open="["> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>γ</mi> <mn>5</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>γ</mi> <mn>3</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>γ</mi> <mn>6</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div></div><div class='html-p'>Its determinant Det(<b><span class='html-italic'>J</span></b>) and trace Tr(<b><span class='html-italic'>J</span></b>) are calculated as <div class='html-disp-formula-info' id='FD7-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mtable> <mtr> <mtd columnalign="left"> <mi>Det</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold-italic">J</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>γ</mi> <mn>5</mn> </msub> <msub> <mi>γ</mi> <mn>6</mn> </msub> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <msub> <mi>γ</mi> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>Tr</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold-italic">J</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>γ</mi> <mn>5</mn> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>γ</mi> <mn>6</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div> where the variables in (7) are represented by <div class='html-disp-formula-info' id='FD8-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mtable> <mtr> <mtd> <msub> <mi>γ</mi> <mn>5</mn> </msub> <mo>=</mo> <mi>y</mi> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>γ</mi> <mn>6</mn> </msub> <mo>=</mo> <mi>x</mi> <msub> <mi>γ</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mn>4</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div></div><div class='html-p'>According to the formula in (4), the internal equilibrium points of the 2P2S-AEG system can be obtained and denoted as E<sub>1</sub>(0, 0), E<sub>2</sub>(0, 1), E<sub>3</sub>(1, 0), E<sub>4</sub>(1, 1), and E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>y</span>*). Among them, the internal equilibrium point E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>y</span>*) cannot always be a long-term evolutionarily stable equilibrium state point, and it is expressed as <div class='html-disp-formula-info' id='FD9-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mtable> <mtr> <mtd> <msup> <mi>x</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>γ</mi> <mn>4</mn> </msub> </mrow> <mrow> <msub> <mi>γ</mi> <mn>3</mn> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>h</mi> <mo>−</mo> <mi>f</mi> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>f</mi> <mo>−</mo> <mi>d</mi> <mo>+</mo> <mi>h</mi> </mrow> </mfrac> </mstyle> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>γ</mi> <mn>1</mn> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>g</mi> <mo>−</mo> <mi>c</mi> </mrow> <mrow> <mi>a</mi> <mo>−</mo> <mi>c</mi> <mo>−</mo> <mi>e</mi> <mo>+</mo> <mi>g</mi> </mrow> </mfrac> </mstyle> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div></div><div class='html-p'>According to the evolutionary stability criterion of Lyapunov’s method, the internal equilibrium points of the system can be classified into three cases based on the determinant and trace of the Jacobian matrix: asymptotically stable points, evolutionarily unstable points, and saddle points. By calculating the determinant and trace values of the aforementioned internal equilibrium points, it can be seen that the final result of the general 2P2S-AEG is determined by the initial payment parameters of the system, with specific results shown in <a href="#mathematics-12-03249-t001" class="html-table">Table 1</a> below.</div><div class='html-p'><a href="#mathematics-12-03249-t001" class="html-table">Table 1</a> indicates that the general two-population asymmetric evolutionary game model has the following characteristics. First, each pure strategy evolutionary equilibrium point corresponds to two mutually exclusive singular points. For example, when the system’s equilibrium points E<sub>1</sub> or E<sub>4</sub> are asymptotically stable points, the corresponding evolutionary stability conditions in <a href="#mathematics-12-03249-t001" class="html-table">Table 1</a> must be satisfied. Since the evolutionary stability conditions of E<sub>1</sub> and E<sub>4</sub> are mutually exclusive to those of E<sub>2</sub> and E<sub>3</sub>, then E<sub>2</sub> and E<sub>3</sub> must be evolutionary unstable equilibrium points, and the system is in an unstable state at E<sub>2</sub> and E<sub>3</sub>, and vice versa. At this time, E<sub>1</sub>, E<sub>4</sub>, and E<sub>2</sub>, E<sub>3</sub> can be referred to as two pairs of mutually exclusive equilibrium points. Therefore, in general, a two-group asymmetric evolutionary game system can have at most two pure strategy evolutionary stable equilibrium points simultaneously. Second, this 2P2S-AEG system can potentially reach an evolutionary stable state at internal equilibrium points E<sub>1</sub>(0, 0), E<sub>2</sub>(0, 1), E<sub>3</sub>(1, 0), and E<sub>4</sub>(1, 1), and can achieve evolutionary stable strategies, which are also strict Nash equilibrium strategies. Third, the internal equilibrium E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>y</span>*) is a saddle point of the system, and the system can never reach an evolutionary stable equilibrium state at E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>y</span>*), remaining in a critically stable state.</div></section><section id='sec4dot2dot2-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 4.2.2. Numerical Simulation</h4><div class='html-p'>Now set the payment distribution parameters (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>, <span class='html-italic'>g</span>, <span class='html-italic'>h</span>) = (7, 1, 5, 6, 8, 4, 2, 3) in the general 2P2S-AEG model and initialize the values of <span class='html-italic'>x</span> and <span class='html-italic'>y</span> at intervals of 1/10, 1/20, 1/30, 1/40, 1/50, and 1/60, respectively, on the decision space of the system of [0, 1] × [0, 1], so that <span class='html-italic'>x</span> and <span class='html-italic'>y</span> are randomized from 0 to 1. The simulation is carried out at different initial game postures (denoted as Case 1 to Case 6, respectively), and the simulated dynamic evolutionary trends or paths of <span class='html-italic'>x</span>, <span class='html-italic'>y,</span> and (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>) at time <span class='html-italic'>t</span> ∈ [0, 10] can be obtained, and the results are shown in <a href="#mathematics-12-03249-f002" class="html-fig">Figure 2</a>, where the green, blue, and red points represent an evolutionarily stable, unstable, and saddle point in the long-term evolution game process, respectively. In the simulations, <span class='html-italic'>x</span> represents the proportion of individuals in the government adopting strategy S<sub>G1</sub>, while <span class='html-italic'>y</span> denotes the proportion of individuals in the power grid adopting strategy S<sub>PG1</sub>. Both <span class='html-italic'>x</span> and <span class='html-italic'>y</span> were initialized within the interval [0, 1] and were incremented by 1/50. This approach allows for the exploration of various initial conditions, ensuring that the system’s evolution can be tracked from multiple starting points. This setup highlights the asymmetric dynamics between the two parties, as changes in <span class='html-italic'>x</span> and <span class='html-italic'>y</span> directly affect the other’s strategy through the payoff structure.</div><div class='html-p'>In <a href="#mathematics-12-03249-f002" class="html-fig">Figure 2</a>c, the green dots indicate the evolutionarily stable equilibrium points, the blue dots indicate the unstable equilibrium points of the system, and the red dots indicate the saddle points of the system. According to the simulation results, it can be seen that the phase trajectories of the system converge to the equilibrium points E<sub>2</sub>(0, 1) and E<sub>3</sub>(1, 0) in the decision space [0, 1] × [0, 1], respectively, then the system spontaneously forms the evolutionary stable equilibrium points at E<sub>2</sub>(0, 1) and E<sub>3</sub>(1, 0), and finally obtains the evolutionary stable strategy. Since this point is a pure strategy evolutionarily stable equilibrium, it is also a strictly refined Nash equilibrium. The system can achieve an evolutionarily unstable equilibrium at E<sub>1</sub>(0, 0) and E<sub>4</sub>(1, 1), so they are unstable equilibrium points of the system. Besides, this 2P2S-AEG system is always critically stable at E<sub>5</sub>(1/6, 3/4), i.e., the equilibrium point E<sub>5</sub> is a system saddle point and remains an unstable evolutionary state during the long-term evolution process.</div></section></section></section><section id='sec5-mathematics-12-03249' type=''><h2 data-nested='1'> 5. The Government-Power Grid-User Tripartite Evolution Game Model</h2><section id='sec5dot1-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 5.1. Analysis of the Game Relationship between Government, Power Grid, and Users</h4><div class='html-p'>The aim of this paper is to thoroughly investigate the dynamic evolutionary game behaviors and strategies of government departments, power grid companies, and power users under the incentive process of DSM policies. This game situation is modeled as a government-power grid-user tripartite evolution game, abbreviated as GPU-EG. Based on this, this section develops a tripartite evolutionary game involving the government, grid companies, and users. The strategies available to each party are clearly defined, and the dynamics of their interactions are explored using replicator dynamics. This study focuses on a tripartite game involving the government, power grid companies, and users. The government is positioned as the dominant player and is responsible for designing DSM policies. Grid companies can choose whether to implement these policies, while users can decide whether to participate. The interactions among these three groups form the basis for the model. The dynamic evolution of strategies between the government, grid companies, and users is modeled using a system of replicator dynamics equations. Each player’s strategy evolves over time based on their payoffs, which are influenced by the actions of the other two stakeholders. The government’s incentives create feedback loops with both the grid companies and users, who adjust their behavior accordingly. The stability of each equilibrium point depends on the alignment of these strategies.</div><div class='html-p'>Incorporating insights from Carbonaro and Menale (2024) [<a href="#B54-mathematics-12-03249" class="html-bibr">54</a>], we also recognize the potential application of Markov chains and kinetic theory in analyzing dynamic socio-economic models. In their work, Carbonaro and Menale [<a href="#B54-mathematics-12-03249" class="html-bibr">54</a>] explore how Markovian processes can be applied to model evolving systems with socio-economic implications. In line with their approach, our study similarly models the evolution of strategies in demand-side management (DSM) as a dynamic process influenced by the decisions of multiple stakeholders. The inclusion of evolutionary game theory in our work allows for a nuanced understanding of strategy evolution in a context where rationality is bounded, and decisions are made over time. By framing DSM strategies within this kinetic and evolutionary context, we identify stable equilibrium points that reflect long-term cooperation among government, power grid enterprises, and power users, further supporting the utility of game theory in socio-economic models.</div><div class='html-p'>In this tripartite evolutionary game model, the government sector is the dominant player in promoting the development of DSM. The government’s strategic choices involve either incentivizing or refraining from incentivizing DSM policies, while the power grid enterprises face the decision of whether to implement these policies, and the power users decide on their level of participation in DSM services. The game framework reflects dynamic strategy adjustments made by each participant over time in response to the evolving decisions of the other two players. The objective of this game is to identify evolutionarily stable strategies (ESS) that maximize the collective benefits of DSM implementation across the three groups, ensuring an optimal balance of interests.</div><div class='html-p'>The payoffs for each party depend on the strategies chosen by the others. The government, for example, gains from increased energy efficiency and reduced environmental impact when DSM is successfully implemented but incurs costs associated with policy incentives. Similarly, power grid companies benefit from operational stability and reduced peak loads but must consider the implementation costs. Power users weigh the financial savings from DSM against the upfront cost of participation. This interdependence of payoffs defines the strategic complexity of the evolutionary game presented in <a href="#mathematics-12-03249-f003" class="html-fig">Figure 3</a>.</div><div class='html-p'>At the same time, government departments also need to pay attention to the degree of acceptance of electricity demand-side management by electricity users. To be truly effective, the demand-side management policy inevitably requires the cooperation of electricity users. Government departments can let power users understand that the demand-side management policy will bring them direct benefits, including savings in electricity costs and a cleaner electricity environment. At the same time, in order to increase the participation of power users, government departments can take measures such as providing concessions to incentivize power users. Due to the different interests of the two, there are many incentive games between government departments and electricity users until a win-win strategy is determined.</div><div class='html-p'>As a result, the government is the dominant party in the incentive mechanism, which incentivizes the grid company and the power users to promote the practice of demand-side management. Based on this relationship, this paper will study and analyze the dynamic evolution of the game behavior of the government-grid evolution game system and the government-users evolution game system, respectively. Based on this, this paper assumes that in the demand-side management of the electricity market, the government, the power grid company, and the electricity users adopt the following strategies: the government can choose whether to incentivize DSM policies, denoted as “incentivize DSM policy” as strategy S<sub>G1</sub>, and “do not incentivize DSM policy” as strategy S<sub>G2</sub>, then its strategy set is {S<sub>G1</sub>, S<sub>G2</sub>}. Similarly, the power grid company’s strategy set is {S<sub>PG1</sub>, S<sub>PG2</sub>}, where strategy S<sub>PG1</sub> means that the power grid implements DSM policies, and strategy S<sub>PG2</sub> means that the power grid does not implement DSM policies. For the power user’s strategy, the strategy of accepting DSM services is denoted as S<sub>EC1</sub>, and the strategy of not accepting DSM services is denoted as S<sub>EC2</sub>, and its strategy set is {S<sub>EC1</sub>, S<sub>EC2</sub>}. Therefore, the game relationship framework among the government, the power grid company, and the power users can be constructed as shown in <a href="#mathematics-12-03249-f003" class="html-fig">Figure 3</a>.</div><div class='html-p'>In the game pattern shown in <a href="#mathematics-12-03249-f003" class="html-fig">Figure 3</a>, the benefits that can be obtained from implementing DSM policies include policy benefits, system benefits, user electricity benefits, and social environmental benefits. The system benefits refer to the improvements in power supply stability achieved by the grid company through the implementation of DSM projects, effectively promoting peak shaving and valley filling, extending the lifespan of power equipment, and optimizing the balance of power supply and demand. The user electricity benefits are the reductions in electricity costs and improvements in the electricity environment that power users experience by utilizing DSM services. The social environmental benefits refer to the optimization of electricity resource allocation, improvements in the electricity efficiency of end users, and energy conservation and emission reduction resulting from the implementation of DSM policies. The policy benefits are financial subsidies provided by the government to grid companies implementing DSM projects, as well as tax reductions to offset grid cost expenditures. For the power users, the government offers financial subsidies and loan incentives based on the efficient energy-saving equipment purchased by users.</div><div class='html-p'>Despite the importance of DSM, there remains a gap in the literature regarding the application of evolutionary game theory to model these multi-stakeholder dynamics. Recent studies, such as Carbonaro and Menale (2024) [<a href="#B54-mathematics-12-03249" class="html-bibr">54</a>], explore the application of Markov chains and kinetic theory in socio-economic models, further supporting the idea that dynamic, evolving systems can be effectively modeled using such approaches. This insight aligns with our use of evolutionary game theory to capture the ongoing interactions and strategic decision-making processes in DSM.</div></section><section id='sec5dot2-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 5.2. Government-Power Grid Evolution Game Model</h4><section id='sec5dot2dot1-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 5.2.1. Model Construction</h4><div class='html-p'>The game process between the government and the power grid enterprises regarding DSM is a two-population, two-strategy asymmetric evolutionary game (2P2S-AEG) process. In this two-party evolution game model, the government contains two pure strategies, i.e., S<sub>G1</sub> and S<sub>G2</sub>, which represent that the government tends to adopt and tends not to adopt incentive strategies for DSM, respectively. Thus, the strategy set of the government is denoted by S<sub>gove</sub> = {S<sub>G1</sub>, S<sub>G2</sub>}. Similarly, the strategy set of the power grid enterprises is shown as S<sub>pgen</sub> = {S<sub>PG1</sub>, S<sub>PG2</sub>}, where the first and second strategies represent that power grid enterprises tend to implement and tend not to implement DSM in the demand-side electricity market, respectively. Based on this, we assume in this paper that the parameters of <math display='inline'><semantics> <mi>α</mi> </semantics></math> represents cost, <math display='inline'><semantics> <mi>β</mi> </semantics></math> represents revenue, and <math display='inline'><semantics> <mi>η</mi> </semantics></math> represents loss. Then, the parameters used in this paper can be summarized as demonstrated in <a href="#mathematics-12-03249-t002" class="html-table">Table 2</a>.</div><div class='html-p'>As demonstrated in <a href="#mathematics-12-03249-t002" class="html-table">Table 2</a>, under the condition of implementing DSM in the power grid enterprises, when the government incentivizes the power grid companies, the benefits obtained will inevitably be greater than those brought by not providing incentives. Thus, we can obtain <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>></mo> <msub> <mi>β</mi> <mn>3</mn> </msub> </mrow> </semantics></math>. Assuming the probability of the government adopting strategy S<sub>G1</sub> is <span class='html-italic'>x</span>, then the probability of adopting strategy S<sub>G2</sub> is 1 − <span class='html-italic'>x</span>, where <math display='inline'><semantics> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math>. Similarly, the probability of the power grid enterprises choosing strategy S<sub>PG1</sub> is <span class='html-italic'>y</span>, and the probability of choosing strategy S<sub>PG2</sub> is 1 − <span class='html-italic'>y</span>, where <math display='inline'><semantics> <mrow> <mi>y</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math>. Based on this, the government-power grid evolution game’s payoff distribution matrix can be obtained as follows. <div class='html-disp-formula-info' id='FD10-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mover> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>y</mi> </mrow> </mtd> <mtd> <mrow> <mtext> </mtext> <mn>1</mn> <mo>−</mo> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext> </mtext> <msub> <mi mathvariant="normal">S</mi> <mrow> <mrow> <mi>PG</mi> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtext> </mtext> <msub> <mi mathvariant="normal">S</mi> <mrow> <mrow> <mi>PG</mi> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo stretchy="true">︷</mo> </mover> <mrow> <mrow> <mi>The</mi> <mtext> </mtext> <mi>power</mi> <mtext> </mtext> <mi>grid</mi> <mtext> </mtext> <mi>enterprises</mi> </mrow> </mrow> </mover> <mtext> </mtext> </mrow> </mtd> <mtd> <mrow/> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>The</mi> <mtext> </mtext> <mi>government</mi> <mfenced close="" open="{"> <mrow> <mtable> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mrow> <msub> <mi mathvariant="normal">S</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi mathvariant="normal">S</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mfenced close="]" open="["> <mrow> <mtable> <mtr> <mtd> <mrow> <mfenced> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </mfenced> </mrow> </mtd> <mtd> <mrow> <mfenced> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>η</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>4</mn> </msub> </mrow> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfenced> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </mfenced> </mrow> </mtd> <mtd> <mrow> <mfenced> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>η</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>4</mn> </msub> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div></div><div class='html-p'>Based on (10), assuming that the expected benefit when the government sector chooses to adopt the incentive strategy S<sub>G1</sub> is <span class='html-italic'>U</span><sub>SG1</sub>, and relatively, the expected benefit when the government sector chooses not to adopt the incentive strategy S<sub>G2</sub> is <span class='html-italic'>U</span><sub>SG2</sub>, and then the average benefit when the government partially adopts these two pure strategies can be obtained, denoted by <span class='html-italic'>U</span><sub>G-AVE</sub>. They are calculated as follows. <div class='html-disp-formula-info' id='FD11-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>U</mi> <mrow> <mrow> <mi>SG</mi> <mn>1</mn> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>η</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>U</mi> <mrow> <mrow> <mi>SG</mi> <mn>2</mn> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>+</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>η</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>U</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>AVE</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>+</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>η</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div></div><div class='html-p'>Based on (11), the corresponding replicator dynamics equation for the government is obtained as <div class='html-disp-formula-info' id='FD12-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div></div><div class='html-p'>We take <math display='inline'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, and then we can obtain <math display='inline'><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>y</mi> <mo>*</mo> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>, i.e., we can obtain two internal equilibrium points: <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> </mrow> </mfrac> </mstyle> <mo stretchy="false">)</mo> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> </mrow> </mfrac> </mstyle> <mo stretchy="false">)</mo> </mrow> </semantics></math>. From this, we can obtain that when <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics></math>, it indicates that the proportion of individuals in the government group adopting the incentive strategy S<sub>G1</sub> is asymptotically stable, i.e., a long-term evolutionarily stable state. Conversely, when <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo><</mo> <mn>0</mn> </mrow> </semantics></math>, the incentive strategy S<sub>G2</sub> adopted by the government will be an evolutionarily unstable equilibrium strategy. Furthermore, we assume that the expected benefit when the power grid company chooses to adopt the incentive strategy S<sub>PG1</sub> is <span class='html-italic'>U</span><sub>SPG1</sub>, and relatively, the expected benefit when the power grid company chooses not to adopt the incentive strategy S<sub>PG2</sub> is <span class='html-italic'>U</span><sub>SPG2</sub>, and then the average benefit when the power grid enterprise partially adopts these two pure strategies can be obtained, denoted by <span class='html-italic'>U</span><sub>PG-AVE</sub>. The calculation results are <div class='html-disp-formula-info' id='FD13-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>U</mi> <mrow> <mrow> <mi>SPG</mi> <mn>1</mn> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>β</mi> <mn>4</mn> </msub> <mo>+</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>U</mi> <mrow> <mrow> <mi>SPG</mi> <mn>2</mn> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>β</mi> <mn>4</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>U</mi> <mrow> <mrow> <mi>PG</mi> <mtext>-</mtext> <mi>AVE</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>β</mi> <mn>4</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(13)</label> </div> </div></div><div class='html-p'>Based on (13), the corresponding replicator dynamics equation for the power grid enterprises is obtained as <div class='html-disp-formula-info' id='FD14-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(14)</label> </div> </div></div><div class='html-p'>We take <math display='inline'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, and then we can obtain <math display='inline'><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>x</mi> <mo>*</mo> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>5</mn> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>, i.e., we can obtain two internal equilibrium points: <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>5</mn> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mstyle> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>5</mn> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mstyle> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>. Thus, combining the two replicator dynamics equations shown in (12) and (14), we can obtain the final replicator dynamics equations of the government-power grid two-party evolution game system can be obtained as follows. <div class='html-disp-formula-info' id='FD15-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mtable columnalign="left"> <mtr> <mtd> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(15)</label> </div> </div></div><div class='html-p'>Based on (15), we can further obtain the Jacobian matrix of this evolution game system, denoted by <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PG</mi> </mrow> </mrow> </msub> </mrow> </semantics></math>, as well as the corresponding determinant and trace of this Jacobian matrix, denoted by <math display='inline'><semantics> <mrow> <mi>D</mi> <mi>e</mi> <mi>t</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PG</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>t</mi> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PG</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math>, respectively, which are calculated as follows. <div class='html-disp-formula-info' id='FD16-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mtable> <mtr> <mtd> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PG</mi> </mrow> </mrow> </msub> <mo>=</mo> <mfenced close="" open="["> <mtable> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced close="]" open=""> <mtable> <mtr> <mtd> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mi>D</mi> <mi>e</mi> <mi>t</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PG</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mtext> </mtext> <mo>−</mo> <mi>x</mi> <mi>y</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PG</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(16)</label> </div> </div></div><div class='html-p'>As seen from formula (16), the internal equilibrium points of the government-power grid two-party evolution game system within the decision-making space of <math display='inline'><semantics> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math> are calculated as E<sub>1</sub>(0, 0), E<sub>2</sub>(0, 1), E<sub>3</sub>(1, 0), E<sub>4</sub>(1, 1), and E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>y</span>*), where <math display='inline'><semantics> <mrow> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msup> <mi>y</mi> <mo>∗</mo> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>. Based on this, the <math display='inline'><semantics> <mrow> <mi>D</mi> <mi>e</mi> <mi>t</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PG</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>t</mi> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PG</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> in (16) can be calculated as shown in <a href="#mathematics-12-03249-t003" class="html-table">Table 3</a>.</div><div class='html-p'>From <a href="#mathematics-12-03249-t003" class="html-table">Table 3</a> and the Lyapunov’s stability theory, it can be seen that the internal equilibrium point E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>y</span>*) is a saddle point, and the asymptotically stability of the other four pure strategy equilibrium points is related to the size relationships between the parameters shown in <a href="#mathematics-12-03249-t002" class="html-table">Table 2</a>. Based on this, the evolutionary stability conditions of this government-power grid two-party evolution game system at the internal equilibrium point and the mutually exclusive equilibrium points are shown in <a href="#mathematics-12-03249-t004" class="html-table">Table 4</a>. As shown in <a href="#mathematics-12-03249-t004" class="html-table">Table 4</a>, it can be seen that the long-term evolutionarily stable equilibrium characteristics of each pure strategy equilibrium point are strictly related to the signs of <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; thus, there are a total of 6 cases that can be discussed, which will be explained in the next subsection.</div></section><section id='sec5dot2dot2-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 5.2.2. Dynamic Numerical Simulation</h4><div class='html-p'>For the sake of discussion, the payoff distribution parameters are set as <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>a</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>b</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>=</mo> <mi>c</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>=</mo> <mi>d</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>=</mo> <mi>e</mi> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>f</mi> </mrow> </semantics></math>, where <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>></mo> <mi>c</mi> </mrow> </semantics></math>, and the saddle point can be expressed as <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="normal">E</mi> <mn>5</mn> </msub> <mo stretchy="false">(</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>f</mi> <mo>−</mo> <mi>e</mi> </mrow> <mi>d</mi> </mfrac> </mstyle> <mo>,</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>b</mi> <mrow> <mi>a</mi> <mo>−</mo> <mi>c</mi> </mrow> </mfrac> </mstyle> <mo stretchy="false">)</mo> </mrow> </semantics></math>. Based on this, we set the system decision space as [0, 1] × [1, 0], so that the initial values <span class='html-italic'>x</span> and <span class='html-italic'>y</span> are taken from 0 to 1, with 1/50 as the interval, a total of 2601 rounds of dynamic evolution game simulations with different initial value scenarios are carried out for verifying the results of the aforementioned theoretical analyses, and the following six kinds of results can be obtained, as demonstrated in <a href="#mathematics-12-03249-f004" class="html-fig">Figure 4</a>, in which Case 1 represents the evolutionary trend of the proportion of individuals <span class='html-italic'>x</span> adopting strategy S<sub>G1</sub> in the governmental group as a result of the change of the strategy with time <span class='html-italic'>t</span> evolution trend, Case 2 represents the evolution trend of the proportion of individuals <span class='html-italic'>y</span> adopting strategy S<sub>PG1</sub> in the grid group over time <span class='html-italic'>t</span>, and Case 3 represents the evolution trend of (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>) developing over time <span class='html-italic'>t</span>. Besides, in <a href="#mathematics-12-03249-f004" class="html-fig">Figure 4</a>, the green dots represent the stable equilibrium point of evolution, the blue dots represent the unstable equilibrium point of the system, and the red dots represent the saddle point of the system. Note that all the simulation result plots in this paper take this representation.</div><div class='html-p'>The simulation results demonstrated In <a href="#mathematics-12-03249-f004" class="html-fig">Figure 4</a> are analyzed as follows.</div><div class='html-p'>In <a href="#mathematics-12-03249-f004" class="html-fig">Figure 4</a>a, the government-power grid two-party asymmetric evolution game system converges at equilibrium points E<sub>1</sub>(0, 0) and E<sub>4</sub>(1, 1), where E<sub>1</sub> and E<sub>4</sub> are asymptotically stable equilibrium points as pure evolution game strategies. Thus, the system spontaneously forms a stable evolution strategy combination, ultimately reaching a long-term evolutionary stable equilibrium state, which is also a highly refined Nash equilibrium state. In addition, the system evolves in an evolutionarily unstable equilibrium state at points E<sub>2</sub>(0, 1) and E<sub>3</sub>(1, 0), making E<sub>2</sub> and E<sub>3</sub> become unstable equilibrium points. The system remains in a critical equilibrium state at equilibrium point E<sub>5</sub>(2/7, 1/3), which is still an unstable evolutionary equilibrium state, meaning this point is a saddle point. In this case, the total revenue obtained by the government via adopting strategy S<sub>G1</sub> is higher than its expenditure costs, while at the same time, the relationship between the revenue and expenditure costs obtained by the power grid enterprises from implementing strategy S<sub>PG1</sub> is uncertain. Assuming that when the grid company receives a relatively high benefit under strategy S<sub>PG1</sub> or spends a relatively low cost, the area enclosed by the four equilibrium points of E<sub>2</sub>, E<sub>5</sub>, E<sub>3</sub>, and E<sub>4</sub> takes up a larger proportion of the decision space. At this time, the equilibrium state of the system will converge to the equilibrium point E<sub>4</sub>, which means that the power grid company will tend to adopt strategy S<sub>PG1</sub>, while the government will adopt strategy S<sub>G1</sub>. Besides, if the power grid company receives less profit or incurs higher costs from adopting strategy S<sub>PG1</sub>, the area enclosed by the four equilibrium points E<sub>1</sub>, E<sub>2</sub>, E<sub>5</sub>, and E<sub>3</sub> will occupy a larger proportion of the entire decision space [0, 1] × [0, 1]. The system’s equilibrium state will converge to the equilibrium point E<sub>1</sub>. In this case, the power grid company will tend to choose strategy S<sub>PG2</sub>, and the government will adopt strategy S<sub>G2</sub>. In addition, in this case, the government needs to formulate more favorable incentive measures to reduce the power grid’s costs or increase the system’s revenue. This can be achieved by appropriately adjusting the position of the saddle point, effectively increasing the probability of the power grid company choosing the S<sub>PG1</sub> strategy.</div><div class='html-p'>In <a href="#mathematics-12-03249-f004" class="html-fig">Figure 4</a>b,c, the system has only one evolutionary stable equilibrium point, E<sub>1</sub>, and the system will achieve an evolutionary stable state at this point. At this time, the cost of implementing strategy S<sub>PG1</sub> for the power grid is higher than the sum of the system revenue and policy revenue. No matter which strategy is chosen by the government, the power grid enterprises will tend to choose strategy S<sub>PG2</sub>. At this time, the power grid’s behavior will affect the government’s tendency to adopt policy S<sub>G2</sub>.</div><div class='html-p'>In <a href="#mathematics-12-03249-f004" class="html-fig">Figure 4</a>d,e, the system’s evolutionary stable equilibrium state converges only to the equilibrium point E<sub>1</sub>, where it achieves an evolutionary stable strategy. When adopting strategy S<sub>G1</sub>, the return to the government is relatively low, and the relationship between the costs required for adopting strategy S<sub>PG1</sub> and the benefits obtained is uncertain; the long-term evolutionary stable equilibrium state of the system will converge to the equilibrium point E<sub>1</sub>, and the evolutionary stable equilibrium strategies formed in the government-power grid two-party evolution game system will ultimately be the strategies of S<sub>G2</sub> and S<sub>PG2</sub>.</div><div class='html-p'>In <a href="#mathematics-12-03249-f004" class="html-fig">Figure 4</a>f, the system’s equilibrium state converges to the equilibrium point E<sub>4</sub>, which is an ESS point. In this case, the points E<sub>1</sub>, E<sub>2</sub>, and E<sub>3</sub> are all evolutionarily unstable equilibrium points for the system. The corresponding actual situation is that the returns of strategies SG1 and SPG1 are relatively high, with lower expenditures. The game results reveal that the government incentivizes DSM, and the power grid implements DSM, at which point the government’s incentives are effective.</div><div class='html-p'>Overall, the simulation results demonstrate that when the government offers substantial incentives, grid companies are more likely to adopt DSM practices, leading to long-term stability. However, if the incentives are insufficient or the costs are too high, grid companies may choose not to participate, leading to a failure in policy implementation. This underscores the importance of carefully calibrated incentives that align the interests of all stakeholders.</div></section></section><section id='sec5dot3-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 5.3. Government-Power User Evolution Game Model</h4><section id='sec5dot3dot1-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 5.3.1. Model Construction</h4><div class='html-p'>Similarly, this paper still adopts the variables of <math display='inline'><semantics> <mrow> <mi>α</mi> <mo>,</mo> <mtext> </mtext> <mi>β</mi> <mo>,</mo> <mtext> </mtext> <mi>η</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mi>κ</mi> </semantics></math> to represent the cost, profit, social gain, and user expenditure, respectively. Based on this, the main parameters used in the government-power user two-party evolution game model are demonstrated in <a href="#mathematics-12-03249-t005" class="html-table">Table 5</a>.</div><div class='html-p'>The mathematical impact of the government’s decisions is captured through the replicator dynamics equations, which describe how the proportion of individuals (or institutions) adopting a particular strategy evolves over time. In this two-party evolutionary game between the government and power users, the government’s choice of incentives (S<sub>G1</sub> or S<sub>G2</sub>) directly influences the stability of the system. When the government adopts the strategy S<sub>G1</sub> (incentivizing DSM), it increases the likelihood of reaching an Evolutionary Stable Strategy (ESS), where no party has an incentive to unilaterally deviate. This ESS is confirmed through the Jacobian analysis, where the system converges to stable equilibrium points (such as E<sub>1</sub> or E<sub>4</sub>), representing long-term stability.</div><div class='html-p'>Mathematically, the stability conditions are derived from the determinant and trace of the Jacobian matrix associated with the system. For example, when the government incentivizes power users, the system exhibits an asymptotically stable equilibrium (E<sub>4</sub>), which means that both the government and power users find it in their best interest to continue adopting the strategies S<sub>G1</sub> and S<sub>PU1</sub>. In contrast, if the government does not incentivize DSM (S<sub>G2</sub>), the system may converge to a less favorable equilibrium (E<sub>1</sub> or E<sub>3</sub>) or remain at an unstable saddle point (E<sub>5</sub>), depending on the payoffs and relative costs of the strategies.</div><div class='html-p'>In summary, the government’s decisions have a direct impact on the stability of the system, and when the system is stable in a dynamic sense, this corresponds to an Evolutionary Stable Strategy (ESS) where neither the government nor the power users benefit from changing their strategies unilaterally. This provides a Nash equilibrium in the evolutionary game.</div><div class='html-p'>As demonstrated in <a href="#mathematics-12-03249-t005" class="html-table">Table 5</a>, due to the condition that electricity users utilize DSM services, the government’s gains from incentivizing users are necessarily greater than the gains without incentivization, so <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>7</mn> </msub> </mrow> </semantics></math> is greater than <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>8</mn> </msub> </mrow> </semantics></math>. Based on this, we assume that the probability of the government adopting strategy S<sub>G1</sub> is <span class='html-italic'>x</span>, then the probability of adopting strategy S<sub>G2</sub> is 1 – <span class='html-italic'>x</span>, where <math display='inline'><semantics> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math>. Similarly, the probability of the power users choosing strategy S<sub>PU1</sub> is <span class='html-italic'>z</span>, and the probability of choosing strategy S<sub>PU2</sub> is 1 – <span class='html-italic'>z</span>, where <math display='inline'><semantics> <mrow> <mi>z</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math>. To this end, the government-power user two-party evolution game’s payoff distribution matrix can be obtained as follows. <div class='html-disp-formula-info' id='FD17-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mover> <mrow> <mtext> </mtext> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd> <mrow> <mi>z</mi> </mrow> </mtd> <mtd> <mrow> <mtext> </mtext> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext> </mtext> <msub> <mi mathvariant="normal">S</mi> <mrow> <mrow> <mi>PU</mi> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtext> </mtext> <msub> <mi mathvariant="normal">S</mi> <mrow> <mrow> <mi>PU</mi> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo stretchy="true">︷</mo> </mover> <mrow> <mrow> <mi>The</mi> <mtext> </mtext> <mi>power</mi> <mtext> </mtext> <mi>users</mi> </mrow> </mrow> </mover> </mrow> </mtd> <mtd> <mrow/> </mtd> </mtr> </mtable> <mtext> </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>The</mi> <mtext> </mtext> <mi>government</mi> <mfenced close="" open="{"> <mrow> <mtable> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mrow> <msub> <mi mathvariant="normal">S</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi mathvariant="normal">S</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mfenced close="]" open="["> <mrow> <mtable> <mtr> <mtd> <mrow> <mfenced> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>κ</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> </mrow> </mfenced> </mrow> </mtd> <mtd> <mrow> <mfenced> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>η</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>κ</mi> <mn>1</mn> </msub> </mrow> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfenced> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>κ</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> </mrow> </mfenced> </mrow> </mtd> <mtd> <mrow> <mfenced> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>η</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>κ</mi> <mn>1</mn> </msub> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> <label >(17)</label> </div> </div></div><div class='html-p'>As assumed earlier, the expected benefit when the government sector chooses to adopt the incentive strategy S<sub>G1</sub> is <span class='html-italic'>U</span><sub>SG1</sub>, and relatively, the expected benefit when the government sector chooses not to adopt the incentive strategy S<sub>G2</sub> is <span class='html-italic'>U</span><sub>SG2</sub>, and then the average benefit when the government partially adopts these two pure strategies can be obtained, denoted by <span class='html-italic'>U</span><sub>G-AVE</sub>. Based on this and the payoff distribution matrix in (17), the joint replicator dynamics equations formed by the government and electricity users are obtained as follows. <div class='html-disp-formula-info' id='FD18-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mtable columnalign="left"> <mtr> <mtd> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>z</mi> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>−</mo> <mi>z</mi> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>z</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(18)</label> </div> </div></div><div class='html-p'>Based on (18), we can further obtain the Jacobian matrix of the government-power user two-party evolution game system, denoted by <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PU</mi> </mrow> </mrow> </msub> </mrow> </semantics></math>, as well as the corresponding determinant and trace of this Jacobian matrix, denoted by <math display='inline'><semantics> <mrow> <mi>D</mi> <mi>e</mi> <mi>t</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PU</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>t</mi> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PU</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math>, respectively, which are calculated as follows. <div class='html-disp-formula-info' id='FD19-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mtable> <mtr> <mtd> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PU</mi> </mrow> </mrow> </msub> <mo>=</mo> <mfenced close="" open="["> <mtable> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>z</mi> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>−</mo> <mi>z</mi> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>β</mi> <mn>9</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced close="]" open=""> <mtable> <mtr> <mtd> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mi>D</mi> <mi>e</mi> <mi>t</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PU</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>z</mi> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>−</mo> <mi>z</mi> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mtext> </mtext> <mo>−</mo> <mi>x</mi> <mi>z</mi> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PU</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>z</mi> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>−</mo> <mi>z</mi> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(19)</label> </div> </div></div><div class='html-p'>The stability analysis is based on the Jacobian matrix obtained from the replicator dynamics equations. The matrix provides the determinant and trace values necessary to classify equilibrium points into stable, unstable, or saddle points. Specifically, we calculate the determinant and trace for various internal equilibrium points (e.g., E<sub>1</sub>, E<sub>2</sub>, E<sub>3</sub>, E<sub>4</sub>, E<sub>5</sub>) and conclude that the government’s incentives can move the system toward a stable equilibrium point (E<sub>4</sub>) when the determinant is positive and the trace is negative.</div><div class='html-p'>Conversely, if the government fails to incentivize DSM (adopting strategy S<sub>G2</sub>), the system may end up at an unstable equilibrium (E<sub>2</sub> or E<sub>3</sub>) or a saddle point (E<sub>5</sub>), as indicated by negative or zero determinant values in the stability matrix. This analysis demonstrates that government decisions play a crucial role in whether the system remains stable or not, as defined by the dynamical systems’ sense of asymptotic stability.</div><div class='html-p'>From a dynamical systems perspective, an Evolutionary Stable Strategy (ESS) is characterized by an equilibrium that is stable under small perturbations. In other words, once the system reaches this state, small deviations in strategy by either the government or the power users will not lead to a significant shift away from the equilibrium. Mathematically, this is equivalent to finding a stable fixed point in the system’s replicator dynamics. Stability is confirmed if the Jacobian matrix at the equilibrium point satisfies the conditions of a negative trace and a positive determinant, which ensures that the system converges to the ESS over time.</div><div class='html-p'>In this model, the government’s role is critical because its policy decisions (e.g., adopting S<sub>G1</sub>) affect whether the system will converge to a stable ESS. If the government provides sufficient incentives, the system gravitates toward a stable ESS (such as E<sub>4</sub>), promoting cooperation and DSM participation among power users. Without these incentives, the system risks instability or settling into a less favorable equilibrium point.</div><div class='html-p'>As seen from the replicator dynamics equations in (18), the internal equilibrium points of the government-power user two-party evolution game system within the decision-making space of <math display='inline'><semantics> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math> are calculated as E<sub>1</sub>(0, 0), E<sub>2</sub>(0, 1), E<sub>3</sub>(1, 0), E<sub>4</sub>(1, 1), and E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>z</span>*), where <math display='inline'><semantics> <mrow> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>9</mn> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msup> <mi>z</mi> <mo>∗</mo> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>3</mn> </msub> </mrow> <mrow> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>. Based on this, the <math display='inline'><semantics> <mrow> <mi>D</mi> <mi>e</mi> <mi>t</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PU</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>t</mi> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mtext>-</mtext> <mi>PU</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> in (19) can be calculated as shown in <a href="#mathematics-12-03249-t006" class="html-table">Table 6</a>.</div><div class='html-p'>From <a href="#mathematics-12-03249-t006" class="html-table">Table 6</a> and the Lyapunov’s stability theory, it can be seen that the internal equilibrium point E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>z</span>*) is a saddle point, and the asymptotically stability of the other four pure strategy equilibrium points is related to the size relationships between the parameters shown in <a href="#mathematics-12-03249-t005" class="html-table">Table 5</a>. Based on this, the evolutionary stability conditions of this government-power user two-party evolution game system at the internal equilibrium point and the mutually exclusive equilibrium points are shown in <a href="#mathematics-12-03249-t007" class="html-table">Table 7</a>. As shown in <a href="#mathematics-12-03249-t006" class="html-table">Table 6</a> and <a href="#mathematics-12-03249-t007" class="html-table">Table 7</a>, it can be seen that E<sub>1</sub> and E<sub>4</sub>, as well as E<sub>2</sub> and E<sub>3</sub>, each represent a pair of mutually exclusive evolutionarily stable equilibrium points, and besides, the long-term evolutionarily stable equilibrium characteristics of each pure strategy equilibrium point is strictly related to the signs of <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> </mrow> </semantics></math>, thus there are a total of 6 cases that can be discussed, which will be explained in the next subsection.</div></section><section id='sec5dot3dot2-mathematics-12-03249' type=''><h4 class='' data-nested='3'> 5.3.2. Dynamic Numerical Simulation</h4><div class='html-p'>For the sake of discussion, the payoff distribution parameters are set as <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>=</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo>=</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>=</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, where <math display='inline'><semantics> <mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>></mo> <msup> <mi>c</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, and the saddle point can be expressed as <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="normal">E</mi> <mn>5</mn> </msub> <mo>:</mo> <mfenced> <mrow> <mi>x</mi> <mo>*</mo> <mo>,</mo> <mi>z</mi> <mo>*</mo> </mrow> </mfenced> <mo>=</mo> <mo stretchy="false">(</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> </mrow> <msup> <mi>d</mi> <mo>′</mo> </msup> </mfrac> </mstyle> <mo>,</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>b</mi> <mo>′</mo> </msup> <mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mstyle> <mo stretchy="false">)</mo> </mrow> </semantics></math>. Based on this, we set the system decision space as [0, 1] × [1, 0], so that the initial values <span class='html-italic'>x</span> and <span class='html-italic'>z</span> are taken from 0 to 1, with 1/50 as the interval, a total of 2601 rounds of dynamic evolution game simulations with different initial value scenarios are carried out for verifying the results of the aforementioned theoretical analysis results in the government-power user two-party evolution game system, and the following six kinds of results can be obtained, as demonstrated in <a href="#mathematics-12-03249-f005" class="html-fig">Figure 5</a>, in which Case 1 represents the evolutionary trend of the proportion of individuals <span class='html-italic'>x</span> adopting strategy S<sub>G1</sub> in the governmental group as a result of the change of the strategy with time <span class='html-italic'>t</span> evolution trend, Case 2 represents the evolution trend of the proportion of individuals <span class='html-italic'>z</span> adopting strategy S<sub>PG1</sub> in the grid group over time <span class='html-italic'>t</span>, and Case 3 represents the evolution trend of (<span class='html-italic'>x</span>, <span class='html-italic'>z</span>) developing over time <span class='html-italic'>t</span>.</div><div class='html-p'>The simulation results demonstrated in <a href="#mathematics-12-03249-f005" class="html-fig">Figure 5</a> are analyzed as follows.</div><div class='html-p'>In <a href="#mathematics-12-03249-f005" class="html-fig">Figure 5</a>a, the pure strategy equilibrium points E<sub>1</sub> and E<sub>4</sub> are spontaneously formed as asymptotically stability equilibria in this asymmetric two-group evolution game among the government and power users, which eventually allows the system to reach a long-term evolutionary stable equilibrium state and be in a strictly Nash equilibrium state. The system is in critical equilibrium at the saddle point E<sub>5</sub>(1/6, 5/6) and can never achieve a long-term evolutionary stable equilibrium state. At this time, the government adopts strategy S<sub>G1</sub> with relatively low cost or high social and environmental benefits, but the total benefits obtained by the power user’s choice of strategy S<sub>PU1</sub> are uncertain in relation to the expenditures. Equilibrium points E<sub>2</sub> and E<sub>3</sub> are unstable equilibrium points of the system, and the system is in an evolutionary unstable equilibrium at E<sub>2</sub> and E<sub>3</sub>. When the overall return of strategy S<sub>PU1</sub> is high, the area enclosed by the four equilibrium points E<sub>2</sub>, E<sub>5</sub>, E<sub>3</sub>, and E<sub>4</sub> is larger, and the system will converge to E<sub>4</sub>, with the strategy ultimately being {S<sub>G1</sub>, S<sub>PU1</sub>}. At this time, the DSM project achieves significant results on the demand side. Conversely, the multi-group evolutionary stable equilibrium strategy will ultimately be {S<sub>G2</sub>, S<sub>PU2</sub>}.</div><div class='html-p'>In <a href="#mathematics-12-03249-f005" class="html-fig">Figure 5</a>b,c, the systems always converge to the equilibrium point E<sub>1</sub>, and there exists only a unique evolutionarily stable equilibrium point within the system where the system achieves an evolutionarily stable equilibrium. When the power users adopt the strategy S<sub>PU1</sub> with the benefit lower than the cost expenditure, the user does not change the strategy regardless of the government’s strategy selection. Thus, the final multi-group evolutionary stable equilibrium strategy will be formed at {S<sub>G2</sub>, S<sub>PU2</sub>}.</div><div class='html-p'>In <a href="#mathematics-12-03249-f005" class="html-fig">Figure 5</a>d, the system achieves an evolutionarily stable equilibrium state at E<sub>1</sub>, where the benefits of choosing the strategies S<sub>G1</sub> and S<sub>PU1</sub> are both far below their costs. Thus, the final multi-group evolutionary stable equilibrium strategy will be formed at {S<sub>G2</sub>, S<sub>PU2</sub>}. However, in <a href="#mathematics-12-03249-f005" class="html-fig">Figure 5</a>e, E<sub>4</sub> becomes the asymptotic stabilization point of the system, where the system eventually achieves a long-term evolutionary stable equilibrium state. At this point, the government and the power users are more motivated because both strategies, S<sub>G1</sub> and S<sub>PU1</sub>, have higher benefits and lower costs. Thus, the final multi-group evolutionary stable equilibrium strategy will be formed at {S<sub>G1</sub>, S<sub>PU1</sub>}. Besides, <a href="#mathematics-12-03249-f005" class="html-fig">Figure 5</a>f reveals that the system can finally achieve an evolutionarily stable equilibrium state at E<sub>2</sub>, which is the asymptotic stabilization point of the system. At this point, the benefits to the power users via choosing strategy S<sub>PU1</sub> are higher than the costs, but the expenses to the government are larger. Thus, the final multi-group evolutionary stable equilibrium strategy will be formed at {S<sub>G2</sub>, S<sub>PU1</sub>}.</div></section></section><section id='sec5dot4-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 5.4. Modeling and Numerical Simulation of the Three-Group Evolutionary Game</h4><div class='html-p'>In the context of modern power systems, the interaction among various stakeholders, such as government entities, power grid enterprises, and power consumers, is critical to achieving demand-side management (DSM) objectives. This interaction involves decisions related to energy consumption, policy incentives, and grid management strategies and can be effectively modeled as an evolutionary game.</div><div class='html-p'>The three-group evolutionary game presented here involves:</div><div class='html-p'>Group 1 (Government entities): The government is responsible for providing incentives for DSM strategies and may choose between two pure strategies: providing incentives (SG1) or not providing incentives (SG2).</div><div class='html-p'>Group 2 (Power grid enterprises): The grid enterprises can either implement DSM (SPG1) or not implement DSM (SPG2), depending on the strategy adopted by the government and power users.</div><div class='html-p'>Group 3 (Power users): Power users can choose to either participate in DSM (SPU1) or decline participation (SPU2).</div><div class='html-p'>The aim of the game is to model how each group’s strategic decisions evolve over time, based on their interactions and resulting payoffs, and to simulate these dynamics numerically. We utilize replicator dynamics, a key concept in evolutionary game theory, to describe how the probability of each strategy being selected evolves based on the payoff differences among competing strategies.</div><div class='html-p'>The evolutionary dynamics are governed by replicator dynamic equations, Ih describe how the probability of adopting a particular strategy changes over time. In this model, <span class='html-italic'>x</span> represents the probability that the government adopts strategy S<sub>G1</sub> (provide incentives), with 1 − <span class='html-italic'>x</span> representing the probability of adopting S<sub>G2</sub> (no incentives). <span class='html-italic'>Y</span> represents the probability that the power grid enterprises adopt strategy S<sub>PG1</sub> (implement DSM), with 1 − <span class='html-italic'>y</span> representing the probability of adopting S<sub>PG2</sub> (no DSM). <span class='html-italic'>Z</span> represents the probability that the power users adopt strategy S<sub>PU1</sub> (participate in DSM), with 1 − <span class='html-italic'>z</span> representing the probability of adopting S<sub>PU2</sub> (no participation). The replicator dynamics for each group are derived from the payoffs associated with each strategy. Based on this, the replicator dynamics model of this three-group evolutionary game is shown as <div class='html-disp-formula-info' id='FD20-mathematics-12-03249'> <div class='f'> <math display='block'><semantics> <mrow> <mfenced close="" open="{"> <mrow> <mtable equalrows="true" equalcolumns="true"> <mtr> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <mi>y</mi> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>z</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(20)</label> </div> </div></div><div class='html-p'>The payoffs are derived from the interaction between the different groups and depend on various cost and benefit parameters, such as:</div><div class='html-p'><span class='html-italic'>α</span><sub>1</sub>, <span class='html-italic'>α</span><sub>2</sub>, <span class='html-italic'>α</span><sub>3</sub>: Costs associated with providing incentives or implementing DSM.</div><div class='html-p'><span class='html-italic'>β</span><sub>1</sub>, <span class='html-italic'>β</span><sub>2</sub>, …, <span class='html-italic'>β</span><sub>11</sub>: Benefits related to the outcomes of DSM adoption, such as reduced electricity expenses, policy benefits, and social welfare gains.</div><div class='html-p'><span class='html-italic'>η</span><sub>1</sub>, <span class='html-italic'>η</span><sub>2</sub>: Social welfare losses when DSM is not adopted.</div><div class='html-p'><span class='html-italic'>κ</span><sub>1</sub>, <span class='html-italic'>κ</span><sub>2</sub>: Initial expenditures and investments in energy-saving equipment by power users.</div><div class='html-p'>To simulate the evolutionary dynamics of the three-group system, we solve the system of differential equations using MATLAB’s ode45 function. This function numerically integrates the replicator dynamics over a specified time horizon, starting from an initial probability distribution for each group.</div><div class='html-p'>Initial Conditions: The initial strategy probabilities for the government, power grid enterprises, and power users are set to 0.5, indicating an equal likelihood of adopting either strategy.</div><div class='html-p'>Time Horizon: The simulation runs over a time period from <span class='html-italic'>t</span> = 0 to <span class='html-italic'>t</span> = 50, providing sufficient time for the strategy adoption probabilities to evolve. The time interval <span class='html-italic'>t</span> ∈ [0, 50] was selected to ensure that the full dynamics of the evolutionary game could be observed. Simulations over this range capture the complete evolution of strategies toward either stable equilibrium points or saddle points. After testing with both shorter and longer time intervals, we found that this time span sufficiently encapsulates the critical phases of strategy adjustment and convergence.</div><div class='html-p'>Visualization: The results are visualized through a 3D phase trajectory plot that shows the evolution of (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>, <span class='html-italic'>z</span>) in three-dimensional space, demonstrating how the strategies of the government, power grid enterprises, and power users co-evolve over time. The 2D plots of each group’s strategy adoption probability as a function of time illustrate how each group’s strategy evolves independently.</div><div class='html-p'>The simulation results provide valuable Insights Into the Interaction among the three groups, including:</div><div class='html-p'>3D Phase Trajectory: The three-dimensional trajectory illustrates the co-evolution of the strategies of the government, power grid enterprises, and power users. The trajectory converges towards an equilibrium point, indicating a stable strategy distribution where no group has an incentive to deviate unilaterally.</div><div class='html-p'>Time-Evolution Plots: The plots of <span class='html-italic'>x</span>, <span class='html-italic'>y</span>, and <span class='html-italic'>z</span> versus time show the temporal evolution of each group’s strategy adoption probability. These plots reveal the rate at which each group converges towards its equilibrium strategy, as well as the influence of the interactions with other groups.</div><div class='html-p'>We conducted simulation experiments 1000 and 2000 times and obtained the phase trajectory diagrams, as shown in <a href="#mathematics-12-03249-f006" class="html-fig">Figure 6</a> and <a href="#mathematics-12-03249-f007" class="html-fig">Figure 7</a>, respectively, where the red dot in <a href="#mathematics-12-03249-f006" class="html-fig">Figure 6</a>a and <a href="#mathematics-12-03249-f007" class="html-fig">Figure 7</a>a represents a long-term evolutionarily stable equilibrium point. Seen from these figures, the evolutionary game involving three key entities—government, power grid enterprises, and power users—is modeled to understand how their interactions and strategies evolve over time. The government is responsible for providing incentives to encourage demand-side management (DSM) initiatives, while the power grid enterprises decide whether to implement DSM, and power users decide whether to participate in DSM programs. The simulation results reveal a complex dynamic relationship among these three groups, driven by the payoffs each group receives based on their strategies and the strategies of others.</div><div class='html-p'>The following sections provide a detailed explanation of the simulation results, considering the behavior of each group, the underlying mechanisms of the evolutionary game, and the physical implications in the context of power system management.</div><dl class='html-order'><dt id=''>(1)</dt><dd><div class='html-p'>Convergence of Strategies to Stable Equilibria</div></dd></dl><div class='html-p'>In the three-dimensional scatter plot, representing the evolution of strategies for the government (<span class='html-italic'>x</span>), power grid enterprises (<span class='html-italic'>y</span>), and power users (<span class='html-italic'>z</span>), we observe that most trajectories eventually converge towards specific stable points. This convergence is seen across the majority of simulation runs, indicating that over time, the system tends towards equilibrium, where each group’s strategy stabilizes.</div><div class='html-p'>Interpretation: The convergence to equilibrium represents a stable Nash equilibrium in this evolutionary game. At this equilibrium, none of the groups—government, power grid, or power users—find it beneficial to unilaterally deviate from their current strategies. Each group’s chosen strategy maximizes their respective payoffs, given the strategies of the other two groups. In a power system context, the stable equilibria reflect a state where the government consistently provides incentives for DSM, power grid enterprises consistently implement DSM strategies, and power users consistently participate in these DSM programs. This is an ideal scenario for demand-side management, as it leads to optimized energy consumption, reduced peak loads, and enhanced grid stability.</div><dl class='html-order'><dt id=''>(2)</dt><dd><div class='html-p'>Government Strategy (<span class='html-italic'>x</span>) Evolution</div></dd></dl><div class='html-p'>The simulation results for the government’s strategy xxx over time show that it generally converges to a value close to 1. This indicates that the government almost always chooses the strategy of providing incentives (S<sub>G1</sub>) to encourage DSM implementation.</div><div class='html-p'>Interpretation: The government’s decision to consistently provide incentives can be attributed to the long-term benefits it perceives from widespread DSM adoption. By providing incentives, the government can reduce energy consumption during peak times, decrease grid instability, and improve overall societal welfare. This is reflected in the payoff structure, where the government gains more by offering incentives if both the grid and power users respond positively (implement and participate in DSM). The simulation shows that the cost associated with providing these incentives (<span class='html-italic'>α</span><sub>1</sub>) is outweighed by the societal and environmental benefits (<span class='html-italic'>β</span><sub>7</sub>, <span class='html-italic'>β</span><sub>8</sub>), making the incentive strategy (S<sub>G1</sub>) the dominant strategy in the long run.</div><dl class='html-order'><dt id=''>(3)</dt><dd><div class='html-p'>Power Grid Enterprises’ Strategy (<span class='html-italic'>y</span>) Evolution</div></dd></dl><div class='html-p'>Similarly, the power grid enterprises’ strategy <span class='html-italic'>y</span> evolves over time, with the majority of trajectories converging towards a value close to 1. This indicates that the grid enterprises frequently opt to implement DSM strategies (S<sub>PG1</sub>).</div><div class='html-p'>Interpretation: The grid enterprises’ decision to consistently implement DSM reflects their recognition of the economic and operational benefits associated with load management. DSM allows for a more balanced distribution of energy consumption, reducing the need for expensive peak-time generation and mitigating the risks of system overload. The incentives provided by the government (<span class='html-italic'>β</span><sub>5</sub>) and the internal benefits derived from implementing DSM (<span class='html-italic'>β</span><sub>6</sub>) contribute to the grid’s preference for DSM. The costs of implementation (<span class='html-italic'>α</span><sub>2</sub>) are outweighed by the financial and operational benefits, leading to the widespread adoption of DSM strategies in the simulated system.</div><dl class='html-order'><dt id=''>(4)</dt><dd><div class='html-p'>Power Users’ Strategy (<span class='html-italic'>z</span>) Evolution</div></dd></dl><div class='html-p'>For power users, the strategy <span class='html-italic'>z</span>, which represents their participation in DSM programs, also converges towards 1 in most cases, indicating a preference for participation in DSM programs.</div><div class='html-p'>Interpretation: Power users are motivated to participate in DSM due to financial savings (reduced electricity bills) and social incentives (environmental benefits). The government’s incentives (<span class='html-italic'>β</span><sub>9</sub>) and the reduction in electricity costs (<span class='html-italic'>β</span><sub>10</sub>) provide strong motivation for users to engage in DSM. Additionally, the environmental and social benefits (<span class='html-italic'>β</span><sub>11</sub>) associated with sustainable energy consumption further reinforce their preference for DSM.</div><div class='html-p'>The convergence of user behavior towards full participation also reflects a collective behavioral shift where users see long-term benefits from DSM participation. This is important in the context of grid stability and efficiency, as user participation helps reduce overall energy demand during peak periods, thus improving system reliability.</div><dl class='html-order'><dt id=''>(5)</dt><dd><div class='html-p'>Physical Implications and Deeper Significance</div></dd></dl><div class='html-p'>The evolutionary dynamics observed in the simulation have significant physical implications for the power system. The convergence of strategies reflects the following deep insights.</div><div class='html-p'>System Stability: The convergence to a stable equilibrium where all parties adopt DSM-friendly strategies indicates that the power system can reach a stable state of operation where demand and supply are more closely aligned. This reduces the likelihood of blackouts or brownouts and enhances the resilience of the grid.</div><div class='html-p'>Energy Efficiency: The widespread adoption of DSM by both the grid and users leads to more efficient energy usage, reducing waste and optimizing the use of available resources. This contributes to lowering operational costs for the grid and decreasing energy bills for consumers.</div><div class='html-p'>Environmental Impact: The adoption of DSM contributes to a reduction in peak demand, which in turn reduces the need to activate costly and often environmentally damaging peak-load power plants. This leads to a greener energy system, as fewer emissions are produced, and more sustainable energy sources can be integrated into the grid.</div><div class='html-p'>Policy Design: The simulation results underscore the importance of well-designed governmental policies. By providing consistent and well-calibrated incentives, the government can guide the grid and power users toward behaviors that improve grid efficiency and sustainability. The results suggest that the costs of implementing incentive programs are justified by the long-term societal and economic benefits.</div><div class='html-p'>Overall, the simulation results in <a href="#mathematics-12-03249-f006" class="html-fig">Figure 6</a> and <a href="#mathematics-12-03249-f007" class="html-fig">Figure 7</a> show that the interactions between the government, power grid enterprises, and power users lead to stable equilibria in the context of DSM adoption. The dominant strategies for each group (government providing incentives, grid implementing DSM, and users participating in DSM) reflect a system where all players recognize the mutual benefits of collaboration. These results highlight the importance of coordinated policies and strategies in managing energy consumption and promoting grid stability. Through this evolutionary game framework, we can better understand how different entities in the power system influence each other’s strategies and contribute to an optimized, sustainable, and resilient energy infrastructure.</div></section><section id='sec5dot5-mathematics-12-03249' type=''><h4 class='html-italic' data-nested='2'> 5.5. Policy Implications</h4><div class='html-p'>In this section, we first analyze the interests of the government, grid, and users in DSM. Secondly, based on the general 2P2S-AEG model, we set the relevant parameters and constructed the asymmetric two-party government-power grid and government-power user evolution game models, respectively. Then, based on the replicator dynamics equation, the Jacobi matrix, and its determinant and trace, the evolutionary stability conditions of the system are derived. Since the system has different evolutionary stable equilibria under different conditions, it is necessary to use MATLAB to simulate the evolutionary trend of the game under different scenarios. Finally, the long-term evolutionary stability of the government and the grid, as well as the government and the users in the DSM incentive mechanism, is analyzed to provide some reference for the implementation and promotion of the DSM. Based on the theoretical analysis and dynamic simulation verification, we can obtain some policy implications, as explained below. When the government chooses strategy S<sub>G1</sub> in the game, the costs required for electricity users to purchase relevant equipment for saving electricity may be lower, or the savings from the use of energy-saving technologies are obvious, at which time the likelihood of electricity users choosing the strategy S<sub>PU1</sub> is elevated. On the contrary, when the expenditure of power users is relatively high, but the actual benefits obtained are low, power users will choose the strategy S<sub>PU2</sub>. For large-sized industrial and commercial electricity users, the government departments can take the approach of giving appropriate financial incentives to users who purchase or remodel energy-efficient equipment. For ordinary residents, the mechanism of electricity price should be improved, and users should be guided to avoid peak loads so as to achieve the purpose of diverting electricity consumption. Government departments can also promote the development and improvement of the DSM platform to improve the electricity environment of power users and enhance the convenience of electricity consumption.</div><div class='html-p'>From a mathematical perspective, the government’s role in ensuring the stability of DSM strategies can be understood through the lens of dynamical systems theory and evolutionary game theory. Specifically, the government’s incentives (or lack thereof) can push the system toward an Evolutionary Stable Strategy (ESS), where no single player benefits from deviating from the current strategy set. The government influences this stability by adjusting payoffs in response to the behavior of other stakeholders. For example, by increasing incentives for energy-saving behaviors (SG1), the government shifts the system toward a stable equilibrium, as demonstrated through replicator dynamics equations.</div><div class='html-p'>The stability of these equilibria is confirmed through the analysis of the Jacobian matrix derived from the replicator dynamics [<a href="#B54-mathematics-12-03249" class="html-bibr">54</a>]. When the government’s strategy leads to a positive determinant and negative trace, this indicates asymptotic stability in the dynamical system sense. Thus, the government not only plays a role in promoting DSM but also ensures the long-term stability of solutions by steering the system toward equilibrium points where cooperative strategies dominate.</div><div class='html-p'>In line with studies such as Poletti et al. (2015) [<a href="#B55-mathematics-12-03249" class="html-bibr">55</a>], where adaptive behaviors during epidemics lead to long-term behavioral equilibria, the government’s interventions in DSM similarly result in adaptive strategies by other stakeholders. Furthermore, Della Marca et al. (2023) [<a href="#B56-mathematics-12-03249" class="html-bibr">56</a>] highlight the impact of rigid versus flexible strategies in epidemics, which mirrors how government incentives in DSM can create flexibility in stakeholder behaviors, leading to more robust and stable outcomes.</div><div class='html-p'>Overall, to ensure that demand-side management of electricity fully plays its role in reducing energy waste rates, optimizing resource allocation, and protecting the environment, the government needs to adopt suitable incentive mechanisms to enhance the enthusiasm of grid companies and electricity users, thereby promoting the implementation of policies. Based on the above research and analysis, this article provides the following two specific policy recommendations.</div><dl class='html-roman-lower'><dt id=''>(i)</dt><dd><div class='html-p'>Regarding the power grid companies: the government can provide certain policy incentives, offer financial support, and encourage power grid companies to research power demand-side management technologies, promote the joint development of related technologies, facilitate the marketization of power demand-side management, and establish economic rewards to advance load and energy efficiency management technologies. At the same time, by setting annual mandatory targets, reasonable penalties can be imposed on companies that fail to meet these targets.</div></dd><dt id=''>(ii)</dt><dd><div class='html-p'>Regarding electricity users: the government departments can guide electricity users to avoid using electricity during peak load periods, adjust and improve the electricity pricing mechanism, and provide subsidies to electricity users who purchase DSM equipment, among other methods, to encourage electricity users to accept DSM services and promote the development of DSM in the open and emerging power market.</div></dd></dl></section></section><section id='sec6-mathematics-12-03249' type='conclusions'><h2 data-nested='1'> 6. Conclusions</h2><div class='html-p'>The implementation of a power demand-side management program has a positive effect on the development of electric energy, which can accelerate the process of resource allocation optimization, save electricity consumption, promote the development of a new energy market, reduce environmental pollution, and facilitate high-quality economic development. To this end, this paper uses evolutionary game theory to study the interactions between government departments, grid companies, and electricity users in the demand-side management process. <a href="#sec3-mathematics-12-03249" class="html-sec">Section 3</a> and <a href="#sec4-mathematics-12-03249" class="html-sec">Section 4</a> play an integral role in the structure of the paper, serving as both a theoretical foundation and a demonstration of the application of the model. <a href="#sec3-mathematics-12-03249" class="html-sec">Section 3</a> introduces key concepts necessary for understanding the later sections, especially for readers new to evolutionary game theory, while <a href="#sec4-mathematics-12-03249" class="html-sec">Section 4</a> provides a practical example that bridges the gap between theory and real-world application. These sections ensure that the subsequent analysis of the tripartite game between government, grid companies, and users is both accessible and grounded in well-established theoretical principles.</div><div class='html-p'>This paper contributes to the evolving literature on DSM by presenting a novel tripartite evolutionary game model that captures the complex, asymmetric interactions among government, grid companies, and users. The findings provide actionable insights for policymakers seeking to enhance DSM adoption through well-calibrated incentives. Future research could extend this model by incorporating additional market participants or using real-world data to refine the payoff structures.</div><div class='html-p'>Overall, this article is conducted based on evolutionary game theory and analyzes the game behaviors of the government, power grid, and users in the process of DSM from the perspective of incentives. First, it introduces the research background and current domestic and international research trends. Next, it analyzes the game process regarding power demand-side management among government departments, power grid companies, and electricity users using the RD equation and stable equilibrium points. Then, it applies Matlab to simulate and analyze the game process. Finally, it proposes suggestions for the power demand-side management mechanism. Based on this, the main research conclusions of this article are summarized as follows.</div><dl class='html-roman-lower'><dt id=''>(i)</dt><dd><div class='html-p'>According to the simulation results, it is necessary for government departments, grid companies, and electricity users to jointly seek the best behavior strategies during the implementation of DSM to achieve a win-win situation.</div></dd><dt id=''>(ii)</dt><dd><div class='html-p'>When implementing DSM projects, the power grid company needs to invest human and financial resources to develop project plans that meet the requirements of demand-side management, ensuring system benefits while reducing the likelihood of risks. Then, according to the requirements of DSM policies, new equipment is procured, or old equipment is upgraded to create necessary conditions for project implementation. In the government-power grid two-party asymmetric evolution game system, both the government and the power grid enterprises will only choose to incentivize or implement DSM projects when the benefits received exceed the costs incurred.</div></dd><dt id=''>(iii)</dt><dd><div class='html-p'>In the government-power user two-group asymmetric evolution game system, the electricity users will only choose to use DSM services when the benefits from reduced electricity expenses due to demand-side management services, the policy benefits from using the services, and the gains from the service experience exceed the costs of purchasing demand-side management equipment. Meanwhile, the government may choose to incentivize DSM when the returns are low or the costs are high.</div></dd></dl><div class='html-p'>Finally, it should be noted that, although this paper conducts a detailed theoretical investigation, it focuses solely on the incentive mechanisms of DSM and does not address other issues that exist in the actual implementation of DSM. Future research can delve deeper into this aspect. Additionally, due to the confidentiality of actual data, it is not possible to obtain real data for research, and when setting parameters, not all relevant parameters may have been considered. Therefore, further research can be conducted on these issues in the future. In this article, evolutionary game theory provides an analytical method for addressing the problems present in DSM. With the rapid development of smart grids, the field of electrical engineering is constantly changing, and we believe that evolutionary game theory will be applied more widely in the research and analysis of various issues in this field.</div></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>Conceptualization, X.S., J.T., Y.Z. (Yijing Zhang) and B.Q.; methodology, X.S., J.T., Y.Z. (Yijing Zhang), B.Q., J.L., M.Z., Y.Z. (Yitao Zhao) and Y.Y.; formal analysis, X.S., J.T., Y.Z. (Yijing Zhang) and B.Q.; investigation, X.S., J.T., J.L., M.Z., Y.Z. (Yitao Zhao) and Y.Y.; writing—original draft preparation, X.S., J.T., Y.Z. (Yijing Zhang) and B.Q.; writing—review and editing, X.S., J.T., Y.Z. (Yijing Zhang) and B.Q.; funding acquisition, X.S. and J.T. All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2>Funding</h2><div class='html-p'>This research was funded by the China Southern Power Grid Technology Project, grant number YNKJXM20222402 (funder: X.S.).</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>Data are contained within the article.</div></section><section id='html-ack' class='html-ack'><h2 >Acknowledgments</h2><div class='html-p'>We would like to express our deepest gratitude to Lefeng Cheng from the School of Mechanical and Electrical Engineering at Guangzhou University and his team of students for their invaluable assistance, insights, and suggestions throughout the preparation of this paper. In particular, their support in building the theoretical model and conducting numerical simulation verifications was immensely helpful. We are profoundly grateful to Cheng and all the students involved in his team for their significant contributions to the success of this research. Besides, we sincerely thank the associate editor and invited anonymous reviewers for their kind and helpful comments on our paper.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>Authors Xin Shen, Yijing Zhang, Jiahao Li, Yitao Zhao and Yujun Yin were employed by Yunnan Power Grid Co., Ltd. Authors Jianlin Tang, Bin Qian, and Mi Zhou were employed by China Southern Power Grid Co., Ltd. The remaining 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.</div></section><section id='html-references_list'><h2>References</h2><ol class='html-xx'><li id='B1-mathematics-12-03249' class='html-x' data-content='1.'>Hsu, H.-P.; Wang, C.-N.; Nguyen, T.T.T.; Dang, T.-T.; Pan, Y.-J. Hybridizing WOA with PSO for coordinating material handling equipment in an automated container terminal considering energy consumption. <span class='html-italic'>Adv. Eng. Inform.</span> <b>2024</b>, <span class='html-italic'>60</span>, 102410. [<a href="https://scholar.google.com/scholar_lookup?title=Hybridizing+WOA+with+PSO+for+coordinating+material+handling+equipment+in+an+automated+container+terminal+considering+energy+consumption&author=Hsu,+H.-P.&author=Wang,+C.-N.&author=Nguyen,+T.T.T.&author=Dang,+T.-T.&author=Pan,+Y.-J.&publication_year=2024&journal=Adv.+Eng.+Inform.&volume=60&pages=102410&doi=10.1016/j.aei.2024.102410" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.aei.2024.102410" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B2-mathematics-12-03249' class='html-x' data-content='2.'>Gungor, V.C.; Sahin, D.; Kocak, T.; Ergut, S.; Buccella, C.; Cecati, C.; Hancke, G.P. Smart grid technologies: Communication technologies and standards. <span class='html-italic'>IEEE Trans. Ind. Inform.</span> <b>2011</b>, <span class='html-italic'>7</span>, 529–539. [<a href="https://scholar.google.com/scholar_lookup?title=Smart+grid+technologies:+Communication+technologies+and+standards&author=Gungor,+V.C.&author=Sahin,+D.&author=Kocak,+T.&author=Ergut,+S.&author=Buccella,+C.&author=Cecati,+C.&author=Hancke,+G.P.&publication_year=2011&journal=IEEE+Trans.+Ind.+Inform.&volume=7&pages=529%E2%80%93539&doi=10.1109/TII.2011.2166794" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/TII.2011.2166794" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B3-mathematics-12-03249' class='html-x' data-content='3.'>Esenogho, E.; Djouani, K.; Kurien, A.M. Integrating artificial intelligence internet of things and 5G for next-generation smartgrid: A survey of trends challenges and prospect. <span class='html-italic'>IEEE Access</span> <b>2022</b>, <span class='html-italic'>10</span>, 4794–4831. [<a href="https://scholar.google.com/scholar_lookup?title=Integrating+artificial+intelligence+internet+of+things+and+5G+for+next-generation+smartgrid:+A+survey+of+trends+challenges+and+prospect&author=Esenogho,+E.&author=Djouani,+K.&author=Kurien,+A.M.&publication_year=2022&journal=IEEE+Access&volume=10&pages=4794%E2%80%934831&doi=10.1109/ACCESS.2022.3140595" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ACCESS.2022.3140595" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B4-mathematics-12-03249' class='html-x' data-content='4.'>Cheng, L.F.; Yu, T. A new generation of AI: A review and perspective on machine learning technologies applied to smart energy and electric power systems. <span class='html-italic'>Int. J. Energy Res.</span> <b>2019</b>, <span class='html-italic'>43</span>, 1928–1973. [<a href="https://scholar.google.com/scholar_lookup?title=A+new+generation+of+AI:+A+review+and+perspective+on+machine+learning+technologies+applied+to+smart+energy+and+electric+power+systems&author=Cheng,+L.F.&author=Yu,+T.&publication_year=2019&journal=Int.+J.+Energy+Res.&volume=43&pages=1928%E2%80%931973&doi=10.1002/er.4333" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1002/er.4333" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B5-mathematics-12-03249' class='html-x' data-content='5.'>Cheng, L.F.; Yu, T. Game-theoretic approaches applied to transactions in the open and ever-growing electricity markets from the perspective of power demand response: An overview. <span class='html-italic'>IEEE Access</span> <b>2019</b>, <span class='html-italic'>7</span>, 25727–25762. [<a href="https://scholar.google.com/scholar_lookup?title=Game-theoretic+approaches+applied+to+transactions+in+the+open+and+ever-growing+electricity+markets+from+the+perspective+of+power+demand+response:+An+overview&author=Cheng,+L.F.&author=Yu,+T.&publication_year=2019&journal=IEEE+Access&volume=7&pages=25727%E2%80%9325762&doi=10.1109/ACCESS.2019.2900356" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ACCESS.2019.2900356" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B6-mathematics-12-03249' class='html-x' data-content='6.'>Zhang, C.; Wang, Q.; Wang, J.; Pinson, P.; Morales, J.M.; Østergaard, J. Real-time procurement strategies of a proactive distribution company with aggregator-based demand response. <span class='html-italic'>IEEE Trans. Smart Grid</span> <b>2018</b>, <span class='html-italic'>9</span>, 766–776. [<a href="https://scholar.google.com/scholar_lookup?title=Real-time+procurement+strategies+of+a+proactive+distribution+company+with+aggregator-based+demand+response&author=Zhang,+C.&author=Wang,+Q.&author=Wang,+J.&author=Pinson,+P.&author=Morales,+J.M.&author=%C3%98stergaard,+J.&publication_year=2018&journal=IEEE+Trans.+Smart+Grid&volume=9&pages=766%E2%80%93776&doi=10.1109/TSG.2016.2565383" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/TSG.2016.2565383" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B7-mathematics-12-03249' class='html-x' data-content='7.'>Kalpana, D.R.; NageshaRao, S.H.; Siddaiah, R.; Mala, R. Case study on demand side management-based cost optimized battery integrated hybrid renewable energy system for remote rural electrification. <span class='html-italic'>Energy Storage</span> <b>2023</b>, <span class='html-italic'>5</span>, e410. [<a href="https://scholar.google.com/scholar_lookup?title=Case+study+on+demand+side+management-based+cost+optimized+battery+integrated+hybrid+renewable+energy+system+for+remote+rural+electrification&author=Kalpana,+D.R.&author=NageshaRao,+S.H.&author=Siddaiah,+R.&author=Mala,+R.&publication_year=2023&journal=Energy+Storage&volume=5&pages=e410&doi=10.1002/est2.410" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1002/est2.410" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B8-mathematics-12-03249' class='html-x' data-content='8.'>Erenoğlu, A.K.; Şengör, I.; Erdinç, O.; Taşcıkaraoğlu, A.; Catalão, J.P. Optimal energy management system for microgrids considering energy storage, demand response and renewable power generation. <span class='html-italic'>Int. J. Electr. Power Energy Syst.</span> <b>2022</b>, <span class='html-italic'>136</span>, 107714. [<a href="https://scholar.google.com/scholar_lookup?title=Optimal+energy+management+system+for+microgrids+considering+energy+storage,+demand+response+and+renewable+power+generation&author=Ereno%C4%9Flu,+A.K.&author=%C5%9Eeng%C3%B6r,+I.&author=Erdin%C3%A7,+O.&author=Ta%C5%9Fc%C4%B1karao%C4%9Flu,+A.&author=Catal%C3%A3o,+J.P.&publication_year=2022&journal=Int.+J.+Electr.+Power+Energy+Syst.&volume=136&pages=107714&doi=10.1016/j.ijepes.2021.107714" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.ijepes.2021.107714" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B9-mathematics-12-03249' class='html-x' data-content='9.'>Cao, Z.; Lin, J.; Wan, C.; Song, Y.; Zhang, Y.; Wang, X. Optimal cloud computing resource allocation for demand side management in smart grid. <span class='html-italic'>IEEE Trans. Smart Grid</span> <b>2017</b>, <span class='html-italic'>8</span>, 1943–1955. [<a href="https://scholar.google.com/scholar_lookup?title=Optimal+cloud+computing+resource+allocation+for+demand+side+management+in+smart+grid&author=Cao,+Z.&author=Lin,+J.&author=Wan,+C.&author=Song,+Y.&author=Zhang,+Y.&author=Wang,+X.&publication_year=2017&journal=IEEE+Trans.+Smart+Grid&volume=8&pages=1943%E2%80%931955&doi=10.1109/TSG.2015.2512712" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/TSG.2015.2512712" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B10-mathematics-12-03249' class='html-xx' data-content='10.'>Pati, U.; Mistry, K.D. A novel demand-side management strategy for residential load with an integrated deep-learning based forecasting engine. In Proceedings of the 2022 22nd National Power Systems Conference (NPSC), New Delhi, India, 17–19 December 2022; pp. 542–547. [<a href="https://scholar.google.com/scholar_lookup?title=A+novel+demand-side+management+strategy+for+residential+load+with+an+integrated+deep-learning+based+forecasting+engine&conference=Proceedings+of+the+2022+22nd+National+Power+Systems+Conference+(NPSC)&author=Pati,+U.&author=Mistry,+K.D.&publication_year=2022&pages=542%E2%80%93547&doi=10.1109/NPSC57038.2022.10069201" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/NPSC57038.2022.10069201" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B11-mathematics-12-03249' class='html-xx' data-content='11.'>Premkumar, R.; Shanmugasundaram, R.; Thilak, K.R.; Balaji, A.N.V.S.S. A review on demand side management: Definition, scope, challenges and benefits. In Proceedings of the 2022 8th International Conference on Smart Structures and Systems (ICSSS), Chennai, India, 21–22 April 2022; pp. 1–4. [<a href="https://scholar.google.com/scholar_lookup?title=A+review+on+demand+side+management:+Definition,+scope,+challenges+and+benefits&conference=Proceedings+of+the+2022+8th+International+Conference+on+Smart+Structures+and+Systems+(ICSSS)&author=Premkumar,+R.&author=Shanmugasundaram,+R.&author=Thilak,+K.R.&author=Balaji,+A.N.V.S.S.&publication_year=2022&pages=1%E2%80%934&doi=10.1109/ICSSS54381.2022.9782161" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ICSSS54381.2022.9782161" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B12-mathematics-12-03249' class='html-xx' data-content='12.'>Jazaeri, J. Demand side management in China: Market, policies, and future trends. In Proceedings of the 2022 IEEE Sustainable Power and Energy Conference (iSPEC), Perth, Australia, 4–7 December 2022; pp. 1–4. [<a href="https://scholar.google.com/scholar_lookup?title=Demand+side+management+in+China:+Market,+policies,+and+future+trends&conference=Proceedings+of+the+2022+IEEE+Sustainable+Power+and+Energy+Conference+(iSPEC)&author=Jazaeri,+J.&publication_year=2022&pages=1%E2%80%934&doi=10.1109/iSPEC54162.2022.10033042" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/iSPEC54162.2022.10033042" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B13-mathematics-12-03249' class='html-xx' data-content='13.'>Zhang, L.; Zhou, S.; An, J.; Kang, Q. Demand-side management optimization in electric vehicles battery swapping service. <span class='html-italic'>IEEE Access</span> <b>2019</b>, <span class='html-italic'>7</span>, 95224–95232. [<a href="https://scholar.google.com/scholar_lookup?title=Demand-side+management+optimization+in+electric+vehicles+battery+swapping+service&author=Zhang,+L.&author=Zhou,+S.&author=An,+J.&author=Kang,+Q.&publication_year=2019&journal=IEEE+Access&volume=7&pages=95224%E2%80%9395232&doi=10.1109/ACCESS.2019.2928312" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ACCESS.2019.2928312" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B14-mathematics-12-03249' class='html-xx' data-content='14.'>Wang, W.; Cai, D.; Liu, H.; Chen, R.; Chen, Q.; Liu, C. Review of the development of Hubei power grid during the 13th Five-Year Plan period and prospect of the 14th Five-Year Plan. In Proceedings of the 2022 IEEE 5th International Electrical and Energy Conference (CIEEC), Nangjing, China, 27–29 May 2022; pp. 578–583. [<a href="https://scholar.google.com/scholar_lookup?title=Review+of+the+development+of+Hubei+power+grid+during+the+13th+Five-Year+Plan+period+and+prospect+of+the+14th+Five-Year+Plan&conference=Proceedings+of+the+2022+IEEE+5th+International+Electrical+and+Energy+Conference+(CIEEC)&author=Wang,+W.&author=Cai,+D.&author=Liu,+H.&author=Chen,+R.&author=Chen,+Q.&author=Liu,+C.&publication_year=2022&pages=578%E2%80%93583&doi=10.1109/CIEEC54735.2022.9846721" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/CIEEC54735.2022.9846721" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B15-mathematics-12-03249' class='html-xx' data-content='15.'>Cheng, L.; Liu, G.; Huang, H.; Wang, X.; Chen, Y.; Zhang, J.; Meng, A.; Yang, R.; Yu, T. Equilibrium analysis of general N-population multi-strategy games for generation-side long-term bidding: An evolutionary game perspective. <span class='html-italic'>J. Clean. Prod.</span> <b>2020</b>, <span class='html-italic'>276</span>, 124123. [<a href="https://scholar.google.com/scholar_lookup?title=Equilibrium+analysis+of+general+N-population+multi-strategy+games+for+generation-side+long-term+bidding:+An+evolutionary+game+perspective&author=Cheng,+L.&author=Liu,+G.&author=Huang,+H.&author=Wang,+X.&author=Chen,+Y.&author=Zhang,+J.&author=Meng,+A.&author=Yang,+R.&author=Yu,+T.&publication_year=2020&journal=J.+Clean.+Prod.&volume=276&pages=124123&doi=10.1016/j.jclepro.2020.124123" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.jclepro.2020.124123" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B16-mathematics-12-03249' class='html-xx' data-content='16.'>He, J.; Li, Y.; Li, H.; Tong, H.; Yuan, Z.; Yang, X.; Huang, W. Application of game theory in integrated energy system systems: A review. <span class='html-italic'>IEEE Access</span> <b>2020</b>, <span class='html-italic'>8</span>, 93380–93397. [<a href="https://scholar.google.com/scholar_lookup?title=Application+of+game+theory+in+integrated+energy+system+systems:+A+review&author=He,+J.&author=Li,+Y.&author=Li,+H.&author=Tong,+H.&author=Yuan,+Z.&author=Yang,+X.&author=Huang,+W.&publication_year=2020&journal=IEEE+Access&volume=8&pages=93380%E2%80%9393397&doi=10.1109/ACCESS.2020.2994133" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ACCESS.2020.2994133" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B17-mathematics-12-03249' class='html-xx' data-content='17.'>Yamaguchi, S.; Iwasa, Y. Evolutionary game in an androdioecious population: Coupling of outcrossing and male production. <span class='html-italic'>J. Theor. Biol.</span> <b>2021</b>, <span class='html-italic'>513</span>, 110594. [<a href="https://scholar.google.com/scholar_lookup?title=Evolutionary+game+in+an+androdioecious+population:+Coupling+of+outcrossing+and+male+production&author=Yamaguchi,+S.&author=Iwasa,+Y.&publication_year=2021&journal=J.+Theor.+Biol.&volume=513&pages=110594&doi=10.1016/j.jtbi.2021.110594&pmid=33460652" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.jtbi.2021.110594" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>] [<a href="https://www.ncbi.nlm.nih.gov/pubmed/33460652" class='cross-ref pub_med' data-typ='pmid' target='_blank' rel='noopener noreferrer'>PubMed</a>]</li><li id='B18-mathematics-12-03249' class='html-xx' data-content='18.'>Nekouei, E.; Alpcan, T.; Chattopadhyay, D. Game-theoretic frameworks for demand response in electricity markets. <span class='html-italic'>IEEE Trans. Smart Grid</span> <b>2015</b>, <span class='html-italic'>6</span>, 748–758. [<a href="https://scholar.google.com/scholar_lookup?title=Game-theoretic+frameworks+for+demand+response+in+electricity+markets&author=Nekouei,+E.&author=Alpcan,+T.&author=Chattopadhyay,+D.&publication_year=2015&journal=IEEE+Trans.+Smart+Grid&volume=6&pages=748%E2%80%93758&doi=10.1109/TSG.2014.2367494" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/TSG.2014.2367494" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B19-mathematics-12-03249' class='html-xx' data-content='19.'>Ning, N.; Wang, G.; Xie, M.; Tao, G.; Huang, B.; Lu, Y.; Li, Y. Two-stage robust optimization for generation-transmission collaborative expansion planning based on potential game theory. In Proceedings of the 2023 IEEE 6th International Electrical and Energy Conference (CIEEC), Hefei, China, 12–14 May 2023; pp. 177–182. [<a href="https://scholar.google.com/scholar_lookup?title=Two-stage+robust+optimization+for+generation-transmission+collaborative+expansion+planning+based+on+potential+game+theory&conference=Proceedings+of+the+2023+IEEE+6th+International+Electrical+and+Energy+Conference+(CIEEC)&author=Ning,+N.&author=Wang,+G.&author=Xie,+M.&author=Tao,+G.&author=Huang,+B.&author=Lu,+Y.&author=Li,+Y.&publication_year=2023&pages=177%E2%80%93182&doi=10.1109/CIEEC58067.2023.10166142" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/CIEEC58067.2023.10166142" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B20-mathematics-12-03249' class='html-xx' data-content='20.'>Xu, N.; Chu, X.; Ye, H. Active power coordination for a large population of grid-forming and grid-following inverters based on mean field games theory. <span class='html-italic'>IEEE Access</span> <b>2023</b>, <span class='html-italic'>11</span>, 90052–90064. [<a href="https://scholar.google.com/scholar_lookup?title=Active+power+coordination+for+a+large+population+of+grid-forming+and+grid-following+inverters+based+on+mean+field+games+theory&author=Xu,+N.&author=Chu,+X.&author=Ye,+H.&publication_year=2023&journal=IEEE+Access&volume=11&pages=90052%E2%80%9390064&doi=10.1109/ACCESS.2023.3307634" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ACCESS.2023.3307634" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B21-mathematics-12-03249' class='html-xx' data-content='21.'>Liao, Z.; Liu, Y.; Huang, M.; Huang, X. Research on optimal scheduling algorithm of microgrid energy based on game theory. In Proceedings of the 2023 IEEE 3rd International Conference on Data Science and Computer Application (ICDSCA), Dalian, China, 27–29 October 2023; pp. 1195–1201. [<a href="https://scholar.google.com/scholar_lookup?title=Research+on+optimal+scheduling+algorithm+of+microgrid+energy+based+on+game+theory&conference=Proceedings+of+the+2023+IEEE+3rd+International+Conference+on+Data+Science+and+Computer+Application+(ICDSCA)&author=Liao,+Z.&author=Liu,+Y.&author=Huang,+M.&author=Huang,+X.&publication_year=2023&pages=1195%E2%80%931201&doi=10.1109/ICDSCA59871.2023.10392430" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ICDSCA59871.2023.10392430" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B22-mathematics-12-03249' class='html-xx' data-content='22.'>Quijano, N.; Ocampo-Martinez, C.; Barreiro-Gomez, J.; Obando, G.; Pantoja, A.; Mojica-Nava, E. The role of population games and evolutionary dynamics in distributed control systems: The advantages of evolutionary game theory. <span class='html-italic'>IEEE Control Syst. Mag.</span> <b>2017</b>, <span class='html-italic'>37</span>, 70–97. [<a href="https://scholar.google.com/scholar_lookup?title=The+role+of+population+games+and+evolutionary+dynamics+in+distributed+control+systems:+The+advantages+of+evolutionary+game+theory&author=Quijano,+N.&author=Ocampo-Martinez,+C.&author=Barreiro-Gomez,+J.&author=Obando,+G.&author=Pantoja,+A.&author=Mojica-Nava,+E.&publication_year=2017&journal=IEEE+Control+Syst.+Mag.&volume=37&pages=70%E2%80%9397" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B23-mathematics-12-03249' class='html-xx' data-content='23.'>Cheng, L.F.; Yu, T. Nash equilibrium-based asymptotic stability analysis of multi-group asymmetric evolutionary games in typical scenario of electricity market. <span class='html-italic'>IEEE Access</span> <b>2018</b>, <span class='html-italic'>6</span>, 32064–32086. [<a href="https://scholar.google.com/scholar_lookup?title=Nash+equilibrium-based+asymptotic+stability+analysis+of+multi-group+asymmetric+evolutionary+games+in+typical+scenario+of+electricity+market&author=Cheng,+L.F.&author=Yu,+T.&publication_year=2018&journal=IEEE+Access&volume=6&pages=32064%E2%80%9332086&doi=10.1109/ACCESS.2018.2842469" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ACCESS.2018.2842469" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B24-mathematics-12-03249' class='html-xx' data-content='24.'>Liu, X.; Zhang, Z.; Qi, W.; Wang, D. Behavior analysis with an evolutionary game theory approach for procurement bid evaluation involving technical and business experts. <span class='html-italic'>IEEE Syst. J.</span> <b>2019</b>, <span class='html-italic'>13</span>, 4374–4385. [<a href="https://scholar.google.com/scholar_lookup?title=Behavior+analysis+with+an+evolutionary+game+theory+approach+for+procurement+bid+evaluation+involving+technical+and+business+experts&author=Liu,+X.&author=Zhang,+Z.&author=Qi,+W.&author=Wang,+D.&publication_year=2019&journal=IEEE+Syst.+J.&volume=13&pages=4374%E2%80%934385&doi=10.1109/JSYST.2019.2925773" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/JSYST.2019.2925773" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B25-mathematics-12-03249' class='html-xx' data-content='25.'>Cheng, L.F.; Chen, Y.; Liu, G.Y. 2PnS-EG: A general two-population <span class='html-italic'>n</span>-strategy evolutionary game for strategic long-term bidding in a deregulated market under different market clearing mechanisms. <span class='html-italic'>Int. J. Electr. Power Energy Syst.</span> <b>2022</b>, <span class='html-italic'>142 (Pt A)</span>, 108182. [<a href="https://scholar.google.com/scholar_lookup?title=2PnS-EG:+A+general+two-population+n-strategy+evolutionary+game+for+strategic+long-term+bidding+in+a+deregulated+market+under+different+market+clearing+mechanisms&author=Cheng,+L.F.&author=Chen,+Y.&author=Liu,+G.Y.&publication_year=2022&journal=Int.+J.+Electr.+Power+Energy+Syst.&volume=142+(Pt+A)&pages=108182&doi=10.1016/j.ijepes.2022.108182" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.ijepes.2022.108182" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B26-mathematics-12-03249' class='html-xx' data-content='26.'>Wang, J.; Zhang, G.; Geng, Y.; Song, Z. The latest technology research and application prospects of the intelligent electrical apparatus. <span class='html-italic'>Trans. China Electrotech. Soc.</span> <b>2015</b>, <span class='html-italic'>30</span>, 2–11. [<a href="https://scholar.google.com/scholar_lookup?title=The+latest+technology+research+and+application+prospects+of+the+intelligent+electrical+apparatus&author=Wang,+J.&author=Zhang,+G.&author=Geng,+Y.&author=Song,+Z.&publication_year=2015&journal=Trans.+China+Electrotech.+Soc.&volume=30&pages=2%E2%80%9311&doi=10.19595/j.cnki.1000-6753.tces.2015.09.001" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.19595/j.cnki.1000-6753.tces.2015.09.001" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B27-mathematics-12-03249' class='html-xx' data-content='27.'>Zhou, J.; Hu, Z.; Tian, J.; Xiao, X. A power integrated resource strategic planning model and its application. <span class='html-italic'>Autom. Electr. Power Syst.</span> <b>2010</b>, <span class='html-italic'>34</span>, 19–22. [<a href="https://scholar.google.com/scholar_lookup?title=A+power+integrated+resource+strategic+planning+model+and+its+application&author=Zhou,+J.&author=Hu,+Z.&author=Tian,+J.&author=Xiao,+X.&publication_year=2010&journal=Autom.+Electr.+Power+Syst.&volume=34&pages=19%E2%80%9322" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B28-mathematics-12-03249' class='html-xx' data-content='28.'>Wang, Y.L.; Li, Y.; Ye, J.W. Research on development model of power demand side management during electric power system reform. <span class='html-italic'>Electr. Power</span> <b>2016</b>, <span class='html-italic'>49</span>, 75–81. [<a href="https://scholar.google.com/scholar_lookup?title=Research+on+development+model+of+power+demand+side+management+during+electric+power+system+reform&author=Wang,+Y.L.&author=Li,+Y.&author=Ye,+J.W.&publication_year=2016&journal=Electr.+Power&volume=49&pages=75%E2%80%9381" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B29-mathematics-12-03249' class='html-xx' data-content='29.'>Zhou, F.Q. Upgrading orientations of power demand side management in a new era of high quality development. <span class='html-italic'>Power DSM</span> <b>2020</b>, <span class='html-italic'>22</span>, 1–2. [<a href="https://scholar.google.com/scholar_lookup?title=Upgrading+orientations+of+power+demand+side+management+in+a+new+era+of+high+quality+development&author=Zhou,+F.Q.&publication_year=2020&journal=Power+DSM&volume=22&pages=1%E2%80%932&doi=10.3969/j.issn.1009-1831.2020.01.001" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.3969/j.issn.1009-1831.2020.01.001" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B30-mathematics-12-03249' class='html-xx' data-content='30.'>Jiang, J.L.; Shen, Y.J. Analysis on regulation of demand side management based on evolutionary game. <span class='html-italic'>J. South China Univ. Technol. (Soc. Sci. Ed.)</span> <b>2012</b>, <span class='html-italic'>14</span>, 50–56. [<a href="https://scholar.google.com/scholar_lookup?title=Analysis+on+regulation+of+demand+side+management+based+on+evolutionary+game&author=Jiang,+J.L.&author=Shen,+Y.J.&publication_year=2012&journal=J.+South+China+Univ.+Technol.+(Soc.+Sci.+Ed.)&volume=14&pages=50%E2%80%9356&doi=10.19366/j.cnki.1009-055x.2012.05.007" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.19366/j.cnki.1009-055x.2012.05.007" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B31-mathematics-12-03249' class='html-xx' data-content='31.'>Sun, Y.; Jia, M.; Lu, J.; Sun, C. Interactive evaluation model for residential electricity in demand side management. automation of electric power systems. <span class='html-italic'>Autom. Electr. Power Syst.</span> <b>2017</b>, <span class='html-italic'>41</span>, 62–69. [<a href="https://scholar.google.com/scholar_lookup?title=Interactive+evaluation+model+for+residential+electricity+in+demand+side+management.+automation+of+electric+power+systems&author=Sun,+Y.&author=Jia,+M.&author=Lu,+J.&author=Sun,+C.&publication_year=2017&journal=Autom.+Electr.+Power+Syst.&volume=41&pages=62%E2%80%9369&doi=10.7500/AEPS20160826012" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.7500/AEPS20160826012" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B32-mathematics-12-03249' class='html-xx' data-content='32.'>Mishra, M.K.; Jaafari, K.A.; Sumaiti, A.A.; Parida, S.K. Demand-side management via game theory considering distributed energy storage and generation with discomfort as objective. In Proceedings of the 2023 IEEE PES Conference on Innovative Smart Grid Technologies-Middle East (ISGT Middle East), Abu Dhabi, United Arab Emirates, 12–15 March 2023; pp. 1–5. [<a href="https://scholar.google.com/scholar_lookup?title=Demand-side+management+via+game+theory+considering+distributed+energy+storage+and+generation+with+discomfort+as+objective&conference=Proceedings+of+the+2023+IEEE+PES+Conference+on+Innovative+Smart+Grid+Technologies-Middle+East+(ISGT+Middle+East)&author=Mishra,+M.K.&author=Jaafari,+K.A.&author=Sumaiti,+A.A.&author=Parida,+S.K.&publication_year=2023&pages=1%E2%80%935&doi=10.1109/ISGTMiddleEast56437.2023.10078594" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ISGTMiddleEast56437.2023.10078594" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B33-mathematics-12-03249' class='html-xx' data-content='33.'>Ji, Z.; Liu, X.; Tang, D. Game-theoretic applications for decision-making behavior on the energy demand side: A systematic review. <span class='html-italic'>Prot. Control Mod. Power Syst.</span> <b>2024</b>, <span class='html-italic'>9</span>, 1–20. [<a href="https://scholar.google.com/scholar_lookup?title=Game-theoretic+applications+for+decision-making+behavior+on+the+energy+demand+side:+A+systematic+review&author=Ji,+Z.&author=Liu,+X.&author=Tang,+D.&publication_year=2024&journal=Prot.+Control+Mod.+Power+Syst.&volume=9&pages=1%E2%80%9320&doi=10.23919/PCMP.2023.000219" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.23919/PCMP.2023.000219" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B34-mathematics-12-03249' class='html-xx' data-content='34.'>Belhaiza, S.; Baroudi, U.; Elhallaoui, I. A game theoretic model for the multiperiodic smart grid demand response problem. <span class='html-italic'>IEEE Syst. J.</span> <b>2020</b>, <span class='html-italic'>14</span>, 1147–1158. [<a href="https://scholar.google.com/scholar_lookup?title=A+game+theoretic+model+for+the+multiperiodic+smart+grid+demand+response+problem&author=Belhaiza,+S.&author=Baroudi,+U.&author=Elhallaoui,+I.&publication_year=2020&journal=IEEE+Syst.+J.&volume=14&pages=1147%E2%80%931158&doi=10.1109/JSYST.2019.2918172" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/JSYST.2019.2918172" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B35-mathematics-12-03249' class='html-xx' data-content='35.'>Mishra, M.K.; Parida, S.K. A game theoretic approach for demand-side management using real-time variable peak pricing considering distributed energy resources. <span class='html-italic'>IEEE Syst. J.</span> <b>2022</b>, <span class='html-italic'>16</span>, 144–154. [<a href="https://scholar.google.com/scholar_lookup?title=A+game+theoretic+approach+for+demand-side+management+using+real-time+variable+peak+pricing+considering+distributed+energy+resources&author=Mishra,+M.K.&author=Parida,+S.K.&publication_year=2022&journal=IEEE+Syst.+J.&volume=16&pages=144%E2%80%93154&doi=10.1109/JSYST.2020.3033128" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/JSYST.2020.3033128" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B36-mathematics-12-03249' class='html-xx' data-content='36.'>Mei, S.; Wang, Y.; Liu, F.; Zhang, X.; Sun, Z. Game approaches for hybrid power system planning. <span class='html-italic'>IEEE Trans. Sustain. Energy</span> <b>2012</b>, <span class='html-italic'>3</span>, 506–517. [<a href="https://scholar.google.com/scholar_lookup?title=Game+approaches+for+hybrid+power+system+planning&author=Mei,+S.&author=Wang,+Y.&author=Liu,+F.&author=Zhang,+X.&author=Sun,+Z.&publication_year=2012&journal=IEEE+Trans.+Sustain.+Energy&volume=3&pages=506%E2%80%93517&doi=10.1109/TSTE.2012.2192299" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/TSTE.2012.2192299" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B37-mathematics-12-03249' class='html-xx' data-content='37.'>Voropai, N.I.; Ivanova, E.Y. Shapley game for expansion planning of generating companies at many non-coincident criteria. <span class='html-italic'>IEEE Trans. Power Syst.</span> <b>2006</b>, <span class='html-italic'>21</span>, 1630–1637. [<a href="https://scholar.google.com/scholar_lookup?title=Shapley+game+for+expansion+planning+of+generating+companies+at+many+non-coincident+criteria&author=Voropai,+N.I.&author=Ivanova,+E.Y.&publication_year=2006&journal=IEEE+Trans.+Power+Syst.&volume=21&pages=1630%E2%80%931637&doi=10.1109/TPWRS.2006.873053" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/TPWRS.2006.873053" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B38-mathematics-12-03249' class='html-xx' data-content='38.'>Xie, J.D.; Wang, S.Y.; Zhou, X.M.; Sun, B.; Sun, X. Electricity market supervision strategy based on tripartite game. <span class='html-italic'>Sci. Technol. Eng.</span> <b>2021</b>, <span class='html-italic'>21</span>, 15072–15083. [<a href="https://scholar.google.com/scholar_lookup?title=Electricity+market+supervision+strategy+based+on+tripartite+game&author=Xie,+J.D.&author=Wang,+S.Y.&author=Zhou,+X.M.&author=Sun,+B.&author=Sun,+X.&publication_year=2021&journal=Sci.+Technol.+Eng.&volume=21&pages=15072%E2%80%9315083" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B39-mathematics-12-03249' class='html-xx' data-content='39.'>Ekneligoda, N.C.; Weaver, W.W. Optimal transient control of microgrids using a game theoretic approach. In Proceedings of the 2011 IEEE Energy Conversion Congress and Exposition (ECCE), Phoenix, AZ, USA, 17–22 September 2011; pp. 935–942. [<a href="https://scholar.google.com/scholar_lookup?title=Optimal+transient+control+of+microgrids+using+a+game+theoretic+approach&conference=Proceedings+of+the+2011+IEEE+Energy+Conversion+Congress+and+Exposition+(ECCE)&author=Ekneligoda,+N.C.&author=Weaver,+W.W.&publication_year=2011&pages=935%E2%80%93942&doi=10.1109/ECCE.2011.6063872" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ECCE.2011.6063872" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B40-mathematics-12-03249' class='html-xx' data-content='40.'>Gao, J.; Sheng, Z.H. Elementary groping for evolutionary game theory and its application in electricity market. <span class='html-italic'>Autom. Electr. Power Syst.</span> <b>2003</b>, <span class='html-italic'>27</span>, 18–21. [<a href="https://scholar.google.com/scholar_lookup?title=Elementary+groping+for+evolutionary+game+theory+and+its+application+in+electricity+market&author=Gao,+J.&author=Sheng,+Z.H.&publication_year=2003&journal=Autom.+Electr.+Power+Syst.&volume=27&pages=18%E2%80%9321" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B41-mathematics-12-03249' class='html-xx' data-content='41.'>Li, X.R.; Liu, C.Q.; Zhong, J.J.; Zhang, C.; Lu, Y. Evolutionary stability analysis of generation side bidding ini day-ahead market based on evolutionary game theory. <span class='html-italic'>Electrotech. Electr.</span> <b>2020</b>, <span class='html-italic'>10</span>, 8–12. [<a href="https://scholar.google.com/scholar_lookup?title=Evolutionary+stability+analysis+of+generation+side+bidding+ini+day-ahead+market+based+on+evolutionary+game+theory&author=Li,+X.R.&author=Liu,+C.Q.&author=Zhong,+J.J.&author=Zhang,+C.&author=Lu,+Y.&publication_year=2020&journal=Electrotech.+Electr.&volume=10&pages=8%E2%80%9312" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B42-mathematics-12-03249' class='html-xx' data-content='42.'>Sun, Y.T.; Song, Y.Q. Study on two-level game model between retailers and users in electric selling market. <span class='html-italic'>Water Resour. Power</span> <b>2018</b>, <span class='html-italic'>36</span>, 210–214. [<a href="https://scholar.google.com/scholar_lookup?title=Study+on+two-level+game+model+between+retailers+and+users+in+electric+selling+market&author=Sun,+Y.T.&author=Song,+Y.Q.&publication_year=2018&journal=Water+Resour.+Power&volume=36&pages=210%E2%80%93214" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B43-mathematics-12-03249' class='html-xx' data-content='43.'>Xu, P. Research on Generation Expansion Planning under the Background of Electricity Market Based on Evolutionary Game. Master’ Thesis, North China Electric Power University, Beijing, China, March 2017. [<a href="https://scholar.google.com/scholar_lookup?title=Research+on+Generation+Expansion+Planning+under+the+Background+of+Electricity+Market+Based+on+Evolutionary+Game&author=Xu,+P.&publication_year=2017" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B44-mathematics-12-03249' class='html-xx' data-content='44.'>Zhu, B.; Xia, X.H.; Wu, Z. Evolutionary game theoretic demand-side management and control for a class of networked smart grid. <span class='html-italic'>Automatica</span> <b>2016</b>, <span class='html-italic'>70</span>, 94–100. [<a href="https://scholar.google.com/scholar_lookup?title=Evolutionary+game+theoretic+demand-side+management+and+control+for+a+class+of+networked+smart+grid&author=Zhu,+B.&author=Xia,+X.H.&author=Wu,+Z.&publication_year=2016&journal=Automatica&volume=70&pages=94%E2%80%93100&doi=10.1016/j.automatica.2016.03.027" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.automatica.2016.03.027" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B45-mathematics-12-03249' class='html-xx' data-content='45.'>Miorandi, D.; Pellegrini, D.F. Demand-side management in smart grids: An evolutionary games perspective. In Proceedings of the 6th International ICST Conference on Performance Evaluation Methodologies and Tools, Cargese, France, 9–12 October 2012; IEEE: New York, NY, USA, 2012; pp. 178–187. [<a href="https://scholar.google.com/scholar_lookup?title=Demand-side+management+in+smart+grids:+An+evolutionary+games+perspective&conference=Proceedings+of+the+6th+International+ICST+Conference+on+Performance+Evaluation+Methodologies+and+Tools&author=Miorandi,+D.&author=Pellegrini,+D.F.&publication_year=2012&pages=178%E2%80%93187&doi=10.4108/valuetools.2012.250351" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.4108/valuetools.2012.250351" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B46-mathematics-12-03249' class='html-xx' data-content='46.'>Cheng, L.F.; Yang, R.; Liu, G.Y.; Wang, J. Multi-population asymmetric evolutionary game dynamics and its applications in power demand-side response in smart grid. <span class='html-italic'>Proc. CSEE</span> <b>2020</b>, <span class='html-italic'>40</span>, 20–36. [<a href="https://scholar.google.com/scholar_lookup?title=Multi-population+asymmetric+evolutionary+game+dynamics+and+its+applications+in+power+demand-side+response+in+smart+grid&author=Cheng,+L.F.&author=Yang,+R.&author=Liu,+G.Y.&author=Wang,+J.&publication_year=2020&journal=Proc.+CSEE&volume=40&pages=20%E2%80%9336&doi=10.13334/j.0258-8013.pcsee.200930" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.13334/j.0258-8013.pcsee.200930" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B47-mathematics-12-03249' class='html-xx' data-content='47.'>Zhu, Z.Y.; Gao, B.T. Research on the participation degree of smart power consumption based on evolutionary game theory. <span class='html-italic'>Power DSM</span> <b>2017</b>, <span class='html-italic'>19</span>, 9–13. [<a href="https://scholar.google.com/scholar_lookup?title=Research+on+the+participation+degree+of+smart+power+consumption+based+on+evolutionary+game+theory&author=Zhu,+Z.Y.&author=Gao,+B.T.&publication_year=2017&journal=Power+DSM&volume=19&pages=9%E2%80%9313&doi=10.3969/j.issn.1009-1831.2017.02.003" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.3969/j.issn.1009-1831.2017.02.003" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B48-mathematics-12-03249' class='html-xx' data-content='48.'>Chai, B.; Chen, J.; Yang, Z.; Zhang, Y. Demand response management with multiple utility companies: A two-level game approach. <span class='html-italic'>IEEE Trans. Smart Grid</span> <b>2014</b>, <span class='html-italic'>5</span>, 722–731. [<a href="https://scholar.google.com/scholar_lookup?title=Demand+response+management+with+multiple+utility+companies:+A+two-level+game+approach&author=Chai,+B.&author=Chen,+J.&author=Yang,+Z.&author=Zhang,+Y.&publication_year=2014&journal=IEEE+Trans.+Smart+Grid&volume=5&pages=722%E2%80%93731&doi=10.1109/TSG.2013.2295024" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/TSG.2013.2295024" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B49-mathematics-12-03249' class='html-xx' data-content='49.'>Neumann, J.L.V.; Morgenstern, O.V. <span class='html-italic'>The Theory of Games and Economic Behavior</span>; Princeton University Press: Princeton, NJ, USA, 1972. [<a href="https://scholar.google.com/scholar_lookup?title=The+Theory+of+Games+and+Economic+Behavior&author=Neumann,+J.L.V.&author=Morgenstern,+O.V.&publication_year=1972" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.geb.2009.09.008" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B50-mathematics-12-03249' class='html-xx' data-content='50.'>Cheng, L.F.; Yin, L.F.; Wang, J.H.; Shen, T.; Chen, Y.; Liu, G.Y.; Yu, T. Behavioral decision-making in power demand-side response management: A multi-population evolutionary game dynamics perspective. <span class='html-italic'>Int. J. Electr. Power Energy Syst.</span> <b>2021</b>, <span class='html-italic'>129</span>, 106743. [<a href="https://scholar.google.com/scholar_lookup?title=Behavioral+decision-making+in+power+demand-side+response+management:+A+multi-population+evolutionary+game+dynamics+perspective&author=Cheng,+L.F.&author=Yin,+L.F.&author=Wang,+J.H.&author=Shen,+T.&author=Chen,+Y.&author=Liu,+G.Y.&author=Yu,+T.&publication_year=2021&journal=Int.+J.+Electr.+Power+Energy+Syst.&volume=129&pages=106743&doi=10.1016/j.ijepes.2020.106743" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.ijepes.2020.106743" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B51-mathematics-12-03249' class='html-xx' data-content='51.'>Cheng, L.F.; Zhang, J.; Yin, L.F.; Chen, Y.; Wang, J.H.; Liu, G.Y.; Wang, X.G.; Zhang, D.X. General three-population multi-strategy evolutionary games for long-term on-grid bidding of generation-side electricity market. <span class='html-italic'>IEEE Access</span> <b>2021</b>, <span class='html-italic'>9</span>, 5177–5198. [<a href="https://scholar.google.com/scholar_lookup?title=General+three-population+multi-strategy+evolutionary+games+for+long-term+on-grid+bidding+of+generation-side+electricity+market&author=Cheng,+L.F.&author=Zhang,+J.&author=Yin,+L.F.&author=Chen,+Y.&author=Wang,+J.H.&author=Liu,+G.Y.&author=Wang,+X.G.&author=Zhang,+D.X.&publication_year=2021&journal=IEEE+Access&volume=9&pages=5177%E2%80%935198&doi=10.1109/ACCESS.2020.3046327" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/ACCESS.2020.3046327" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B52-mathematics-12-03249' class='html-xx' data-content='52.'>Mojica-Nava, E.; Macana, C.A.; Quijano, N. Dynamic population games for optimal dispatch on hierarchical microgrid control. <span class='html-italic'>IEEE Trans. Syst. Man Cybern. Syst.</span> <b>2014</b>, <span class='html-italic'>44</span>, 306–317. [<a href="https://scholar.google.com/scholar_lookup?title=Dynamic+population+games+for+optimal+dispatch+on+hierarchical+microgrid+control&author=Mojica-Nava,+E.&author=Macana,+C.A.&author=Quijano,+N.&publication_year=2014&journal=IEEE+Trans.+Syst.+Man+Cybern.+Syst.&volume=44&pages=306%E2%80%93317&doi=10.1109/TSMCC.2013.2266117" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/TSMCC.2013.2266117" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B53-mathematics-12-03249' class='html-xx' data-content='53.'>Ovalle, A.; Hably, A.; Bacha, S.; Ramos, G.; Hossain, J.M. Escort evolutionary game dynamics approach for integral load management of electric vehicle fleets. <span class='html-italic'>IEEE Trans. Ind. Electron.</span> <b>2017</b>, <span class='html-italic'>64</span>, 1358–1369. [<a href="https://scholar.google.com/scholar_lookup?title=Escort+evolutionary+game+dynamics+approach+for+integral+load+management+of+electric+vehicle+fleets&author=Ovalle,+A.&author=Hably,+A.&author=Bacha,+S.&author=Ramos,+G.&author=Hossain,+J.M.&publication_year=2017&journal=IEEE+Trans.+Ind.+Electron.&volume=64&pages=1358%E2%80%931369&doi=10.1109/TIE.2016.2615042" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1109/TIE.2016.2615042" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B54-mathematics-12-03249' class='html-xx' data-content='54.'>Della Marca, R.; Loy, N.; Menale, M. Intransigent vs. volatile opinions in a kinetic epidemic model with imitation game dynamics. <span class='html-italic'>Math. Med. Biol. A J. IMA</span> <b>2023</b>, <span class='html-italic'>40</span>, 111–140. [<a href="https://scholar.google.com/scholar_lookup?title=Intransigent+vs.+volatile+opinions+in+a+kinetic+epidemic+model+with+imitation+game+dynamics&author=Della+Marca,+R.&author=Loy,+N.&author=Menale,+M.&publication_year=2023&journal=Math.+Med.+Biol.+A+J.+IMA&volume=40&pages=111%E2%80%93140&doi=10.1093/imammb/dqac018&pmid=36482506" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1093/imammb/dqac018" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>] [<a href="https://www.ncbi.nlm.nih.gov/pubmed/36482506" class='cross-ref pub_med' data-typ='pmid' target='_blank' rel='noopener noreferrer'>PubMed</a>]</li><li id='B55-mathematics-12-03249' class='html-xx' data-content='55.'>Carbonaro, B.; Menale, M. Markov Chains and Kinetic Theory: A Possible Application to Socio-Economic Problems. <span class='html-italic'>Mathematics</span> <b>2024</b>, <span class='html-italic'>12</span>, 1571. [<a href="https://scholar.google.com/scholar_lookup?title=Markov+Chains+and+Kinetic+Theory:+A+Possible+Application+to+Socio-Economic+Problems&author=Carbonaro,+B.&author=Menale,+M.&publication_year=2024&journal=Mathematics&volume=12&pages=1571&doi=10.3390/math12101571" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.3390/math12101571" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B56-mathematics-12-03249' class='html-xx' data-content='56.'>Poletti, P.; Ajelli, M.; Merler, S. Behavioral changes and adaptation induced by epidemics. In <span class='html-italic'>Social Phenomena: From Data Analysis to Models</span>; Springer International Publishing: Cham, Switzerland, 2015; pp. 155–175. [<a href="https://scholar.google.com/scholar_lookup?title=Behavioral+changes+and+adaptation+induced+by+epidemics&author=Poletti,+P.&author=Ajelli,+M.&author=Merler,+S.&publication_year=2015&pages=155%E2%80%93175" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li></ol></section><section id='FiguresandTables' type='display-objects'><div class="html-fig-wrap" id="mathematics-12-03249-f001"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f001"> <img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001.png" alt="Mathematics 12 03249 g001" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f001"></a> </div> </div> <div class="html-fig_description"> <b>Figure 1.</b> Schematic diagram of symmetric and asymmetric evolutionary game structures [<a href="#B46-mathematics-12-03249" class="html-bibr">46</a>]. In the figure (<b>a</b>), it illustrates the structure of a symmetric evolutionary game. In figure (<b>b</b>), it demonstrates the structure of an asymmetric evolutionary game.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_mathematics-12-03249-f001"> <div class="html-caption"> <b>Figure 1.</b> Schematic diagram of symmetric and asymmetric evolutionary game structures [<a href="#B46-mathematics-12-03249" class="html-bibr">46</a>]. In the figure (<b>a</b>), it illustrates the structure of a symmetric evolutionary game. In figure (<b>b</b>), it demonstrates the structure of an asymmetric evolutionary game.</div> <div class="html-img"><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001.png" alt="Mathematics 12 03249 g001" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001.png" /></div> </div> <div class="html-fig-wrap" id="mathematics-12-03249-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f002"> <img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002a.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002a.png" alt="Mathematics 12 03249 g002a" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002a-550.jpg" /><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002b.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002b.png" alt="Mathematics 12 03249 g002b" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002b-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> The simulation results of the 2P2S-AEG system. In figure (<b>a</b>), it illustrates the evolution trend of the proportion <span class='html-italic'>x</span> of individuals selecting strategy S<sub>A1</sub> in group A over time <span class='html-italic'>t</span>. In figure (<b>b</b>), it demonstrates the evolution trend of the proportion <span class='html-italic'>y</span> of individuals selecting strategy S<sub>B1</sub> in group B over time <span class='html-italic'>t</span>. In figure (<b>c</b>), it reveals the evolution trend of (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>) over time <span class='html-italic'>t</span> in the game system.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_mathematics-12-03249-f002"> <div class="html-caption"> <b>Figure 2.</b> The simulation results of the 2P2S-AEG system. In figure (<b>a</b>), it illustrates the evolution trend of the proportion <span class='html-italic'>x</span> of individuals selecting strategy S<sub>A1</sub> in group A over time <span class='html-italic'>t</span>. In figure (<b>b</b>), it demonstrates the evolution trend of the proportion <span class='html-italic'>y</span> of individuals selecting strategy S<sub>B1</sub> in group B over time <span class='html-italic'>t</span>. In figure (<b>c</b>), it reveals the evolution trend of (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>) over time <span class='html-italic'>t</span> in the game system.</div> <div class="html-img"><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002a.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002a.png" alt="Mathematics 12 03249 g002a" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002a.png" /><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002b.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002b.png" alt="Mathematics 12 03249 g002b" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002b.png" /></div> </div> <div class="html-fig-wrap" id="mathematics-12-03249-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f003"> <img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g003.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g003.png" alt="Mathematics 12 03249 g003" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> The tripartite evolutionary game framework among the government, the power grid company, and power users in demand-side management (DSM). This figure illustrates the strategic interactions among the three parties, where each entity’s strategy selection impacts the others’ payoffs. The government incentivizes or refrains from incentivizing DSM policies, the grid company decides whether to implement these policies, and the power users choose to participate or opt out of DSM services. The figure reflects the dynamic evolution of these strategic choices over time, with the goal of achieving a stable evolutionary equilibrium among all participants.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_mathematics-12-03249-f003"> <div class="html-caption"> <b>Figure 3.</b> The tripartite evolutionary game framework among the government, the power grid company, and power users in demand-side management (DSM). This figure illustrates the strategic interactions among the three parties, where each entity’s strategy selection impacts the others’ payoffs. The government incentivizes or refrains from incentivizing DSM policies, the grid company decides whether to implement these policies, and the power users choose to participate or opt out of DSM services. The figure reflects the dynamic evolution of these strategic choices over time, with the goal of achieving a stable evolutionary equilibrium among all participants.</div> <div class="html-img"><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g003.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g003.png" alt="Mathematics 12 03249 g003" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g003.png" /></div> </div> <div class="html-fig-wrap" id="mathematics-12-03249-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f004"> <img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004a.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004a.png" alt="Mathematics 12 03249 g004a" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004a-550.jpg" /><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004b.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004b.png" alt="Mathematics 12 03249 g004b" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004b-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (6, 1, 3, 7, 2, 4) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> > 0 and −<span class='html-italic'>d</span> < <span class='html-italic'>e</span> − <span class='html-italic'>f</span> < 0. In figure (<b>b</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (6, 1, 3, 4, 2, 7) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> < 0 and <span class='html-italic'>e</span> − <span class='html-italic'>f</span> + <span class='html-italic'>d</span> < 0. In figure (<b>c</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (3, 4, 1, 4, 2, 7) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> > 0 and <span class='html-italic'>e</span> − <span class='html-italic'>f</span> + <span class='html-italic'>d</span> < 0. In figure (<b>d</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (3, 4, 1, 7, 2, 4) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> < 0 and −<span class='html-italic'>d</span> < <span class='html-italic'>e</span> − <span class='html-italic'>f</span> < 0. In figure (<b>e</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (5, 2, 4, 3, 1, 6) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> < 0 and <span class='html-italic'>e</span> − <span class='html-italic'>f</span> > 0. In figure (<b>f</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (7, 2, 4, 3, 6, 5) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> > 0 and <span class='html-italic'>e</span> − <span class='html-italic'>f</span> > 0.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_mathematics-12-03249-f004"> <div class="html-caption"> <b>Figure 4.</b> The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (6, 1, 3, 7, 2, 4) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> > 0 and −<span class='html-italic'>d</span> < <span class='html-italic'>e</span> − <span class='html-italic'>f</span> < 0. In figure (<b>b</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (6, 1, 3, 4, 2, 7) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> < 0 and <span class='html-italic'>e</span> − <span class='html-italic'>f</span> + <span class='html-italic'>d</span> < 0. In figure (<b>c</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (3, 4, 1, 4, 2, 7) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> > 0 and <span class='html-italic'>e</span> − <span class='html-italic'>f</span> + <span class='html-italic'>d</span> < 0. In figure (<b>d</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (3, 4, 1, 7, 2, 4) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> < 0 and −<span class='html-italic'>d</span> < <span class='html-italic'>e</span> − <span class='html-italic'>f</span> < 0. In figure (<b>e</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (5, 2, 4, 3, 1, 6) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> < 0 and <span class='html-italic'>e</span> − <span class='html-italic'>f</span> > 0. In figure (<b>f</b>), it assumes (<span class='html-italic'>a</span>, <span class='html-italic'>b</span>, <span class='html-italic'>c</span>, <span class='html-italic'>d</span>, <span class='html-italic'>e</span>, <span class='html-italic'>f</span>) = (7, 2, 4, 3, 6, 5) for meeting <span class='html-italic'>a</span> − <span class='html-italic'>b</span> − <span class='html-italic'>c</span> > 0 and <span class='html-italic'>e</span> − <span class='html-italic'>f</span> > 0.</div> <div class="html-img"><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004a.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004a.png" alt="Mathematics 12 03249 g004a" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004a.png" /><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004b.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004b.png" alt="Mathematics 12 03249 g004b" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004b.png" /></div> </div> <div class="html-fig-wrap" id="mathematics-12-03249-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f005"> <img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005a.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005a.png" alt="Mathematics 12 03249 g005a" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005a-550.jpg" /><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005b.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005b.png" alt="Mathematics 12 03249 g005b" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005b-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (7, 5, 1, 6, 2, 1, 4) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ > 0 and −<span class='html-italic'>d</span>′ < <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ < 0. In figure (<b>b</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (7, 4, 1, 3, 2, 5, 12) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ > 0 and <span class='html-italic'>d</span>′ + <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ < 0. In figure (<b>c</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (6, 5, 2, 3, 2, 1, 9) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ < 0 and <span class='html-italic'>d</span>′ + <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ < 0. In figure (<b>d</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (6, 5, 2, 7, 3, 1, 6) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ < 0 and −<span class='html-italic'>d</span>′ < <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ < 0. In figure (<b>e</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (6, 1, 2, 7, 3, 4, 5) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ > 0 and <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ > 0. In figure (<b>f</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (3, 4, 1, 7, 2, 6, 5) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ < 0 and <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ > 0.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_mathematics-12-03249-f005"> <div class="html-caption"> <b>Figure 5.</b> The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (7, 5, 1, 6, 2, 1, 4) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ > 0 and −<span class='html-italic'>d</span>′ < <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ < 0. In figure (<b>b</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (7, 4, 1, 3, 2, 5, 12) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ > 0 and <span class='html-italic'>d</span>′ + <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ < 0. In figure (<b>c</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (6, 5, 2, 3, 2, 1, 9) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ < 0 and <span class='html-italic'>d</span>′ + <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ < 0. In figure (<b>d</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (6, 5, 2, 7, 3, 1, 6) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ < 0 and −<span class='html-italic'>d</span>′ < <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ < 0. In figure (<b>e</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (6, 1, 2, 7, 3, 4, 5) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ > 0 and <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ > 0. In figure (<b>f</b>), it assumes (<span class='html-italic'>a</span>′, <span class='html-italic'>b</span>′, <span class='html-italic'>c</span>′, <span class='html-italic'>d</span>′, <span class='html-italic'>e</span>′, <span class='html-italic'>f</span>′) = (3, 4, 1, 7, 2, 6, 5) for meeting <span class='html-italic'>a</span>′ − <span class='html-italic'>b</span>′ − <span class='html-italic'>c</span>′ < 0 and <span class='html-italic'>e</span>′ + <span class='html-italic'>f</span>′ − <span class='html-italic'>g</span>′ > 0.</div> <div class="html-img"><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005a.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005a.png" alt="Mathematics 12 03249 g005a" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005a.png" /><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005b.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005b.png" alt="Mathematics 12 03249 g005b" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005b.png" /></div> </div> <div class="html-fig-wrap" id="mathematics-12-03249-f006"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f006"> <img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g006.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g006.png" alt="Mathematics 12 03249 g006" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g006-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f006"></a> </div> </div> <div class="html-fig_description"> <b>Figure 6.</b> The simulation results of the phase trajectory of this government-power grid-power user three-group asymmetric evolutionary game system when taking 1000 times of repeated evolution game. Figure (<b>a</b>) shows the phase trajectory of (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>, <span class='html-italic'>z</span>), and Figure (<b>b</b>) demonstrates the phase trajectories of (<span class='html-italic'>x</span>, <span class='html-italic'>t</span>), (<span class='html-italic'>y</span>, <span class='html-italic'>t</span>) and (<span class='html-italic'>z</span>, <span class='html-italic'>t</span>).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_mathematics-12-03249-f006"> <div class="html-caption"> <b>Figure 6.</b> The simulation results of the phase trajectory of this government-power grid-power user three-group asymmetric evolutionary game system when taking 1000 times of repeated evolution game. Figure (<b>a</b>) shows the phase trajectory of (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>, <span class='html-italic'>z</span>), and Figure (<b>b</b>) demonstrates the phase trajectories of (<span class='html-italic'>x</span>, <span class='html-italic'>t</span>), (<span class='html-italic'>y</span>, <span class='html-italic'>t</span>) and (<span class='html-italic'>z</span>, <span class='html-italic'>t</span>).</div> <div class="html-img"><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g006.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g006.png" alt="Mathematics 12 03249 g006" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g006.png" /></div> </div> <div class="html-fig-wrap" id="mathematics-12-03249-f007"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f007"> <img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g007.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g007.png" alt="Mathematics 12 03249 g007" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g007-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#fig_body_display_mathematics-12-03249-f007"></a> </div> </div> <div class="html-fig_description"> <b>Figure 7.</b> The simulation results of the phase trajectory of this government-power grid-power user three-group asymmetric evolutionary game system when taking 2000 times of repeated evolution game. Figure (<b>a</b>) shows the phase trajectory of (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>, <span class='html-italic'>z</span>), and Figure (<b>b</b>) demonstrates the phase trajectories of (<span class='html-italic'>x</span>, <span class='html-italic'>t</span>), (<span class='html-italic'>y</span>, <span class='html-italic'>t</span>) and (<span class='html-italic'>z</span>, <span class='html-italic'>t</span>).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_mathematics-12-03249-f007"> <div class="html-caption"> <b>Figure 7.</b> The simulation results of the phase trajectory of this government-power grid-power user three-group asymmetric evolutionary game system when taking 2000 times of repeated evolution game. Figure (<b>a</b>) shows the phase trajectory of (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>, <span class='html-italic'>z</span>), and Figure (<b>b</b>) demonstrates the phase trajectories of (<span class='html-italic'>x</span>, <span class='html-italic'>t</span>), (<span class='html-italic'>y</span>, <span class='html-italic'>t</span>) and (<span class='html-italic'>z</span>, <span class='html-italic'>t</span>).</div> <div class="html-img"><img data-large="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g007.png" data-original="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g007.png" alt="Mathematics 12 03249 g007" data-lsrc="/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g007.png" /></div> </div> <div class="html-table-wrap" id="mathematics-12-03249-t001"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href='#table_body_display_mathematics-12-03249-t001'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#table_body_display_mathematics-12-03249-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> Calculations of the 2P2S-AEG system at 5 internal equilibrium points for the general case. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_mathematics-12-03249-t001"> <div class="html-caption"><b>Table 1.</b> Calculations of the 2P2S-AEG system at 5 internal equilibrium points for the general case.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Internal Equilibrium Points</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Det(<span class='html-italic'>J</span>)</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Tr(<span class='html-italic'>J</span>)</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Asymptotic Stability Conditions</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>1</sub>(0, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>2</sub>(0, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>3</sub>(1, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>4</sub>(1, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>y</span>*)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <msub> <mi>γ</mi> <mn>4</mn> </msub> <msubsup> <mi>γ</mi> <mn>1</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msubsup> <mi>γ</mi> <mn>3</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >/</td></tr></tbody> </table> </div> <div class="html-table-wrap" id="mathematics-12-03249-t002"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href='#table_body_display_mathematics-12-03249-t002'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#table_body_display_mathematics-12-03249-t002"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 2.</b> The definitions of the main parameters used in the government-power grid two-party evolution game model. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_mathematics-12-03249-t002"> <div class="html-caption"><b>Table 2.</b> The definitions of the main parameters used in the government-power grid two-party evolution game model.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Main Parameters</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Definitions</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The costs incurred by the government in adopting an incentive strategy for the power grid enterprises</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The costs incurred by the power grid enterprises when they implement the DSM strategy</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The government’s initial revenue</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The benefits received by the government when it adopts an incentive strategy for the power grid enterprises who tend to implement DSM at the same time</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>3</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The benefits received by the government when it does not adopt an incentive strategy for the power grid enterprises who implement the DSM on their own initiative at the same time</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>4</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The initial revenues of the power grid enterprises</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>5</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The policy benefits received by power grid enterprises when the government adopts an incentive strategy for power grid companies who implement DSM at the same time</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>6</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The system revenues gained by power grid enterprises when implementing a DSM strategy</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>η</mi> <mn>1</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >The social gains lost to the government when the power grid enterprises do not implement a DSM strategy</td></tr></tbody> </table> </div> <div class="html-table-wrap" id="mathematics-12-03249-t003"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href='#table_body_display_mathematics-12-03249-t003'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#table_body_display_mathematics-12-03249-t003"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 3.</b> The calculation results of determinant and trace for the government-power grid two-party evolution game model. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_mathematics-12-03249-t003"> <div class="html-caption"><b>Table 3.</b> The calculation results of determinant and trace for the government-power grid two-party evolution game model.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Internal Equilibrium Points</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <mi mathvariant="bold">The</mi> <mtext> </mtext> <mi mathvariant="bold">Determinant</mi> <mtext> </mtext> <mi mathvariant="bold">Value</mi> <mtext> </mtext> <mi mathvariant="bold-italic">D</mi> <mi mathvariant="bold-italic">e</mi> <mi mathvariant="bold-italic">t</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mi mathvariant="bold">G</mi> <mtext>-</mtext> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">G</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <mi mathvariant="bold">The</mi> <mtext> </mtext> <mi mathvariant="bold">Trace</mi> <mtext> </mtext> <mi mathvariant="bold">Value</mi> <mtext> </mtext> <mi mathvariant="bold-italic">T</mi> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mi mathvariant="bold">G</mi> <mtext>-</mtext> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">G</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </semantics> </math></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>1</sub>(0, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>2</sub>(0, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>3</sub>(1, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>4</sub>(1, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>y</span>*)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td></tr></tbody> </table> </div> <div class="html-table-wrap" id="mathematics-12-03249-t004"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href='#table_body_display_mathematics-12-03249-t004'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#table_body_display_mathematics-12-03249-t004"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 4.</b> Evolutionary stability conditions and mutually exclusive equilibrium points of the government-power grid two-party evolution game system at various internal equilibrium points. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_mathematics-12-03249-t004"> <div class="html-caption"><b>Table 4.</b> Evolutionary stability conditions and mutually exclusive equilibrium points of the government-power grid two-party evolution game system at various internal equilibrium points.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Internal Equilibrium Points</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Conditions of Achieving Evolutionarily Stable</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Mutually Exclusive Equilibrium Points</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>1</sub>(0, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo><</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>2</sub>(0, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo><</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>></mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>3</sub>(1, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>4</sub>(1, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>β</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>y</span>*)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >/</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >/</td></tr></tbody> </table> </div> <div class="html-table-wrap" id="mathematics-12-03249-t005"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href='#table_body_display_mathematics-12-03249-t005'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#table_body_display_mathematics-12-03249-t005"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 5.</b> The definitions of the main parameters used in the government-power user two-party evolution game model. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_mathematics-12-03249-t005"> <div class="html-caption"><b>Table 5.</b> The definitions of the main parameters used in the government-power user two-party evolution game model.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Main Parameters</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Definitions</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>3</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The costs incurred by the government in adopting an incentive strategy for the power users</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The government’s initial revenue</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>7</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The social gains to the government when the government incentivizes the power users, and the users receive the DSM services.</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>8</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The social gains to the government when the government does not incentivize power users, but the power users independently accept the DSM services.</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>9</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The policy benefits received by the power users when they accept incentives.</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The reduced electricity expenses for the power users when they choose to accept the DSM services.</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The equivalent benefits of the power users obtaining a clean and environmentally friendly electricity usage environment through accepting the DSM services.</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>η</mi> <mn>2</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The social gains are lost to the government when the power users do not accept the DSM services.</td></tr><tr ><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>κ</mi> <mn>1</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' >The initial expenditure of the power users.</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>κ</mi> <mn>2</mn> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >The expenditures by the power users for purchasing energy-saving equipment, etc.</td></tr></tbody> </table> </div> <div class="html-table-wrap" id="mathematics-12-03249-t006"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href='#table_body_display_mathematics-12-03249-t006'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#table_body_display_mathematics-12-03249-t006"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 6.</b> The calculation results of determinant and trace for the government-power user two-party evolution game model. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_mathematics-12-03249-t006"> <div class="html-caption"><b>Table 6.</b> The calculation results of determinant and trace for the government-power user two-party evolution game model.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Internal Equilibrium Points</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <mi mathvariant="bold">The</mi> <mtext> </mtext> <mi mathvariant="bold">Determinant</mi> <mtext> </mtext> <mi mathvariant="bold">Value</mi> <mtext> </mtext> <mi mathvariant="bold-italic">D</mi> <mi mathvariant="bold-italic">e</mi> <mi mathvariant="bold-italic">t</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mi mathvariant="bold-italic">G</mi> <mtext>-</mtext> <mi mathvariant="bold-italic">P</mi> <mi mathvariant="bold-italic">U</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <mi mathvariant="bold">The</mi> <mtext> </mtext> <mi mathvariant="bold">Trace</mi> <mtext> </mtext> <mi mathvariant="bold">Value</mi> <mtext> </mtext> <mi mathvariant="bold-italic">T</mi> <mi mathvariant="bold-italic">r</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="bold-italic">J</mi> <mrow> <mi mathvariant="bold-italic">G</mi> <mtext>-</mtext> <mi mathvariant="bold-italic">P</mi> <mi mathvariant="bold-italic">U</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </semantics> </math></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>1</sub>(0, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>2</sub>(0, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>3</sub>(1, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>4</sub>(1, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>z</span>*)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>α</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td></tr></tbody> </table> </div> <div class="html-table-wrap" id="mathematics-12-03249-t007"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href='#table_body_display_mathematics-12-03249-t007'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/12/20/3249/display" href="#table_body_display_mathematics-12-03249-t007"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 7.</b> Evolutionary stability conditions and mutually exclusive equilibrium points of the government-power user two-party evolution game system at various internal equilibrium points. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_mathematics-12-03249-t007"> <div class="html-caption"><b>Table 7.</b> Evolutionary stability conditions and mutually exclusive equilibrium points of the government-power user two-party evolution game system at various internal equilibrium points.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Internal Equilibrium Points</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Conditions of Achieving Evolutionarily Stable</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Mutually Exclusive Equilibrium Points</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>1</sub>(0, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>β</mi> <mn>6</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo><</mo> <mn>0</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>2</sub>(0, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo><</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>3</sub>(1, 0)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo><</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo><</mo> <mn>0</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >E<sub>4</sub>(1, 1)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>β</mi> <mn>7</mn> </msub> <mo>−</mo> <msub> <mi>β</mi> <mn>8</mn> </msub> <mo>−</mo> <msub> <mi>α</mi> <mn>3</mn> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>β</mi> <mn>9</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >E<sub>5</sub>(<span class='html-italic'>x</span>*, <span class='html-italic'>z</span>*)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >/</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >/</td></tr></tbody> </table> </div> </section><section class='html-fn_group'><table><tr id=''><td></td><td><div class='html-p'><b>Disclaimer/Publisher’s Note:</b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). 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