CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Median voter theorem - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"86ca9919-e303-422a-a509-c56b551d8003","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Median_voter_theorem","wgTitle":"Median voter theorem","wgCurRevisionId":1276142150,"wgRevisionId":1276142150,"wgArticleId":1233715,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description matches Wikidata","Political science theories","Public choice theory","Voting theory","Game theory","Mathematical economics"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Median_voter_theorem","wgRelevantArticleId":1233715,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject": "wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q648511","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={ "ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.tablesorter.styles":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.tablesorter","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader", "ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cjquery.tablesorter.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.18"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/1200px-Electoral-systems-gears.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1200"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/800px-Electoral-systems-gears.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="800"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/640px-Electoral-systems-gears.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="640"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Median voter theorem - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Median_voter_theorem"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Median_voter_theorem&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Median_voter_theorem"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Median_voter_theorem rootpage-Median_voter_theorem skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Median+voter+theorem" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Median+voter+theorem" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Median+voter+theorem" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Median+voter+theorem" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Statement_and_proof_of_the_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Statement_and_proof_of_the_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Statement and proof of the theorem</span> </div> </a> <button aria-controls="toc-Statement_and_proof_of_the_theorem-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Statement and proof of the theorem subsection</span> </button> <ul id="toc-Statement_and_proof_of_the_theorem-sublist" class="vector-toc-list"> <li id="toc-The_median_voter_property" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_median_voter_property"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>The median voter property</span> </div> </a> <ul id="toc-The_median_voter_property-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions_to_higher_dimensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extensions_to_higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Extensions to higher dimensions</span> </div> </a> <button aria-controls="toc-Extensions_to_higher_dimensions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Extensions to higher dimensions subsection</span> </button> <ul id="toc-Extensions_to_higher_dimensions-sublist" class="vector-toc-list"> <li id="toc-Omnidirectional_medians" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Omnidirectional_medians"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Omnidirectional medians</span> </div> </a> <ul id="toc-Omnidirectional_medians-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_between_the_median_in_all_directions_and_the_geometric_median" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_between_the_median_in_all_directions_and_the_geometric_median"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Relation between the median in all directions and the geometric median</span> </div> </a> <ul id="toc-Relation_between_the_median_in_all_directions_and_the_geometric_median-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Hotelling–Downs_model" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hotelling–Downs_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Hotelling–Downs model</span> </div> </a> <ul id="toc-Hotelling–Downs_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uses_of_the_median_voter_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Uses_of_the_median_voter_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Uses of the median voter theorem</span> </div> </a> <ul id="toc-Uses_of_the_median_voter_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Empirical_evidence_and_contradictions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Empirical_evidence_and_contradictions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Empirical evidence and contradictions</span> </div> </a> <ul id="toc-Empirical_evidence_and_contradictions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Median voter theorem</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 15 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-15" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">15 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AE%E0%A6%BE%E0%A6%A7%E0%A7%8D%E0%A6%AF%E0%A6%BF%E0%A6%95%E0%A6%BE_%E0%A6%AE%E0%A6%A4%E0%A6%A6%E0%A6%BE%E0%A6%A4%E0%A6%BE_%E0%A6%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="মাধ্যিকা মতদাতা উপপাদ্য – Assamese" lang="as" hreflang="as" data-title="মাধ্যিকা মতদাতা উপপাদ্য" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Medianv%C3%A6lgere" title="Medianvælgere – Danish" lang="da" hreflang="da" data-title="Medianvælgere" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Medianw%C3%A4hlertheorem" title="Medianwählertheorem – German" lang="de" hreflang="de" data-title="Medianwählertheorem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_del_votante_mediano" title="Teorema del votante mediano – Spanish" lang="es" hreflang="es" data-title="Teorema del votante mediano" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D8%B1%D8%A3%DB%8C%E2%80%8C%D8%AF%D9%87%D9%86%D8%AF%D9%87_%D9%85%DB%8C%D8%A7%D9%86%D9%87" title="قضیه رأی‌دهنده میانه – Persian" lang="fa" hreflang="fa" data-title="قضیه رأی‌دهنده میانه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_l%27%C3%A9lecteur_m%C3%A9dian" title="Théorème de l'électeur médian – French" lang="fr" hreflang="fr" data-title="Théorème de l'électeur médian" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A4%91%EC%9C%84_%ED%88%AC%ED%91%9C%EC%9E%90_%EC%A0%95%EB%A6%AC" title="중위 투표자 정리 – Korean" lang="ko" hreflang="ko" data-title="중위 투표자 정리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_dell%27elettore_mediano" title="Teorema dell'elettore mediano – Italian" lang="it" hreflang="it" data-title="Teorema dell'elettore mediano" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%91%D7%95%D7%97%D7%A8_%D7%94%D7%97%D7%A6%D7%99%D7%95%D7%A0%D7%99" title="הבוחר החציוני – Hebrew" lang="he" hreflang="he" data-title="הבוחר החציוני" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%AD%E4%BD%8D%E6%8A%95%E7%A5%A8%E8%80%85%E5%AE%9A%E7%90%86" title="中位投票者定理 – Japanese" lang="ja" hreflang="ja" data-title="中位投票者定理" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_o_medianowym_wyborcy" title="Twierdzenie o medianowym wyborcy – Polish" lang="pl" hreflang="pl" data-title="Twierdzenie o medianowym wyborcy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_do_eleitor_mediano" title="Teorema do eleitor mediano – Portuguese" lang="pt" hreflang="pt" data-title="Teorema do eleitor mediano" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Median_voter_theorem" title="Median voter theorem – Simple English" lang="en-simple" hreflang="en-simple" data-title="Median voter theorem" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Medianv%C3%A4ljarteorin" title="Medianväljarteorin – Swedish" lang="sv" hreflang="sv" data-title="Medianväljarteorin" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%AD%E9%97%B4%E9%80%89%E6%B0%91%E5%AE%9A%E7%90%86" title="中间选民定理 – Chinese" lang="zh" hreflang="zh" data-title="中间选民定理" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q648511#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Median_voter_theorem" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Median_voter_theorem" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Median_voter_theorem"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Median_voter_theorem&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Median_voter_theorem&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Median_voter_theorem"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Median_voter_theorem&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Median_voter_theorem&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Median_voter_theorem" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Median_voter_theorem" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Median_voter_theorem&oldid=1276142150" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Median_voter_theorem&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Median_voter_theorem&id=1276142150&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMedian_voter_theorem"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMedian_voter_theorem"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Median_voter_theorem&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Median_voter_theorem&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q648511" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Theorem in political science</div><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">A joint <a href="/wiki/Portal:Politics" title="Portal:Politics">Politics</a> and <a href="/wiki/Portal:Economics" title="Portal:Economics">Economics</a> series</td></tr><tr><th class="sidebar-title-with-pretitle" style="border-top:1px #fafafa solid; border-bottom:1px #fafafa solid; background:#efefef; background: var(--background-color-interactive, #efefef); color: var(--color-base, #000); padding:0.2em;"><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice</a> and <a href="/wiki/Electoral_system" title="Electoral system">electoral systems</a></th></tr><tr><td class="sidebar-image"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Electoral-systems-gears.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/128px-Electoral-systems-gears.svg.png" decoding="async" width="128" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/192px-Electoral-systems-gears.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/256px-Electoral-systems-gears.svg.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption></figcaption></figure></td></tr><tr><td class="sidebar-above"> <div class="hlist"><ul><li><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice</a></li><li><a href="/wiki/Mechanism_design" title="Mechanism design">Mechanism design</a></li><li><a href="/wiki/Comparative_politics" title="Comparative politics">Comparative politics</a></li><li><a href="/wiki/Comparison_of_voting_rules" title="Comparison of voting rules">Comparison</a></li><li><a href="/wiki/List_of_electoral_systems" title="List of electoral systems">List</a><span class="nowrap"> </span>(<a href="/wiki/List_of_electoral_systems_by_country" title="List of electoral systems by country">By country</a>)</li></ul></div></td></tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Single-member_district" title="Single-member district">Single-winner methods</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Single vote - <a href="/wiki/Plurality_voting" title="Plurality voting">plurality</a> methods</b> <ul><li><a href="/wiki/First-past-the-post_voting" title="First-past-the-post voting">First preference plurality (FPP)</a></li> <li><a href="/wiki/Two-round_system" title="Two-round system">Two-round</a> (<abbr style="font-size:85%" title=""><a href="/wiki/American_English" title="American English">US</a>:</abbr> <a href="/wiki/Nonpartisan_primary" title="Nonpartisan primary">Jungle primary</a>) <ul><li><a href="/wiki/Partisan_primary" class="mw-redirect" title="Partisan primary">Partisan primary</a></li></ul></li> <li><a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff</a> <ul><li><abbr style="font-size:85%" title=""><a href="/wiki/British_English" title="British English">UK</a>:</abbr> Alternative vote (AV)</li> <li><abbr style="font-size:85%" title=""><a href="/wiki/American_English" title="American English">US</a>:</abbr> Ranked-choice (RCV)</li></ul></li></ul> <hr /> <p><b><a href="/wiki/Condorcet_method" title="Condorcet method">Condorcet methods</a></b> </p> <ul><li><a href="/wiki/Tideman_alternative_method" title="Tideman alternative method">Condorcet-IRV</a></li> <li><a href="/wiki/Round-robin_voting" title="Round-robin voting">Round-robin voting</a> <ul><li><a href="/wiki/Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax</a></li> <li><a href="/wiki/Schulze_method" title="Schulze method">Schulze</a></li> <li><a href="/wiki/Ranked_pairs" title="Ranked pairs">Ranked pairs</a></li> <li><a href="/wiki/Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Positional_voting" title="Positional voting">Positional voting</a></b> </p> <ul><li><a href="/wiki/First-preference_plurality" class="mw-redirect" title="First-preference plurality">Plurality</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">IRV</a>)</li> <li><a href="/wiki/Borda_count" title="Borda count">Borda count</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Baldwin%27s_method" class="mw-redirect" title="Baldwin's method">Baldwin</a>)</li> <li><a href="/wiki/Anti-plurality_voting" title="Anti-plurality voting">Antiplurality</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Coombs_method" class="mw-redirect" title="Coombs method">Coombs</a>)</li></ul> <hr /> <p><b><a href="/wiki/Rated_voting" title="Rated voting">Cardinal voting</a></b> </p> <ul><li><a href="/wiki/Score_voting" title="Score voting">Score voting</a></li> <li><a href="/wiki/Approval_voting" title="Approval voting">Approval voting</a></li> <li><a href="/wiki/Highest_median_voting_rules" title="Highest median voting rules">Majority judgment</a></li> <li><a href="/wiki/STAR_voting" title="STAR voting">STAR voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Proportional_representation" title="Proportional representation">Proportional representation</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="/wiki/Party-list_proportional_representation" title="Party-list proportional representation">Party-list</a></b> <ul><li><a href="/wiki/Apportionment_(politics)" title="Apportionment (politics)">Apportionment</a> <ul><li><a href="/wiki/Highest_averages_method" title="Highest averages method">Highest averages</a></li> <li><a href="/wiki/Largest_remainder_method" class="mw-redirect" title="Largest remainder method">Largest remainders</a></li> <li><a href="/wiki/National_remnant" title="National remnant">National remnant</a></li> <li><a href="/wiki/Biproportional_apportionment" title="Biproportional apportionment">Biproportional</a></li></ul></li> <li><a href="/wiki/Electoral_list" title="Electoral list">List type</a> <ul><li><a href="/wiki/Closed_list" title="Closed list">Closed list</a></li> <li><a href="/wiki/Open_list" title="Open list">Open list</a></li> <li><a href="/wiki/Panachage" title="Panachage">Panachage</a></li> <li><a href="/wiki/Justified_representation" title="Justified representation">List-free PR</a></li> <li><a href="/wiki/Localized_list" title="Localized list">Localized list</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Electoral_quota" title="Electoral quota">Quota-remainder methods</a></b> </p> <ul><li><a href="/wiki/Single_transferable_vote" title="Single transferable vote">Hare STV</a></li> <li><a href="/wiki/Schulze_STV" title="Schulze STV">Schulze STV</a></li> <li><a href="/wiki/CPO-STV" title="CPO-STV">CPO-STV</a></li> <li><a href="/wiki/Quota_Borda_system" title="Quota Borda system">Quota Borda</a></li></ul> <hr /> <p><b><a href="/wiki/Approval-based_committee" class="mw-redirect" title="Approval-based committee">Approval-based committees</a></b> </p> <ul><li><a href="/wiki/Proportional_approval_voting" title="Proportional approval voting">Thiele's method</a></li> <li><a href="/wiki/Phragmen%27s_voting_rules" title="Phragmen's voting rules">Phragmen's method</a></li> <li><a href="/wiki/Expanding_approvals_rule" title="Expanding approvals rule">Expanding approvals rule</a></li> <li><a href="/wiki/Method_of_equal_shares" title="Method of equal shares">Method of equal shares</a></li></ul> <hr /> <p><b><a href="/wiki/Fractional_social_choice" title="Fractional social choice">Fractional social choice</a></b> </p> <ul><li><a href="/wiki/Direct_representation" title="Direct representation">Direct representation</a> <ul><li><a href="/wiki/Interactive_representation" title="Interactive representation">Interactive representation</a></li> <li><a href="/wiki/Liquid_democracy" title="Liquid democracy">Liquid democracy</a></li></ul></li> <li><a href="/wiki/Fractional_approval_voting" title="Fractional approval voting">Fractional approval voting</a></li> <li><a href="/wiki/Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li> <li><a href="/wiki/Random_ballot" title="Random ballot">Random ballot</a></li></ul> <hr /> <p><b><a href="/wiki/Semi-proportional_representation" title="Semi-proportional representation">Semi-proportional representation</a></b> </p> <ul><li><a href="/wiki/Cumulative_voting" title="Cumulative voting">Cumulative</a> <ul><li><a href="/wiki/Single_non-transferable_vote" title="Single non-transferable vote">SNTV</a></li></ul></li> <li><a href="/wiki/Limited_voting" title="Limited voting">Limited voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Mixed systems</a></div><div class="sidebar-list-content mw-collapsible-content"><b>By results of combination</b> <ul><li><a href="/wiki/Mixed-member_majoritarian_representation" title="Mixed-member majoritarian representation">Mixed-member majoritarian</a></li> <li><a href="/wiki/Mixed-member_proportional_representation" title="Mixed-member proportional representation">Mixed-member proportional</a></li></ul> <hr /><b>By mechanism of combination</b> <ul><li><b>Non-<a href="/wiki/Compensation_(electoral_systems)" title="Compensation (electoral systems)">compensatory</a></b> <ul><li><a href="/wiki/Parallel_voting" title="Parallel voting">Parallel (superposition)</a></li> <li><a href="/wiki/Coexistence_(electoral_systems)" title="Coexistence (electoral systems)">Coexistence</a></li> <li><a href="/w/index.php?title=Conditional_electoral_system&action=edit&redlink=1" class="new" title="Conditional electoral system (page does not exist)">Conditional</a></li> <li><a href="/wiki/Majority_bonus_system" title="Majority bonus system">Fusion (majority bonus)</a></li></ul></li> <li><b><a href="/wiki/Compensation_(electoral_systems)" title="Compensation (electoral systems)">Compensatory</a></b> <ul><li><a href="/w/index.php?title=Seat_linkage_mixed_system&action=edit&redlink=1" class="new" title="Seat linkage mixed system (page does not exist)">Seat linkage system</a> <ul><li><abbr style="font-size:85%" title=""><a href="/wiki/British_English" title="British English">UK</a>:</abbr> <a href="/wiki/Additional_member_system" class="mw-redirect" title="Additional member system">'AMS'</a></li> <li><abbr style="font-size:85%" title=""><a href="/wiki/New_Zealand_English" title="New Zealand English">NZ</a>:</abbr> <a href="/wiki/Mixed-member_proportional" class="mw-redirect" title="Mixed-member proportional">'MMP'</a></li></ul></li> <li><a href="/wiki/Vote_linkage_mixed_system" class="mw-redirect" title="Vote linkage mixed system">Vote linkage system</a> <ul><li><a href="/wiki/Scorporo" title="Scorporo">Negative vote transfer</a></li> <li><a href="/wiki/Mixed_ballot_transferable_vote" title="Mixed ballot transferable vote">Mixed ballot</a></li></ul></li></ul></li> <li><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Supermixed systems</a> <ul><li><a href="/wiki/Dual-member_proportional_representation" class="mw-redirect" title="Dual-member proportional representation">Dual-member proportional</a></li> <li><a href="/wiki/Rural%E2%80%93urban_proportional_representation" title="Rural–urban proportional representation">Rural–urban proportional</a></li> <li><a href="/wiki/Majority_jackpot_system" title="Majority jackpot system">Majority jackpot</a></li></ul></li></ul> <hr /> <p><b>By ballot type</b> </p> <ul><li><a href="/wiki/Mixed_single_vote" title="Mixed single vote">Single vote</a> <ul><li><a href="/wiki/Double_simultaneous_vote" title="Double simultaneous vote">Double simultaneous vote</a></li></ul></li> <li><a href="/wiki/Mixed_electoral_systems" class="mw-redirect" title="Mixed electoral systems">Dual-vote</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Pathological_(mathematics)#Voting" title="Pathological (mathematics)">Paradoxes and pathologies</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Spoiler effects</b> <ul><li><a href="/wiki/Spoiler_effect" title="Spoiler effect">Spoiler effect</a></li> <li><a href="/wiki/Independence_of_clones" class="mw-redirect" title="Independence of clones">Cloning paradox</a></li> <li><a href="/wiki/Condorcet_winner_criterion" title="Condorcet winner criterion">Frustrated majorities paradox</a></li> <li><a href="/wiki/Center_squeeze" title="Center squeeze">Center squeeze</a></li></ul> <hr /> <p><b>Pathological response</b> </p> <ul><li><a href="/wiki/Perverse_response" class="mw-redirect" title="Perverse response">Perverse response</a></li> <li><a href="/wiki/Best-is-worst_paradox" title="Best-is-worst paradox">Best-is-worst paradox</a></li> <li><a href="/wiki/No-show_paradox" title="No-show paradox">No-show paradox</a> <ul><li><a href="/wiki/Multiple_districts_paradox" title="Multiple districts paradox">Multiple districts paradox</a></li></ul></li> <li><a href="/wiki/Wasted_vote" title="Wasted vote">Wasted vote</a></li></ul> <hr /> <p><b><a href="/wiki/Strategic_voting" title="Strategic voting">Strategic voting</a></b> </p> <ul><li><a href="/wiki/Sincere_favorite_criterion" title="Sincere favorite criterion">Lesser evil voting</a></li> <li><a href="/wiki/Strategic_voting#Exaggeration" title="Strategic voting">Exaggeration</a></li> <li><a href="/wiki/Truncation_(voting)" class="mw-redirect" title="Truncation (voting)">Truncation</a></li> <li><a href="/wiki/Turkey-raising" class="mw-redirect" title="Turkey-raising">Turkey-raising</a></li></ul> <hr /> <p><b>Paradoxes of <a href="/wiki/Majority_rule" title="Majority rule">majority rule</a></b> </p> <ul><li><a href="/wiki/Tyranny_of_the_majority" title="Tyranny of the majority">Tyranny of the majority</a></li> <li><a href="/wiki/Discursive_dilemma" title="Discursive dilemma">Discursive dilemma</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Conflicting majorities paradox</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Social_choice_theory" title="Social choice theory">Social and collective choice</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Impossibility theorems</a></b> <ul><li><a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's theorem</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Majority impossibility</a></li> <li><a href="/wiki/Moulin%27s_impossibility_theorem" class="mw-redirect" title="Moulin's impossibility theorem">Moulin's impossibility theorem</a></li> <li><a href="/wiki/McKelvey%E2%80%93Schofield_chaos_theorem" title="McKelvey–Schofield chaos theorem">McKelvey–Schofield chaos theorem</a></li> <li><a href="/wiki/Gibbard%27s_theorem" title="Gibbard's theorem">Gibbard's theorem</a></li></ul> <hr /> <p><b>Positive results</b> </p> <ul><li><a class="mw-selflink selflink">Median voter theorem</a></li> <li><a href="/wiki/Condorcet%27s_jury_theorem" title="Condorcet's jury theorem">Condorcet's jury theorem</a></li> <li><a href="/wiki/May%27s_theorem" title="May's theorem">May's theorem</a></li> <li><a href="/wiki/Arrow%27s_theorem#Minimizing" class="mw-redirect" title="Arrow's theorem">Condorcet dominance theorems</a></li> <li><a href="/w/index.php?title=Harsanyi%27s_utilitarian_theorem&action=edit&redlink=1" class="new" title="Harsanyi's utilitarian theorem (page does not exist)">Harsanyi's utilitarian theorem</a></li> <li><a href="/wiki/Vickrey-Clarke-Groves_mechanism" class="mw-redirect" title="Vickrey-Clarke-Groves mechanism">VCG mechanism</a></li> <li><a href="/wiki/Quadratic_voting" title="Quadratic voting">Quadratic voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="background: var(--background-color-interactive, #efefef); color: inherit; padding-top:0.2em;"> <div class="hlist"><ul><li><span class="nowrap"><span class="mw-image-border noviewer" typeof="mw:File"><a href="/wiki/File:A_coloured_voting_box.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/en/thumb/0/01/A_coloured_voting_box.svg/16px-A_coloured_voting_box.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/01/A_coloured_voting_box.svg/24px-A_coloured_voting_box.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/01/A_coloured_voting_box.svg/32px-A_coloured_voting_box.svg.png 2x" data-file-width="160" data-file-height="160" /></a></span> </span><a href="/wiki/Portal:Politics" title="Portal:Politics">Politics portal</a></li><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Emblem-money.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/16px-Emblem-money.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/24px-Emblem-money.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/32px-Emblem-money.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span> </span><a href="/wiki/Portal:Economics" title="Portal:Economics">Economics portal</a></li></ul></div><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electoral_systems_sidebar" title="Template:Electoral systems sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electoral_systems_sidebar" title="Template talk:Electoral systems sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electoral_systems_sidebar" title="Special:EditPage/Template:Electoral systems sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Median_mechanism" class="mw-redirect" title="Median mechanism">median mechanism</a>.</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">For a discussion on applicability of the median voter theorem to different electoral systems, see <a href="/wiki/Center_squeeze" title="Center squeeze">center squeeze</a>.</div> <p>In <a href="/wiki/Political_science" title="Political science">political science</a> and <a href="/wiki/Social_choice_theory" title="Social choice theory">social choice</a>, the <b>median voter theorem</b> states that if voters and candidates are distributed along a one-dimensional <a href="/wiki/Political_spectrum" title="Political spectrum">spectrum</a> and voters have <a href="/wiki/Single-peaked_preferences" class="mw-redirect" title="Single-peaked preferences">single-peaked preferences</a>, any voting method that is <a href="/wiki/Condorcet_criterion" class="mw-redirect" title="Condorcet criterion">compatible with majority-rule</a> will elect the candidate preferred by the <a href="/wiki/Median" title="Median">median</a> voter. </p><p>The theorem was first set out by <a href="/wiki/Duncan_Black" title="Duncan Black">Duncan Black</a> in 1948.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> He wrote that he saw a large gap in economic theory concerning how voting determines the outcome of decisions, including political decisions. Black's paper triggered research on how economics can explain voting systems. </p><p>A different argument due to <a href="/wiki/Anthony_Downs" title="Anthony Downs">Anthony Downs</a> and <a href="/wiki/Harold_Hotelling" title="Harold Hotelling">Harold Hotelling</a> is only loosely-related to Black's median voter theorem, but is often confused with it. This model argues that politicians in a <a href="/wiki/Representative_democracy" title="Representative democracy">representative democracy</a> will converge to the viewpoint of the median voter,<sup id="cite_ref-holcombe-2006_2-0" class="reference"><a href="#cite_note-holcombe-2006-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> because the median voter theorem implies that a candidate who wishes to win will adopt the positions of the median voter.<sup id="cite_ref-holcombe-2006_2-1" class="reference"><a href="#cite_note-holcombe-2006-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-hotelling_harold-1929_3-0" class="reference"><a href="#cite_note-hotelling_harold-1929-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-downs-1957_4-0" class="reference"><a href="#cite_note-downs-1957-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> However, this argument only applies to systems satisfying the median voter property, and cannot be applied to systems like <a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">ranked choice voting</a> (RCV) or <a href="/wiki/Plurality_voting" title="Plurality voting">plurality voting</a> outside of limited conditions (see <a href="#Hotelling–Downs_model">§ Hotelling–Downs model</a>).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-myerson-1993_6-0" class="reference"><a href="#cite_note-myerson-1993-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Statement_and_proof_of_the_theorem">Statement and proof of the theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=1" title="Edit section: Statement and proof of the theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Median_voter.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Median_voter.png/443px-Median_voter.png" decoding="async" width="443" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Median_voter.png/665px-Median_voter.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Median_voter.png/886px-Median_voter.png 2x" data-file-width="910" data-file-height="440" /></a><figcaption>A <a href="/wiki/Proof_without_words" title="Proof without words">proof without words</a> of the median voter theorem.</figcaption></figure> <p>Say there is an election where candidates and voters have opinions distributed along a one-dimensional <a href="/wiki/Political_spectrum" title="Political spectrum">political spectrum</a>. Voters rank candidates by proximity, i.e. the closest candidate is their first preference, the second-closest is their second preference, and so on. Then, the median voter theorem says that the candidate closest to the median voter is a <a href="/wiki/Condorcet_winner" class="mw-redirect" title="Condorcet winner"><i>majority-preferred</i> (or <i>Condorcet</i>) candidate</a>. In other words, this candidate preferred to any one of their opponents by a majority of voters. When there are only two candidates, a simple <a href="/wiki/Majority_vote" class="mw-redirect" title="Majority vote">majority vote</a> satisfies this condition, while for multi-candidate votes any majority-rule (Condorcet) method will satisfy it. </p><p><b>Proof sketch:</b> Let the <a href="/wiki/Median" title="Median">median</a> voter be Marlene. The candidate who is closest to her will receive her first preference vote. Suppose that this candidate is Charles and that he lies to her left. Marlene and all voters to her left (by definition a majority of the electorate) will prefer Charles to all candidates to his right, and Marlene and all voters to her right (also a majority) will prefer Charles to all candidates to his left. ∎ </p> <ul><li>The assumption that preferences are cast in order of proximity can be relaxed to say merely that they are <a href="/wiki/Single_peaked_preferences" title="Single peaked preferences">single-peaked</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li>The assumption that opinions lie along a real line can be relaxed to allow more general topologies.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li><i><b>Spatial / valence models:</b></i> Suppose that each candidate has a <i><a href="/wiki/Valence_politics" class="mw-redirect" title="Valence politics">valence</a></i> (attractiveness) in addition to his or her position in space, and suppose that voter <i>i</i> ranks candidates <i>j</i> in decreasing order of <i>v<sub>j</sub></i> – <i>d<sub>ij</sub></i> where <i>v<sub>j</sub></i> is <i>j</i> 's valence and <i>d<sub>ij</sub></i> is the distance from <i>i</i> to <i>j</i>. Then the median voter theorem still applies: Condorcet methods will elect the candidate voted for by the median voter.</li></ul> <div class="mw-heading mw-heading3"><h3 id="The_median_voter_property">The median voter property</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=2" title="Edit section: The median voter property"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We will say that a voting method has the "median voter property in one dimension" if it always elects the candidate closest to the median voter under a one-dimensional spatial model. We may summarize the median voter theorem as saying that all Condorcet methods possess the median voter property in one dimension. </p><p>It turns out that Condorcet methods are not unique in this: <a href="/wiki/Coombs%27_method" title="Coombs' method">Coombs' method</a> is not Condorcet-consistent but nonetheless satisfies the median voter property in one dimension.<sup id="cite_ref-b-2004_10-0" class="reference"><a href="#cite_note-b-2004-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Approval voting satisfies the same property under several models of strategic voting. </p> <div class="mw-heading mw-heading2"><h2 id="Extensions_to_higher_dimensions">Extensions to higher dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=3" title="Edit section: Extensions to higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is impossible to fully generalize the median voter theorem to <a href="/wiki/Spatial_model_of_voting" class="mw-redirect" title="Spatial model of voting">spatial models</a> in more than one dimension, as there is no longer a single unique "median" for all possible distributions of voters. However, it is still possible to demonstrate similar theorems under some limited conditions. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SaariExample.png" class="mw-file-description"><img alt="Saari's example" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/SaariExample.png/220px-SaariExample.png" decoding="async" width="220" height="242" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/SaariExample.png/330px-SaariExample.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/82/SaariExample.png 2x" data-file-width="400" data-file-height="440" /></a><figcaption>Saari's example of a domain where the Condorcet winner is not the socially-optimal candidate.</figcaption></figure> <table class="wikitable sortable floatleft"> <tbody><tr> <th>Ranking </th> <th>Votes </th></tr> <tr> <td style="background:white">A-B-C </td> <td style="background:white">30 </td></tr> <tr> <td style="background:white">B-A-C </td> <td style="background:white">29 </td></tr> <tr> <td style="background:#fadadd">C-A-B </td> <td style="background:#fadadd">10 </td></tr> <tr> <td style="background:#fadadd">B-C-A </td> <td style="background:#fadadd">10 </td></tr> <tr> <td style="background:#cfeeee">A-C-B </td> <td style="background:#cfeeee">1 </td></tr> <tr> <td style="background:#cfeeee">C-B-A </td> <td style="background:#cfeeee">1 </td></tr></tbody></table> <table class="wikitable floatleft"> <tbody><tr> <th> </th> <th>Number of voters </th></tr> <tr> <th>A > B </th> <td>41:40 </td></tr> <tr> <th>A > C </th> <td>60:21 </td></tr> <tr> <th>B > C </th> <td>69:12 </td></tr> <tr> <th>Total </th> <td>81 </td></tr></tbody></table> <p>The table shows an example of an election given by the <a href="/wiki/Marquis_de_Condorcet" title="Marquis de Condorcet">Marquis de Condorcet</a>, who concluded it showed a problem with the <a href="/wiki/Borda_count" title="Borda count">Borda count</a>.<sup id="cite_ref-george_g-2010_11-0" class="reference"><a href="#cite_note-george_g-2010-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 90">: 90 </span></sup> The Condorcet winner on the left is A, who is preferred to B by 41:40 and to C by 60:21. The Borda winner is instead B. However, <a href="/wiki/Donald_Saari" class="mw-redirect" title="Donald Saari">Donald Saari</a> constructs an example in two dimensions where the Borda count (but not the Condorcet winner) correctly identifies the candidate closest to the center (as determined by the <a href="/wiki/Geometric_median" title="Geometric median">geometric median</a>).<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The diagram shows a possible configuration of the voters and candidates consistent with the ballots, with the voters positioned on the circumference of a unit circle. In this case, A's <a href="/wiki/Mean_absolute_deviation" class="mw-redirect" title="Mean absolute deviation">mean absolute deviation</a> is 1.15, whereas B's is 1.09 (and C's is 1.70), making B the spatial winner. </p><p>Thus the election is ambiguous in that two different spatial representations imply two different optimal winners. This is the ambiguity we sought to avoid earlier by adopting a median metric for spatial models; but although the median metric achieves its aim in a single dimension, the property does not fully generalize to higher dimensions. </p> <div class="mw-heading mw-heading3"><h3 id="Omnidirectional_medians">Omnidirectional medians</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=4" title="Edit section: Omnidirectional medians"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Median_Voter_Theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Median_Voter_Theorem.svg/220px-Median_Voter_Theorem.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Median_Voter_Theorem.svg/330px-Median_Voter_Theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Median_Voter_Theorem.svg/440px-Median_Voter_Theorem.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>The median voter theorem in two dimensions</figcaption></figure><p>Despite this result, the median voter theorem can be applied to distributions that are rotationally symmetric, e.g. <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Gaussians</a>, which have a single median that is the same in all directions. Whenever the distribution of voters has a unique median in all directions, and voters rank candidates in order of proximity, the median voter theorem applies: the candidate closest to the median will have a majority preference over all his or her rivals, and will be elected by any voting method satisfying the median voter property in one dimension.<sup id="cite_ref-dotti-2016_13-0" class="reference"><a href="#cite_note-dotti-2016-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>It follows that all <a href="/wiki/Median_voter_criterion" class="mw-redirect" title="Median voter criterion">median voter methods</a> satisfy the same property in spaces of any dimension, for voter distributions with omnidirectional medians. </p><p>It is easy to construct voter distributions which do not have a median in all directions. The simplest example consists of a distribution limited to 3 points not lying in a straight line, such as 1, 2 and 3 in the second diagram. Each voter location coincides with the median under a certain set of one-dimensional projections. If A, B and C are the candidates, then '1' will vote A-B-C, '2' will vote B-C-A, and '3' will vote C-A-B, giving a Condorcet cycle. This is the subject of the <a href="/wiki/McKelvey%E2%80%93Schofield_chaos_theorem" title="McKelvey–Schofield chaos theorem">McKelvey–Schofield theorem</a>. </p><p><i><b>Proof</b></i>. See the diagram, in which the grey disc represents the voter distribution as uniform over a circle and M is the median in all directions. Let A and B be two candidates, of whom A is the closer to the median. Then the voters who rank A above B are precisely the ones to the left (i.e. the 'A' side) of the solid red line; and since A is closer than B to M, the median is also to the left of this line. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Voting_Paradox_example.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Voting_Paradox_example.png/220px-Voting_Paradox_example.png" decoding="async" width="220" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Voting_Paradox_example.png/330px-Voting_Paradox_example.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Voting_Paradox_example.png/440px-Voting_Paradox_example.png 2x" data-file-width="942" data-file-height="1017" /></a><figcaption>A distribution with no median in all directions</figcaption></figure><p>Now, since M is a median in all directions, it coincides with the one-dimensional median in the particular case of the direction shown by the blue arrow, which is perpendicular to the solid red line. Thus if we draw a broken red line through M, perpendicular to the blue arrow, then we can say that half the voters lie to the left of this line. But since this line is itself to the left of the solid red line, it follows that more than half of the voters will rank A above B. </p><div class="mw-heading mw-heading3"><h3 id="Relation_between_the_median_in_all_directions_and_the_geometric_median">Relation between the median in all directions and the geometric median</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=5" title="Edit section: Relation between the median in all directions and the geometric median"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Whenever a unique omnidirectional median exists, it determines the result of Condorcet voting methods. At the same time the <a href="/wiki/Median#Multivariate_median" title="Median">geometric median</a> can arguably be identified as the ideal winner of a ranked preference election. It is therefore important to know the relationship between the two. In fact whenever a median in all directions exists (at least for the case of discrete distributions), it coincides with the geometric median. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Median_Plott.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Median_Plott.svg/220px-Median_Plott.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Median_Plott.svg/330px-Median_Plott.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Median_Plott.svg/440px-Median_Plott.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Diagram for the lemma</figcaption></figure><p><i><b>Lemma</b></i>. Whenever a discrete distribution has a median <i>M</i>  in all directions, the data points not located at <i>M</i>  must come in balanced pairs (<i>A</i>,<i>A</i> ' ) on either side of <i>M</i>  with the property that <i>A</i> – <i>M</i> – <i>A</i> ' is a straight line (ie. <i>not</i> like <i>A</i><sub> 0 </sub>– <i>M</i> – <i>A</i><sub> 2</sub> in the diagram). </p><p><i><b>Proof</b></i>. This result was proved algebraically by Charles Plott in 1967.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Here we give a simple geometric proof by contradiction in two dimensions. </p><p>Suppose, on the contrary, that there is a set of points <i>A<sub>i</sub></i> which have <i>M</i>  as median in all directions, but for which the points not coincident with <i>M</i>  do not come in balanced pairs. Then we may remove from this set any points at <i>M</i>, and any balanced pairs about <i>M</i>, without <i>M</i>  ceasing to be a median in any direction; so <i>M</i>  remains an omnidirectional median. </p><p>If the number of remaining points is odd, then we can easily draw a line through <i>M</i>  such that the majority of points lie on one side of it, contradicting the median property of <i>M</i>. </p><p>If the number is even, say 2<i>n</i>, then we can label the points <i>A</i><sub> 0</sub>, <i>A</i><sub>1</sub>,... in clockwise order about <i>M</i>  starting at any point (see the diagram). Let θ be the angle subtended by the arc from <i>M</i> –<i>A</i><sub> 0</sub> to <i>M</i> –<i>A</i><sub> <i>n</i> </sub>. Then if θ < 180° as shown, we can draw a line similar to the broken red line through <i>M</i>  which has the majority of data points on one side of it, again contradicting the median property of <i>M</i> ; whereas if θ > 180° the same applies with the majority of points on the other side. And if θ = 180°, then <i>A</i><sub> 0</sub> and <i>A</i><sub> <i>n</i></sub> form a balanced pair, contradicting another assumption. </p><p><i><b>Theorem</b></i>. Whenever a discrete distribution has a median <i>M</i>  in all directions, it coincides with its geometric median. </p><p><i><b>Proof</b></i>. The sum of distances from any point <i>P</i>  to a set of data points in balanced pairs (<i>A</i>,<i>A</i> ' ) is the sum of the lengths <i>A</i> – <i>P</i> – <i>A</i> '. Each individual length of this form is minimized over <i>P</i> when the line is straight, as happens when <i>P</i>  coincides with <i>M</i>. The sum of distances from <i>P</i> to any data points located at <i>M</i> is likewise minimized when <i>P</i>  and <i>M</i>  coincide. Thus the sum of distances from the data points to <i>P</i> is minimized when <i>P</i> coincides with <i>M</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Hotelling–Downs_model"><span id="Hotelling.E2.80.93Downs_model"></span>Hotelling–Downs model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=6" title="Edit section: Hotelling–Downs model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A related observation was discussed by <a href="/wiki/Harold_Hotelling" title="Harold Hotelling">Harold Hotelling</a> as his 'principle of minimum differentiation', also known as '<a href="/wiki/Hotelling%27s_law" title="Hotelling's law">Hotelling's law</a>'. It states that if: </p> <ol><li>Candidates can choose ideological positions without consequence,</li> <li>Candidates only care about winning the election (not their actual beliefs),</li> <li>All other criteria of the median voter theorem are met (i.e. voters rank candidates by ideological distance),</li> <li>The voting system satisfies the median voter criterion,</li></ol> <p>Then all politicians will converge to the median voter. As a special case, this law applies to the situation where there are exactly two candidates in the race, if it is impossible or implausible that any more candidates will join the race, because a simple majority vote between two alternatives satisfies the <a href="/wiki/Condorcet_winner_criterion" title="Condorcet winner criterion">Condorcet criterion</a>. </p><p>This theorem was first described by Hotelling in 1929.<sup id="cite_ref-hotelling_harold-1929_3-1" class="reference"><a href="#cite_note-hotelling_harold-1929-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In practice, none of these conditions hold for modern American elections, though they may have held in Hotelling's time (when nominees were often <a href="/wiki/Dark_horse" title="Dark horse">previously-unknown</a> and chosen by closed <a href="/wiki/Party_caucus" class="mw-redirect" title="Party caucus">party caucuses</a> in ideologically diverse parties). Most importantly, politicians must win <a href="/wiki/Partisan_primary" class="mw-redirect" title="Partisan primary">primary elections</a>, which often include challengers or competitors, to be chosen as major-party nominees. As a result, politicians must compromise between appealing to the median voter in the primary and general electorates. Similar effects imply candidates do not converge to the median voter under <a href="/wiki/Electoral_system" title="Electoral system">electoral systems</a> that do not satisfy the median voter theorem, including <a href="/wiki/First-past-the-post_voting" title="First-past-the-post voting">plurality voting</a>, <a href="/wiki/Two-round_system" title="Two-round system">plurality-with-primaries</a>, <a href="/wiki/Two-round_system" title="Two-round system">plurality-with-runoff</a>, or <a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">ranked-choice runoff (RCV)</a>.<sup id="cite_ref-myerson-1993_6-1" class="reference"><a href="#cite_note-myerson-1993-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Uses_of_the_median_voter_theorem">Uses of the median voter theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=7" title="Edit section: Uses of the median voter theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The theorem is valuable for the light it sheds on the optimality (and the limits to the optimality) of certain voting systems. </p><p>Valerio Dotti points out broader areas of application: </p> <blockquote><p>The <i>Median Voter Theorem</i> proved extremely popular in the Political Economy literature. The main reason is that it can be adopted to derive testable implications about the relationship between some characteristics of the voting population and the policy outcome, abstracting from other features of the political process.<sup id="cite_ref-dotti-2016_13-1" class="reference"><a href="#cite_note-dotti-2016-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>He adds that... </p> <blockquote><p>The median voter result has been applied to an incredible variety of questions. Examples are the analysis of the relationship between income inequality and size of governmental intervention in redistributive policies (Meltzer and Richard, 1981),<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> the study of the determinants of immigration policies (Razin and Sadka, 1999),<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> of the extent of taxation on different types of income (Bassetto and Benhabib, 2006),<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> and many more.</p></blockquote> <div class="mw-heading mw-heading2"><h2 id="Empirical_evidence_and_contradictions">Empirical evidence and contradictions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=8" title="Edit section: Empirical evidence and contradictions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the <a href="/wiki/United_States" title="United States">United States</a> <a href="/wiki/United_States_Senate" title="United States Senate">Senate</a>, each <a href="/wiki/U.S._state" title="U.S. state">state</a> is allocated two seats. <a href="/wiki/Steven_Levitt" title="Steven Levitt">Levitt</a> (1996) examined the voting patterns of pairs of senators from the same state when one belonged to the Democratic Party and the other to the Republican Party. According to the <b>Median Voter Theorem</b>, the voting patterns of two senators representing the same state should be identical, regardless of party affiliation. However, reality differs. Moreover, Levitt found that the similarity in their voting patterns was only slightly higher than that of randomly paired senators. This finding suggests that senators' ideological leanings have a stronger influence on their decisions than voters' preferences, contradicting the prediction of the Median Voter Theorem.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Pande (2003) studied political changes in <a href="/wiki/India" title="India">India</a> between 1960 and 1992 that increased political representation for marginalized groups. The data she collected showed that as a result of these changes, transfer payments to these populations increased even though the overall electorate (which had already included these groups) remained unchanged. This finding contradicts the Median Voter Theorem, as the model predicts that such a political shift should not alter the political equilibrium.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Chattopadhyay and <a href="/wiki/Esther_Duflo" title="Esther Duflo">Duflo</a> (2004) examined another political change in India, which mandated that women lead one-third of village councils. These councils are responsible for providing various public goods to rural communities. According to the <b>Median Voter Theorem</b>, this policy should not have affected the composition of <a href="/wiki/Public_good" title="Public good">public goods</a> supplied by local governments, as a female candidate still needs to be elected by a majority vote. As long as the median voter's preferences remain unchanged, the allocation of public goods should remain stable. However, empirical data showed that in villages where a woman was elected, the distribution of public goods shifted toward those preferred by women. Furthermore, in districts where women were elected for a second term, the allocation of public goods continued to reflect women's preferences. It is important to note, however, that while the composition of public goods changed when a woman led the village council, this does not necessarily imply an improvement or decline in overall social welfare.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Similar findings were reported by Miller (2008), who analyzed the impact of <a href="/wiki/Women%27s_suffrage" title="Women's suffrage">granting women the right to vote across the United States</a> in 1920. Miller built on previous research indicating that women prioritize child welfare more than men and demonstrated that extending voting rights to women led to an immediate shift in federal policy. This change resulted in a significant increase in healthcare spending and a consequent reduction in child mortality rates by 8%–15%. However, unlike previous cases, Miller's findings actually <b>support</b> the Median Voter Theorem. This is because granting women suffrage altered the composition of the electorate, shifting the median voter’s position toward the preferences of the new female voters.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>Lee, Moretti, and Butler (2004) investigated whether voters influence politicians' positions or merely choose from existing policy stances. They found that an <b><a href="/wiki/Exogeny" title="Exogeny">exogenous</a></b> shift in the voter base does not alter candidates' positions. For instance, an increase in <a href="/wiki/Democratic_Party_(United_States)" title="Democratic Party (United States)">Democratic</a> voters in a given area does not push a <a href="/wiki/Republican_Party_(United_States)" title="Republican Party (United States)">Republican</a> candidate’s stance further to the left, and vice versa. This finding suggests that the electorate selects from the positions that politicians already hold, rather than shaping those positions, contradicting the prediction of the Median Voter Theorem, which assumes candidates are ideologically neutral.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>  </p><p>Gerber and Lewis (2015) analyzed voting data from a series of referendums in <a href="/wiki/California" title="California">California</a> to estimate the preferences of the median voter. They found that elected officials are constrained by the preferences of the median voter in homogeneous regions but less so in heterogeneous ones.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>In contrast, Brunner and Ross (2010), who also studied voter data from two referendums in California, found that the decisive voter in votes concerning public expenditure was not the median voter, but rather a voter from the fourth income <a href="/wiki/Decile" title="Decile">decile</a>. This finding aligns with other studies suggesting that low-income voters often form coalitions with high-income voters to oppose increases in public spending.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>Referendum data from <a href="/wiki/Switzerland" title="Switzerland">Switzerland</a> was used by Stadelmann, Portmann, and Eichenberger (2012) to examine the degree to which legislators' votes align with the preferences of the median voter in their districts. Their research showed that the <b>Median Voter Model</b> explains legislative voting behavior better than an alternative <b>random voting hypothesis,</b> but only by a modest margin of 17.6%. Additionally, they found that support from the median voter in a senator’s district increases the likelihood of the senator supporting a given proposal by 8.4% in parliament.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Milanovic (2000), using data from 79 countries, concluded that the greater the inequality in a country's pre-tax income distribution, the more aggressive the redistributive policies of the winning government. This finding supports the <b>Median Voter Theorem</b>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's impossibility theorem</a></li> <li><a href="/wiki/McKelvey%E2%80%93Schofield_chaos_theorem" title="McKelvey–Schofield chaos theorem">McKelvey–Schofield chaos theorem</a></li> <li><a href="/wiki/Median_mechanism" class="mw-redirect" title="Median mechanism">Median mechanism</a></li> <li><a href="/wiki/Ranked_voting" title="Ranked voting">Ranked voting</a></li> <li><a href="/wiki/Median_voting_rule" title="Median voting rule">Median voting rule</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=10" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBlack1948" class="citation journal cs1">Black, Duncan (1948-02-01). <a rel="nofollow" class="external text" href="https://www.journals.uchicago.edu/doi/10.1086/256633">"On the Rationale of Group Decision-making"</a>. <i>Journal of Political Economy</i>. <b>56</b> (1): <span class="nowrap">23–</span>34. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F256633">10.1086/256633</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-3808">0022-3808</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:153953456">153953456</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Political+Economy&rft.atitle=On+the+Rationale+of+Group+Decision-making&rft.volume=56&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E23-%3C%2Fspan%3E34&rft.date=1948-02-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A153953456%23id-name%3DS2CID&rft.issn=0022-3808&rft_id=info%3Adoi%2F10.1086%2F256633&rft.aulast=Black&rft.aufirst=Duncan&rft_id=https%3A%2F%2Fwww.journals.uchicago.edu%2Fdoi%2F10.1086%2F256633&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-holcombe-2006-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-holcombe-2006_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-holcombe-2006_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHolcombe2006" class="citation book cs1">Holcombe, Randall G. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aOntAAAAMAAJ"><i>Public Sector Economics: The Role of Government in the American Economy</i></a>. Pearson Education. p. 155. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780131450424" title="Special:BookSources/9780131450424"><bdi>9780131450424</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Public+Sector+Economics%3A+The+Role+of+Government+in+the+American+Economy&rft.pages=155&rft.pub=Pearson+Education&rft.date=2006&rft.isbn=9780131450424&rft.aulast=Holcombe&rft.aufirst=Randall+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaOntAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-hotelling_harold-1929-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-hotelling_harold-1929_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hotelling_harold-1929_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHotelling,_Harold1929" class="citation journal cs1">Hotelling, Harold (1929). "Stability in Competition". <i>The Economic Journal</i>. <b>39</b> (153): <span class="nowrap">41–</span>57. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2224214">10.2307/2224214</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2224214">2224214</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Economic+Journal&rft.atitle=Stability+in+Competition&rft.volume=39&rft.issue=153&rft.pages=%3Cspan+class%3D%22nowrap%22%3E41-%3C%2Fspan%3E57&rft.date=1929&rft_id=info%3Adoi%2F10.2307%2F2224214&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2224214%23id-name%3DJSTOR&rft.au=Hotelling%2C+Harold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-downs-1957-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-downs-1957_4-0">^</a></b></span> <span class="reference-text">Anthony Downs, "<a href="/wiki/An_Economic_Theory_of_Democracy" title="An Economic Theory of Democracy">An Economic Theory of Democracy</a>" (1957).</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMcGannKoetzleGrofman2002" class="citation journal cs1">McGann, Anthony J.; Koetzle, William; Grofman, Bernard (2002). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3088418?casa_token=0NcBsoSqKiUAAAAA:1CY5Ul4Tahqhsnupv_ZwTG3G83TDl2y7wYLsj7Lw9rkav4pTVWSodtmq7dy2jVpP8VZTd5FpUss19CupKzVvZvSU9LXP4ZlrMowV3QoFlUxr9FTRm8c&seq=3">"How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Run-off, and Sequential Elimination Elections"</a>. <i>American Journal of Political Science</i>. <b>46</b> (1): <span class="nowrap">134–</span>147. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3088418">10.2307/3088418</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0092-5853">0092-5853</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Political+Science&rft.atitle=How+an+Ideologically+Concentrated+Minority+Can+Trump+a+Dispersed+Majority%3A+Nonmedian+Voter+Results+for+Plurality%2C+Run-off%2C+and+Sequential+Elimination+Elections&rft.volume=46&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E134-%3C%2Fspan%3E147&rft.date=2002&rft_id=info%3Adoi%2F10.2307%2F3088418&rft.issn=0092-5853&rft.aulast=McGann&rft.aufirst=Anthony+J.&rft.au=Koetzle%2C+William&rft.au=Grofman%2C+Bernard&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3088418%3Fcasa_token%3D0NcBsoSqKiUAAAAA%3A1CY5Ul4Tahqhsnupv_ZwTG3G83TDl2y7wYLsj7Lw9rkav4pTVWSodtmq7dy2jVpP8VZTd5FpUss19CupKzVvZvSU9LXP4ZlrMowV3QoFlUxr9FTRm8c%26seq%3D3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-myerson-1993-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-myerson-1993_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-myerson-1993_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMyersonWeber1993" class="citation journal cs1">Myerson, Roger B.; Weber, Robert J. (March 1993). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/journals/american-political-science-review/article/abs/theory-of-voting-equilibria/64F7B2BB2FEED712B1E3A375AF86504E">"A Theory of Voting Equilibria"</a>. <i>American Political Science Review</i>. <b>87</b> (1): <span class="nowrap">102–</span>114. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2938959">10.2307/2938959</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10419%2F221141">10419/221141</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1537-5943">1537-5943</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2938959">2938959</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Political+Science+Review&rft.atitle=A+Theory+of+Voting+Equilibria&rft.volume=87&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E102-%3C%2Fspan%3E114&rft.date=1993-03&rft_id=info%3Ahdl%2F10419%2F221141&rft.issn=1537-5943&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2938959%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2938959&rft.aulast=Myerson&rft.aufirst=Roger+B.&rft.au=Weber%2C+Robert+J.&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fjournals%2Famerican-political-science-review%2Farticle%2Fabs%2Ftheory-of-voting-equilibria%2F64F7B2BB2FEED712B1E3A375AF86504E&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMusselSchlechta2023" class="citation journal cs1">Mussel, Johanan D.; Schlechta, Henry (2023-07-21). <a rel="nofollow" class="external text" href="http://journals.sagepub.com/doi/10.1177/13540688231189363">"Australia: No party convergence where we would most expect it"</a>. <i>Party Politics</i>. <b>30</b> (6): <span class="nowrap">1040–</span>1050. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F13540688231189363">10.1177/13540688231189363</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1354-0688">1354-0688</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Party+Politics&rft.atitle=Australia%3A+No+party+convergence+where+we+would+most+expect+it&rft.volume=30&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1040-%3C%2Fspan%3E1050&rft.date=2023-07-21&rft_id=info%3Adoi%2F10.1177%2F13540688231189363&rft.issn=1354-0688&rft.aulast=Mussel&rft.aufirst=Johanan+D.&rft.au=Schlechta%2C+Henry&rft_id=http%3A%2F%2Fjournals.sagepub.com%2Fdoi%2F10.1177%2F13540688231189363&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">See Black's paper.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Berno Buechel, "Condorcet winners on median spaces" (2014).</span> </li> <li id="cite_note-b-2004-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-b-2004_10-0">^</a></b></span> <span class="reference-text">B. Grofman and S. L. Feld, "If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs rule" (2004).</span> </li> <li id="cite_note-george_g-2010-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-george_g-2010_11-0">^</a></b></span> <span class="reference-text">George G. Szpiro, "Numbers Rule" (2010).</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Eric Pacuit, <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/fall2019/entries/voting-methods/">"Voting Methods"</a>, The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.).</span> </li> <li id="cite_note-dotti-2016-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-dotti-2016_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dotti-2016_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">See Valerio Dotti's thesis <a rel="nofollow" class="external text" href="https://discovery.ucl.ac.uk/id/eprint/1516004/1/thesis_Valerio_Dotti_final.pdf">"Multidimensional Voting Models"</a> (2016).</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">C. R. Plott, "A Notion of Equilibrium and its Possibility Under Majority Rule" (1967).</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRobinette2023" class="citation journal cs1">Robinette, Robbie (2023-09-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/s10602-022-09378-6">"Implications of strategic position choices by candidates"</a>. <i>Constitutional Political Economy</i>. <b>34</b> (3): <span class="nowrap">445–</span>457. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10602-022-09378-6">10.1007/s10602-022-09378-6</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1572-9966">1572-9966</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Constitutional+Political+Economy&rft.atitle=Implications+of+strategic+position+choices+by+candidates&rft.volume=34&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E445-%3C%2Fspan%3E457&rft.date=2023-09-01&rft_id=info%3Adoi%2F10.1007%2Fs10602-022-09378-6&rft.issn=1572-9966&rft.aulast=Robinette&rft.aufirst=Robbie&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2Fs10602-022-09378-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">A. H. Meltzer and S. F. Richard, "A Rational Theory of the Size of Government" (1981).</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">A. Razin and E. Sadka "Migration and Pension with International Capital Mobility" (1999).</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">M. Bassetto and J. Benhabib, "Redistribution, Taxes, and the Median Voter" (2006).</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLevitt1996" class="citation journal cs1">Levitt, Steven D. (1996). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2118205">"How Do Senators Vote? Disentangling the Role of Voter Preferences, Party Affiliation, and Senator Ideology"</a>. <i>The American Economic Review</i>. <b>86</b> (3): <span class="nowrap">425–</span>441. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-8282">0002-8282</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Economic+Review&rft.atitle=How+Do+Senators+Vote%3F+Disentangling+the+Role+of+Voter+Preferences%2C+Party+Affiliation%2C+and+Senator+Ideology&rft.volume=86&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E425-%3C%2Fspan%3E441&rft.date=1996&rft.issn=0002-8282&rft.aulast=Levitt&rft.aufirst=Steven+D.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2118205&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPande2003" class="citation journal cs1">Pande, Rohini (September 2003). <a rel="nofollow" class="external text" href="https://www.aeaweb.org/articles?id=10.1257/000282803769206232">"Can Mandated Political Representation Increase Policy Influence for Disadvantaged Minorities? Theory and Evidence from India"</a>. <i>American Economic Review</i>. <b>93</b> (4): <span class="nowrap">1132–</span>1151. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1257%2F000282803769206232">10.1257/000282803769206232</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-8282">0002-8282</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Economic+Review&rft.atitle=Can+Mandated+Political+Representation+Increase+Policy+Influence+for+Disadvantaged+Minorities%3F+Theory+and+Evidence+from+India&rft.volume=93&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1132-%3C%2Fspan%3E1151&rft.date=2003-09&rft_id=info%3Adoi%2F10.1257%2F000282803769206232&rft.issn=0002-8282&rft.aulast=Pande&rft.aufirst=Rohini&rft_id=https%3A%2F%2Fwww.aeaweb.org%2Farticles%3Fid%3D10.1257%2F000282803769206232&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChattopadhyayDuflo2004" class="citation journal cs1">Chattopadhyay, Raghabendra; Duflo, Esther (2004). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1468-0262.2004.00539.x">"Women as Policy Makers: Evidence from a Randomized Policy Experiment in India"</a>. <i>Econometrica</i>. <b>72</b> (5): <span class="nowrap">1409–</span>1443. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1468-0262.2004.00539.x">10.1111/j.1468-0262.2004.00539.x</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10.1111%2Fj.1468-0262.2004.00539.x">10.1111/j.1468-0262.2004.00539.x</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1468-0262">1468-0262</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Econometrica&rft.atitle=Women+as+Policy+Makers%3A+Evidence+from+a+Randomized+Policy+Experiment+in+India&rft.volume=72&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1409-%3C%2Fspan%3E1443&rft.date=2004&rft_id=info%3Ahdl%2F10.1111%2Fj.1468-0262.2004.00539.x&rft.issn=1468-0262&rft_id=info%3Adoi%2F10.1111%2Fj.1468-0262.2004.00539.x&rft.aulast=Chattopadhyay&rft.aufirst=Raghabendra&rft.au=Duflo%2C+Esther&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2Fabs%2F10.1111%2Fj.1468-0262.2004.00539.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMiller2008" class="citation journal cs1">Miller, Grant (2008-08-01). <a rel="nofollow" class="external text" href="https://academic.oup.com/qje/article-abstract/123/3/1287/1928181?redirectedFrom=fulltext">"Women's Suffrage, Political Responsiveness, and Child Survival in American History*"</a>. <i>The Quarterly Journal of Economics</i>. <b>123</b> (3): <span class="nowrap">1287–</span>1327. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2Fqjec.2008.123.3.1287">10.1162/qjec.2008.123.3.1287</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0033-5533">0033-5533</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3046394">3046394</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21373369">21373369</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Quarterly+Journal+of+Economics&rft.atitle=Women%27s+Suffrage%2C+Political+Responsiveness%2C+and+Child+Survival+in+American+History%2A&rft.volume=123&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1287-%3C%2Fspan%3E1327&rft.date=2008-08-01&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3046394%23id-name%3DPMC&rft.issn=0033-5533&rft_id=info%3Apmid%2F21373369&rft_id=info%3Adoi%2F10.1162%2Fqjec.2008.123.3.1287&rft.aulast=Miller&rft.aufirst=Grant&rft_id=https%3A%2F%2Facademic.oup.com%2Fqje%2Farticle-abstract%2F123%2F3%2F1287%2F1928181%3FredirectedFrom%3Dfulltext&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLeeMorettiButler2004" class="citation journal cs1">Lee, David S.; Moretti, Enrico; Butler, Matthew J. (2004-08-01). <a rel="nofollow" class="external text" href="https://academic.oup.com/qje/article-abstract/119/3/807/1938834?redirectedFrom=fulltext">"Do Voters Affect or Elect Policies? Evidence from the U. S. House*"</a>. <i>The Quarterly Journal of Economics</i>. <b>119</b> (3): <span class="nowrap">807–</span>859. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2F0033553041502153">10.1162/0033553041502153</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0033-5533">0033-5533</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Quarterly+Journal+of+Economics&rft.atitle=Do+Voters+Affect+or+Elect+Policies%3F+Evidence+from+the+U.+S.+House%2A&rft.volume=119&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E807-%3C%2Fspan%3E859&rft.date=2004-08-01&rft_id=info%3Adoi%2F10.1162%2F0033553041502153&rft.issn=0033-5533&rft.aulast=Lee&rft.aufirst=David+S.&rft.au=Moretti%2C+Enrico&rft.au=Butler%2C+Matthew+J.&rft_id=https%3A%2F%2Facademic.oup.com%2Fqje%2Farticle-abstract%2F119%2F3%2F807%2F1938834%3FredirectedFrom%3Dfulltext&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGerberLewis2004" class="citation journal cs1">Gerber, Elisabeth R.; Lewis, Jeffrey B. (December 2004). <a rel="nofollow" class="external text" href="https://www.journals.uchicago.edu/doi/10.1086/424737">"Beyond the Median: Voter Preferences, District Heterogeneity, and Political Representation"</a>. <i>Journal of Political Economy</i>. <b>112</b> (6): <span class="nowrap">1364–</span>1383. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F424737">10.1086/424737</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-3808">0022-3808</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Political+Economy&rft.atitle=Beyond+the+Median%3A+Voter+Preferences%2C+District+Heterogeneity%2C+and+Political+Representation&rft.volume=112&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1364-%3C%2Fspan%3E1383&rft.date=2004-12&rft_id=info%3Adoi%2F10.1086%2F424737&rft.issn=0022-3808&rft.aulast=Gerber&rft.aufirst=Elisabeth+R.&rft.au=Lewis%2C+Jeffrey+B.&rft_id=https%3A%2F%2Fwww.journals.uchicago.edu%2Fdoi%2F10.1086%2F424737&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrunnerRoss2010" class="citation journal cs1">Brunner, Eric J.; Ross, Stephen L. (2010-12-01). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S004727271000126X">"Is the median voter decisive? Evidence from referenda voting patterns"</a>. <i>Journal of Public Economics</i>. <b>94</b> (11): <span class="nowrap">898–</span>910. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jpubeco.2010.09.009">10.1016/j.jpubeco.2010.09.009</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0047-2727">0047-2727</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Public+Economics&rft.atitle=Is+the+median+voter+decisive%3F+Evidence+from+referenda+voting+patterns&rft.volume=94&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E898-%3C%2Fspan%3E910&rft.date=2010-12-01&rft_id=info%3Adoi%2F10.1016%2Fj.jpubeco.2010.09.009&rft.issn=0047-2727&rft.aulast=Brunner&rft.aufirst=Eric+J.&rft.au=Ross%2C+Stephen+L.&rft_id=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS004727271000126X&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStadelmannPortmannEichenberger2012" class="citation journal cs1">Stadelmann, David; Portmann, Marco; Eichenberger, Reiner (2012-03-01). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S0165176511003958">"Evaluating the median voter model's explanatory power"</a>. <i>Economics Letters</i>. <b>114</b> (3): <span class="nowrap">312–</span>314. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.econlet.2011.10.015">10.1016/j.econlet.2011.10.015</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0165-1765">0165-1765</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Economics+Letters&rft.atitle=Evaluating+the+median+voter+model%E2%80%99s+explanatory+power&rft.volume=114&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E312-%3C%2Fspan%3E314&rft.date=2012-03-01&rft_id=info%3Adoi%2F10.1016%2Fj.econlet.2011.10.015&rft.issn=0165-1765&rft.aulast=Stadelmann&rft.aufirst=David&rft.au=Portmann%2C+Marco&rft.au=Eichenberger%2C+Reiner&rft_id=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS0165176511003958&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMilanovic2000" class="citation journal cs1">Milanovic, Branko (September 2000). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S0176268000000148">"The median-voter hypothesis, income inequality, and income redistribution: an empirical test with the required data"</a>. <i>European Journal of Political Economy</i>. <b>16</b> (3): <span class="nowrap">367–</span>410. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0176-2680%2800%2900014-8">10.1016/S0176-2680(00)00014-8</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10419%2F160928">10419/160928</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Political+Economy&rft.atitle=The+median-voter+hypothesis%2C+income+inequality%2C+and+income+redistribution%3A+an+empirical+test+with+the+required+data&rft.volume=16&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E367-%3C%2Fspan%3E410&rft.date=2000-09&rft_id=info%3Ahdl%2F10419%2F160928&rft_id=info%3Adoi%2F10.1016%2FS0176-2680%2800%2900014-8&rft.aulast=Milanovic&rft.aufirst=Branko&rft_id=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS0176268000000148&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBuchananTollison1984" class="citation book cs1">Buchanan, James M.; Tollison, Robert D. (1984). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bX3TbknHJj8C"><i>The Theory of Public Choice</i></a>. Vol. II. Ann Arbor: University of Michigan Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0472080415" title="Special:BookSources/0472080415"><bdi>0472080415</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Public+Choice&rft.place=Ann+Arbor&rft.pub=University+of+Michigan+Press&rft.date=1984&rft.isbn=0472080415&rft.aulast=Buchanan&rft.aufirst=James+M.&rft.au=Tollison%2C+Robert+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbX3TbknHJj8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFClinton2006" class="citation journal cs1">Clinton, Joshua D. (2006). "Representation in Congress: Constituents and the Roll Calls in the 106th House". <i><a href="/wiki/Journal_of_Politics" class="mw-redirect" title="Journal of Politics">Journal of Politics</a></i>. <b>68</b> (2): <span class="nowrap">397–</span>409. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1468-2508.2006.00415.x">10.1111/j.1468-2508.2006.00415.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Politics&rft.atitle=Representation+in+Congress%3A+Constituents+and+the+Roll+Calls+in+the+106th+House&rft.volume=68&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E397-%3C%2Fspan%3E409&rft.date=2006&rft_id=info%3Adoi%2F10.1111%2Fj.1468-2508.2006.00415.x&rft.aulast=Clinton&rft.aufirst=Joshua+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCongleton2003" class="citation book cs1">Congleton, Roger (2003). <a rel="nofollow" class="external text" href="http://rdc1.net/forthcoming/medianvt.pdf">"The Median Voter Model"</a> <span class="cs1-format">(PDF)</span>. In Rowley, C. K.; Schneider, F. (eds.). <i>The Encyclopedia of Public Choice</i>. Kluwer Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-8607-0" title="Special:BookSources/978-0-7923-8607-0"><bdi>978-0-7923-8607-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Median+Voter+Model&rft.btitle=The+Encyclopedia+of+Public+Choice&rft.pub=Kluwer+Academic+Press&rft.date=2003&rft.isbn=978-0-7923-8607-0&rft.aulast=Congleton&rft.aufirst=Roger&rft_id=http%3A%2F%2Frdc1.net%2Fforthcoming%2Fmedianvt.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li>Dasgupta, Partha and Eric Maskin, "On the Robustness of Majority Rule", Journal of the European Economic Association, 2008.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDowns1957" class="citation journal cs1">Downs, Anthony (1957). "An Economic Theory of Political Action in a Democracy". <i><a href="/wiki/Journal_of_Political_Economy" title="Journal of Political Economy">Journal of Political Economy</a></i>. <b>65</b> (2): <span class="nowrap">135–</span>150. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F257897">10.1086/257897</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:154363730">154363730</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Political+Economy&rft.atitle=An+Economic+Theory+of+Political+Action+in+a+Democracy&rft.volume=65&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E135-%3C%2Fspan%3E150&rft.date=1957&rft_id=info%3Adoi%2F10.1086%2F257897&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A154363730%23id-name%3DS2CID&rft.aulast=Downs&rft.aufirst=Anthony&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHolcombe1980" class="citation journal cs1">Holcombe, Randall G. (1980). "An Empirical Test of the Median Voter Model". <i><a href="/wiki/Economic_Inquiry" title="Economic Inquiry">Economic Inquiry</a></i>. <b>18</b> (2): <span class="nowrap">260–</span>275. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1465-7295.1980.tb00574.x">10.1111/j.1465-7295.1980.tb00574.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Economic+Inquiry&rft.atitle=An+Empirical+Test+of+the+Median+Voter+Model&rft.volume=18&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E260-%3C%2Fspan%3E275&rft.date=1980&rft_id=info%3Adoi%2F10.1111%2Fj.1465-7295.1980.tb00574.x&rft.aulast=Holcombe&rft.aufirst=Randall+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHolcombeSobel1995" class="citation journal cs1">Holcombe, Randall G.; Sobel, Russell S. (1995). "Empirical Evidence on the Publicness of State Legislative Activities". <i><a href="/wiki/Public_Choice_(journal)" title="Public Choice (journal)">Public Choice</a></i>. <b>83</b> (<span class="nowrap">1–</span>2): <span class="nowrap">47–</span>58. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01047682">10.1007/BF01047682</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:44831293">44831293</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Public+Choice&rft.atitle=Empirical+Evidence+on+the+Publicness+of+State+Legislative+Activities&rft.volume=83&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E47-%3C%2Fspan%3E58&rft.date=1995&rft_id=info%3Adoi%2F10.1007%2FBF01047682&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A44831293%23id-name%3DS2CID&rft.aulast=Holcombe&rft.aufirst=Randall+G.&rft.au=Sobel%2C+Russell+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHustedKenny1997" class="citation journal cs1">Husted, Thomas A.; Kenny, Lawrence W. (1997). "The Effect of the Expansion of the Voting Franchise on the Size of Government". <i><a href="/wiki/Journal_of_Political_Economy" title="Journal of Political Economy">Journal of Political Economy</a></i>. <b>105</b> (1): <span class="nowrap">54–</span>82. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F262065">10.1086/262065</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:41897793">41897793</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Political+Economy&rft.atitle=The+Effect+of+the+Expansion+of+the+Voting+Franchise+on+the+Size+of+Government&rft.volume=105&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E54-%3C%2Fspan%3E82&rft.date=1997&rft_id=info%3Adoi%2F10.1086%2F262065&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A41897793%23id-name%3DS2CID&rft.aulast=Husted&rft.aufirst=Thomas+A.&rft.au=Kenny%2C+Lawrence+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKrehbiel2004" class="citation journal cs1">Krehbiel, Keith (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1257%2F089533004773563467">"Legislative Organization"</a>. <i><a href="/wiki/Journal_of_Economic_Perspectives" title="Journal of Economic Perspectives">Journal of Economic Perspectives</a></i>. <b>18</b> (1): <span class="nowrap">113–</span>128. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1257%2F089533004773563467">10.1257/089533004773563467</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:249607866">249607866</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Economic+Perspectives&rft.atitle=Legislative+Organization&rft.volume=18&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E113-%3C%2Fspan%3E128&rft.date=2004&rft_id=info%3Adoi%2F10.1257%2F089533004773563467&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A249607866%23id-name%3DS2CID&rft.aulast=Krehbiel&rft.aufirst=Keith&rft_id=https%3A%2F%2Fdoi.org%2F10.1257%252F089533004773563467&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMcKelvey1976" class="citation journal cs1">McKelvey, Richard D. (1976). "Intransitives in Multidimensional Voting Models and Some Implications for Agenda Control". <i><a href="/wiki/Journal_of_Economic_Theory" title="Journal of Economic Theory">Journal of Economic Theory</a></i>. <b>12</b> (3): <span class="nowrap">472–</span>482. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-0531%2876%2990040-5">10.1016/0022-0531(76)90040-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Economic+Theory&rft.atitle=Intransitives+in+Multidimensional+Voting+Models+and+Some+Implications+for+Agenda+Control&rft.volume=12&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E472-%3C%2Fspan%3E482&rft.date=1976&rft_id=info%3Adoi%2F10.1016%2F0022-0531%2876%2990040-5&rft.aulast=McKelvey&rft.aufirst=Richard+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchummerVohra2013" class="citation book cs1">Schummer, James; Vohra, Rakesh V. (2013). "Mechanism Design Without Money". In Nisan, Noam; Roughgarden, Tim; Tardos, Eva; Vazirani, Vijay (eds.). <i>Algorithmic Game Theory</i>. New York: Cambridge University Press. pp. <span class="nowrap">246–</span>252. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-87282-9" title="Special:BookSources/978-0-521-87282-9"><bdi>978-0-521-87282-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mechanism+Design+Without+Money&rft.btitle=Algorithmic+Game+Theory&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E246-%3C%2Fspan%3E252&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-0-521-87282-9&rft.aulast=Schummer&rft.aufirst=James&rft.au=Vohra%2C+Rakesh+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRice1985" class="citation journal cs1">Rice, Tom W. (1985). "An Examination of the Median Voter Hypothesis". <i><a href="/wiki/Western_Political_Quarterly" class="mw-redirect" title="Western Political Quarterly">Western Political Quarterly</a></i>. <b>38</b> (2): <span class="nowrap">211–</span>223. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F448625">10.2307/448625</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/448625">448625</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Western+Political+Quarterly&rft.atitle=An+Examination+of+the+Median+Voter+Hypothesis&rft.volume=38&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E211-%3C%2Fspan%3E223&rft.date=1985&rft_id=info%3Adoi%2F10.2307%2F448625&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F448625%23id-name%3DJSTOR&rft.aulast=Rice&rft.aufirst=Tom+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRomerRosenthal1979" class="citation journal cs1">Romer, Thomas; Rosenthal, Howard (1979). "The Elusive Median Voter". <i><a href="/wiki/Journal_of_Public_Economics" title="Journal of Public Economics">Journal of Public Economics</a></i>. <b>12</b> (2): <span class="nowrap">143–</span>170. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0047-2727%2879%2990010-0">10.1016/0047-2727(79)90010-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Public+Economics&rft.atitle=The+Elusive+Median+Voter&rft.volume=12&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E143-%3C%2Fspan%3E170&rft.date=1979&rft_id=info%3Adoi%2F10.1016%2F0047-2727%2879%2990010-0&rft.aulast=Romer&rft.aufirst=Thomas&rft.au=Rosenthal%2C+Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSobelHolcombe2001" class="citation journal cs1">Sobel, Russell S.; Holcombe, Randall G. (2001). "The Unanimous Voting Rule is not the Political Equivalent to Market Exchange". <i><a href="/wiki/Public_Choice_(journal)" title="Public Choice (journal)">Public Choice</a></i>. <b>106</b> (<span class="nowrap">3–</span>4): <span class="nowrap">233–</span>242. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1005298607876">10.1023/A:1005298607876</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16736216">16736216</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Public+Choice&rft.atitle=The+Unanimous+Voting+Rule+is+not+the+Political+Equivalent+to+Market+Exchange&rft.volume=106&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E233-%3C%2Fspan%3E242&rft.date=2001&rft_id=info%3Adoi%2F10.1023%2FA%3A1005298607876&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16736216%23id-name%3DS2CID&rft.aulast=Sobel&rft.aufirst=Russell+S.&rft.au=Holcombe%2C+Randall+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWaldfogel2008" class="citation journal cs1">Waldfogel, Joel (2008). "The Median Voter and the Median Consumer: Local <i>Private</i> Goods and Population Composition". <i><a href="/wiki/Journal_of_Urban_Economics" title="Journal of Urban Economics">Journal of Urban Economics</a></i>. <b>63</b> (2): <span class="nowrap">567–</span>582. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jue.2007.04.002">10.1016/j.jue.2007.04.002</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:152378898">152378898</a>. <a href="/wiki/SSRN_(identifier)" class="mw-redirect" title="SSRN (identifier)">SSRN</a> <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=878059">878059</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Urban+Economics&rft.atitle=The+Median+Voter+and+the+Median+Consumer%3A+Local+Private+Goods+and+Population+Composition&rft.volume=63&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E567-%3C%2Fspan%3E582&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A152378898%23id-name%3DS2CID&rft_id=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D878059%23id-name%3DSSRN&rft_id=info%3Adoi%2F10.1016%2Fj.jue.2007.04.002&rft.aulast=Waldfogel&rft.aufirst=Joel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMedian+voter+theorem" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Median_voter_theorem&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://rdc1.net/forthcoming/medianvt.pdf">The Median Voter Model</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Median_voter_theorem&oldid=1276142150">https://en.wikipedia.org/w/index.php?title=Median_voter_theorem&oldid=1276142150</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Political_science_theories" title="Category:Political science theories">Political science theories</a></li><li><a href="/wiki/Category:Public_choice_theory" title="Category:Public choice theory">Public choice theory</a></li><li><a href="/wiki/Category:Voting_theory" title="Category:Voting theory">Voting theory</a></li><li><a href="/wiki/Category:Game_theory" title="Category:Game theory">Game theory</a></li><li><a href="/wiki/Category:Mathematical_economics" title="Category:Mathematical economics">Mathematical economics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 17 February 2025, at 03:31<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Median_voter_theorem&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Median voter theorem</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>15 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-64b5bb4b79-m86v8","wgBackendResponseTime":179,"wgPageParseReport":{"limitreport":{"cputime":"0.477","walltime":"0.604","ppvisitednodes":{"value":2556,"limit":1000000},"postexpandincludesize":{"value":105123,"limit":2097152},"templateargumentsize":{"value":1069,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":6,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":116957,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 507.615 1 -total"," 33.86% 171.867 26 Template:Cite_journal"," 31.14% 158.094 2 Template:Reflist"," 28.33% 143.810 1 Template:Electoral_systems_sidebar"," 27.89% 141.558 1 Template:Sidebar_with_collapsible_lists"," 15.15% 76.922 2 Template:Hlist"," 14.09% 71.540 1 Template:Short_description"," 8.91% 45.247 2 Template:Pagetype"," 6.17% 31.343 1 Template:Confused"," 5.39% 27.373 3 Template:Portal-inline"]},"scribunto":{"limitreport-timeusage":{"value":"0.309","limit":"10.000"},"limitreport-memusage":{"value":8107974,"limit":52428800},"limitreport-logs":"table#1 {\n [\"size\"] = \"tiny\",\n}\ntable#1 {\n [\"size\"] = \"tiny\",\n}\ntable#1 {\n [\"size\"] = \"tiny\",\n}\n"},"cachereport":{"origin":"mw-api-int.codfw.main-67794785f5-vw9pm","timestamp":"20250228165927","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Median voter theorem","url":"https:\/\/en.wikipedia.org\/wiki\/Median_voter_theorem","sameAs":"http:\/\/www.wikidata.org\/entity\/Q648511","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q648511","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-12-02T10:36:09Z","dateModified":"2025-02-17T03:31:37Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/82\/Electoral-systems-gears.svg","headline":"theorem in political science"}</script> </body> </html>