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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Implications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Implications</span> </div> </a> <button aria-controls="toc-Implications-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 Implications subsection</span> </button> <ul id="toc-Implications-sublist" class="vector-toc-list"> <li id="toc-Two-stage_voting_processes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-stage_voting_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Two-stage voting processes</span> </div> </a> <ul id="toc-Two-stage_voting_processes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spoiler_effects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spoiler_effects"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Spoiler effects</span> </div> </a> <ul id="toc-Spoiler_effects-sublist" class="vector-toc-list"> </ul> </li> </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">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">6</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">7</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">8</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" 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Available in 25 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-25" 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">25 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Paradoxa_de_Condorcet" title="Paradoxa de Condorcet – Catalan" lang="ca" hreflang="ca" data-title="Paradoxa de Condorcet" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Condorcet-Paradoxon" title="Condorcet-Paradoxon – German" lang="de" hreflang="de" data-title="Condorcet-Paradoxon" 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/Paradoja_de_Condorcet" title="Paradoja de Condorcet – Spanish" lang="es" hreflang="es" data-title="Paradoja de Condorcet" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Balotada_paradokso" title="Balotada paradokso – Esperanto" lang="eo" hreflang="eo" data-title="Balotada paradokso" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%A7%D8%B1%D8%A7%D8%AF%D9%88%DA%A9%D8%B3_%DA%A9%D9%86%D8%AF%D9%88%D8%B1%D8%B3%D9%87" title="پارادوکس کندورسه – Persian" lang="fa" hreflang="fa" data-title="پارادوکس کندورسه" 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/Paradoxe_de_Condorcet" title="Paradoxe de Condorcet – French" lang="fr" hreflang="fr" data-title="Paradoxe de Condorcet" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Paradoxo_de_Condorcet" title="Paradoxo de Condorcet – Galician" lang="gl" hreflang="gl" data-title="Paradoxo de Condorcet" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%88%AC%ED%91%9C%EC%9D%98_%EC%97%AD%EC%84%A4" title="투표의 역설 – Korean" lang="ko" hreflang="ko" data-title="투표의 역설" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B6%D5%A4%D5%B8%D6%80%D5%BD%D5%A5%D5%AB_%D5%BA%D5%A1%D6%80%D5%A1%D5%A4%D5%B8%D6%84%D5%BD" title="Կոնդորսեի պարադոքս – Armenian" lang="hy" hreflang="hy" data-title="Կոնդորսեի պարադոքս" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%9Evers%C3%B6gn_Condorcets" title="Þversögn Condorcets – Icelandic" lang="is" hreflang="is" data-title="Þversögn Condorcets" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Paradosso_di_Condorcet" title="Paradosso di Condorcet – Italian" lang="it" hreflang="it" data-title="Paradosso di Condorcet" 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%A4%D7%A8%D7%93%D7%95%D7%A7%D7%A1_%D7%94%D7%94%D7%A6%D7%91%D7%A2%D7%94" title="פרדוקס ההצבעה – Hebrew" lang="he" hreflang="he" data-title="פרדוקס ההצבעה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Balsavimo_paradoksas" title="Balsavimo paradoksas – Lithuanian" lang="lt" hreflang="lt" data-title="Balsavimo paradoksas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Szavaz%C3%A1si_paradoxon" title="Szavazási paradoxon – Hungarian" lang="hu" hreflang="hu" data-title="Szavazási paradoxon" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Paradox_van_Condorcet" title="Paradox van Condorcet – Dutch" lang="nl" hreflang="nl" data-title="Paradox van Condorcet" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%8A%95%E7%A5%A8%E3%81%AE%E9%80%86%E7%90%86" title="投票の逆理 – Japanese" lang="ja" hreflang="ja" data-title="投票の逆理" 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/Paradoks_g%C5%82osowania" title="Paradoks głosowania – Polish" lang="pl" hreflang="pl" data-title="Paradoks głosowania" 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/Paradoxo_de_Condorcet" title="Paradoxo de Condorcet – Portuguese" lang="pt" hreflang="pt" data-title="Paradoxo de Condorcet" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Paradoxul_lui_Condorcet" title="Paradoxul lui Condorcet – Romanian" lang="ro" hreflang="ro" data-title="Paradoxul lui Condorcet" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%9A%D0%BE%D0%BD%D0%B4%D0%BE%D1%80%D1%81%D0%B5" title="Парадокс Кондорсе – Russian" lang="ru" hreflang="ru" data-title="Парадокс Кондорсе" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Condorcet%E2%80%99n_paradoksi" title="Condorcet’n paradoksi – Finnish" lang="fi" hreflang="fi" data-title="Condorcet’n paradoksi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Condorcetparadoxen" title="Condorcetparadoxen – Swedish" lang="sv" hreflang="sv" data-title="Condorcetparadoxen" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Se%C3%A7im_paradoksu" title="Seçim paradoksu – Turkish" lang="tr" hreflang="tr" data-title="Seçim paradoksu" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%9A%D0%BE%D0%BD%D0%B4%D0%BE%D1%80%D1%81%D0%B5" title="Парадокс Кондорсе – Ukrainian" lang="uk" hreflang="uk" data-title="Парадокс Кондорсе" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8A%95%E7%A5%A8%E6%82%96%E8%AE%BA" title="投票悖论 – Chinese" lang="zh" hreflang="zh" data-title="投票悖论" 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/Q745768#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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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></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 class="mw-selflink selflink">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 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/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 class="mw-selflink selflink">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 href="/wiki/Median_voter_theorem" title="Median voter theorem">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" 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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> <p>In <a href="/wiki/Social_choice_theory" title="Social choice theory">social choice theory</a>, <b>Condorcet's voting paradox</b> is a fundamental discovery by the <a href="/wiki/Marquis_de_Condorcet" title="Marquis de Condorcet">Marquis de Condorcet</a> that <a href="/wiki/Majority_rule" title="Majority rule">majority rule</a> is inherently <a href="/wiki/Contradiction" title="Contradiction">self-contradictory</a>. The result implies that it is logically impossible for any voting system to guarantee that a winner will have support from a majority of voters: for example there can be rock-paper-scissors scenario where a majority of voters will prefer A to B, B to C, and also C to A, even if every voter's individual preferences are rational and avoid self-contradiction. Examples of Condorcet's paradox are called <b>Condorcet cycles</b> or <b>cyclic ties</b>. </p><p>In such a cycle, every possible choice is rejected by the electorate in favor of another alternative, who is preferred by more than half of all voters. Thus, any attempt to ground social decision-making in <a href="/wiki/Majoritarianism" title="Majoritarianism">majoritarianism</a> must accept such self-contradictions (commonly called <a href="/wiki/Spoiler_effect" title="Spoiler effect">spoiler effects</a>). Systems that attempt to do so, while minimizing the rate of such self-contradictions, are called <a href="/wiki/Condorcet_method" title="Condorcet method">Condorcet methods</a>. </p><p>Condorcet's paradox is a special case of <a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's paradox</a>, which shows that <i>any</i> kind of social decision-making process is either self-contradictory, a <a href="/wiki/Dictatorship_mechanism" title="Dictatorship mechanism">dictatorship</a>, or incorporates information about the strength of different voters' preferences (e.g. <a href="/wiki/Cardinal_utility" title="Cardinal utility">cardinal utility</a> or <a href="/wiki/Rated_voting" title="Rated voting">rated voting</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Condorcet's paradox was first discovered by <a href="/wiki/Catalonia" title="Catalonia">Catalan</a> <a href="/wiki/Philosopher" class="mw-redirect" title="Philosopher">philosopher</a> and <a href="/wiki/Theology" title="Theology">theologian</a> <a href="/wiki/Ramon_Llull" title="Ramon Llull">Ramon Llull</a> in the 13th century, during his investigations into <a href="/wiki/Church_governance" class="mw-redirect" title="Church governance">church governance</a>, but his work was lost until the 21st century. The mathematician and political philosopher <a href="/wiki/Marquis_de_Condorcet" title="Marquis de Condorcet">Marquis de Condorcet</a> rediscovered the paradox in the late 18th century.<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Condorcet's discovery means he arguably identified the key result of <a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's impossibility theorem</a>, albeit under stronger conditions than required by Arrow: Condorcet cycles create situations where any ranked voting system <a href="/wiki/May%27s_theorem" title="May's theorem">that respects majorities</a> must have a <a href="/wiki/Spoiler_effect" title="Spoiler effect">spoiler effect</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows: </p> <table class="wikitable" style="text-align: center;"> <tbody><tr> <th>Voter</th> <th>First preference</th> <th>Second preference</th> <th>Third preference </th></tr> <tr> <th>Voter 1 </th> <td>A</td> <td>B</td> <td>C </td></tr> <tr> <th>Voter 2 </th> <td>B</td> <td>C</td> <td>A </td></tr> <tr> <th>Voter 3 </th> <td>C</td> <td>A</td> <td>B </td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Voting_Paradox_example.png" class="mw-file-description"><img alt="3 blue dots in a triangle. 3 red dots in a triangle, connected by arrows that point counterclockwise." 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>Voters (blue) and candidates (red) plotted in a 2-dimensional preference space. Each voter prefers a closer candidate over a farther. Arrows show the order in which voters prefer the candidates.</figcaption></figure> <p>If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. Thus the society's preferences show cycling: A is preferred over B which is preferred over C which is preferred over A. </p><p>As a result, any attempt to appeal to the principle of <a href="/wiki/Majority_rule" title="Majority rule">majority rule</a> will lead to logical <a href="/wiki/Contradiction" title="Contradiction">self-contradiction</a>. Regardless of which alternative we select, we can find another alternative that would be preferred by most voters. </p> <div class="mw-heading mw-heading3"><h3 id="Practical_scenario">Practical scenario</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=3" title="Edit section: Practical scenario"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The voters in Cactus County prefer the incumbent <a href="/wiki/County_executive" title="County executive">county executive</a> <b>Alex</b> of the Farmers' Party over rival <b>Beatrice</b> of the Solar Panel Party by about a 2-to-1 margin. This year a third candidate, <b>Charlie</b>, is running as an independent. Charlie is a wealthy and outspoken businessman, of whom the voters hold polarized views. </p><p>The voters divide into three groups: </p> <ul><li>Group 1 revere Charlie for saving the high school football team. They rank Charlie first, and then Alex above Beatrice as usual (<b>CAB</b>).</li> <li>Group 2 despise Charlie for his sharp business practices. They rank Charlie <i>last</i>, and then Alex above Beatrice as usual (<b>ABC</b>).</li> <li>Group 3 are Beatrice's core supporters. They want the Farmers' Party out of office in favor of the Solar Panel Party, and regard Charlie's candidacy as a sideshow. They rank Beatrice first and Alex last as usual, and Charlie second by default (<b>BCA</b>).</li></ul> <p>Therefore a majority of voters prefer Alex to Beatrice (A > B), as they always have. A majority of voters are either Beatrice-lovers or Charlie-haters, so prefer Beatrice to Charlie (B > C). And a majority of voters are either Charlie-lovers or Alex-haters, so prefer Charlie to Alex (C > A). Combining the three preferences gives us A > B > C > A, a Condorcet cycle. </p> <div class="mw-heading mw-heading2"><h2 id="Likelihood">Likelihood</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=4" title="Edit section: Likelihood"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is possible to estimate the probability of the paradox by extrapolating from real election data, or using mathematical models of voter behavior, though the results depend strongly on which model is used. </p> <div class="mw-heading mw-heading3"><h3 id="Impartial_culture_model">Impartial culture model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=5" title="Edit section: Impartial culture model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We can calculate the probability of seeing the paradox for the special case where voter preferences are uniformly distributed among the candidates. (This is the "<a href="/wiki/Impartial_culture" title="Impartial culture">impartial culture</a>" model, which is known to be a "worst-case scenario"<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_5-0" class="reference"><a href="#cite_note-:1-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 40">: 40 </span></sup><sup id="cite_ref-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 320">: 320 </span></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>—most models show substantially lower probabilities of Condorcet cycles.) </p><p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> voters providing a preference list of three candidates A, B, C, we write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> (resp. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f19a1b3bf39298aacb7e2daeab9320130a986fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.569ex; height:2.509ex;" alt="{\displaystyle Y_{n}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5073995dface6fb94824a8bec0075e65205fc64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.806ex; height:2.509ex;" alt="{\displaystyle Z_{n}}"></span>) the random variable equal to the number of voters who placed A in front of B (respectively B in front of C, C in front of A). The sought probability is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}=2P(X_{n}>n/2,Y_{n}>n/2,Z_{n}>n/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}=2P(X_{n}>n/2,Y_{n}>n/2,Z_{n}>n/2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ad187be48c1db226c1ebab5be29fb34c6eaa18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:41.333ex; height:2.843ex;" alt="{\displaystyle p_{n}=2P(X_{n}>n/2,Y_{n}>n/2,Z_{n}>n/2)}"></span> (we double because there is also the symmetric case A> C> B> A). We show that, for odd <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}=3q_{n}-1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}=3q_{n}-1/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d541db536fd8a8bfbadc66b2eaa6287688551c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:15.322ex; height:2.843ex;" alt="{\displaystyle p_{n}=3q_{n}-1/2}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{n}=P(X_{n}>n/2,Y_{n}>n/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{n}=P(X_{n}>n/2,Y_{n}>n/2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14662045fb37dc7f8fc711ed745924f76528fee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.291ex; height:2.843ex;" alt="{\displaystyle q_{n}=P(X_{n}>n/2,Y_{n}>n/2)}"></span> which makes one need to know only the joint distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f19a1b3bf39298aacb7e2daeab9320130a986fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.569ex; height:2.509ex;" alt="{\displaystyle Y_{n}}"></span>. </p><p>If we put <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n,i,j}=P(X_{n}=i,Y_{n}=j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n,i,j}=P(X_{n}=i,Y_{n}=j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38f0a458cd3bf645f8f6f686a56658d0e7d5536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:25.993ex; height:3.009ex;" alt="{\displaystyle p_{n,i,j}=P(X_{n}=i,Y_{n}=j)}"></span>, we show the relation which makes it possible to compute this distribution by recurrence: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n+1,i,j}={1 \over 6}p_{n,i,j}+{1 \over 3}p_{n,i-1,j}+{1 \over 3}p_{n,i,j-1}+{1 \over 6}p_{n,i-1,j-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n+1,i,j}={1 \over 6}p_{n,i,j}+{1 \over 3}p_{n,i-1,j}+{1 \over 3}p_{n,i,j-1}+{1 \over 6}p_{n,i-1,j-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a36a7da370b144f3089fc26476929511f4e30fa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:52.944ex; height:5.176ex;" alt="{\displaystyle p_{n+1,i,j}={1 \over 6}p_{n,i,j}+{1 \over 3}p_{n,i-1,j}+{1 \over 3}p_{n,i,j-1}+{1 \over 6}p_{n,i-1,j-1}}"></span>. </p><p>The following results are then obtained: </p> <table class="wikitable"> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> </th> <th>3 </th> <th>101 </th> <th>201 </th> <th>301 </th> <th>401 </th> <th>501 </th> <th>601 </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"></span> </td> <td>5.556% </td> <td>8.690% </td> <td>8.732% </td> <td>8.746% </td> <td>8.753% </td> <td>8.757% </td> <td>8.760% </td></tr></tbody></table> <p>The sequence seems to be tending towards a finite limit. </p><p>Using the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>, we show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4de60376835c9362d759da9c14e5aad398f9d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.256ex; height:2.009ex;" alt="{\displaystyle q_{n}}"></span> tends to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\frac {1}{4}}P\left(|T|>{\frac {\sqrt {2}}{4}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\frac {1}{4}}P\left(|T|>{\frac {\sqrt {2}}{4}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbfc193adbf050347aa632e077682e4f0918cd72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.717ex; height:6.509ex;" alt="{\displaystyle q={\frac {1}{4}}P\left(|T|>{\frac {\sqrt {2}}{4}}\right),}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is a variable following a <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>, which gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\dfrac {1}{2\pi }}\int _{{\sqrt {2}}/4}^{+\infty }{\frac {dt}{1+t^{2}}}={\dfrac {\arctan 2{\sqrt {2}}}{2\pi }}={\dfrac {\arccos {\frac {1}{3}}}{2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\dfrac {1}{2\pi }}\int _{{\sqrt {2}}/4}^{+\infty }{\frac {dt}{1+t^{2}}}={\dfrac {\arctan 2{\sqrt {2}}}{2\pi }}={\dfrac {\arccos {\frac {1}{3}}}{2\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710bf78afda4338f25529b5747148ca23d6b6fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.87ex; height:7.343ex;" alt="{\displaystyle q={\dfrac {1}{2\pi }}\int _{{\sqrt {2}}/4}^{+\infty }{\frac {dt}{1+t^{2}}}={\dfrac {\arctan 2{\sqrt {2}}}{2\pi }}={\dfrac {\arccos {\frac {1}{3}}}{2\pi }}}"></span> (constant <a href="//oeis.org/A289505" class="extiw" title="oeis:A289505">quoted in the OEIS</a>). </p><p>The asymptotic probability of encountering the Condorcet paradox is therefore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{3\arccos {1 \over 3}} \over {2\pi }}-{1 \over 2}={\arcsin {{\sqrt {6}} \over 9} \over \pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mn>9</mn> </mfrac> </mrow> </mrow> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{3\arccos {1 \over 3}} \over {2\pi }}-{1 \over 2}={\arcsin {{\sqrt {6}} \over 9} \over \pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14020cd706f960a8c2e17dbfd5d22333862230bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.798ex; height:7.009ex;" alt="{\displaystyle {{3\arccos {1 \over 3}} \over {2\pi }}-{1 \over 2}={\arcsin {{\sqrt {6}} \over 9} \over \pi }}"></span> which gives the value 8.77%.<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><sup id="cite_ref-:2_9-0" class="reference"><a href="#cite_note-:2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Some results for the case of more than three candidates have been calculated<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and simulated.<sup id="cite_ref-:4_11-0" class="reference"><a href="#cite_note-:4-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The simulated likelihood for an impartial culture model with 25 voters increases with the number of candidates:<sup id="cite_ref-:4_11-1" class="reference"><a href="#cite_note-:4-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">: <span title="Page: 28
Quotation: "% Condorcet winners 100.0 91.6 83.4 75.8 64.3 52.5"" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">28</span> </sup> </p> <table class="wikitable"> <caption> </caption> <tbody><tr> <th>3 </th> <th>4 </th> <th>5 </th> <th>7 </th> <th>10 </th></tr> <tr> <td>8.4% </td> <td>16.6% </td> <td>24.2% </td> <td>35.7% </td> <td>47.5% </td></tr></tbody></table> <p>The likelihood of a Condorcet cycle for related models approach these values for three-candidate elections with large electorates:<sup id="cite_ref-:2_9-1" class="reference"><a href="#cite_note-:2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Impartial_culture#Impartial_Anonymous_Culture_(IAC)" title="Impartial culture">Impartial anonymous culture</a> (IAC): 6.25%</li> <li>Uniform culture (UC): 6.25%</li> <li>Maximal culture condition (MC): 9.17%</li></ul> <p>All of these models are unrealistic, but can be investigated to establish an upper bound on the likelihood of a cycle.<sup id="cite_ref-:2_9-2" class="reference"><a href="#cite_note-:2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Group_coherence_models">Group coherence models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=6" title="Edit section: Group coherence models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When modeled with more realistic voter preferences, Condorcet paradoxes in elections with a small number of candidates and a large number of voters become very rare.<sup id="cite_ref-:1_5-1" class="reference"><a href="#cite_note-:1-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 78">: 78 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Spatial_model">Spatial model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=7" title="Edit section: Spatial model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A study of three-candidate elections analyzed 12 different models of voter behavior, and found the <a href="/wiki/Spatial_model_of_voting" class="mw-redirect" title="Spatial model of voting">spatial model of voting</a> to be the most accurate to real-world <a href="/wiki/Ranked_voting" title="Ranked voting">ranked-ballot</a> election data. Analyzing this spatial model, they found the likelihood of a cycle to decrease to zero as the number of voters increases, with likelihoods of 5% for 100 voters, 0.5% for 1000 voters, and 0.06% for 10,000 voters.<sup id="cite_ref-:3_12-0" class="reference"><a href="#cite_note-:3-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Another spatial model found likelihoods of 2% or less in all simulations of 201 voters and 5 candidates, whether two or four-dimensional, with or without correlation between dimensions, and with two different dispersions of candidates.<sup id="cite_ref-:4_11-2" class="reference"><a href="#cite_note-:4-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">: <span title="Page: 31
Quotation: "% Condorcet winners 99+ 99 99+ 99+ 98 98 98 99"" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">31</span> </sup> </p> <div class="mw-heading mw-heading3"><h3 id="Empirical_studies">Empirical studies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=8" title="Edit section: Empirical studies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many attempts have been made at finding empirical examples of the paradox.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Empirical identification of a Condorcet paradox presupposes extensive data on the decision-makers' preferences over all alternatives—something that is only very rarely available. </p><p>While examples of the paradox seem to occur occasionally in small settings (e.g., parliaments) very few examples have been found in larger groups (e.g. electorates), although some have been identified.<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> </p><p>A summary of 37 individual studies, covering a total of 265 real-world elections, large and small, found 25 instances of a Condorcet paradox, for a total likelihood of 9.4%<sup id="cite_ref-:0_6-1" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 325">: 325 </span></sup> (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without).<sup id="cite_ref-:1_5-2" class="reference"><a href="#cite_note-:1-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 47">: 47 </span></sup> </p><p>An analysis of 883 three-candidate elections extracted from 84 real-world ranked-ballot elections of the <a href="/wiki/Electoral_Reform_Society" title="Electoral Reform Society">Electoral Reform Society</a> found a Condorcet cycle likelihood of 0.7%. These derived elections had between 350 and 1,957 voters.<sup id="cite_ref-:3_12-1" class="reference"><a href="#cite_note-:3-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> A similar analysis of data from the 1970–2004 <a href="/wiki/American_National_Election_Studies" title="American National Election Studies">American National Election Studies</a> <a href="/wiki/Thermometer_scale" class="mw-redirect" title="Thermometer scale">thermometer scale</a> surveys found a Condorcet cycle likelihood of 0.4%. These derived elections had between 759 and 2,521 "voters".<sup id="cite_ref-:3_12-2" class="reference"><a href="#cite_note-:3-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Andrew Myers, who operates the <a href="/wiki/Online_poll" class="mw-redirect" title="Online poll">Condorcet Internet Voting Service</a>, analyzed 10,354 nonpolitical CIVS elections and found cycles in 17% of elections with at least 10 votes, with the figure dropping to 2.1% for elections with at least 100 votes, and 1.2% for ≥300 votes.<sup id="cite_ref-CIVS_15-0" class="reference"><a href="#cite_note-CIVS-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Real_world_instances">Real world instances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=9" title="Edit section: Real world instances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A database of 189 ranked United States elections from 2004 to 2022 contained only one Condorcet cycle: the <a href="/wiki/2021_Minneapolis_City_Council_election#Ward_2" title="2021 Minneapolis City Council election">2021 Minneapolis City Council election in Ward 2</a>, with a narrow circular tie between candidates of the <a href="/wiki/Green_Party_of_Minnesota" title="Green Party of Minnesota">Green Party</a> (<a href="/wiki/Cam_Gordon" title="Cam Gordon">Cam Gordon</a>), the <a href="/wiki/Minnesota_Democratic%E2%80%93Farmer%E2%80%93Labor_Party" title="Minnesota Democratic–Farmer–Labor Party">Minnesota Democratic–Farmer–Labor Party</a>, (Yusra Arab) and an independent <a href="/wiki/Democratic_socialism" title="Democratic socialism">democratic socialist</a> (<a href="/wiki/Robin_Wonsley" title="Robin Wonsley">Robin Wonsley</a>).<sup id="cite_ref-GSM2023_16-0" class="reference"><a href="#cite_note-GSM2023-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Voters' preferences were non-transitive: Arab was preferred over Gordon, Gordon over Wonsley, and Wonsley over Arab, creating a cyclical pattern with no clear winner. Additionally, the election exhibited a <a href="/wiki/Negative_responsiveness" title="Negative responsiveness">downward monotonicity</a> paradox, as well as a paradox akin to <a href="/wiki/Simpson%E2%80%99s_paradox" class="mw-redirect" title="Simpson’s paradox">Simpson’s paradox</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Implications">Implications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=10" title="Edit section: Implications"><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:Mexican_Standoff.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Mexican_Standoff.jpg/220px-Mexican_Standoff.jpg" decoding="async" width="220" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Mexican_Standoff.jpg/330px-Mexican_Standoff.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Mexican_Standoff.jpg/440px-Mexican_Standoff.jpg 2x" data-file-width="5270" data-file-height="3450" /></a><figcaption>Three men portraying a <a href="/wiki/Mexican_standoff" title="Mexican standoff">Mexican standoff</a>. Just as there is no winner in a Mexican standoff with certain combinations of gun-pointings, there is sometimes no <a href="/wiki/Condorcet_winner" class="mw-redirect" title="Condorcet winner">majority-preferred winner</a> in a ranked-ballot election.</figcaption></figure><p>When a <a href="/wiki/Condorcet_method" title="Condorcet method">Condorcet method</a> is used to determine an election, the voting paradox of cyclical societal preferences implies that the election has no <a href="/wiki/Condorcet_winner" class="mw-redirect" title="Condorcet winner">Condorcet winner</a>: no candidate who can win a one-on-one election against each other candidate. There will still be a smallest group of candidates, known as the <a href="/wiki/Smith_set" title="Smith set">Smith set</a>, such that each candidate in the group can win a one-on-one election against each of the candidates outside the group. The several variants of the Condorcet method differ on how they <a href="/wiki/Condorcet_method#Circular_ambiguities" title="Condorcet method">resolve such ambiguities</a> when they arise to determine a winner.<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> The Condorcet methods which always elect someone from the Smith set when there is no Condorcet winner are known as <a href="/wiki/Smith_criterion" class="mw-redirect" title="Smith criterion">Smith-efficient</a>. Note that using only rankings, there is no fair and deterministic resolution to the trivial example given earlier because each candidate is in an exactly symmetrical situation. </p><p>Situations having the voting paradox can cause voting mechanisms to violate the axiom of <a href="/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">independence of irrelevant alternatives</a>—the choice of winner by a voting mechanism could be influenced by whether or not a losing candidate is available to be voted for. </p> <div class="mw-heading mw-heading3"><h3 id="Two-stage_voting_processes">Two-stage voting processes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=11" title="Edit section: Two-stage voting processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One important implication of the possible existence of the voting paradox in a practical situation is that in a paired voting process like those of standard <a href="/wiki/Parliamentary_procedure" title="Parliamentary procedure">parliamentary procedure</a>, the eventual winner will depend on the way the majority votes are ordered. For example, say a popular bill is set to pass, before some other group offers an amendment; this amendment passes by majority vote. This may result in a majority of a <a href="/wiki/Legislature" title="Legislature">legislature</a> rejecting the bill as a whole, thus creating a paradox (where a popular amendment to a popular bill has made it unpopular). This logical inconsistency is the origin of the <a href="/wiki/Poison_pill_amendment" class="mw-redirect" title="Poison pill amendment">poison pill amendment</a>, which deliberately engineers a false Condorcet cycle to kill a bill. Likewise, the order of votes in a legislature can be manipulated by the person arranging them to ensure their preferred outcome wins. </p><p>Despite frequent objections by <a href="/wiki/Social_choice_theory" title="Social choice theory">social choice theorists</a> about the logically incoherent results of such procedures, and the existence of better alternatives for choosing between multiple versions of a bill, the procedure of pairwise majority-rule is widely-used and is codified into the <a href="/wiki/By-law" title="By-law">by-laws</a> or parliamentary procedures of almost every kind of <a href="/wiki/Deliberative_assembly" title="Deliberative assembly">deliberative assembly</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Spoiler_effects">Spoiler effects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Condorcet_paradox&action=edit&section=12" title="Edit section: Spoiler effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Condorcet paradoxes imply <a href="/wiki/Majority_rule" title="Majority rule">majoritarian methods</a> fail independent of irrelevant alternatives. Label the three candidates in a race <a href="/wiki/Rock_paper_scissors" title="Rock paper scissors"><i>Rock</i>, <i>Paper</i>, and <i>Scissors</i></a>. In a one-on-one race, Rock loses to Paper, Paper to Scissors, etc. </p><p><a href="/wiki/Without_loss_of_generality" title="Without loss of generality">Without loss of generality</a>, say that Rock wins the election with a certain method. Then, Scissors is a spoiler candidate for Paper: if Scissors were to drop out, Paper would win the only one-on-one race (Paper defeats Rock). The same reasoning applies regardless of the winner. </p><p>This example also shows why Condorcet elections are rarely (if ever) spoiled: spoilers can <i>only</i> happen when there is no Condorcet winner. Condorcet cycles are rare in large elections,<sup id="cite_ref-:53_18-0" class="reference"><a href="#cite_note-:53-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:63_19-0" class="reference"><a href="#cite_note-:63-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Median_voter_theorem" title="Median voter theorem">median voter theorem</a> shows cycles are impossible whenever candidates are arrayed on a <a href="/wiki/Political_spectrum" title="Political spectrum">left-right spectrum</a>. </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=Condorcet_paradox&action=edit&section=13" 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/Discursive_dilemma" title="Discursive dilemma">Discursive dilemma</a></li> <li><a href="/wiki/Spoiler_effect" title="Spoiler effect">Spoiler effect</a></li> <li><a href="/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">Independence of irrelevant alternatives</a></li> <li><a href="/wiki/Nakamura_number" title="Nakamura number">Nakamura number</a></li> <li><a href="/wiki/Quadratic_voting" title="Quadratic voting">Quadratic voting</a></li> <li><a href="/wiki/Rock_paper_scissors" title="Rock paper scissors">Rock paper scissors</a></li> <li><a href="/wiki/Smith_set" title="Smith set">Smith set</a></li></ul> <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=Condorcet_paradox&action=edit&section=14" title="Edit section: References"><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 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="CITEREFMarquis_de_Condorcet1785" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Marquis_de_Condorcet" title="Marquis de Condorcet">Marquis de Condorcet</a> (1785). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k417181"><i>Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix</i></a> <span class="cs1-format">(PNG)</span> (in French)<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-03-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essai+sur+l%27application+de+l%27analyse+%C3%A0+la+probabilit%C3%A9+des+d%C3%A9cisions+rendues+%C3%A0+la+pluralit%C3%A9+des+voix&rft.date=1785&rft.au=Marquis+de+Condorcet&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k417181&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCondorcetSommerladMcLean1989" class="citation book cs1">Condorcet, Jean-Antoine-Nicolas de Caritat; Sommerlad, Fiona; McLean, Iain (1989-01-01). <i>The political theory of Condorcet</i>. Oxford: University of Oxford, Faculty of Social Studies. pp. <span class="nowrap">69–</span>80, <span class="nowrap">152–</span>166. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/20408445">20408445</a>. <q>Clearly, if anyone's vote was self-contradictory (having cyclic preferences), it would have to be discounted, and we should therefore establish a form of voting which makes such absurdities impossible</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+political+theory+of+Condorcet&rft.place=Oxford&rft.pages=%3Cspan+class%3D%22nowrap%22%3E69-%3C%2Fspan%3E80%2C+%3Cspan+class%3D%22nowrap%22%3E152-%3C%2Fspan%3E166&rft.pub=University+of+Oxford%2C+Faculty+of+Social+Studies&rft.date=1989-01-01&rft_id=info%3Aoclcnum%2F20408445&rft.aulast=Condorcet&rft.aufirst=Jean-Antoine-Nicolas+de+Caritat&rft.au=Sommerlad%2C+Fiona&rft.au=McLean%2C+Iain&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGehrlein2002" class="citation journal cs1">Gehrlein, William V. (2002). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". <i>Theory and Decision</i>. <b>52</b> (2): <span class="nowrap">171–</span>199. <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%3A1015551010381">10.1023/A:1015551010381</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/0040-5833">0040-5833</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:118143928">118143928</a>. <q>Here, Condorcet notes that we have a 'contradictory system' that represents what has come to be known as Condorcet's Paradox.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theory+and+Decision&rft.atitle=Condorcet%27s+paradox+and+the+likelihood+of+its+occurrence%3A+different+perspectives+on+balanced+preferences%2A&rft.volume=52&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E171-%3C%2Fspan%3E199&rft.date=2002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118143928%23id-name%3DS2CID&rft.issn=0040-5833&rft_id=info%3Adoi%2F10.1023%2FA%3A1015551010381&rft.aulast=Gehrlein&rft.aufirst=William+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTsetlinRegenwetterGrofman2003" class="citation journal cs1">Tsetlin, Ilia; Regenwetter, Michel; Grofman, Bernard (2003-12-01). "The impartial culture maximizes the probability of majority cycles". <i>Social Choice and Welfare</i>. <b>21</b> (3): <span class="nowrap">387–</span>398. <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%2Fs00355-003-0269-z">10.1007/s00355-003-0269-z</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/0176-1714">0176-1714</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:15488300">15488300</a>. <q>it is widely acknowledged that the impartial culture is unrealistic ... the impartial culture is the worst case scenario</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Social+Choice+and+Welfare&rft.atitle=The+impartial+culture+maximizes+the+probability+of+majority+cycles&rft.volume=21&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E387-%3C%2Fspan%3E398&rft.date=2003-12-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15488300%23id-name%3DS2CID&rft.issn=0176-1714&rft_id=info%3Adoi%2F10.1007%2Fs00355-003-0269-z&rft.aulast=Tsetlin&rft.aufirst=Ilia&rft.au=Regenwetter%2C+Michel&rft.au=Grofman%2C+Bernard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></span> </li> <li id="cite_note-:1-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGehrleinLepelley2011" class="citation book cs1">Gehrlein, William V.; Lepelley, Dominique (2011). <i>Voting paradoxes and group coherence : the condorcet efficiency of voting rules</i>. Berlin: Springer. <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%2F978-3-642-03107-6">10.1007/978-3-642-03107-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783642031076" title="Special:BookSources/9783642031076"><bdi>9783642031076</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/695387286">695387286</a>. <q>most election results do not correspond to anything like any of DC, IC, IAC or MC ... empirical studies ... indicate that some of the most common paradoxes are relatively unlikely to be observed in actual elections. ... it is easily concluded that Condorcet's Paradox should very rarely be observed in any real elections on a small number of candidates with large electorates, as long as voters' preferences reflect any reasonable degree of group mutual coherence</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Voting+paradoxes+and+group+coherence+%3A+the+condorcet+efficiency+of+voting+rules&rft.place=Berlin&rft.pub=Springer&rft.date=2011&rft_id=info%3Aoclcnum%2F695387286&rft_id=info%3Adoi%2F10.1007%2F978-3-642-03107-6&rft.isbn=9783642031076&rft.aulast=Gehrlein&rft.aufirst=William+V.&rft.au=Lepelley%2C+Dominique&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></span> </li> <li id="cite_note-:0-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_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="CITEREFVan_Deemen2014" class="citation journal cs1">Van Deemen, Adrian (2014). "On the empirical relevance of Condorcet's paradox". <i>Public Choice</i>. <b>158</b> (<span class="nowrap">3–</span>4): <span class="nowrap">311–</span>330. <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%2Fs11127-013-0133-3">10.1007/s11127-013-0133-3</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/0048-5829">0048-5829</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:154862595">154862595</a>. <q>small departures of the impartial culture assumption may lead to large changes in the probability of the paradox. It may lead to huge declines or, just the opposite, to huge increases.</q></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=On+the+empirical+relevance+of+Condorcet%27s+paradox&rft.volume=158&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E311-%3C%2Fspan%3E330&rft.date=2014&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A154862595%23id-name%3DS2CID&rft.issn=0048-5829&rft_id=info%3Adoi%2F10.1007%2Fs11127-013-0133-3&rft.aulast=Van+Deemen&rft.aufirst=Adrian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" 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="CITEREFMay1971" class="citation journal cs1">May, Robert M. (1971). "Some mathematical remarks on the paradox of voting". <i>Behavioral Science</i>. <b>16</b> (2): <span class="nowrap">143–</span>151. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fbs.3830160204">10.1002/bs.3830160204</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/0005-7940">0005-7940</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Behavioral+Science&rft.atitle=Some+mathematical+remarks+on+the+paradox+of+voting&rft.volume=16&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E143-%3C%2Fspan%3E151&rft.date=1971&rft_id=info%3Adoi%2F10.1002%2Fbs.3830160204&rft.issn=0005-7940&rft.aulast=May&rft.aufirst=Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuilbaud2012" class="citation journal cs1">Guilbaud, Georges-Théodule (2012). <a rel="nofollow" class="external text" href="https://doi.org/10.3917%2Freco.634.0659">"Les théories de l'intérêt général et le problème logique de l'agrégation"</a>. <i>Revue économique</i>. <b>63</b> (4): <span class="nowrap">659–</span>720. <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/10.3917%2Freco.634.0659">10.3917/reco.634.0659</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/0035-2764">0035-2764</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Revue+%C3%A9conomique&rft.atitle=Les+th%C3%A9ories+de+l%27int%C3%A9r%C3%AAt+g%C3%A9n%C3%A9ral+et+le+probl%C3%A8me+logique+de+l%27agr%C3%A9gation&rft.volume=63&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E659-%3C%2Fspan%3E720&rft.date=2012&rft_id=info%3Adoi%2F10.3917%2Freco.634.0659&rft.issn=0035-2764&rft.aulast=Guilbaud&rft.aufirst=Georges-Th%C3%A9odule&rft_id=https%3A%2F%2Fdoi.org%2F10.3917%252Freco.634.0659&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></span> </li> <li id="cite_note-:2-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:2_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGehrlein2002" class="citation journal cs1">Gehrlein, William V. 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"An Examination of Ranked Choice Voting in the United States, 2004-2022". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2301.12075v2">2301.12075v2</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/econ.GN">econ.GN</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=An+Examination+of+Ranked+Choice+Voting+in+the+United+States%2C+2004-2022&rft.date=2023-01-28&rft_id=info%3Aarxiv%2F2301.12075v2&rft.aulast=Graham-Squire&rft.aufirst=Adam&rft.au=McCune%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLippman2014" class="citation book cs1">Lippman, David (2014). <a rel="nofollow" class="external text" href="http://www.opentextbookstore.com/mathinsociety/">"Voting Theory"</a>. <i>Math in society</i>. CreateSpace Independent Publishing Platform. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1479276530" title="Special:BookSources/978-1479276530"><bdi>978-1479276530</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/913874268">913874268</a>. <q>There are many Condorcet methods, which vary primarily in how they deal with ties, which are very common when a Condorcet winner does not exist.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Voting+Theory&rft.btitle=Math+in+society&rft.pub=CreateSpace+Independent+Publishing+Platform&rft.date=2014&rft_id=info%3Aoclcnum%2F913874268&rft.isbn=978-1479276530&rft.aulast=Lippman&rft.aufirst=David&rft_id=http%3A%2F%2Fwww.opentextbookstore.com%2Fmathinsociety%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></span> </li> <li id="cite_note-:53-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-:53_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGehrlein2002" class="citation journal cs1">Gehrlein, William V. (2002-03-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1023/A:1015551010381">"Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*"</a>. <i>Theory and Decision</i>. <b>52</b> (2): <span class="nowrap">171–</span>199. <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%3A1015551010381">10.1023/A:1015551010381</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/1573-7187">1573-7187</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theory+and+Decision&rft.atitle=Condorcet%27s+paradox+and+the+likelihood+of+its+occurrence%3A+different+perspectives+on+balanced+preferences%2A&rft.volume=52&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E171-%3C%2Fspan%3E199&rft.date=2002-03-01&rft_id=info%3Adoi%2F10.1023%2FA%3A1015551010381&rft.issn=1573-7187&rft.aulast=Gehrlein&rft.aufirst=William+V.&rft_id=https%3A%2F%2Fdoi.org%2F10.1023%2FA%3A1015551010381&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></span> </li> <li id="cite_note-:63-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-:63_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Deemen2014" class="citation journal cs1">Van Deemen, Adrian (2014-03-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/s11127-013-0133-3">"On the empirical relevance of Condorcet's paradox"</a>. <i>Public Choice</i>. <b>158</b> (3): <span class="nowrap">311–</span>330. <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%2Fs11127-013-0133-3">10.1007/s11127-013-0133-3</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/1573-7101">1573-7101</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=On+the+empirical+relevance+of+Condorcet%27s+paradox&rft.volume=158&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E311-%3C%2Fspan%3E330&rft.date=2014-03-01&rft_id=info%3Adoi%2F10.1007%2Fs11127-013-0133-3&rft.issn=1573-7101&rft.aulast=Van+Deemen&rft.aufirst=Adrian&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2Fs11127-013-0133-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" 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=Condorcet_paradox&action=edit&section=15" 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="CITEREFGarmanKamien1968" class="citation journal cs1">Garman, M. B.; Kamien, M. I. (1968). "The paradox of voting: Probability calculations". <i>Behavioral Science</i>. <b>13</b> (4): <span class="nowrap">306–</span>316. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fbs.3830130405">10.1002/bs.3830130405</a>. <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/5663897">5663897</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Behavioral+Science&rft.atitle=The+paradox+of+voting%3A+Probability+calculations&rft.volume=13&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E306-%3C%2Fspan%3E316&rft.date=1968&rft_id=info%3Adoi%2F10.1002%2Fbs.3830130405&rft_id=info%3Apmid%2F5663897&rft.aulast=Garman&rft.aufirst=M.+B.&rft.au=Kamien%2C+M.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNiemiWeisberg1968" class="citation journal cs1">Niemi, R. G.; Weisberg, H. (1968). "A mathematical solution for the probability of the paradox of voting". <i>Behavioral Science</i>. <b>13</b> (4): <span class="nowrap">317–</span>323. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fbs.3830130406">10.1002/bs.3830130406</a>. <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/5663898">5663898</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Behavioral+Science&rft.atitle=A+mathematical+solution+for+the+probability+of+the+paradox+of+voting&rft.volume=13&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E317-%3C%2Fspan%3E323&rft.date=1968&rft_id=info%3Adoi%2F10.1002%2Fbs.3830130406&rft_id=info%3Apmid%2F5663898&rft.aulast=Niemi&rft.aufirst=R.+G.&rft.au=Weisberg%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNiemiWright1987" class="citation journal cs1">Niemi, R. G.; Wright, J. R. (1987). "Voting cycles and the structure of individual preferences". <i>Social Choice and Welfare</i>. <b>4</b> (3): <span class="nowrap">173–</span>183. <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%2FBF00433943">10.1007/BF00433943</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/41105865">41105865</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:145654171">145654171</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Social+Choice+and+Welfare&rft.atitle=Voting+cycles+and+the+structure+of+individual+preferences&rft.volume=4&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E173-%3C%2Fspan%3E183&rft.date=1987&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A145654171%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F41105865%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1007%2FBF00433943&rft.aulast=Niemi&rft.aufirst=R.+G.&rft.au=Wright%2C+J.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACondorcet+paradox" 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=Condorcet_paradox&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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portal-bar-bordered" role="navigation" aria-label="Portals"><span class="portal-bar-header"><a href="/wiki/Wikipedia:Contents/Portals" title="Wikipedia:Contents/Portals">Portal</a>:</span><ul class="portal-bar-content"><li class="portal-bar-item"><span class="nowrap"><span class="mw-image-border" 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/19px-A_coloured_voting_box.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/01/A_coloured_voting_box.svg/29px-A_coloured_voting_box.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/01/A_coloured_voting_box.svg/38px-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</a></li></ul></div><style 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.sister-bar-content{flex:1;border-top:1px solid #a2a9b1;margin:0;list-style:none}.mw-parser-output .sister-bar-item{flex:0 0 20em;min-width:20em}}.mw-parser-output .navbox+link+.sister-bar,.mw-parser-output .navbox+style+.sister-bar,.mw-parser-output .portal-bar+link+.sister-bar,.mw-parser-output .portal-bar+style+.sister-bar,.mw-parser-output .sister-bar+.navbox-styles+.navbox,.mw-parser-output .sister-bar+.navbox-styles+.portal-bar{margin-top:-1px}@media print{body.ns-0 .mw-parser-output .sister-bar{display:none!important}}</style><div class="noprint metadata sister-bar" role="navigation" aria-label="sister-projects"><div class="sister-bar-header"><b>Condorcet paradox</b> at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects"><span id="sister-projects" style="white-space:nowrap;">sister projects</span></a>:</div><ul class="sister-bar-content"><li class="sister-bar-item"><span class="sister-bar-logo"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/14px-Commons-logo.svg.png" decoding="async" width="14" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/21px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/28px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></span><span class="sister-bar-link"><b><a href="https://commons.wikimedia.org/wiki/Category:Condorcet_paradox" class="extiw" title="c:Category:Condorcet paradox">Media</a></b> from Commons</span></li><li class="sister-bar-item"><span class="sister-bar-logo"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/21px-Wikidata-logo.svg.png" decoding="async" width="21" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/32px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/42px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></span></span></span><span class="sister-bar-link"><b><a href="https://www.wikidata.org/wiki/Q745768" class="extiw" title="d:Q745768">Data</a></b> from Wikidata</span></li></ul></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output 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.navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Notable_paradoxes289" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Paradoxes" title="Template:Paradoxes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Paradoxes" title="Template talk:Paradoxes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Paradoxes" title="Special:EditPage/Template:Paradoxes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Notable_paradoxes289" style="font-size:114%;margin:0 4em">Notable <a href="/wiki/Paradox" title="Paradox">paradoxes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Philosophical</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paradox_of_analysis" title="Paradox of analysis">Analysis</a></li> <li><a href="/wiki/Buridan%27s_bridge" title="Buridan's bridge">Buridan's bridge</a></li> <li><a href="/wiki/Dream_argument" title="Dream argument">Dream argument</a></li> <li><a href="/wiki/Epicurean_paradox" title="Epicurean paradox">Epicurean</a></li> <li><a href="/wiki/Paradox_of_fiction" title="Paradox of fiction">Fiction</a></li> <li><a href="/wiki/Fitch%27s_paradox_of_knowability" title="Fitch's paradox of knowability">Fitch's knowability</a></li> <li><a href="/wiki/Argument_from_free_will" title="Argument from free will">Free will</a></li> <li><a href="/wiki/New_riddle_of_induction" title="New riddle of induction">Goodman's</a></li> <li><a href="/wiki/Paradox_of_hedonism" title="Paradox of hedonism">Hedonism</a></li> <li><a href="/wiki/Liberal_paradox" title="Liberal paradox">Liberal</a></li> <li><a href="/wiki/Meno" title="Meno">Meno's</a></li> <li><a href="/wiki/Mere_addition_paradox" title="Mere addition paradox">Mere addition</a></li> <li><a href="/wiki/Moore%27s_paradox" title="Moore's paradox">Moore's</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Paradox_of_nihilism" title="Paradox of nihilism">Nihilism</a></li> <li><a href="/wiki/Omnipotence_paradox" title="Omnipotence paradox">Omnipotence</a></li> <li><a href="/wiki/Preface_paradox" title="Preface paradox">Preface</a></li> <li><a href="/wiki/Wittgenstein_on_Rules_and_Private_Language" title="Wittgenstein on Rules and Private Language">Rule-following</a></li> <li><a href="/wiki/Sorites_paradox" title="Sorites paradox">Sorites</a></li> <li><a href="/wiki/Ship_of_Theseus" title="Ship of Theseus">Theseus' ship</a></li> <li><a href="/wiki/White_Horse_Dialogue" title="White Horse Dialogue">White horse</a></li> <li><a href="/wiki/Zeno%27s_paradoxes" title="Zeno's paradoxes">Zeno's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Logical</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barber_paradox" title="Barber paradox">Barber</a></li> <li><a href="/wiki/Berry_paradox" title="Berry paradox">Berry</a></li> <li><a href="/wiki/Bhartrhari%27s_paradox" class="mw-redirect" title="Bhartrhari's paradox">Bhartrhari's</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti</a></li> <li><a href="/wiki/Paradox_of_the_Court" title="Paradox of the Court">Court</a></li> <li><a href="/wiki/Crocodile_dilemma" title="Crocodile dilemma">Crocodile</a></li> <li><a href="/wiki/Curry%27s_paradox" title="Curry's paradox">Curry's</a></li> <li><a href="/wiki/Epimenides_paradox" title="Epimenides paradox">Epimenides</a></li> <li><a href="/wiki/Free_choice_inference" title="Free choice inference">Free choice paradox</a></li> <li><a href="/wiki/Grelling%E2%80%93Nelson_paradox" title="Grelling–Nelson paradox">Grelling–Nelson</a></li> <li><a href="/wiki/Kleene%E2%80%93Rosser_paradox" title="Kleene–Rosser paradox">Kleene–Rosser</a></li> <li><a href="/wiki/Liar_paradox" title="Liar paradox">Liar</a> <ul><li><a href="/wiki/Card_paradox" title="Card paradox">Card</a></li> <li><a href="/wiki/No%E2%80%93no_paradox" title="No–no paradox">No-no</a></li> <li><a href="/wiki/Pinocchio_paradox" title="Pinocchio paradox">Pinocchio</a></li> <li><a href="/wiki/Quine%27s_paradox" title="Quine's paradox">Quine's</a></li> <li><a href="/wiki/Yablo%27s_paradox" class="mw-redirect" title="Yablo's paradox">Yablo's</a></li></ul></li> <li><a href="/wiki/Opposite_Day" title="Opposite Day">Opposite Day</a></li> <li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes of set theory</a></li> <li><a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's</a></li> <li><a href="/wiki/I_know_that_I_know_nothing" title="I know that I know nothing">Socratic</a></li> <li><a href="/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel" title="Hilbert's paradox of the Grand Hotel">Hilbert's Hotel</a></li> <li><a href="/wiki/Temperature_paradox" title="Temperature paradox">Temperature paradox</a></li> <li><a href="/wiki/Barbershop_paradox" title="Barbershop paradox">Barbershop</a></li> <li><a href="/wiki/Catch-22_(logic)" title="Catch-22 (logic)">Catch-22</a></li> <li><a href="/wiki/Chicken_or_the_egg" title="Chicken or the egg">Chicken or the egg</a></li> <li><a href="/wiki/Drinker_paradox" title="Drinker paradox">Drinker</a></li> <li><a href="/wiki/Paradoxes_of_material_implication" title="Paradoxes of material implication">Entailment</a></li> <li><a href="/wiki/Lottery_paradox" title="Lottery paradox">Lottery</a></li> <li><a href="/wiki/Plato%27s_beard" title="Plato's beard">Plato's beard</a></li> <li><a href="/wiki/Raven_paradox" title="Raven paradox">Raven</a></li> <li><a href="/wiki/Imperative_logic#Ross's_paradox" title="Imperative logic">Ross's</a></li> <li><a href="/wiki/Unexpected_hanging_paradox" title="Unexpected hanging paradox">Unexpected hanging</a></li> <li>"<a href="/wiki/What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a>"</li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Olbers%27s_paradox" title="Olbers's paradox">Olbers's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Economic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Allais_paradox" title="Allais paradox">Allais</a></li> <li><a href="/wiki/The_Antitrust_Paradox" title="The Antitrust Paradox">Antitrust</a></li> <li><a href="/wiki/Arrow_information_paradox" title="Arrow information paradox">Arrow information</a></li> <li><a href="/wiki/Bertrand_paradox_(economics)" title="Bertrand paradox (economics)">Bertrand</a></li> <li><a href="/wiki/Braess%27s_paradox" title="Braess's paradox">Braess's</a></li> <li><a href="/wiki/Paradox_of_competition" title="Paradox of competition">Competition</a></li> <li><a href="/wiki/Income_and_fertility" title="Income and fertility">Income and fertility</a></li> <li><a href="/wiki/Downs%E2%80%93Thomson_paradox" title="Downs–Thomson paradox">Downs–Thomson</a></li> <li><a href="/wiki/Easterlin_paradox" title="Easterlin paradox">Easterlin</a></li> <li><a href="/wiki/Edgeworth_paradox" title="Edgeworth paradox">Edgeworth</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/European_paradox" title="European paradox">European</a></li> <li><a href="/wiki/Gibson%27s_paradox" title="Gibson's paradox">Gibson's</a></li> <li><a href="/wiki/Giffen_good" title="Giffen good">Giffen good</a></li> <li><a href="/wiki/Icarus_paradox" title="Icarus paradox">Icarus</a></li> <li><a href="/wiki/Jevons_paradox" title="Jevons paradox">Jevons</a></li> <li><a href="/wiki/Leontief_paradox" title="Leontief paradox">Leontief</a></li> <li><a href="/wiki/Lerner_paradox" title="Lerner paradox">Lerner</a></li> <li><a href="/wiki/Lucas_paradox" title="Lucas paradox">Lucas</a></li> <li><a href="/wiki/Mandeville%27s_paradox" title="Mandeville's paradox">Mandeville's</a></li> <li><a href="/wiki/Mayfield%27s_paradox" title="Mayfield's paradox">Mayfield's</a></li> <li><a href="/wiki/Metzler_paradox" title="Metzler paradox">Metzler</a></li> <li><a href="/wiki/Resource_curse" title="Resource curse">Plenty</a></li> <li><a href="/wiki/Productivity_paradox" title="Productivity paradox">Productivity</a></li> <li><a href="/wiki/Paradox_of_prosperity" title="Paradox of prosperity">Prosperity</a></li> <li><a href="/wiki/Scitovsky_paradox" title="Scitovsky paradox">Scitovsky</a></li> <li><a href="/wiki/Service_recovery_paradox" title="Service recovery paradox">Service recovery</a></li> <li><a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg</a></li> <li><a href="/wiki/Paradox_of_thrift" title="Paradox of thrift">Thrift</a></li> <li><a href="/wiki/Paradox_of_toil" title="Paradox of toil">Toil</a></li> <li><a href="/wiki/Tullock_paradox" class="mw-redirect" title="Tullock paradox">Tullock</a></li> <li><a href="/wiki/Paradox_of_value" title="Paradox of value">Value</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decision theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abilene_paradox" title="Abilene paradox">Abilene</a></li> <li><a href="/wiki/Apportionment_paradox" title="Apportionment paradox">Apportionment</a> <ul><li><a href="/wiki/House_monotonicity" title="House monotonicity">Alabama</a></li> <li><a href="/wiki/Coherence_(fairness)" title="Coherence (fairness)">New states</a></li> <li><a href="/wiki/State-population_monotonicity" class="mw-redirect" title="State-population monotonicity">Population</a></li></ul></li> <li><a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's</a></li> <li><a href="/wiki/Buridan%27s_ass" title="Buridan's ass">Buridan's ass</a></li> <li><a href="/wiki/Chainstore_paradox" title="Chainstore paradox">Chainstore</a></li> <li><a class="mw-selflink selflink">Condorcet's</a></li> <li><a href="/wiki/Decision-making_paradox" title="Decision-making paradox">Decision-making</a></li> <li><a href="/wiki/Paradox_of_voting" title="Paradox of voting">Downs</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/Fenno%27s_paradox" title="Fenno's paradox">Fenno's</a></li> <li><a href="/wiki/Fredkin%27s_paradox" title="Fredkin's paradox">Fredkin's</a></li> <li><a href="/wiki/The_Green_Paradox" title="The Green Paradox">Green</a></li> <li><a href="/wiki/Hedgehog%27s_dilemma" title="Hedgehog's dilemma">Hedgehog's</a></li> <li><a href="/wiki/Inventor%27s_paradox" title="Inventor's paradox">Inventor's</a></li> <li><a href="/wiki/Kavka%27s_toxin_puzzle" title="Kavka's toxin puzzle">Kavka's toxin puzzle</a></li> <li><a href="/wiki/Morton%27s_fork" title="Morton's fork">Morton's fork</a></li> <li><a href="/wiki/Navigation_paradox" title="Navigation paradox">Navigation</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Parrondo%27s_paradox" title="Parrondo's paradox">Parrondo's</a></li> <li><a href="/wiki/Preparedness_paradox" title="Preparedness paradox">Preparedness</a></li> <li><a href="/wiki/Prevention_paradox" title="Prevention paradox">Prevention</a></li> <li><a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a></li> <li><a href="/wiki/Paradox_of_tolerance" title="Paradox of tolerance">Tolerance</a></li> <li><a href="/wiki/Willpower_paradox" title="Willpower paradox">Willpower</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_paradoxes" title="List of 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