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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ranked pairs</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://eo.wikipedia.org/wiki/Paroranga_metodo" title="Paroranga metodo – Esperanto" lang="eo" hreflang="eo" data-title="Paroranga metodo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9thode_Condorcet_avec_rangement_des_paires_par_ordre_d%C3%A9croissant" title="Méthode Condorcet avec rangement des paires par ordre décroissant – French" lang="fr" hreflang="fr" data-title="Méthode Condorcet avec rangement des paires par ordre décroissant" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rangsorolt_p%C3%A1rok" title="Rangsorolt párok – Hungarian" lang="hu" hreflang="hu" data-title="Rangsorolt párok" 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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"><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 class="mw-selflink selflink">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 href="/wiki/Condorcet_paradox" title="Condorcet paradox">Conflicting majorities paradox</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible 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 href="/wiki/Condorcet_paradox" title="Condorcet paradox">Majority impossibility</a></li> <li><a href="/wiki/Moulin%27s_impossibility_theorem" class="mw-redirect" title="Moulin's impossibility theorem">Moulin's impossibility theorem</a></li> <li><a href="/wiki/McKelvey%E2%80%93Schofield_chaos_theorem" title="McKelvey–Schofield chaos theorem">McKelvey–Schofield chaos theorem</a></li> <li><a href="/wiki/Gibbard%27s_theorem" title="Gibbard's theorem">Gibbard's theorem</a></li></ul> <hr /> <p><b>Positive results</b> </p> <ul><li><a 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" 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template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Ranked Pairs</b> (<b>RP</b>), also known as the <b>Tideman method</b>, is a <a href="/wiki/Round-robin_voting" title="Round-robin voting">tournament-style</a> system of <a href="/wiki/Ranked_voting" title="Ranked voting">ranked voting</a> first proposed by <a href="/wiki/Nicolaus_Tideman" title="Nicolaus Tideman">Nicolaus Tideman</a> in 1987.<sup id="cite_ref-Tideman2_1-0" class="reference"><a href="#cite_note-Tideman2-1"><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> </p><p>If there is a candidate who is preferred over the other candidates, when compared in turn with each of the others, the ranked-pairs procedure guarantees that candidate will win. Therefore, the ranked-pairs procedure complies with the <a href="/wiki/Condorcet_winner_criterion" title="Condorcet winner criterion">Condorcet winner criterion</a> (and as a result is considered to be a <a href="/wiki/Condorcet_method" title="Condorcet method">Condorcet method</a>).<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>Ranked pairs begins with a <a href="/wiki/Round-robin_tournament" title="Round-robin tournament">round-robin tournament</a>, where the one-on-one margins of victory for each possible pair of candidates are compared to find a <a href="/wiki/Condorcet_winner_criterion" title="Condorcet winner criterion">majority-preferred candidate</a>; if such a candidate exists, they are immediately elected. Otherwise, if there is a <a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Condorcet cycle</a>—a rock-paper-scissors-like sequence A > B > C > A—the cycle is broken by dropping the "weakest" elections in the cycle, i.e. the ones that are closest to being tied.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Procedure">Procedure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=1" title="Edit section: Procedure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The ranked pairs procedure is as follows: </p> <ol><li>Consider each pair of candidates round-robin style, and <a href="/wiki/Pairwise_counting" class="mw-redirect" title="Pairwise counting">calculate the pairwise margin of victory</a> for each in a one-on-one matchup.</li> <li>Sort the pairs by the (<a href="/wiki/Absolute_value" title="Absolute value">absolute</a>) margin of victory, going from largest to smallest.</li> <li>Going down the list, check whether adding each matchup would create a <a href="/wiki/Condorcet_cycle" class="mw-redirect" title="Condorcet cycle">cycle</a>. If it would, <a href="/wiki/Cross_out" class="mw-redirect" title="Cross out">cross out</a> the election; this will be the election(s) in the cycle with the smallest margin of victory (near-ties).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup></li></ol> <p>At the end of this procedure, all cycles will be eliminated, leaving a unique winner who wins all of the remaining one-on-one matchups. The lack of cycles means that candidates can be ranked directly based on the matchups that have been left behind. </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=Ranked_pairs&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="The_situation">The situation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=3" title="Edit section: The situation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Tennessee"></span> </p> <div style="float: left;"> <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:Tenn_voting_example" title="Template:Tenn voting example"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tenn_voting_example" title="Template talk:Tenn voting example"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tenn_voting_example" title="Special:EditPage/Template:Tenn voting example"><abbr title="Edit this template">e</abbr></a></li></ul></div></div> <p><span typeof="mw:File"><a href="/wiki/File:Tennessee_map_for_voting_example.svg" class="mw-file-description"><img alt="Tennessee and its four major cities: Memphis in the far west; Nashville in the center; Chattanooga in the east; and Knoxville in the far northeast" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Tennessee_map_for_voting_example.svg/500px-Tennessee_map_for_voting_example.svg.png" decoding="async" width="500" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Tennessee_map_for_voting_example.svg/750px-Tennessee_map_for_voting_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Tennessee_map_for_voting_example.svg/1000px-Tennessee_map_for_voting_example.svg.png 2x" data-file-width="780" data-file-height="190" /></a></span> </p><p>Suppose that <a href="/wiki/Tennessee" title="Tennessee">Tennessee</a> is holding an election on the location of its <a href="/wiki/Capital_city" title="Capital city">capital</a>. The population is concentrated around four major cities. <a href="/wiki/Spatial_voting" title="Spatial voting">All voters want the capital to be as close to them as possible.</a> The options are: </p> <ul><li><a href="/wiki/Memphis,_Tennessee" title="Memphis, Tennessee">Memphis</a>, the largest city, but far from the others (42% of voters)</li> <li><a href="/wiki/Nashville,_Tennessee" title="Nashville, Tennessee">Nashville</a>, near the center of the state (26% of voters)</li> <li><a href="/wiki/Chattanooga,_Tennessee" title="Chattanooga, Tennessee">Chattanooga</a>, somewhat east (15% of voters)</li> <li><a href="/wiki/Knoxville,_Tennessee" title="Knoxville, Tennessee">Knoxville</a>, far to the northeast (17% of voters)</li></ul> <p>The preferences of each region's voters are: </p> <table class="wikitable"> <tbody><tr> <th width="25%" style="background-color: #ffdddd">42% of voters<br /><small>Far-West</small> </th> <th width="25%" style="background-color: #ccffcc">26% of voters<br /><small>Center</small> </th> <th width="25%" style="background-color: #ddddff">15% of voters<br /><small>Center-East</small> </th> <th width="25%" style="background-color: #ffeedd">17% of voters<br /><small>Far-East</small> </th></tr> <tr> <td> <ol style="margin-left: 1.5em;"> <li> <b>Memphis</b> </li><li> Nashville </li><li> Chattanooga </li><li> Knoxville </li></ol> </td> <td> <ol style="margin-left: 1.5em;"> <li> <b>Nashville</b> </li><li> Chattanooga </li><li> Knoxville </li><li> Memphis </li></ol> </td> <td> <ol style="margin-left: 1.5em;"> <li> <b>Chattanooga</b> </li><li> Knoxville </li><li> Nashville </li><li> Memphis </li></ol> </td> <td> <ol style="margin-left: 1.5em;"> <li> <b>Knoxville</b> </li><li> Chattanooga </li><li> Nashville </li><li> Memphis </li></ol> </td></tr></tbody></table> <p><br /> The results are tabulated as follows: </p> <table class="wikitable"> <caption>Pairwise election results </caption> <tbody><tr> <th style="background:#ffe0e0;color:inherit;background:linear-gradient(to top right,#ffe0e0 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#e0e0ff 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">A</div><div style="margin-right:2em;text-align:left">B</div> </th> <td bgcolor="#c0c0ff"><b>Memphis</b> </td> <td bgcolor="#c0c0ff"><b>Nashville</b> </td> <td bgcolor="#c0c0ff"><b>Chattanooga</b> </td> <td bgcolor="#c0c0ff"><b>Knoxville</b> </td></tr> <tr> <td bgcolor="#ffc0c0"><b>Memphis</b> </td> <td> </td> <td style="background:#e0e0ff;color:inherit;background:linear-gradient(to top right,#e0e0ff 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#e0e0ff 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">58%</div><div style="margin-right:2em;text-align:left">42%</div> </td> <td style="background:#e0e0ff;color:inherit;background:linear-gradient(to top right,#e0e0ff 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#e0e0ff 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">58%</div><div style="margin-right:2em;text-align:left">42%</div> </td> <td style="background:#e0e0ff;color:inherit;background:linear-gradient(to top right,#e0e0ff 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#e0e0ff 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">58%</div><div style="margin-right:2em;text-align:left">42%</div> </td></tr> <tr> <td bgcolor="#ffc0c0"><b>Nashville</b> </td> <td style="background:#ffe0e0;color:inherit;background:linear-gradient(to top right,#ffe0e0 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#ffe0e0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">42%</div><div style="margin-right:2em;text-align:left">58%</div> </td> <td> </td> <td style="background:#ffe0e0;color:inherit;background:linear-gradient(to top right,#ffe0e0 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#ffe0e0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">32%</div><div style="margin-right:2em;text-align:left">68%</div> </td> <td style="background:#ffe0e0;color:inherit;background:linear-gradient(to top right,#ffe0e0 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#ffe0e0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">32%</div><div style="margin-right:2em;text-align:left">68%</div> </td></tr> <tr> <td bgcolor="#ffc0c0"><b>Chattanooga</b> </td> <td style="background:#ffe0e0;color:inherit;background:linear-gradient(to top right,#ffe0e0 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#ffe0e0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">42%</div><div style="margin-right:2em;text-align:left">58%</div> </td> <td style="background:#e0e0ff;color:inherit;background:linear-gradient(to top right,#e0e0ff 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#e0e0ff 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">68%</div><div style="margin-right:2em;text-align:left">32%</div> </td> <td> </td> <td style="background:#ffe0e0;color:inherit;background:linear-gradient(to top right,#ffe0e0 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#ffe0e0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">17%</div><div style="margin-right:2em;text-align:left">83%</div> </td></tr> <tr> <td bgcolor="#ffc0c0"><b>Knoxville</b> </td> <td style="background:#ffe0e0;color:inherit;background:linear-gradient(to top right,#ffe0e0 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#ffe0e0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">42%</div><div style="margin-right:2em;text-align:left">58%</div> </td> <td style="background:#e0e0ff;color:inherit;background:linear-gradient(to top right,#e0e0ff 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#e0e0ff 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">68%</div><div style="margin-right:2em;text-align:left">32%</div> </td> <td style="background:#e0e0ff;color:inherit;background:linear-gradient(to top right,#e0e0ff 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,#e0e0ff 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">83%</div><div style="margin-right:2em;text-align:left">17%</div> </td> <td> </td></tr></tbody></table> <ul><li>[A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption</li> <li>[B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption</li></ul> <div class="mw-heading mw-heading3"><h3 id="Tally">Tally</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=4" title="Edit section: Tally"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>First, list every pair, and determine the winner: </p> <table class="wikitable sortable"> <tbody><tr> <th>Pair </th> <th>Winner </th></tr> <tr> <td>Memphis (42%) vs. Nashville (58%) </td> <td>Nashville 58% </td></tr> <tr> <td>Memphis (42%) vs. Chattanooga (58%) </td> <td>Chattanooga 58% </td></tr> <tr> <td>Memphis (42%) vs. Knoxville (58%) </td> <td>Knoxville 58% </td></tr> <tr> <td>Nashville (68%) vs. Chattanooga (32%) </td> <td>Nashville 68% </td></tr> <tr> <td>Nashville (68%) vs. Knoxville (32%) </td> <td>Nashville 68% </td></tr> <tr> <td>Chattanooga (83%) vs. Knoxville (17%) </td> <td>Chattanooga 83% </td></tr></tbody></table> <p>The votes are then sorted. The largest majority is "Chattanooga over Knoxville"; 83% of the voters prefer Chattanooga. Thus, the pairs from above would be sorted this way: </p> <table class="wikitable"> <tbody><tr> <th>Pair </th> <th>Winner </th></tr> <tr> <td>Chattanooga (83%) vs. Knoxville (17%) </td> <td>Chattanooga 83% </td></tr> <tr> <td>Nashville (68%) vs. Knoxville (32%) </td> <td>Nashville 68% </td></tr> <tr> <td>Nashville (68%) vs. Chattanooga (32%) </td> <td>Nashville 68% </td></tr> <tr> <td>Memphis (42%) vs. Nashville (58%) </td> <td>Nashville 58% </td></tr> <tr> <td>Memphis (42%) vs. Chattanooga (58%) </td> <td>Chattanooga 58% </td></tr> <tr> <td>Memphis (42%) vs. Knoxville (58%) </td> <td>Knoxville 58% </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Lock">Lock</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=5" title="Edit section: Lock"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The pairs are then locked in order, skipping any pairs that would create a cycle: </p> <ul><li>Lock Chattanooga over Knoxville.</li> <li>Lock Nashville over Knoxville.</li> <li>Lock Nashville over Chattanooga.</li> <li>Lock Nashville over Memphis.</li> <li>Lock Chattanooga over Memphis.</li> <li>Lock Knoxville over Memphis.</li></ul> <p>In this case, no cycles are created by any of the pairs, so every single one is locked in. </p><p>Every "lock in" would add another arrow to the graph showing the relationship between the candidates. Here is the final graph (where arrows point away from the winner). </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Tennessee-vote.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Tennessee-vote.svg/350px-Tennessee-vote.svg.png" decoding="async" width="350" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Tennessee-vote.svg/525px-Tennessee-vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Tennessee-vote.svg/700px-Tennessee-vote.svg.png 2x" data-file-width="350" data-file-height="240" /></a></span> </p><p>In this example, Nashville is the winner using the ranked-pairs procedure. Nashville is followed by Chattanooga, Knoxville, and Memphis in second, third, and fourth places respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Summary">Summary</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=6" title="Edit section: Summary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the example election, the winner is Nashville. This would be true for any <a href="/wiki/Condorcet_method" title="Condorcet method">Condorcet method</a>. </p><p>Under <a href="/wiki/First-past-the-post_voting" title="First-past-the-post voting">first-past-the-post</a> and some other systems, Memphis would have won the election by having the most people, even though Nashville won every simulated pairwise election outright. Using <a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">instant-runoff voting</a> in this example would result in Knoxville winning even though more people preferred Nashville over Knoxville. </p> <div class="mw-heading mw-heading2"><h2 id="Criteria">Criteria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=7" title="Edit section: Criteria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Of the formal <a href="/wiki/Voting_criteria" title="Voting criteria">voting criteria</a>, the ranked pairs method passes the <a href="/wiki/Majority_favorite_criterion" class="mw-redirect" title="Majority favorite criterion">majority criterion</a>, the <a href="/wiki/Monotonicity_criterion" class="mw-redirect" title="Monotonicity criterion">monotonicity criterion</a>, the <a href="/wiki/Smith_criterion" class="mw-redirect" title="Smith criterion">Smith criterion</a> (which implies the <a href="/wiki/Condorcet_criterion" class="mw-redirect" title="Condorcet criterion">Condorcet criterion</a>), the <a href="/wiki/Condorcet_loser_criterion" title="Condorcet loser criterion">Condorcet loser criterion</a>, and the <a href="/wiki/Independence_of_clones_criterion" title="Independence of clones criterion">independence of clones criterion</a>. Ranked pairs fails the <a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion">consistency criterion</a> and the <a href="/wiki/Participation_criterion" class="mw-redirect" title="Participation criterion">participation criterion</a>. While ranked pairs is not fully <a href="/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">independent of irrelevant alternatives</a>, it still satisfies <a href="/wiki/Independence_of_irrelevant_alternatives#Local_independence" title="Independence of irrelevant alternatives">local independence of irrelevant alternatives</a> and <a href="/wiki/Independence_of_Smith-dominated_alternatives" title="Independence of Smith-dominated alternatives">independence of Smith-dominated alternatives</a>, meaning it is likely to roughly satisfy IIA "in practice." </p> <div class="mw-heading mw-heading3"><h3 id="Independence_of_irrelevant_alternatives">Independence of irrelevant alternatives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=8" title="Edit section: Independence of irrelevant alternatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ranked pairs fails <a href="/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">independence of irrelevant alternatives</a>, like all other <a href="/wiki/Ranked_voting" title="Ranked voting">ranked voting systems</a>. However, the method adheres to a less strict property, sometimes called <a href="/wiki/Independence_of_Smith-dominated_alternatives" title="Independence of Smith-dominated alternatives">independence of Smith-dominated alternatives</a> (ISDA). It says that if one candidate (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the <a href="/wiki/Smith_set" title="Smith set">Smith set</a>. ISDA implies the Condorcet criterion. </p> <div class="mw-heading mw-heading3"><h3 id="Comparison_table">Comparison table</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=9" title="Edit section: Comparison table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following table compares ranked pairs with other single-winner election methods:<style data-mw-deduplicate="TemplateStyles:r1245584064">@media screen{html.client-js .mw-parser-output .sort-under.sortable.wikitable th.headerSort,html.client-js .mw-parser-output .sort-under-right.sortable.wikitable th.headerSort,html.client-js .mw-parser-output .sort-under-center.sortable.wikitable th.headerSort{padding-right:0.4em}html.client-js .mw-parser-output .sort-under.sortable:not(.wikitable) th.headerSort,html.client-js .mw-parser-output .sort-under-right.sortable:not(.wikitable) th.headerSort,html.client-js 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table{display:table}body.skin-minerva .mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) table>caption{display:table-caption}}.mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) .jquery-tablesorter>thead,.mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) .mw-sticky-header>thead{top:0!important}.mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) .jquery-tablesorter>tfoot,.mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) .mw-sticky-header>tfoot{position:static;bottom:auto}}@media all{.mw-parser-output .sticky-table-collapsible.mw-collapsed .sticky-table-scroll{max-height:none!important;max-width:none!important}}</style><div class="sticky-table-collapsible mw-collapsible" data-expandtext="enable" data-collapsetext="disable"><div class="sticky-table-scroll"> <table class="wikitable sortable sort-under mw-collapsible sticky-table-row1 sticky-table-col1" style="text-align:center;"> <caption>Comparison of single-winner voting systems </caption> <tbody><tr> <th style="border-bottom:2px solid #a0a0a0;"><div style="margin-left:2em;text-align:right">Criterion</div><div style="margin-right:2em;text-align:left"><br /><br />Method</div> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Majority_winner_criterion" title="Majority winner criterion">Majority winner</a> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Majority_loser_criterion" title="Majority loser criterion">Majority loser</a> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Mutual_majority_criterion" title="Mutual majority criterion">Mutual majority</a> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Condorcet_winner" class="mw-redirect" title="Condorcet winner">Condorcet winner</a><wbr /><sup id="cite_ref-condorcet-iia-incompatibility_6-0" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-6"><span class="cite-bracket">[</span>Tn 1<span class="cite-bracket">]</span></a></sup> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Condorcet_loser" class="mw-redirect" title="Condorcet loser">Condorcet loser</a> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Smith_criterion" class="mw-redirect" title="Smith criterion">Smith</a><wbr /><sup id="cite_ref-condorcet-iia-incompatibility_6-1" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-6"><span class="cite-bracket">[</span>Tn 1<span class="cite-bracket">]</span></a></sup> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Independence_of_Smith-dominated_alternatives" title="Independence of Smith-dominated alternatives">Smith-IIA</a><wbr /><sup id="cite_ref-condorcet-iia-incompatibility_6-2" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-6"><span class="cite-bracket">[</span>Tn 1<span class="cite-bracket">]</span></a></sup> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">IIA</a>/<a href="/wiki/Independence_of_irrelevant_alternatives#Local_independence" title="Independence of irrelevant alternatives">LIIA</a><wbr /><sup id="cite_ref-condorcet-iia-incompatibility_6-3" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-6"><span class="cite-bracket">[</span>Tn 1<span class="cite-bracket">]</span></a></sup> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Independence_of_clones" class="mw-redirect" title="Independence of clones">Cloneproof</a> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Monotonicity_criterion" class="mw-redirect" title="Monotonicity criterion">Monotone</a> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Participation_criterion" class="mw-redirect" title="Participation criterion">Participation</a> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Later-no-harm" class="mw-redirect" title="Later-no-harm">Later-no-harm</a><wbr /><sup id="cite_ref-condorcet-iia-incompatibility_6-4" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-6"><span class="cite-bracket">[</span>Tn 1<span class="cite-bracket">]</span></a></sup> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/Later-no-help" class="mw-redirect" title="Later-no-help">Later-no-help</a><wbr /><sup id="cite_ref-condorcet-iia-incompatibility_6-5" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-6"><span class="cite-bracket">[</span>Tn 1<span class="cite-bracket">]</span></a></sup> </th> <th style="border-bottom:2px solid #a0a0a0;"><a href="/wiki/No_favorite_betrayal" class="mw-redirect" title="No favorite betrayal">No favorite betrayal</a><wbr /><sup id="cite_ref-condorcet-iia-incompatibility_6-6" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-6"><span class="cite-bracket">[</span>Tn 1<span class="cite-bracket">]</span></a></sup> </th> <th style="border-bottom:2px solid #a0a0a0; border-left:2px solid #a0a0a0;">Ballot <p>type </p> </th></tr> <tr> <th style="font-weight:bold"><a href="/wiki/First-past-the-post_voting" title="First-past-the-post voting">First-past-the-post voting</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="6" style="border-left:2px solid #a0a0a0;">Single mark </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Anti-plurality_voting" title="Anti-plurality voting">Anti-plurality</a> </th> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td data-sort-value="6" style="border-left:2px solid #a0a0a0;">Single mark </td></tr> <tr> <th><a href="/wiki/Two-round_system" title="Two-round system">Two round system</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="6" style="border-left:2px solid #a0a0a0;">Single mark </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Coombs%27_method" title="Coombs' method">Coombs</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Nanson%27s_method" title="Nanson's method">Nanson</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Baldwin%27s_method" class="mw-redirect" title="Baldwin's method">Baldwin</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Tideman_alternative_method" title="Tideman alternative method">Tideman alternative</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Minimax_Condorcet" class="mw-redirect" title="Minimax Condorcet">Minimax</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes<wbr /><sup id="cite_ref-MinimaxVarient_7-0" class="reference"><a href="#cite_note-MinimaxVarient-7"><span class="cite-bracket">[</span>Tn 2<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No<wbr /><sup id="cite_ref-MinimaxVarient_7-1" class="reference"><a href="#cite_note-MinimaxVarient-7"><span class="cite-bracket">[</span>Tn 2<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Copeland%27s_method" title="Copeland's method">Copeland</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Black%27s_method" title="Black's method">Black</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Kemeny-Young_method" class="mw-redirect" title="Kemeny-Young method">Kemeny–Young</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">LIIA Only </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a class="mw-selflink selflink">Ranked pairs</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">LIIA Only </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No<wbr /><sup id="cite_ref-semipc_8-0" class="reference"><a href="#cite_note-semipc-8"><span class="cite-bracket">[</span>Tn 3<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Schulze_method" title="Schulze method">Schulze</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No<wbr /><sup id="cite_ref-semipc_8-1" class="reference"><a href="#cite_note-semipc-8"><span class="cite-bracket">[</span>Tn 3<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Borda_count" title="Borda count">Borda</a> </th> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Bucklin_voting" title="Bucklin voting">Bucklin</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Ranking </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Approval_voting" title="Approval voting">Approval</a> </th> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes<wbr /><sup id="cite_ref-IIA_rating_methods_9-0" class="reference"><a href="#cite_note-IIA_rating_methods-9"><span class="cite-bracket">[</span>Tn 4<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td data-sort-value="2" style="border-left:2px solid #a0a0a0;">Approvals </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Majority_judgment" title="Majority judgment">Majority Judgement</a> </th> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No<wbr /><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>Tn 5<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No<wbr /><sup id="cite_ref-mjmmc_11-0" class="reference"><a href="#cite_note-mjmmc-11"><span class="cite-bracket">[</span>Tn 6<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes<wbr /><sup id="cite_ref-IIA_rating_methods_9-1" class="reference"><a href="#cite_note-IIA_rating_methods-9"><span class="cite-bracket">[</span>Tn 4<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No<wbr /><sup id="cite_ref-semipc_8-2" class="reference"><a href="#cite_note-semipc-8"><span class="cite-bracket">[</span>Tn 3<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td data-sort-value="1" style="border-left:2px solid #a0a0a0;">Scores </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Score_voting" title="Score voting">Score</a> </th> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes<wbr /><sup id="cite_ref-IIA_rating_methods_9-2" class="reference"><a href="#cite_note-IIA_rating_methods-9"><span class="cite-bracket">[</span>Tn 4<span class="cite-bracket">]</span></a></sup> </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td data-sort-value="1" style="border-left:2px solid #a0a0a0;">Scores </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/STAR_voting" title="STAR voting">STAR</a> </th> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="1" style="border-left:2px solid #a0a0a0;">Scores </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Quadratic_voting" title="Quadratic voting">Quadratic</a> </th> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle;">N/A </td> <td style="vertical-align:middle;">N/A </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td data-sort-value="1" style="border-left:2px solid #a0a0a0;">Credits </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Random_ballot" title="Random ballot">Random ballot</a><wbr /><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>Tn 7<span class="cite-bracket">]</span></a></sup> </th> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td data-sort-value="6" style="border-left:2px solid #a0a0a0;">Single mark </td></tr> <tr> <th style="font-weight:bold"><a href="/wiki/Sortition" title="Sortition">Sortition</a><wbr /><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>Tn 8<span class="cite-bracket">]</span></a></sup> </th> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#FFC7C7;">No </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td style="vertical-align:middle; background-color:#9EFF9E;">Yes </td> <td data-sort-value="6" style="border-left:2px solid #a0a0a0;">None </td></tr> <tr class="sortbottom"> <th>Table Notes </th> <td colspan="16" style="text-align: left;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-condorcet-iia-incompatibility-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-condorcet-iia-incompatibility_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_6-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_6-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_6-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_6-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Condorcet_criterion" class="mw-redirect" title="Condorcet criterion">Condorcet's criterion</a> is incompatible with the <a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion">consistency</a>, <a href="/wiki/Participation_criterion" class="mw-redirect" title="Participation criterion">participation</a>, <a href="/wiki/Later-no-harm" class="mw-redirect" title="Later-no-harm">later-no-harm</a>, <a href="/wiki/Later-no-help_criterion" title="Later-no-help criterion">later-no-help</a>, and <a href="/wiki/Sincere_favorite_criterion" title="Sincere favorite criterion">sincere favorite</a> criteria.</span> </li> <li id="cite_note-MinimaxVarient-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-MinimaxVarient_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MinimaxVarient_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.</span> </li> <li id="cite_note-semipc-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-semipc_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-semipc_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-semipc_8-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">In Highest median, Ranked Pairs, and Schulze voting, there is always a regret-free, semi-honest ballot for any voter, holding all other ballots constant and assuming they know enough about how others will vote. Under such circumstances, there is always at least one way for a voter to participate without grading any less-preferred candidate above any more-preferred one.</span> </li> <li id="cite_note-IIA_rating_methods-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-IIA_rating_methods_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IIA_rating_methods_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-IIA_rating_methods_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Approval voting, score voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates independently using their own <a href="/wiki/Approval_voting#Dichotomous_cutoff" title="Approval voting">absolute scale</a>. For this to hold, in some elections, some voters must use less than their full voting power despite having meaningful preferences among viable candidates.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters.</span> </li> <li id="cite_note-mjmmc-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-mjmmc_11-0">^</a></b></span> <span class="reference-text">Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">A randomly chosen ballot determines winner. This and closely related methods are of mathematical interest and included here to demonstrate that even unreasonable methods can pass voting method criteria.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Where a winner is randomly chosen from the candidates, sortition is included to demonstrate that even non-voting methods can pass some criteria.</span> </li> </ol></div> </td></tr></tbody></table> </div></div> <p><br /> </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=10" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Rather than crossing out near-ties, step 3 is sometimes described as going down the list and confirming ("locking in") the largest victories that do not create a cycle, then ignoring any victories that are not locked-in.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ranked_pairs&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Tideman2-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tideman2_1-0">^</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 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(2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10602-022-09382-w">"The best Condorcet-compatible election method: Ranked Pairs"</a>. <i><a href="/wiki/Constitutional_Political_Economy" title="Constitutional Political Economy">Constitutional Political Economy</a></i>. <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/10.1007%2Fs10602-022-09382-w">10.1007/s10602-022-09382-w</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Constitutional+Political+Economy&rft.atitle=The+best+Condorcet-compatible+election+method%3A+Ranked+Pairs&rft.date=2022&rft_id=info%3Adoi%2F10.1007%2Fs10602-022-09382-w&rft.aulast=Munger&rft.aufirst=Charles+T.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs10602-022-09382-w&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARanked+pairs" 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="CITEREFMunger2022" class="citation journal cs1">Munger, Charles T. (2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10602-022-09382-w">"The best Condorcet-compatible election method: Ranked Pairs"</a>. <i><a href="/wiki/Constitutional_Political_Economy" title="Constitutional Political Economy">Constitutional Political Economy</a></i>. <b>34</b> (3): <span class="nowrap">434–</span>444. <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/10.1007%2Fs10602-022-09382-w">10.1007/s10602-022-09382-w</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Constitutional+Political+Economy&rft.atitle=The+best+Condorcet-compatible+election+method%3A+Ranked+Pairs&rft.volume=34&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E434-%3C%2Fspan%3E444&rft.date=2022&rft_id=info%3Adoi%2F10.1007%2Fs10602-022-09382-w&rft.aulast=Munger&rft.aufirst=Charles+T.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs10602-022-09382-w&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARanked+pairs" class="Z3988"></span></span> </li> </ol></div></div> <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=Ranked_pairs&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://accuratedemocracy.com/download/software/rbvote/desc.html">Descriptions of ranked-ballot voting methods</a> by Rob LeGrand</li> <li><a rel="nofollow" class="external text" href="https://gist.github.com/asafh/a8e9af7a3e5282cbba27">Example JS implementation</a> by Asaf Haddad</li> <li><a rel="nofollow" class="external text" href="https://bitbucket.org/bparanj/rasam">Pair Ranking Ruby Gem</a> by Bala Paranj</li> <li><a rel="nofollow" class="external text" href="https://github.com/pivot-libre/tideman/">A margin-based PHP Implementation of Tideman's Ranked Pairs</a></li> <li><a rel="nofollow" class="external text" href="https://github.com/corydickson/pairwise-voting">Rust implementation of Ranked Pairs</a> by Cory Dickson</li></ul> <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 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Electoral_systems390" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed 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:Electoral_systems_footer" title="Template:Electoral systems footer"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electoral_systems_footer" title="Template talk:Electoral systems footer"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electoral_systems_footer" title="Special:EditPage/Template:Electoral systems footer"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Electoral_systems390" style="font-size:114%;margin:0 4em"><a href="/wiki/Electoral_system" title="Electoral system">Electoral systems</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>Part of the <a href="/wiki/Portal:Politics" title="Portal:Politics">politics</a> and <a href="/wiki/Portal:Economics" title="Portal:Economics">Economics</a> series</i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Single-winner_voting_system" class="mw-redirect" title="Single-winner voting system">Single-winner</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approval_voting" title="Approval voting">Approval voting</a> <ul><li><a href="/wiki/Combined_approval_voting" title="Combined approval voting">Combined approval voting</a></li> <li><a href="/wiki/Unified_primary" title="Unified primary">Unified primary</a></li></ul></li> <li><a href="/wiki/Borda_count" title="Borda count">Borda count</a></li> <li><a href="/wiki/Bucklin_voting" title="Bucklin voting">Bucklin voting</a></li> <li><a href="/wiki/Condorcet_methods" class="mw-redirect" title="Condorcet methods">Condorcet methods</a> <ul><li><a href="/wiki/Copeland%27s_method" title="Copeland's method">Copeland's method</a></li> <li><a href="/wiki/Dodgson%27s_method" title="Dodgson's method">Dodgson's method</a></li> <li><a href="/wiki/Kemeny%E2%80%93Young_method" title="Kemeny–Young method">Kemeny–Young method</a></li> <li><a href="/wiki/Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax Condorcet method</a></li> <li><a href="/wiki/Nanson%27s_method" title="Nanson's method">Nanson's method</a></li> <li><a class="mw-selflink selflink">Ranked pairs</a></li> <li><a href="/wiki/Schulze_method" title="Schulze method">Schulze method</a></li></ul></li> <li><a href="/wiki/Exhaustive_ballot" title="Exhaustive ballot">Exhaustive ballot</a></li> <li><a href="/wiki/First-past-the-post_voting" title="First-past-the-post voting">First-past-the-post voting</a></li> <li><a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff voting</a> <ul><li><a href="/wiki/Coombs%27_method" title="Coombs' method">Coombs' method</a></li> <li><a href="/wiki/Contingent_vote" title="Contingent vote">Contingent vote</a></li> <li><a href="/wiki/Supplementary_vote" class="mw-redirect" title="Supplementary vote">Supplementary vote</a></li></ul></li> <li><a href="/wiki/Majority_rule" title="Majority rule">Simple majoritarianism</a></li> <li><a href="/wiki/Plurality_voting_system" class="mw-redirect" title="Plurality voting system">Plurality</a></li> <li><a href="/wiki/Positional_voting_system" class="mw-redirect" title="Positional voting system">Positional voting system</a></li> <li><a href="/wiki/Score_voting" title="Score voting">Score voting</a></li> <li><a href="/wiki/STAR_voting" title="STAR voting">STAR voting</a></li> <li><a href="/wiki/Two-round_system" title="Two-round system">Two-round system</a></li> <li><a href="/wiki/Graduated_majority_judgment" title="Graduated majority judgment">Graduated majority judgment</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proportional_representation" title="Proportional representation">Proportional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Systems</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/Mixed-member_proportional_representation" title="Mixed-member proportional representation">Mixed-member</a></li> <li><a href="/wiki/Mixed_single_vote#Proportional_systems" title="Mixed single vote">Mixed single vote</a></li> <li><a href="/wiki/Party-list_proportional_representation" title="Party-list proportional representation">Party-list</a></li> <li><a href="/wiki/Proportional_approval_voting" title="Proportional approval voting">Proportional approval voting</a></li> <li><a href="/wiki/Rural%E2%80%93urban_proportional_representation" title="Rural–urban proportional representation">Rural–urban</a></li> <li><a href="/wiki/Sequential_proportional_approval_voting" title="Sequential proportional approval voting">Sequential proportional approval voting</a></li> <li><a href="/wiki/Single_transferable_vote" title="Single transferable vote">Single transferable vote</a> <ul><li><a href="/wiki/CPO-STV" title="CPO-STV">CPO-STV</a></li> <li><a href="/wiki/Hare%E2%80%93Clark_electoral_system" title="Hare–Clark electoral system">Hare–Clark</a></li> <li><a href="/wiki/Schulze_STV" title="Schulze STV">Schulze STV</a></li></ul></li> <li><a href="/wiki/Spare_vote" title="Spare vote">Spare vote</a></li> <li><a href="/wiki/Indirect_single_transferable_voting" title="Indirect single transferable voting">Indirect single transferable voting</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Allocation</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/Highest_averages_method" title="Highest averages method">Highest averages method</a> <ul><li><a href="/wiki/Sainte-Lagu%C3%AB_method" title="Sainte-Laguë method">Webster/Sainte-Laguë</a></li> <li><a href="/wiki/D%27Hondt_method" title="D'Hondt method">D'Hondt</a></li></ul></li> <li><a href="/wiki/Largest_remainders_method" class="mw-redirect" title="Largest remainders method">Largest remainders method</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quotas</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/Droop_quota" title="Droop quota">Droop quota</a></li> <li><a href="/wiki/Hagenbach-Bischoff_quota" class="mw-redirect" title="Hagenbach-Bischoff quota">Hagenbach-Bischoff quota</a></li> <li><a href="/wiki/Hare_quota" title="Hare quota">Hare quota</a></li> <li><a href="/wiki/Imperiali_quota" title="Imperiali quota">Imperiali quota</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Mixed</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Parallel_voting" title="Parallel voting">Parallel voting</a></li> <li><a href="/wiki/Mixed-member_proportional_representation" title="Mixed-member proportional representation">MMP</a></li> <li><a href="/wiki/Additional_member_system" class="mw-redirect" title="Additional member system">Additional member system</a></li> <li><a href="/wiki/Alternative_vote_plus" title="Alternative vote plus">Alternative vote plus</a></li> <li><a href="/wiki/Mixed_single_vote" title="Mixed single vote">Mixed single vote</a></li> <li><a href="/wiki/Mixed_ballot_transferable_vote" title="Mixed ballot transferable vote">Mixed ballot transferable vote</a></li> <li><a href="/wiki/Scorporo" title="Scorporo">Scorporo</a></li> <li><a href="/wiki/Vote_linkage_mixed_system" class="mw-redirect" title="Vote linkage mixed system">Vote linkage mixed system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Semi-proportional_representation" title="Semi-proportional representation">Semi-proportional</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Single_non-transferable_vote" title="Single non-transferable vote">Single non-transferable vote</a></li> <li><a href="/wiki/Limited_voting" title="Limited voting">Limited voting</a></li> <li><a href="/wiki/Cumulative_voting" title="Cumulative voting">Cumulative voting</a></li> <li><a href="/wiki/Satisfaction_approval_voting" title="Satisfaction approval voting">Satisfaction approval voting</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Criteria</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Condorcet_winner_criterion" title="Condorcet winner criterion">Condorcet winner criterion</a></li> <li><a href="/wiki/Condorcet_loser_criterion" title="Condorcet loser criterion">Condorcet loser criterion</a></li> <li><a href="/wiki/Consistency_criterion" class="mw-redirect" title="Consistency criterion">Consistency criterion</a></li> <li><a href="/wiki/Independence_of_clones_criterion" title="Independence of clones criterion">Independence of clones</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/Independence_of_Smith-dominated_alternatives" title="Independence of Smith-dominated alternatives">Independence of Smith-dominated alternatives</a></li> <li><a href="/wiki/Later-no-harm_criterion" title="Later-no-harm criterion">Later-no-harm criterion</a></li> <li><a href="/wiki/Majority_favorite_criterion" class="mw-redirect" title="Majority favorite criterion">Majority criterion</a></li> <li><a href="/wiki/Majority_loser_criterion" title="Majority loser criterion">Majority loser criterion</a></li> <li><a href="/wiki/Monotonicity_criterion" class="mw-redirect" title="Monotonicity criterion">Monotonicity criterion</a></li> <li><a href="/wiki/Mutual_majority_criterion" title="Mutual majority criterion">Mutual majority criterion</a></li> <li><a href="/wiki/Participation_criterion" class="mw-redirect" title="Participation criterion">Participation criterion</a></li> <li><a href="/wiki/Plurality_criterion" class="mw-redirect" title="Plurality criterion">Plurality criterion</a></li> <li><a href="/wiki/Resolvability_criterion" title="Resolvability criterion">Resolvability criterion</a></li> <li><a href="/wiki/Reversal_symmetry" class="mw-redirect" title="Reversal symmetry">Reversal symmetry</a></li> <li><a href="/wiki/Smith_criterion" class="mw-redirect" title="Smith criterion">Smith criterion</a></li> <li><a href="/wiki/Seats-to-votes_ratio" title="Seats-to-votes ratio">Seats-to-votes ratio</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ballot" title="Ballot">Ballot</a></li> <li><a href="/wiki/Election_threshold" class="mw-redirect" title="Election threshold">Election threshold</a></li> <li><a href="/wiki/First-preference_votes" title="First-preference votes">First-preference votes</a></li> <li><a href="/wiki/Liquid_democracy" title="Liquid democracy">Liquid democracy</a></li> <li><a href="/wiki/Spoilt_vote" title="Spoilt vote">Spoilt vote</a></li> <li><a href="/wiki/Sortition" title="Sortition">Sortition</a></li> <li><a href="/wiki/Unseating" title="Unseating">Unseating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Comparison</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Comparison_of_electoral_systems" class="mw-redirect" title="Comparison of electoral systems">Comparison of voting systems</a></li> <li><a href="/wiki/List_of_electoral_systems_by_country" title="List of electoral systems by country">Voting systems by country</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><a href="/wiki/Portal:Politics" title="Portal:Politics">Portal</a></b> — <b><a href="/wiki/Wikipedia:WikiProject_Politics" title="Wikipedia:WikiProject Politics">Project</a></b></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Ranked_pairs&oldid=1275676392">https://en.wikipedia.org/w/index.php?title=Ranked_pairs&oldid=1275676392</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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