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href="#Examples_of_the_Copeland_Method"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Examples of the Copeland Method</span> </div> </a> <button aria-controls="toc-Examples_of_the_Copeland_Method-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples of the Copeland Method subsection</span> </button> <ul id="toc-Examples_of_the_Copeland_Method-sublist" class="vector-toc-list"> <li id="toc-Example_with_Condorcet_winner" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_with_Condorcet_winner"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Example with Condorcet winner</span> </div> </a> <ul id="toc-Example_with_Condorcet_winner-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_without_Condorcet_winner" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_without_Condorcet_winner"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Example without Condorcet winner</span> </div> </a> <ul id="toc-Example_without_Condorcet_winner-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Use_for_producing_a_tabulation_in_other_methods" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Use_for_producing_a_tabulation_in_other_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Use for producing a tabulation in other methods</span> </div> </a> <ul id="toc-Use_for_producing_a_tabulation_in_other_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> 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id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">A joint <a href="/wiki/Portal:Politics" title="Portal:Politics">Politics</a> and <a href="/wiki/Portal:Economics" title="Portal:Economics">Economics</a> series</td></tr><tr><th class="sidebar-title-with-pretitle" style="border-top:1px #fafafa solid; border-bottom:1px #fafafa solid; background:#efefef; background: var(--background-color-interactive, #efefef); color: var(--color-base, #000); padding:0.2em;"><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice</a> and <a href="/wiki/Electoral_system" title="Electoral system">electoral systems</a></th></tr><tr><td class="sidebar-image"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Electoral-systems-gears.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/128px-Electoral-systems-gears.svg.png" decoding="async" width="128" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/192px-Electoral-systems-gears.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Electoral-systems-gears.svg/256px-Electoral-systems-gears.svg.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption></figcaption></figure></td></tr><tr><td class="sidebar-above"> <div class="hlist"><ul><li><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice</a></li><li><a href="/wiki/Mechanism_design" title="Mechanism design">Mechanism design</a></li><li><a href="/wiki/Comparative_politics" title="Comparative politics">Comparative politics</a></li><li><a href="/wiki/Comparison_of_voting_rules" title="Comparison of voting rules">Comparison</a></li><li><a href="/wiki/List_of_electoral_systems" title="List of electoral systems">List</a> (<a href="/wiki/List_of_electoral_systems_by_country" title="List of electoral systems by country">By country</a>)</li></ul></div></td></tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Single-member_district" title="Single-member district">Single-winner methods</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Single vote - <a href="/wiki/Plurality_voting" title="Plurality voting">plurality</a> methods</b> <ul><li><a href="/wiki/First-past-the-post_voting" title="First-past-the-post voting">First preference plurality (FPP)</a></li> <li><a href="/wiki/Two-round_system" title="Two-round system">Two-round</a> (<abbr style="font-size:85%" title=""><a href="/wiki/American_English" title="American English">US</a>:</abbr> <a href="/wiki/Nonpartisan_blanket_primary" title="Nonpartisan blanket primary">Jungle primary</a>) <ul><li><a href="/wiki/Partisan_primary" class="mw-redirect" title="Partisan primary">Partisan primary</a></li></ul></li> <li><a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff</a> <ul><li><abbr style="font-size:85%" title=""><a href="/wiki/British_English" title="British English">UK</a>:</abbr> Alternative vote (AV)</li> <li><abbr style="font-size:85%" title=""><a href="/wiki/American_English" title="American English">US</a>:</abbr> Ranked-choice (RCV)</li></ul></li></ul> <hr /> <p><b><a href="/wiki/Condorcet_method" title="Condorcet method">Condorcet methods</a></b> </p> <ul><li><a href="/wiki/Tideman_alternative_method" title="Tideman alternative method">Condorcet-IRV</a></li> <li><a href="/wiki/Round-robin_voting" title="Round-robin voting">Round-robin voting</a> <ul><li><a href="/wiki/Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax</a></li> <li><a href="/wiki/Schulze_method" title="Schulze method">Schulze</a></li> <li><a href="/wiki/Ranked_pairs" title="Ranked pairs">Ranked pairs</a></li> <li><a href="/wiki/Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Positional_voting" title="Positional voting">Positional voting</a></b> </p> <ul><li><a href="/wiki/First-preference_plurality" class="mw-redirect" title="First-preference plurality">Plurality</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">IRV</a>)</li> <li><a href="/wiki/Borda_count" title="Borda count">Borda count</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Baldwin%27s_method" class="mw-redirect" title="Baldwin's method">Baldwin</a>)</li> <li><a href="/wiki/Anti-plurality_voting" title="Anti-plurality voting">Antiplurality</a> (<abbr style="font-size:85%" title=""><a href="/wiki/Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="/wiki/Coombs_method" class="mw-redirect" title="Coombs method">Coombs</a>)</li></ul> <hr /> <p><b><a href="/wiki/Rated_voting" title="Rated voting">Cardinal voting</a></b> </p> <ul><li><a href="/wiki/Score_voting" title="Score voting">Score voting</a></li> <li><a href="/wiki/Approval_voting" title="Approval voting">Approval voting</a></li> <li><a href="/wiki/Highest_median_voting_rules" title="Highest median voting rules">Majority judgment</a></li> <li><a href="/wiki/STAR_voting" title="STAR voting">STAR voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Proportional_representation" title="Proportional representation">Proportional representation</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="/wiki/Party-list_proportional_representation" title="Party-list proportional representation">Party-list</a></b> <ul><li><a href="/wiki/Apportionment_(politics)" title="Apportionment (politics)">Apportionment</a> <ul><li><a href="/wiki/Highest_averages_method" title="Highest averages method">Highest averages</a></li> <li><a href="/wiki/Largest_remainder_method" class="mw-redirect" title="Largest remainder method">Largest remainders</a></li> <li><a href="/wiki/National_remnant" title="National remnant">National remnant</a></li> <li><a href="/wiki/Biproportional_apportionment" title="Biproportional apportionment">Biproportional</a></li></ul></li> <li><a href="/wiki/Electoral_list" title="Electoral list">List type</a> <ul><li><a href="/wiki/Closed_list" title="Closed list">Closed list</a></li> <li><a href="/wiki/Open_list" title="Open list">Open list</a></li> <li><a href="/wiki/Panachage" title="Panachage">Panachage</a></li> <li><a href="/wiki/Justified_representation" title="Justified representation">List-free PR</a></li> <li><a href="/wiki/Localized_list" title="Localized list">Localized list</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Electoral_quota" title="Electoral quota">Quota-remainder methods</a></b> </p> <ul><li><a href="/wiki/Single_transferable_vote" title="Single transferable vote">Hare STV</a></li> <li><a href="/wiki/Schulze_STV" title="Schulze STV">Schulze STV</a></li> <li><a href="/wiki/CPO-STV" title="CPO-STV">CPO-STV</a></li> <li><a href="/wiki/Quota_Borda_system" title="Quota Borda system">Quota Borda</a></li></ul> <hr /> <p><b><a href="/wiki/Approval-based_committee" class="mw-redirect" title="Approval-based committee">Approval-based committees</a></b> </p> <ul><li><a href="/wiki/Proportional_approval_voting" title="Proportional approval voting">Thiele's method</a></li> <li><a href="/wiki/Phragmen%27s_voting_rules" title="Phragmen's voting rules">Phragmen's method</a></li> <li><a href="/wiki/Expanding_approvals_rule" title="Expanding approvals rule">Expanding approvals rule</a></li> <li><a href="/wiki/Method_of_equal_shares" title="Method of equal shares">Method of equal shares</a></li></ul> <hr /> <p><b><a href="/wiki/Fractional_social_choice" title="Fractional social choice">Fractional social choice</a></b> </p> <ul><li><a href="/wiki/Direct_representation" title="Direct representation">Direct representation</a> <ul><li><a href="/wiki/Interactive_representation" title="Interactive representation">Interactive representation</a></li> <li><a href="/wiki/Liquid_democracy" title="Liquid democracy">Liquid democracy</a></li></ul></li> <li><a href="/wiki/Fractional_approval_voting" title="Fractional approval voting">Fractional approval voting</a></li> <li><a href="/wiki/Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li> <li><a href="/wiki/Random_ballot" title="Random ballot">Random ballot</a></li></ul> <hr /> <p><b><a href="/wiki/Semi-proportional_representation" title="Semi-proportional representation">Semi-proportional representation</a></b> </p> <ul><li><a href="/wiki/Cumulative_voting" title="Cumulative voting">Cumulative</a> <ul><li><a href="/wiki/Single_non-transferable_vote" title="Single non-transferable vote">SNTV</a></li></ul></li> <li><a href="/wiki/Limited_voting" title="Limited voting">Limited voting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Mixed systems</a></div><div class="sidebar-list-content mw-collapsible-content"><b>By results of combination</b> <ul><li><a href="/wiki/Mixed-member_majoritarian_representation" title="Mixed-member majoritarian representation">Mixed-member majoritarian</a></li> <li><a href="/wiki/Mixed-member_proportional_representation" title="Mixed-member proportional representation">Mixed-member proportional</a></li></ul> <hr /><b>By mechanism of combination</b> <ul><li><b>Non-<a href="/wiki/Compensation_(electoral_systems)" title="Compensation (electoral systems)">compensatory</a></b> <ul><li><a href="/wiki/Parallel_voting" title="Parallel voting">Parallel (superposition)</a></li> <li><a href="/wiki/Coexistence_(electoral_systems)" title="Coexistence (electoral systems)">Coexistence</a></li> <li><a href="/w/index.php?title=Conditional_electoral_system&action=edit&redlink=1" class="new" title="Conditional electoral system (page does not exist)">Conditional</a></li> <li><a href="/wiki/Majority_bonus_system" title="Majority bonus system">Fusion (majority bonus)</a></li></ul></li> <li><b><a href="/wiki/Compensation_(electoral_systems)" title="Compensation (electoral systems)">Compensatory</a></b> <ul><li><a href="/w/index.php?title=Seat_linkage_mixed_system&action=edit&redlink=1" class="new" title="Seat linkage mixed system (page does not exist)">Seat linkage system</a> <ul><li><abbr style="font-size:85%" title=""><a href="/wiki/British_English" title="British English">UK</a>:</abbr> <a href="/wiki/Additional_member_system" class="mw-redirect" title="Additional member system">'AMS'</a></li> <li><abbr style="font-size:85%" title=""><a href="/wiki/New_Zealand_English" title="New Zealand English">NZ</a>:</abbr> <a href="/wiki/Mixed-member_proportional" class="mw-redirect" title="Mixed-member proportional">'MMP'</a></li></ul></li> <li><a href="/wiki/Vote_linkage_mixed_system" class="mw-redirect" title="Vote linkage mixed system">Vote linkage system</a> <ul><li><a href="/wiki/Scorporo" title="Scorporo">Negative vote transfer</a></li> <li><a href="/wiki/Mixed_ballot_transferable_vote" title="Mixed ballot transferable vote">Mixed ballot</a></li></ul></li></ul></li> <li><a href="/wiki/Mixed_electoral_system" title="Mixed electoral system">Supermixed systems</a> <ul><li><a href="/wiki/Dual-member_proportional_representation" class="mw-redirect" title="Dual-member proportional representation">Dual-member proportional</a></li> <li><a href="/wiki/Rural%E2%80%93urban_proportional_representation" title="Rural–urban proportional representation">Rural–urban proportional</a></li> <li><a href="/wiki/Majority_jackpot_system" title="Majority jackpot system">Majority jackpot</a></li></ul></li></ul> <hr /> <p><b>By ballot type</b> </p> <ul><li><a href="/wiki/Mixed_single_vote" title="Mixed single vote">Single vote</a> <ul><li><a href="/wiki/Double_simultaneous_vote" title="Double simultaneous vote">Double simultaneous vote</a></li></ul></li> <li><a href="/wiki/Mixed_electoral_systems" class="mw-redirect" title="Mixed electoral systems">Dual-vote</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="text-align:left;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="/wiki/Pathological_(mathematics)#Voting" title="Pathological (mathematics)">Paradoxes and pathologies</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Spoiler effects</b> <ul><li><a href="/wiki/Spoiler_effect" title="Spoiler effect">Spoiler effect</a></li> <li><a href="/wiki/Independence_of_clones" class="mw-redirect" title="Independence of clones">Cloning paradox</a></li> <li><a href="/wiki/Condorcet_winner_criterion" title="Condorcet winner criterion">Frustrated majorities paradox</a></li> <li><a href="/wiki/Center_squeeze" title="Center squeeze">Center squeeze</a></li></ul> <hr /> <p><b>Pathological response</b> </p> <ul><li><a href="/wiki/Perverse_response" class="mw-redirect" title="Perverse response">Perverse response</a></li> <li><a href="/wiki/Best-is-worst_paradox" title="Best-is-worst paradox">Best-is-worst paradox</a></li> <li><a href="/wiki/No-show_paradox" title="No-show paradox">No-show paradox</a> <ul><li><a href="/wiki/Multiple_districts_paradox" title="Multiple districts paradox">Multiple districts paradox</a></li></ul></li></ul> <hr /> <p><b><a href="/wiki/Strategic_voting" title="Strategic voting">Strategic voting</a></b> </p> <ul><li><a href="/wiki/Sincere_favorite_criterion" title="Sincere favorite criterion">Lesser evil voting</a></li> <li><a href="/wiki/Strategic_voting#Exaggeration" title="Strategic voting">Exaggeration</a></li> <li><a href="/wiki/Truncation_(voting)" class="mw-redirect" title="Truncation (voting)">Truncation</a></li> <li><a href="/wiki/Turkey-raising" class="mw-redirect" title="Turkey-raising">Turkey-raising</a></li></ul> <hr /> <p><b>Paradoxes of <a href="/wiki/Majority_rule" title="Majority rule">majority rule</a></b> </p> <ul><li><a href="/wiki/Tyranny_of_the_majority" title="Tyranny of the majority">Tyranny of the majority</a></li> <li><a href="/wiki/Discursive_dilemma" title="Discursive dilemma">Discursive dilemma</a></li> <li><a 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></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 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//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electoral_systems_sidebar" title="Template:Electoral systems sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electoral_systems_sidebar" title="Template talk:Electoral systems sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electoral_systems_sidebar" title="Special:EditPage/Template:Electoral systems sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>Copeland</b> or <b>Llull method</b> is a <a href="/wiki/Ranked_voting" title="Ranked voting">ranked-choice voting system</a> based on counting each candidate's pairwise wins and losses. </p><p>In the system, voters rank candidates from best to worst on their ballot. Candidates then compete in a <a href="/wiki/Round-robin_tournament" title="Round-robin tournament">round-robin tournament</a>, where the ballots are used to determine which candidate would be preferred by a majority of voters in each matchup. The candidate is the one who wins the most matchups (with ties winning half a point). </p><p>Copeland's method falls in the class of <a href="/wiki/Condorcet_method" title="Condorcet method">Condorcet methods</a>, as any candidate who wins every one-on-one election will clearly have the most victories overall.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Copeland's method has the advantage of being likely the simplest Condorcet method to explain and of being easy to administer by hand. On the other hand, if there is no Condorcet winner, the procedure frequently results in ties. As a result, it is typically only used for low-stakes elections. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Copeland's method was devised by <a href="/wiki/Ramon_Llull" title="Ramon Llull">Ramon Llull</a> in his 1299 treatise <i>Ars Electionis,</i> which was discussed by <a href="/wiki/Nicholas_of_Cusa" title="Nicholas of Cusa">Nicholas of Cusa</a> in the fifteenth century.<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> However, it is frequently named after <a href="/wiki/Arthur_Herbert_Copeland" title="Arthur Herbert Copeland">Arthur Herbert Copeland</a>, who advocated it independently in a 1951 lecture.<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Ranked_robin_vote.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ranked_robin_vote.png/292px-Ranked_robin_vote.png" decoding="async" width="292" height="339" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ranked_robin_vote.png/438px-Ranked_robin_vote.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ranked_robin_vote.png/584px-Ranked_robin_vote.png 2x" data-file-width="1794" data-file-height="2084" /></a><figcaption>A simple description of Copeland's method.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Voting_mechanism">Voting mechanism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=2" title="Edit section: Voting mechanism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ballot">Ballot</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=3" title="Edit section: Ballot"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The input is the same as for other ranked voting systems: each voter must furnish an ordered preference list on candidates where <a href="/wiki/Tie_(draw)" title="Tie (draw)">ties</a> are allowed (<a href="/wiki/Weak_ordering" title="Weak ordering">a strict weak order</a>). </p><p>This can be done by providing each voter with a list of candidates on which to write a "1" against the most preferred candidate, a "2" against the second preference, and so forth. A voter who leaves some candidates' rankings blank is assumed to be indifferent between them but to prefer all ranked candidates to them. </p> <div class="mw-heading mw-heading3"><h3 id="Computation">Computation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=4" title="Edit section: Computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A results matrix <i>r</i> is constructed as follows:<sup id="cite_ref-saari_4-0" class="reference"><a href="#cite_note-saari-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <i>r<sub>ij</sub></i> is </p> <ul><li>1 if more voters strictly prefer candidate <i>i</i> to candidate <i>j</i> than prefer <i>j</i> to <i>i</i></li> <li><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> if the numbers are equal</li> <li>0 if more voters prefer <i>j</i> to <i>i</i> than prefer <i>i</i> to <i>j</i>.</li></ul> <p>This may be called the "1/<style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>/0" method (one number for wins, ties, and losses, respectively). </p><p>By convention, <i>r<sub>ii</sub></i> is 0. </p><p>The Copeland score for candidate <i>i</i> is the sum over <i>j</i> of the <i>r<sub>ij</sub></i>. If there is a candidate with a score of <span class="nowrap"><i>n</i> − 1</span> (where <i>n</i> is the number of candidates) then this candidate is the (necessarily unique) Condorcet and Copeland winner. Otherwise the Condorcet method produces no decision and the candidate with greatest score is the Copeland winner (but may not be unique). </p><p>An alternative (and equivalent) way to construct the results matrix is by letting <i>r<sub>ij</sub></i> be 1 if more voters strictly prefer candidate <i>i</i> to candidate <i>j</i> than prefer <i>j</i> to <i>i</i>, 0 if the numbers are equal, and −1 if more voters prefer <i>j</i> to <i>i</i> than prefer <i>i</i> to <i>j</i>. In this case the matrix <i>r</i> is <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">antisymmetric</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Tied_preferences">Tied preferences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=5" title="Edit section: Tied preferences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The method as initially described above is sometimes called the "1/<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>/0" method. Llull himself put forward a 1/1/0 method, so that two candidates with equal support would both get the same credit as if they had beaten the other.<sup id="cite_ref-judge_5-0" class="reference"><a href="#cite_note-judge-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Preference ties become increasingly unlikely as the number of voters increases. </p> <div class="mw-heading mw-heading3"><h3 id="Use_in_sporting_tournaments">Use in sporting tournaments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=6" title="Edit section: Use in sporting tournaments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A method related to Copeland's is commonly used in <a href="/wiki/Round-robin_tournament" title="Round-robin tournament">round-robin tournaments</a>. Generally it is assumed that each pair of competitors plays the same number of games against each other. <i>r<sub>ij</sub></i> is the number of times competitor <i>i</i> won against competitor <i>j</i> plus half the number of draws between them. </p><p>It was adopted in precisely this form in international chess in the middle of the nineteenth century.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> It was adopted in the first season of the <a href="/wiki/English_Football_League" title="English Football League">English Football League</a> (1888–1889), the organisers having initially considered using a 1/0/0 system. For convenience the numbers were doubled, i.e. the system was written as 2/1/0 rather than as 1/<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>/0. </p><p>(The <a href="/wiki/Borda_count" title="Borda count">Borda count</a> has also been used to judge sporting tournaments. The Borda count is analogous to a tournament in which every completed ballot determines the result of a game between every pair of competitors.) </p> <div class="mw-heading mw-heading2"><h2 id="Rationale">Rationale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=7" title="Edit section: Rationale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In many cases decided by Copeland's method the winner is the unique candidate satisfying the Condorcet criterion; in these cases, the arguments for that criterion (which are powerful, but not universally accepted<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>) apply equally to Copeland's method. </p><p>When there is no Condorcet winner, Copeland's method seeks to make a decision by a natural extension of the Condorcet method, combining preferences by simple addition. The justification for this lies more in its simplicity than in logical arguments. </p><p>The <a href="/wiki/Borda_count" title="Borda count">Borda count</a> is another method which combines preferences additively. The salient difference is that a voter's preference for one candidate over another has a weight in the Borda system which increases with the number of candidates ranked between them. The argument from the viewpoint of the Borda count is that the number of intervening candidates gives an indication of the strength of the preference; the counter-argument is that it depends to a worrying degree on which candidates stood in the election. </p><p><a href="/wiki/Partha_Dasgupta" title="Partha Dasgupta">Partha Dasgupta</a> and <a href="/wiki/Eric_Maskin" title="Eric Maskin">Eric Maskin</a> sought to justify Copeland's method in a popular journal, where they compare it with the Borda count and plurality voting.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Their argument turns on the merits of the Condorcet criterion, paying particular attention to opinions lying on a spectrum. The use of Copeland's method in the first instance, and then of a tie-break, to decide elections with no Condorcet winner is presented as "perhaps the simplest modification" to the Condorcet method. </p> <div class="mw-heading mw-heading2"><h2 id="Tied_results">Tied results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=8" title="Edit section: Tied results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Like any voting method, Copeland's may give rise to tied results if two candidates receive equal numbers of votes; but unlike most methods, it may also lead to ties for causes which do not disappear as the electorate becomes larger. This may happen whenever there are Condorcet cycles in the voting preferences, as illustrated by the following example. </p><p>Suppose that there are four candidates, Able, Baker, Charlie and Drummond, and five voters, of whom two vote A-B-C-D, two vote B-C-D-A, and one votes D-A-B-C. The results between pairs of candidates are shown in the main part of the following table, with the Copeland score for the first candidate in the additional column. </p> <table class="wikitable" style="border:none"> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">2nd</div><div style="margin-right:2em;text-align:left">1st</div> </th> <th>A</th> <th>B</th> <th>C</th> <th>D </th> <td rowspan="5" style="border: none; background: white;"> </td> <th>score </th></tr> <tr> <th>A </th> <td data-sort-value="" style="vertical-align:middle; text-align:center" class="table-na">—</td> <td>3:2</td> <td>3:2</td> <td>2:3</td> <td>2 </td></tr> <tr> <th>B </th> <td>2:3</td> <td data-sort-value="" style="vertical-align:middle; text-align:center" class="table-na">—</td> <td>5:0</td> <td>4:1</td> <td>2 </td></tr> <tr> <th>C </th> <td>2:3</td> <td>0:5</td> <td data-sort-value="" style="vertical-align:middle; text-align:center" class="table-na">—</td> <td>4:1</td> <td>1 </td></tr> <tr> <th>D </th> <td>3:2</td> <td>1:4</td> <td>1:4</td> <td data-sort-value="" style="vertical-align:middle; text-align:center" class="table-na">—</td> <td>1 </td></tr></tbody></table> <p>No candidate satisfies the Condorcet criterion, and there is a Copeland tie between A and B. If there were 100 times as many voters, but they voted in roughly the same proportions (subject to sampling fluctuations), then the numbers of ballots would scale up but the Copeland scores would stay the same; for instance the 'A' row might read: </p> <table class="wikitable" style="border:none"> <tbody><tr> <th>A </th> <td data-sort-value="" style="vertical-align:middle; text-align:center" class="table-na">—</td> <td>317:183</td> <td>296:204</td> <td>212:288 </td> <td style="border: none; background: white;"> </td> <td>2 </td></tr></tbody></table> <p>The risk of ties is particularly concerning because the main aim of Copeland's method is to produce a winner in cases when no candidate satisfies the Condorcet criterion. A simulation performed by Richard Darlington implies that for fields of up to 10 candidates, it will succeed in this task less than half the time.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>In general, if voters vote according to preferences along a <a href="/wiki/Political_spectrum" title="Political spectrum">spectrum</a>, the <a href="/wiki/Median_voter_theorem" title="Median voter theorem">median voter theorem</a> guarantees the absence of Condorcet cycles. Consequently such cycles can only arise either because voters' preferences do not lie along a spectrum or because voters do not vote according to their preferences (eg. for tactical reasons). </p><p><a href="/wiki/Nicolaus_Tideman" title="Nicolaus Tideman">Nicolaus Tideman</a> and Florenz Plassman conducted a large study of reported electoral preferences.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> They found a significant number of cycles in the subelections, but remarked that they could be attributed wholly or largely to the smallness of the numbers of voters. They concluded that it was consistent with their data to suppose that "voting cycles will occur very rarely, if at all, in elections with many voters". </p> <div class="mw-heading mw-heading3"><h3 id="Proposed_tie_breaks">Proposed tie breaks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=9" title="Edit section: Proposed tie breaks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Instant-runoff_voting" title="Instant-runoff voting">Instant runoff (IRV)</a>, <a href="/wiki/Minimax_Condorcet_method" title="Minimax Condorcet method">minimax</a> and the Borda count are natural tie-breaks. The first two are not frequently advocated for this use but are sometimes discussed in connection with <a href="/wiki/Smith_set#Smith's_method" title="Smith set">Smith's method</a> where similar considerations apply. </p><p>Dasgupta and Maskin proposed the Borda count as a Copeland tie-break: this is known as the <b>Dasgupta-Maskin method</b>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> It had previously been used in figure-skating under the name of the 'OBO' (=one-by-one) rule.<sup id="cite_ref-judge_5-1" class="reference"><a href="#cite_note-judge-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The alternatives can be illustrated in the 'Able-Baker' example above, in which Able and Baker are joint Copeland winners. Charlie and Drummond are eliminated, reducing the ballots to 3 A-Bs and 2 B-As. Any tie-break will then elect Able.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=10" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Copeland's method has many of the standard desirable properties (see the table below). Most importantly it satisfies the <a href="/wiki/Condorcet_criterion" class="mw-redirect" title="Condorcet criterion">Condorcet criterion</a>, i.e. if a candidate would win against each of their rivals in a one-on-one vote, this candidate is the winner. Copeland's method therefore satisfies the median voter theorem, which states that if views lie along a spectrum, then <a href="/wiki/Median_voter_theorem" title="Median voter theorem">the winning candidate will be the one preferred by the median voter</a>. </p><p>Copeland's method also satisfies the <a href="/wiki/Smith_criterion" class="mw-redirect" title="Smith criterion">Smith criterion</a>.<sup id="cite_ref-:moulin_13-0" class="reference"><a href="#cite_note-:moulin-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>The analogy between Copeland's method and sporting tournaments, and the overall simplicity of Copeland's method, has been argued to make it more acceptable to voters than other Condorcet algorithms.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 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.mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) .overflowed{margin-top:0;margin-bottom:0}body.skin-timeless .mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) .scroll-right.overflowed .content-table-right{box-shadow:none;border-left:none}body.skin-timeless .mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) .wikitable{padding:0}@media(max-width:639px){body.skin-minerva .mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) table,body.skin-monobook .mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) table,body.skin-vector-legacy .mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) table,body.skin-vector-2022 .mw-parser-output .sticky-table-collapsible:not(.mw-collapsed) 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_15-0" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-15"><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_15-1" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-15"><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_15-2" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-15"><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_15-3" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-15"><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_15-4" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-15"><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_15-5" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-15"><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_15-6" class="reference"><a href="#cite_note-condorcet-iia-incompatibility-15"><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_16-0" class="reference"><a href="#cite_note-MinimaxVarient-16"><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_16-1" class="reference"><a href="#cite_note-MinimaxVarient-16"><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 class="mw-selflink selflink">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 href="/wiki/Ranked_pairs" title="Ranked pairs">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_17-0" class="reference"><a href="#cite_note-semipc-17"><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_17-1" class="reference"><a href="#cite_note-semipc-17"><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_18-0" class="reference"><a href="#cite_note-IIA_rating_methods-18"><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-19" class="reference"><a href="#cite_note-19"><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_20-0" class="reference"><a href="#cite_note-mjmmc-20"><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_18-1" class="reference"><a href="#cite_note-IIA_rating_methods-18"><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_17-2" class="reference"><a href="#cite_note-semipc-17"><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_18-2" class="reference"><a href="#cite_note-IIA_rating_methods-18"><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/Random_ballot" title="Random ballot">Random ballot</a><wbr /><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><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-22" class="reference"><a href="#cite_note-22"><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-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-condorcet-iia-incompatibility_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_15-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_15-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_15-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_15-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-condorcet-iia-incompatibility_15-6"><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-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-MinimaxVarient_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MinimaxVarient_16-1"><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-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-semipc_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-semipc_17-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-semipc_17-2"><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-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-IIA_rating_methods_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IIA_rating_methods_18-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-IIA_rating_methods_18-2"><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-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</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-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-mjmmc_20-0">^</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-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</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-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</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="Examples_of_the_Copeland_Method">Examples of the Copeland Method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=12" title="Edit section: Examples of the Copeland Method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Example_with_Condorcet_winner">Example with Condorcet winner</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=13" title="Edit section: Example with Condorcet winner"><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_(political)" class="mw-redirect" title="Capital (political)">capital</a>. The population is concentrated around four major cities. <a href="/wiki/Spatial_model_of_voting" class="mw-redirect" title="Spatial model of 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 /> To find the Condorcet winner, every candidate must be matched against every other candidate in a series of imaginary one-on-one contests. In each pairing, each voter will choose the city physically closest to their location. In each pairing the winner is the candidate preferred by a majority of voters. When results for every possible pairing have been found they are as follows: </p> <table class="wikitable"> <tbody><tr> <th>Comparison</th> <th>Result</th> <th>Winner </th></tr> <tr> <td>Memphis vs Nashville</td> <td>42 v 58</td> <td>Nashville </td></tr> <tr> <td>Memphis vs Knoxville</td> <td>42 v 58</td> <td>Knoxville </td></tr> <tr> <td>Memphis vs Chattanooga</td> <td>42 v 58</td> <td>Chattanooga </td></tr> <tr> <td>Nashville vs Knoxville</td> <td>68 v 32</td> <td>Nashville </td></tr> <tr> <td>Nashville vs Chattanooga</td> <td>68 v 32</td> <td>Nashville </td></tr> <tr> <td>Knoxville vs Chattanooga</td> <td>17 v 83</td> <td>Chattanooga </td></tr></tbody></table> <p>The wins and losses of each candidate sum as follows: </p> <table class="wikitable"> <tbody><tr> <th>Candidate</th> <th>Wins</th> <th>Losses</th> <th>Net</th> <th><i>r</i> </th></tr> <tr> <td>Memphis</td> <td>0</td> <td>3</td> <td>−3 </td> <td>0 0 0 0 </td></tr> <tr> <td>Nashville</td> <td>3</td> <td>0</td> <td>3 </td> <td>1 0 1 1 </td></tr> <tr> <td>Knoxville</td> <td>1</td> <td>2</td> <td>−1 </td> <td>1 0 0 0 </td></tr> <tr> <td>Chattanooga</td> <td>2</td> <td>1</td> <td>1 </td> <td>1 0 1 0 </td></tr></tbody></table> <p><b>Nashville</b>, with no defeats, is the Condorcet winner. The Copeland score under the 1/0/−1 method is the number of net wins, maximized by Nashville. Since the voters expressed a preference one way or the other between every pair of candidates, the score under the 1/<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>/0 method is just the number of wins, likewise maximized by Nashville. The <i>r</i> matrix for this scoring system is shown in the final column. </p> <div class="mw-heading mw-heading3"><h3 id="Example_without_Condorcet_winner">Example without Condorcet winner</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=14" title="Edit section: Example without Condorcet winner"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In an election with five candidates competing for one seat, the following votes were cast using a <a href="/wiki/Ranked_voting_systems" class="mw-redirect" title="Ranked voting systems">ranked voting method</a> (100 votes with four distinct sets): </p> <table class="wikitable"> <tbody><tr> <td>31: A > E > C > D > B </td> <td>30: B > A > E </td> <td>29: C > D > B </td> <td>10: D > A > E </td></tr></tbody></table> <p>In this example there are some tied votes: for instance 10% of the voters assigned no position to B or C in their rankings; they are therefore considered to have tied these candidates with each other while ranking them below D, A and E. </p><p>The results of the 10 possible pairwise comparisons between the candidates are as follows: </p> <table class="wikitable"> <tbody><tr> <th>Comparison </th> <th>Result </th> <th>Winner </th> <th>Comparison </th> <th>Result </th> <th>Winner </th></tr> <tr> <th>A v B </th> <td>41 v 59 </td> <td>B </td> <th>B v D </th> <td>30 v 70 </td> <td>D </td></tr> <tr> <th>A v C </th> <td>71 v 29 </td> <td>A </td> <th>B v E </th> <td>59 v 41 </td> <td>B </td></tr> <tr> <th>A v D </th> <td>61 v 39 </td> <td>A </td> <th>C v D </th> <td>60 v 10 </td> <td>C </td></tr> <tr> <th>A v E </th> <td>71 v 0 </td> <td>A </td> <th>C v E </th> <td>29 v 71 </td> <td>E </td></tr> <tr> <th>B v C </th> <td>30 v 60 </td> <td>C </td> <th>D v E </th> <td>39 v 61 </td> <td>E </td></tr></tbody></table> <p>The wins and losses of each candidate sum as follows: </p> <table class="wikitable"> <tbody><tr> <th>Candidate </th> <th>Wins </th> <th>Losses </th> <th>Net</th> <th><i>r</i> </th></tr> <tr> <th>A </th> <td>3 </td> <td>1 </td> <td>2 </td> <td>0 0 1 1 1 </td></tr> <tr> <th>B </th> <td>2 </td> <td>2 </td> <td>0 </td> <td>1 0 0 0 1 </td></tr> <tr> <th>C </th> <td>2 </td> <td>2 </td> <td>0 </td> <td>0 1 0 1 0 </td></tr> <tr> <th>D </th> <td>1 </td> <td>3 </td> <td>−2 </td> <td>0 1 0 0 0 </td></tr> <tr> <th>E </th> <td>2 </td> <td>2 </td> <td>0 </td> <td>0 0 1 1 0 </td></tr> </tbody></table> <p>No <a href="/wiki/Condorcet_winner" class="mw-redirect" title="Condorcet winner">Condorcet winner</a> (candidate who beats all other candidates in pairwise comparisons) exists. Candidate A is the Copeland winner. Again there is no pair of candidates between whom the voters express no preference. </p> <div class="mw-heading mw-heading2"><h2 id="Use_for_producing_a_tabulation_in_other_methods">Use for producing a tabulation in other methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=15" title="Edit section: Use for producing a tabulation in other methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since Copeland's method produces a total ordering of candidates by score and is simple to compute, it is often useful for producing a sorted list of candidates in conjunction with another voting method which does not produce a total order. For example, the <a href="/wiki/Schulze_method" title="Schulze method">Schulze</a> and <a href="/wiki/Ranked_pairs" title="Ranked pairs">Ranked pairs</a> methods produce a transitive partial ordering of candidates, which generally produces a single winner, but not a unique way of tabulating runner-ups. Applying Copeland's method according to the respective method's partial ordering will yield a total order (topological ordering) guaranteed to be compatible with the method's partial order, and is simpler than a depth-first search when the partial order is given by an <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a>. </p><p>More generally, the Copeland score has the useful property that if there is a subset S of candidates such that every candidate in S will beat every candidate not in S, then there exists a threshold θ such that every candidate with a Copeland score above θ is in S while every candidate with a Copeland score below θ is not in S. This makes the Copeland score practical for finding various subsets of candidates that may be of interest, such as the <a href="/wiki/Smith_set" title="Smith set">Smith set</a> or the dominant mutual third set. </p> <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=Copeland%27s_method&action=edit&section=16" 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://plato.stanford.edu/archives/fall2019/entries/voting-methods/">Eric Pacuit, "Voting Methods", The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.)</a></li> <li><a rel="nofollow" class="external text" href="https://github.com/julien-boudry/Condorcet">Condorcet Class</a> <a href="/wiki/PHP" title="PHP">PHP</a> <a href="/wiki/Library_(computing)" title="Library (computing)">library</a> supporting multiple Condorcet methods, including Copeland method.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ranked_voting" title="Ranked voting">Ranked voting</a></li> <li><a href="/wiki/Comparison_of_electoral_systems" class="mw-redirect" title="Comparison of electoral systems">Comparison of electoral systems</a></li> <li><a href="/wiki/List_of_democracy_and_elections-related_topics" class="mw-redirect" title="List of democracy and elections-related topics">List of democracy and elections-related topics</a></li> <li><a href="/wiki/Voting_system" class="mw-redirect" title="Voting system">Voting systems</a></li> <li><a href="/wiki/Multiwinner_voting" title="Multiwinner voting">Multiwinner voting</a> – contains information on some multiwinner variants of Copeland.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPomerolSergio_Barba-Romero2000" class="citation book cs1">Pomerol, Jean-Charles; Sergio Barba-Romero (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mNOKayvMqH4C"><i>Multicriterion decision in management: principles and practice</i></a>. Springer. p. 122. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-7756-7" title="Special:BookSources/0-7923-7756-7"><bdi>0-7923-7756-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multicriterion+decision+in+management%3A+principles+and+practice&rft.pages=122&rft.pub=Springer&rft.date=2000&rft.isbn=0-7923-7756-7&rft.aulast=Pomerol&rft.aufirst=Jean-Charles&rft.au=Sergio+Barba-Romero&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmNOKayvMqH4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACopeland%27s+method" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">George G. Szpiro, "Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present" (2010).</span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCopeland1951" class="citation cs2">Copeland, Arthur Herbert (1951), <i>A 'reasonable' social welfare function</i>, Seminar on Mathematics in Social Sciences, University of Michigan</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+%27reasonable%27+social+welfare+function&rft.series=Seminar+on+Mathematics+in+Social+Sciences&rft.pub=University+of+Michigan&rft.date=1951&rft.aulast=Copeland&rft.aufirst=Arthur+Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACopeland%27s+method" class="Z3988"></span> (unpublished).</span> </li> <li id="cite_note-saari-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-saari_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaariMerlin1996" class="citation journal cs1">Saari, Donald G.; Merlin, Vincent R. (1996). "The Copeland Method: I.: Relationships and the Dictionary". <i>Economic Theory</i>. <b>8</b> (1): 51–76. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/25054952">25054952</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Economic+Theory&rft.atitle=The+Copeland+Method%3A+I.%3A+Relationships+and+the+Dictionary&rft.volume=8&rft.issue=1&rft.pages=51-76&rft.date=1996&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25054952%23id-name%3DJSTOR&rft.aulast=Saari&rft.aufirst=Donald+G.&rft.au=Merlin%2C+Vincent+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACopeland%27s+method" class="Z3988"></span></span> </li> <li id="cite_note-judge-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-judge_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-judge_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Balinski, Michel, and Rida Laraki, "Judge: Don't vote!" (2014), esp. footnote 4.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.thesprucecrafts.com/tournament-scoring-systems-611116">Scoring Systems in Chess Tournaments</a>. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources"><span title="The material near this tag may rely on an unreliable source. (January 2021)">unreliable source?</span></a></i>]</sup></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/fall2019/entries/voting-methods/">Eric Pacuit, "Voting Methods", The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.)</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">P. Dasgupta and E. Maskin, "The fairest vote of all" (2004).</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">R. B. Darlington, "Minimax Is the Best Electoral System After All" (2016).</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">T. N. Tideman and F. Plassman, "Modeling the Outcomes of Vote-Casting in Actual Elections" (2012).</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">P. Dasgupta and E. Maskin, "The fairest vote of all" (2004). The specification of their method is on p. 97, where they write "If no one [candidate] obtains a majority against all opponents, then among those candidates who defeat the most opponents in head-to-head comparisons, select as winner the one with the highest rank-order score".</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">An alternative method of applying a tie-break suggests itself for the Borda count, which is to compute the scores for each candidate – in this case (8,11,6,5) – and elect the Copeland winner with the highest Borda score, who in this case would be Baker. This has the drawback that the Borda winner may not lie within the set of Copeland winners, and it might be seen as delegitimising the result if the Borda count was the final arbiter without the associated Borda winner being elected.</span> </li> <li id="cite_note-:moulin-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-:moulin_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoulin1986" class="citation journal cs1">Moulin, H. (1986). "Choosing from a Tournament". <i>Social Choice and Welfare</i>. <b>3</b> (4): 271–191. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00292732">10.1007/BF00292732</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Social+Choice+and+Welfare&rft.atitle=Choosing+from+a+Tournament&rft.volume=3&rft.issue=4&rft.pages=271-191&rft.date=1986&rft_id=info%3Adoi%2F10.1007%2FBF00292732&rft.aulast=Moulin&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACopeland%27s+method" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">J.-F. Laslier, "And the loser is... Plurality Voting" (2012).</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Copeland%27s_method&action=edit&section=19" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>E Stensholt, "<a rel="nofollow" class="external text" href="http://www.votingmatters.org.uk/ISSUE15/P2.HTM">Nonmonotonicity in AV</a>"; <i><a href="/wiki/Voting_matters" title="Voting matters">Voting matters</a></i>; Issue 15, June 2002 (online).</li> <li>V.R. Merlin, and D.G. Saari, "Copeland Method. II. Manipulation, Monotonicity, and Paradoxes"; Journal of Economic Theory; Vol. 72, No. 1; January, 1997; 148–172.</li> <li>D.G. Saari. and V.R. Merlin, "The Copeland Method. I. Relationships and the Dictionary"; Economic Theory; Vol. 8, No. l; June, 1996; 51–76.</li></ol> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output 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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 class="mw-selflink selflink">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 href="/wiki/Ranked_pairs" title="Ranked pairs">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" 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?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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